Sample Size for Count/Poisson Outcomes

Introduction

Count outcomes occur frequently in clinical research. Examples include the number of disease exacerbations, hospitalizations, seizures, migraine attacks, infections, rescue-medication uses, or other recurrent events observed during a specified period of follow-up.

Unlike a binary endpoint, where each participant contributes an outcome such as yes or no, a count endpoint can take values such as 0, 1, 2, 3, and so forth. This creates a different statistical structure and requires a different approach to sample size planning.

A common starting point is the Poisson distribution, particularly when events are relatively uncommon and the variance is approximately equal to the mean.

What Is a Count Outcome?

Let \(Y_i\) denote the number of events observed for participant \(i\) during a defined period. For example:

The resulting endpoint is not simply whether an event occurred. A participant can contribute multiple events during follow-up.

Count outcomes can therefore contain substantially more information than binary endpoints when recurrent events are common and appropriately analyzed.

The Poisson Distribution

The Poisson distribution is commonly used to model the number of events occurring during a fixed amount of exposure time. If:

then:

The equality between the mean and variance is the defining feature of the Poisson model.

For example, if the expected number of events per participant is 2:

This assumption is convenient mathematically, but it is frequently violated in clinical data. That issue becomes important later when we discuss overdispersion.

Event Rates and Person-Time

Count outcomes are often expressed as an event rate:

For example, suppose a treatment group experiences 120 events over 600 person-years of observation. The estimated event rate is:

Thus, the rate is 0.20 events per person-year, or equivalently 20 events per 100 person-years.

Using person-time is particularly useful when participants have different amounts of follow-up.

Poisson Regression

A common analysis is Poisson regression. The expected event count can be modeled as:

When exposure time differs between participants, an offset is commonly included:

where \(t_i\) is the participant's exposure time. Equivalently:

The logarithm of exposure time is therefore called an offset because its coefficient is fixed at one.

The Rate Ratio

Suppose the control-group event rate is:

and the treatment-group event rate is:

The rate ratio is:

A rate ratio below 1 indicates fewer events per unit of person-time in the treatment group.

For example:

corresponds to a 30% reduction in the event rate.

Rate Ratio and the Log Scale

Poisson regression operates naturally on the logarithmic scale. The treatment effect is:

Therefore, if:

The logarithmic scale is important because rate ratios are multiplicative, whereas regression coefficients are additive on the log-rate scale.

Basic Sample Size Principle

For a simple two-group Poisson comparison with equal exposure time, sample size can be understood primarily through the expected number of events.

Let:

• \(R_C\) = control event rate
• \(R_T\) = treatment event rate
• \(RR=R_T/R_C\) = rate ratio
• \(T\) = follow-up time per participant
• \(n\) = participants per group

The expected number of events in each group is approximately:

The total expected number of events is therefore:

This illustrates why longer follow-up or a higher event rate can substantially reduce the number of participants required.

Approximate Power Formula for Comparing Two Poisson Rates

For two independent Poisson rates with equal exposure, a useful large-sample approximation for testing:

can be expressed in terms of the expected event rates. For equal follow-up time, an approximate number of participants per group is:

This expression is useful for understanding the major drivers of sample size, although software-based calculations are generally preferable for final protocol planning.

A more intuitive approach is often to formulate the design in terms of the expected event counts and then convert the required exposure into participants.

A Complete Worked Example

Suppose a randomized trial evaluates a treatment intended to reduce recurrent hospitalizations. The primary endpoint is the number of hospitalizations during one year.

The anticipated rate ratio is:

Thus, the treatment is assumed to reduce the hospitalization rate by 30%.

Step 1: Determine the Relevant Rate Difference

The absolute difference in event rates is:

The treatment group is therefore expected to experience 0.15 fewer events per person-year.

Step 2: Determine the Critical Values

For a two-sided α of 0.05:

For 90% power:

Step 3: Approximate the Required Participants

Using the large-sample approximation:

This produces an approximate requirement of about:

Thus, approximately 332 participants would be required before allowing for non-evaluable participants or other design complications.

Expected Number of Events

With 166 participants per group and one year of follow-up, the expected number of events is approximately:

and:

The total expected number of events is therefore approximately:

This provides a useful way to sanity-check the design. The study is expected to observe roughly 141 events across both treatment groups.

