Wald test for BNB ratio of means

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

Usage

wald_test_bnb(
  data,
  ci_level = NULL,
  link = "log",
  ratio_null = 1,
  distribution = asymptotic(),
  ...
)

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().
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". See 'Details' for additional information.
ratio_null: (Scalar numeric: 1; (0, Inf))
The (pre-transformation) ratio of means assumed under the null hypothesis (sample 2 / sample 1). Typically ratio_null = 1 (no difference). See 'Details' for additional information.
distribution: (function: asymptotic() or simulated())
The method used to define the distribution of the $\chi^2$ Wald test statistic under the null hypothesis. See 'Details' and asymptotic() or simulated() for additional information.
...: Optional arguments passed to the MLE function mle_bnb().

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 sample 1.
6		`mean2`	Estimated mean of sample 2.
7		`dispersion`	Estimated dispersion.
8		`n1`	The sample size of sample 1.
9		`n2`	The sample size of sample 2.
10		`method`	Method used for the results.
11		`ci_level`	The confidence level.
12		`link`	Link function used to transform the ratio of means in the test hypotheses.
13		`ratio_null`	Assumed ratio of means under the null hypothesis.
14		`mle_code`	Integer indicating why the optimization process terminated.
15		`mle_message`	Information from the optimizer.

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 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 sample 2 with respect to sample 1, and $r_{null}$ is a constant for the assumed null population ratio of means (typically $r_{null} = 1$).

Rettiganti and Nagaraja (2012) found that $f(r) = r^2$, $f(r) = r$, and $f(r) = r^{0.5}$ had greatest power when $r < 1$. However, when $r > 1$, $f(r) = \ln r$, the likelihood ratio test, and $f(r) = r^{0.5}$ had greatest power. $f(r) = r^2$ was biased when $r > 1$. Both $f(r) = \ln r$ and $f(r) = r^{0.5}$ 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

$$ \hat{\sigma}^{2}_{\hat{r}} = \frac{\hat{r} (1 + \hat{r}) (\hat{\mu} + \hat{r}\hat{\mu} + \hat{\theta})}{n \left[ \hat{\mu} (1 + \hat{r}) (\hat{\mu} + \hat{\theta}) - \hat{\theta}\hat{r} \right]} $$

Under $H_{null}$, the Wald test statistic is asymptotically distributed as $\chi^2_1$. The approximate level $\alpha$ test rejects $H_{null}$ if $W(f(\hat{r})) \geq \chi^2_1(1 - \alpha)$. However, the asymptotic critical value is known to underestimate the exact critical value and the nominal significance level may not be achieved for small sample sizes. 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 should be used. Argument distribution allows control of the distribution of the $\chi^2_1$ test statistic under the null hypothesis by use of functions asymptotic() and simulated().

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