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class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Quantitative analysis of residuals</span> </div> </a> <ul id="toc-Quantitative_analysis_of_residuals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Out-of-sample_evaluation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Out-of-sample_evaluation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Out-of-sample evaluation</span> </div> </a> <ul id="toc-Out-of-sample_evaluation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 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.sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></th></tr><tr><th class="sidebar-heading"> Models</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></li> <li><a href="/wiki/Simple_linear_regression" title="Simple linear regression">Simple regression</a></li> <li><a href="/wiki/Polynomial_regression" title="Polynomial regression">Polynomial regression</a></li> <li><a href="/wiki/General_linear_model" title="General linear model">General linear model</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></li> <li><a href="/wiki/Vector_generalized_linear_model" title="Vector generalized linear model">Vector generalized linear model</a></li> <li><a href="/wiki/Discrete_choice" title="Discrete choice">Discrete choice</a></li> <li><a href="/wiki/Binomial_regression" title="Binomial regression">Binomial regression</a></li> <li><a href="/wiki/Binary_regression" title="Binary regression">Binary regression</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic regression</a></li> <li><a href="/wiki/Multinomial_logistic_regression" title="Multinomial logistic regression">Multinomial logistic regression</a></li> <li><a href="/wiki/Mixed_logit" title="Mixed logit">Mixed logit</a></li> <li><a href="/wiki/Probit_model" title="Probit model">Probit</a></li> <li><a href="/wiki/Multinomial_probit" title="Multinomial probit">Multinomial probit</a></li> <li><a href="/wiki/Ordered_logit" title="Ordered logit">Ordered logit</a></li> <li><a href="/wiki/Ordered_probit" class="mw-redirect" title="Ordered probit">Ordered probit</a></li> <li><a href="/wiki/Poisson_regression" title="Poisson regression">Poisson</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Multilevel_model" title="Multilevel model">Multilevel model</a></li> <li><a href="/wiki/Fixed_effects_model" title="Fixed effects model">Fixed effects</a></li> <li><a href="/wiki/Random_effects_model" title="Random effects model">Random effects</a></li> <li><a href="/wiki/Mixed_model" title="Mixed model">Linear mixed-effects model</a></li> <li><a href="/wiki/Nonlinear_mixed-effects_model" title="Nonlinear mixed-effects model">Nonlinear mixed-effects model</a></li></ul></td> </tr><tr><td class="sidebar-content"> <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/Robust_regression" title="Robust regression">Robust</a></li> <li><a href="/wiki/Quantile_regression" title="Quantile regression">Quantile</a></li> <li><a href="/wiki/Isotonic_regression" title="Isotonic regression">Isotonic</a></li> <li><a href="/wiki/Principal_component_regression" title="Principal component regression">Principal components</a></li> <li><a href="/wiki/Least-angle_regression" title="Least-angle regression">Least angle</a></li> <li><a href="/wiki/Local_regression" title="Local regression">Local</a></li> <li><a href="/wiki/Segmented_regression" title="Segmented regression">Segmented</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Errors-in-variables_models" title="Errors-in-variables models">Errors-in-variables</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Estimation</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Least_squares" title="Least squares">Least squares</a></li> <li><a href="/wiki/Linear_least_squares" title="Linear least squares">Linear</a></li> <li><a href="/wiki/Non-linear_least_squares" title="Non-linear least squares">Non-linear</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary</a></li> <li><a href="/wiki/Weighted_least_squares" title="Weighted least squares">Weighted</a></li> <li><a href="/wiki/Generalized_least_squares" title="Generalized least squares">Generalized</a></li> <li><a href="/wiki/Generalized_estimating_equation" title="Generalized estimating equation">Generalized estimating equation</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Partial_least_squares_regression" title="Partial least squares regression">Partial</a></li> <li><a href="/wiki/Total_least_squares" title="Total least squares">Total</a></li> <li><a href="/wiki/Non-negative_least_squares" title="Non-negative least squares">Non-negative</a></li> <li><a href="/wiki/Tikhonov_regularization" class="mw-redirect" title="Tikhonov regularization">Ridge regression</a></li> <li><a href="/wiki/Regularized_least_squares" title="Regularized least squares">Regularized</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Least_absolute_deviations" title="Least absolute deviations">Least absolute deviations</a></li> <li><a href="/wiki/Iteratively_reweighted_least_squares" title="Iteratively reweighted least squares">Iteratively reweighted</a></li> <li><a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Bayesian</a></li> <li><a href="/wiki/Bayesian_multivariate_linear_regression" title="Bayesian