Grouped Binomial Score Equations

Introduction

A grouped binomial response records a number of successes out of a known number of trials for each covariate pattern. For example, a clinic might report that 8 of 12 patients responded to a treatment at one dose and 3 of 5 responded at another. These are not two equally precise measurements of a response proportion: the groups have different numbers of trials. The likelihood, score, and curvature must retain that difference.

This tutorial derives the regression score and observed Hessian for independent grouped binomial counts under a logit link. The central feature is simple but consequential: a group contributes its success count minus its expected success count, \(y_i-m_i p_i\), to the score. Equivalently, its residual in proportion units is weighted by its trial count \(m_i\).

Grouped Binomial Logit Model

The count \(Y_i\) takes values \(0,1,\ldots,m_i\). Conditional on the fixed design and trial counts, the probability of observing \(Y_i=y_i\) is

Independence makes the likelihood the product of these probabilities. Writing \(\eta_i=x_i^\mathsf{T}\beta\), the logit model gives \(\log(p_i)=\eta_i-\log(1+\exp(\eta_i))\) and \(\log(1-p_i)=-\log(1+\exp(\eta_i))\). Therefore the log-likelihood is

The binomial coefficient is part of the grouped-data likelihood, but it does not depend on \(\beta\). It contributes no terms to either the score or the Hessian.

Deriving the Score Equation

For each group, \(\partial\eta_i/\partial\beta=x_i\), while differentiating the log-likelihood contribution with respect to \(\eta_i\) gives \(y_i-m_i p_i\). The chain rule then gives the vector score.

A finite maximum-likelihood estimate \(\hat\beta\), if it exists in the parameter space, satisfies the score equations \(U(\hat\beta)=0\). In proportion units, when \(r_i=y_i/m_i\) is the observed group proportion, the same score is

Thus a group with a larger trial count contributes more for the same discrepancy between its observed and fitted proportions. This weighting is not an additional modeling choice: it follows from the binomial likelihood.

Observed Hessian and Likelihood Curvature

Differentiate the score once more. Since \(p_i=\exp(\eta_i)/\{1+\exp(\eta_i)\}\), its derivative with respect to \(\eta_i\) is \(p_i(1-p_i)\). Therefore the derivative of \(y_i-m_i p_i\) with respect to \(\beta\) is \(-m_i p_i(1-p_i)x_i\). The resulting observed Hessian is

In this model the Hessian does not depend on the observed counts \(y_i\); for a fixed \(\beta\) it is also the expected Hessian. The negative sign describes downward likelihood curvature. More precisely, for any vector \(a\in\mathbb{R}^p\),

Thus the Hessian is negative semidefinite at every finite \(\beta\). Because \(m_i>0\) and finite logit probabilities satisfy \(0<p_i<1\), it is negative definite exactly when the design vectors span \(\mathbb{R}^p\). If they do not, some nonzero direction \(a\) has \(x_i^\mathsf{T}a=0\) for every group, and the quadratic form in that direction is zero. If they do span \(\mathbb{R}^p\), every nonzero \(a\) has at least one nonzero \(x_i^\mathsf{T}a\), making the sum strictly negative. This is a statement about curvature; it does not by itself guarantee that a finite maximum-likelihood estimate exists.

Why Aggregation and Trial Counts Matter

A group count can also be understood as a compressed record of \(m_i\) Bernoulli outcomes that share the same success probability. If those outcomes were observed separately, their likelihood contribution would be \(p_i^{y_i}(1-p_i)^{m_i-y_i}\). The grouped count likelihood multiplies this by \(\binom{m_i}{y_i}\), the number of distinct binary sequences producing that count. This factor is independent of \(\beta\), so expanding or grouping the observations gives the same score and Hessian when all trials in a group have the same covariates and probability.

This equivalence concerns the likelihood for \(\beta\); it does not mean that one may discard the trial counts. They determine how many Bernoulli outcomes each group represents and therefore appear in both the expected counts \(m_i p_i\) and the curvature weights \(m_i p_i(1-p_i)\).

Common Pitfall

A common error is to replace the count residual \(y_i-m_i p_i\) with the proportion residual \(y_i/m_i-p_i\) and then sum those residuals without weights. The two forms are equivalent only when the proportion residual for each group is multiplied by \(m_i\). Omitting this factor treats a group based on two trials as equally informative as one based on twenty trials.