Comparing Ordinary and Quasi-GLM Results

What Changes—and What Does Not

An ordinary binomial or Poisson GLM and its quasi-GLM counterpart can look different in a results table even when they describe the same fitted mean pattern. The key distinction is between the mean model and the variance assumption. With the same observations, predictors, link, offsets, and weights, the standard matched fits have the same coefficient estimates and fitted means. Their uncertainty calculations can differ because the quasi-GLM estimates a dispersion parameter rather than fixing it at one.

This comparison is useful when reviewing software output: it prevents a change in standard errors from being mistaken for a change in the estimated effects. It also makes clear that “ordinary” does not mean the data have been shown to have dispersion exactly one. Rather, the conventional ordinary binomial or Poisson fit uses its standard variance specification for model-based inference; a dispersion diagnostic can still be calculated separately.

Definition: A matched ordinary and quasi-GLM comparison uses the same response data and mean-model specification—including predictors, link, offset, and observation or prior weights—in both fits. The ordinary binomial or Poisson GLM uses its standard dispersion value of one for model-based covariance calculations. The corresponding quasi-GLM allows a dispersion parameter \(\phi\) in the variance, and commonly estimates it from the fitted residuals. Under the usual matched setup, dispersion changes the estimated covariance and inference, not the fitted mean.

Why the Coefficients and Fitted Means Agree

The equality of point estimates follows from the estimating equations. For an observation with linear predictor \(\eta_i\), mean \(\mu_i\), and covariate vector \(x_i\), the quasi-likelihood estimating equation has contributions proportional to \(\{y_i-\mu_i\}(d\mu_i/d\eta_i)x_i/V(\mu_i)\), with a factor \(1/\phi\). For the logit link, \(d\mu_i/d\eta_i=\mu_i(1-\mu_i)\), the binomial variance function. For the log link, \(d\mu_i/d\eta_i=\mu_i\), the Poisson variance function. Thus, for these canonical links, the factors involving the derivative and variance cancel. The remaining quasi-GLM equation is the ordinary likelihood score equation, multiplied by a positive scalar.

For grouped binomial data, let \(m_i\) be the number of trials and \(y_i\) the successes. The mean count is \(m_i p_i\), and the same conclusion holds: the grouped binomial score and the quasi-binomial estimating equation have the same roots. For Poisson regression with an offset, the offset is part of \(\eta_i\); if it is identical in both fits, it does not break the match.

Result: If the two fits use the same observations, mean-model specification, link, offsets, and weights, and the ordinary score has a unique finite solution, the matched ordinary binomial or Poisson GLM and quasi-GLM have the same coefficient estimates and fitted means. This statement concerns the standard matched models; it does not apply automatically when a model, data subset, weight interpretation, or fitting method differs.

Derivation. For binomial and Poisson models with their canonical links, write the quasi-GLM estimating equation as

U_Q(\beta) = \frac{1}{\phi} \sum_i x_i\{y_i-\mu_i(\beta)\}.

Here the displayed form is for individual responses; in grouped binomial data the residual contribution is \(y_i-m_i p_i\), with the corresponding trial count included in the mean. The ordinary canonical-link likelihood score is

U_L(\beta) = \sum_i x_i\{y_i-\mu_i(\beta)\}.

For any positive \(\phi\), \(U_Q(\beta)=0\) if and only if \(U_L(\beta)=0\). Therefore the two procedures have the same score roots. If the ordinary score has a unique finite solution, the coefficient estimates agree. Since both fits apply the same inverse link to the same linear predictors, their fitted means agree as well. A shared root alone does not guarantee uniqueness; that is why the uniqueness condition matters. \(\square\)

What Dispersion Changes

For the standard model-based GLM covariance calculation, let \(A\) denote the information matrix formed from the fitted means, design, link, and variance function, before multiplying by dispersion. The ordinary fit uses \(A^{-1}\); the quasi-GLM uses \(\phi A^{-1}\), or \(\hat{\phi}A^{-1}\) when dispersion is estimated. This scalar rescaling leaves the fitted coefficients unchanged. It multiplies each coefficient variance by \(\hat{\phi}\) and each standard error by \(\sqrt{\hat{\phi}}\). The same factor applies to the model-based covariance between coefficients.

Consequently, an estimated dispersion above one increases standard errors and generally widens intervals; a value below one decreases them. The estimate of \(\phi\) is not a universal fact independent of the analysis: it depends on the dispersion estimator and residual degrees of freedom used. The comparisons below use the Pearson estimate \(\hat{\phi}=X_P^2/(n-p)\), where \(X_P^2\) is the Pearson statistic and \(p\) is the number of fitted coefficients.