multivariate linear regression">Bayesian multivariate</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Background</th></tr><tr><td class="sidebar-content"> <ul><li><a class="mw-selflink selflink">Regression validation</a></li> <li><a href="/wiki/Mean_and_predicted_response" class="mw-redirect" title="Mean and predicted response">Mean and predicted response</a></li> <li><a href="/wiki/Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li> <li><a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></li> <li><a href="/wiki/Studentized_residual" title="Studentized residual">Studentized residual</a></li> <li><a href="/wiki/Gauss%E2%80%93Markov_theorem" title="Gauss–Markov theorem">Gauss–Markov theorem</a></li></ul></td> </tr><tr><td class="sidebar-below"> <ul><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/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-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/56px-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></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Regression_bar" title="Template:Regression bar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Regression_bar" title="Template talk:Regression bar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Regression_bar" title="Special:EditPage/Template:Regression bar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, <b>regression validation</b> is the process of deciding whether the numerical results quantifying hypothesized relationships between variables, obtained from <a href="/wiki/Regression_analysis" title="Regression analysis">regression analysis</a>, are acceptable as descriptions of the data. The validation process can involve analyzing the <a href="/wiki/Goodness_of_fit" title="Goodness of fit">goodness of fit</a> of the regression, analyzing whether the <a href="/wiki/Residual_(statistics)" class="mw-redirect" title="Residual (statistics)">regression residuals</a> are random, and checking whether the model's predictive performance deteriorates substantially when applied to data that were not used in model estimation. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Goodness_of_fit">Goodness of fit</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regression_validation&action=edit&section=1" title="Edit section: Goodness of fit"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></div> <p>One measure of goodness of fit is the <a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">coefficient of determination</a>, often denoted, <i>R</i><sup>2</sup>. In <a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">ordinary least squares</a> with an intercept, it ranges between 0 and 1. However, an <i>R</i><sup>2</sup> close to 1 does not guarantee that the model fits the data well. For example, if the functional form of the model does not match the data, <i>R</i><sup>2</sup> can be high despite a poor model fit. <a href="/wiki/Anscombe%27s_quartet" title="Anscombe's quartet">Anscombe's quartet</a> consists of four example data sets with similarly high <i>R</i><sup>2</sup> values, but data that sometimes clearly does not fit the regression line. Instead, the data sets include <a href="/wiki/Outlier" title="Outlier">outliers</a>, <a href="/wiki/High-leverage_point" class="mw-redirect" title="High-leverage point">high-leverage points</a>, or non-linearities. </p><p>One problem with the <i>R</i><sup>2</sup> as a measure of model validity is that it can always be increased by adding more variables into the model, except in the unlikely event that the additional variables are exactly uncorrelated with the dependent variable in the data sample being used. This problem can be avoided by doing an <a href="/wiki/F-test" title="F-test">F-test</a> of the statistical significance of the increase in the <i>R</i><sup>2</sup>, or by instead using the <a href="/wiki/Adjusted_R-squared" class="mw-redirect" title="Adjusted R-squared">adjusted <i>R</i>2</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Analysis_of_residuals">Analysis of residuals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regression_validation&action=edit&section=2" title="Edit section: Analysis of residuals"><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/Residual_analysis" class="mw-redirect" title="Residual analysis">residual analysis</a></div> <p>The <a href="/wiki/Errors_and_residuals_in_statistics" class="mw-redirect" title="Errors and residuals in statistics">residuals</a> from a fitted model are the differences between the responses observed at each combination of values of the <a href="/wiki/Explanatory_variable" class="mw-redirect" title="Explanatory variable">explanatory variables</a> and the corresponding prediction of the response computed using the regression function. Mathematically, the definition of the residual for the <i>i</i><sup>th</sup> observation in the <a href="/wiki/Data_set" title="Data set">data set</a> is written </p> <dl><dd><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 e_{i}=y_{i}-f(x_{i};{\hat {\beta }}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{i}=y_{i}-f(x_{i};{\hat {\beta }}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7659e93da3adcba5712b6f8cc1edc86a5825ff79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.11ex; height:3.343ex;" alt="{\displaystyle e_{i}=y_{i}-f(x_{i};{\hat {\beta }}),}"></span></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Plot_of_noisy_data_%2B_Gaussian_fit_%2B_plot_ressduals.svg" class="mw-file-description"><img alt="plot of a fit and residuals to illustrate how plotting residuals allows us to evaluate how good a fit is" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Plot_of_noisy_data_%2B_Gaussian_fit_%2B_plot_ressduals.svg/220px-Plot_of_noisy_data_%2B_Gaussian_fit_%2B_plot_ressduals.svg.png" decoding="async" width="220" height="330" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Plot_of_noisy_data_%2B_Gaussian_fit_%2B_plot_ressduals.svg/330px-Plot_of_noisy_data_%2B_Gaussian_fit_%2B_plot_ressduals.