Count and Recurrent-Event Endpoints Under Overdispersion

Why count endpoints need exposure-aware planning

A recurrent-event endpoint records how many times an event occurs for each participant during follow-up. Examples include exacerbations, hospitalizations, or rescue-medication uses. The same count can represent different event risks if participants have different time at risk: two events over six months is not the same observed rate as two events over two years. Sample-size planning therefore needs to specify both the count model and how exposure time enters the analysis.

A Poisson model is a natural starting point, but its variance assumption can be too restrictive when participants differ in their underlying event propensity or when events cluster. A negative-binomial model allows this extra variability, often called overdispersion. If it is ignored, a design may appear more precise than it is and enroll too few participants.

Definition: Let \(Y_i\) be participant \(i\)'s event count and \(T_i\) their time at risk. Under a Poisson rate model, \(Y_i\sim\text{Poisson}(\lambda T_i)\), so \(E(Y_i)=\lambda T_i\) and \(\operatorname{Var}(Y_i)=\lambda T_i\). Under the negative-binomial model used here, \(E(Y_i)=\mu_i=\lambda T_i\) and \(\operatorname{Var}(Y_i)=\mu_i+\kappa\mu_i^2\), where \(\kappa\geq 0\) is the dispersion parameter. The Poisson model is the special case \(\kappa=0\).

For a treatment comparison, let \(\lambda_0\) and \(\lambda_1\) be the control and treatment event rates, respectively, measured in events per unit of time at risk. The rate ratio is \(RR=\lambda_1/\lambda_0\), and the treatment contrast on the log scale is \(\log(RR)\). A value below 1 indicates a lower rate in the treatment arm, under the model. It is a rate contrast, not by itself evidence that treatment caused the difference.

The formulas below assume independent participants, a common rate within each arm, a common negative-binomial dispersion parameter when that model is used, and a prespecified two-sided type I error rate. They also assume that exposure time is correctly measured and that the planned exposure distribution is appropriate. In particular, follow-up should not be shortened in a way related to event risk unless the estimand and analysis explicitly accommodate that process.

Approximate variance and sample-size formula

Within one arm, estimate the rate by total events divided by total exposure, \(\hat{\lambda}=C/E\), where \(C=\sum_iY_i\) and \(E=\sum_iT_i\). For independent participants, the variance of the total count is the sum of the participant variances. Applying the delta method to \(\log(\hat{\lambda})=\log(C)-\log(E)\), treating planned exposure as fixed, gives the following approximate variance:

\operatorname{Var}\{\log(\hat{\lambda})\} \approx \frac{\operatorname{Var}(C)}{E(C)^2} = \frac{1}{\lambda\sum_iT_i} + \kappa\frac{\sum_iT_i^2}{(\sum_iT_i)^2}.

The first term is the Poisson contribution. The second is the extra variance from negative-binomial overdispersion. It depends on the squared exposure times, not just total exposure: participants with more follow-up contribute more expected events, but unequal exposure also changes the dispersion contribution.

With \(n\) participants per arm and the same planned exposure-time distribution in each arm, write \(\overline{T}\) for the mean exposure and \(\overline{T^2}\) for the mean squared exposure. The approximate variance of the log rate ratio is the sum of the two independent arm variances:

\operatorname{Var}\{\log(\hat{\lambda}_1/\hat{\lambda}_0)\} \approx \frac{1}{n} \left[ \frac{1}{\lambda_0\overline{T}} + \frac{1}{\lambda_1\overline{T}} + \kappa\frac{\overline{T^2}}{\overline{T}^{\,2}} + \kappa\frac{\overline{T^2}}{\overline{T}^{\,2}} \right].

For a two-sided Wald test at level \(\alpha\), an approximate sample size per arm for power \(1-\beta\) under a target rate ratio \(RR\) is:

n\approx \frac{ \left(z_{1-\alpha/2}+z_{1-\beta}\right)^2 }{ \{\log(RR)\}^2 } \left[ \frac{1}{\lambda_0\overline{T}} + \frac{1}{\lambda_1\overline{T}} + 2\kappa\frac{\overline{T^2}}{\overline{T}^{\,2}} \right].

Here \(z_q\) is the \(q\)-quantile of the standard normal distribution. This is an asymptotic planning approximation, not an exact guarantee of power. When everyone has the same follow-up \(T\), the dispersion term simplifies to \(2\kappa\), because \(\overline{T^2}/\overline{T}^{\,2}=1\). For a Poisson analysis, set \(\kappa=0\). In unequal allocation or with different exposure distributions by arm, calculate each arm's variance using its own size and exposure distribution, then add the two variances.

