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<span>Minimizing the variance of estimators</span> </div> </a> <button aria-controls="toc-Minimizing_the_variance_of_estimators-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Minimizing the variance of estimators subsection</span> </button> <ul id="toc-Minimizing_the_variance_of_estimators-sublist" class="vector-toc-list"> <li id="toc-Contrasts" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Contrasts"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Contrasts</span> </div> </a> <ul id="toc-Contrasts-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Implementation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Implementation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Implementation</span> </div> </a> <ul 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id="toc-Model_uncertainty_and_Bayesian_approaches" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Model_uncertainty_and_Bayesian_approaches"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Model uncertainty and Bayesian approaches</span> </div> </a> <ul id="toc-Model_uncertainty_and_Bayesian_approaches-sublist" class="vector-toc-list"> <li id="toc-Model_selection" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Model_selection"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.1</span> <span>Model selection</span> </div> </a> <ul id="toc-Model_selection-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bayesian_experimental_design" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Bayesian_experimental_design"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.2</span> <span>Bayesian experimental design</span> </div> 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id="toc-Using_a_computer_to_find_a_good_design-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discretizing_probability-measure_designs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Discretizing_probability-measure_designs"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Discretizing probability-measure designs</span> </div> </a> <ul id="toc-Discretizing_probability-measure_designs-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See 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class="vector-toc-list"> </ul> </li> <li id="toc-Articles_and_chapters" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Articles_and_chapters"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.3</span> <span>Articles and chapters</span> </div> </a> <ul id="toc-Articles_and_chapters-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Historical" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Historical"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.4</span> <span>Historical</span> </div> </a> <ul id="toc-Historical-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc 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class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Optimal_design&redirect=no" class="mw-redirect" title="Optimal design">Optimal design</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Experimental design that is optimal with respect to some statistical criterion</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about optimal <a href="/wiki/Design_of_experiments" title="Design of experiments">design of experiments</a>. For optimal design in control theory, see <a href="/wiki/Shape_optimization" title="Shape optimization">shape optimization</a>.</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Theb1982.jpg" class="mw-file-description"><img alt="Picture of a man taking measurements with a theodolite in a frozen environment." src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Theb1982.jpg/220px-Theb1982.jpg" decoding="async" width="220" height="331" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Theb1982.jpg/330px-Theb1982.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Theb1982.jpg/440px-Theb1982.jpg 2x" data-file-width="1088" data-file-height="1636" /></a><figcaption><a href="/wiki/Gustav_Elfving" title="Gustav Elfving">Gustav Elfving</a> developed the optimal design of experiments, and so minimized surveyors' need for <a href="/wiki/Theodolite" title="Theodolite">theodolite measurements</a> <i>(pictured)</i>, while trapped in his tent in storm-ridden <a href="/wiki/Greenland" title="Greenland">Greenland</a>.<sup id="cite_ref-Nord76_1-0" class="reference"><a href="#cite_note-Nord76-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>In the <a href="/wiki/Design_of_experiments" title="Design of experiments">design of experiments</a>, <b>optimal experimental designs</b> (or <b>optimum designs</b><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>) are a class of <a href="/wiki/Design_of_experiments" title="Design of experiments">experimental designs</a> that are <a href="/wiki/Optimization_(mathematics)" class="mw-redirect" title="Optimization (mathematics)">optimal</a> with respect to some <a href="/wiki/Statistical_theory" title="Statistical theory">statistical</a> <a href="/wiki/Objective_function" class="mw-redirect" title="Objective function">criterion</a>. The creation of this field of statistics has been credited to Danish statistician <a href="/wiki/Kirstine_Smith" title="Kirstine Smith">Kirstine Smith</a>.<sup id="cite_ref-GL2009_3-0" class="reference"><a href="#cite_note-GL2009-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-KS1918_4-0" class="reference"><a href="#cite_note-KS1918-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>In the <a href="/wiki/Design_of_experiments" title="Design of experiments">design of experiments</a> for <a href="/wiki/Estimation_theory" title="Estimation theory">estimating</a> <a href="/wiki/Statistical_model" title="Statistical model">statistical models</a>, <b>optimal designs</b> allow parameters to be <a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">estimated without bias</a> and with <a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">minimum variance</a>. A non-optimal design requires a greater number of <a href="/wiki/Replication_(statistics)" title="Replication (statistics)">experimental runs</a> to <a href="/wiki/Estimation_theory" title="Estimation theory">estimate</a> the <a href="/wiki/Parametric_model" title="Parametric model">parameters</a> with the same <a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">precision</a> as an optimal design. In practical terms, optimal experiments can reduce the costs of experimentation. </p><p>The optimality of a design depends on the <a href="/wiki/Statistical_model" title="Statistical model">statistical model</a> and is assessed with respect to a statistical criterion, which is related to the variance-matrix of the estimator. Specifying an appropriate model and specifying a suitable criterion function both require understanding of <a href="/wiki/Statistical_theory" title="Statistical theory">statistical theory</a> and practical knowledge with <a href="/wiki/Design_of_experiments" title="Design of experiments">designing experiments</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Advantages">Advantages</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=1" title="Edit section: Advantages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Optimal designs offer three advantages over sub-optimal <a href="/wiki/Design_of_experiments" title="Design of experiments">experimental designs</a>:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <ol><li>Optimal designs reduce the costs of experimentation by allowing <a href="/wiki/Statistical_model" title="Statistical model">statistical models</a> to be estimated with fewer experimental runs.</li> <li>Optimal designs can accommodate multiple types of factors, such as process, mixture, and discrete factors.</li> <li>Designs can be optimized when the design-space is constrained, for example, when the mathematical process-space contains factor-settings that are practically infeasible (e.g. due to safety concerns).</li></ol> <div class="mw-heading mw-heading2"><h2 id="Minimizing_the_variance_of_estimators">Minimizing the variance of estimators</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=2" title="Edit section: Minimizing the variance of estimators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Experimental designs are evaluated using statistical criteria.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>It is known that the <a href="/wiki/Least_squares" title="Least squares">least squares</a> estimator minimizes the <a href="/wiki/Variance" title="Variance">variance</a> of <a href="/wiki/Expected_value" title="Expected value">mean</a>-<a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">unbiased</a> <a href="/wiki/Estimators" class="mw-redirect" title="Estimators">estimators</a> (under the conditions of the <a href="/wiki/Gauss%E2%80%93Markov_theorem" title="Gauss–Markov theorem">Gauss–Markov theorem</a>). In the <a href="/wiki/Estimation" title="Estimation">estimation</a> theory for <a href="/wiki/Statistical_model" title="Statistical model">statistical models</a> with one <a href="/wiki/Real_number" title="Real number">real</a> <a href="/wiki/Parameter" title="Parameter">parameter</a>, the <a href="/wiki/Reciprocal_(mathematics)" class="mw-redirect" title="Reciprocal (mathematics)">reciprocal</a> of the variance of an (<a href="/wiki/Statistical_efficiency" class="mw-redirect" title="Statistical efficiency">"efficient"</a>) estimator is called the "<a href="/wiki/Fisher_information" title="Fisher information">Fisher information</a>" for that estimator.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Because of this reciprocity, <b><i>minimizing</i> the <i>variance</i></b> corresponds to <b><i>maximizing</i> the <i>information</i></b>. </p><p>When the <a href="/wiki/Statistical_model" title="Statistical model">statistical model</a> has several <a href="/wiki/Parameter" title="Parameter">parameters</a>, however, the <a href="/wiki/Expected_value" title="Expected value">mean</a> of the parameter-estimator is a <a href="/wiki/Coordinate_vector" title="Coordinate vector">vector</a> and its <a href="/wiki/Covariance_matrix" title="Covariance matrix">variance</a> is a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>. The <a href="/wiki/Inverse_matrix" class="mw-redirect" title="Inverse matrix">inverse matrix</a> of the variance-matrix is called the "information matrix". Because the variance of the estimator of a parameter vector is a matrix, the problem of "minimizing the variance" is complicated. Using <a href="/wiki/Statistical_theory" title="Statistical theory">statistical theory</a>, statisticians compress the information-matrix using real-valued <a href="/wiki/Summary_statistics" title="Summary statistics">summary statistics</a>; being real-valued functions, these "information criteria" can be maximized.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The traditional optimality-criteria are <a href="/wiki/Invariant_theory" title="Invariant theory">invariants</a> of the <a href="/wiki/Fisher_information" title="Fisher information">information</a> matrix; algebraically, the traditional optimality-criteria are <a href="/wiki/Functional_(mathematics)" title="Functional (mathematics)">functionals</a> of the <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of the information matrix. </p> <ul><li><b>A</b>-optimality ("<b>average</b>" or <b>trace</b>) <ul><li>One criterion is <b>A-optimality</b>, which seeks to minimize the <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> of the <a href="/wiki/Invertible_matrix" title="Invertible matrix">inverse</a> of the information matrix. This criterion results in minimizing the average variance of the estimates of the regression coefficients.</li></ul></li> <li><b>C</b>-optimality <ul><li>This criterion minimizes the variance of a <a href="/wiki/Best_linear_unbiased_estimator" class="mw-redirect" title="Best linear unbiased estimator">best linear unbiased estimator</a> of a predetermined linear combination of model parameters.</li></ul></li> <li><span class="anchor" id="D-optimality"></span><b>D</b>-optimality (<b>determinant</b>) <ul><li>A popular criterion is <b>D-optimality</b>, which seeks to minimize |(X'X)<sup>−1</sup>|, or equivalently maximize the <a href="/wiki/Determinant" title="Determinant">determinant</a> of the <a href="/wiki/Information_matrix" class="mw-redirect" title="Information matrix">information matrix</a> X'X of the design. This criterion results in maximizing the <a href="/wiki/Differential_entropy" title="Differential entropy">differential Shannon information</a> content of the parameter estimates.</li></ul></li> <li><b>E</b>-optimality (<b>eigenvalue</b>) <ul><li>Another design is <b>E-optimality</b>, which maximizes the minimum <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> of the information matrix.</li></ul></li> <li><b>S</b>-optimality<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <ul><li>This criterion maximizes a quantity measuring the mutual column orthogonality of X and the <a href="/wiki/Determinant" title="Determinant">determinant</a> of the information matrix.</li></ul></li> <li><b>T</b>-optimality <ul><li>This criterion maximizes the discrepancy between two proposed models at the design locations.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li></ul></li></ul> <p>Other optimality-criteria are concerned with the variance of <a href="/wiki/Predictive_inference" class="mw-redirect" title="Predictive inference">predictions</a>: </p> <ul><li><b>G</b>-optimality <ul><li>A popular criterion is <b>G-optimality</b>, which seeks to minimize the maximum entry in the <a href="/wiki/Diagonal#Matrices" title="Diagonal">diagonal</a> of the <a href="/wiki/Hat_matrix" class="mw-redirect" title="Hat matrix">hat matrix</a> X(X'X)<sup>−1</sup>X'. This has the effect of minimizing the maximum variance of the predicted values.</li></ul></li> <li><b>I</b>-optimality (<b>integrated</b>) <ul><li>A second criterion on prediction variance is <b>I-optimality</b>, which seeks to minimize the average prediction variance <i>over the design space</i>.</li></ul></li> <li><b>V</b>-optimality (<b>variance</b>) <ul><li>A third criterion on prediction variance is <b>V-optimality</b>, which seeks to minimize the average prediction variance over a set of m specific points.