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/09/Plot_of_noisy_data_%2B_Gaussian_fit_%2B_plot_ressduals.svg/440px-Plot_of_noisy_data_%2B_Gaussian_fit_%2B_plot_ressduals.svg.png 2x" data-file-width="540" data-file-height="810" /></a><figcaption>An illustrative plot of a fit to data (green curve in top panel, data in red) plus a plot of residuals: red points in bottom plot. Dashed curve in bottom panel is a straight line fit to the residuals. If the functional form is correct then there should be little or no trend to the residuals - as seen here.</figcaption></figure> <p>with <i>y<sub>i</sub></i> denoting the <i>i</i><sup>th</sup> response in the data set and <i>x<sub>i</sub></i> the vector of explanatory variables, each set at the corresponding values found in the <i>i</i><sup>th</sup> observation in the data set. </p><p>If the model fit to the data were correct, the residuals would approximate the random errors that make the relationship between the explanatory variables and the response variable a statistical relationship. Therefore, if the residuals appear to behave randomly, it suggests that the model fits the data well. On the other hand, if non-random structure is evident in the residuals, it is a clear sign that the model fits the data poorly. The next section details the types of plots to use to test different aspects of a model and gives the correct interpretations of different results that could be observed for each type of plot. </p> <div class="mw-heading mw-heading3"><h3 id="Graphical_analysis_of_residuals">Graphical analysis of residuals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regression_validation&action=edit&section=3" title="Edit section: Graphical analysis of residuals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Statistical_graphics" title="Statistical graphics">Statistical graphics</a></div> <p>A basic, though not quantitatively precise, way to check for problems that render a model inadequate is to conduct a visual examination of the residuals (the mispredictions of the data used in quantifying the model) to look for obvious deviations from randomness. If a visual examination suggests, for example, the possible presence of <a href="/wiki/Heteroscedasticity" class="mw-redirect" title="Heteroscedasticity">heteroscedasticity</a> (a relationship between the variance of the model errors and the size of an independent variable's observations), then statistical tests can be performed to confirm or reject this hunch; if it is confirmed, different modeling procedures are called for. </p><p>Different types of plots of the residuals from a fitted model provide information on the adequacy of different aspects of the model. </p> <ol><li>sufficiency of the functional part of the model: <a href="/wiki/Scatter_plot" title="Scatter plot">scatter plots</a> of residuals versus predictors</li> <li>non-constant variation across the data: <a href="/wiki/Scatter_plot" title="Scatter plot">scatter plots</a> of residuals versus predictors; for data collected over time, also plots of residuals against time</li> <li>drift in the errors (data collected over time): <a href="/wiki/Run_chart" title="Run chart">run charts</a> of the response and errors versus time</li> <li>independence of errors: <a href="/wiki/Lag_plot" class="mw-redirect" title="Lag plot">lag plot</a></li> <li>normality of errors: <a href="/wiki/Histogram" title="Histogram">histogram</a> and <a href="/wiki/Normal_probability_plot" title="Normal probability plot">normal probability plot</a></li></ol> <p>Graphical methods have an advantage over numerical methods for model validation because they readily illustrate a broad range of complex aspects of the relationship between the model and the data. </p> <div class="mw-heading mw-heading3"><h3 id="Quantitative_analysis_of_residuals">Quantitative analysis of residuals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regression_validation&action=edit&section=4" title="Edit section: Quantitative analysis of residuals"><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/Regression_diagnostic" title="Regression diagnostic">Regression diagnostic</a></div> <p>Numerical methods also play an important role in model validation. For example, the <a href="/wiki/Goodness_of_fit" title="Goodness of fit">lack-of-fit test</a> for assessing the correctness of the functional part of the model can aid in interpreting a borderline residual plot. One common situation when numerical validation methods take precedence over