Histograms And Quantile Plots Of Residuals

Why Examine the Distribution of Residuals?

Residual plots against time, predictions, and other variables can reveal systematic structure, but they do not summarize the overall distributional shape of residuals. A histogram and a normal quantile plot (also called a normal Q–Q plot) offer complementary views: the histogram shows the broad shape, while the quantile plot compares observed residual quantiles with those expected under a reference normal distribution. They can help identify skewness, unusually heavy or light tails, and departures concentrated among the most extreme values.

In an NLME analysis, the choice of residual matters. A population-level diagnostic and an individual-level diagnostic answer different questions. The earlier material on raw, weighted, and conditional residuals, standardization, and reference distributions distinguishes these targets. Here, the key requirement is to state which residual is plotted and interpret its reference distribution accordingly. For example, CWRES are commonly assessed against an approximate standard normal reference for population-model adequacy; this is not a claim that every finite-sample set of estimated CWRES must be exactly independent standard normal observations.

These are descriptive diagnostics, not automatic pass-or-fail tests. A roughly symmetric histogram and an approximately straight normal quantile plot are consistent with the selected normal reference, but do not prove the model is correct. Conversely, departures may arise from model misspecification, residual dependence, a small sample, heterogeneous residual behavior across subjects, or influential observations. The plots must be read alongside the model assumptions and other diagnostics.

Reading a Residual Histogram

A histogram’s shape depends partly on bin width and bin boundaries. Wide bins can hide features; narrow bins can make ordinary sampling variation look like multiple peaks. Always inspect the residual scale and units, state whether the vertical axis shows counts or proportions, and consider whether the broad pattern persists under reasonable bin choices.

For residuals standardized to a standard normal reference, zero is the reference center and values are measured in standard deviation units of that reference. A longer right tail than left tail suggests positive skewness; a longer left tail suggests negative skewness. More observations far from zero than expected under the reference suggest heavy tails, while fewer suggest light tails. A histogram alone does not reliably distinguish a smooth tail departure from a few isolated extreme values.

Constructing and Interpreting a Normal Quantile Plot

Let \(e_1,\ldots,e_m\) denote the residuals under examination, and write their ordered values as \(e_{(1)}\leq \cdots \leq e_{(m)}\). For a standard normal reference, a common plotting-position convention pairs \(e_{(i)}\) with \(z_i=\Phi^{-1}((i-0.5)/m)\), where \(\Phi\) is the standard normal cumulative distribution function. Other plotting-position conventions are also used; the exact positions vary slightly, especially in small samples, but the interpretation is the same.

If the residuals follow the reference distribution reasonably well, the plotted points should lie approximately along a straight line. In a standard normal Q–Q plot with theoretical quantiles on the horizontal axis and residuals on the vertical axis, the reference pattern is near the line \(y=x\). A fitted line may have a different intercept or slope if location and scale are not fixed by standardization; systematic curvature is more informative about shape than a small shift or scale difference.

Interpret departures by where they occur. A smooth bend through much of the plot can indicate skewness. If both ends bend away from the central trend, the tails may be heavier than the reference; if the extremes bend inward, they may be lighter. A departure confined to one or two end points may indicate an unusually extreme residual, but the plot alone cannot tell whether it reflects a data error, an unusual subject, or model inadequacy.

Pooled Residuals and Within-Subject Behavior

Pooling residuals means combining observations from multiple subjects into one set before making the histogram or quantile plot. A pooled display describes the empirical distribution across the observations included; it does not show whether the same distributional behavior occurs within each subject. Subjects with more observations contribute more points and therefore more weight to an observation-level pooled plot.

This distinction is important in NLME models. Even if a pooled histogram is symmetric and the pooled quantile plot looks nearly straight, residuals could have subject-specific shifts, different spreads, or patterns over time. The reverse can happen too: a pooled distribution may depart from normality because subjects contribute residuals with differing locations or scales, even when no single subject’s limited set of residuals clearly shows the pooled feature. The shape of a mixture of distributions need not match the shape of each component.

Practical Interpretation, Alternatives, and Limitations

The main alternative to a histogram is the normal quantile plot, not because one replaces the other, but because each emphasizes different features. A histogram makes overall shape and possible multimodality easy to see, but depends on bin choices. A Q–Q plot avoids bins and highlights departures in the tails, but can look irregular when there are few observations and cannot reveal time ordering. Use both when the sample allows, and relate their evidence to plots that address time, predictions, and subject-level structure.

Several limitations deserve attention. Standardized residuals may have an approximate, rather than exact, normal reference because model parameters are estimated and the residuals can be correlated. Some residual definitions also condition on subject-specific estimates; the reference distribution and diagnostic target must match the chosen residual. Unequal numbers of observations per subject affect pooled displays. Sparse within-subject sampling makes individual histograms or quantile plots unstable. Finally, a single extreme residual can strongly affect the tails without establishing a general distributional departure.

Do not use these plots as a substitute for checking residuals against time or predictions. A normal-looking pooled histogram cannot rule out serial structure, and a normal quantile plot does not assess whether residual variance changes with time or fitted value. Nor do these displays alone identify influential subjects or establish that a particular observation should be removed.

A Focused Diagnostic Workflow