Sample Size Calculation For Negative Binomial Regression

Negative Binomial Sample Size Formula:

\[ n = \frac{(Z_{1-\alpha/2} + Z_{1-\beta})^2 (\mu_1 (1 + k_1) + \mu_2 (1 + k_2))}{(\mu_1 - \mu_2)^2} \]

Z-score for Significance (Z_1-α/2):

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Z-score for Power (Z_1-β):

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Mean Count Group 1 (μ₁):

counts

Mean Count Group 2 (μ₂):

counts

Dispersion Parameter Group 1 (k₁):

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Dispersion Parameter Group 2 (k₂):

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Sample Size Per Group (n):

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1. What is Negative Binomial Sample Size Calculation?

The negative binomial sample size calculation determines the number of participants needed per group to detect a specified difference in count outcomes between two groups, accounting for overdispersion commonly found in count data.

2. How Does the Calculator Work?

The calculator uses the negative binomial sample size formula:

\[ n = \frac{(Z_{1-\alpha/2} + Z_{1-\beta})^2 (\mu_1 (1 + k_1) + \mu_2 (1 + k_2))}{(\mu_1 - \mu_2)^2} \]

Where:

\( Z_{1-\alpha/2} \) — Z-score for significance level (e.g., 1.96 for α=0.05)
\( Z_{1-\beta} \) — Z-score for statistical power (e.g., 0.84 for 80% power)
\( \mu_1, \mu_2 \) — Mean counts for groups 1 and 2
\( k_1, k_2 \) — Dispersion parameters for groups 1 and 2
\( n \) — Sample size per group

Explanation: This formula accounts for the extra variability (overdispersion) in count data that cannot be adequately modeled by Poisson regression.

3. Importance of Sample Size Calculation

Details: Proper sample size calculation ensures studies have sufficient power to detect meaningful differences while avoiding unnecessary resource expenditure on overly large studies.

4. Using the Calculator

Tips: Enter appropriate z-scores for your desired significance level and power, provide realistic mean counts based on pilot data or literature, and estimate dispersion parameters from previous studies.

5. Frequently Asked Questions (FAQ)

Q1: When should I use negative binomial instead of Poisson?
A: Use negative binomial when your count data shows overdispersion (variance > mean), which is common in biomedical and ecological count data.

Q2: How do I estimate dispersion parameters?
A: Dispersion parameters can be estimated from pilot studies, previous similar research, or by fitting negative binomial models to existing data.

Q3: What are typical z-score values?
A: For α=0.05 (two-tailed), Z=1.96; for 80% power, Z=0.84; for 90% power, Z=1.28.

Q4: Can this be used for more than two groups?
A: This formula is specifically for comparing two groups. Multi-group comparisons require different sample size calculations.

Q5: What if my means are very similar?
A: Smaller differences between means require larger sample sizes. If means are identical, the sample size becomes undefined.