Grouped Outcomes and Binomial Logit Models

Why Grouped Outcomes Need Their Denominators

Many data sets report outcomes by group rather than by individual: a hospital may report how many of its patients were readmitted, or a laboratory may report how many of a batch of specimens tested positive. A count such as “8 successes” is not enough to specify the response. Eight successes among 10 trials represents a different success rate, and a different amount of information, from 8 among 100.

A grouped binomial logit model uses both the success count and the number of trials in each group. Its regression structure models the probability of success for a trial with the group’s covariates. The resulting likelihood accounts for the number of successes and failures, rather than treating a group’s proportion as if it were a single binary observation.

Here \(x_j\) is the vector of predictors for group \(j\), typically including an intercept, and \(\beta\) is the coefficient vector. The model says that trials in a group share a probability \(p_j\) conditional on the predictors. The group’s expected success count is \(m_jp_j\), and its expected success proportion is \(p_j\).

Likelihood for Grouped Counts

The binomial probability of observing exactly \(y_j\) successes among \(m_j\) trials is the number of possible arrangements of those successes times the probability of any one such arrangement. Therefore,

If groups are independent conditional on their covariates, the likelihood for \(J\) groups is the product of these probabilities. Substituting the logit model gives a likelihood in terms of \(\beta\):

Taking logarithms yields the log-likelihood. Its binomial-coefficient term depends on the observed data but not on \(\beta\). It must be included when calculating the probability of the observed grouped counts; it can be omitted when maximizing the log-likelihood over \(\beta\), because it does not change which value maximizes it.

What Grouping Preserves—and What It Changes

Suppose that, within a group, each individual trial has the same covariates and the same success probability, and trials are conditionally independent. Then the individual outcomes are Bernoulli observations with probability \(p_j\). Their sum has a binomial distribution with denominator \(m_j\). Grouping those outcomes retains the count of successes and failures needed for the likelihood of \(\beta\).

The individual-data likelihood for a particular ordering of the outcomes is \(p_j^{y_j}(1-p_j)^{m_j-y_j}\). The grouped-count probability is larger by the factor \(\binom{m_j}{y_j}\), because the count can arise from that many orderings. This factor does not depend on \(\beta\); consequently, with the same covariates for every trial in a group, maximizing the individual-data likelihood or the grouped-data likelihood gives the same coefficient estimates.

A proportion \(y_j/m_j\) can summarize a group, but it does not by itself retain the denominator. For example, a proportion of \(0.5\) could represent 1 success out of 2 trials or 50 out of 100. Those groups have the same observed proportion but different binomial likelihood contributions. A grouped model therefore needs both \(y_j\) and \(m_j\), or an equivalent representation that explicitly preserves the denominator.

How the Coefficients Are Estimated

The assumptions behind the likelihood are substantive. Conditional on the predictors, trials are Bernoulli with a common group probability; the group counts are independent across groups; and that probability follows the specified logit form. In particular, this binomial likelihood does not allow trials in a group to share additional dependence beyond what is represented by the predictors. These conditions justify using the product of binomial probabilities; they should be considered before interpreting fitted coefficients.

The likelihood generally has no closed-form maximizer when predictors vary. Numerical maximum likelihood methods find coefficient values that maximize \(\ell(\beta)\). The score equation can be obtained directly by differentiating the log-likelihood. Write \(p_j=\exp(x_j^\top\beta)/\{1+\exp(x_j^\top\beta)\}\). Since the derivative of \(\log\{1+\exp(\eta)\}\) with respect to \(\eta\) is \(p\), the contribution from group \(j\) has derivative \(x_j(y_j-m_jp_j)\). Summing gives:

At an interior maximum, this score is zero. In words, the fitted model balances covariate-weighted observed success counts against covariate-weighted expected counts. The second derivative is

This matrix is negative semidefinite, as \(m_jp_j(1-p_j)\geq0\) and each \(x_jx_j^\top\) is positive semidefinite. Thus the log-likelihood is concave. Under suitable identification and finite-estimate conditions, its maximum is unique. In particular, if the predictors perfectly separate groups with successes from groups with failures, a finite maximum may not exist; software warnings or coefficients that grow extremely large should not be treated as ordinary finite estimates.

Interpreting Effects on Group Probabilities

A logit coefficient describes a change in log-odds, not a constant change in probability. If \(x_k\) is a continuous predictor and the other predictors are held fixed, increasing \(x_k\) by one unit changes the log-odds by \(\beta_k\) and multiplies the odds by \(\exp(\beta_k)\). The corresponding probability change depends on the starting probability. In the grouped model, these are statements about a trial’s group-conditional success probability \(p_j\), not about the group count independently of its denominator.

For a continuous predictor, the local probability slope is \(\beta_kp_j(1-p_j)\). A group with denominator \(m_j\) has expected count \(m_jp_j\), so the local change in expected successes per unit increase in \(x_k\) is \(m_j\beta_kp_j(1-p_j)\). Both quantities depend on the group covariates; the latter also depends on the denominator.

Common Mistake: Treating a Group Proportion as One Outcome

A common error is to enter \(y_j/m_j\) as though each group supplied one binary observation, or to analyze the proportions without their denominators. That reasoning discards how many trials produced each proportion: it treats 1 success out of 2 as equally informative as 50 out of 100. The correction is to use the binomial count and denominator in the likelihood. If the data are entered as proportions in software, the number of trials must still be supplied as the binomial denominator or equivalent weight.

A related mistake is to interpret \(\beta_k\) as the change in probability for every group. The coefficient is a log-odds change; the probability change depends on the group’s starting probability. Report predicted probabilities or probability differences at specified covariate values when the question concerns changes in success probability.