Count Outcomes Across Dropout Patterns

Count outcomes within dropout patterns

A repeated count outcome might record the number of symptoms, hospital admissions, or episodes of a behavior at each visit. Counts are nonnegative integers, and their variability often increases with their mean. In a pattern-mixture analysis, the count distribution is specified separately for each dropout pattern. This allows, for example, the expected count trajectory among participants who complete follow-up to differ from the trajectory among those who drop out earlier.

Let \(Y_{it}\) be the count for participant \(i\) at visit \(t\), and let \(G_i=g\) denote that participant’s dropout pattern. Write \(\mu_{git}=E(Y_{it}\mid G_i=g,\mathbf{x}_{it})\), where \(\mathbf{x}_{it}\) contains the covariates used to model the count. A pattern-specific model has its own parameters for each \(g\), unless the analyst deliberately imposes constraints tying patterns together.

The count distribution is only one part of the repeated-outcome model. Repeated counts from the same participant may be associated. Specifying each visit’s conditional distribution does not, by itself, specify that within-participant dependence. A likelihood that treats observations as independent needs an appropriate independence assumption; otherwise, the joint model or the method used for uncertainty estimation must account for dependence.

Pattern-specific Poisson and negative-binomial models

For a Poisson model, a common specification is a log-linear mean:

The vector \(\boldsymbol{\beta}_g\) can include an intercept, visit indicators, treatment, baseline covariates, and interactions that vary by dropout pattern. The log link guarantees a positive fitted mean. If counts are measured over different lengths of observation time, an exposure \(e_{it}>0\) can be included as an offset:

Here the model is for the expected count over exposure \(e_{it}\); the exponential of the linear predictor excluding the offset is the expected count per unit exposure. The offset coefficient is fixed at one, so this specification assumes expected counts are proportional to exposure.

Under the Poisson model, \(E(Y_{it}\mid G_i=g,\mathbf{x}_{it})=\mu_{git}\) and \(\operatorname{Var}(Y_{it}\mid G_i=g,\mathbf{x}_{it})=\mu_{git}\). This equality is a substantive distributional restriction, not just a computational convenience. If the data vary more than this restriction allows, estimated uncertainty and fitted probabilities can be unreliable.

A common alternative is the NB2 negative-binomial model. It retains a log-linear mean but adds a dispersion parameter \(\alpha_g\):

The parameter \(\alpha_g\) can be shared across patterns or estimated separately, if the data support that choice. As \(\alpha_g\) approaches zero, the variance approaches the Poisson variance. One equivalent parameterization uses size \(r_g=1/\alpha_g\) and success probability \(q_{git}=r_g/(r_g+\mu_{git})\); then \(Y_{it}\) has a negative-binomial distribution with mean \(\mu_{git}\) and variance \(\mu_{git}+\mu_{git}^2/r_g\).

Estimating and checking the count model

In a pattern-specific analysis, estimate the model using the observations available in that pattern and at the visits they actually contribute. The usual Poisson likelihood assumes the specified conditional Poisson distributions and, if it is formed as a product of visit-level probabilities, conditional independence of those observations. The analogous product likelihood for negative-binomial observations makes the corresponding independence assumption. If repeated outcomes are associated, that assumption must be justified or the dependence addressed in the analysis.

A basic check for extra-Poisson variation compares observed variation with the Poisson mean-variance restriction. For fitted means \(\hat{\mu}_{it}\), one summary is the Pearson dispersion ratio, the Pearson statistic divided by its residual degrees of freedom. A value notably above one suggests more variation than the Poisson model predicts, though it is not a stand-alone test and can be affected by an incorrect mean model or dependence. Also compare observed and fitted count frequencies, including zeros and large counts. A negative-binomial model can accommodate extra variation, but it does not automatically fix a misspecified mean, unmodeled serial association, or an unsuitable pattern definition.

Pattern-specific estimation can be unstable when a pattern has few participants, few observed counts, or very little variation. Warning signs include large standard errors, estimates that change substantially under small modeling changes, or difficulty obtaining a finite estimate of dispersion. A more flexible model is not necessarily better when the pattern-specific data are sparse.

Calculating fitted count summaries

Once a model has been fitted, its mean gives an expected count, not a guarantee that any individual will have that count. Under a Poisson model, the probability of a particular count \(k\) is \(e^{-\mu}\mu^k/k!\). Under the NB2 parameterization above, \(r=1/\alpha\) and \(q=r/(r+\mu)\), so the probability of zero is \(q^r\). These probabilities describe the fitted distribution conditional on the specified pattern and covariates.

Pattern weights can be used to combine fitted means for a visit where the relevant pattern-specific outcomes are observed. If \(w_g=P(G_i=g)\), then the population mean at visit \(t\) is \(\sum_g w_g E(Y_{it}\mid G_i=g)\), provided each conditional mean refers to that same visit and target population. This is the standard mixture identity: partition the expectation by pattern, then sum the conditional expectations weighted by pattern probabilities. The earlier material on pattern weights and the population mixture develops this identity; here it is applied to counts.

Assumptions, alternatives, and limitations

The pattern-specific Poisson model is most defensible when its conditional mean structure is appropriate and the Poisson variance restriction is reasonable. The negative-binomial model is an alternative when there is extra-Poisson variation that is adequately represented by its variance form. Neither distribution automatically captures every source of heterogeneity. For example, a negative-binomial variance does not itself specify how counts at different visits from the same participant are associated.

Compared with Poisson regression, negative-binomial regression adds a dispersion parameter and can produce wider predictive distributions when counts vary substantially around their fitted means. In return, it estimates an additional parameter and can be difficult to fit reliably in small dropout patterns. A Poisson fit may still be useful for its mean structure when the variance assumption is imperfect, but uncertainty calculations must then account for the discrepancy and any within-participant dependence; the ordinary independent Poisson likelihood alone does not do so.

A crucial limitation is that pattern membership does not reveal outcomes after dropout. The conditional distribution of counts observed in a pattern can be estimated from available data, but a pattern-specific fitted value at an unobserved post-dropout visit is not learned merely by extending a fitted curve. Such a prediction needs an explicit extrapolation assumption. Pattern-specific count modeling describes the observed pattern structure; it does not, by itself, solve the missing-data identification problem.