Why Event Counts Matter

Imagine two studies with the same 300 participants.

Study B contains substantially more events and therefore substantially more information for estimating and comparing event rates.

This is one of the fundamental differences between count-outcome studies and binary-outcome studies.

Follow-Up Time and Sample Size

Suppose the same trial can follow participants for two years instead of one. Ignoring losses to follow-up and other complications, the available person-time approximately doubles.

Therefore:

Longer follow-up can consequently reduce the number of participants required, although the relationship is not always as simple in real clinical trials.

For recurrent-event outcomes, longer follow-up can also change the event process, treatment effect, censoring distribution, and risk of informative dropout.

Unequal Follow-Up

Participants in clinical studies frequently contribute different amounts of person-time. For example, one participant may contribute 12 months while another contributes only 7 months.

The total exposure is:

If the average exposure is approximately 0.8 years per participant, then 500 participants provide approximately:

The expected event count can then be approximated as:

Accounting for Loss to Follow-Up

Suppose the study requires 332 evaluable participants and investigators expect 10% to become non-evaluable. A simple enrollment inflation is:

Therefore:

However, for recurrent-event studies, dropout may reduce person-time rather than simply remove participants. That distinction can matter substantially.

Person-Time Inflation Is Often More Appropriate

Suppose participants are expected to contribute an average of 12 months of follow-up, but the expected average is only 10.8 months because of dropout. The effective exposure is:

or approximately 90% of the planned exposure.

If the original calculation requires 300 participants under complete follow-up, a rough exposure-based adjustment would therefore be:

giving approximately 334 participants.

Poisson Assumption: Mean Equals Variance

The Poisson model assumes:

Clinical count data often violate this assumption. A common situation is:

This is called overdispersion.

For example, suppose the mean number of events is 1.5 but the observed variance is 4.5.

The data exhibit substantial overdispersion relative to the Poisson assumption.

Why Overdispersion Matters for Sample Size

If the true variance is larger than assumed under the Poisson model, standard errors are larger than expected. A study designed under the strict Poisson assumption may therefore have less power than planned.

A simple approximation is to apply a dispersion inflation factor to the Poisson sample size:

where \(\phi\) is the dispersion factor.

For example, if:

and the original Poisson calculation requires 332 participants:

Approximately 498 participants would therefore be required under this simple inflation approach.

Negative Binomial Models

When substantial overdispersion is anticipated, the negative binomial model is often a more appropriate primary analysis than a Poisson model.

One common parameterization gives:

where:

• \(\mu\) is the mean count
• \(\alpha\) is an overdispersion parameter

When \(\alpha>0\), the variance exceeds the Poisson variance.

The negative binomial distribution can therefore accommodate heterogeneity in event rates across participants that is not captured by the Poisson model.

Why Clinical Count Data Are Often Overdispersed

Suppose two patients have the same nominal follow-up. One patient experiences no events, while another experiences repeated events because of a much higher underlying susceptibility.

This patient-to-patient heterogeneity can create greater variability than the Poisson model permits.

Overdispersion can arise from:

• Heterogeneity in baseline event risk
• Unmeasured prognostic factors
• Clustering of recurrent events
• Subgroups with different event rates
• Time-varying event intensity
• Dependence between recurrent events

Excess Zeros

Another common feature of clinical count data is a large number of zero counts. For example, many participants may never experience a hospitalization during follow-up, while a smaller subgroup experiences several.

A high proportion of zeros does not automatically imply that a zero-inflated model is required.

The investigator should first determine whether the observed zero frequency can reasonably arise from the chosen count distribution and study design.

Potential models include:

• Poisson regression
• Negative binomial regression
• Zero-inflated Poisson models
• Zero-inflated negative binomial models
• Hurdle models

Rate Ratio vs. Mean Ratio

For count outcomes, investigators sometimes describe treatment effects using different terminology. With an exposure offset, the primary parameter from Poisson or negative binomial regression is naturally interpreted as a rate ratio.

If every participant has exactly the same follow-up duration, a ratio of expected counts can be numerically equivalent to a rate ratio.

When exposure times differ, however, the rate interpretation becomes particularly important.

Equal vs. Unequal Allocation

The most statistically efficient allocation for a simple two-group comparison is often close to equal allocation.

