lrt_bnb: Likelihood ratio test for BNB ratio of means

In depower: Power Analysis for Differential Expression Studies

Likelihood ratio test for BNB ratio of means

Description

Likelihood ratio test for the ratio of means from bivariate negative binomial outcomes.

Usage

lrt_bnb(data, ratio_null = 1, ...)

Arguments

data	(list) A list whose first element is the vector of negative binomial values from sample 1 and the second element is the vector of negative binomial values from sample 2. Each vector must be sorted by the subject/item index and must be the same sample size. NAs are silently excluded. The default output from sim_bnb().
ratio_null	(Scalar numeric: 1; ⁠(0, Inf)⁠) The ratio of means assumed under the null hypothesis (sample 2 / sample 1). Typically, ratio_null = 1 (no difference). See 'Details' for additional information.
...	Optional arguments passed to the MLE function mle_bnb().

Details

This function is primarily designed for speed in simulation. Missing values are silently excluded.

Suppose X_1 \mid G = g \sim \text{Poisson}(\mu g) and X_2 \mid G = g \sim \text{Poisson}(r \mu g) where G \sim \text{Gamma}(\theta, \theta^{-1}) is the random item (subject) effect. Then X_1, X_2 \sim \text{BNB}(\mu, r, \theta) is the joint distribution where X_1 and X_2 are dependent (though conditionally independent), X_1 is the count outcome for sample 1 of the items (subjects), X_2 is the count outcome for sample 2 of the items (subjects), \mu is the conditional mean of sample 1, r is the ratio of the conditional means of sample 2 with respect to sample 1, and \theta is the gamma distribution shape parameter which controls the dispersion and the correlation between sample 1 and 2.

The hypotheses for the LRT of r are

\begin{aligned} H_{null} &: r = r_{null} \\ H_{alt} &: r \neq r_{null} \end{aligned}

where r = \frac{\bar{X}_2}{\bar{X}_1} is the population ratio of arithmetic means for sample 2 with respect to sample 1 and r_{null} is a constant for the assumed null population ratio of means (typically r_{null} = 1).

The LRT statistic is

\begin{aligned} \lambda &= -2 \ln \frac{\text{sup}_{\Theta_{null}} L(r, \mu, \theta)}{\text{sup}_{\Theta} L(r, \mu, \theta)} \\ &= -2 \left[ \ln \text{sup}_{\Theta_{null}} L(r, \mu, \theta) - \ln \text{sup}_{\Theta} L(r, \mu, \theta) \right] \\ &= -2(l(r_{null}, \tilde{\mu}, \tilde{\theta}) - l(\hat{r}, \hat{\mu}, \hat{\theta})) \end{aligned}

Under H_{null}, the LRT test statistic is asymptotically distributed as \chi^2_1. The approximate level \alpha test rejects H_{null} if \lambda \geq \chi^2_1(1 - \alpha). Note that the asymptotic critical value is known to underestimate the exact critical value. Hence, the nominal significance level may not be achieved for small sample sizes (possibly n \leq 10 or n \leq 50).

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 (sample 2 / sample 1).
5		alternative	Point estimates under the alternative hypothesis.
5	1	mean1	Estimated mean of sample 1.
5	2	mean2	Estimated mean of sample 2.
5	3	dispersion	Estimated dispersion.
6		null	Point estimates under the null hypothesis.
6	1	mean1	Estimated mean of sample 1.
6	2	mean2	Estimated mean of sample 2.
6	3	dispersion	Estimated dispersion.
7		n1	The sample size of sample 1.
8		n2	The sample size of sample 2.
9		method	Method used for the results.
10		ratio_null	Assumed population ratio of means.
11		mle_code	Integer indicating why the optimization process terminated.
12		mle_message	Information from the optimizer.

References

\insertRef

rettiganti_2012depower

\insertRef

aban_2009depower

Examples

#----------------------------------------------------------------------------
# lrt_bnb() examples
#----------------------------------------------------------------------------
library(depower)

set.seed(1234)
sim_bnb(
  n = 40,
  mean1 = 10,
  ratio = 1.2,
  dispersion = 2
) |>
  lrt_bnb()

depower documentation built on April 3, 2025, 9:23 p.m.