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mw-list-item"><a href="https://ast.wikipedia.org/wiki/Dise%C3%B1u_esperimental" title="Diseñu esperimental – Asturian" lang="ast" hreflang="ast" data-title="Diseñu esperimental" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Vollst%C3%A4ndiger_Versuchsplan" title="Vollständiger Versuchsplan – German" lang="de" hreflang="de" data-title="Vollständiger Versuchsplan" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Dise%C3%B1o_factorial" title="Diseño factorial – Spanish" lang="es" hreflang="es" data-title="Diseño factorial" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A2%D8%B2%D9%85%D8%A7%DB%8C%D8%B4_%D8%B9%D8%A7%D9%85%D9%84%DB%8C" title="آزمایش عاملی – Persian" lang="fa" hreflang="fa" data-title="آزمایش عاملی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Plan_factoriel" title="Plan factoriel – French" lang="fr" hreflang="fr" data-title="Plan factoriel" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Eksperimen_faktorial" title="Eksperimen faktorial – Indonesian" lang="id" hreflang="id" data-title="Eksperimen faktorial" 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class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B8_%D0%BD%D0%B0%D1%86%D1%80%D1%82%D0%B8" title="Факторијални нацрти – Serbian" lang="sr" hreflang="sr" data-title="Факторијални нацрти" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B2%D0%BD%D0%B8%D0%B9_%D1%84%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%BD%D0%B8%D0%B9_%D0%B5%D0%BA%D1%81%D0%BF%D0%B5%D1%80%D0%B8%D0%BC%D0%B5%D0%BD%D1%82" title="Повний факторний експеримент – Ukrainian" lang="uk" hreflang="uk" data-title="Повний факторний експеримент" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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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 factorial design. For factor loadings, see <a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a>. For factorial numbers (n!), see <a href="/wiki/Factorial" title="Factorial">Factorial</a>.</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Response_surface_metodology.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Response_surface_metodology.jpg/313px-Response_surface_metodology.jpg" decoding="async" width="313" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Response_surface_metodology.jpg/470px-Response_surface_metodology.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Response_surface_metodology.jpg/626px-Response_surface_metodology.jpg 2x" data-file-width="1920" data-file-height="1200" /></a><figcaption>Designed experiments with full factorial design (left), response surface with second-degree polynomial (right)</figcaption></figure> <p>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, a full <b>factorial experiment</b> is an experiment whose design consists of two or more factors, each with discrete possible values or "levels", and whose <a href="/wiki/Experimental_unit" class="mw-redirect" title="Experimental unit">experimental units</a> take on all possible combinations of these levels across all such factors. A full <b>factorial design</b> may also be called a <b>fully crossed design</b>. Such an experiment allows the investigator to study the effect of each factor on the <a href="/wiki/Response_variable" class="mw-redirect" title="Response variable">response variable</a>, as well as the effects of <a href="/wiki/Interaction_(statistics)" title="Interaction (statistics)">interactions</a> between factors on the response variable. </p><p>For the vast majority of factorial experiments, each factor has only two levels. For example, with two factors each taking two levels, a factorial experiment would have four treatment combinations in total, and is usually called a <i>2×2 factorial design</i>. In such a design, the interaction between the variables is often the most important. This applies even to scenarios where a main effect and an interaction are present. </p><p>If the number of combinations in a full factorial design is too high to be logistically feasible, a <a href="/wiki/Fractional_factorial_design" title="Fractional factorial design">fractional factorial design</a> may be done, in which some of the possible combinations (usually at least half) are omitted. </p><p>Other terms for "treatment combinations" are often used, such as <i>runs</i> (of an experiment), <i>points</i> (viewing the combinations as <a href="/wiki/Vertex_(graph_theory)" title="Vertex (graph theory)">vertices of a graph</a>, and <i>cells</i> (arising as intersections of rows and columns). </p> <meta property="mw:PageProp/toc" /> <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=Factorial_experiment&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Factorial designs were used in the 19th century by <a href="/wiki/John_Bennet_Lawes" title="John Bennet Lawes">John Bennet Lawes</a> and <a href="/wiki/Joseph_Henry_Gilbert" title="Joseph Henry Gilbert">Joseph Henry Gilbert</a> of the <a href="/wiki/Rothamsted_Experimental_Station" class="mw-redirect" title="Rothamsted Experimental Station">Rothamsted Experimental Station</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <p><a href="/wiki/Ronald_Fisher" title="Ronald Fisher">Ronald Fisher</a> argued in 1926 that "complex" designs (such as factorial designs) were more efficient than studying one factor at a time.<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> Fisher wrote, <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></p><blockquote class="templatequote"><p>"No aphorism is more frequently repeated in connection with field trials, than that we must ask Nature few questions, or, ideally, one question, at a time. The writer is convinced that this view is wholly mistaken. Nature, he suggests, will best respond to a logical and carefully thought out questionnaire; indeed, if we ask her a single question, she will often refuse to answer until some other topic has been discussed."</p></blockquote> <p>A factorial design allows the effect of several factors and even interactions between them to be determined with the same number of trials as are necessary to determine any one of the effects by itself with the same degree of accuracy. </p><p><a href="/wiki/Frank_Yates" title="Frank Yates">Frank Yates</a> made significant contributions, particularly in the analysis of designs, by the <a href="/wiki/Yates_analysis" title="Yates analysis">Yates analysis</a>. </p><p>The term "factorial" may not have been used in print before 1935, when Fisher used it in his book <i><a href="/wiki/The_Design_of_Experiments" title="The Design of Experiments">The Design of Experiments</a></i>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Advantages_and_disadvantages_of_factorial_experiments">Advantages and disadvantages of factorial experiments</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorial_experiment&action=edit&section=2" title="Edit section: Advantages and disadvantages of factorial experiments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many people examine the effect of only a single factor or variable. Compared to such <a href="/wiki/One-factor-at-a-time_method" title="One-factor-at-a-time method">one-factor-at-a-time</a> (OFAT) experiments, factorial experiments offer several advantages<sup id="cite_ref-Montgomery_4-0" class="reference"><a href="#cite_note-Montgomery-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Oehlert_5-0" class="reference"><a href="#cite_note-Oehlert-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <ul><li>Factorial designs are more efficient than OFAT experiments. They provide more information at similar or lower cost. They can find optimal conditions faster than OFAT experiments.</li> <li>When the effect of one factor is different for different levels of another factor, it cannot be detected by an OFAT experiment design. Factorial designs are required to detect such <a href="/wiki/Interaction_(statistics)" title="Interaction (statistics)">interactions</a>. Use of OFAT when interactions are present can lead to serious misunderstanding of how the response changes with the factors.</li> <li>Factorial designs allow the effects of a factor to be estimated at several levels of the other factors, yielding conclusions that are valid over a range of experimental conditions.</li></ul> <p>The main disadvantage of the full factorial design is its sample size requirement, which grows exponentially with the number of factors or inputs considered.<sup id="cite_ref-Tong2006_6-0" class="reference"><a href="#cite_note-Tong2006-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Alternative strategies with improved computational efficiency include <a href="/wiki/Fractional_factorial_design" title="Fractional factorial design">fractional factorial designs</a>, <a href="/wiki/Latin_hypercube_sampling" title="Latin hypercube sampling">Latin hypercube sampling</a>, and <a href="/wiki/Quasi-Monte_Carlo_method" title="Quasi-Monte Carlo method">quasi-random sampling techniques</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Example_of_advantages_of_factorial_experiments">Example of advantages of factorial experiments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorial_experiment&action=edit&section=3" title="Edit section: Example of advantages of factorial experiments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In his book, <i>Improving Almost Anything: Ideas and Essays</i>, statistician <a href="/wiki/George_E._P._Box" title="George E. P. Box">George Box</a> gives many examples of the benefits of factorial experiments. Here is one.