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li></ul></li></ul> <div class="mw-heading mw-heading3"><h3 id="Contrasts">Contrasts</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=3" title="Edit section: Contrasts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Contrast_(statistics)" title="Contrast (statistics)">Contrast (statistics)</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Nuisance_parameter" title="Nuisance parameter">Nuisance parameter</a></div> <p>In many applications, the statistician is most concerned with a <a href="/wiki/Nuisance_parameter" title="Nuisance parameter">"parameter of interest"</a> rather than with <a href="/wiki/Nuisance_parameter" title="Nuisance parameter">"nuisance parameters"</a>. More generally, statisticians consider <a href="/wiki/Linear_combination" title="Linear combination">linear combinations</a> of parameters, which are estimated via linear combinations of treatment-means in the <a href="/wiki/Design_of_experiments" title="Design of experiments">design of experiments</a> and in the <a href="/wiki/Analysis_of_variance" title="Analysis of variance">analysis of variance</a>; such linear combinations are called <a href="/wiki/Contrast_(statistics)" title="Contrast (statistics)">contrasts</a>. Statisticians can use appropriate optimality-criteria for such <a href="/wiki/Nuisance_parameter" title="Nuisance parameter">parameters of interest</a> and for <a href="/wiki/Contrast_(statistics)" title="Contrast (statistics)">contrasts</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Implementation">Implementation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=4" title="Edit section: Implementation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Catalogs of optimal designs occur in books and in software libraries. </p><p>In addition, major <a href="/wiki/List_of_statistical_packages" class="mw-redirect" title="List of statistical packages">statistical systems</a> like <a href="/wiki/SAS_System" class="mw-redirect" title="SAS System">SAS</a> and <a href="/wiki/R_(programming_language)" title="R (programming language)">R</a> have procedures for optimizing a design according to a user's specification. The experimenter must specify a <a href="/wiki/Statistical_model" title="Statistical model">model</a> for the design and an optimality-criterion before the method can compute an optimal design.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Practical_considerations">Practical considerations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=5" title="Edit section: Practical considerations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some advanced topics in optimal design require more <a href="/wiki/Statistical_theory" title="Statistical theory">statistical theory</a> and practical knowledge in designing experiments. </p> <div class="mw-heading mw-heading3"><h3 id="Model_dependence_and_robustness">Model dependence and robustness</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=6" title="Edit section: Model dependence and robustness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since the optimality criterion of most optimal designs is based on some function of the information matrix, the 'optimality' of a given design is <i><a href="/wiki/Statistical_model" title="Statistical model">model</a> dependent</i>: While an optimal design is best for that <a href="/wiki/Statistical_model" title="Statistical model">model</a>, its performance may deteriorate on other <a href="/wiki/Statistical_model" title="Statistical model">models</a>. On other <a href="/wiki/Statistical_model" title="Statistical model">models</a>, an <i>optimal</i> design can be either better or worse than a non-optimal design.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Therefore, it is important to <a href="/wiki/Benchmarking" title="Benchmarking">benchmark</a> the performance of designs under alternative <a href="/wiki/Statistical_model" title="Statistical model">models</a>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Choosing_an_optimality_criterion_and_robustness">Choosing an optimality criterion and robustness</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=7" title="Edit section: Choosing an optimality criterion and robustness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The choice of an appropriate optimality criterion requires some thought, and it is useful to benchmark the performance of designs with respect to several optimality criteria. Cornell writes that </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>since the [traditional optimality] criteria . . . are variance-minimizing criteria, . . . a design that is optimal for a given model using one of the . . . criteria is usually near-optimal for the same model with respect to the other criteria.</p><div class="templatequotecite">— <cite><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup></cite></div></blockquote> <p>Indeed, there are several classes of designs for which all the traditional optimality-criteria agree, according to the theory of "universal optimality" of <a href="/wiki/Jack_Kiefer_(mathematician)" class="mw-redirect" title="Jack Kiefer (mathematician)">Kiefer</a>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> The experience of practitioners like Cornell and the "universal optimality" theory of Kiefer suggest that robustness with respect to changes in the <i>optimality-criterion</i> is much greater than is robustness with respect to changes in the <i>model</i>. </p> <div class="mw-heading mw-heading4"><h4 id="Flexible_optimality_criteria_and_convex_analysis">Flexible optimality criteria and convex analysis</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=8" title="Edit section: Flexible optimality criteria and convex analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>High-quality statistical software provide a combination of libraries of optimal designs or iterative methods for constructing approximately optimal designs, depending on the model specified and the optimality criterion. Users may use a standard optimality-criterion or may program a custom-made criterion. </p><p>All of the traditional optimality-criteria are <a href="/wiki/Convex_function" title="Convex function">convex (or concave) functions</a>, and therefore optimal-designs are amenable to the mathematical theory of <a href="/wiki/Convex_analysis" title="Convex analysis">convex analysis</a> and their computation can use specialized methods of <a href="/wiki/Convex_optimization" title="Convex optimization">convex minimization</a>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> The practitioner need not select <i>exactly one</i> traditional, optimality-criterion, but can specify a custom criterion. In particular, the practitioner can specify a convex criterion using the maxima of convex optimality-criteria and <a href="/wiki/Conical_combination" title="Conical combination">nonnegative combinations</a> of optimality criteria (since these operations preserve <a href="/wiki/Convex_functions" class="mw-redirect" title="Convex functions">convex functions</a>). For <i>convex</i> optimality criteria, the <a href="/wiki/Jack_Kiefer_(mathematician)" class="mw-redirect" title="Jack Kiefer (mathematician)">Kiefer</a>-<a href="/wiki/Jacob_Wolfowitz" title="Jacob Wolfowitz">Wolfowitz</a> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5ZcfDZUJ4F8C&dq=Kiefer+Wolfowitz&pg=PA212">equivalence theorem</a> allows the practitioner to verify that a given design is globally optimal.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Jack_Kiefer_(mathematician)" class="mw-redirect" title="Jack Kiefer (mathematician)">Kiefer</a>-<a href="/wiki/Jacob_Wolfowitz" title="Jacob Wolfowitz">Wolfowitz</a> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5ZcfDZUJ4F8C&dq=Kiefer+Wolfowitz&pg=PA212">equivalence theorem</a> is related with the <a href="/wiki/Legendre_transformation" title="Legendre transformation">Legendre</a>-<a href="/wiki/Fenchel%27s_duality_theorem" title="Fenchel's duality theorem">Fenchel</a> <a href="/wiki/Convex_conjugate" title="Convex conjugate">conjugacy</a> for <a href="/wiki/Convex_function" title="Convex function">convex functions</a>.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>If an optimality-criterion lacks <a href="/wiki/Quasiconvex_function" title="Quasiconvex function">convexity</a>, then finding a <a href="/wiki/Global_optimization" title="Global optimization">global optimum</a> and verifying its optimality often are difficult. </p> <div class="mw-heading mw-heading3"><h3 id="Model_uncertainty_and_Bayesian_approaches">Model uncertainty and Bayesian approaches</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=9" title="Edit section: Model uncertainty and Bayesian approaches"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Model_selection">Model selection</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=10" title="Edit section: Model selection"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Model_selection" title="Model selection">Model selection</a></div> <p>When scientists wish to test several theories, then a statistician can design an experiment that allows optimal tests between specified models. Such "discrimination experiments" are especially important in the <a href="/wiki/Biostatistics" title="Biostatistics">biostatistics</a> supporting <a href="/wiki/Pharmacokinetics" title="Pharmacokinetics">pharmacokinetics</a> and <a href="/wiki/Pharmacodynamics" title="Pharmacodynamics">pharmacodynamics</a>, following the work of <a href="/wiki/David_R._Cox" class="mw-redirect" title="David R. Cox">Cox</a> and Atkinson.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Bayesian_experimental_design">Bayesian experimental design</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=11" title="Edit section: Bayesian experimental design"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Bayesian_experimental_design" title="Bayesian experimental design">Bayesian experimental design</a></div> <p>When practitioners need to consider multiple <a href="/wiki/Statistical_model" title="Statistical model">models</a>, they can specify a <a href="/wiki/Probability_measure" title="Probability measure">probability-measure</a> on the models and then select any design maximizing the <a href="/wiki/Expected_value" title="Expected value">expected value</a> of such an experiment. Such probability-based optimal-designs are called optimal <a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian</a> <a href="/wiki/Bayesian_experimental_design" title="Bayesian experimental design">designs</a>. Such <a href="/wiki/Bayesian_experimental_design" title="Bayesian experimental design">Bayesian designs</a> are used especially for <a href="/wiki/Generalized_linear_models" class="mw-redirect" title="Generalized linear models">generalized linear models</a> (where the response follows an <a href="/wiki/Exponential_family" title="Exponential family">exponential-family</a> distribution).<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>The use of a <a href="/wiki/Bayesian_experimental_design" title="Bayesian experimental design">Bayesian design</a> does not force statisticians to use <a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian methods</a> to analyze the data, however. Indeed, the "Bayesian" label for probability-based experimental-designs is disliked by some researchers.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Alternative terminology for "Bayesian" optimality includes "on-average" optimality or "population" optimality. </p> <div class="mw-heading mw-heading2"><h2 id="Iterative_experimentation">Iterative experimentation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=12" title="Edit section: Iterative experimentation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Scientific experimentation is an iterative process, and statisticians have developed several approaches to the optimal design of sequential experiments. </p> <div class="mw-heading mw-heading3"><h3 id="Sequential_analysis">Sequential analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=13" title="Edit section: Sequential analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Sequential_analysis" title="Sequential analysis">Sequential analysis</a></div> <p><a href="/wiki/Sequential_analysis" title="Sequential analysis">Sequential analysis</a> was pioneered by <a href="/wiki/Abraham_Wald" title="Abraham Wald">Abraham Wald</a>.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> In 1972, <a href="/wiki/Herman_Chernoff" title="Herman Chernoff">Herman Chernoff</a> wrote an overview of optimal sequential designs,<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> while <a href="/wiki/Adaptive_design_(medicine)" title="Adaptive design (medicine)">adaptive designs</a> were surveyed later by S. Zacks.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> Of course, much work on the optimal design of experiments is related to the theory of <a href="/wiki/Optimal_decision" title="Optimal decision">optimal decisions</a>, especially the <a href="/wiki/Statistical_decision_theory" class="mw-redirect" title="Statistical decision theory">statistical decision theory</a> of <a href="/wiki/Abraham_Wald" title="Abraham Wald">Abraham Wald</a>.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Response-surface_methodology">Response-surface methodology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=14" title="Edit section: Response-surface methodology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Response_surface_methodology" title="Response surface methodology">Response surface methodology</a></div> <p>Optimal designs for <a href="/wiki/Response_surface_methodology" title="Response surface methodology">response-surface models</a> are discussed in the textbook by Atkinson, Donev and Tobias, and in the survey of Gaffke and Heiligers and in the mathematical text of Pukelsheim. The <a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">blocking</a> of optimal designs is discussed in the textbook of Atkinson, Donev and Tobias and also in the monograph by Goos. </p><p>The earliest optimal designs were developed to estimate the parameters of regression models with continuous variables, for example, by <a href="/wiki/Joseph_Diaz_Gergonne" class="mw-redirect" title="Joseph Diaz Gergonne">J. D. Gergonne</a> in 1815 (Stigler). In English, two early contributions were made by <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles S. Peirce</a> and <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090810040534/http://www.webdoe.cc/publications/kirstine.php">Kirstine Smith</a>. </p><p>Pioneering designs for multivariate <a href="/wiki/Response_surface_methodology" title="Response surface methodology">response-surfaces</a> were proposed by <a href="/wiki/George_E._P._Box" title="George E. P. Box">George E. P. Box</a>. However, Box's designs have few optimality properties. Indeed, the <a href="/wiki/Box%E2%80%93Behnken_design" title="Box–Behnken design">Box–Behnken design</a> requires excessive experimental runs when the number of variables exceeds three.