wald_test_nb: Wald test for NB ratio of means
In depower: Power Analysis for Differential Expression Studies
wald_test_nb 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,
distribution = asymptotic(),
...
)
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". See 'Details' for additional information.
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.
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_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).
\insertCiterettiganti_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). 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().
Note that standalone use of this function with equal_dispersion = FALSE
and distribution = simulated(), e.g.
data |>
wald_test_nb(
equal_dispersion = FALSE,
distribution = simulated()
)
results in a nonparametric randomization test based on label permutation.
This violates the assumption of exchangeability for the randomization test
because the labels are not exchangeable when the null hypothesis assumes
unequal dispersions. However, used inside power(), e.g.
data |>
power(
wald_test_nb(
equal_dispersion = FALSE,
distribution = simulated()
)
)
results in parametric resampling and no label permutation in performed.
Thus, setting equal_dispersion = FALSE and distribution = simulated() is
only recommended when wald_test_nb() is used inside of
power(). See also, simulated().
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
\insertRefrettiganti_2012depower
\insertRefaban_2009depower
See Also
lrt_nb()
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 Nov. 5, 2025, 5:21 p.m.
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| wald_test_nb | 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,
distribution = asymptotic(),
...
)
Arguments
data |
(list) |
equal_dispersion |
(Scalar logical: |
ci_level |
(Scalar numeric: |
link |
(Scalar string: |
ratio_null |
(Scalar numeric: |
distribution |
(function: |
... |
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 \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_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). 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().
Note that standalone use of this function with equal_dispersion = FALSE
and distribution = simulated(), e.g.
data |>
wald_test_nb(
equal_dispersion = FALSE,
distribution = simulated()
)
results in a nonparametric randomization test based on label permutation.
This violates the assumption of exchangeability for the randomization test
because the labels are not exchangeable when the null hypothesis assumes
unequal dispersions. However, used inside power(), e.g.
data |>
power(
wald_test_nb(
equal_dispersion = FALSE,
distribution = simulated()
)
)
results in parametric resampling and no label permutation in performed.
Thus, setting equal_dispersion = FALSE and distribution = simulated() is
only recommended when wald_test_nb() is used inside of
power(). See also, simulated().
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
\insertRefrettiganti_2012depower
\insertRefaban_2009depower
See Also
lrt_nb()
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()
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