Residual Plots for Unequal Variance

Why Plot Residuals When Spread May Change?

A fitted regression line describes the mean response, but it does not tell us whether individual responses vary around that mean by a similar amount at every level of the predictors. If that variability grows or shrinks, a plot of residuals can make the change visible. The key is to look at the vertical spread of residuals across the plot, not just whether points lie above or below zero.

A widening band can be evidence of unequal variance, but it is not the only explanation for a changing-looking pattern. Curvature in the mean relationship, one unusual observation, or features of how the sample was collected can also affect what the plot looks like. A careful diagnosis checks these alternatives before describing a pattern as evidence of unequal variance.

The Model and the Diagnostic Plots

For clarity, consider a simple linear regression with predictor \(x_i\) and response \(y_i\), for observations \(i=1,\ldots,n\). The mean model is \(y_i=\beta_0+\beta_1x_i+\varepsilon_i\), where the errors have conditional mean zero. For this diagnostic, we allow their conditional variances to differ: \(\operatorname{Var}(\varepsilon_i\mid x_i)=\sigma_i^2\). Assume observations are independent, so their errors are uncorrelated. Constant variance is not assumed; assessing whether the spread appears to change is the point of these plots.

The fitted value is the response predicted by the least-squares line, \(\hat{y}_i=b_0+b_1x_i\). The residual is the observed response minus its fitted value, \(e_i=y_i-\hat{y}_i\). A residual estimates the unobserved error \(\varepsilon_i\), but it is not that error. The least-squares coefficients \(b_0\) and \(b_1\) are calculated from the same observations whose residuals are plotted.

A residual-versus-fitted plot is a natural first view: fitted values index the model's predicted response level. A residual-versus-predictor plot can help identify which predictor values are associated with changing spread. In simple regression, fitted values are a linear function of \(x_i\), so the two plots have the same horizontal ordering (or the reverse ordering if the fitted slope is negative). With several predictors, a residual-versus-fitted plot may not reveal which predictor is related to the spread; plotting residuals against each predictor can add useful detail.

Look for a roughly centered band whose width changes, such as a narrow band at small fitted values that widens at large fitted values. This is often called a funnel or fan pattern. Also inspect the pattern of the residuals' signs and their central tendency: a curved trend in the residuals can indicate that the fitted mean relationship is not adequately linear, rather than a pure change in variance. These visual cues are not formal tests and do not prove that the conditional variance changes.

Why Residual Spread Is Only an Approximation

Residuals are constrained by the fitted regression. For example, in a regression with an intercept their sum is zero, and they cannot vary freely like the unobserved errors. This matters because a residual plot is a diagnostic view of estimated errors, not a direct plot of the errors themselves.

A short derivation shows why residual variation need not match error variation point for point. Write the regression in matrix form as \(y=X\beta+\varepsilon\), where the columns of \(X\) contain the intercept and predictor values. Let \(H=X(X^\mathsf{T}X)^{-1}X^\mathsf{T}\) be the projection matrix for least squares, and let \(M=I-H\). Since \(HX=X\), the residual vector is \(e=My=M\varepsilon\). If \(\Omega=\operatorname{Var}(\varepsilon\mid X)\), then

For independent errors, \(\Omega\) is diagonal with entries \(\sigma_j^2\). Since \(M\) is symmetric, the variance of residual \(e_i\) is therefore

Each residual's variability can depend on several errors and the regression's projection, not just on the variance of the error at that observation. This is one reason to interpret an apparent funnel as evidence to investigate, rather than as a precise measurement of the error variance at every fitted value.

Separate Changing Spread from Curvature

Before interpreting a fan-shaped appearance as unequal variance, check whether residuals have a systematic trend in their location. A curved sequence of positive and negative residuals points to a possible mismatch in the mean model. The following hypothetical data illustrate the distinction.

Recognize the Influence of an Isolated Point

A single large residual can stretch the vertical scale and make the rest of the points look artificially compressed. That visual effect is different from a broad, systematic increase in spread across the horizontal axis. Look for whether many points support the change in width or whether one observation dominates it.

A Careful Reading Process

A disciplined reading helps keep mean patterns, isolated points, and changes in spread separate. It also makes a diagnostic report clearer: describe what the plot shows and how strongly it supports an interpretation.

Common Mistakes

• Calling every large residual a variance pattern. One isolated point is not the same as a sustained change in spread across fitted values or predictor values. Check whether multiple observations support the pattern.
• Calling a U-shape a funnel. A systematic sequence of positive and negative residuals suggests a location pattern such as curvature. Check the signs and center as well as the width.
• Treating residuals as the unobserved errors. Residuals are calculated after fitting and are constrained by the regression. Use them as diagnostic estimates, not as direct observations of each error.
• Declaring unequal variance proven by a plot. A visible fan is evidence worth reporting, but plots are affected by sample size, individual observations, and the fitted mean model. Describe what is visible without claiming more than the evidence supports.