Interpreting Effects When Dispersion Changes

Effects and uncertainty are different parts of a model

A regression coefficient describes how the model's mean changes with a predictor. Its interpretation comes from the mean structure and link function. In a quasi-binomial or quasi-Poisson model, a dispersion parameter also describes variability around that mean. When the dispersion is a common scalar, adjusting for it changes the estimated uncertainty of coefficients, but not the coefficient estimates themselves. Keeping these two roles separate is essential: the effect may be the same while the evidence about its precision changes.

For a quasi-binomial model, the logit link makes coefficients interpretable on the log-odds scale. For a quasi-Poisson model, the log link makes them interpretable on the log-mean, or log-rate, scale. Exponentiating a coefficient or a coefficient contrast gives a multiplicative effect. The interpretation is conditional on the other predictors in the linear predictor; it does not by itself describe a constant change in probability or count.

From coefficients to odds ratios and rate ratios

For a single predictor \(x_j\) with no interaction, increasing \(x_j\) by one unit changes the linear predictor by \(\beta_j\). Thus, \(\exp(\beta_j)\) is the odds ratio per unit increase in a quasi-binomial logit model, or the mean ratio per unit increase in a quasi-Poisson log-link model. A change of \(d\) units has linear-predictor contrast \(d\beta_j\), so its ratio is \(\exp(d\beta_j)\). For a contrast involving several predictors, replace \(d\beta_j\) with the corresponding change in the full linear predictor.

These are multiplicative interpretations. In the binomial case, an odds ratio is not a probability ratio or a difference in probabilities. In the Poisson case, a mean ratio is not a fixed difference in expected counts. The probability or expected-count difference depends on the starting covariate values as well as the coefficient contrast.

Why the coefficient estimates can stay the same

The shared-coefficient result follows from the form of the quasi-score. Consider independent observations with mean \(\mu_i\), linear predictor \( \eta_i=x_i^\mathsf{T}\beta \), and variance \( \operatorname{Var}(Y_i)=\phi V(\mu_i) \), where \(\phi>0\) is a common dispersion parameter. For a differentiable link and mean function, the quasi-score for \(\beta\) is

Here \(U_0(\beta)\) is the same score equation with the common factor \(1/\phi\) removed. Since multiplication by a nonzero scalar does not change the zeros, the dispersion value does not change the set of solutions to the coefficient estimating equations.

For the two models in focus, this cancellation can be seen directly. In a grouped quasi-binomial model, let \(Y_i\) be the number of successes in \(m_i\) trials, with mean \(m_i p_i\), variance \(\phi m_i p_i(1-p_i)\), and logit \( \operatorname{logit}(p_i)=x_i^\mathsf{T}\beta \). Since \(d(m_i p_i)/d\eta_i=m_i p_i(1-p_i)\), its quasi-score contribution is \(x_i(Y_i-m_i p_i)/\phi\). In a quasi-Poisson model, \( \mu_i=\exp(x_i^\mathsf{T}\beta) \) and \( \operatorname{Var}(Y_i)=\phi\mu_i \). Its contribution is \(x_i(Y_i-\mu_i)/\phi\). In both cases the common factor changes the score's scale, not its roots.

Dispersion changes uncertainty, not the effect scale

Under the conventional model-based quasi-GLM covariance calculation, with design matrix \(X\) of full column rank, the estimated covariance matrix has the form \( \hat{\phi}(X^\mathsf{T}WX)^{-1} \), where \(W\) is the diagonal matrix of GLM weights evaluated at the fitted means. The weights depend on the mean and variance function, not on the common scalar dispersion. The coefficient estimates, and hence the exponentiated coefficient contrasts, are therefore unchanged by this scalar factor, while the covariance matrix is multiplied by \(\hat{\phi}\).

In particular, if a coefficient's unscaled standard error is \(s\), its scaled standard error is \(\sqrt{\hat{\phi}}s\). The point estimate still gives the same odds ratio or rate ratio. But a Wald statistic, confidence interval, or test based on that standard error will generally change. If dispersion is estimated rather than known, the estimated value is used in this scaling. The effective degrees of freedom or reference distribution used for a test can also matter; the key distinction here is that these affect uncertainty assessment, not the coefficient's link-scale meaning.

The covariance scaling follows directly from the matrix form. The matrix \(X^\mathsf{T}WX\) is unchanged when only the common dispersion multiplier changes, while its inverse is unchanged as well. Multiplying that inverse by \(\hat{\phi}\) therefore multiplies every variance and covariance by \(\hat{\phi}\). A standard error is the square root of a variance, so it is multiplied by \(\sqrt{\hat{\phi}}\). This is a statement about the conventional quasi-GLM covariance calculation, not a claim that every possible uncertainty procedure has this form.

Common pitfall: treating a ratio as a probability or count difference

A frequent error is to say that a logit coefficient of \(0.60\) means the probability rises by 60%, or that a quasi-Poisson coefficient of \(0.4055\) means the count rises by \(0.4055\). Neither statement follows from the model. The logit coefficient is an additive change in log odds, and its exponential is an odds ratio. The log-link coefficient is an additive change in log mean, and its exponential is a mean ratio. The corresponding absolute changes depend on the starting probability or mean.

A second error is to say that increasing dispersion makes the odds ratio or rate ratio closer to 1. Under the common scalar-dispersion assumptions above, it does not change the estimated coefficient, so it does not change its exponentiation. It increases the model-based standard error when \(\hat{\phi}>1\), widening uncertainty intervals and often reducing the strength of evidence against a null contrast. This conclusion relies on the shared mean model and a common dispersion multiplier; if the mean structure, link, data, or variance weighting changes, the coefficient equations need not have the same solutions.