Predicting Effects with Unequal Binomial Denominators

Why Unequal Denominators Matter

A binomial group with 8 successes in 10 trials and one with 80 successes in 100 trials have the same observed proportion. They do not, however, provide the same amount of likelihood information about the group’s underlying probability. In a generalized linear mixed model, that difference matters: the group-specific random-intercept prediction balances the group’s binomial likelihood against the random-effects distribution. A larger denominator generally gives the group’s own outcomes more influence on that balance.

The comparison must be made on the conditional scale. A random intercept describes a group’s deviation from the fixed part of the linear predictor, conditional on the model and covariates. It is not itself a marginal effect or a probability. The focus here is how the binomial denominator changes the prediction of that latent deviation.

Definition: For group $j$, let $Y_j$ be the number of successes out of $m_j$ trials, and let $b_j$ be its random intercept. The binomial random-intercept model is $$ Y_j\mid b_j \sim \operatorname{Binomial}(m_j,p_j),\qquad \operatorname{logit}(p_j)=\eta_j+b_j,\qquad b_j\sim N(0,\tau^2), $$ where $\eta_j=\mathbf{x}_j^\top\boldsymbol\beta$ is the fixed part of the linear predictor and $\tau^2>0$ is the random-intercept variance. Conditional on the random effects, groups’ outcomes are assumed independent; the trials within each group are conditionally independent with a common probability $p_j$. The prediction considered below is the conditional posterior mode of $b_j$, given the group’s outcomes and fixed model parameters.

For clarity, examples treat $\eta_j$ and $\tau^2$ as known. This isolates the effect of the denominator; in an application they are estimated from the full model. The conditional mode is one standard point prediction, not the only possible summary of the conditional distribution. In a non-Gaussian model, a posterior mean and a mode need not coincide.

Deriving the Group-Specific Prediction

Write the observed number of successes in group $j$ as $y_j$. Conditional on a proposed value $b$, the binomial likelihood is proportional to $p_j^{y_j}(1-p_j)^{m_j-y_j}$, where $p_j=\operatorname{logit}^{-1}(\eta_j+b)$. Multiplying this likelihood by the normal prior density gives the conditional density of $b_j$, up to a normalizing constant.

\log \pi(b\mid y_j) = y_j(\eta_j+b)-m_j\log\{1+\exp(\eta_j+b)\} -\frac{b^2}{2\tau^2}+C

Here $C$ does not depend on $b$. Differentiating with respect to $b$, and setting the derivative to zero, gives the equation that the conditional mode $\hat b_j$ must satisfy.

\frac{\partial}{\partial b}\log \pi(b\mid y_j) = y_j-m_jp_j-\frac{b}{\tau^2}=0, \qquad \text{so}\qquad y_j-m_j\operatorname{logit}^{-1}(\eta_j+\hat b_j)-\frac{\hat b_j}{\tau^2}=0.

The first two terms are the binomial likelihood score for the group intercept; the final term is the contribution of the normal prior. Thus the prediction is not generally the group’s raw log-odds minus the fixed predictor. That difference maximizes the likelihood alone when $0<y_j/m_j<1$, whereas the conditional mode also accounts for the prior.

The denominator’s role is explicit in both the score and the curvature. Differentiating once more gives

\frac{\partial^2}{\partial b^2}\log \pi(b\mid y_j) =-m_jp_j(1-p_j)-\frac{1}{\tau^2}.

This curvature is strictly negative for every finite $b$, so the log conditional density is strictly concave and has at most one mode. It tends downward in both tails because the normal prior contributes $-b^2/(2\tau^2)$; therefore a finite, unique mode exists. The term $m_jp_j(1-p_j)$ is the observed likelihood information about the group intercept at that value of $b$. Holding the probability fixed, it grows in proportion to the denominator. The prior contributes information $1/\tau^2$, which does not grow with $m_j$.

