Observation Weights and Quasi-Dispersion

Why Weights and Dispersion Must Be Kept Separate

A quasi-GLM can give an observation more influence because it represents repeated records, because its response is measured with greater precision, or because it contains more binomial trials. These are different reasons for weighting, even when they produce similar-looking multipliers in a score equation. The common dispersion parameter has another role: it scales the assumed variance across observations. Keeping these roles distinct is essential for interpreting both the fitted coefficients and dispersion summaries.

Consider independent responses \(Y_i\) with means \(\mu_i\), linear predictors \(\eta_i=x_i^\mathsf{T}\beta+o_i\), and a specified variance function \(V(\mu_i)\). Here \(x_i\) is a vector of predictors, \(\beta\) is the coefficient vector, and \(o_i\) is an optional offset. We use the quasi-GLM variance convention \(\operatorname{Var}(Y_i)=\phi V(\mu_i)/a_i\), where \(a_i\) is an observation-level weight and \(\phi>0\) is the common dispersion multiplier.

Terminology varies across software and statistical fields. A “prior weight” in GLM software often enters as an inverse-variance or precision weight; “variance weight” is also used for this role. A frequency weight is typically a nonnegative integer when it literally represents replicated records. The safest practice is to state what a weight means and show how it enters the variance or estimating equation, rather than relying on its label alone.

The Weighted Quasi-Score

Let \(g(\mu_i)=\eta_i\), and write \(d\mu_i/d\eta_i\) for the derivative of the mean with respect to the linear predictor. With variance \(\phi V(\mu_i)/a_i\), the quasi-score for \(\beta\) is

This formula makes the roles explicit. The observation weight \(a_i\) changes that observation’s contribution relative to the others. The common \(\phi\) divides the whole score by the same positive constant. For a frequency weight, the same multiplication arises by adding the score contribution once for each represented record. For a variance weight, it arises because the observation is assigned variance \(\phi V(\mu_i)/a_i\).

For a Poisson variance function \(V(\mu)=\mu\) and a log link, \(d\mu/d\eta=\mu\), so the score simplifies to \(U(\beta)=\phi^{-1}\sum_i a_i x_i(Y_i-\mu_i)\). For grouped binomial data, if \(Y_i\) is the observed success proportion and \(m_i\) is the number of trials, then \(V(\mu_i)=\mu_i(1-\mu_i)\) and the trial total enters as \(a_i=m_i\).

Frequency Weights and Variance Weights

A frequency weight represents repeated identical rows. If a row with response \(y\) and predictor vector \(x\) occurs \(f\) times, its total score is \(f\) times the score for one such row. This follows by summing the same contribution \(f\) times. A variance weight, by contrast, says that the recorded response has a specified precision relative to the variance function. It need not mean that there were \(a_i\) independent copies of that response.

Grouped binomial trial totals offer a familiar precision interpretation: a proportion based on more trials has smaller variance, since \(\operatorname{Var}(Y_i)=\phi p_i(1-p_i)/m_i\). This resembles a variance weight \(a_i=m_i\). But the trial total also has a specific sampling meaning: it is the number of Bernoulli outcomes summarized by the proportion. It should not be confused with a frequency weight that repeats an entire group-level observation.

Two Consequences for Fitting and Dispersion

The common dispersion and the observation weights both appear in the score, but they affect fitting differently. Multiplying every score contribution by the same positive constant does not change the equation \(U(\beta)=0\). Changing the observation weights, however, generally changes the relative contributions and can change the solution. The first result is about the coefficient estimates; it does not say that dispersion is irrelevant to uncertainty.

Proof. Define the score without the common multiplier as \(U_0(\beta)=\sum_i a_i x_i(d\mu_i/d\eta_i)(Y_i-\mu_i)/V(\mu_i)\). Then \(U(\beta)=U_0(\beta)/\phi\). Since \(\phi\) is finite and positive, \(U(\beta)=0\) if and only if \(U_0(\beta)=0\). Therefore the root sets are identical, and so are the solutions when the root is unique. \(\square\)

The same distinction appears in dispersion estimation. For variance weights \(a_i\), the Pearson statistic has the form

A common Pearson-based estimate divides this quantity by an appropriate residual degrees of freedom. The weights alter the observation-level terms in the numerator; they do not replace the common dispersion being estimated. For literal frequency weights, aggregating rows also represents repeated residual contributions, and the corresponding residual degrees of freedom should reflect the represented records under that interpretation. With general precision weights, simply treating the sum of weights as a sample size is not automatically justified. The intended weight meaning and the fitting convention must be clear before interpreting a dispersion estimate.

Proof. Let the individual outcomes in group \(i\) be \(Z_{ij}\in\{0,1\}\), for \(j=1,\ldots,m_i\), with common fitted probability \(p_i\) and predictor vector \(x_i\). Under the logit link, the Bernoulli score contribution for outcome \(Z_{ij}\) is \(x_i(Z_{ij}-p_i)/\phi\). Summing within the group gives

The grouped quasi-binomial score for the proportion \(y_i\) uses variance function \(p_i(1-p_i)\), derivative \(dp_i/d\eta_i=p_i(1-p_i)\), and trial weight \(m_i\). Its contribution is therefore \(\phi^{-1}m_i x_i(y_i-p_i)\), equal to the sum above. Thus grouping preserves this estimating-equation contribution under the stated common-probability condition. \(\square\)

Common Mistake: Treating Every Weight as Dispersion

A common error is to see a multiplier in a quasi-score and conclude that it can be absorbed into \(\phi\). That reasoning fails when the multiplier varies by observation. A common positive \(\phi\) scales every score contribution equally; changing one observation’s weight relative to another changes their balance and can change the fitted coefficients. Conversely, increasing all observation weights by the same constant can leave the score roots unchanged, just as changing a common scale can, while still changing how variance and dispersion summaries are interpreted.

Another mistake is to read a frequency weight as a dispersion estimate, or to multiply the common \(\phi\) by a trial total when the grouped-binomial variance already uses the trial total as a precision multiplier. For a proportion, the roles are explicit in \(\operatorname{Var}(Y_i)=\phi p_i(1-p_i)/m_i\): \(m_i\) describes the group’s trial count, and \(\phi\) scales the variance beyond the binomial variance convention.

Offsets and exposure terms have a separate role: they enter the linear predictor and hence determine the fitted mean, not the observation’s precision weight. A large exposure can change the mean in a quasi-Poisson model without being a frequency weight or a common dispersion multiplier.