<sup id="cite_ref-Box2006_7-0" class="reference"><a href="#cite_note-Box2006-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Engineers at the bearing manufacturer SKF wanted to know if changing to a less expensive "cage" design would affect bearing life. The engineers asked Christer Hellstrand, a statistician, for help in designing the experiment.<sup id="cite_ref-Hellstrand1989_8-0" class="reference"><a href="#cite_note-Hellstrand1989-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-right" typeof="mw:File/Frameless"><a href="/wiki/File:Cube_plot_for_bearing_life.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Cube_plot_for_bearing_life.svg/300px-Cube_plot_for_bearing_life.svg.png" decoding="async" width="300" height="190" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Cube_plot_for_bearing_life.svg/450px-Cube_plot_for_bearing_life.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/58/Cube_plot_for_bearing_life.svg/600px-Cube_plot_for_bearing_life.svg.png 2x" data-file-width="538" data-file-height="341" /></a><figcaption></figcaption></figure> <p>Box reports the following. "The results were assessed by an accelerated life test. … The runs were expensive because they needed to be made on an actual production line and the experimenters were planning to make four runs with the standard cage and four with the modified cage. Christer asked if there were other factors they would like to test. They said there were, but that making added runs would exceed their budget. Christer showed them how they could test two additional factors "for free" – without increasing the number of runs and without reducing the accuracy of their estimate of the cage effect. In this arrangement, called a 2×2×2 factorial design, each of the three factors would be run at two levels and all the eight possible combinations included. The various combinations can conveniently be shown as the vertices of a cube ... " "In each case, the standard condition is indicated by a minus sign and the modified condition by a plus sign. The factors changed were heat treatment, outer ring osculation, and cage design. The numbers show the relative lengths of lives of the bearings. If you look at [the cube plot], you can see that the choice of cage design did not make a lot of difference. … But, if you average the pairs of numbers for cage design, you get the [table below], which shows what the two other factors did. … It led to the extraordinary discovery that, in this particular application, the life of a bearing can be increased fivefold if the two factor(s) outer ring osculation and inner ring heat treatments are increased together." </p> <table class="wikitable" style="margin:0 0 10px 1em; text-align:center; width:9em;"> <caption>Bearing life vs. heat and osculation </caption> <tbody><tr> <th></th> <th>Osculation −</th> <th>Osculation + </th></tr> <tr> <th>Heat − </th> <td>18</td> <td>23 </td></tr> <tr> <th>Heat + </th> <td>21</td> <td>106 </td></tr></tbody></table> <p>"Remembering that bearings like this one have been made for decades, it is at first surprising that it could take so long to discover so important an improvement. A likely explanation is that, because most engineers have, until recently, employed only one factor at a time experimentation, <a href="/wiki/Interaction_(statistics)" title="Interaction (statistics)">interaction</a> effects have been missed." </p> <div class="mw-heading mw-heading2"><h2 id="Example">Example</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorial_experiment&action=edit&section=4" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The simplest factorial experiment contains two levels for each of two factors. Suppose an engineer wishes to study the total power used by each of two different motors, A and B, running at each of two different speeds, 2000 or 3000 RPM. The factorial experiment would consist of four experimental units: motor A at 2000 RPM, motor B at 2000 RPM, motor A at 3000 RPM, and motor B at 3000 RPM. Each combination of a single level selected from every factor is present once. </p><p>This experiment is an example of a 2<sup>2</sup> (or 2×2) factorial experiment, so named because it considers two levels (the base) for each of two factors (the power or superscript), or #levels<sup>#factors</sup>, producing 2<sup>2</sup>=4 factorial points. </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Factorial_Design.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Factorial_Design.svg/250px-Factorial_Design.svg.png" decoding="async" width="250" height="127" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Factorial_Design.svg/375px-Factorial_Design.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/94/Factorial_Design.svg/500px-Factorial_Design.svg.png 2x" data-file-width="969" data-file-height="494" /></a><figcaption> Cube plot for factorial design </figcaption></figure> <p>Designs can involve many independent variables. As a further example, the effects of three input variables can be evaluated in eight experimental conditions shown as the corners of a cube. </p><p>This can be conducted with or without replication, depending on its intended purpose and available resources. It will provide the effects of the three independent variables on the dependent variable and possible interactions. </p> <div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorial_experiment&action=edit&section=5" title="Edit section: Notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Factorial experiments are described by two things: the number of factors, and the number of levels of each factor. For example, a 2×3 factorial experiment has two factors, the first at 2 levels and the second at 3 levels. Such an experiment has 2×3=6 treatment combinations or cells. Similarly, a 2×2×3 experiment has three factors, two at 2 levels and one at 3, for a total of 12 treatment combinations. If every factor has <i>s</i> levels (a so-called <i>fixed-level</i> or <i>symmetric</i> design), the experiment is typically denoted by <i>s<sup>k</sup></i>, where <i>k</i> is the number of factors. Thus a 2<sup>5</sup> experiment has 5 factors, each at 2 levels. Experiments that are not fixed-level are said to be <i>mixed-level</i> or <i>asymmetric</i>. </p><p>There are various traditions to denote the levels of each factor. If a factor already has natural units, then those are used. For example, a shrimp aquaculture experiment<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> might have factors <i>temperature</i> at 25 °C and 35 °C, <i>density</i> at 80 or 160 shrimp/40 liters, and <i>salinity</i> at 10%, 25% and 40%. In many cases, though, the factor levels are simply categories, and the coding of levels is somewhat arbitrary. For example, the levels of an 6-level factor might simply be denoted 1, 2, ..., 6. </p> <table class="wikitable" style="float:right; center; margin: 0px; margin-top: 5px"> <caption>The cells in a 2 × 3 experiment </caption> <tbody><tr> <td style="background:#EAECF0;background:linear-gradient(to top right,#EAECF0 49%,#AAA 49.5%,#AAA 50.5%,#EAECF0 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right"><i>B</i></div><div style="margin-right:2em;text-align:left"><i>A</i></div> </td> <td>1 </td> <td>2 </td> <td>3 </td></tr> <tr> <td>1 </td> <td>11 </td> <td>12 </td> <td>13 </td></tr> <tr> <td>2 </td> <td>21 </td> <td>22 </td> <td>23 </td></tr></tbody></table> <p>Treatment combinations are denoted by ordered pairs or, more generally, ordered <a href="/wiki/Tuple" title="Tuple">tuples</a>. In the aquaculture experiment, the ordered triple (25, 80, 10) represents the treatment combination having the lowest level of each factor. In a general 2×3 experiment the ordered pair (2, 1) would indicate the cell in which factor <i>A</i> is at level 2 and factor <i>B</i> at level 1. The parentheses are often dropped, as shown in the accompanying table. </p> <table class="wikitable" style="float:right; margin:0 0 10px 1em; text-align:center; width:9em;"> <caption>Cell notation in a 2×2 experiment </caption> <tbody><tr> <td>Both low</td> <td>00</td> <td>−−</td> <td>(1) </td></tr> <tr> <td><i>A</i> low</td> <td>01</td> <td>−+</td> <td>a </td></tr> <tr> <td><i>B</i> low</td> <td>10</td> <td>+−</td> <td>b </td></tr> <tr> <td>Both high</td> <td>11</td> <td>++</td> <td>ab </td></tr></tbody></table> <p>To denote factor levels in 2<sup><i>k</i></sup> experiments, three particular systems appear in the literature: </p> <ul><li>The values 1 and 0;</li> <li>the values 1 and −1, often simply abbreviated by + and −;</li> <li>A lower-case letter with the exponent 0 or 1.</li></ul> <p>If these values represent "low" and "high" settings of a treatment, then it is natural to have 1 represent "high", whether using 0 and 1 or −1 and 1. This is illustrated in the accompanying table for a 2×2 experiment. If the factor levels are simply categories, the correspondence might be different; for example, it is natural to represent "control" and "experimental" conditions by coding "control" as 0 if using 0 and 1, and as 1 if using 1 and −1.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> An example of the latter is given <a href="#2x2x2-ex">below</a>. That example illustrates another use of the coding +1 and −1. </p><p>For other fixed-level (<i>s<sup>k</sup></i>) experiments, the values 0, 1, ..., <i>s</i>−1 are often used to denote factor levels. These are the values of the integers <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> <i>s</i> when <i>s</i> is prime.