However, unequal allocation may be appropriate when:

• One treatment is substantially more expensive
• One treatment is difficult to administer
• The control group is ethically or scientifically prioritized
• The event rates differ substantially
• External controls are available

Changing the allocation ratio changes the amount of information contributed by each group and should therefore be incorporated into the sample-size calculation.

Unequal Allocation and Exposure

Let the allocation ratio be:

where \(r\) is the ratio of treatment to control participants.

The expected event counts become:

A design with very unequal allocation may therefore require more total participants to obtain the same statistical information.

Testing a Rate Ratio

A typical hypothesis is:

For a treatment expected to reduce the event rate:

The analysis is generally performed on the logarithmic scale:

and the estimated effect can be converted back to a rate ratio using:

Precision-Based Sample Size for a Rate Ratio

Not every study is designed around statistical power. A study may instead seek a specified confidence interval width for the rate ratio.

Because the rate ratio is analyzed on the logarithmic scale, precision is often specified for:

Suppose the standard error of the log rate ratio is approximately:

Then the approximate 95% confidence interval is:

Exponentiating gives the confidence interval for the rate ratio itself.

Therefore, precision-based planning can be performed by specifying an acceptable width on either the log scale or the rate-ratio scale.

Count Outcomes With Exposure Offsets

Suppose the study records the number of infections and each participant has different follow-up time. The model may be:

The expected count is therefore:

The offset ensures that the model compares event rates rather than raw event counts.

Recurrent Events Are Not Always Simple Poisson Events

A recurrent event can occur multiple times within the same participant. Those observations are therefore not necessarily independent.

For example, if one patient experiences three exacerbations, those events may share underlying patient-level characteristics.

Depending on the scientific question, the analysis might use:

• Poisson regression
• Negative binomial regression
• Generalized estimating equations
• Frailty models
• Andersen–Gill models
• Prentice–Williams–Peterson models
• Other recurrent-event survival models

The sample-size calculation should correspond to the planned primary analysis.

Event Rate Assumptions Are Critical

Suppose the control event rate is assumed to be 0.50 events per person-year. If the true rate is only 0.20, substantially fewer events will be observed than anticipated.

For 300 participants followed for one year:

under the original assumption, but:

under the lower rate.

The difference is substantial.

Event Rate Uncertainty

Historical studies may provide several plausible estimates of the control event rate. For example:

A useful planning strategy is to calculate sample size under all plausible scenarios rather than relying on a single point estimate.

This creates a sensitivity analysis of the sample-size assumptions.

Sample Size Sensitivity to the Rate Ratio

The treatment effect is another major driver. Suppose the control rate is 0.50 events/person-year. Consider several possible treatment effects:

Smaller treatment effects require substantially more information. A trial designed to detect a 10% reduction may therefore require many more participants than a trial designed to detect a 30% reduction.

Count Outcomes With a Clinically Meaningful Effect

The rate ratio used for sample size should correspond to a clinically meaningful difference rather than simply the largest effect that appears plausible.

For example, if a treatment reducing exacerbations by 20% would be considered clinically worthwhile, the sample size should generally be designed around:

rather than an optimistic assumption such as:

unless there is strong evidence supporting the larger effect.

Poisson vs. Negative Binomial Planning

When a Simple Poisson Calculation Is Appropriate

A Poisson-based calculation may be reasonable when:

• The event process is approximately Poisson
• Overdispersion is minimal
• Exposure time is well characterized
• The expected event rate is reasonably known
• The planned analysis is Poisson-based
• The event counts are not dominated by structural zeros

Even then, assumptions should be examined using historical or pilot data when available.

When a Negative Binomial Calculation Is Preferable

A negative binomial design is generally more appropriate when:

• Historical data demonstrate overdispersion
• There is substantial heterogeneity in event rates
• Recurrent events cluster within participants
• The planned primary analysis is negative binomial regression
• A Poisson design would clearly underestimate variability

The sample-size method should incorporate the anticipated dispersion parameter rather than simply assuming the Poisson variance structure.

