Why a Proportion Does Not Specify the Response
A cluster-level proportion is a summary of binary trials, not a complete description of how many trials produced that summary. A proportion of 0.80 could mean 4 successes in 5 eligible trials or 80 successes in 100 eligible trials. Those clusters have the same observed proportion, but the second proportion is based on many more trials and is usually much more precise. In a generalized linear mixed model (GLMM), that difference belongs in the response likelihood.
The earlier material on grouped binomial likelihoods and matching denominators to eligible trials gives the basic count representation. Here the central point is that the proportion alone does not specify that representation: both the number of successes and the number of eligible trials are needed. A cluster identifier or a count of clusters does not supply the missing trial denominator.
For a simple random-intercept model, one might write \(\operatorname{logit}(p_j(b_j))=\mathbf{x}_j^\top\boldsymbol{\beta}+b_j\), with \(b_j\sim N(0,\sigma_b^2)\). The binomial statement is conditional on the random effect: given \(b_j\), the \(m_j\) trials in cluster \(j\) are independent and share the same success probability \(p_j(b_j)\). The denominator must count the trials eligible for the success event, and the common-probability assumption requires that the grouped trials have the same modeled covariates and conditional probability. If those conditions do not hold, the group cannot automatically be represented by one binomial count.
What the Denominator Contributes
Conditional on \(b_j\), the binomial count has mean \(m_jp_j(b_j)\) and variance \(m_jp_j(b_j)\{1-p_j(b_j)\}\). Dividing the count by its denominator gives the mean and variance of the observed proportion:
The denominator therefore affects the sampling variation even when the observed proportion is unchanged. It also appears directly in the likelihood. Conditional on \(b_j\), the likelihood contribution of cluster \(j\) is
The proportion \(y_j/m_j\) cannot replace \(y_j\) and \(m_j\) in this expression. Knowing the exact proportion and the denominator lets us recover the count; knowing the proportion alone generally does not. If a proportion has been rounded, even a supplied denominator may not identify a unique count.
Worked Example: Same Proportion, Different Precision
Suppose two hypothetical clusters have observed success proportion 0.60. Cluster A has 3 successes among 5 eligible trials; cluster B has 30 successes among 50 eligible trials. For a transparent conditional comparison, suppose the model probability for both clusters is \(p_j(b_j)=0.60\). The binomial assumptions are that trials are eligible, conditionally independent given the random effect, and have this common conditional probability within each cluster.
For cluster A, the estimated proportion is \(3/5=0.60\), and the conditional standard deviation of the proportion under the stated model is \(\sqrt{0.60(0.40)/5}=\sqrt{0.048}=0.2191\). For cluster B it is \(\sqrt{0.60(0.40)/50}=\sqrt{0.0048}=0.06928\). Thus the second cluster’s proportion has less conditional sampling variation because it is based on ten times as many trials.
The likelihoods also retain the denominators. At \(p=0.60\), the probability of observing exactly 3 successes in 5 trials is \(\binom{5}{3}(0.60)^3(0.40)^2=10(0.216)(0.16)=0.3456\). For 30 successes in 50 trials it is \(\binom{50}{30}(0.60)^{30}(0.40)^{20}\), approximately 0.1146. These are probabilities of the respective exact counts, not measures to compare as if they were proportions; the variance calculation directly shows the difference in precision.
Why the Result Holds
The variance result follows from scaling a count by a fixed denominator. Conditional on \(b_j\), \(Q_j=Y_j/m_j\). For any fixed constant \(c\), \(\operatorname{Var}(cY)=c^2\operatorname{Var}(Y)\). Applying this fact and the binomial variance gives
This is a conditional variance, not the full marginal variance across clusters. A random effect makes the conditional success probability vary across clusters. For fixed \(m_j\), the law of total variance gives
The first term is within-cluster binomial sampling variation; the second is variation in the cluster probabilities induced by the random effects. The denominator determines the first term. Knowing only \(Q_j\) does not tell us how much of the observed cluster-to-cluster pattern reflects trial-level sampling versus variation in underlying cluster probabilities.
Worked Example: Unequal Denominators Change the Combined Estimate
Consider two hypothetical clusters with common covariates and a common success probability, so that a pooled binomial estimate is appropriate. Cluster A has 2 successes in 4 eligible trials, and cluster B has 80 successes in 100 eligible trials. The respective proportions are \(2/4=0.50\) and \(80/100=0.80\).
The pooled success count is \(2+80=82\), and the pooled denominator is \(4+100=104\). The grouped binomial maximum-likelihood estimate of the common probability is therefore \(\hat p=82/104=0.7885\) (to four significant figures).
By contrast, the unweighted average of the two cluster proportions is \((0.50+0.80)/2=0.6500\). That calculation gives the four-trial cluster the same weight as the 100-trial cluster; it is not the pooled binomial estimate. The discrepancy is not a numerical nuisance: the two calculations target different summaries. The binomial likelihood uses the counts and denominators, while the unweighted average treats clusters equally regardless of their trial totals.
