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Explained variation

Explained variation, also known as the explained sum of squares (ESS), is a statistical measure in regression analysis that quantifies the portion of the total variation in the dependent variable that can be attributed to the independent variable(s) through the fitted model.^[1] It is calculated as the sum of the squared differences between each predicted value (ŷ) and the mean of the observed dependent variable (ȳ), given by the formula ESS = Σ(ŷ_i - ȳ)^2.^[2] This component contrasts with the unexplained or residual variation, which represents deviations between observed and predicted values, and together they partition the total sum of squares (TSS = Σ(y_i - ȳ)^2), such that TSS = ESS + RSS, where RSS is the residual sum of squares.^[3] The concept is central to assessing model fit, particularly through the coefficient of determination (R²), which is the ratio of explained variation to total variation (R² = ESS / TSS), indicating the proportion of variance in the dependent variable accounted for by the model, ranging from 0 (no explanation) to 1 (perfect explanation).^[1] In simple linear regression, R² equals the square of the correlation coefficient (r), providing a standardized measure of predictive power.^[2] Explained variation extends to multiple regression and other models like logistic regression, where analogous pseudo-R² measures adapt the concept to non-linear or binary outcomes, though interpretations require caution due to differing formulations.^[4] Key applications include evaluating the effectiveness of predictive models in fields such as economics, social sciences, and natural sciences, where high explained variation signals strong relationships between variables, while low values may indicate omitted factors or model misspecification.^[3] Assumptions underlying its use, such as linearity, independence of errors, and homoscedasticity, must hold for valid inference, and it is often reported alongside hypothesis tests like the F-statistic to confirm statistical significance.^[2]

Fundamentals

Core Definition

Explained variation refers to the portion of the total variation in a dataset that can be attributed to a specific model, variable, or explanatory factor, quantifying how much of the observed dispersion is accounted for by the predictors. It is commonly measured either as an absolute quantity, such as the sum of squared deviations explained by the model, or as a proportion relative to the overall variability in the data.^[5] The concept of explained variation emerged from foundational work in regression analysis by Karl Pearson in the early 1900s, who developed methods to fit lines and planes to data points, laying the groundwork for partitioning variability between explained and unexplained components. The term gained popularity in statistics during the mid-20th century, particularly through extensions in analysis of variance techniques introduced by Ronald A. Fisher in the 1920s, which formalized the decomposition of variation in experimental designs. Intuitively, the total variation in a dataset can be viewed as comprising a systematic "signal" from the explanatory factors and an unsystematic "noise" component, with explained variation representing the captured signal attributable to the model. The basic proportion of explained variation is calculated as the ratio of explained variation to total variation:

\text{Proportion of Explained Variation} = \frac{\text{Explained Variation}}{\text{Total Variation}}

This proportion indicates the model's effectiveness in accounting for the data's variability, while the complementary residual variation captures what remains unexplained.^[6]

Decomposition of Total Variation

Total variation represents the overall dispersion or spread in a dataset of observed values, quantifying the total uncertainty or variability present before any modeling is applied. In statistical analysis, it is typically measured using the total sum of squares (TSS), defined as the sum of the squared deviations of each observation from the sample mean.^[7]^[8] The core decomposition identity partitions this total variation into two orthogonal components: the explained variation, which accounts for the portion attributable to a model's predictions, and the residual (unexplained) variation, which remains after accounting for the model. Mathematically, for a dataset with observations y_i and model-fitted values \hat{y}_i, the identity is expressed as:

\sum_{i=1}^n (y_i - \bar{y})^2 = \sum_{i=1}^n (\hat{y}_i - \bar{y})^2 + \sum_{i=1}^n (y_i - \hat{y}_i)^2,