Sample Size Planning in R

A simple Poisson approximation can be implemented directly in R. For the worked example:

alpha <- 0.05
power <- 0.90

z.alpha <- qnorm(1 - alpha / 2)
z.beta  <- qnorm(power)

rate.control <- 0.50
rate.treatment <- 0.35
followup <- 1

n <- (
  z.alpha * sqrt(rate.treatment + rate.control) +
  z.beta * sqrt(
    rate.treatment / (rate.treatment / rate.control)^2 +
    rate.control
  )
)^2 / (
  followup *
  (rate.treatment - rate.control)^2
)

ceiling(n)

The calculation gives an approximate sample size per group of:

# Approximately
# 166 participants per group

The exact implementation used for a protocol should reflect the planned analysis, allocation ratio, exposure assumptions, and any dispersion adjustment.

Applying a Dispersion Inflation Factor in R

Suppose the Poisson calculation gives 332 total participants and historical data suggest a dispersion factor of 1.5:

N.poisson <- 332
dispersion <- 1.5

N.adjusted <- ceiling(
  N.poisson * dispersion
)

N.adjusted
# 498

This is a planning approximation rather than a universal negative-binomial sample-size formula. For important confirmatory studies, the final calculation should preferably use software or simulation that matches the planned negative-binomial model.

Simulation-Based Sample Size Planning

Simulation is particularly valuable when the planned analysis is more complicated than a simple Poisson comparison.

For example, simulation can incorporate:

• Negative binomial dispersion
• Unequal follow-up
• Dropout
• Staggered enrollment
• Unequal allocation
• Covariate adjustment
• Excess zeros
• Variable event rates
• Recurrent-event dependence

A simulation-based calculation can repeatedly generate trial datasets under the assumed data-generating mechanism and determine the proportion of simulated trials that achieve statistical significance.

That proportion estimates the empirical power.

Simulation becomes especially attractive when the analytical assumptions become too restrictive to represent the actual protocol.

Example Simulation Structure in R

set.seed(123)

nsim <- 2000
n.per.group <- 250

rate.control <- 0.50
rate.treatment <- 0.35

p.values <- numeric(nsim)

for(i in seq_len(nsim)) {

  y.control <- rpois(
    n.per.group,
    lambda = rate.control
  )

  y.treatment <- rpois(
    n.per.group,
    lambda = rate.treatment
  )

  test <- poisson.test(
    c(sum(y.treatment), sum(y.control)),
    T = c(n.per.group, n.per.group)
  )

  p.values[i] <- test$p.value
}

mean(p.values < 0.05)

The resulting value is an empirical estimate of the statistical power under the specified assumptions.

For a more realistic study, the simulation should generate participant-level exposure times, dropout, overdispersion, and the exact analysis model specified in the statistical analysis plan.

Exposure Time as an Offset in R

A Poisson regression with exposure time can be fit using:

fit <- glm(
  events ~ treatment + offset(log(followup)),
  family = poisson,
  data = dat
)

summary(fit)

exp(coef(fit)["treatment"])

The exponentiated treatment coefficient estimates the rate ratio.

For example, if:

coef(fit)["treatment"]
# -0.357

then:

corresponding to an estimated 30% lower event rate.

Negative Binomial Regression in R

A negative binomial model can be fit using the MASS package:

library(MASS)

fit.nb <- glm.nb(
  events ~ treatment + offset(log(followup)),
  data = dat
)

summary(fit.nb)

exp(coef(fit.nb)["treatment"])

The exponentiated treatment coefficient is again interpreted as a rate ratio, but the negative binomial model allows the variance to exceed the mean.

Common Sample Size Mistakes

1. Using a binary-outcome sample-size formula. Count outcomes have a different variance structure and often require event-rate and exposure assumptions.
2. Ignoring follow-up time. Person-time is a fundamental source of information for rate-based outcomes.
3. Assuming the Poisson variance without checking overdispersion. This can produce an overly optimistic sample size.
4. Using an unrealistic baseline event rate. If the control event rate is overestimated, the study may observe fewer events than planned.
5. Choosing an unrealistically large treatment effect. A smaller true effect can require substantially more participants.
6. Ignoring unequal exposure. Different follow-up durations should generally be handled through person-time or an exposure offset.
7. Ignoring recurrent-event dependence. Multiple events within the same participant are not necessarily independent.
8. Automatically choosing a zero-inflated model because many observations are zero. The underlying data-generating process should determine the model.
9. Inflating only for participant dropout. For rate outcomes, loss to follow-up reduces person-time and expected events.
10. Using a Poisson sample-size calculation for a negative-binomial analysis without accounting for dispersion. The resulting design may be underpowered.
11. Failing to align the sample-size calculation with the primary analysis. The design should reflect the actual statistical model that will be used.

