wald_test_bnb: Wald test for BNB ratio of means
In depower: Power Analysis for Differential Expression Studies
View source: R/wald_test_bnb.r
wald_test_bnb R Documentation
Wald test for BNB ratio of means
Description
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, ...)
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.
...
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 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).
\insertCiterettiganti_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.
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.
References
\insertRefrettiganti_2012depower
\insertRefaban_2009depower
Examples
#----------------------------------------------------------------------------
# wald_test_bnb() examples
#----------------------------------------------------------------------------
library(depower)
set.seed(1234)
sim_bnb(
n = 40,
mean1 = 10,
ratio = 1.2,
dispersion = 2
) |>
wald_test_bnb()
depower documentation built on April 3, 2025, 9:23 p.m.
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View source: R/wald_test_bnb.r
| wald_test_bnb | R Documentation |
Wald test for BNB ratio of means
Description
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, ...)
Arguments
data |
(list) |
ci_level |
(Scalar numeric: |
link |
(Scalar string: |
ratio_null |
(Scalar numeric: |
... |
Optional arguments passed to the MLE function |
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_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.
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. |
References
\insertRefrettiganti_2012depower
\insertRefaban_2009depower
Examples
#----------------------------------------------------------------------------
# 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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