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>note 2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Contrasts,_main_effects_and_interactions"><span id="Contrasts.2C_main_effects_and_interactions"></span>Contrasts, main effects and interactions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorial_experiment&action=edit&section=6" title="Edit section: Contrasts, main effects and interactions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" style="float:right; center; margin: 0px; margin-top: 5px"> <caption>Cell means in a 2 × 3 factorial experiment </caption> <tbody><tr> <td style="background:#EAECF0;background:linear-gradient(to top right,#EAECF0 49%,#AAA 49.5%,#AAA 50.5%,#EAECF0 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right"><i>B</i></div><div style="margin-right:2em;text-align:left"><i>A</i></div> </td> <td>1 </td> <td>2 </td> <td>3 </td></tr> <tr> <td>1 </td> <td>μ<sub>11</sub> </td> <td>μ<sub>12</sub> </td> <td>μ<sub>13</sub> </td></tr> <tr> <td>2 </td> <td>μ<sub>21</sub> </td> <td>μ<sub>22</sub> </td> <td>μ<sub>23</sub> </td></tr></tbody></table><p>The <a href="/wiki/Expected_value" title="Expected value"><i>expected response</i></a> to a given treatment combination is called a <i>cell mean</i>,<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> usually denoted using the Greek letter μ. (The term <i>cell</i> is borrowed from its use in <a href="/wiki/Table_(information)" title="Table (information)">tables of data</a>.) This notation is illustrated here for the 2 × 3 experiment. </p><p>A <i>contrast in cell means</i> is a <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of cell means in which the coefficients sum to 0. Contrasts are of interest in themselves, and are the building blocks by which main effects and interactions are defined. </p><p>In the 2 × 3 experiment illustrated here, the expression </p> <div style="text-align: center;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{11}-\mu _{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{11}-\mu _{12}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88dccdf410eba7b7c87aebd9e1d045d8c583c6f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.396ex; height:2.509ex;" alt="{\displaystyle \mu _{11}-\mu _{12}}"></span> </p> </div> <p>is a contrast that compares the mean responses of the treatment combinations 11 and 12. (The coefficients here are 1 and –1.) The contrast </p> <div style="text-align: center;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{11}+\mu _{12}+\mu _{13}-\mu _{21}-\mu _{22}-\mu _{23}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{11}+\mu _{12}+\mu _{13}-\mu _{21}-\mu _{22}-\mu _{23}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bdf3df4dc3639894e70b075876ce1a739ca0139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.869ex; height:2.509ex;" alt="{\displaystyle \mu _{11}+\mu _{12}+\mu _{13}-\mu _{21}-\mu _{22}-\mu _{23}}"></span> </p> </div> <p>is said to <i>belong to the main effect of factor A</i> as it contrasts the responses to the "1" level of factor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> with those for the "2" level. The main effect of <i>A</i> is said to be <i>absent</i> if this expression equals 0. </p><p><i>Interaction</i> in a factorial experiment is the lack of <a href="/wiki/Interaction_(statistics)#In_ANOVA" title="Interaction (statistics)">additivity</a> between factors, and is also expressed by contrasts. In the 2 × 3 experiment, the contrasts </p> <div style="text-align: center;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{11}-\mu _{12}-\mu _{21}+\mu _{22}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{11}-\mu _{12}-\mu _{21}+\mu _{22}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/090655b946621e3f1db60eb6f0d4e3eded83699b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.633ex; height:2.509ex;" alt="{\displaystyle \mu _{11}-\mu _{12}-\mu _{21}+\mu _{22}}"></span>   and   <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{11}-\mu _{13}-\mu _{21}+\mu _{23}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{11}-\mu _{13}-\mu _{21}+\mu _{23}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e57b2d5a4c4b8529189899d147e35d23256671ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.633ex; height:2.509ex;" alt="{\displaystyle \mu _{11}-\mu _{13}-\mu _{21}+\mu _{23}}"></span> </p> </div> <p><i>belong to the A × B interaction</i>; interaction is <i>absent</i> (additivity is <i>present</i>) if these expressions equal 0.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Additivity may be viewed as a kind of parallelism between factors, as illustrated in the Analysis section <a href="#Analysis_example">below</a>. </p><p>Since it is the coefficients of these contrasts that carry the essential information, they are often displayed as <a href="/wiki/Row_and_column_vectors" title="Row and column vectors">column vectors</a>. For the example above, such a table might look like this:<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> <span class="anchor" id="2x3-ex"></span> </p> <table class="wikitable defaultright col1center col2center col3center" style="text-align: center; margin: auto; margin-top: 5px"> <caption>Contrast vectors for the <br /> 2 × 3 factorial experiment </caption> <tbody><tr> <th>cell</th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span></th> <th colspan="2"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span></th> <th colspan="2"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\times B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>×</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\times B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65f31ae45b0098f06b5d22c38d317eb097a88fa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.348ex; height:2.176ex;" alt="{\displaystyle A\times B}"></span> </th></tr> <tr> <td>11</td> <td>1</td> <td>1</td> <td>0</td> <td>1</td> <td>1 </td></tr> <tr> <td>12</td> <td>1</td> <td>−1</td> <td>1</td> <td>-1</td> <td>0 </td></tr> <tr> <td>13</td> <td>1</td> <td>0</td> <td>−1</td> <td>0</td> <td>−1 </td></tr> <tr> <td>21</td> <td>−1</td> <td>1</td> <td>0</td> <td>−1</td> <td>-1 </td></tr> <tr> <td>22</td> <td>−1</td> <td>−1</td> <td>1</td> <td>1</td> <td>0 </td></tr> <tr> <td>23</td> <td>−1</td> <td>0</td> <td>−1</td> <td>0</td> <td>1 </td></tr></tbody></table> <p>The columns of such a table are called <i>contrast vectors</i>: their components add up to 0. Each effect is determined by both the <i>pattern of components</i> in its columns and the <i>number of columns</i>. </p><p>The <b>patterns of components</b> of these columns reflect the general definitions given by <a href="/wiki/Raj_Chandra_Bose" title="Raj Chandra Bose">Bose</a>:<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <ul><li>A contrast vector <i>belongs to the main effect of a particular factor</i> if the values of its components depend only on the level of that factor.</li> <li>A contrast vector <i>belongs to the interaction of two factors</i>, say <i>A</i> and <i>B</i>, if (i) the values of its components depend only on the levels of <i>A</i> and <i>B</i>, and (ii) it is <a href="/wiki/Orthogonality_(mathematics)#Definitions" title="Orthogonality (mathematics)">orthogonal</a> (perpendicular) to the contrast vectors representing the main effects of <i>A</i> and <i>B</i>. <sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>note 3<span class="cite-bracket">]</span></a></sup></li></ul> <p>Similar definitions hold for interactions of more than two factors. In the 2 × 3 example, for instance, the pattern of the <i>A</i> column follows the pattern of the levels of factor <i>A</i>, indicated by the first component of each cell. </p><p>The <b>number of columns</b> needed to specify each effect is the <i>degrees of freedom</i> for the effect,<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>note 4<span class="cite-bracket">]</span></a></sup> and is an essential quantity in the <a href="/wiki/Analysis_of_variance#Algorithm" title="Analysis of variance">analysis of variance</a>. The formula is as follows:<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <ul><li>A main effect for a factor with <i>s</i> levels has <i>s</i>−1 degrees of freedom.</li> <li>The interaction of two factors with <i>s</i><sub>1</sub> and <i>s</i><sub>2</sub> levels, respectively, has (<i>s</i><sub>1</sub>−1)(<i>s</i><sub>2</sub>−1) degrees of freedom.</li></ul> <p>The formula for more than two factors follows this pattern. In the 2 × 3 example above, the degrees of freedom for the two main effects and the interaction — the number of columns for each — are 1, 2 and 2, respectively. </p> <div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorial_experiment&action=edit&section=7" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the tables in the following examples, the entries in the "cell" column are treatment combinations: The first component of each combination is the level of factor <i>A</i>, the second for factor <i>B</i>, and the third (in the 2 × 2 × 2 example) the level of factor <i>C</i>. The entries in each of the other columns sum to 0, so that each column is a contrast vector. </p> <table class="wikitable sortable" style="float:right;"> <caption>Contrast vectors in a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\times 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\times 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc0d4d6106875f8006be1d898512ca5843bad8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 3\times 3}"></span> experiment </caption> <tbody><tr> <th>cell</th> <th colspan="2"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span></th> <th colspan="2"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span></th> <th colspan="4"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\times B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>×</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\times B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65f31ae45b0098f06b5d22c38d317eb097a88fa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.348ex; height:2.176ex;" alt="{\displaystyle A\times