wald_test_nb: Wald test for NB ratio of means

In depower: Power Analysis for Differential Expression Studies

Wald test for NB ratio of means

Description

Wald test for the ratio of means from two independent negative binomial outcomes.

Usage

wald_test_nb(
  data,
  equal_dispersion = FALSE,
  ci_level = NULL,
  link = "log",
  ratio_null = 1,
  ...
)

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 Wald test is calculated assuming both groups have the same population dispersion parameter. If FALSE (default), the Wald test is calculated assuming different dispersions.
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.
link	(Scalar string: "log") The one-to-one link function for transformation of the ratio in the test hypotheses. Must be one of "log" (default), "sqrt", "squared", or "⁠identity"⁠.
ratio_null	(Scalar numeric: 1; ⁠(0, Inf)⁠) The (pre-transformation) ratio of means assumed under the null hypothesis (group 2 / group 1). Typically ratio_null = 1 (no difference). See 'Details' for additional information.
...	Optional arguments passed to the MLE function mle_nb().

Details

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

Suppose X_1 \sim NB(\mu, \theta_1) and X_2 \sim NB(r\mu, \theta_2) where X_1 and X_2 are independent, X_1 is the count outcome for items in group 1, X_2 is the count outcome for items in group 2, \mu is the arithmetic mean count in group 1, r is the ratio of arithmetic means for 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 Wald test of r are

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

where f(\cdot) is a one-to-one link function with nonzero derivative, r = \frac{\bar{X}_2}{\bar{X}_1} is the population ratio of arithmetic means for group 2 with respect to group 1, and r_{null} is a constant for the assumed null population ratio of means (typically r_{null} = 1).

\insertCite

rettiganti_2012;textualdepower found that f(r) = r^2 and f(r) = r had greatest power when r < 1. However, when r > 1, f(r) = \ln r, the likelihood ratio test, and the Rao score test have greatest power. Note that f(r) = \ln r, LRT, and RST were unbiased tests while the f(r) = r and f(r) = r^2 tests were biased when r > 1. The f(r) = \ln r, LRT, and RST produced acceptable results for any r value. These results depend on the use of asymptotic vs. exact critical values.

The Wald test statistic is

W(f(\hat{r})) = \left( \frac{f \left( \frac{\bar{x}_2}{\bar{x}_1} \right) - f(r_{null})}{f^{\prime}(\hat{r}) \hat{\sigma}_{\hat{r}}} \right)^2

where

\hat{\sigma}^{2}_{\hat{r}} = \frac{\hat{r} \left[ n_1 \hat{\theta}_1 (\hat{r} \hat{\mu} + \hat{\theta}_2) + n_2 \hat{\theta}_2 \hat{r} (\hat{\mu} + \hat{\theta}_1) \right]}{n_1 n_2 \hat{\theta}_1 \hat{\theta}_2 \hat{\mu}}

Under H_{null}, the Wald test statistic is asymptotically distributed as \chi^2_1. The approximate level \alpha test rejects H_{null} if W(f(\hat{r})) \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). The level of significance inflation also depends on f(\cdot) and is most severe for f(r) = r^2, where only the exact critical value is recommended.

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.
6		mean2	Estimated mean of group 2.
7		dispersion1	Estimated dispersion of group 1.
8		dispersion2	Estimated dispersion of group 2.
9		n1	Sample size of group 1.
10		n2	Sample size of group 2.
11		method	Method used for the results.
12		ci_level	The confidence level.
13		equal_dispersion	Whether or not equal dispersions were assumed.
14		link	Link function used to transform the ratio of means in the test hypotheses.
15		ratio_null	Assumed ratio of means under the null hypothesis.
16		mle_code	Integer indicating why the optimization process terminated.
17		mle_message	Information from the optimizer.

References

\insertRef

rettiganti_2012depower

\insertRef

aban_2009depower

Examples

#----------------------------------------------------------------------------
# wald_test_nb() examples
#----------------------------------------------------------------------------
library(depower)

set.seed(1234)
sim_nb(
  n1 = 60,
  n2 = 40,
  mean1 = 10,
  ratio = 1.5,
  dispersion1 = 2,
  dispersion2 = 8
) |>
  wald_test_nb()

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