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> Box's <a href="/wiki/Central_composite_design" title="Central composite design">"central-composite" designs</a> require more experimental runs than do the optimal designs of Kôno.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="System_identification_and_stochastic_approximation">System identification and stochastic approximation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=15" title="Edit section: System identification and stochastic approximation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/System_identification" title="System identification">System identification</a> and <a href="/wiki/Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></div> <p>The optimization of sequential experimentation is studied also in <a href="/wiki/Stochastic_programming" title="Stochastic programming">stochastic programming</a> and in <a href="/wiki/Systems_analysis" title="Systems analysis">systems</a> and <a href="/wiki/Control_theory" title="Control theory">control</a>. Popular methods include <a href="/wiki/Stochastic_approximation" title="Stochastic approximation">stochastic approximation</a> and other methods of <a href="/wiki/Stochastic_optimization" title="Stochastic optimization">stochastic optimization</a>. Much of this research has been associated with the subdiscipline of <a href="/wiki/System_identification" title="System identification">system identification</a>.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> In computational <a href="/wiki/Optimal_control" title="Optimal control">optimal control</a>, D. Judin & A. Nemirovskii and <a rel="nofollow" class="external text" href="https://web.archive.org/web/20071031081747/http://www.ipu.ru/labs/lab7/eng/staff/polyak.htm">Boris Polyak</a> has described methods that are more efficient than the (<a href="/wiki/Armijo_rule" class="mw-redirect" title="Armijo rule">Armijo-style</a>) <a href="/wiki/Subgradient_method#Step_size_rules" title="Subgradient method">step-size rules</a> introduced by <a href="/wiki/George_E._P._Box" title="George E. P. Box">G. E. P. Box</a> in <a href="/wiki/Response_surface_methodology" title="Response surface methodology">response-surface methodology</a>.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Adaptive_design_(medicine)" title="Adaptive design (medicine)">Adaptive designs</a> are used in <a href="/wiki/Clinical_trials" class="mw-redirect" title="Clinical trials">clinical trials</a>, and optimal <a href="/wiki/Adaptive_design_(medicine)" title="Adaptive design (medicine)">adaptive designs</a> are surveyed in the <i>Handbook of Experimental Designs</i> chapter by Shelemyahu Zacks. </p> <div class="mw-heading mw-heading2"><h2 id="Specifying_the_number_of_experimental_runs">Specifying the number of experimental runs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=16" title="Edit section: Specifying the number of experimental runs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Using_a_computer_to_find_a_good_design">Using a computer to find a good design</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=17" title="Edit section: Using a computer to find a good design"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are several methods of finding an optimal design, given an <i>a priori</i> restriction on the number of experimental runs or replications. Some of these methods are discussed by Atkinson, Donev and Tobias and in the paper by Hardin and <a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane</a>. Of course, fixing the number of experimental runs <i>a priori</i> would be impractical. Prudent statisticians examine the other optimal designs, whose number of experimental runs differ. </p> <div class="mw-heading mw-heading3"><h3 id="Discretizing_probability-measure_designs">Discretizing probability-measure designs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=18" title="Edit section: Discretizing probability-measure designs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the mathematical theory on optimal experiments, an optimal design can be a <a href="/wiki/Probability_measure" title="Probability measure">probability measure</a> that is <a href="/wiki/Support_(measure_theory)" title="Support (measure theory)">supported</a> on an infinite set of observation-locations. Such optimal probability-measure designs solve a mathematical problem that neglected to specify the cost of observations and experimental runs. Nonetheless, such optimal probability-measure designs can be <a href="/wiki/Discretization" title="Discretization">discretized</a> to furnish <a href="/wiki/Approximation" title="Approximation">approximately</a> optimal designs.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p><p>In some cases, a finite set of observation-locations suffices to <a href="/wiki/Support_(measure_theory)" title="Support (measure theory)">support</a> an optimal design. Such a result was proved by Kôno and <a href="/wiki/Jack_Kiefer_(mathematician)" class="mw-redirect" title="Jack Kiefer (mathematician)">Kiefer</a> in their works on <a href="/wiki/Response_surface_methodology" title="Response surface methodology">response-surface designs</a> for quadratic models. The Kôno–Kiefer analysis explains why optimal designs for response-surfaces can have discrete supports, which are very similar as do the less efficient designs that have been traditional in <a href="/wiki/Response_surface_methodology" title="Response surface methodology">response surface methodology</a>.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=19" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1815, an article on optimal designs for <a href="/wiki/Polynomial_regression" title="Polynomial regression">polynomial regression</a> was published by <a href="/wiki/Joseph_Diaz_Gergonne" class="mw-redirect" title="Joseph Diaz Gergonne">Joseph Diaz Gergonne</a>, according to <a href="/wiki/Stephen_M._Stigler" class="mw-redirect" title="Stephen M. Stigler">Stigler</a>. </p><p><a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles S. Peirce</a> proposed an economic theory of scientific experimentation in 1876, which sought to maximize the precision of the estimates. Peirce's optimal allocation immediately improved the accuracy of gravitational experiments and was used for decades by Peirce and his colleagues. In his 1882 published lecture at <a href="/wiki/Johns_Hopkins_University" title="Johns Hopkins University">Johns Hopkins University</a>, Peirce introduced experimental design with these words: </p> <blockquote> <p>Logic will not undertake to inform you what kind of experiments you ought to make in order best to determine the acceleration of gravity, or the value of the Ohm; but it will tell you how to proceed to form a plan of experimentation.<br /> <br />[....] Unfortunately practice generally precedes theory, and it is the usual fate of mankind to get things done in some boggling way first, and find out afterward how they could have been done much more easily and perfectly.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> </blockquote> <p><a href="/wiki/Kirstine_Smith" title="Kirstine Smith">Kirstine Smith</a> proposed optimal designs for polynomial models in 1918. (Kirstine Smith had been a student of the Danish statistician <a href="/wiki/Thorvald_N._Thiele" title="Thorvald N. Thiele">Thorvald N. Thiele</a> and was working with <a href="/wiki/Karl_Pearson" title="Karl Pearson">Karl Pearson</a> in London.) </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=20" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Bayesian_experimental_design" title="Bayesian experimental design">Bayesian experimental design</a></li> <li><a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">Blocking (statistics)</a></li> <li><a href="/wiki/Computer_experiment" title="Computer experiment">Computer experiment</a></li> <li><a href="/wiki/Convex_function" title="Convex function">Convex function</a></li> <li><a href="/wiki/Convex_optimization" title="Convex optimization">Convex minimization</a></li> <li><a href="/wiki/Design_of_experiments" title="Design of experiments">Design of experiments</a></li> <li><a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency (statistics)</a></li> <li><a href="/wiki/Entropy_(information_theory)" title="Entropy (information theory)">Entropy (information theory)</a></li> <li><a href="/wiki/Fisher_information" title="Fisher information">Fisher information</a></li> <li><a href="/wiki/Glossary_of_experimental_design" title="Glossary of experimental design">Glossary of experimental design</a></li> <li><a href="/wiki/Hadamard%27s_maximal_determinant_problem" title="Hadamard's maximal determinant problem">Hadamard's maximal determinant problem</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Jack_Kiefer_(mathematician)" class="mw-redirect" title="Jack Kiefer (mathematician)">Kiefer, Jack</a></li> <li><a href="/wiki/Replication_(statistics)" title="Replication (statistics)">Replication (statistics)</a></li> <li><a href="/wiki/Response_surface_methodology" title="Response surface methodology">Response surface methodology</a></li> <li><a href="/wiki/Statistical_model" title="Statistical model">Statistical model</a></li> <li><a href="/wiki/Abraham_Wald" title="Abraham Wald">Wald, Abraham</a></li> <li><a href="/wiki/Jacob_Wolfowitz" title="Jacob Wolfowitz">Wolfowitz, Jacob</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=21" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-Nord76-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Nord76_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFNordström1999">Nordström (1999</a>, p. 176)</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">The adjective "optimum" (and not "optimal") "is the slightly older form in English and avoids the construction 'optim(um) + al´—there is no 'optimalis' in Latin" (page x in <i>Optimum Experimental Designs, with SAS</i>, by Atkinson, Donev, and Tobias).</span> </li> <li id="cite_note-GL2009-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-GL2009_3-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFGuttorpLindgren2009" class="citation journal cs1">Guttorp, P.; Lindgren, G. (2009). "Karl Pearson and the Scandinavian school of statistics". <i>International Statistical Review</i>. <b>77</b>: 64. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.368.8328">10.1.1.368.8328</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1751-5823.2009.00069.x">10.1111/j.1751-5823.2009.00069.x</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121294724">121294724</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Statistical+Review&rft.atitle=Karl+Pearson+and+the+Scandinavian+school+of+statistics&rft.volume=77&rft.pages=64&rft.date=2009&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.368.8328%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121294724%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1111%2Fj.1751-5823.2009.00069.x&rft.aulast=Guttorp&rft.aufirst=P.&rft.au=Lindgren%2C+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></span> </li> <li id="cite_note-KS1918-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-KS1918_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith1918" class="citation journal cs1">Smith, Kirstine (1918). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1431591">"On the standard deviations of adjusted and interpolated values of an observed polynomial function and its constants and the guidance they give towards a proper choice of the distribution of observations"</a>. <i>Biometrika</i>. <b>12</b> (1/2): 1–85. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2331929">10.2307/2331929</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2331929">2331929</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrika&rft.atitle=On+the+standard+deviations+of+adjusted+and+interpolated+values+of+an+observed+polynomial+function+and+its+constants+and+the+guidance+they+give+towards+a+proper+choice+of+the+distribution+of+observations&rft.volume=12&rft.issue=1%2F2&rft.pages=1-85&rft.date=1918&rft_id=info%3Adoi%2F10.2307%2F2331929&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2331929%23id-name%3DJSTOR&rft.aulast=Smith&rft.aufirst=Kirstine&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1431591&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">These three advantages (of optimal designs) are documented in the textbook by Atkinson, Donev, and Tobias.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Such criteria are called <a href="/wiki/Objective_function" class="mw-redirect" title="Objective function">objective functions</a> in <a href="/wiki/Optimization_(mathematics)" class="mw-redirect" title="Optimization (mathematics)">optimization theory</a>.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">The <a href="/wiki/Fisher_information" title="Fisher information">Fisher information</a> and other "<a href="/wiki/Entropy_(information_theory)" title="Entropy (information theory)">information</a>" <a href="/wiki/Functional_(mathematics)" title="Functional (mathematics)">functionals</a> are fundamental concepts in <a href="/wiki/Statistical_theory" title="Statistical theory">statistical theory</a>.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Traditionally, statisticians have evaluated estimators and designs by considering some <a href="/wiki/Summary_statistics" title="Summary statistics">summary statistic</a> of the covariance matrix (of a <a href="/wiki/Expected_value" title="Expected value">mean</a>-<a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">unbiased estimator</a>), usually with positive real values (like the <a href="/wiki/Determinant" title="Determinant">determinant</a> or <a href="/wiki/Matrix_trace" class="mw-redirect" title="Matrix trace">matrix trace</a>). Working with positive real-numbers brings several advantages: If the estimator of a single parameter has a positive variance, then the variance and the Fisher information are both positive real numbers; hence they are members of the convex cone of nonnegative real numbers (whose nonzero members have reciprocals in this same cone). <br /> For several parameters, the covariance-matrices and information-matrices are elements of the convex cone of nonnegative-definite symmetric matrices in a <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partially</a> <a href="/wiki/Ordered_vector_space" title="Ordered vector space">ordered vector space</a>, under the <a href="/wiki/Charles_Loewner" title="Charles Loewner">Loewner</a> (Löwner) order. This cone is closed under matrix-matrix addition, under matrix-inversion, and under the multiplication of positive real-numbers and matrices. An exposition of matrix theory and the Loewner-order appears in Pukelsheim.