Wald test for the ratio of means from bivariate negative binomial outcomes.
wald_test_bnb(data, ci_level = NULL, link = "log", ratio_null = 1, ...)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. |
(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().
(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.
(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.
(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.
Optional arguments passed to the MLE function mle_bnb().
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_2012;textualdepower 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 \(W(f(\hat{r})) \geq \chi^2_1(1 - \alpha)\). 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.
rettiganti_2012depower
aban_2009depower
#----------------------------------------------------------------------------
# wald_test_bnb() examples
#----------------------------------------------------------------------------
library(depower)
set.seed(1234)
sim_bnb(
n = 40,
mean1 = 10,
ratio = 1.2,
dispersion = 2
) |>
wald_test_bnb()
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