wald_test_nb {depower}R Documentation

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).

Rettiganti and Nagaraja (2012) 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

Rettiganti M, Nagaraja HN (2012). “Power Analyses for Negative Binomial Models with Application to Multiple Sclerosis Clinical Trials.” Journal of Biopharmaceutical Statistics, 22(2), 237–259. ISSN 1054-3406, 1520-5711, doi:10.1080/10543406.2010.528105.

Aban IB, Cutter GR, Mavinga N (2009). “Inferences and power analysis concerning two negative binomial distributions with an application to MRI lesion counts data.” Computational Statistics & Data Analysis, 53(3), 820–833. ISSN 01679473, doi:10.1016/j.csda.2008.07.034.

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()


[Package depower version 2025.1.20 Index]