Worked Data Forms and Assumptions
There are two standard ways to preserve the binomial information. Individual Bernoulli records retain one binary outcome per eligible trial. Grouped binomial data compress those records into a success count and trial denominator when the trials in a group share the relevant modeled probability. Either representation can support a GLMM when its assumptions and covariates are represented correctly. A proportion with its denominator can encode the same grouped information; a proportion without its denominator cannot.
Worked Example: Recovering a Count From a Proportion
Suppose a hypothetical clinic reports that 18% of 50 eligible patients experienced a specified outcome. Assume the percentage is exact rather than rounded, and that each patient contributes one eligible binary trial. The count is \(Y=0.18(50)=9\), so the grouped response is 9 successes out of 50 trials, not a binomial response of 0.18.
If the model assigns this clinic conditional probability \(p=0.20\), its likelihood contribution is \(\binom{50}{9}(0.20)^9(0.80)^{41}\). The model’s conditional mean count is \(50(0.20)=10\), and its conditional variance is \(50(0.20)(0.80)=8\). For the proportion, the corresponding conditional mean is 0.20 and variance is \(0.20(0.80)/50=0.0032\).
If the report instead says “about 18%” without giving the denominator, it does not establish that there were 9 successes in 50 trials. The same reported percentage could arise from another count and denominator, and rounding can make several combinations compatible with a reported value. The count and denominator must come from the underlying eligible-trial data, not from guessing based on the displayed proportion.
Diagnostics, Alternatives, and Limitations
Before fitting a grouped binomial GLMM, check that every row has an integer success count \(y_j\) and an eligible-trial denominator \(m_j\), with \(0\leq y_j\leq m_j\) and \(m_j>0\). Verify that the reported proportion agrees with \(y_j/m_j\), allowing only documented rounding. Confirm that the denominator counts the opportunities for the defined success event, rather than all people in a cluster regardless of eligibility. Also check whether grouped trials share the covariates needed for a common conditional probability. The earlier material on aggregation under shared covariates explains why within-group covariate variation can prevent a simple grouped representation.
A common warning sign is a data file containing only one proportion per cluster, with no trial totals. In that case, the binomial likelihood and the denominators’ contribution to precision cannot be reconstructed from those values. A second warning sign is that denominators have been replaced by a constant or by the number of clusters. Such substitutions change the implied likelihood. A third is an apparent proportion outside the range from 0 to 1, or a count larger than its denominator; these indicate a coding or definition problem, not a valid grouped binomial row.
The main alternatives differ in what information they retain. Individual Bernoulli data preserve each trial and its covariates. Grouped binomial data preserve the success count and denominator while reducing the data to a group when the conditional-probability assumption is justified. A proportion with a known denominator can be converted back to that grouped representation if the count is recoverable. Modeling proportions as ordinary continuous measurements makes different assumptions about their sampling variation; it is not a substitute that becomes justified simply because the data are displayed as percentages.
Even a correctly specified denominator does not guarantee a good model. The binomial form assumes conditional independence of trials given the modeled random effects and a suitable conditional probability for each grouped set. Dependence not captured by the model, incorrect eligibility rules, or unmodeled differences among grouped trials can make the assumed likelihood inadequate. Conversely, a small denominator does not make the proportion invalid; it makes its binomial sampling uncertainty larger. The denominator is information about precision, not a reason to discard the cluster.
Check Your Understanding
Answer each question, then compare with the solutions below.
- Two clusters both have a proportion of 0.25. One has 1 success in 4 trials; the other has 25 successes in 100 trials. At conditional probability 0.25, calculate the conditional variance of each proportion.
- A cluster reports 12 successes and 30 eligible trials. State its proportion and the grouped binomial response.
- Why can a reported proportion of 0.40 not, by itself, identify a binomial likelihood contribution?
- Two clusters have counts 3/10 and 45/50 and share a common probability and covariates. Calculate the pooled binomial estimate and the unweighted mean of the proportions.
- Give one reason why a cluster’s recorded denominator might not be the correct binomial denominator.
Solutions. (1) For 1/4, the variance is \(0.25(0.75)/4=0.04688\). For 25/100, it is \(0.25(0.75)/100=0.001875\). (2) The proportion is \(12/30=0.40\); the grouped response is 12 successes out of 30 eligible trials. (3) The proportion does not specify the number of trials or successes: for example, 2/5 and 40/100 both equal 0.40 but produce different likelihoods and sampling variances. (4) The pooled estimate is \((3+45)/(10+50)=48/60=0.8000\). The unweighted mean is \((0.30+0.90)/2=0.6000\). (5) For example, the denominator may count all people in the cluster even though some were not eligible for the defined event; the binomial denominator must count eligible trials.