Worked Example: Same Proportion, Different Denominators

Suppose two hypothetical groups have the same fixed predictor $\eta=0$ and share $\tau^2=1$. Group A has $y=4$ successes in $m=5$ trials; group B has $y=40$ successes in $m=50$ trials. Both observed proportions are $0.8$. Find and compare their conditional modes.

For either group, the score equation is $y-m\operatorname{logit}^{-1}(b)-b=0$. For group A, solving $4-5\operatorname{logit}^{-1}(b)-b=0$ gives $\hat b_A\approx0.681$. Substitution checks the result: $\operatorname{logit}^{-1}(0.681)\approx0.664$, so $4-5(0.664)-0.681\approx-0.001$, with the small discrepancy due to rounding.

For group B, solving $40-50\operatorname{logit}^{-1}(b)-b=0$ gives $\hat b_B\approx1.238$. Substitution gives $\operatorname{logit}^{-1}(1.238)\approx0.775216$, and $40-50(0.775216)-1.238\approx0.0012$, again close to zero after rounding. The implied conditional probabilities are about $0.664$ and $0.775$, respectively.

The likelihood-only estimate in each group would be $\operatorname{logit}(0.8)-0=\log(4)\approx1.386$. The prior pulls each mode toward zero, but the larger group’s likelihood score carries more weight relative to the same prior contribution. Its prediction is consequently closer to the likelihood-only value. The observed rates are identical; the difference comes from the denominators.

Unequal Denominators at the Boundaries and at the Center

Unequal denominators matter even when the observed proportion is at a boundary. A group with no successes has a likelihood that favors an increasingly negative intercept, so its likelihood-only estimate is not finite. The normal prior nevertheless yields a finite conditional mode. The denominator determines how strongly the zero-success observation pulls that mode below zero.

Worked Example: Zero Successes With Two Trial Totals

Consider hypothetical groups with $\eta=0$, $\tau^2=1$, and zero successes. Group C has $y=0,m=2$; group D has $y=0,m=30$. Their score equation is $-m\operatorname{logit}^{-1}(b)-b=0$.

For group C, solving $-2\operatorname{logit}^{-1}(b)-b=0$ gives $\hat b_C\approx-0.675$. Indeed, $\operatorname{logit}^{-1}(-0.675)\approx0.337$, and $-2(0.337)-(-0.675)\approx0.001$. The predicted conditional probability is about $0.337$.

For group D, solving $-30\operatorname{logit}^{-1}(b)-b=0$ gives $\hat b_D\approx-2.429$. Here $\operatorname{logit}^{-1}(-2.429)\approx0.0810$, and $-30(0.0810)-(-2.429)\approx-0.001$. The predicted conditional probability is about $0.0810$.

Both groups observed no successes, but 30 failures provide stronger evidence of a low group probability than two failures. The prior still prevents the predicted random intercept from becoming negative infinity. This is not an arbitrary correction: it follows from combining the finite binomial likelihood with the proper normal prior.

A larger denominator does not guarantee a different mode. In a symmetric case, the likelihood and prior can balance at the same point regardless of the trial total, even though the amount of information differs.

Worked Example: Equal Predictions but Different Information

Suppose two more hypothetical groups have $\eta=0$, $\tau^2=1$, and exactly half their trials are successes: group E has $y=5,m=10$, and group F has $y=20,m=40$. At $b=0$, the probability is $0.5$, so the score in group E is $5-10(0.5)-0=0$, and in group F it is $20-40(0.5)-0=0$. Strict concavity ensures that $b=0$ is the unique mode for both.

Their local posterior curvatures differ. At $b=0$, the negative second derivative is $m(0.5)(0.5)+1$. It equals $3.5$ for group E and $11$ for group F. Thus the modes coincide, but group F’s conditional density is more sharply concentrated near its mode. A point prediction alone does not report this difference in information.

