depower (version 2025.1.20)

glm_nb: GLM for NB ratio of means

Description

Generalized linear model for two independent negative binomial outcomes.

Usage

glm_nb(data, equal_dispersion = FALSE, test = "wald", ci_level = NULL, ...)

Value

A list with the following elements:

Slot	Subslot	Name	Description
1		chisq	\(\chi^2\) test statistic for the ratio of means.
2		df	Degrees of freedom.
3		p	p-value.
4		ratio	Estimated ratio of means (group 2 / group 1).
4	1	estimate	Point estimate.
4	2	lower	Confidence interval lower bound.
4	3	upper	Confidence interval upper bound.
5		mean1	Estimated mean of group 1.
5	1	estimate	Point estimate.
5	2	lower	Confidence interval lower bound.
5	3	upper	Confidence interval upper bound.
6		mean2	Estimated mean of group 2.
6	1	estimate	Point estimate.
6	2	lower	Confidence interval lower bound.
6	3	upper	Confidence interval upper bound.
7		dispersion1	Estimated dispersion of group 1.
7	1	estimate	Point estimate.
7	2	lower	Confidence interval lower bound.
7	3	upper	Confidence interval upper bound.
8		dispersion2	Estimated dispersion of group 2.
8	1	estimate	Point estimate.
8	2	lower	Confidence interval lower bound.
8	3	upper	Confidence interval upper bound.
9		n1	Sample size of group 1.
10		n2	Sample size of group 2.
11		method	Method used for the results.
12		test	Type of hypothesis test.
13		alternative	The alternative hypothesis.
14		equal_dispersion	Whether or not equal dispersions were assumed.
15		ci_level	Confidence level of the intervals.
16		hessian	Information about the Hessian matrix.
17		convergence	Information about convergence.

Arguments

data

(list)
A list whose first element is the vector of negative binomial values from group 1 and the second element is the vector of negative binomial values from group 2. NAs are silently excluded. The default output from sim_nb().

equal_dispersion

(Scalar logical: FALSE)
If TRUE, the model is fit assuming both groups have the same population dispersion parameter. If FALSE (default), the model is fit assuming different dispersions.

test

(String: "wald"; "c("wald", "lrt"))
The statistical method used for the test results. test = "wald" performs a Wald test and optionally the Wald confidence intervals. test = "lrt" performs a likelihood ratio test and optionally the profile likelihood confidence intervals.

ci_level

(Scalar numeric: NULL; (0, 1))
If NULL, confidence intervals are set as NA. If in (0, 1), confidence intervals are calculated at the specified level. Profile likelihood intervals are computationally intensive, so intervals from test = "lrt" may be slow.

...

Optional arguments passed to glmmTMB::glmmTMB().

Details

Uses glmmTMB::glmmTMB() in the form

glmmTMB(
  formula = value ~ condition,
  data = data,
  dispformula = ~ condition,
  family = nbinom2
)

to model independent negative binomial outcomes \(X_1 \sim \text{NB}(\mu, \theta_1)\) and \(X_2 \sim \text{NB}(r\mu, \theta_2)\) where \(\mu\) is the mean of group 1, \(r\) is the ratio of the means of group 2 with respect to group 1, \(\theta_1\) is the dispersion parameter of group 1, and \(\theta_2\) is the dispersion parameter of group 2.

The hypotheses for the LRT and Wald test of \(r\) are

$$ \begin{aligned} H_{null} &: log(r) = 0 \\ H_{alt} &: log(r) \neq 0 \end{aligned} $$

where \(r = \frac{\bar{X}_2}{\bar{X}_1}\) is the population ratio of arithmetic means for group 2 with respect to group 1 and \(log(r_{null}) = 0\) assumes the population means are identical.

References

hilbe_2011depower

hilbe_2014depower

See Also

glmmTMB::glmmTMB()

Examples

#----------------------------------------------------------------------------
# glm_nb() examples
#----------------------------------------------------------------------------
library(depower)

set.seed(1234)
d <- sim_nb(
  n1 = 60,
  n2 = 40,
  mean1 = 10,
  ratio = 1.5,
  dispersion1 = 2,
  dispersion2 = 8
)

lrt <- glm_nb(d, equal_dispersion = FALSE, test = "lrt", ci_level = 0.95)
lrt

wald <- glm_nb(d, equal_dispersion = FALSE, test = "wald", ci_level = 0.95)
wald

#----------------------------------------------------------------------------
# Compare results to manual calculation of chi-square statistic
#----------------------------------------------------------------------------
# Use the same data, but as a data frame instead of list
set.seed(1234)
d <- sim_nb(
  n1 = 60,
  n2 = 40,
  mean1 = 10,
  ratio = 1.5,
  dispersion1 = 2,
  dispersion2 = 8,
  return_type = "data.frame"
)

mod_alt <- glmmTMB::glmmTMB(
  formula = value ~ condition,
  data = d,
  dispformula = ~ condition,
  family = glmmTMB::nbinom2,
)
mod_null <- glmmTMB::glmmTMB(
  formula = value ~ 1,
  data = d,
  dispformula = ~ condition,
  family = glmmTMB::nbinom2,
)

lrt_chisq <- as.numeric(-2 * (logLik(mod_null) - logLik(mod_alt)))
lrt_chisq
wald_chisq <- summary(mod_alt)$coefficients$cond["condition2", "z value"]^2
wald_chisq

anova(mod_null, mod_alt)