B}"></span> </th></tr> <tr> <td>00</td> <td>1</td> <td>1</td> <td>1</td> <td>1</td> <td>1</td> <td>1</td> <td>1</td> <td>1 </td></tr> <tr> <td>01</td> <td>1</td> <td>1</td> <td>-1</td> <td>0</td> <td>-1</td> <td>0</td> <td>-1</td> <td>0 </td></tr> <tr> <td>02</td> <td>1</td> <td>1</td> <td>0</td> <td>-1</td> <td>0</td> <td>-1</td> <td>0</td> <td>-1 </td></tr> <tr> <td>10</td> <td>-1</td> <td>0</td> <td>1</td> <td>1</td> <td>-1</td> <td>-1</td> <td>0</td> <td>0 </td></tr> <tr> <td>11</td> <td>-1</td> <td>0</td> <td>-1</td> <td>0</td> <td>1</td> <td>0</td> <td>0</td> <td>0 </td></tr> <tr> <td>12</td> <td>-1</td> <td>0</td> <td>0</td> <td>-1</td> <td>0</td> <td>1</td> <td>0</td> <td>0 </td></tr> <tr> <td>20</td> <td>0</td> <td>-1</td> <td>1</td> <td>1</td> <td>0</td> <td>0</td> <td>-1</td> <td>-1 </td></tr> <tr> <td>21</td> <td>0</td> <td>-1</td> <td>-1</td> <td>0</td> <td>0</td> <td>0</td> <td>1</td> <td>0 </td></tr> <tr> <td>22</td> <td>0</td> <td>-1</td> <td>0</td> <td>-1</td> <td>0</td> <td>0</td> <td>0</td> <td>1 </td></tr></tbody></table> <p><b>A 3 × 3 experiment:</b> Here we expect 3-1 = 2 degrees of freedom each for the main effects of factors <i>A</i> and <i>B</i>, and (3-1)(3-1) = 4 degrees of freedom for the <i>A × B</i> interaction. This accounts for the number of columns for each effect in the accompanying table. </p><p>The two contrast vectors for <i>A</i> depend only on the level of factor <i>A</i>. This can be seen by noting that the pattern of entries in each <i>A</i> column is the same as the pattern of the first component of "cell". (If necessary, sorting the table on <i>A</i> will show this.) Thus these two vectors belong to the main effect of <i>A</i>. Similarly, the two contrast vectors for <i>B</i> depend only on the level of factor <i>B</i>, namely the second component of "cell", so they belong to the main effect of <i>B</i>. </p><p>The last four column vectors belong to the <i>A × B</i> interaction, as their entries depend on the values of both factors, and as all four columns are orthogonal to the columns for <i>A</i> and <i>B</i>. The latter can be verified by taking <a href="/wiki/Dot_product" title="Dot product">dot products</a>. </p><p><b>A 2 × 2 × 2 experiment:</b> This will have 1 degree of freedom for every main effect and interaction. For example, a two-factor interaction will have (2-1)(2-1) = 1 degree of freedom. Thus just a single column is needed to specify each of the seven effects.<span class="anchor" id="2x2x2-ex"></span> </p> <table class="wikitable defaultright col1center col2center col3center sortable" style="text-align: center; margin: auto; margin-top: 5px"> <caption>Contrast vectors in a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2\times 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/128c392aacbd66d7eebc7f18ddd6256ca69e327f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.168ex; height:2.176ex;" alt="{\displaystyle 2\times 2\times 2}"></span> experiment </caption> <tbody><tr> <th>cell</th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04153f9681e5b06066357774475c04aaef3a8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:2.176ex;" alt="{\displaystyle AB}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b930d133ca536a071bec52a9acc4b05482890d53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.509ex; height:2.176ex;" alt="{\displaystyle AC}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74e0f24a49061dcd63874f7d81f395b5f38800f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.53ex; height:2.176ex;" alt="{\displaystyle BC}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}"></span> </th></tr> <tr> <td>000</td> <td>1</td> <td>1</td> <td>1</td> <td>1</td> <td>1</td> <td>1</td> <td>1 </td></tr> <tr> <td>001</td> <td>1</td> <td>1</td> <td>−1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>−1 </td></tr> <tr> <td>010</td> <td>1</td> <td>−1</td> <td>1</td> <td>−1</td> <td>1</td> <td>−1</td> <td>−1 </td></tr> <tr> <td>011</td> <td>1</td> <td>−1</td> <td>−1</td> <td>−1</td> <td>−1</td> <td>1</td> <td>1 </td></tr> <tr> <td>100</td> <td>−1</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>1</td> <td>−1 </td></tr> <tr> <td>101</td> <td>−1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>1</td> <td>−1</td> <td>1 </td></tr> <tr> <td>110</td> <td>−1</td> <td>−1</td> <td>1</td> <td>1</td> <td>−1</td> <td>−1</td> <td>1 </td></tr> <tr> <td>111</td> <td>−1</td> <td>−1</td> <td>−1</td> <td>1</td> <td>1</td> <td>1</td> <td>−1 </td></tr></tbody></table> <p>The columns for <i>A</i>, <i>B</i> and <i>C</i> represent the corresponding main effects, as the entries in each column depend only on the level of the corresponding factor. For example, the entries in the <i>B</i> column follow the same pattern as the middle component of "cell", as can be seen by sorting on <i>B</i>. </p><p>The columns for <i>AB</i>, <i>AC</i> and <i>BC</i> represent the corresponding two-factor interactions. For example, (i) the entries in the <i>BC</i> column depend on the second and third (<i>B</i> and <i>C</i>) components of <i>cell</i>, and are independent of the first (<i>A</i>) component, as can be seen by sorting on <i>BC</i>; and (ii) the <i>BC</i> column is orthogonal to columns <i>B</i> and <i>C</i>, as can be verified by computing dot products. </p><p>Finally, the <i>ABC</i> column represents the three-factor interaction: its entries depend on the levels of all three factors, and it is orthogonal to the other six contrast vectors. </p><p>Combined and read row-by-row, columns <i>A, B, C</i> give an alternate notation, mentioned above, for the treatment combinations (cells) in this experiment: cell 000 corresponds to +++, 001 to ++−, etc. </p><p>In columns <i>A</i> through <i>ABC</i>, the number 1 may be replaced by any constant, because the resulting columns will still be contrast vectors. For example, it is common to use the number 1/4 in 2 × 2 × 2 experiments <sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>note 5<span class="cite-bracket">]</span></a></sup> to define each main effect or interaction, and to declare, for example, that the contrast </p> <div style="text-align: center;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mu _{000}+\mu _{001}+\mu _{010}+\mu _{011})/4-(\mu _{100}+\mu _{101}+\mu _{110}+\mu _{111})/4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>000</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>001</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>010</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>011</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>100</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>101</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>110</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>111</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mu _{000}+\mu _{001}+\mu _{010}+\mu _{011})/4-(\mu _{100}+\mu _{101}+\mu _{110}+\mu _{111})/4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ed38f148baa501f6e6eebedb9b6ebbcbb747a44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.95ex; height:2.843ex;" alt="{\displaystyle (\mu _{000}+\mu _{001}+\mu _{010}+\mu _{011})/4-(\mu _{100}+\mu _{101}+\mu _{110}+\mu _{111})/4}"></span> </p> </div> <p>is "the" main effect of factor <i>A</i>, a numerical quantity that can be estimated.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>20<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=Factorial_experiment&action=edit&section=8" title="Edit section: Implementation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For more than two factors, a 2<sup><i>k</i></sup> factorial experiment can usually be recursively designed from a 2<sup><i>k</i>−1</sup> factorial experiment by replicating the 2<sup><i>k</i>−1</sup> experiment, assigning the first replicate to the first (or low) level of the new factor, and the second replicate to the second (or high) level. This framework can be generalized to, <i>e.g.</i>, designing three replicates for three level factors, <i>etc</i>. </p><p>A factorial experiment allows for estimation of <a href="/wiki/Experimental_error" class="mw-redirect" title="Experimental error">experimental error</a> in two ways. The experiment can be <a href="/wiki/Reproducibility" title="Reproducibility">replicated</a>, or the <a href="/wiki/Sparsity-of-effects_principle" title="Sparsity-of-effects principle">sparsity-of-effects principle</a> can often be exploited. Replication is more common for small experiments and is a very reliable way of assessing experimental error. When the number of factors is large (typically more than about 5 factors, but this does vary by application), replication of the design can become operationally difficult. In these cases, it is common to only run a single replicate of the design, and to assume that factor interactions of more than a certain order (say, between three or more factors) are negligible. Under this assumption, estimates of such high order interactions are estimates of an exact zero, thus really an estimate of experimental error. </p><p>When there are many factors, many experimental runs will be necessary, even without replication. For example, experimenting with 10 factors at two levels each produces 2<sup>10</sup>=1024 combinations. At some point this becomes infeasible due to high cost or insufficient resources. In this case, <a href="/wiki/Fractional_factorial_designs" class="mw-redirect" title="Fractional factorial designs">fractional factorial designs</a> may be used. </p><p>As with any statistical experiment, the experimental runs in a factorial experiment should be randomized to reduce the impact that <a href="/wiki/Biased_sample" class="mw-redirect" title="Biased sample">bias</a> could have on the experimental results. In practice, this can be a large operational challenge. </p><p>Factorial experiments can be used when there are more than two levels of each factor. However, the number of experimental runs required for three-level (or more) factorial designs will be considerably greater than for their two-level counterparts. Factorial designs are therefore less attractive if a researcher wishes to consider more than two levels. </p> <div class="mw-heading mw-heading2"><h2 id="Analysis">Analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorial_experiment&action=edit&section=9" title="Edit section: 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/Yates_analysis" title="Yates analysis">Yates analysis</a></div> <p>A factorial experiment can be analyzed using <a href="/wiki/ANOVA" class="mw-redirect" title="ANOVA">ANOVA</a> or <a href="/wiki/Regression_analysis" title="Regression analysis">regression analysis</a>.