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShinXiu2016" class="citation journal cs1">Shin, Yeonjong; Xiu, Dongbin (2016). "Nonadaptive quasi-optimal points selection for least squares linear regression". <i>SIAM Journal on Scientific Computing</i>. <b>38</b> (1): A385–A411. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016SJSC...38A.385S">2016SJSC...38A.385S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F15M1015868">10.1137/15M1015868</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIAM+Journal+on+Scientific+Computing&rft.atitle=Nonadaptive+quasi-optimal+points+selection+for+least+squares+linear+regression&rft.volume=38&rft.issue=1&rft.pages=A385-A411&rft.date=2016&rft_id=info%3Adoi%2F10.1137%2F15M1015868&rft_id=info%3Abibcode%2F2016SJSC...38A.385S&rft.aulast=Shin&rft.aufirst=Yeonjong&rft.au=Xiu%2C+Dongbin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAtkinsonFedorov1975" class="citation journal cs1">Atkinson, A. C.; Fedorov, V. V. (1975). <a rel="nofollow" class="external text" href="https://academic.oup.com/biomet/article-lookup/doi/10.1093/biomet/62.1.57">"The design of experiments for discriminating between two rival models"</a>. <i>Biometrika</i>. <b>62</b> (1): 57–70. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbiomet%2F62.1.57">10.1093/biomet/62.1.57</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0006-3444">0006-3444</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrika&rft.atitle=The+design+of+experiments+for+discriminating+between+two+rival+models&rft.volume=62&rft.issue=1&rft.pages=57-70&rft.date=1975&rft_id=info%3Adoi%2F10.1093%2Fbiomet%2F62.1.57&rft.issn=0006-3444&rft.aulast=Atkinson&rft.aufirst=A.+C.&rft.au=Fedorov%2C+V.+V.&rft_id=https%3A%2F%2Facademic.oup.com%2Fbiomet%2Farticle-lookup%2Fdoi%2F10.1093%2Fbiomet%2F62.1.57&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">The above optimality-criteria are convex functions on domains of <a href="/wiki/Semidefinite_programming" title="Semidefinite programming">symmetric positive-semidefinite matrices</a>: See an on-line textbook for practitioners, which has many illustrations and statistical applications: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoydVandenberghe2004" class="citation book cs1">Boyd, Stephen P.; Vandenberghe, Lieven (2004). <a rel="nofollow" class="external text" href="https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf"><i>Convex Optimization</i></a> <span class="cs1-format">(PDF)</span>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-83378-3" title="Special:BookSources/978-0-521-83378-3"><bdi>978-0-521-83378-3</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">October 15,</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Convex+Optimization&rft.pub=Cambridge+University+Press&rft.date=2004&rft.isbn=978-0-521-83378-3&rft.aulast=Boyd&rft.aufirst=Stephen+P.&rft.au=Vandenberghe%2C+Lieven&rft_id=https%3A%2F%2Fweb.stanford.edu%2F~boyd%2Fcvxbook%2Fbv_cvxbook.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span> (book in pdf)</li></ul> Boyd and Vandenberghe discuss optimal experimental designs on pages 384–396.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Optimality criteria for <a href="/wiki/Nuisance_parameter" title="Nuisance parameter">"parameters of interest"</a> and for <a href="/wiki/Contrast_(statistics)" title="Contrast (statistics)">contrasts</a> are discussed by Atkinson, Donev and Tobias.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Iterative methods and approximation algorithms are surveyed in the textbook by Atkinson, Donev and Tobias and in the monographs of Fedorov (historical) and Pukelsheim, and in the survey article by Gaffke and Heiligers.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">See Kiefer ("Optimum Designs for Fitting Biased Multiresponse Surfaces" pages 289–299).</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Such benchmarking is discussed in the textbook by Atkinson et al. and in the papers of Kiefer. <i><a href="/wiki/Statistical_model" title="Statistical model">Model</a>-<a href="/wiki/Robust_statistics" title="Robust statistics">robust</a></i> designs (including "Bayesian" designs) are surveyed by Chang and Notz.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCornell,_John2002" class="citation book cs1">Cornell, John (2002). <i>Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data</i> (third ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-07916-3" title="Special:BookSources/978-0-471-07916-3"><bdi>978-0-471-07916-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Experiments+with+Mixtures%3A+Designs%2C+Models%2C+and+the+Analysis+of+Mixture+Data&rft.edition=third&rft.pub=Wiley&rft.date=2002&rft.isbn=978-0-471-07916-3&rft.au=Cornell%2C+John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span> (Pages 400-401)</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">An introduction to "universal optimality" appears in the textbook of Atkinson, Donev, and Tobias. More detailed expositions occur in the advanced textbook of Pukelsheim and the papers of Kiefer.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Computational methods are discussed by Pukelsheim and by Gaffke and Heiligers.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">The <a href="/wiki/Jack_Kiefer_(mathematician)" class="mw-redirect" title="Jack Kiefer (mathematician)">Kiefer</a>-<a href="/wiki/Jacob_Wolfowitz" title="Jacob Wolfowitz">Wolfowitz</a> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5ZcfDZUJ4F8C&dq=Kiefer+Wolfowitz&pg=PA212">equivalence theorem</a> is discussed in Chapter 9 of Atkinson, Donev, and Tobias.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Pukelsheim uses <a href="/wiki/Convex_analysis" title="Convex analysis">convex analysis</a> to study <a href="/wiki/Jack_Kiefer_(mathematician)" class="mw-redirect" title="Jack Kiefer (mathematician)">Kiefer</a>-<a href="/wiki/Jacob_Wolfowitz" title="Jacob Wolfowitz">Wolfowitz</a> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5ZcfDZUJ4F8C&dq=Kiefer+Wolfowitz&pg=PA212">equivalence theorem</a> in relation to the <a href="/wiki/Legendre_transformation" title="Legendre transformation">Legendre</a>-<a href="/wiki/Fenchel%27s_duality_theorem" title="Fenchel's duality theorem">Fenchel</a> <a href="/wiki/Convex_conjugate" title="Convex conjugate">conjugacy</a> for <a href="/wiki/Convex_function" title="Convex function">convex functions</a> The <a href="/wiki/Convex_optimization" title="Convex optimization">minimization</a> of <a href="/wiki/Convex_function" title="Convex function">convex functions</a> on domains of <a href="/wiki/Semidefinite_programming" title="Semidefinite programming">symmetric positive-semidefinite matrices</a> is explained in an on-line textbook for practitioners, which has many illustrations and statistical applications: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://web.stanford.edu/~boyd/cvxbook/"><i>Convex Optimization</i></a>. Cambridge University Press. 2004.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Convex+Optimization&rft.pub=Cambridge+University+Press&rft.date=2004&rft_id=https%3A%2F%2Fweb.stanford.edu%2F~boyd%2Fcvxbook%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span> (book in pdf)</li></ul> Boyd and Vandenberghe discuss optimal experimental designs on pages 384–396.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">See Chapter 20 in Atkinison, Donev, and Tobias.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><a href="/wiki/Bayesian_experimental_design" title="Bayesian experimental design">Bayesian designs</a> are discussed in Chapter 18 of the textbook by Atkinson, Donev, and Tobias. More advanced discussions occur in the monograph by Fedorov and Hackl, and the articles by Chaloner and Verdinelli and by DasGupta. <a href="/wiki/Bayesian_experimental_design" title="Bayesian experimental design">Bayesian designs</a> and other aspects of "model-robust" designs are discussed by Chang and Notz.</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">As an alternative to "<i>Bayesian</i> optimality", "<i>on-average</i> optimality" is advocated in Fedorov and Hackl.</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWald1945" class="citation journal cs1"><a href="/wiki/Abraham_Wald" title="Abraham Wald">Wald, Abraham</a> (June 1945). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177731118">"Sequential Tests of Statistical Hypotheses"</a>. <i>The Annals of Mathematical Statistics</i>. <b>16</b> (2): 117–186. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177731118">10.1214/aoms/1177731118</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2235829">2235829</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Annals+of+Mathematical+Statistics&rft.atitle=Sequential+Tests+of+Statistical+Hypotheses&rft.volume=16&rft.issue=2&rft.pages=117-186&rft.date=1945-06&rft_id=info%3Adoi%2F10.1214%2Faoms%2F1177731118&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2235829%23id-name%3DJSTOR&rft.aulast=Wald&rft.aufirst=Abraham&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faoms%252F1177731118&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">Chernoff, H. (1972) <i>Sequential Analysis and Optimal Design,</i> SIAM Monograph.</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text">Zacks, S. (1996) "Adaptive Designs for Parametric Models". In: Ghosh, S. and Rao, C. R., (Eds) (1996). <i>Design and Analysis of Experiments,</i> Handbook of Statistics, Volume 13. North-Holland. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-82061-2" title="Special:BookSources/0-444-82061-2">0-444-82061-2</a>. (pages 151–180)</span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"> <a rel="nofollow" class="external text" href="http://www.lse.ac.uk/collections/cats/People/HenryPage.htm">Henry P. Wynn</a> wrote, "the modern theory of optimum design has its roots in the decision theory school of U.S. statistics founded by <a href="/wiki/Abraham_Wald" title="Abraham Wald">Abraham Wald</a>" in his introduction "Jack Kiefer's Contributions to Experimental Design", which is pages xvii–xxiv in the following volume: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKiefer,_Jack_Carl1985" class="citation book cs1"><a href="/wiki/Jack_Kiefer_(mathematician)" class="mw-redirect" title="Jack Kiefer (mathematician)">Kiefer, Jack Carl</a> (1985). <a href="/wiki/Lawrence_D._Brown" title="Lawrence D. Brown">Brown, Lawrence D.</a>; Olkin, Ingram; Jerome Sacks; Wynn, Henry P (eds.). <a href="/wiki/Jack_Kiefer_(mathematician)" class="mw-redirect" title="Jack Kiefer (mathematician)"><i>Jack Carl Kiefer Collected Papers III Design of Experiments</i></a>. Springer-Verlag and the <a href="/wiki/Institute_of_Mathematical_Statistics" title="Institute of Mathematical Statistics">Institute of Mathematical Statistics</a>. pp. 718+xxv. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-96004-3" title="Special:BookSources/978-0-387-96004-3"><bdi>978-0-387-96004-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Jack+Carl+Kiefer+Collected+Papers+III+Design+of+Experiments&rft.pages=718%2Bxxv&rft.pub=Springer-Verlag+and+the+Institute+of+Mathematical+Statistics&rft.date=1985&rft.isbn=978-0-387-96004-3&rft.au=Kiefer%2C+Jack+Carl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li></ul> <a href="/wiki/Jack_Kiefer_(mathematician)" class="mw-redirect" title="Jack Kiefer (mathematician)">Kiefer</a> acknowledges Wald's influence and results on many pages – 273 (page 55 in the reprinted volume), 280 (62), 289-291 (71-73), 294 (76), 297 (79), 315 (97) 319 (101) – in this article: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKiefer1959" class="citation journal cs1"><a href="/wiki/Jack_Kiefer_(mathematician)" class="mw-redirect" title="Jack Kiefer (mathematician)">Kiefer, J.</a> (1959). "Optimum Experimental Designs". <i>Journal of the Royal Statistical Society, Series B</i>. <b>21</b>: 272–319.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Royal+Statistical+Society%2C+Series+B&rft.atitle=Optimum+Experimental+Designs&rft.volume=21&rft.pages=272-319&rft.date=1959&rft.aulast=Kiefer&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li></ul> </span></li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text">In the field of <a href="/wiki/Response_surface_methodology" title="Response surface methodology">response surface methodology</a>, the <a href="/wiki/Statistical_efficiency" class="mw-redirect" title="Statistical efficiency">inefficiency</a> of the <a href="/wiki/Box%E2%80%93Behnken_design" title="Box–Behnken design">Box–Behnken design</a> is noted by Wu and Hamada (page 422). <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWu,_C._F._JeffHamada,_Michael2002" class="citation book cs1">Wu, C. F. Jeff & Hamada, Michael (2002). <i>Experiments: Planning, Analysis, and Parameter Design Optimization</i>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-25511-6" title="Special:BookSources/978-0-471-25511-6"><bdi>978-0-471-25511-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Experiments%3A+Planning%2C+Analysis%2C+and+Parameter+Design+Optimization&rft.pub=Wiley&rft.date=2002&rft.isbn=978-0-471-25511-6&rft.au=Wu%2C+C.+F.+Jeff&rft.au=Hamada%2C+Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li></ul> Optimal designs for "follow-up" experiments are discussed by Wu and Hamada.</span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text">The <a href="/wiki/Statistical_efficiency" class="mw-redirect" title="Statistical efficiency">inefficiency</a> of <a href="/wiki/George_E._P._Box" title="George E. P. Box">Box</a>'s <a href="/wiki/Central_composite_design" title="Central composite design">"central-composite" designs</a> are discussed by according to Atkinson, Donev, and Tobias (page 165). These authors also discuss the <a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">blocking</a> of Kôno-type designs for quadratic <a href="/wiki/Response_surface_methodology" title="Response surface methodology">response-surfaces</a>.