Negative Binomial Regression Sample Size

Methodology

Negative binomial regression is useful for count outcomes whose variance exceeds the mean. this method describes the model with a negative binomial dispersion parameter κ and uses a log-link regression model in which the treatment coefficient represents the log of the event-rate ratio.

This calculator implements the two-group sample-size calculation described by Zhu and Lakkis (2014) and reproduced in the this method procedure Tests for the Ratio of Two Negative Binomial Rates. The calculation is for a Wald or likelihood-ratio test of the treatment regression coefficient.

Negative Binomial Regression Model

Here, λ₁ is the control event rate, λ₂ is the treatment event rate, and RR = λ₂/λ₁. The average exposure time is denoted by μ_t. The allocation ratio is R = N₂/N₁.

Variance Components

The alternative variance component used by Zhu and Lakkis is V_A:

For the maximum-likelihood estimate of the null variance, the corresponding component is:

Sample Size Formula

For a two-sided test, the required number of subjects in Group 1 is:

For a one-sided test, z_1−α/2 is replaced by z_1−α. Group 2 is then obtained from N₂ = R N₁. The displayed group sample sizes are rounded upward to whole subjects.

Dispersion and Overdispersion

The negative binomial dispersion parameter is κ. Under the parameterization used by this method, κ is the additional variance component beyond the Poisson mean-variance relationship. Larger κ values increase the variance component used in the sample-size calculation and therefore generally increase the required sample size.

Worked Validation Example

this method provides a validation example based on Zhu and Lakkis (2014). The design is two-sided with 80% power and α = 0.05, equal allocation, average exposure time 0.75 years, control event rate 0.8 events per patient-year, treatment rate ratio 0.85, and κ = 0.4.

The implementation reproduces the this method validation result: 1,311 subjects per group, with calculated power approximately 0.800.

Interpretation

• The rate ratio is the multiplicative treatment effect on the expected event rate.
• An RR below 1 indicates a lower expected treatment event rate than the control rate; an RR above 1 indicates a higher expected treatment rate.
• Smaller absolute departures of RR from 1 require larger sample sizes because the denominator [log(RR)]² becomes smaller.
• Greater overdispersion generally increases the required sample size.
• Longer average exposure provides more information per subject and generally reduces the required sample size.

Important Approximation

Note that these asymptotic Wald and likelihood-ratio calculations are most appropriate when the sample size exceeds approximately 50 subjects per group. When the resulting sample size is smaller, the calculation should be regarded as a rough planning estimate rather than a substitute for simulation or small-sample methods.

References