Choosing Pearson or Deviance Residuals for GLM Diagnostics

Why GLM Residuals Need a Model-Based Scale

A generalized linear model (GLM) predicts a conditional mean, not usually the observed response exactly. For a binomial outcome, a group with a fitted success probability of 0.40 will not necessarily have exactly 40% successes. For a count outcome, a fitted mean of 4 does not require the observed count to equal 4. A useful diagnostic therefore compares the observation with its fitted mean on a scale that accounts for the response distribution.

Two common choices are the Pearson residual and the deviance residual. Both are signed: positive values mean the observed response exceeds the fitted mean, and negative values mean it falls below the fitted mean. Their magnitudes measure discrepancy in different ways. Neither residual, by itself, proves that the model is wrong.

For the ordinary binomial and Poisson models, the dispersion parameter is fixed at 1. Thus the variance used here is the variance specified by the distribution and fitted mean. If a model instead uses an estimated dispersion parameter, residual scaling must reflect that model’s convention. The formulas below are for the standard binomial and Poisson GLMs.

Binomial Residuals

Suppose observation \(i\) records \(y_i\) successes in \(m_i\) trials, with a fitted success probability \(\hat{p}_i\). The model assumes a binomial response conditional on the predictors, so \(\hat{\mu}_i=m_i\hat{p}_i\) and its variance is \(m_i\hat{p}_i(1-\hat{p}_i)\). The Pearson residual is therefore

The binomial deviance contribution compares the fitted model’s likelihood for the observation with the likelihood under a saturated model, which fits that observation exactly. Its signed square root is

A term of the form \(0\ln(0/c)\) is defined to be 0 by continuity. This convention is necessary when an observation has no successes or no failures. The formula assumes \(0<\hat{p}_i<1\); a fitted probability at a boundary requires care because the likelihood comparison can be infinite or undefined.

Count Residuals

For a Poisson GLM, assume the observed count \(Y_i\), conditional on its predictors and exposure, has a Poisson distribution with fitted mean \(\hat{\mu}_i>0\). Its variance is also \(\hat{\mu}_i\). The Pearson residual is

The Poisson deviance residual is

When \(y_i=0\), define \(y_i\ln(y_i/\hat{\mu}_i)=0\), again by continuity. An exposure offset changes the fitted mean, not the residual formula: if the fitted rate is \(\hat{\lambda}_i\) per unit exposure \(e_i\), then \(\hat{\mu}_i=e_i\hat{\lambda}_i\).

Why the Deviance Formula Has This Form

The deviance measures how much log-likelihood is lost by using the fitted model rather than a model that fits the observed response exactly. For a single observation, twice that log-likelihood difference gives its deviance contribution. Taking a square root puts the contribution on a residual-like scale, and assigning the sign of \(y_i-\hat{\mu}_i\) records the direction of the discrepancy.

For a Poisson observation, its log-likelihood, ignoring a constant that is the same for both models, is \(y_i\ln(\mu_i)-\mu_i\). The saturated model sets its mean to the observed count \(y_i\); the fitted model uses \(\hat{\mu}_i\). Their log-likelihood difference, multiplied by two, is

This is nonnegative because the saturated model can fit the observation at least as well as the restricted fitted model. The signed square root retains this magnitude information while distinguishing overprediction from underprediction. For binomial data, the same likelihood comparison uses the saturated probability \(y_i/m_i\); the binomial combinatorial term cancels between the two likelihoods, leaving the two logarithmic terms in the binomial deviance formula.

How to Compare and Interpret the Two

Pearson residuals divide the observed-minus-fitted difference by the model standard deviation. Deviance residuals instead reflect the likelihood penalty for the discrepancy. For observations close to their fitted means, these scales are often similar. They need not agree closely for sparse counts, small expected binomial cell counts, or observations near the response boundaries. The examples above illustrate that the formulas respond differently to the same kind of discrepancy.

A positive residual indicates an observation larger than predicted; a negative residual indicates an observation smaller than predicted. A larger absolute residual signals a more pronounced discrepancy on that residual’s scale. It does not automatically mean the model is inadequate: chance variation can produce unusual observations, and model parameters were estimated from data that may include the observation itself. Residuals are not generally independent exact standard normal variables.

The sum of squared Pearson residuals is the Pearson goodness-of-fit statistic, and the sum of the deviance contributions is the residual deviance. A signed deviance residual’s square is its individual deviance contribution; summing the signed residuals themselves is not the residual deviance. These quantities summarize fit differently, so a large individual residual is a reason to examine the observation and model assumptions, not a standalone formal test.