</span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text">In system identification, the following books have chapters on optimal experimental design: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoodwin,_Graham_C.Payne,_Robert_L.1977" class="citation book cs1">Goodwin, Graham C. & Payne, Robert L. (1977). <a href="/wiki/System_identification" title="System identification"><i>Dynamic System Identification: Experiment Design and Data Analysis</i></a>. Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-289750-4" title="Special:BookSources/978-0-12-289750-4"><bdi>978-0-12-289750-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dynamic+System+Identification%3A+Experiment+Design+and+Data+Analysis&rft.pub=Academic+Press&rft.date=1977&rft.isbn=978-0-12-289750-4&rft.au=Goodwin%2C+Graham+C.&rft.au=Payne%2C+Robert+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWalter,_ÉricPronzato,_Luc1997" class="citation book cs1">Walter, Éric & Pronzato, Luc (1997). <a href="/wiki/System_identification" title="System identification"><i>Identification of Parametric Models from Experimental Data</i></a>. Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Identification+of+Parametric+Models+from+Experimental+Data&rft.pub=Springer&rft.date=1997&rft.au=Walter%2C+%C3%89ric&rft.au=Pronzato%2C+Luc&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li></ul> </span></li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text">Some step-size rules for of Judin & Nemirovskii and of <a rel="nofollow" class="external text" href="http://www.ipu.ru/labs/lab7/eng/staff/polyak.htm">Polyak</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20071031081747/http://www.ipu.ru/labs/lab7/eng/staff/polyak.htm">Archived</a> 2007-10-31 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> are explained in the textbook by Kushner and Yin: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKushner,_Harold_J.Yin,_G._George2003" class="citation book cs1"><a href="/wiki/Harold_J._Kushner" title="Harold J. Kushner">Kushner, Harold J.</a>; Yin, G. George (2003). <a href="/wiki/Stochastic_approximation" title="Stochastic approximation"><i>Stochastic Approximation and Recursive Algorithms and Applications</i></a> (Second ed.). Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-00894-3" title="Special:BookSources/978-0-387-00894-3"><bdi>978-0-387-00894-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+Approximation+and+Recursive+Algorithms+and+Applications&rft.edition=Second&rft.pub=Springer&rft.date=2003&rft.isbn=978-0-387-00894-3&rft.au=Kushner%2C+Harold+J.&rft.au=Yin%2C+G.+George&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li></ul> </span></li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text">The <a href="/wiki/Discretization" title="Discretization">discretization</a> of optimal probability-measure designs to provide <a href="/wiki/Approximation" title="Approximation">approximately</a> optimal designs is discussed by Atkinson, Donev, and Tobias and by Pukelsheim (especially Chapter 12).</span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text">Regarding designs for quadratic <a href="/wiki/Response_surface_methodology" title="Response surface methodology">response-surfaces</a>, the results of Kôno and <a href="/wiki/Jack_Kiefer_(mathematician)" class="mw-redirect" title="Jack Kiefer (mathematician)">Kiefer</a> are discussed in Atkinson, Donev, and Tobias. Mathematically, such results are associated with <a href="/wiki/Chebyshev_polynomials" title="Chebyshev polynomials">Chebyshev polynomials</a>, "Markov systems", and "moment spaces": See <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKarlin,_SamuelShapley,_Lloyd1953" class="citation journal cs1"><a href="/wiki/Samuel_Karlin" title="Samuel Karlin">Karlin, Samuel</a>; <a href="/wiki/Lloyd_Shapley" title="Lloyd Shapley">Shapley, Lloyd</a> (1953). "Geometry of moment spaces". <i>Mem. Amer. Math. Soc</i>. <b>12</b>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mem.+Amer.+Math.+Soc.&rft.atitle=Geometry+of+moment+spaces&rft.volume=12&rft.date=1953&rft.au=Karlin%2C+Samuel&rft.au=Shapley%2C+Lloyd&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKarlin,_SamuelStudden,_William_J.1966" class="citation book cs1"><a href="/wiki/Samuel_Karlin" title="Samuel Karlin">Karlin, Samuel</a>; Studden, William J. (1966). <a href="/wiki/Chebyshev_polynomials" title="Chebyshev polynomials"><i>Tchebycheff systems: With applications in analysis and statistics</i></a>. Wiley-Interscience.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tchebycheff+systems%3A+With+applications+in+analysis+and+statistics&rft.pub=Wiley-Interscience&rft.date=1966&rft.au=Karlin%2C+Samuel&rft.au=Studden%2C+William+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDette,_HolgerStudden,_William_J.1997" class="citation book cs1">Dette, Holger & Studden, William J. (1997). <a href="/wiki/Moment_problem" title="Moment problem"><i>The Theory of canonical moments with applications in statistics, probability, and analysis</i></a>. John Wiley & Sons Inc.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+canonical+moments+with+applications+in+statistics%2C+probability%2C+and+analysis&rft.pub=John+Wiley+%26+Sons+Inc.&rft.date=1997&rft.au=Dette%2C+Holger&rft.au=Studden%2C+William+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li></ul> </span></li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text">Peirce, C. S. (1882), "Introductory Lecture on the Study of Logic" delivered September 1882, published in <i>Johns Hopkins University Circulars</i>, v. 2, n. 19, pp. 11–12, November 1882, see p. 11, <i>Google Books</i> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=E0YFAAAAQAAJ&dq=%22Logic+will+not+undertake+to+inform%22+%22unfortunately+practice+generally%22&pg=PA11">Eprint</a>. Reprinted in <i>Collected Papers</i> v. 7, paragraphs 59–76, see 59, 63, <i>Writings of Charles S. Peirce</i> v. 4, pp. 378–82, see 378, 379, and <i>The Essential Peirce</i> v. 1, pp. 210–14, see 210–1, also lower down on 211.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=22" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAtkinsonDonevTobias2007" class="citation book cs1">Atkinson, A. C.; Donev, A. N.; Tobias, R. D. (2007). <a rel="nofollow" class="external text" href="http://www.us.oup.com/us/catalog/general/subject/Mathematics/ProbabilityStatistics/~~/dmlldz11c2EmY2k9OTc4MDE5OTI5NjYwNg"><i>Optimum experimental designs, with <b>SAS</b><span></span></i></a>. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. pp. 511+xvi. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-929660-6" title="Special:BookSources/978-0-19-929660-6"><bdi>978-0-19-929660-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Optimum+experimental+designs%2C+with+SAS&rft.pages=511%2Bxvi&rft.pub=Oxford+University+Press&rft.date=2007&rft.isbn=978-0-19-929660-6&rft.aulast=Atkinson&rft.aufirst=A.+C.&rft.au=Donev%2C+A.+N.&rft.au=Tobias%2C+R.+D.&rft_id=http%3A%2F%2Fwww.us.oup.com%2Fus%2Fcatalog%2Fgeneral%2Fsubject%2FMathematics%2FProbabilityStatistics%2F~~%2Fdmlldz11c2EmY2k9OTc4MDE5OTI5NjYwNg&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChernoff1972" class="citation book cs1"><a href="/wiki/Herman_Chernoff" title="Herman Chernoff">Chernoff, Herman</a> (1972). <i>Sequential analysis and optimal design</i>. Society for Industrial and Applied Mathematics. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-006-9" title="Special:BookSources/978-0-89871-006-9"><bdi>978-0-89871-006-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sequential+analysis+and+optimal+design&rft.pub=Society+for+Industrial+and+Applied+Mathematics&rft.date=1972&rft.isbn=978-0-89871-006-9&rft.aulast=Chernoff&rft.aufirst=Herman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFedorov1972" class="citation book cs1">Fedorov, V. V. (1972). <i>Theory of Optimal Experiments</i>. Academic Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Optimal+Experiments&rft.pub=Academic+Press&rft.date=1972&rft.aulast=Fedorov&rft.aufirst=V.+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFedorovHackl1997" class="citation book cs1">Fedorov, Valerii V.; Hackl, Peter (1997). <i>Model-Oriented Design of Experiments</i>. Lecture Notes in Statistics. Vol. 125. Springer-Verlag.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Model-Oriented+Design+of+Experiments&rft.series=Lecture+Notes+in+Statistics&rft.pub=Springer-Verlag&rft.date=1997&rft.aulast=Fedorov&rft.aufirst=Valerii+V.&rft.au=Hackl%2C+Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoos2002" class="citation book cs1">Goos, Peter (2002). <a rel="nofollow" class="external text" href="http://users.telenet.be/peter.goos/springer.htm"><i>The Optimal Design of Blocked and Split-plot Experiments</i></a>. Lecture Notes in Statistics. Vol. 164. Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Optimal+Design+of+Blocked+and+Split-plot+Experiments&rft.series=Lecture+Notes+in+Statistics&rft.pub=Springer&rft.date=2002&rft.aulast=Goos&rft.aufirst=Peter&rft_id=http%3A%2F%2Fusers.telenet.be%2Fpeter.goos%2Fspringer.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKiefer1985" class="citation book cs1"><a href="/wiki/Jack_Kiefer_(mathematician)" class="mw-redirect" title="Jack Kiefer (mathematician)">Kiefer, Jack Carl</a> (1985). <a href="/wiki/Lawrence_D._Brown" title="Lawrence D. Brown">Brown</a>; <a href="/wiki/Ingram_Olkin" title="Ingram Olkin">Olkin, Ingram</a>; <a href="/w/index.php?title=Jerome_Sacks&action=edit&redlink=1" class="new" title="Jerome Sacks (page does not exist)">Sacks, Jerome</a>; et al. (eds.). <i>Jack Carl Kiefer: Collected papers III—Design of experiments</i>. Springer-Verlag and the Institute of Mathematical Statistics. pp. 718+xxv. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-96004-3" title="Special:BookSources/978-0-387-96004-3"><bdi>978-0-387-96004-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Jack+Carl+Kiefer%3A+Collected+papers+III%E2%80%94Design+of+experiments&rft.pages=718%2Bxxv&rft.pub=Springer-Verlag+and+the+Institute+of+Mathematical+Statistics&rft.date=1985&rft.isbn=978-0-387-96004-3&rft.aulast=Kiefer&rft.aufirst=Jack+Carl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLogothetisWynn1989" class="citation book cs1">Logothetis, N.; <a href="/w/index.php?title=Henry_P._Wynn&action=edit&redlink=1" class="new" title="Henry P. Wynn (page does not exist)">Wynn, H. P.</a> (1989). <i>Quality through design: Experimental design, off-line quality control, and Taguchi's contributions</i>. Oxford U. P. pp. 464+xi. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-851993-5" title="Special:BookSources/978-0-19-851993-5"><bdi>978-0-19-851993-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quality+through+design%3A+Experimental+design%2C+off-line+quality+control%2C+and+Taguchi%27s+contributions&rft.pages=464%2Bxi&rft.pub=Oxford+U.+P.&rft.date=1989&rft.isbn=978-0-19-851993-5&rft.aulast=Logothetis&rft.aufirst=N.&rft.au=Wynn%2C+H.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNordström1999" class="citation journal cs1">Nordström, Kenneth (May 1999). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Fss%2F1009212244">"The life and work of Gustav Elfving"</a>. <i>Statistical Science</i>. <b>14</b> (2): 174–196. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Fss%2F1009212244">10.1214/ss/1009212244</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2676737">2676737</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1722074">1722074</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistical+Science&rft.atitle=The+life+and+work+of+Gustav+Elfving&rft.volume=14&rft.issue=2&rft.pages=174-196&rft.date=1999-05&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1722074%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2676737%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1214%2Fss%2F1009212244&rft.aulast=Nordstr%C3%B6m&rft.aufirst=Kenneth&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Fss%252F1009212244&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPukelsheim2006" class="citation book cs1"><a href="/w/index.php?title=Friedrich_Pukelsheim&action=edit&redlink=1" class="new" title="Friedrich Pukelsheim (page does not exist)">Pukelsheim, Friedrich</a> (2006). <a rel="nofollow" class="external text" href="http://www.ec-securehost.com/SIAM/CL50.html"><i>Optimal design of experiments</i></a>. Classics in Applied Mathematics. Vol. 50 (republication with errata-list and new preface of Wiley (0-471-61971-X) 1993 ed.). <a href="/wiki/Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">Society for Industrial and Applied Mathematics</a>. pp. 454+xxxii. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-604-7" title="Special:BookSources/978-0-89871-604-7"><bdi>978-0-89871-604-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Optimal+design+of+experiments&rft.series=Classics+in+Applied+Mathematics&rft.pages=454%2Bxxxii&rft.edition=republication+with+errata-list+and+new+preface+of+Wiley+%280-471-61971-X%29+1993&rft.pub=Society+for+Industrial+and+Applied+Mathematics&rft.date=2006&rft.isbn=978-0-89871-604-7&rft.aulast=Pukelsheim&rft.aufirst=Friedrich&rft_id=http%3A%2F%2Fwww.ec-securehost.com%2FSIAM%2FCL50.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShah,_Kirti_R.Sinha,_Bikas_K.1989" class="citation book cs1">Shah, Kirti R. & Sinha, Bikas K. (1989). <i>Theory of Optimal Designs</i>. Lecture Notes in Statistics. Vol. 54. Springer-Verlag. pp. 171+viii. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-96991-6" title="Special:BookSources/978-0-387-96991-6"><bdi>978-0-387-96991-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Optimal+Designs&rft.series=Lecture+Notes+in+Statistics&rft.pages=171%2Bviii&rft.pub=Springer-Verlag&rft.date=1989&rft.isbn=978-0-387-96991-6&rft.au=Shah%2C+Kirti+R.&rft.au=Sinha%2C+Bikas+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=23" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Textbooks_for_practitioners_and_students">Textbooks for practitioners and students</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=24" title="Edit section: Textbooks for practitioners and students"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Textbooks_emphasizing_regression_and_response-surface_methodology">Textbooks emphasizing regression and response-surface methodology</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=25" title="Edit section: Textbooks emphasizing regression and response-surface methodology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The textbook by Atkinson, Donev and Tobias has been used for short courses for industrial practitioners as well as university courses. </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAtkinsonDonevTobias2007" class="citation book cs1">Atkinson, A. C.; Donev, A. N.; Tobias, R. D. (2007). <a rel="nofollow" class="external text" href="http://www.us.oup.com/us/catalog/general/subject/Mathematics/ProbabilityStatistics/~~/dmlldz11c2EmY2k9OTc4MDE5OTI5NjYwNg"><i>Optimum experimental designs, with <b>SAS</b><span></span></i></a>. Oxford University Press. pp. 511+xvi. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-929660-6" title="Special:BookSources/978-0-19-929660-6"><bdi>978-0-19-929660-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Optimum+experimental+designs%2C+with+SAS&rft.pages=511%2Bxvi&rft.pub=Oxford+University+Press&rft.date=2007&rft.isbn=978-0-19-929660-6&rft.aulast=Atkinson&rft.aufirst=A.+C.&rft.au=Donev%2C+A.+N.&rft.au=Tobias%2C+R.+D.&rft_id=http%3A%2F%2Fwww.us.oup.com%2Fus%2Fcatalog%2Fgeneral%2Fsubject%2FMathematics%2FProbabilityStatistics%2F~~%2Fdmlldz11c2EmY2k9OTc4MDE5OTI5NjYwNg&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLogothetisWynn1989" class="citation book cs1">Logothetis, N.; <a href="/w/index.php?title=Henry_P._Wynn&action=edit&redlink=1" class="new" title="Henry P. Wynn (page does not exist)">Wynn, H. P.</a> (1989). <i>Quality through design: Experimental design, off-line quality control, and Taguchi's contributions</i>. Oxford U. P. pp. 464+xi. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-851993-5" title="Special:BookSources/978-0-19-851993-5"><bdi>978-0-19-851993-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quality+through+design%3A+Experimental+design%2C+off-line+quality+control%2C+and+Taguchi%27s+contributions&rft.pages=464%2Bxi&rft.pub=Oxford+U.+P.&rft.date=1989&rft.isbn=978-0-19-851993-5&rft.aulast=Logothetis&rft.aufirst=N.&rft.au=Wynn%2C+H.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading4"><h4 id="Textbooks_emphasizing_block_designs">Textbooks emphasizing block designs</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=26" title="Edit section: Textbooks emphasizing block designs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Optimal <a href="/wiki/Randomized_block_design" class="mw-redirect" title="Randomized block design">block designs</a> are discussed by Bailey and by Bapat. The first chapter of Bapat's book reviews the <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a> used by Bailey (or the advanced books below). Bailey's exercises and discussion of <a href="/wiki/Random_assignment" title="Random assignment">randomization</a> both emphasize statistical concepts (rather than algebraic computations). </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBailey2008" class="citation book cs1"><a href="/wiki/Rosemary_A._Bailey" title="Rosemary A. Bailey">Bailey, R. A.</a> (2008). <a rel="nofollow" class="external text" href="http://www.maths.qmul.ac.uk/~rab/DOEbook"><i>Design of Comparative Experiments</i></a>. Cambridge U. P. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-68357-9" title="Special:BookSources/978-0-521-68357-9"><bdi>978-0-521-68357-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Design+of+Comparative+Experiments&rft.pub=Cambridge+U.+P.&rft.date=2008&rft.isbn=978-0-521-68357-9&rft.aulast=Bailey&rft.aufirst=R.+A.&rft_id=http%3A%2F%2Fwww.maths.qmul.ac.uk%2F~rab%2FDOEbook&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span> Draft available on-line. (Especially Chapter 11.8 "Optimality")</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBapat,_R._B.2000" class="citation book cs1">Bapat, R. B. (2000). <a rel="nofollow" class="external text" href="https://www.springer.com/math/algebra/book/978-0-387-98871-9?cm_mmc=Google-_-Book%20Search-_-Springer-_-0"><i>Linear Algebra and Linear Models</i></a> (Second ed.). Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98871-9" title="Special:BookSources/978-0-387-98871-9"><bdi>978-0-387-98871-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+and+Linear+Models&rft.edition=Second&rft.pub=Springer&rft.date=2000&rft.isbn=978-0-387-98871-9&rft.au=Bapat%2C+R.+B.&rft_id=https%3A%2F%2Fwww.springer.com%2Fmath%2Falgebra%2Fbook%2F978-0-387-98871-9%3Fcm_mmc%3DGoogle-_-Book%2520Search-_-Springer-_-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span> (Chapter 5 "Block designs and optimality", pages 99–111)</li></ul> <p>Optimal <a href="/wiki/Randomized_block_design" class="mw-redirect" title="Randomized block design">block designs</a> are discussed in the advanced monograph by Shah and Sinha and in the survey-articles by Cheng and by Majumdar. </p> <div class="mw-heading mw-heading3"><h3 id="Books_for_professional_statisticians_and_researchers">Books for professional statisticians and researchers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=27" title="Edit section: Books for professional statisticians and researchers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChernoff,_Herman1972" class="citation book cs1"><a href="/wiki/Herman_Chernoff" title="Herman Chernoff">Chernoff, Herman</a> (1972). <a href="/wiki/Sequential_analysis" title="Sequential analysis"><i>Sequential Analysis and Optimal Design</i></a>. <a href="/wiki/Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">SIAM</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-006-9" title="Special:BookSources/978-0-89871-006-9"><bdi>978-0-89871-006-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sequential+Analysis+and+Optimal+Design&rft.pub=SIAM&rft.date=1972&rft.isbn=978-0-89871-006-9&rft.au=Chernoff%2C+Herman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFedorov,_V._V.1972" class="citation book cs1">Fedorov, V. V. (1972). <i>Theory of Optimal Experiments</i>. Academic Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Optimal+Experiments&rft.pub=Academic+Press&rft.date=1972&rft.au=Fedorov%2C+V.+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFedorov,_Valerii_V.Hackl,_Peter1997" class="citation book cs1">Fedorov, Valerii V.; Hackl, Peter (1997). <i>Model-Oriented Design of Experiments</i>. Vol. 125. Springer-Verlag.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Model-Oriented+Design+of+Experiments&rft.pub=Springer-Verlag&rft.date=1997&rft.au=Fedorov%2C+Valerii+V.&rft.au=Hackl%2C+Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoos,_Peter2002" class="citation book cs1">Goos, Peter (2002). <i>The Optimal Design of Blocked and Split-plot Experiments</i>. Vol. 164. Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Optimal+Design+of+Blocked+and+Split-plot+Experiments&rft.pub=Springer&rft.date=2002&rft.au=Goos%2C+Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoos,_PeterJones,_Bradley2011" class="citation book cs1">Goos, Peter & Jones, Bradley (2011). <i>Optimal design of experiments: a case study approach</i>. Chichester Wiley. p. 304. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-74461-1" title="Special:BookSources/978-0-470-74461-1"><bdi>978-0-470-74461-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Optimal+design+of+experiments%3A+a+case+study+approach&rft.pages=304&rft.pub=Chichester+Wiley&rft.date=2011&rft.isbn=978-0-470-74461-1&rft.au=Goos%2C+Peter&rft.au=Jones%2C+Bradley&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKiefer,_Jack_Carl.1985" class="citation book cs1"><a href="/wiki/Jack_Kiefer_(mathematician)" class="mw-redirect" title="Jack Kiefer (mathematician)">Kiefer, Jack Carl</a>. (1985). <a href="/wiki/Lawrence_D._Brown" title="Lawrence D. Brown">Brown, Lawrence D.</a>; <a href="/wiki/Ingram_Olkin" title="Ingram Olkin">Olkin, Ingram</a>; Jerome Sacks; Wynn, Henry P (eds.). <a href="/wiki/Jack_Kiefer_(mathematician)" class="mw-redirect" title="Jack Kiefer (mathematician)"><i>Jack Carl Kiefer Collected Papers III Design of Experiments</i></a>. Springer-Verlag and the <a href="/wiki/Institute_of_Mathematical_Statistics" title="Institute of Mathematical Statistics">Institute of Mathematical Statistics</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-96004-3" title="Special:BookSources/978-0-387-96004-3"><bdi>978-0-387-96004-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Jack+Carl+Kiefer+Collected+Papers+III+Design+of+Experiments&rft.pub=Springer-Verlag+and+the+Institute+of+Mathematical+Statistics&rft.date=1985&rft.isbn=978-0-387-96004-3&rft.au=Kiefer%2C+Jack+Carl.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPukelsheim,_Friedrich2006" class="citation book cs1">Pukelsheim, Friedrich (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5ZcfDZUJ4F8C"><i>Optimal Design of Experiments</i></a>. Vol. 50. <a href="/wiki/Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">Society for Industrial and Applied Mathematics</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-604-7" title="Special:BookSources/978-0-89871-604-7"><bdi>978-0-89871-604-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Optimal+Design+of+Experiments&rft.pub=Society+for+Industrial+and+Applied+Mathematics&rft.date=2006&rft.isbn=978-0-89871-604-7&rft.au=Pukelsheim%2C+Friedrich&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5ZcfDZUJ4F8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span> Republication with errata-list and new preface of Wiley (0-471-61971-X) 1993</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShah,_Kirti_R.Sinha,_Bikas_K.1989" class="citation book cs1">Shah, Kirti R. & Sinha, Bikas K. (1989). <i>Theory of Optimal Designs</i>. Vol. 54. Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-96991-6" title="Special:BookSources/978-0-387-96991-6"><bdi>978-0-387-96991-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Optimal+Designs&rft.pub=Springer-Verlag&rft.date=1989&rft.isbn=978-0-387-96991-6&rft.au=Shah%2C+Kirti+R.&rft.au=Sinha%2C+Bikas+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Articles_and_chapters">Articles and chapters</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=28" title="Edit section: Articles and chapters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChaloner,_KathrynVerdinelli,_Isabella1995" class="citation journal cs1">Chaloner, Kathryn & Verdinelli, Isabella (1995). <a href="/wiki/Bayesian_experimental_design" title="Bayesian experimental design">"Bayesian Experimental Design: A Review"</a>. <i><a href="/wiki/Statistical_Science" title="Statistical Science">Statistical Science</a></i>. <b>10</b> (3): 273–304. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.5355">10.1.1.29.5355</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Fss%2F1177009939">10.1214/ss/1177009939</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistical+Science&rft.atitle=Bayesian+Experimental+Design%3A+A+Review&rft.volume=10&rft.issue=3&rft.pages=273-304&rft.date=1995&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.29.5355%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1214%2Fss%2F1177009939&rft.au=Chaloner%2C+Kathryn&rft.au=Verdinelli%2C+Isabella&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGhosh,_S.Rao,_C._R.1996" class="citation book cs1">Ghosh, S.; <a href="/wiki/Calyampudi_Radhakrishna_Rao" class="mw-redirect" title="Calyampudi Radhakrishna Rao">Rao, C. R.</a>, eds. (1996). <i>Design and Analysis of Experiments</i>. Handbook of Statistics. Vol. 13. North-Holland. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-82061-7" title="Special:BookSources/978-0-444-82061-7"><bdi>978-0-444-82061-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Design+and+Analysis+of+Experiments&rft.series=Handbook+of+Statistics&rft.pub=North-Holland&rft.date=1996&rft.isbn=978-0-444-82061-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1">"<a href="/wiki/Statistical_model" title="Statistical model">Model</a> <a href="/wiki/Robust_statistics" title="Robust statistics">Robust</a> Designs". <i>Design and Analysis of Experiments</i>. Handbook of Statistics. pp. 1055–1099.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Model+Robust+Designs&rft.btitle=Design+and+Analysis+of+Experiments&rft.series=Handbook+of+Statistics&rft.pages=1055-1099&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCheng,_C.-S" class="citation book cs1">Cheng, C.-S. "Optimal Design: Exact Theory". <i>Design and Analysis of Experiments</i>. Handbook of Statistics. pp. 977–1006.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Optimal+Design%3A+Exact+Theory&rft.btitle=Design+and+Analysis+of+Experiments&rft.series=Handbook+of+Statistics&rft.pages=977-1006&rft.au=Cheng%2C+C.-S&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDasGupta,_A" class="citation book cs1">DasGupta, A. "Review of Optimal <a href="/wiki/Bayesian_experimental_design" title="Bayesian experimental design">Bayesian Designs</a>". <i>Design and Analysis of Experiments</i>. Handbook of Statistics. pp. 1099–1148.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Review+of+Optimal+Bayesian+Designs&rft.btitle=Design+and+Analysis+of+Experiments&rft.series=Handbook+of+Statistics&rft.pages=1099-1148&rft.au=DasGupta%2C+A&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGaffke,_N.Heiligers,_B" class="citation book cs1">Gaffke, N. & Heiligers, B. "Approximate Designs for <a href="/wiki/Response_surface_methodology" title="Response surface methodology">Polynomial Regression</a>: <a href="/wiki/Invariant_estimator" title="Invariant estimator">Invariance</a>, <a href="/wiki/Admissible_decision_rule" title="Admissible decision rule">Admissibility</a>, and Optimality". <i>Design and Analysis of Experiments</i>. Handbook of Statistics. pp. 1149–1199.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Approximate+Designs+for+Polynomial+Regression%3A+Invariance%2C+Admissibility%2C+and+Optimality&rft.btitle=Design+and+Analysis+of+Experiments&rft.series=Handbook+of+Statistics&rft.pages=1149-1199&rft.au=Gaffke%2C+N.&rft.au=Heiligers%2C+B&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMajumdar,_D" class="citation book cs1">Majumdar, D. "Optimal and Efficient Treatment-Control Designs". <i>Design and Analysis of Experiments</i>. Handbook of Statistics. pp. 1007–1054.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Optimal+and+Efficient+Treatment-Control+Designs&rft.btitle=Design+and+Analysis+of+Experiments&rft.series=Handbook+of+Statistics&rft.pages=1007-1054&rft.au=Majumdar%2C+D&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStufken,_J" class="citation book cs1">Stufken, J. "Optimal <a href="/wiki/Crossover_study" title="Crossover study">Crossover Designs</a>". <i>Design and Analysis of Experiments</i>. Handbook of Statistics. pp. 63–90.