Derivation: Under the negative-binomial model, \(\operatorname{Var}(C)=\sum_i(\lambda T_i+\kappa\lambda^2T_i^2)=\lambda\sum_iT_i+\kappa\lambda^2\sum_iT_i^2\), while \(E(C)=\lambda\sum_iT_i\). The delta-method variance for \(\log(C/E)\) is therefore approximately \(\operatorname{Var}(C)/E(C)^2=1/(\lambda\sum_iT_i)+\kappa\sum_iT_i^2/(\sum_iT_i)^2\). Independence between arms makes the variance of the log rate ratio the sum of those two terms. Setting the expected standardized contrast to \(z_{1-\alpha/2}+z_{1-\beta}\) and solving for \(n\) gives the stated formula.

Planning with equal follow-up

Worked Example: Rate-Ratio Sample Size With Overdispersion

Suppose a hypothetical trial plans equal randomization and 1.5 years of follow-up per participant. The assumed control rate is \(0.40\) events per person-year, and the target treatment rate is \(0.28\) events per person-year. Plan for a rate ratio of \(0.28/0.40=0.70\), dispersion \(\kappa=0.25\), two-sided \(\alpha=0.05\), and 80% power.

The expected counts per participant are \(\mu_0=0.40(1.5)=0.60\) and \(\mu_1=0.28(1.5)=0.42\). With equal follow-up, the bracketed variance term is:

\frac{1}{0.60}+\frac{1}{0.42}+2(0.25) =1.6667+2.3810+0.5000 =4.5476.

Using \(z_{0.975}=1.960\), \(z_{0.80}=0.8416\), and \(\log(0.70)=-0.3567\):

n\approx \frac{(1.960+0.8416)^2(4.5476)}{(-0.3567)^2} = \frac{7.849(4.5476)}{0.1272} \approx 280.6.

Rounding up gives 281 participants per arm, or 562 total before any allowance for loss of exposure or other design features. At the assumed follow-up this corresponds to 421.5 person-years per arm and expected totals of \(0.40(421.5)=168.6\) control events and \(0.28(421.5)=118.0\) treatment events. If the Poisson assumption were used instead, the bracket would be \(1.6667+2.3810=4.0476\), giving about 250 participants per arm. That smaller number reflects the assumption of no overdispersion; it should not be chosen merely because it is more convenient.

Unequal follow-up and recurrent events

Unequal follow-up may arise from staggered enrollment, varying planned observation windows, or participants entering the risk set for different durations. If the intended analysis uses total events and total person-time, the exposure-aware variance formula can represent a prespecified distribution of \(T_i\). It does not, however, automatically correct informative dropout or account for time-varying event rates. The design should specify whether treatment could change the amount of time at risk, and whether the target is an event rate over observed follow-up or another estimand.

Worked Example: Sample Size With Two Planned Follow-Up Durations

Consider a hypothetical trial in which half of participants are expected to contribute 1 year and half 2 years, in each arm. Thus \(\overline{T}=(1+2)/2=1.5\) years and \(\overline{T^2}=(1^2+2^2)/2=2.5\) years squared. Assume rates of \(0.50\) and \(0.35\) events per person-year, \(\kappa=0.40\), a target rate ratio of \(0.70\), two-sided \(\alpha=0.05\), and 80% power.

The term in the sample-size formula is:

\frac{1}{0.50(1.5)}+\frac{1}{0.35(1.5)} +2(0.40)\frac{2.5}{1.5^2} = 1.3333+1.9048+0.8889 =4.1270.

Therefore:

n\approx \frac{(1.960+0.8416)^2(4.1270)}{\{\log(0.70)\}^2} = \frac{7.849(4.1270)}{0.1272} \approx 254.6.

Round up to 255 participants per arm. Each arm has expected total exposure \(255(1.5)=382.5\) person-years, giving expected counts of \(0.50(382.5)=191.3\) and \(0.35(382.5)=133.9\). If everyone instead had exactly 1.5 years of follow-up, the dispersion term would be \(2(0.40)=0.80\), rather than \(0.8889\), and the result would be about 249 participants per arm. Thus, with the same mean exposure, the specified unequal exposure pattern modestly increases the required size under this model.