where the left side is the total sum of squares (TSS), the first term on the right is the explained sum of squares (ESS), and the second term is the residual sum of squares (RSS). This holds due to the additive property of quadratic forms under the orthogonality condition in least-squares estimation, where the residuals e_i = y_i - \hat{y}_i are perpendicular to the fitted values in the vector space sense, making the cross-product term vanish: $2 \sum_{i=1}^n (y_i - \hat{y}_i)(\hat{y}_i - \bar{y}) = 0.^[9] (Searle, 1971, Linear Models) To illustrate, consider a simple dataset with observations y_i and sample mean \bar{y}. The total sum of squares TSS measures the overall deviation from the mean, while applying a model yields fitted values \hat{y}_i. The explained portion \sum ( \hat{y}_i - \bar{y} )^2 captures how much the model's predictions deviate from the mean, and the residual portion \sum ( y_i - \hat{y}_i )^2 quantifies the leftover deviations, ensuring the equality TSS = ESS + RSS. This decomposition is fundamental for model evaluation, as it allows researchers to assess the proportion of total variation captured by the model—via ratios like the coefficient of determination—facilitating comparisons of fit across diverse statistical frameworks from regression to experimental designs. Introduced by Ronald Fisher in the context of agricultural experiments, it underpins variance partitioning in analysis of variance (ANOVA) and broader linear modeling. (Fisher, 1925, Statistical Methods for Research Workers)

Theoretical Frameworks

Variance-Based Measures

Variance-based measures of explained variation quantify the proportion of total variability in a response variable that can be attributed to a model's predictors, primarily through the lens of variance decomposition in statistical modeling. Explained variance is defined as the reduction in the residual error variance achieved by incorporating the model, relative to a baseline model that uses only the mean of the response variable. This reduction is often expressed as a ratio, where the explained variance represents the portion of the total variance accounted for by the predictors.^[6] The key metric in this framework is the coefficient of determination, denoted R^2, which is calculated as

R^2 = 1 - \frac{\mathrm{SS_{res}}}{\mathrm{SS_{tot}}} = \frac{\mathrm{SS_{expl}}}{\mathrm{SS_{tot}}},

where \mathrm{SS_{res}} is the residual sum of squares (sum of squared differences between observed and predicted values), \mathrm{SS_{tot}} is the total sum of squares (sum of squared differences between observed values and the mean), and \mathrm{SS_{expl}} is the explained sum of squares (sum of squared differences between predicted values and the mean). This formula arises directly from the principles of ordinary least squares (OLS) regression, which minimizes \mathrm{SS_{res}}. Under OLS, the total sum of squares decomposes additively as \mathrm{SS_{tot}} = \mathrm{SS_{expl}} + \mathrm{SS_{res}}, ensuring that R^2 captures the fraction of variance explained by the model. The derivation follows from partitioning the squared deviations around the mean: the model's fitted values orthogonalize the explained and residual components, leading to the identity \sum (y_i - \bar{y})^2 = \sum (\hat{y}_i - \bar{y})^2 + \sum (y_i - \hat{y}_i)^2, where y_i are observations, \hat{y}_i are predictions, and \bar{y} is the mean.^[10] A primary property of R^2 is that it ranges from 0 to 1, where 0 indicates no explanatory power (model no better than the mean) and 1 indicates perfect fit (residuals are zero). However, in multiple regression, R^2 non-decreasingly increases with additional predictors, even if they add no true explanatory value, due to overfitting. To address this, the adjusted R^2 penalizes model complexity:

R^2_{\mathrm{adj}} = 1 - \left(1 - R^2\right) \frac{n-1}{n - k - 1},

where n is the sample size and k is the number of predictors. This adjustment provides an unbiased estimate of the population R^2 by accounting for degrees of freedom lost to estimation.^[10] The foundations of variance-based measures trace back to Karl Pearson's development of the product-moment correlation coefficient in 1896, where for simple linear regression, R^2 = r^2 represents the shared variance between two variables.^[11] The term "coefficient of determination" was introduced by Sewall Wright in 1921 to quantify determination in path analysis models.^[12] These ideas were further expanded in the context of analysis of variance (ANOVA) by Ronald Fisher in his 1925 work, where variance decomposition underpins tests of group differences.^[13]

Information-Theoretic Measures

Information-theoretic measures of explained variation quantify the reduction in uncertainty about a response variable provided by a predictive model relative to a baseline or null model, drawing on concepts from information theory such as entropy and divergence. In this framework, explained variation is interpreted as the information gain achieved by incorporating explanatory variables, which decreases the entropy (a measure of uncertainty) in the conditional distribution of the response given the predictors compared to the marginal distribution under the null model. This approach contrasts with variance-based measures by focusing on distributional divergence rather than second-moment properties, allowing for a more general assessment of model improvement across diverse data types.^[14] A key formulation is the information gain between two parametric models with parameters \theta_1 (the full model) and \theta_0 (the baseline model), defined using the Fraser information function F(\theta), which arises from structural inference and differential geometry in statistics. The gain is given by