Interpreting the Denominator and Checking a Prediction

For a fixed probability, the binomial likelihood information about an intercept is $m_jp_j(1-p_j)$. This is largest near $p_j=0.5$ and smaller near zero or one. Consequently, the denominator’s effect should be understood through the likelihood curvature, not as a rule that every larger group must have a larger numerical prediction or a fixed amount less shrinkage. The observed successes, fixed predictor, and probability at the solution all matter.

A useful check is to substitute a computed mode into its score equation. The residual $y_j-m_j\operatorname{logit}^{-1}(\eta_j+\hat b_j)-\hat b_j/\tau^2$ should be close to zero, with any remaining discrepancy explainable by rounding or numerical tolerance. A materially nonzero residual may indicate a coding error, an incorrect denominator, or failure of the numerical optimizer to converge. Because the score is strictly decreasing, a root-finding method can also bracket and locate the unique solution.

The denominator must represent the number of trials that actually share the modeled probability. If observations within a group have different covariates or probabilities, aggregating them into a single binomial count with one probability is not justified without an appropriate model. Likewise, dependence among trials beyond the specified random intercept can make the binomial likelihood overstate the information. Diagnostics should therefore include checking the outcome definition, the group-level trial totals, model fit, and whether residual patterns suggest unmodeled heterogeneity or dependence.

The comparison with alternatives clarifies what this prediction does and does not say:

Raw group proportion: $y_j/m_j$ describes the observed fraction. It ignores the prior and does not directly estimate the latent intercept.
Likelihood-only group estimate: for an interior proportion, $\operatorname{logit}(y_j/m_j)-\eta_j$ is the unconstrained group-intercept estimate. It ignores the random-effects distribution and can diverge for all-success or all-failure groups.
Conditional random-intercept mode: combines group outcomes with the normal random-effects distribution, so groups with less likelihood information are pulled more strongly toward zero.
Marginal prediction: averages over the random-effects distribution to target a population-level probability. It answers a different question from predicting a particular observed group’s conditional intercept.

The prediction’s limitations follow from its assumptions. It depends on the specified binomial sampling model, the fixed predictor, and the normal random-intercept distribution. It is conditional on model parameters and uses a single point summary; it does not by itself convey uncertainty in the group effect or in the estimated model parameters. Nor does a strong group-level prediction establish a causal group effect. These distinctions are especially important when comparing groups with very different denominators.

Key takeaway: The binomial denominator scales the likelihood score and its information about a group intercept. With the same observed proportion, a larger denominator generally lets the data pull the conditional prediction farther from the prior center, but the actual mode is determined jointly by the successes, the fixed predictor, the probability, and the random-effects variance.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

For a binomial random-intercept model, write the score equation whose root is the conditional mode of a group effect.
Two groups have $y/m=0.7$, the same fixed predictor, and the same random-effects variance, but one has ten times as many trials. Which group generally has more likelihood information about its intercept, and what quantity expresses that information?
With $\eta=0$ and $\tau^2=1$, evaluate the score for a hypothetical group with $y=6,m=10$ at $b=0$. Is zero its conditional mode?
Why can a group with zero successes have a finite conditional-mode prediction even though its likelihood-only estimate is unbounded below?

Solutions. (1) The score equation is $y_j-m_j\operatorname{logit}^{-1}(\eta_j+\hat b_j)-\hat b_j/\tau^2=0$. (2) The group with more trials generally has more likelihood information; the likelihood curvature is $m_jp_j(1-p_j)$ at the fitted probability. (3) At $b=0$, $p=0.5$, so the score is $6-10(0.5)-0=1$, not zero. Therefore zero is not the mode; since the score is strictly decreasing, its unique zero lies above zero. (4) The normal prior contributes a quadratic penalty, $-b^2/(2\tau^2)$, which keeps the conditional log density from increasing without bound as $b$ tends to negative infinity and yields a finite mode.

Predicting Effects with Unequal Binomial Denominators

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