Derivation: In the standard model-based calculation, the quasi-GLM information matrix is \(A/\phi\), because the variance is multiplied by \(\phi\). Its inverse is \((A/\phi)^{-1}=\phi A^{-1}\). The ordinary binomial or Poisson model-based calculation sets \(\phi=1\), giving \(A^{-1}\). Substituting an estimate of dispersion yields \(\hat{\phi}A^{-1}\), so the standard-error multiplier is \(\sqrt{\hat{\phi}}\). This derives the model-based scaling used in a matched comparison; it does not assert that every alternative covariance estimator has this form. \(\square\)

Point estimates, standard errors, intervals, and tests should therefore be compared separately. Under common asymptotic ordinary-GLM practice, coefficient tests use a normal reference distribution. A quasi-GLM with estimated dispersion commonly uses a residual-degrees-of-freedom \(t\) reference for a single coefficient and an \(F\) reference for a joint contrast. These are conventional approximate inferential procedures, not exact finite-sample guarantees. In particular, a changed test result can reflect both the dispersion rescaling and the chosen reference distribution.

Worked Comparisons

Worked Example: Quasi-Poisson Intercept-Only Fit

Suppose four independent counts are \(1,2,3,6\), and fit an intercept-only Poisson model and its matched quasi-Poisson model with a log link. Find the common fit, estimate Pearson dispersion, and compare the intercept uncertainty.

The fitted Poisson mean is the sample mean: \(\hat{\mu}=(1+2+3+6)/4=3\). Thus both fits have \(\hat{\beta}_0=\log(3)\approx1.0986\) and fitted means \(3,3,3,3\). The Pearson statistic is

X_P^2 = \sum_i \frac{(y_i-\hat{\mu})^2}{\hat{\mu}} = \frac{4+1+0+9}{3} = \frac{14}{3}.

There are \(n-p=4-1=3\) residual degrees of freedom, so \(\hat{\phi}=(14/3)/3=14/9\approx1.5556\). The ordinary model-based intercept variance is \(1/12\): for a log-link intercept-only Poisson fit, the information is the sum of the four fitted means, \(4(3)=12\). Hence the ordinary standard error is \(1/\sqrt{12}\approx0.2887\). The quasi standard error is \(\sqrt{14/9}/\sqrt{12}=\sqrt{14/108}\approx0.3600\).

For illustration, an ordinary large-sample 95% normal interval is \(1.0986\pm1.96(0.2887)=[0.5328,1.6644]\). A conventional quasi interval using \(t_3\), with \(t_{3,0.975}\approx3.182\), is \(1.0986\pm3.182(0.3600)\approx[-0.047,2.244]\). The intervals differ even though both fits estimate the same mean of \(3\). The quasi interval is approximate; estimating dispersion from only four observations does not make a \(t\) interval an exact finite-sample result.

Worked Example: Grouped Quasi-Binomial Fit

Four groups each have ten trials, with observed success counts \(1,2,8,9\). Fit an intercept-only grouped binomial model and the matched quasi-binomial model. Calculate the fitted proportion and Pearson dispersion, then state how the standard error changes.

The total number of successes is \(20\) out of \(40\) trials, so both fits estimate \(\hat p=20/40=0.5\). The intercept is \(\hat{\beta}_0=\log\{\hat p/(1-\hat p)\}=0\). Each group has fitted success count \(10(0.5)=5\). The Pearson statistic for grouped binomial data is

X_P^2 = \sum_{i=1}^4 \frac{(y_i-5)^2}{10(0.5)(0.5)} = \frac{16+9+9+16}{2.5} = 20.

With four groups and one fitted coefficient, the residual degrees of freedom are \(3\), giving \(\hat{\phi}=20/3\approx6.667\). The ordinary intercept information is \(40(0.5)(0.5)=10\), so its standard error is \(1/\sqrt{10}\approx0.3162\). The quasi standard error is \(\sqrt{20/3}/\sqrt{10}=\sqrt{2/3}\approx0.8165\). The fitted probability remains \(0.5\) in both models, but the quasi fit reports much greater uncertainty because its estimated dispersion is much larger than one.

Worked Example: A Quasi-Poisson Fit With Estimated Dispersion Below One

Consider Poisson counts \(1,2\) at \(x=0\) and \(4,5\) at \(x=1\), with a log link and an intercept and slope. Compare the fitted slope standard errors using the Pearson dispersion estimate.

This model gives a separate fitted mean for each predictor group. The fitted means are the group averages: \(1.5\) at \(x=0\) and \(4.5\) at \(x=1\). Therefore \(\hat{\beta}_0=\log(1.5)\approx0.4055\), and \(\hat{\beta}_1=\log(4.5)-\log(1.5)=\log(3)\approx1.0986\). These estimates and fitted means are the same for the matched quasi-Poisson fit.