\Gamma(\theta_1 : \theta_0) = 2 \left[ F(\theta_1) - F(\theta_0) \right],

where F(\theta) = \int \log f(\mathbf{x}; \theta) \, g(\mathbf{x}) \, d\mathbf{x} represents the expected log-likelihood under a reference measure g, measuring the alignment between the model density f and the data-generating process. This expression, derived as twice the difference in expected log-likelihoods, quantifies the additional information captured by the more complex model and can be linked to the Kullback-Leibler divergence for asymptotic interpretations. The factor of 2 ensures interpretability akin to squared correlation measures in Gaussian cases.^[15]^[14] Information gain manifests in two primary subtypes relevant to explained variation. The first subtype arises from better modeling through improved parameter estimates within a fixed structure, where the gain reflects enhanced precision in capturing the data's underlying dependencies without altering the model's form. The second subtype emerges from conditional models that account for inter-variable dependencies, such as in multivariate settings, where the gain measures the incremental reduction in conditional entropy upon including covariates. These subtypes enable decomposition of total explained variation into contributions from estimation accuracy and structural enhancements.^[14] Compared to variance-based measures, information-theoretic approaches offer greater flexibility for non-normal distributions and nonlinear relationships, as they rely on general divergence metrics rather than assuming quadratic loss or normality, facilitating application to censored, discrete, or heavy-tailed data without restrictive transformations. This robustness stems from the entropy-based foundation, which inherently accommodates arbitrary dependence structures, unlike variance decompositions that may underestimate nonlinear effects.^[16]^[14]

Applications in Univariate Models

Linear Regression

In simple linear regression, the explained variation is quantified by the regression sum of squares, defined as

SS_{\text{reg}} = \sum_{i=1}^n (\hat{y}_i - \bar{y})^2,

where \hat{y}_i is the predicted value of the dependent variable for the i-th observation from the fitted regression line, and \bar{y} is the mean of the observed dependent variable values. This measure captures the amount of total variation in the dependent variable that is attributable to the linear relationship with the independent variable, as opposed to random error. The total sum of squares, SS_{\text{total}} = \sum_{i=1}^n (y_i - \bar{y})^2, decomposes into SS_{\text{reg}} plus the residual sum of squares, representing the unexplained variation. The coefficient of determination, R^2, expresses the explained variation as a proportion of the total variation in the dependent variable:

R^2 = \frac{SS_{\text{reg}}}{SS_{\text{total}}} = 1 - \frac{SS_{\text{res}}}{SS_{\text{total}}},

where SS_{\text{res}} is the residual sum of squares. In the bivariate case with one independent variable, R^2 equals the square of the Pearson correlation coefficient r between the independent and dependent variables, providing a direct measure of how much variance in the response is explained by the predictor. For example, consider a dataset with n=4 observations where the independent variable x values are 1, 2, 3, 4 and the dependent variable y values are 2, 4, 5, 7; the fitted line yields \hat{y} values of approximately 2.1, 3.7, 5.3, 6.9, with \bar{y} = 4.5. Here, SS_{\text{reg}} = 12.8, SS_{\text{total}} = 13, so R^2 \approx 0.985, indicating that about 98.5% of the variance in y is explained by x. Ronald Fisher's contributions to regression analysis in the 1920s formalized the application of explained variation through the integration of least squares estimation with variance decomposition, laying the groundwork for modern inferential uses in linear models.^[17]

Correlation Coefficient

In the context of bivariate relationships, explained variation is quantified through Pearson's correlation coefficient, denoted r, which assesses the strength and direction of the linear association between two continuous variables X and Y. The square of this coefficient, r^2, directly corresponds to the proportion of variance in Y that is explained by its linear relationship with X, serving as a measure of how much the variability in one variable can be accounted for by the other. This relationship holds in simple linear regression, where the coefficient of determination R^2 equals r^2, providing a standardized indicator of linear dependency.^[18] The formula for Pearson's correlation coefficient is derived from the standardized covariance:

r = \frac{\Cov(X, Y)}{\sigma_X \sigma_Y}

where \Cov(X, Y) is the covariance between X and Y, and \sigma_X and \sigma_Y are the standard deviations of X and Y, respectively. This formulation, introduced by Karl Pearson, normalizes the covariance to range between -1 and 1, with values near 0 indicating weak or no linear association, positive values denoting direct relationships, and negative values indicating inverse ones. The explained variation is then r^2 \cdot \Var(Y), representing the absolute variance in Y attributable to the linear component shared with X. Geometrically, this is visualized in scatterplots, where points tightly clustered along a straight line of best fit yield high |r| values, minimizing perpendicular deviations and maximizing the explained proportion of total variation.^[19]^[18] Despite its utility, the use of r^2 for explained variation assumes a strictly linear relationship and normality in the data distribution, capturing only shared linear variance while ignoring potential nonlinear patterns or confounding factors. For instance, even strong nonlinear associations may yield low r^2 values, leading to underestimation of overall dependency.

Applications in Multivariate Models

Principal Component Analysis

Principal component analysis (PCA) is a dimensionality reduction technique that identifies orthogonal directions, known as principal components (PCs), which capture the maximum variance in a multivariate dataset. The explained variation in PCA refers to the proportion of the total variance accounted for by these components, derived from the eigendecomposition of the data's covariance matrix. This decomposition allows for quantifying how much of the data's variability is preserved when projecting onto a lower-dimensional subspace, aiding in tasks like noise reduction and feature extraction.^[20] The mathematical foundation of explained variation in PCA stems from the eigendecomposition of the covariance matrix \Sigma, expressed as \Sigma = V \Lambda V^T, where V is the matrix of eigenvectors (principal directions) and \Lambda is a diagonal matrix containing the eigenvalues \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_p > 0, with p being the number of variables. Each eigenvalue \lambda_k represents the variance explained by the k-th principal component, and the total variance is the trace of \Sigma, equal to \sum_{k=1}^p \lambda_k. The proportion of explained variation by the k-th component is thus \frac{\lambda_k}{\sum_{j=1}^p \lambda_j}, providing a measure of its relative importance.^[21]^[22] To select the number of components for dimensionality reduction, the scree plot visualizes the eigenvalues in decreasing order, often revealing an "elbow" point where additional components contribute diminishing returns in explained variation. The cumulative explained variation for the first m components is \sum_{k=1}^m \frac{\lambda_k}{\sum_{j=1}^p \lambda_j}, interpreted as the fraction of total variance retained; for instance, retaining components until this sum reaches 80-90% is a common threshold for effective reduction while minimizing information loss.^[23]^[24] In genomics applications, such as analyzing gene expression data, the first few principal components often capture a substantial portion of genetic variation; for example, the first four PCs can explain nearly 80% of the variance in high-dimensional datasets from microarray studies, enabling identification of intrinsic biological clusters without overfitting.^[25]

Analysis of Variance

In analysis of variance (ANOVA), explained variation quantifies the portion of total variability in a response variable attributable to differences among predefined groups, typically defined by categorical factors, as opposed to random error within groups. This approach partitions the total sum of squares (SS_{total}) into between-group and within-group components, where the between-group sum of squares represents the explained variation due to group membership. Developed initially for experimental designs in agriculture, ANOVA uses this partitioning to test hypotheses about group mean equality while providing measures of explanatory power. In one-way ANOVA, which examines the effect of a single categorical factor with k levels on a continuous response variable, the explained variation is captured by the between-group sum of squares:

SS_{between} = \sum_{j=1}^k n_j (\bar{y}_j - \bar{y})^2

where n_j denotes the number of observations in group j, \bar{y}_j is the mean of group j, and \bar{y} is the grand mean across all N observations. This measure reflects the variability in the data explained by deviations of group means from the overall mean, weighted by group sizes. The proportion of total variation explained by the factor is then given by SS_{between} / SS_{total}, indicating how much of the observed differences in the response arise from group effects rather than error. This builds briefly on the general decomposition of total variation into explained and unexplained parts. The F-statistic central to ANOVA hypothesis testing derives from this partitioning, comparing the mean square between groups (MS_{between} = SS_{between} / (k-1)) to the mean square within groups (MS_{within} = SS_{within} / (N-k)), yielding F = MS_{between} / MS_{within}. A related effect size measure, eta-squared (\eta^2), directly quantifies explained variation as \eta^2 = SS_{between} / SS_{total}, ranging from 0 (no group effect) to 1 (all variation explained by groups); values around 0.01, 0.06, and 0.14 are often interpreted as small, medium, and large effects, respectively, though context matters. Eta-squared links the statistical significance of the F-test to practical importance by estimating the proportion of variance accounted for by the factor. Extensions to multi-way or factorial ANOVA incorporate multiple factors, allowing explained variation to include main effects for each factor as well as interaction effects, which assess whether the influence of one factor varies across levels of another. For instance, in a two-way ANOVA, the total explained variation comprises [SS](/page/.ss) for factor A, factor B, their interaction (A×B), and error, with the interaction term SS_{A \times B} capturing joint contributions not reducible to individual main effects. Partial eta-squared (\eta_p^2) extends eta-squared for these designs, measuring the unique explained variation for a specific effect (main or interaction) as \eta_p^2 = SS_{effect} / (SS_{effect} + SS_{error}), isolating its contribution after adjusting for other terms in the model; this is particularly useful for interactions, as it avoids overestimation in complex designs. The framework of ANOVA and its variance partitioning originated with Ronald Fisher in the 1920s at the Rothamsted Experimental Station in England, where he applied it to analyze crop yield data from randomized agricultural trials, formalizing the approach in his 1925 book Statistical Methods for Research Workers. This innovation enabled rigorous inference in experimental settings by linking explained variation to both hypothesis testing and effect quantification.

Extensions and Criticisms

Generalized Forms

Explained variation extends to conditional settings, where the measure quantifies the additional explanatory power of a predictor after accounting for other covariates. This is captured by the partial R², which assesses the incremental contribution of a variable in a multiple regression model by comparing the residual sums of squares (SS) from the full model (including the variable) to the reduced model (excluding it).^[26] The formula for partial R² is given by:

R^2_{\text{partial}} = 1 - \frac{\text{SS}_{\text{res, full}}}{\text{SS}_{\text{res, reduced}}}

This approach isolates the unique variance explained by the focal predictor, net of confounding influences.^[27] In information-theoretic terms, conditional explained variation can analogously employ conditional mutual information, which measures the reduction in uncertainty about the outcome given covariates, building on basic information gain concepts.^[28] In hierarchical or multilevel models, explained variation decomposes across layers, attributing portions of the total variance to individual-level and group-level factors. For instance, in clustered data such as students nested within schools, multilevel modeling partitions the variance into within-group (individual) and between-group (school) components, allowing separate quantification of explained variation at each level.^[29] This layered approach reveals how predictors operate at different scales, such as individual traits explaining within-school variation while school policies account for between-school differences.^[30] Modern adaptations appear in generalized linear models (GLMs), where traditional R² is unsuitable due to non-normal errors; instead, deviance-based pseudo-R² measures adapt the concept of explained variation. Nagelkerke's pseudo-R², proposed in 1991, scales the Cox & Snell pseudo-R² to range from 0 to 1 by dividing by its maximum possible value, providing a normalized assessment of model fit in binary or count outcomes. This metric, widely used in logistic and Poisson regression, quantifies the proportion of deviance explained relative to a null model.^[31]

Limitations and Interpretive Challenges

One common limitation of explained variation measures, particularly the coefficient of determination R^2, is its tendency to increase when irrelevant predictors are added to a model, even if they contribute no meaningful explanatory power. This occurs because R^2 is a non-decreasing function of the number of predictors, leading to overfitting where the metric suggests improved fit without actual enhancement in predictive accuracy.^[32] Additionally, high values of R^2 do not imply causation, as the measure only quantifies association between variables and can be inflated by confounding factors or spurious correlations without establishing directional influence.^[33] Specific criticisms highlight interpretive challenges with R^2, as emphasized by Achen (1982), who argued that it primarily measures dispersion rather than substantive explanation, and its value depends heavily on how variables are measured.^[34] For information-theoretic measures like information gain, a key challenge is computational complexity, as calculating it requires evaluating entropy across all possible splits, which becomes prohibitive for large datasets with high-dimensional features.^[35] In non-stationary or non-i.i.d. data, such as time series, explained variation metrics like R^2 exhibit instability because they assume independent observations, leading to biased estimates when autocorrelation or trends are present. This can result in artificially high R^2 values due to spurious regressions, where non-stationary series appear correlated by chance rather than true explanatory power.^[36] As alternatives for model comparison, criteria like the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) address some of these pitfalls by penalizing model complexity more explicitly than R^2, favoring parsimonious models that balance fit and generalizability without directly measuring explained variation.^[37]

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