Sample Size Calculation For Negative Binomial Distribution

Negative Binomial Sample Size Formula:

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

Significance Level (α):

(e.g., 0.05)

Power (1-β):

(e.g., 0.8)

Mean Count Group 1 (μ₁):

counts

Mean Count Group 2 (μ₂):

counts

Dispersion Parameter Group 1 (k₁):

dimensionless

Dispersion Parameter Group 2 (k₂):

dimensionless

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 subjects needed per group to detect a specified difference in mean counts 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:

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

Where:

\( Z_{\alpha/2} \) — Z-score for significance level (two-tailed)
\( Z_\beta \) — Z-score for power
\( \mu_1, \mu_2 \) — Mean counts for group 1 and group 2
\( k_1, k_2 \) — Dispersion parameters for group 1 and group 2
\( n \) — Sample size per group

Explanation: This formula accounts for the extra variability (overdispersion) in count data that follows a negative binomial distribution rather than a Poisson distribution.

3. Importance of Sample Size Calculation

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

4. Using the Calculator

Tips: Enter significance level (typically 0.05), power (typically 0.8-0.9), mean counts for both groups, and dispersion parameters. Dispersion parameters can be estimated from pilot data or literature.

5. Frequently Asked Questions (FAQ)

Q1: When should I use negative binomial instead of Poisson sample size?
A: Use negative binomial when your count data shows overdispersion (variance > mean). Poisson assumes variance equals mean.

Q2: How do I estimate dispersion parameters?
A: Dispersion parameters can be estimated from pilot data using maximum likelihood estimation or method of moments.

Q3: What if I don't know the dispersion parameters?
A: Use conservative estimates from similar studies. Underestimating dispersion can lead to underpowered studies.

Q4: Can this be used for one-sample tests?
A: This formula is designed for two-group comparisons. Different formulas exist for one-sample scenarios.

Q5: What are typical values for dispersion parameters?
A: Dispersion parameters typically range from 0.1 to 10, with smaller values indicating less overdispersion.