Within each group, the residuals are \(-0.5\) and \(0.5\). The Pearson statistic is

X_P^2 = \frac{0.25+0.25}{1.5} + \frac{0.25+0.25}{4.5} = \frac{1}{3}+\frac{1}{9} = \frac{4}{9}.

There are \(4-2=2\) residual degrees of freedom, so \(\hat{\phi}=(4/9)/2=2/9\). The ordinary information matrix is \(\left(\begin{smallmatrix}12&9\\9&9\end{smallmatrix}\right)\): the four fitted means sum to \(12\), and the two means at \(x=1\) sum to \(9\). Its inverse has slope variance \(4/9\), so the ordinary slope standard error is \(2/3\approx0.6667\). The quasi standard error is \((2/3)\sqrt{2/9}=2\sqrt{2}/9\approx0.3143\). Here the estimate below one reduces the model-based standard error. It is an estimate from a very small residual sample, not proof that underdispersion is a stable feature of the data-generating process.

How to Read a Matched Results Table

A useful comparison is organized by what the quantities represent, rather than by which model is “better.” The ordinary fit is not automatically a wrong fit when a residual dispersion estimate differs from one; the quasi fit changes the variance specification used for uncertainty. Conversely, a quasi fit is not evidence that the mean structure is adequate. Mean-model misspecification can affect both fits.

Quantity	Matched ordinary fit	Matched quasi-GLM fit
Coefficients and fitted means	Obtained from the ordinary score equations	Same under the matched canonical-link setup
Dispersion in model-based covariance	Fixed at one	Estimated value, such as a Pearson estimate
Standard errors	Based on \(A^{-1}\)	Based on \(\hat{\phi}A^{-1}\), hence scaled by \(\sqrt{\hat{\phi}}\)
Intervals and tests	Often use asymptotic normal or chi-square references	Often use residual-df \(t\) or \(F\) references when dispersion is estimated
Likelihood-based comparisons	Available under the specified likelihood model	Not supplied by quasi-likelihood alone

For a particular coefficient, inspect the estimate first, then its standard error, interval, and test. If the coefficient is unchanged but its standard error is multiplied by about \(1.5\), a plausible explanation is \(\hat{\phi}\approx1.5^2=2.25\), subject to the reference distribution and covariance method used. For a joint contrast, the same dispersion scaling enters its estimated covariance; the test calibration also matters. Do not compare p-values without checking both ingredients.

Common Mistake and Practical Checks

Common mistake: “The quasi-Poisson slope is smaller because its standard error is larger, so the quasi model estimates a weaker effect.” This confuses the point estimate with its uncertainty. In a matched fit, the slope is unchanged; the increased standard error reflects the variance assumption and dispersion estimate. The correction is to compare coefficient estimates separately from covariance-based quantities, and then state what the estimated dispersion does to the uncertainty.

Another mistake is to treat an ordinary model’s dispersion as an estimated value of one. In the conventional ordinary binomial or Poisson fit, one is the specified dispersion used in its model-based covariance. A Pearson statistic divided by residual degrees of freedom may be calculated as a diagnostic, but that diagnostic does not automatically rescale the ordinary fit’s standard errors.

Before interpreting a discrepancy, verify that the fits use the same response representation, observations, link, covariates, offset, and weights. For grouped binomial data, check that trial totals and weights have the same meaning in both fits. For Poisson data, check that exposure is represented consistently. Different choices can change the mean fit itself, so the matched-fit conclusions no longer apply. Finally, a quasi-GLM specifies mean and variance relationships but does not provide an ordinary full likelihood; likelihood-ratio conclusions should not be inferred from a quasi fit alone.

Key takeaway: In a matched canonical-link binomial or Poisson comparison, coefficients and fitted means agree because the estimating equations have the same roots. Dispersion changes the model-based covariance, and therefore standard errors, intervals, and tests—not the fitted mean pattern.

Check Your Understanding

Use the matched-fit distinctions to answer the following.

Under what conditions can a quasi-Poisson fit have the same coefficients as an ordinary Poisson fit?
If \(\hat{\phi}=4\), what is the model-based standard-error multiplier relative to the ordinary fit, assuming all other covariance inputs match?
In a grouped binomial comparison, the fitted proportions agree but quasi standard errors are larger. What does this suggest about the estimated dispersion?
Why is an ordinary Poisson dispersion convention of one not the same as a residual estimate of one?
Name two fitting differences that could make ordinary and quasi coefficients disagree.