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> To compute the main effect of a factor "A" in a 2-level experiment, subtract the average response of all experimental runs for which A was at its low (or first) level from the average response of all experimental runs for which A was at its high (or second) level. </p><p>Other useful exploratory analysis tools for factorial experiments include <a href="/wiki/Main_effect" title="Main effect">main effects</a> plots, <a href="/wiki/Interaction_(statistics)" title="Interaction (statistics)">interaction plots</a>, <a href="/wiki/Pareto_chart" title="Pareto chart">Pareto plots</a>, and a <a href="/wiki/Normal_probability_plot" title="Normal probability plot">normal probability plot</a> of the estimated effects. </p><p>When the factors are continuous, two-level factorial designs assume that the effects are <a href="/wiki/Linear" class="mw-redirect" title="Linear">linear</a>. If a <a href="/wiki/Quadratic_function" title="Quadratic function">quadratic</a> effect is expected for a factor, a more complicated experiment should be used, such as a <a href="/wiki/Central_composite_design" title="Central composite design">central composite design</a>. Optimization of factors that could have quadratic effects is the primary goal of <a href="/wiki/Response_surface_methodology" title="Response surface methodology">response surface methodology</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Analysis_example">Analysis example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorial_experiment&action=edit&section=10" title="Edit section: Analysis example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> Montgomery<sup id="cite_ref-Montgomery_4-1" class="reference"><a href="#cite_note-Montgomery-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> gives the following example of analysis of a factorial experiment:.</p><blockquote><p>An engineer would like to increase the filtration rate (output) of a process to produce a chemical, and to reduce the amount of <a href="/wiki/Formaldehyde" title="Formaldehyde">formaldehyde</a> used in the process. Previous attempts to reduce the formaldehyde have lowered the filtration rate. The current filtration rate is 75 gallons per hour. Four factors are considered: temperature (A), pressure (B), formaldehyde concentration (C), and stirring rate (D). Each of the four factors will be tested at two levels.</p></blockquote><p>Onwards, the minus (−) and plus (+) signs will indicate whether the factor is run at a low or high level, respectively. </p><table class="wikitable sortable mw-collapsible"> <caption>Design matrix and resulting filtration rate </caption> <tbody><tr> <th>A </th> <th>B </th> <th>C </th> <th>D </th> <th>Filtration rate </th></tr> <tr> <td>− </td> <td>− </td> <td>− </td> <td>− </td> <td>45 </td></tr> <tr> <td>+ </td> <td>− </td> <td>− </td> <td>− </td> <td>71 </td></tr> <tr> <td>− </td> <td>+ </td> <td>− </td> <td>− </td> <td>48 </td></tr> <tr> <td>+ </td> <td>+ </td> <td>− </td> <td>− </td> <td>65 </td></tr> <tr> <td>− </td> <td>− </td> <td>+ </td> <td>− </td> <td>68 </td></tr> <tr> <td>+ </td> <td>− </td> <td>+ </td> <td>− </td> <td>60 </td></tr> <tr> <td>− </td> <td>+ </td> <td>+ </td> <td>− </td> <td>80 </td></tr> <tr> <td>+ </td> <td>+ </td> <td>+ </td> <td>− </td> <td>65 </td></tr> <tr> <td>− </td> <td>− </td> <td>− </td> <td>+ </td> <td>43 </td></tr> <tr> <td>+ </td> <td>− </td> <td>− </td> <td>+ </td> <td>100 </td></tr> <tr> <td>− </td> <td>+ </td> <td>− </td> <td>+ </td> <td>45 </td></tr> <tr> <td>+ </td> <td>+ </td> <td>− </td> <td>+ </td> <td>104 </td></tr> <tr> <td>− </td> <td>− </td> <td>+ </td> <td>+ </td> <td>75 </td></tr> <tr> <td>+ </td> <td>− </td> <td>+ </td> <td>+ </td> <td>86 </td></tr> <tr> <td>− </td> <td>+ </td> <td>+ </td> <td>+ </td> <td>70 </td></tr> <tr> <td>+ </td> <td>+ </td> <td>+ </td> <td>+ </td> <td>96 </td></tr></tbody></table> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 635px"> <div class="thumb" style="width: 630px; height: 430px;"><span typeof="mw:File"><a href="/wiki/File:Montgomery_filtration_rates.svg" class="mw-file-description" title="Plot of the main effects showing the filtration rates for the low (−) and high (+) settings for each factor."><img alt="Plot of the main effects showing the filtration rates for the low (−) and high (+) settings for each factor." src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Montgomery_filtration_rates.svg/431px-Montgomery_filtration_rates.svg.png" decoding="async" width="431" height="400" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Montgomery_filtration_rates.svg/646px-Montgomery_filtration_rates.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Montgomery_filtration_rates.svg/862px-Montgomery_filtration_rates.svg.png 2x" data-file-width="321" data-file-height="298" /></a></span></div> <div class="gallerytext">Plot of the main effects showing the filtration rates for the low (−) and high (+) settings for each factor.</div> </li> <li class="gallerybox" style="width: 635px"> <div class="thumb" style="width: 630px; height: 430px;"><span typeof="mw:File"><a href="/wiki/File:Interaction_plots_filtration_rate.png" class="mw-file-description" title="Plot of the interaction effects showing the mean filtration rate at each of the four possible combinations of levels for a given pair of factors."><img alt="Plot of the interaction effects showing the mean filtration rate at each of the four possible combinations of levels for a given pair of factors." src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Interaction_plots_filtration_rate.png/600px-Interaction_plots_filtration_rate.png" decoding="async" width="600" height="302" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/0/00/Interaction_plots_filtration_rate.png 1.5x" data-file-width="777" data-file-height="391" /></a></span></div> <div class="gallerytext">Plot of the interaction effects showing the mean filtration rate at each of the four possible combinations of levels for a given pair of factors.</div> </li> </ul><p>The non-parallel lines in the A:C interaction plot indicate that the effect of factor A depends on the level of factor C. A similar results holds for the A:D interaction. The graphs indicate that factor B has little effect on filtration rate. The <a href="/wiki/Analysis_of_variance" title="Analysis of variance">analysis of variance</a> (ANOVA) including all 4 factors and all possible interaction terms between them yields the coefficient estimates shown in the table below. </p><table class="wikitable sortable mw-collapsible"> <caption>ANOVA results </caption> <tbody><tr> <th>Coefficients </th> <th>Estimate </th></tr> <tr> <td>Intercept </td> <td>70.063 </td></tr> <tr> <td>A </td> <td>10.813 </td></tr> <tr> <td>B </td> <td>1.563 </td></tr> <tr> <td>C </td> <td>4.938 </td></tr> <tr> <td>D </td> <td>7.313 </td></tr> <tr> <td>A:B </td> <td>0.063 </td></tr> <tr> <td>A:C </td> <td>−9.063 </td></tr> <tr> <td>B:C </td> <td>1.188 </td></tr> <tr> <td>A:D </td> <td>8.313 </td></tr> <tr> <td>B:D </td> <td>−0.188 </td></tr> <tr> <td>C:D </td> <td>−0.563 </td></tr> <tr> <td>A:B:C </td> <td>0.938 </td></tr> <tr> <td>A:B:D </td> <td>2.063 </td></tr> <tr> <td>A:C:D </td> <td>−0.813 </td></tr> <tr> <td>B:C:D </td> <td>−1.313 </td></tr> <tr> <td>A:B:C:D </td> <td>0.688 </td></tr></tbody></table> <figure class="mw-halign-none" typeof="mw:File/Frame"><a href="/wiki/File:Pareto_plot_filtration_rate.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Pareto_plot_filtration_rate.svg/478px-Pareto_plot_filtration_rate.svg.png" decoding="async" width="478" height="280" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Pareto_plot_filtration_rate.svg/717px-Pareto_plot_filtration_rate.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/75/Pareto_plot_filtration_rate.svg/956px-Pareto_plot_filtration_rate.svg.png 2x" data-file-width="478" data-file-height="280" /></a><figcaption><a href="/wiki/Pareto_chart" title="Pareto chart">Pareto plot</a> showing the relative magnitude of the factor coefficients.</figcaption></figure> <p>Because there are 16 observations and 16 coefficients (intercept, main effects, and interactions), <a href="/wiki/P-value" title="P-value">p-values</a> cannot be calculated for this model. The coefficient values and the graphs suggest that the important factors are A, C, and D, and the interaction terms A:C and A:D. </p><p>The coefficients for A, C, and D are all positive in the ANOVA, which would suggest running the process with all three variables set to the high value. However, the main effect of each variable is the average over the levels of the other variables. The A:C interaction plot above shows that the effect of factor A depends on the level of factor C, and vice versa. Factor A (temperature) has very little effect on filtration rate when factor C is at the + level. But Factor A has a large effect on filtration rate when factor C (formaldehyde) is at the − level. The combination of A at the + level and C at the − level gives the highest filtration rate. This observation indicates how one-factor-at-a-time analyses can miss important interactions. Only by varying both factors A and C at the same time could the engineer discover that the effect of factor A depends on the level of factor C. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Montgomery_filtration_cube_plot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Montgomery_filtration_cube_plot.png/390px-Montgomery_filtration_cube_plot.png" decoding="async" width="390" height="203" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Montgomery_filtration_cube_plot.png/585px-Montgomery_filtration_cube_plot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/d/dd/Montgomery_filtration_cube_plot.png 2x" data-file-width="612" data-file-height="319" /></a><figcaption>Cube plot for the ANOVA using factors A, C, and D, and the interaction terms A:C and A:D. The plot aids in visualizing the result and shows that the best combination is A+, D+, and C−.</figcaption></figure> <p>The best filtration rate is seen when A and D are at the high level, and C is at the low level. This result also satisfies the objective of reducing formaldehyde (factor C). Because B does not appear to be important, it can be dropped from the model. Performing the ANOVA using factors A, C, and D, and the interaction terms A:C and A:D, gives the result shown in the following table, in which all the terms are significant (p-value < 0.05). </p> <table class="wikitable sortable mw-collapsible"> <caption>ANOVA results </caption> <tbody><tr> <th>Coefficient </th> <th>Estimate </th> <th>Standard error </th> <th>t value </th> <th>p-value </th></tr> <tr> <td>Intercept </td> <td>70.062 </td> <td>1.104 </td> <td>63.444 </td> <td>2.3 × 10<sup>−14</sup> </td></tr> <tr> <td>A </td> <td>10.812 </td> <td>1.104 </td> <td>9.791 </td> <td>1.9 × 10<sup>−6</sup> </td></tr> <tr> <td>C </td> <td>4.938 </td> <td>1.104 </td> <td>4.471 </td> <td>1.2 × 10<sup>−3</sup> </td></tr> <tr> <td>D </td> <td>7.313 </td> <td>1.104 </td> <td>6.622 </td> <td>5.9 × 10<sup>−5</sup> </td></tr> <tr> <td>A:C </td> <td>−9.063 </td> <td>1.104 </td> <td>−8.206 </td> <td>9.4 × 10<sup>−6</sup> </td></tr> <tr> <td>A:D </td> <td>8.312 </td> <td>1.104 </td> <td>7.527 </td> <td>2 × 10<sup>−5</sup> </td></tr></tbody></table> <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=Factorial_experiment&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Combinatorial_design" title="Combinatorial design">Combinatorial design</a></li> <li><a href="/wiki/Design_of_experiments" title="Design of experiments">Design of experiments</a></li> <li><a href="/wiki/Orthogonal_array" title="Orthogonal array">Orthogonal array</a></li> <li><a href="/wiki/Plackett%E2%80%93Burman_design" title="Plackett–Burman design">Plackett–Burman design</a></li> <li><a href="/wiki/Taguchi_methods" title="Taguchi methods">Taguchi methods</a></li> <li><a href="/wiki/Welch%27s_t-test" title="Welch's t-test">Welch's t-test</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Explanatory_footnotes">Explanatory footnotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorial_experiment&action=edit&section=12" title="Edit section: Explanatory footnotes"><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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">This choice gives the correspondence 01 ←→ +−, the opposite of that given in the table. There are also algebraic reasons for doing this.<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> The choice of coding via + and − is not important "as long as the labeling is consistent."<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></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">This choice of factor levels facilitates the use of algebra to handle certain issues of experimental design. If <i>s</i> is a power of a prime, the levels may be denoted by the elements of the <a href="/wiki/Finite_field" title="Finite field">finite field</a> <i>GF(s)</i> for the same reason.</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">Orthogonality is determined by computing the <a href="/wiki/Dot_product" title="Dot product">dot product</a> of vectors.</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">The degrees of freedom for an effect is actually the <a href="/wiki/Dimension_(vector_space)" title="Dimension (vector space)">dimension of a vector space</a>, namely the space of all contrast vectors belonging to that effect.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></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">And 1/2<sup>k-1</sup> in 2<sup>k</sup> experiments.</span> </li> </ol></div></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=Factorial_experiment&action=edit&section=13" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</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="CITEREFYatesMather1963" class="citation journal cs1"><a href="/wiki/Frank_Yates" title="Frank Yates">Yates, Frank</a>; <a href="/wiki/Kenneth_Mather" title="Kenneth Mather">Mather, Kenneth</a> (1963). <a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frsbm.1963.0006">"Ronald Aylmer Fisher"</a>. <i><a href="/wiki/Biographical_Memoirs_of_Fellows_of_the_Royal_Society" title="Biographical Memoirs of Fellows of the Royal Society">Biographical Memoirs of Fellows of the Royal Society</a></i>. <b>9</b>. London, England: <a href="/wiki/Royal_Society" title="Royal Society">Royal Society</a>: 91–120. <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/10.1098%2Frsbm.1963.0006">10.1098/rsbm.1963.0006</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biographical+Memoirs+of+Fellows+of+the+Royal+Society&rft.atitle=Ronald+Aylmer+Fisher&rft.volume=9&rft.pages=91-120&rft.date=1963&rft_id=info%3Adoi%2F10.1098%2Frsbm.1963.0006&rft.aulast=Yates&rft.aufirst=Frank&rft.au=Mather%2C+Kenneth&rft_id=https%3A%2F%2Fdoi.org%2F10.1098%252Frsbm.1963.0006&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorial+experiment" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFisher1926" class="citation journal cs1"><a href="/wiki/Ronald_Fisher" title="Ronald Fisher">Fisher, Ronald</a> (1926). <a rel="nofollow" class="external text" href="http://digital.library.adelaide.edu.au/dspace/bitstream/2440/15191/1/48.pdf">"The Arrangement of Field Experiments"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/w/index.php?title=Journal_of_the_Ministry_of_Agriculture_of_Great_Britain&action=edit&redlink=1" class="new" title="Journal of the Ministry of Agriculture of Great Britain (page does not exist)">Journal of the Ministry of Agriculture of Great Britain</a></i>. <b>33</b>. London, England: Ministry of Agriculture and Fisheries: 503–513.</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+Ministry+of+Agriculture+of+Great+Britain&rft.atitle=The+Arrangement+of+Field+Experiments&rft.volume=33&rft.pages=503-513&rft.date=1926&rft.aulast=Fisher&rft.aufirst=Ronald&rft_id=http%3A%2F%2Fdigital.library.adelaide.edu.au%2Fdspace%2Fbitstream%2F2440%2F15191%2F1%2F48.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorial+experiment" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://jeff560.tripod.com/f.html">"Earliest Known Uses of Some of the Words of Mathematics (F)"</a>. <i>jeff560.tripod.com</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=jeff560.tripod.com&rft.atitle=Earliest+Known+Uses+of+Some+of+the+Words+of+Mathematics+%28F%29&rft_id=http%3A%2F%2Fjeff560.tripod.com%2Ff.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorial+experiment" class="Z3988"></span></span> </li> <li id="cite_note-Montgomery-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Montgomery_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Montgomery_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMontgomery2013" class="citation book cs1">Montgomery, Douglas C. (2013). <i>Design and Analysis of Experiments</i> (8th ed.). Hoboken, New Jersey: <a href="/wiki/Wiley_(publisher)" title="Wiley (publisher)">Wiley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-119-32093-7" title="Special:BookSources/978-1-119-32093-7"><bdi>978-1-119-32093-7</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.place=Hoboken%2C+New+Jersey&rft.edition=8th&rft.pub=Wiley&rft.date=2013&rft.isbn=978-1-119-32093-7&rft.aulast=Montgomery&rft.aufirst=Douglas+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorial+experiment" class="Z3988"></span></span> </li> <li id="cite_note-Oehlert-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Oehlert_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOehlert2000" class="citation book cs1">Oehlert, Gary (2000). <a rel="nofollow" class="external text" href="http://users.stat.umn.edu/~gary/Book.html"><i>A First Course in Design and Analysis of Experiments</i></a> (Revised ed.). New York City: <a href="/wiki/W._H._Freeman_and_Company" title="W. H. Freeman and Company">W. H. Freeman and Company</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-3510-6" title="Special:BookSources/978-0-7167-3510-6"><bdi>978-0-7167-3510-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+Design+and+Analysis+of+Experiments&rft.place=New+York+City&rft.edition=Revised&rft.pub=W.+H.+Freeman+and+Company&rft.date=2000&rft.isbn=978-0-7167-3510-6&rft.aulast=Oehlert&rft.aufirst=Gary&rft_id=http%3A%2F%2Fusers.stat.umn.edu%2F~gary%2FBook.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorial+experiment" class="Z3988"></span></span> </li> <li id="cite_note-Tong2006-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Tong2006_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTong2006" class="citation journal cs1">Tong, C. (2006). "Refinement strategies for stratified sampling methods". <i>Reliability Engineering & System Safety</i>. <b>91</b> (10–11): 1257–1265. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ress.2005.11.027">10.1016/j.ress.2005.11.027</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reliability+Engineering+%26+System+Safety&rft.atitle=Refinement+strategies+for+stratified+sampling+methods&rft.volume=91&rft.issue=10%E2%80%9311&rft.pages=1257-1265&rft.date=2006&rft_id=info%3Adoi%2F10.1016%2Fj.ress.2005.11.027&rft.aulast=Tong&rft.aufirst=C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorial+experiment" class="Z3988"></span></span> </li> <li id="cite_note-Box2006-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Box2006_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorge_E.P.2006" class="citation book cs1"><a href="/wiki/George_E.P._Box" class="mw-redirect" title="George E.P. Box">George E.P., Box</a> (2006). <i>Improving Almost Anything: Ideas and Essays</i> (Revised ed.). Hoboken, New Jersey: <a href="/wiki/Wiley_(publisher)" title="Wiley (publisher)">Wiley</a>. <a href="/wiki/ASIN_(identifier)" class="mw-redirect" title="ASIN (identifier)">ASIN</a> <a rel="nofollow" class="external text" href="https://www.amazon.com/dp/B01FKSM9VY">B01FKSM9VY</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Improving+Almost+Anything%3A+Ideas+and+Essays&rft.place=Hoboken%2C+New+Jersey&rft.edition=Revised&rft.pub=Wiley&rft.date=2006&rft_id=https%3A%2F%2Fwww.amazon.com%2Fdp%2FB01FKSM9VY%23id-name%3DASIN&rft.aulast=George+E.P.&rft.aufirst=Box&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorial+experiment" class="Z3988"></span></span> </li> <li id="cite_note-Hellstrand1989-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hellstrand1989_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHellstrandOosterhoornSherwinGerson1989" class="citation journal cs1">Hellstrand, C.; Oosterhoorn, A. D.; Sherwin, D. J.; Gerson, M. (24 February 1989). "The Necessity of Modern Quality Improvement and Some Experience with its Implementation in the Manufacture of Rolling Bearings [and Discussion]". <i>Philosophical Transactions of the Royal Society</i>. <b>327</b> (1596): 529–537. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frsta.1989.0008">10.1098/rsta.1989.0008</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:122252479">122252479</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society&rft.atitle=The+Necessity+of+Modern+Quality+Improvement+and+Some+Experience+with+its+Implementation+in+the+Manufacture+of+Rolling+Bearings+%5Band+Discussion%5D&rft.volume=327&rft.issue=1596&rft.pages=529-537&rft.date=1989-02-24&rft_id=info%3Adoi%2F10.1098%2Frsta.1989.0008&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122252479%23id-name%3DS2CID&rft.aulast=Hellstrand&rft.aufirst=C.&rft.au=Oosterhoorn%2C+A.+D.&rft.au=Sherwin%2C+D.+J.&rft.au=Gerson%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorial+experiment" class="Z3988"></span></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"><a href="#CITEREFKuehl2000">Kuehl (2000</a>, pp. 200–205)</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"><a href="#CITEREFCheng2019">Cheng (2019</a>, Remark 8.1)</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"><a href="#CITEREFBoxHunterHunter1978">Box, Hunter & Hunter (1978</a>, p. 307)</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"><a href="#CITEREFHocking1985">Hocking (1985</a>, p. 73). Hocking and others use the term "population mean" for expected value.</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"><a href="#CITEREFGraybill1976">Graybill (1976</a>, pp. 559–560)</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"><a href="#CITEREFBeder2022">Beder (2022</a>, pp. 29–30)</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"><a href="#CITEREFBeder2022">Beder (2022</a>, Example 5.21)</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"><a href="#CITEREFBose1947">Bose (1947</a>, pp. 110–111)</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"><a href="#CITEREFCheng2019">Cheng (2019</a>, p. 77)</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="#CITEREFKuehl2000">Kuehl (2000</a>, p. 202)</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"><a href="#CITEREFCheng2019">Cheng (2019</a>, p. 78)</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"><a href="#CITEREFBoxHunterHunter2005">Box, Hunter & Hunter (2005</a>, p. 180)</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohen1968" class="citation journal cs1">Cohen, J (1968). "Multiple regression as a general data-analytic system". <i>Psychological Bulletin</i>. <b>70</b> (6): 426–443. <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=10.1.1.476.6180">10.1.1.476.6180</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/10.1037%2Fh0026714">10.1037/h0026714</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Psychological+Bulletin&rft.atitle=Multiple+regression+as+a+general+data-analytic+system&rft.volume=70&rft.issue=6&rft.pages=426-443&rft.date=1968&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.476.6180%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1037%2Fh0026714&rft.aulast=Cohen&rft.aufirst=J&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorial+experiment" class="Z3988"></span></span> </li> </ol></div></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=Factorial_experiment&action=edit&section=14" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeder2022" class="citation book cs1">Beder, Jay H. (2022). <i>Linear Models and Design</i>. Cham, Switzerland: <a href="/wiki/Springer_Publishing" title="Springer Publishing">Springer</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-031-08176-7">10.1007/978-3-031-08176-7</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-031-08175-0" title="Special:BookSources/978-3-031-08175-0"><bdi>978-3-031-08175-0</bdi></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:253542415">253542415</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Models+and+Design&rft.place=Cham%2C+Switzerland&rft.pub=Springer&rft.date=2022&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A253542415%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-3-031-08176-7&rft.isbn=978-3-031-08175-0&rft.aulast=Beder&rft.aufirst=Jay+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorial+experiment" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBose1947" class="citation journal cs1"><a href="/wiki/Raj_Chandra_Bose" title="Raj Chandra Bose">Bose</a>, R. C. (1947). "Mathematical theory of the symmetrical factorial design". <i>Sankhya</i>. <b>8</b>: 107–166.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sankhya&rft.atitle=Mathematical+theory+of+the+symmetrical+factorial+design&rft.volume=8&rft.pages=107-166&rft.date=1947&rft.aulast=Bose&rft.aufirst=R.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorial+experiment" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoxHunterHunter1978" class="citation book cs1"><a href="/wiki/George_E._P._Box" title="George E. P. Box">Box, G. E.</a>; Hunter, W. G.; Hunter, J. S. (1978). <i>Statistics for Experimenters: An Introduction to Design, Data Analysis and Model Building</i>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-09315-2" title="Special:BookSources/978-0-471-09315-2"><bdi>978-0-471-09315-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistics+for+Experimenters%3A+An+Introduction+to+Design%2C+Data+Analysis+and+Model+Building&rft.pub=Wiley&rft.date=1978&rft.isbn=978-0-471-09315-2&rft.aulast=Box&rft.aufirst=G.+E.&rft.au=Hunter%2C+W.+G.&rft.au=Hunter%2C+J.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorial+experiment" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoxHunterHunter2005" class="citation book cs1"><a href="/wiki/George_E._P._Box" title="George E. P. Box">Box, G. E.</a>; Hunter, W. G.; Hunter, J. S. (2005). <i>Statistics for Experimenters: Design, Innovation, and Discovery</i> (2nd ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-71813-0" title="Special:BookSources/978-0-471-71813-0"><bdi>978-0-471-71813-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistics+for+Experimenters%3A+Design%2C+Innovation%2C+and+Discovery&rft.edition=2nd&rft.pub=Wiley&rft.date=2005&rft.isbn=978-0-471-71813-0&rft.aulast=Box&rft.aufirst=G.+E.&rft.au=Hunter%2C+W.+G.&rft.au=Hunter%2C+J.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorial+experiment" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCheng2019" class="citation book cs1">Cheng, Ching-Shui (2019). <i>Theory of Factorial Design: Single- and Multi-Stratum Experiments</i>. Boca Raton, Florida: CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-367-37898-1" title="Special:BookSources/978-0-367-37898-1"><bdi>978-0-367-37898-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Factorial+Design%3A+Single-+and+Multi-Stratum+Experiments&rft.place=Boca+Raton%2C+Florida&rft.pub=CRC+Press&rft.date=2019&rft.isbn=978-0-367-37898-1&rft.aulast=Cheng&rft.aufirst=Ching-Shui&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorial+experiment" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDeanVossDraguljić2017" class="citation book cs1">Dean, Angela; Voss, Daniel; Draguljić, Danel (2017). <i>Design and Analysis of Experiments</i> (2nd ed.). Cham, Switzerland: <a href="/wiki/Springer_Publishing" title="Springer Publishing">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-52250-0" title="Special:BookSources/978-3-319-52250-0"><bdi>978-3-319-52250-0</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.place=Cham%2C+Switzerland&rft.edition=2nd&rft.pub=Springer&rft.date=2017&rft.isbn=978-3-319-52250-0&rft.aulast=Dean&rft.aufirst=Angela&rft.au=Voss%2C+Daniel&rft.au=Dragulji%C4%87%2C+Danel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorial+experiment" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGraybill1976" class="citation book cs1">Graybill, Franklin A. (1976). <i>Fundamental Concepts in the Design of Experiments</i> (3rd ed.). New York: Holt, Rinehart and Winston. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-03-061706-5" title="Special:BookSources/0-03-061706-5"><bdi>0-03-061706-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamental+Concepts+in+the+Design+of+Experiments&rft.place=New+York&rft.edition=3rd&rft.pub=Holt%2C+Rinehart+and+Winston&rft.date=1976&rft.isbn=0-03-061706-5&rft.aulast=Graybill&rft.aufirst=Franklin+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorial+experiment" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHicks1982" class="citation book cs1">Hicks, Charles R. (1982). <i>Theory and Application of the Linear Model</i>. Pacific Grove, CA: Wadsworth & Brooks/Cole. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-87872-108-8" title="Special:BookSources/0-87872-108-8"><bdi>0-87872-108-8</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+and+Application+of+the+Linear+Model&rft.place=Pacific+Grove%2C+CA&rft.pub=Wadsworth+%26+Brooks%2FCole&rft.date=1982&rft.isbn=0-87872-108-8&rft.aulast=Hicks&rft.aufirst=Charles+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorial+experiment" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHocking1985" class="citation book cs1">Hocking, Ronald R. (1985). <i>The Analysis of Linear Models</i>. Pacific Grove, CA: <a href="/wiki/Brooks/Cole" class="mw-redirect" title="Brooks/Cole">Brooks/Cole</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0534036188" title="Special:BookSources/978-0534036188"><bdi>978-0534036188</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Analysis+of+Linear+Models&rft.place=Pacific+Grove%2C+CA&rft.pub=Brooks%2FCole&rft.date=1985&rft.isbn=978-0534036188&rft.aulast=Hocking&rft.aufirst=Ronald+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorial+experiment" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKuehl2000" class="citation book cs1">Kuehl, Robert O. (2000). <i>Design of Experiments: Statistical Principles of Research Design and Analysis</i> (2nd ed.). Pacific Grove, CA: Brooks/Cole. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0534368340" title="Special:BookSources/978-0534368340"><bdi>978-0534368340</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+Experiments%3A+Statistical+Principles+of+Research+Design+and+Analysis&rft.place=Pacific+Grove%2C+CA&rft.edition=2nd&rft.pub=Brooks%2FCole&rft.date=2000&rft.isbn=978-0534368340&rft.aulast=Kuehl&rft.aufirst=Robert+O.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorial+experiment" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWuHamada2021" class="citation book cs1">Wu, C. F. Jeff; Hamada, Michael S. (30 March 2021). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LZEOEAAAQBAJ&pg=PA269"><i>Experiments: Planning, Analysis, and Optimization</i></a>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-119-47010-6" title="Special:BookSources/978-1-119-47010-6"><bdi>978-1-119-47010-6</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+Optimization&rft.pub=John+Wiley+%26+Sons&rft.date=2021-03-30&rft.isbn=978-1-119-47010-6&rft.aulast=Wu&rft.aufirst=C.+F.+Jeff&rft.au=Hamada%2C+Michael+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLZEOEAAAQBAJ%26pg%3DPA269&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorial+experiment" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorial_experiment&action=edit&section=15" title="Edit section: External 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href="/wiki/Restricted_randomization" title="Restricted randomization">Restricted randomization</a></li> <li><a href="/wiki/Replication_(statistics)" title="Replication (statistics)">Replication versus subsampling</a></li> <li><a href="/wiki/Sample_size" class="mw-redirect" title="Sample size">Sample size</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Glossary_of_experimental_design" title="Glossary of experimental design">Treatment</a><br /> and <a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">blocking</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><b><a href="/wiki/Glossary_of_experimental_design" title="Glossary of experimental design">Treatment</a></b></li> <li><a href="/wiki/Effect_size" title="Effect size">Effect size</a></li> <li><a href="/wiki/Contrast_(statistics)" title="Contrast (statistics)">Contrast</a></li> <li><a href="/wiki/Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li> <li><a href="/wiki/Confounding" title="Confounding">Confounding</a></li> <li><a href="/wiki/Orthogonality#Statistics,_econometrics,_and_economics" title="Orthogonality">Orthogonality</a></li> <li><b><a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></b></li> <li><a href="/wiki/Covariate" class="mw-redirect" title="Covariate">Covariate</a></li> <li><a href="/wiki/Nuisance_variable" title="Nuisance variable">Nuisance variable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Statistical_model" title="Statistical model">Models</a> <br /> and <a href="/wiki/Statistical_inference" title="Statistical inference">inference</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/Linear_regression" title="Linear 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variance"><b>Manova</b> (<i>multivariate</i>)</a></li> <li><a href="/wiki/Analysis_of_covariance" title="Analysis of covariance"><b>Ancova</b> (<i>covariance</i>)</a></li></ul> <ul><li><a href="/wiki/Comparing_means" class="mw-redirect" title="Comparing means">Compare means</a></li> <li><a href="/wiki/Multiple_comparison" class="mw-redirect" title="Multiple comparison">Multiple comparison</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Design_of_experiments" title="Design of experiments">Designs</a> <br /> <br /><a href="/wiki/Completely_randomized_design" title="Completely randomized design">Completely<br />randomized</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><b><a class="mw-selflink selflink">Factorial</a></b></li> <li><a href="/wiki/Fractional_factorial_design" title="Fractional factorial design">Fractional factorial</a></li> <li><a href="/wiki/Plackett-Burman_design" class="mw-redirect" title="Plackett-Burman design">Plackett–Burman</a></li> <li><a href="/wiki/Taguchi_methods" title="Taguchi methods">Taguchi</a></li></ul> <ul><li><b><a href="/wiki/Response_surface_methodology" title="Response surface methodology">Response surface methodology</a></b></li> <li><a href="/wiki/Polynomial_and_rational_function_modeling" title="Polynomial and rational function modeling">Polynomial and rational modeling</a></li> <li><a href="/wiki/Box%E2%80%93Behnken_design" title="Box–Behnken design">Box–Behnken</a></li> <li><a href="/wiki/Central_composite_design" title="Central composite design">Central composite</a></li></ul> <ul><li><b><a href="/wiki/Randomized_block_design" class="mw-redirect" title="Randomized block design">Block</a></b></li> <li><a href="/wiki/Generalized_randomized_block_design" title="Generalized randomized block design">Generalized randomized block design</a> (GRBD)</li> <li><a href="/wiki/Latin_square" title="Latin square">Latin square</a></li> <li><a href="/wiki/Graeco-Latin_square" class="mw-redirect" title="Graeco-Latin square">Graeco-Latin square</a></li> <li><a href="/wiki/Orthogonal_array" title="Orthogonal array">Orthogonal array</a></li> <li><a href="/wiki/Latin_hypercube_sampling" title="Latin hypercube sampling">Latin hypercube</a> <br /> <b><a href="/wiki/Repeated_measures_design" title="Repeated measures design">Repeated measures design</a></b></li> <li><a href="/wiki/Crossover_study" title="Crossover study">Crossover study</a></li></ul> <ul><li><b><a href="/wiki/Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></b></li> <li><a href="/wiki/Sequential_analysis" title="Sequential analysis">Sequential analysis</a></li> <li><a href="/wiki/Sequential_probability_ratio_test" title="Sequential probability ratio test">Sequential probability ratio test</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Glossary_of_experimental_design" title="Glossary of experimental design">Glossary</a></li> <li><a href="/wiki/Category:Design_of_experiments" title="Category:Design of experiments">Category</a></li> <li><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li> <li><a href="/wiki/Outline_of_statistics" title="Outline of statistics">Statistical outline</a></li> <li><a href="/wiki/List_of_statistics_articles" title="List of statistics articles">Statistical topics</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Statistics" style="padding:3px"><table class="nowraplinks hlist mw-collapsible uncollapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Statistics" title="Template:Statistics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Statistics" title="Template talk:Statistics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Statistics" title="Special:EditPage/Template:Statistics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Statistics" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistics" title="Statistics">Statistics</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Outline_of_statistics" title="Outline of statistics">Outline</a></li> <li><a href="/wiki/List_of_statistics_articles" title="List of statistics articles">Index</a></li></ul> </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="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 class="mw-selflink selflink">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" 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