Binary Outcomes Versus Event Counts

One Window, Two Possible Responses

Suppose a clinic records whether each patient experienced a particular event during a 30-day follow-up. It could instead record how many times the event occurred during those same 30 days. These outcomes use the same event definition and observation window, but they answer different questions: did anything happen? versus how many times did it happen?

The distinction matters in a generalized linear mixed model (GLMM) because the response determines what the model describes. A binary model targets the conditional probability of the defined outcome. A count model targets a distribution of event counts, often summarized by a conditional mean. An indicator of any event can be calculated from the count, but it generally does not preserve all the information in that count.

The definition requires a clear event, observational unit, and window. For example, “at least one hospital visit per patient during 30 days” is a binary outcome, while “number of hospital visits per patient during 30 days” is a count outcome. A patient with four visits contributes \(B_{ij}=1\) to the first outcome and \(C_{ij}=4\) to the second.

The indicator and count are exactly related only if they refer to the same event, unit, and window and the indicator means “one or more.” If the binary response instead means “a severe event occurred,” it need not be derivable from the total event count. A count of several mild events does not establish that any event was severe.

What the Two Responses Measure

For a nonnegative integer count \(C_{ij}\), the indicator has two possible values, 0 and 1. Its conditional expectation is the conditional probability of at least one event. By contrast, the count can take values \(0,1,2,\ldots\), and its conditional expectation is the average number of events per observation at given covariates and random effects.

Here \(b_j\) denotes the random effects for cluster \(j\). These quantities answer distinct questions. The probability concerns whether the count is zero or positive; the mean count also reflects how large positive counts can be. In general, knowing the probability of any event does not determine the expected count, and knowing the expected count alone does not determine the probability of any event.

A binary GLMM may, for example, specify \( \operatorname{logit}\{P(B_{ij}=1\mid b_j)\}=X_{ij}\beta+Z_{ij}b_j \). Its linear predictor describes the log odds of at least one event, conditional on the random effects. A count GLMM instead specifies a distribution for \(C_{ij}\), conditional on predictors and random effects; a model for its mean may use a linear predictor such as \( \log\{E(C_{ij}\mid b_j)\}=X_{ij}\beta+Z_{ij}b_j \) when that link and distribution are appropriate. The coefficients therefore refer to different response-scale quantities. A binary-model coefficient is not automatically an effect on the mean event count.

A Relationship That Holds for Every Nonnegative Count

The exact relation between the two expected values can be established without choosing a particular count distribution. For a nonnegative integer-valued count, the indicator equals the count when the count is 0 or 1. When the count is at least 2, the count exceeds the indicator.

To see the equality, write \(E(B_{ij}\mid b_j)=\sum_{k=1}^{\infty}P(C_{ij}=k\mid b_j)\), since the indicator is 1 exactly for positive counts. Subtract this from the count mean, \(\sum_{k=1}^{\infty}kP(C_{ij}=k\mid b_j)\). The \(k=1\) terms cancel, leaving the displayed sum for \(k\geq2\). Every term is nonnegative, so the conditional mean count is at least the conditional probability of any event.

Equality holds if and only if \(P(C_{ij}\geq2\mid b_j)=0\): given those random effects, counts can only be 0 or 1. In that special case, the count itself is numerically the binary indicator. When counts greater than 1 are possible, the conditional mean count is strictly larger whenever such counts have positive probability. This result concerns numerical values on their respective scales; it does not make a probability and an event count the same estimand.

Choosing and Checking the Response

Before fitting a GLMM, confirm that the recorded response matches the scientific question. If the question concerns the chance of any occurrence in a defined window, the indicator is a direct response. If the question concerns event burden, frequency, or total number per window, retain the count. It may be useful to study both, but they should be stated as separate outcomes with separate interpretations.

A basic data check is to compare the count and indicator for every observation whenever both are available. The indicator should be 0 for a count of 0 and 1 for every positive count. A mismatch can signal a coding error, different window definitions, inconsistent event criteria, or missing information. Also check whether all observations have comparable windows and whether the event was observable throughout each one. A 30-day window and a 90-day window do not provide the same opportunity to record events.

If only the indicator was collected, do not describe its fitted probability as an expected number of events. The lost information cannot be recovered from the indicator. Conversely, if a count is recorded but the scientific target is any occurrence, a model of the derived indicator is possible; however, it deliberately ignores differences among positive counts.

The response distribution and its support should be checked against the observed records and the measurement process. Indicators must be 0 or 1; counts must be nonnegative integers if they represent numbers of events. Values outside those supports point to a definition or data problem, or to a response that is not actually the stated type. For clustered observations, the GLMM also relies on its specified conditional structure: for example, the random effects represent modeled between-cluster variation, and the assumed conditional response distributions must be suitable for the response being analyzed.

Common Mistakes and Limitations

• Calling any positive count a binary outcome without defining the threshold. “At least one event” is one threshold; “at least two” is another. State the event rule explicitly.
• Interpreting an event probability as an event rate or mean count. A probability concerns whether an observation window has any event; it does not reveal the number of events in positive windows.
• Assuming rare events turn a count into a Bernoulli response. Rarity does not change the recorded support. If counts above 1 remain possible and are recorded, they remain counts; a binary indicator is a deliberate recoding.
• Combining unlike windows without accounting for their definitions. Different observation durations or exposure opportunities can change both the chance of any event and the count. The response must be interpreted in light of how it was observed.

Neither response is universally preferable. The indicator can be easier to interpret when occurrence is the target, but it loses event-frequency information. The count retains that information, but requires a count model suited to the data-generating and recording process. A binary GLMM and a count GLMM are not interchangeable alternatives that differ only in coding: they address different conditional outcomes and require different response distributions.