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Optimal+Crossover+Designs&rft.btitle=Design+and+Analysis+of+Experiments&rft.series=Handbook+of+Statistics&rft.pages=63-90&rft.au=Stufken%2C+J&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZacks,_S" class="citation book cs1">Zacks, S. "Adaptive Designs for Parametric Models". <i>Design and Analysis of Experiments</i>. Handbook of Statistics. pp. 151–180.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Adaptive+Designs+for+Parametric+Models&rft.btitle=Design+and+Analysis+of+Experiments&rft.series=Handbook+of+Statistics&rft.pages=151-180&rft.au=Zacks%2C+S&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li></ul></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKôno,_Kazumasa1962" class="citation journal cs1">Kôno, Kazumasa (1962). <a rel="nofollow" class="external text" href="https://www.ams.org/mathscinet/pdf/153090.pdf">"Optimum designs for quadratic regression on <i>k</i>-cube"</a> <span class="cs1-format">(PDF)</span>. <i>Memoirs of the Faculty of Science. Kyushu University. Series A. Mathematics</i>. <b>16</b> (2): 114–122. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2206%2Fkyushumfs.16.114">10.2206/kyushumfs.16.114</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Memoirs+of+the+Faculty+of+Science.+Kyushu+University.+Series+A.+Mathematics&rft.atitle=Optimum+designs+for+quadratic+regression+on+k-cube&rft.volume=16&rft.issue=2&rft.pages=114-122&rft.date=1962&rft_id=info%3Adoi%2F10.2206%2Fkyushumfs.16.114&rft.au=K%C3%B4no%2C+Kazumasa&rft_id=https%3A%2F%2Fwww.ams.org%2Fmathscinet%2Fpdf%2F153090.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Historical">Historical</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optimal_experimental_design&action=edit&section=29" title="Edit section: Historical"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGergonne,_J._D.1974" class="citation journal cs1"><a href="/wiki/Joseph_Diaz_Gergonne" class="mw-redirect" title="Joseph Diaz Gergonne">Gergonne, J. D.</a> (November 1974) [1815]. <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0315-0860%2874%2990034-2">"The application of the method of least squares to the interpolation of sequences"</a>. <i>Historia Mathematica</i>. <b>1</b> (4) (Translated by Ralph St. John and <a href="/wiki/Stephen_M._Stigler" class="mw-redirect" title="Stephen M. Stigler">S. M. Stigler</a> from the 1815 French ed.): 439–447. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0315-0860%2874%2990034-2">10.1016/0315-0860(74)90034-2</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=The+application+of+the+method+of+least+squares+to+the+interpolation+of+sequences&rft.volume=1&rft.issue=4&rft.pages=439-447&rft.date=1974-11&rft_id=info%3Adoi%2F10.1016%2F0315-0860%2874%2990034-2&rft.au=Gergonne%2C+J.+D.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0315-0860%252874%252990034-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStigler,_Stephen_M.1974" class="citation journal cs1"><a href="/wiki/Stephen_M._Stigler" class="mw-redirect" title="Stephen M. Stigler">Stigler, Stephen M.</a> (November 1974). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0315-0860%2874%2990033-0">"Gergonne's 1815 paper on the design and analysis of polynomial regression experiments"</a>. <i>Historia Mathematica</i>. <b>1</b> (4): 431–439. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0315-0860%2874%2990033-0">10.1016/0315-0860(74)90033-0</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=Gergonne%27s+1815+paper+on+the+design+and+analysis+of+polynomial+regression+experiments&rft.volume=1&rft.issue=4&rft.pages=431-439&rft.date=1974-11&rft_id=info%3Adoi%2F10.1016%2F0315-0860%2874%2990033-0&rft.au=Stigler%2C+Stephen+M.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0315-0860%252874%252990033-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeirce,_C._S1876" class="citation journal cs1"><a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce, C. S</a> (1876). "Note on the Theory of the Economy of Research". <i>Coast Survey Report</i>: 197–201.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Coast+Survey+Report&rft.atitle=Note+on+the+Theory+of+the+Economy+of+Research&rft.pages=197-201&rft.date=1876&rft.au=Peirce%2C+C.+S&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span> (Appendix No. 14). <a rel="nofollow" class="external text" href="http://docs.lib.noaa.gov/rescue/cgs/001_pdf/CSC-0025.PDF#page=222">NOAA PDF Eprint</a>. Reprinted in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Charles_Sanders_Peirce_bibliography#CP" title="Charles Sanders Peirce bibliography"><i>Collected Papers of Charles Sanders Peirce</i></a>. Vol. 7. 1958.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Collected+Papers+of+Charles+Sanders+Peirce&rft.date=1958&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span> paragraphs 139–157, and in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeirce1967" class="citation journal cs1">Peirce, C. S. (July–August 1967). "Note on the Theory of the Economy of Research". <i>Operations Research</i>. <b>15</b> (4): 643–648. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Fopre.15.4.643">10.1287/opre.15.4.643</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/168276">168276</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Operations+Research&rft.atitle=Note+on+the+Theory+of+the+Economy+of+Research&rft.volume=15&rft.issue=4&rft.pages=643-648&rft.date=1967-07%2F1967-08&rft_id=info%3Adoi%2F10.1287%2Fopre.15.4.643&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F168276%23id-name%3DJSTOR&rft.aulast=Peirce&rft.aufirst=C.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith,_Kirstine1918" class="citation journal cs1">Smith, Kirstine (1918). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1431591">"On the Standard Deviations of Adjusted and Interpolated Values of an Observed Polynomial Function and its Constants and the Guidance They Give Towards a Proper Choice of the Distribution of the Observations"</a>. <i>Biometrika</i>. <b>12</b> (1/2): 1–85. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2331929">10.2307/2331929</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2331929">2331929</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrika&rft.atitle=On+the+Standard+Deviations+of+Adjusted+and+Interpolated+Values+of+an+Observed+Polynomial+Function+and+its+Constants+and+the+Guidance+They+Give+Towards+a+Proper+Choice+of+the+Distribution+of+the+Observations&rft.volume=12&rft.issue=1%2F2&rft.pages=1-85&rft.date=1918&rft_id=info%3Adoi%2F10.2307%2F2331929&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2331929%23id-name%3DJSTOR&rft.au=Smith%2C+Kirstine&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1431591&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptimal+experimental+design" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output 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colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Descriptive_statistics" style="font-size:114%;margin:0 4em"><a href="/wiki/Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">Continuous data</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Central_tendency" title="Central tendency">Center</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mean" title="Mean">Mean</a> <ul><li><a href="/wiki/Arithmetic_mean" title="Arithmetic mean">Arithmetic</a></li> <li><a href="/wiki/Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic-Geometric</a></li> <li><a href="/wiki/Contraharmonic_mean" title="Contraharmonic mean">Contraharmonic</a></li> <li><a href="/wiki/Cubic_mean" title="Cubic mean">Cubic</a></li> <li><a href="/wiki/Generalized_mean" title="Generalized mean">Generalized/power</a></li> <li><a href="/wiki/Geometric_mean" title="Geometric mean">Geometric</a></li> <li><a href="/wiki/Harmonic_mean" title="Harmonic mean">Harmonic</a></li> <li><a href="/wiki/Heronian_mean" title="Heronian mean">Heronian</a></li> <li><a href="/wiki/Heinz_mean" title="Heinz mean">Heinz</a></li> <li><a href="/wiki/Lehmer_mean" title="Lehmer mean">Lehmer</a></li></ul></li> <li><a href="/wiki/Median" title="Median">Median</a></li> <li><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_dispersion" title="Statistical dispersion">Dispersion</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Average_absolute_deviation" title="Average absolute deviation">Average absolute deviation</a></li> <li><a href="/wiki/Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li> <li><a href="/wiki/Interquartile_range" title="Interquartile range">Interquartile range</a></li> <li><a href="/wiki/Percentile" title="Percentile">Percentile</a></li> <li><a href="/wiki/Range_(statistics)" title="Range (statistics)">Range</a></li> <li><a href="/wiki/Standard_deviation" title="Standard deviation">Standard deviation</a></li> <li><a href="/wiki/Variance#Sample_variance" title="Variance">Variance</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Shape_of_the_distribution" class="mw-redirect" title="Shape of the distribution">Shape</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li> <li><a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">Moments</a> <ul><li><a href="/wiki/Kurtosis" title="Kurtosis">Kurtosis</a></li> <li><a href="/wiki/L-moment" title="L-moment">L-moments</a></li> <li><a href="/wiki/Skewness" title="Skewness">Skewness</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Count_data" title="Count data">Count data</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Summary tables</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li> <li><a href="/wiki/Grouped_data" title="Grouped data">Grouped data</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Dependence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson product-moment correlation</a></li> <li><a href="/wiki/Rank_correlation" title="Rank correlation">Rank correlation</a> <ul><li><a href="/wiki/Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li> <li><a href="/wiki/Spearman%27s_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li></ul></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_graphics" title="Statistical graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bar_chart" title="Bar chart">Bar chart</a></li> <li><a href="/wiki/Biplot" title="Biplot">Biplot</a></li> <li><a href="/wiki/Box_plot" title="Box plot">Box plot</a></li> <li><a href="/wiki/Control_chart" title="Control chart">Control chart</a></li> <li><a href="/wiki/Correlogram" title="Correlogram">Correlogram</a></li> <li><a href="/wiki/Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li> <li><a href="/wiki/Forest_plot" title="Forest plot">Forest plot</a></li> <li><a href="/wiki/Histogram" title="Histogram">Histogram</a></li> <li><a href="/wiki/Pie_chart" title="Pie chart">Pie chart</a></li> <li><a href="/wiki/Q%E2%80%93Q_plot" title="Q–Q plot">Q–Q plot</a></li> <li><a href="/wiki/Radar_chart" title="Radar chart">Radar chart</a></li> <li><a href="/wiki/Run_chart" title="Run chart">Run chart</a></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li> <li><a href="/wiki/Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li> <li><a href="/wiki/Violin_plot" title="Violin plot">Violin plot</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_collection" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_collection" title="Data collection">Data collection</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Design_of_experiments" title="Design of experiments">Study design</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Effect_size" title="Effect size">Effect size</a></li> <li><a href="/wiki/Missing_data" title="Missing data">Missing data</a></li> <li><a href="/wiki/Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li> <li><a href="/wiki/Statistical_population" title="Statistical population">Population</a></li> <li><a href="/wiki/Replication_(statistics)" title="Replication (statistics)">Replication</a></li> <li><a href="/wiki/Sample_size_determination" title="Sample size determination">Sample size determination</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survey_methodology" title="Survey methodology">Survey methodology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">Sampling</a> <ul><li><a href="/wiki/Cluster_sampling" title="Cluster sampling">Cluster</a></li> <li><a href="/wiki/Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li> <li><a href="/wiki/Opinion_poll" title="Opinion poll">Opinion poll</a></li> <li><a href="/wiki/Questionnaire" title="Questionnaire">Questionnaire</a></li> <li><a href="/wiki/Standard_error" title="Standard error">Standard error</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Experiment" title="Experiment">Controlled experiments</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li> <li><a href="/wiki/Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li> <li><a href="/wiki/Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li> <li><a href="/wiki/Random_assignment" title="Random assignment">Random assignment</a></li> <li><a href="/wiki/Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li> <li><a href="/wiki/Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li> <li><a href="/wiki/Scientific_control" title="Scientific control">Scientific control</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive designs</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li> <li><a href="/wiki/Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li> <li><a href="/wiki/Up-and-Down_Designs" class="mw-redirect" title="Up-and-Down Designs">Up-and-down designs</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Observational_study" title="Observational study">Observational studies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohort_study" title="Cohort study">Cohort study</a></li> <li><a href="/wiki/Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li> <li><a href="/wiki/Natural_experiment" title="Natural experiment">Natural experiment</a></li> <li><a href="/wiki/Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a></li> <li><a href="/wiki/Sampling_distribution" title="Sampling distribution">Sampling distribution</a> <ul><li><a href="/wiki/Order_statistic" title="Order statistic">Order statistic</a></li></ul></li> <li><a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a> <ul><li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li></ul></li> <li><a href="/wiki/Statistical_model" title="Statistical model">Statistical model</a> <ul><li><a href="/wiki/Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li> <li><a href="/wiki/Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li> <li><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameter</a> <ul><li><a