Worked Example: Consequence of Underestimating Dispersion

Return to the first hypothetical design: rates \(0.40\) and \(0.28\) events per person-year, 1.5 years of follow-up, and target rate ratio \(0.70\). Suppose planning assumes \(\kappa=0.25\), yielding 281 participants per arm, but the true dispersion is \(\kappa=0.50\).

With the larger dispersion, the variance coefficient becomes:

\frac{1}{0.60}+\frac{1}{0.42}+2(0.50) =1.6667+2.3810+1.0000 =5.0476.

The revised sample size is:

n\approx \frac{7.849(5.0476)}{0.1272} \approx 311.4,

so the revised approximation is 312 participants per arm. To see the consequence of retaining 281 per arm, the approximate standard error under the true dispersion is \(\sqrt{5.0476/281}=0.1340\). The absolute standardized contrast is \(0.3567/0.1340=2.662\). Approximate two-sided power is \(\Phi(2.662-1.960)+\Phi(-2.662-1.960)\), about \(0.759\), rather than 80%. Uncertainty in dispersion is therefore a design issue, not just a detail of the final analysis.

Assumptions, diagnostics, and alternatives

A rate-based calculation is useful when participants have a reasonably stable event-generating process over the planned observation period and the event count is the endpoint of interest. Its assumptions should be examined using prior data or pilot data, with attention to the definition of an event, risk time, event-rate assumptions, and the distribution of participant-level counts. The negative-binomial model allows more variance than a Poisson model, but it does not make every pattern of recurrent events appropriate for a single rate ratio.

Useful warning signs include a variance substantially exceeding the mean, a small number of participants accounting for a large fraction of all events, event rates that change systematically over time, and an observed zero count far more often than the fitted model predicts. These patterns can signal heterogeneity, clustering, time trends, or a poorly chosen model. Check the planned exposure distribution as well: errors in follow-up assumptions affect both expected event totals and the variance calculation. When informative discontinuation, death, or other changes to time at risk are plausible, a simple fixed-exposure calculation may not preserve the intended estimand.

The Poisson model is simpler and can be efficient when its mean-variance assumption is credible. The negative-binomial model is a common alternative when counts are more variable, with \(\kappa\) providing an interpretable overdispersion term. Other recurrent-event approaches may represent event timing, within-person dependence, or changing hazards more directly. Those approaches need their own design calculations; replacing a count model with a time-to-first-event model changes the endpoint because later events are no longer counted. Simulation is often useful when follow-up is unequal in a complex way, rates vary over time, or the planned analysis is more elaborate than a two-arm rate comparison. It should reproduce the intended analysis and the uncertainty or range of plausible design inputs, rather than merely simulate counts from the most favorable assumptions.

A common mistake is to calculate the sample size from the expected total number of events and then assume that meeting that target guarantees the planned power. The correction is that total events alone do not determine precision under overdispersion: participant-level clustering and the exposure distribution matter. A second mistake is to treat \(\kappa\) as an arbitrary inflation factor without defining its model. Here it specifically means \(\operatorname{Var}(Y_i)=\mu_i+\kappa\mu_i^2\); a dispersion estimate from a different model or unit of analysis may not be interchangeable.

Key takeaway: For count endpoints, plan against a clinically relevant rate ratio, expected rates, exposure-time distribution, and a defensible dispersion assumption. Check the model and follow-up assumptions, and use simulation when they are not adequately represented by the analytic approximation.

Check Your Understanding

Answer each question, then compare with the solutions below.

Under the negative-binomial parameterization in this tutorial, what does \(\kappa=0\) imply?
With equal follow-up and \(n\) participants per arm, write the approximate variance of the estimated log rate ratio.
For \(n=100\) participants, with half contributing 1 year and half 2 years, calculate total exposure and \(\sum_iT_i^2\).
Why can a Poisson-based sample-size calculation be too small for recurrent-event counts?

Solutions. (1) When \(\kappa=0\), \(\operatorname{Var}(Y_i)=\mu_i\), so the model reduces to Poisson variance. (2) The variance is \(\{1/(\lambda_0T)+1/(\lambda_1T)+2\kappa\}/n\), where each participant has the same follow-up \(T\). (3) Total exposure is \(50(1)+50(2)=150\) person-years; the sum of squared exposures is \(50(1^2)+50(2^2)=250\) years squared. (4) If counts are overdispersed, the Poisson model omits the positive \(\kappa\)-dependent variance contribution. It consequently understates uncertainty and can overstate power for a given sample size.