graphical methods is when the number of <a href="/wiki/Statistical_parameter" title="Statistical parameter">parameters</a> being estimated is relatively close to the size of the data set. In this situation residual plots are often difficult to interpret due to constraints on the residuals imposed by the estimation of the unknown parameters. One area in which this typically happens is in optimization applications using <a href="/wiki/Designed_experiment" class="mw-redirect" title="Designed experiment">designed experiments</a>. <a href="/wiki/Logistic_regression" title="Logistic regression">Logistic regression</a> with <a href="/wiki/Binary_data" title="Binary data">binary data</a> is another area in which graphical residual analysis can be difficult. </p><p><a href="/wiki/Serial_correlation" class="mw-redirect" title="Serial correlation">Serial correlation</a> of the residuals can indicate model misspecification, and can be checked for with the <a href="/wiki/Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson statistic</a>. The problem of <a href="/wiki/Heteroskedasticity" class="mw-redirect" title="Heteroskedasticity">heteroskedasticity</a> can be checked for in any of <a href="/wiki/Heteroskedasticity#Detection" class="mw-redirect" title="Heteroskedasticity">several ways</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Out-of-sample_evaluation">Out-of-sample evaluation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regression_validation&action=edit&section=5" title="Edit section: Out-of-sample evaluation"><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/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross-validation</a></div> <p>Cross-validation is the process of assessing how the results of a statistical analysis will generalize to an independent data set. If the model has been estimated over some, but not all, of the available data, then the model using the estimated parameters can be used to predict the held-back data. If, for example, the out-of-sample <a href="/wiki/Mean_squared_error" title="Mean squared error">mean squared error</a>, also known as the <a href="/wiki/Mean_squared_prediction_error" title="Mean squared prediction error">mean squared prediction error</a>, is substantially higher than the in-sample mean square error, this is a sign of deficiency in the model. </p><p>A development in medical statistics is the use of out-of-sample cross validation techniques in meta-analysis. It forms the basis of the <i>validation statistic, Vn</i>, which is used to test the statistical validity of meta-analysis summary estimates. Essentially it measures a type of normalized prediction error and its distribution is a linear combination of <i>χ</i><sup>2</sup> variables of degree 1. <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> <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=Regression_validation&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/All_models_are_wrong" title="All models are wrong">All models are wrong</a></li> <li><a href="/wiki/Model_selection" title="Model selection">Model selection</a></li> <li><a href="/wiki/Prediction_error" class="mw-redirect" title="Prediction error">Prediction error</a></li> <li><a href="/wiki/Prediction_interval" title="Prediction interval">Prediction interval</a></li> <li><a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">Resampling (statistics)</a></li> <li><a href="/wiki/Statistical_conclusion_validity" title="Statistical conclusion validity">Statistical conclusion validity</a></li> <li><a href="/wiki/Statistical_model_specification" title="Statistical model specification">Statistical model specification</a></li> <li><a href="/wiki/Statistical_model_validation" title="Statistical model validation">Statistical model validation</a></li> <li><a href="/wiki/Validity_(statistics)" title="Validity (statistics)">Validity (statistics)</a></li> <li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li> <li><a href="/wiki/Lack-of-fit_sum_of_squares" title="Lack-of-fit sum of squares">Lack-of-fit sum of squares</a></li> <li><a href="/wiki/Reduced_chi-squared" class="mw-redirect" title="Reduced chi-squared">Reduced chi-squared</a></li></ul> <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=Regression_validation&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 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.ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span 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Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Regression+validation%22">"Regression validation"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Regression+validation%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Regression+validation%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Regression+validation%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Regression+validation%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Regression+validation%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">March 2010</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <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 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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="CITEREFWillis_BH,_Riley_RD2017" class="citation journal cs1">Willis BH, Riley RD (2017). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5575530">"Measuring the statistical validity of summary meta-analysis and meta-regression results for use in clinical practice"</a>. <i>Statistics in Medicine</i>. <b>36</b> (21): 3283–3301. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fsim.7372">10.1002/sim.7372</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5575530">5575530</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/28620945">28620945</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistics+in+Medicine&rft.atitle=Measuring+the+statistical+validity+of+summary+meta-analysis+and+meta-regression+results+for+use+in+clinical+practice&rft.volume=36&rft.issue=21&rft.pages=3283-3301&rft.date=2017&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5575530%23id-name%3DPMC&rft_id=info%3Apmid%2F28620945&rft_id=info%3Adoi%2F10.1002%2Fsim.7372&rft.au=Willis+BH%2C+Riley+RD&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5575530&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegression+validation" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regression_validation&action=edit&section=8" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArboretti_GiancristofaroSalmaso2003" class="citation cs2">Arboretti Giancristofaro, R.; Salmaso, L. (2003), <a rel="nofollow" class="external text" href="https://rivista-statistica.unibo.it/article/viewFile/358/351">"Model performance analysis and model validation in logistic regression"</a>, <i><a href="/wiki/Statistica_(journal)" title="Statistica (journal)">Statistica</a></i>, <b>63</b>: 375–396</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistica&rft.atitle=Model+performance+analysis+and+model+validation+in+logistic+regression&rft.volume=63&rft.pages=375-396&rft.date=2003&rft.aulast=Arboretti+Giancristofaro&rft.aufirst=R.&rft.au=Salmaso%2C+L.&rft_id=https%3A%2F%2Frivista-statistica.unibo.it%2Farticle%2FviewFile%2F358%2F351&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegression+validation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKmenta1986" class="citation cs2"><a href="/wiki/Jan_Kmenta" title="Jan Kmenta">Kmenta, Jan</a> (1986), <i>Elements of Econometrics</i> (Second ed.), Macmillan, pp. 593–600</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Econometrics&rft.pages=593-600&rft.edition=Second&rft.pub=Macmillan&rft.date=1986&rft.aulast=Kmenta&rft.aufirst=Jan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegression+validation" class="Z3988"></span>; republished in 1997 by <a href="/wiki/University_of_Michigan_Press" title="University of Michigan Press">University of Michigan Press</a></li></ul> <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=Regression_validation&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.itl.nist.gov/div898/handbook/pmd/section4/pmd44.htm">How can I tell if a model fits my data? (NIST)</a></li> <li><a rel="nofollow" class="external text" href="http://www.itl.nist.gov/div898/handbook/">NIST/SEMATECH e-Handbook of Statistical Methods</a></li> <li><a rel="nofollow" class="external text" href="https://newonlinecourses.science.psu.edu/stat504/node/161/">Model Diagnostics</a> (<a href="/wiki/Eberly_College_of_Science" title="Eberly College of Science">Eberly College of Science</a>)</li></ul> <p><span class="noviewer" typeof="mw:File"><span><img alt="Public Domain" src="//upload.wikimedia.org/wikipedia/en/thumb/6/62/PD-icon.svg/12px-PD-icon.svg.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/6/62/PD-icon.svg/18px-PD-icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/6/62/PD-icon.svg/24px-PD-icon.svg.png 2x" data-file-width="196" data-file-height="196" /></span></span> This article incorporates <a href="/wiki/Copyright_status_of_works_by_the_federal_government_of_the_United_States" title="Copyright status of works by the federal government of the United States">public domain material</a> from the <a rel="nofollow" class="external text" href="https://www.nist.gov">National Institute of Standards and Technology</a> </p>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Regression_validation&oldid=1222104555">https://en.wikipedia.org/w/index.php?title=Regression_validation&oldid=1222104555</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Validity_(statistics)" title="Category:Validity (statistics)">Validity (statistics)</a></li><li><a href="/wiki/Category:Regression_diagnostics" title="Category:Regression diagnostics">Regression diagnostics</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_March_2010" title="Category:Articles needing additional references from March 2010">Articles needing additional references from March 2010</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li><li><a href="/wiki/Category:Wikipedia_articles_incorporating_text_from_the_National_Institute_of_Standards_and_Technology" title="Category:Wikipedia articles incorporating text from the National Institute of Standards and Technology">Wikipedia articles incorporating text from the National Institute of Standards and Technology</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 3 May 2024, at 22:30<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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