href="/wiki/Location_parameter" title="Location parameter">location</a></li> <li><a href="/wiki/Scale_parameter" title="Scale parameter">scale</a></li> <li><a href="/wiki/Shape_parameter" title="Shape parameter">shape</a></li></ul></li> <li><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric family</a> <ul><li><a href="/wiki/Likelihood_function" title="Likelihood function">Likelihood</a> <a href="/wiki/Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size:85%;">(monotone)</span></a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential family</a></li></ul></li> <li><a href="/wiki/Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li> <li><a href="/wiki/Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/U-statistic" title="U-statistic">U</a></li> <li><a href="/wiki/V-statistic" title="V-statistic">V</a></li></ul></li> <li><a href="/wiki/Optimal_decision" title="Optimal decision">Optimal decision</a> <ul><li><a href="/wiki/Loss_function" title="Loss function">loss function</a></li></ul></li> <li><a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li> <li><a href="/wiki/Statistical_distance" title="Statistical distance">Statistical distance</a> <ul><li><a href="/wiki/Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li> <li><a href="/wiki/Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li> <li><a href="/wiki/Robust_statistics" title="Robust statistics">Robustness</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Point_estimation" title="Point estimation">Point estimation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Estimating_equations" title="Estimating equations">Estimating equations</a> <ul><li><a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li> <li><a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li> <li><a href="/wiki/M-estimator" title="M-estimator">M-estimator</a></li> <li><a href="/wiki/Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li> <li><a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a> <ul><li><a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a> <ul><li><a href="/wiki/Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li> <li><a href="/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li> <li><a href="/wiki/Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Plug-in</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Interval_estimation" title="Interval estimation">Interval estimation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Confidence_interval" title="Confidence interval">Confidence interval</a></li> <li><a href="/wiki/Pivotal_quantity" title="Pivotal quantity">Pivot</a></li> <li><a href="/wiki/Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li> <li><a href="/wiki/Prediction_interval" title="Prediction interval">Prediction interval</a></li> <li><a href="/wiki/Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li> <li><a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">Resampling</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Testing hypotheses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li> <li><a href="/wiki/Power_(statistics)" title="Power (statistics)">Power</a> <ul><li><a href="/wiki/Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li> <li><a href="/wiki/Permutation_test" title="Permutation test">Permutation test</a> <ul><li><a href="/wiki/Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li> <li><a href="/wiki/Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li> <li><a href="/wiki/Score_test" title="Score test">Score/Lagrange multiplier</a></li> <li><a href="/wiki/Wald_test" title="Wald test">Wald</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/List_of_statistical_tests" title="List of statistical tests">Specific tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Z-test" title="Z-test"><i>Z</i>-test <span style="font-size:85%;">(normal)</span></a></li> <li><a href="/wiki/Student%27s_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li> <li><a href="/wiki/F-test" title="F-test"><i>F</i>-test</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chi-squared_test" title="Chi-squared test">Chi-squared</a></li> <li><a href="/wiki/G-test" title="G-test"><i>G</i>-test</a></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li> <li><a href="/wiki/Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li> <li><a href="/wiki/Lilliefors_test" title="Lilliefors test">Lilliefors</a></li> <li><a href="/wiki/Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li> <li><a href="/wiki/Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size:85%;">(Shapiro–Wilk)</span></a></li> <li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li> <li><a href="/wiki/Model_selection" title="Model selection">Model selection</a> <ul><li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li> <li><a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">AIC</a></li> <li><a href="/wiki/Bayesian_information_criterion" title="Bayesian information criterion">BIC</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sign_test" title="Sign test">Sign</a> <ul><li><a href="/wiki/Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li> <li><a href="/wiki/Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size:85%;">(Wilcoxon)</span></a> <ul><li><a href="/wiki/Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li> <li><a href="/wiki/Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size:85%;">(Mann–Whitney)</span></a></li> <li><a href="/wiki/Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="/wiki/Analysis_of_variance" title="Analysis of variance">anova</a> <ul><li><a href="/wiki/Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size:85%;">(Kruskal–Wallis)</span></a></li> <li><a href="/wiki/Friedman_test" title="Friedman test">2-way <span style="font-size:85%;">(Friedman)</span></a></li> <li><a href="/wiki/Jonckheere%27s_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size:85%;">(Jonckheere–Terpstra)</span></a></li></ul></li> <li><a href="/wiki/Van_der_Waerden_test" title="Van der Waerden test">Van der Waerden test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian probability</a> <ul><li><a href="/wiki/Prior_probability" title="Prior probability">prior</a></li> <li><a href="/wiki/Posterior_probability" title="Posterior probability">posterior</a></li></ul></li> <li><a href="/wiki/Credible_interval" title="Credible interval">Credible interval</a></li> <li><a href="/wiki/Bayes_factor" title="Bayes factor">Bayes factor</a></li> <li><a href="/wiki/Bayes_estimator" title="Bayes estimator">Bayesian estimator</a> <ul><li><a href="/wiki/Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="CorrelationRegression_analysis" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></li></ul></div></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li> <li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Confounding" title="Confounding">Confounding variable</a></li> <li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li> <li><a href="/wiki/Regression_validation" title="Regression validation">Regression validation</a></li> <li><a href="/wiki/Mixed_model" title="Mixed model">Mixed effects models</a></li> <li><a href="/wiki/Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li> <li><a href="/wiki/Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">Multivariate adaptive regression splines (MARS)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li> <li><a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li> <li><a href="/wiki/General_linear_model" title="General linear model">General linear model</a></li> <li><a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Bayesian regression</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Non-standard predictors</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li> <li><a href="/wiki/Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li> <li><a href="/wiki/Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li> <li><a href="/wiki/Isotonic_regression" title="Isotonic regression">Isotonic</a></li> <li><a href="/wiki/Robust_regression" title="Robust regression">Robust</a></li> <li><a href="/wiki/Heteroscedasticity" class="mw-redirect" title="Heteroscedasticity">Heteroscedasticity</a></li> <li><a href="/wiki/Homoscedasticity" class="mw-redirect" title="Homoscedasticity">Homoscedasticity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Exponential_family" title="Exponential family">Exponential families</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic <span style="font-size:85%;">(Bernoulli)</span></a> / <a href="/wiki/Binomial_regression" title="Binomial regression">Binomial</a> / <a href="/wiki/Poisson_regression" title="Poisson regression">Poisson regressions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Partition_of_sums_of_squares" title="Partition of sums of squares">Partition of variance</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li> <li><a href="/wiki/Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li> <li><a href="/wiki/Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">Degrees of freedom</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_Multivariate_/_Time-series_/_Survival_analysis" style="font-size:114%;margin:0 4em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a> / <a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a> / <a href="/wiki/Time_series" title="Time series">Time-series</a> / <a href="/wiki/Survival_analysis" title="Survival analysis">Survival analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohen%27s_kappa" title="Cohen's kappa">Cohen's kappa</a></li> <li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Graphical_model" title="Graphical model">Graphical model</a></li> <li><a href="/wiki/Poisson_regression" title="Poisson regression">Log-linear model</a></li> <li><a href="/wiki/McNemar%27s_test" title="McNemar's test">McNemar's test</a></li> <li><a href="/wiki/Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics" title="Cochran–Mantel–Haenszel statistics">Cochran–Mantel–Haenszel statistics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_linear_model" title="General linear model">Regression</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li> <li><a href="/wiki/Principal_component_analysis" title="Principal component analysis">Principal components</a></li> <li><a href="/wiki/Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li> <li><a href="/wiki/Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li> <li><a href="/wiki/Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li> <li><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></li> <li><a href="/wiki/Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a> <ul><li><a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li> <li><a href="/wiki/Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a> <ul><li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a> <ul><li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Normal</a></li></ul></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Time_series" title="Time series">Time-series</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li> <li><a href="/wiki/Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationarity</a></li> <li><a href="/wiki/Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li> <li><a href="/wiki/Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li> <li><a href="/wiki/Cointegration" title="Cointegration">Cointegration</a></li> <li><a href="/wiki/Structural_break" title="Structural break">Structural break</a></li> <li><a href="/wiki/Granger_causality" title="Granger causality">Granger causality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Specific tests</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li> <li><a href="/wiki/Johansen_test" title="Johansen test">Johansen</a></li> <li><a href="/wiki/Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size:85%;">(Ljung–Box)</span></a></li> <li><a href="/wiki/Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li> <li><a href="/wiki/Breusch%E2%80%93Godfrey_test" title="Breusch–Godfrey test">Breusch–Godfrey</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Time_domain" title="Time domain">Time domain</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a> <ul><li><a href="/wiki/Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li> <li><a href="/wiki/Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li> <li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li> <li><a href="/wiki/Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size:85%;">(Box–Jenkins)</span></a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li> <li><a href="/wiki/Vector_autoregression" title="Vector autoregression">Vector autoregression (VAR)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Frequency_domain" title="Frequency domain">Frequency domain</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li> <li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li> <li><a href="/wiki/Wavelet" title="Wavelet">Wavelet</a></li> <li><a href="/wiki/Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survival_analysis" title="Survival analysis">Survival</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Survival_function" title="Survival function">Survival function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator (product limit)</a></li> <li><a href="/wiki/Proportional_hazards_model" title="Proportional hazards model">Proportional hazards models</a></li> <li><a href="/wiki/Accelerated_failure_time_model" title="Accelerated failure time model">Accelerated failure time (AFT) model</a></li> <li><a href="/wiki/First-hitting-time_model" title="First-hitting-time model">First hitting time</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Failure_rate" title="Failure rate">Hazard function</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nelson%E2%80%93Aalen_estimator" title="Nelson–Aalen estimator">Nelson–Aalen estimator</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Test</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Log-rank_test" class="mw-redirect" title="Log-rank test">Log-rank test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Applications" style="font-size:114%;margin:0 4em"><a href="/wiki/List_of_fields_of_application_of_statistics" title="List of fields of application of statistics">Applications</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Biostatistics" title="Biostatistics">Biostatistics</a></th><td class="navbox-list-with-group navbox-list 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