A Practical Count-Outcome Sample Size Workflow

A useful workflow is:

1. Define the count endpoint precisely.
2. Determine whether the endpoint represents recurrent events, events per exposure time, or a fixed-period count.
3. Specify the primary estimand, such as a rate ratio.
4. Specify the planned analysis model.
5. Estimate the control event rate.
6. Specify the clinically meaningful rate ratio.
7. Specify the allocation ratio.
8. Specify the planned follow-up and expected person-time.
9. Specify the type I error rate and desired power.
10. Calculate the required sample size or expected event count.
11. Assess overdispersion using historical or pilot data.
12. Consider negative binomial or simulation-based planning when appropriate.
13. Inflate for loss of person-time, dropout, and other sources of non-evaluable exposure.
14. Perform sensitivity analyses across plausible event rates and treatment effects.
15. Ensure that the final calculation matches the planned SAP analysis.

Scenario Analysis

Because event-rate assumptions can be uncertain, a protocol should often examine several scenarios. For example:

The final sample size can then be selected based on a scientifically defensible scenario rather than a single optimistic assumption.

Fixed Follow-Up vs. Event-Driven Designs

Some clinical studies specify a fixed number of participants and follow-up period. Others are effectively event-driven.

In an event-driven design, the study continues until a prespecified number of events has accumulated.

This can be attractive when the event rate is uncertain because statistical information is more directly tied to the number of observed events.

However, event-driven designs require assumptions about:

• Event accrual rate
• Enrollment rate
• Follow-up duration
• Dropout
• Administrative censoring
• Potential treatment effects

The number of participants and calendar duration are therefore still important even when the statistical target is expressed in terms of events.

When an Event-Driven Approach Is Attractive

An event-driven design can be particularly useful when:

• The primary analysis is strongly event-based
• The event rate is relatively stable
• Enrollment can continue until the required event count is reached
• The sponsor can tolerate uncertainty in the total number enrolled
• The event endpoint is clinically meaningful

For fixed-duration studies, however, the event rate and expected person-time remain central to sample-size planning.

What a Protocol Should State

A count-outcome sample-size section should clearly document:

• Primary count endpoint
• Primary estimand
• Planned statistical model
• Control-group event rate
• Expected treatment-group event rate
• Expected rate ratio
• Allocation ratio
• Follow-up duration
• Expected person-time
• Type I error rate
• Target power
• Expected dispersion parameter, if applicable
• Expected dropout or loss of exposure
• Required total sample size
• Expected total number of events
• Method used to account for overdispersion
• Any simulation methodology used

These assumptions should be sufficiently explicit that another statistician can reconstruct the sample-size calculation.

How the Major Inputs Affect Sample Size

The Central Planning Relationship

The fundamental relationship for count outcomes can be summarized conceptually as:

and:

Therefore:

The sample-size problem is then to determine how much exposure is required to distinguish the treatment and control event rates with the desired statistical power or precision.

Summary

Sample-size planning for count outcomes requires a different framework from binary or continuous endpoints. The most common starting point is the Poisson model, in which the number of events is related directly to the amount of exposure time.

The central relationship is:

where \(\lambda\) is the event rate and \(T\) is total person-time.

When comparing two treatment groups, the treatment effect is commonly expressed as a rate ratio:

Sample size is driven by the expected event rates, the clinically meaningful rate ratio, follow-up duration, allocation ratio, type I error rate, and desired power.

A crucial complication is overdispersion. The Poisson model assumes that the variance equals the mean, but clinical recurrent-event data frequently exhibit greater variability. When this occurs, a negative binomial model or an appropriate dispersion adjustment may be required.

Another important consideration is exposure. Participant dropout does not merely reduce the number of participants—it can reduce the amount of person-time and therefore the number of events available for analysis.

For simple designs, analytical approximations can provide useful planning calculations. For complex recurrent-event studies involving negative binomial models, substantial overdispersion, unequal follow-up, staggered enrollment, or other features, simulation is often the most defensible approach.

References

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