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Method of Moments Estimation for Negative Binomial Parameters

Source: R/estimate_mom.R
estimate_nb_mom.Rd

Estimates the event rate(s) and common dispersion parameter (k) for negative binomial count data using the method of moments. This is a robust alternative to Maximum Likelihood Estimation (MLE), especially when MLE fails to converge or produces boundary estimates.

Usage

estimate_nb_mom(data, group = NULL)

Arguments

data

A data frame containing the data. Must include columns events (number of events) and tte (total exposure/follow-up time).

group

Optional character string specifying the grouping column name (e.g., "treatment"). If provided, rates are estimated separately for each group, while a common dispersion parameter is estimated across groups. If NULL (default), a single rate and dispersion are estimated (blinded case).

Value

A list containing:

lambda

Estimated event rate(s). A single numeric value if group is NULL, or a named vector if group is provided.

dispersion

Estimated common dispersion parameter (k).

Details

The method of moments estimator for the dispersion parameter \(k\) is derived by equating the theoretical variance to the observed second central moment, accounting for varying exposure times.

For a given group with rate \(\lambda\), the expected count for subject \(i\) is \(\mu_i = \lambda t_i\). The variance is \(V_i = \mu_i + k \mu_i^2\). The estimator is calculated as: $$\hat{k} = \max\left(0, \frac{\sum (y_i - \hat{\mu}_i)^2 - \sum y_i}{\sum \hat{\mu}_i^2}\right)$$ where \(y_i\) is the number of events, \(t_i\) is the exposure time, and \(\hat{\mu}_i = \hat{\lambda} t_i\) is the estimated expected count.

When multiple groups are present, the numerator and denominator are summed across all groups to estimate a common \(k\).

Examples

# Blinded estimation (single group)
df <- data.frame(events = c(1, 2, 0, 3), tte = c(1, 1.2, 0.5, 1.5))
estimate_nb_mom(df)
#> $lambda
#> [1] 1.428571
#> 
#> $dispersion
#> [1] 0
#> 

# Unblinded estimation (two groups)
df_group <- df
df_group$group <- c("A", "A", "B", "B")
estimate_nb_mom(df_group, group = "group")
#> $lambda
#>        A        B 
#> 1.363636 1.500000 
#> 
#> $dispersion
#> [1] 0
#>