Likelihood ratio test for NB ratio of means

Likelihood ratio test for the ratio of means from two independent negative binomial outcomes.

Usage

lrt_nb(
  data,
  equal_dispersion = FALSE,
  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 LRT is calculated assuming both groups have the same population dispersion parameter. If FALSE (default), the LRT is calculated assuming different dispersions.

ratio_null

(Scalar numeric: 1; (0, Inf))
The 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 \(\chi^2\) likelihood ratio test statistic under the null hypothesis. See 'Details' and asymptotic() or simulated() for additional information.

...

Optional arguments passed to the MLE function mle_nb().

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).
5		alternative	Point estimates under the alternative hypothesis.
5	1	mean1	Estimated mean of group 1.
5	2	mean2	Estimated mean of group 2.
5	3	dispersion1	Estimated dispersion of group 1.
5	4	dispersion2	Estimated dispersion of group 2.
6		null	Point estimates under the null hypothesis.
6	1	mean1	Estimated mean of group 1.
6	2	mean2	Estimated mean of group 2.
6	3	dispersion1	Estimated dispersion of group 1.
6	4	dispersion2	Estimated dispersion of group 2.
7		n1	Sample size of group 1.
8		n2	Sample size of group 2.
9		method	Method used for the results.
10		equal_dispersion	Whether or not equal dispersions were assumed.
11		ratio_null	Assumed population ratio of means.
12		mle_code	Integer indicating why the optimization process terminated.
13		mle_message	Information from the optimizer.

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 LRT of \(r\) are

$$ \begin{aligned} H_{null} &: r = r_{null} \\ H_{alt} &: r \neq r_{null} \end{aligned} $$

where \(r = \frac{\bar{X}_2}{\bar{X}_1}\) 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 \(r_{null} = 1\)).

The LRT statistic is

$$ \begin{aligned} \lambda &= -2 \ln \frac{\text{sup}_{\Theta_{null}} L(r, \mu, \theta_1, \theta_2)}{\text{sup}_{\Theta} L(r, \mu, \theta_1, \theta_2)} \\ &= -2 \left[ \ln \text{sup}_{\Theta_{null}} L(r, \mu, \theta_1, \theta_2) - \ln \text{sup}_{\Theta} L(r, \mu, \theta_1, \theta_2) \right] \\ &= -2(l(r_{null}, \tilde{\mu}, \tilde{\theta}_1, \tilde{\theta}_2) - l(\hat{r}, \hat{\mu}, \hat{\theta}_1, \hat{\theta}_2)) \end{aligned} $$

Under \(H_{null}\), the LRT test statistic is asymptotically distributed as \(\chi^2_1\). The approximate level \(\alpha\) test rejects \(H_{null}\) if \(\lambda \geq \chi^2_1(1 - \alpha)\). 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. 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 |>
  lrt_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(
    lrt_nb(
      equal_dispersion = FALSE,
      distribution = simulated()
    )
  )

results in parametric resampling and no label permutation is performed. Thus, setting equal_dispersion = FALSE and distribution = simulated() is only recommended when lrt_nb() is used inside of power(). See also, 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 .

See also

wald_test_nb()

Examples

#----------------------------------------------------------------------------
# lrt_nb() examples
#----------------------------------------------------------------------------
library(depower)

set.seed(1234)
sim_nb(
  n1 = 60,
  n2 = 40,
  mean1 = 10,
  ratio = 1.5,
  dispersion1 = 2,
  dispersion2 = 8
) |>
  lrt_nb()
#> $chisq
#> [1] 10.02464
#> 
#> $df
#> [1] 1
#> 
#> $p
#> [1] 0.001544595
#> 
#> $ratio
#> [1] 1.542934
#> 
#> $alternative
#> $alternative$mean1
#> [1] 9.316667
#> 
#> $alternative$mean2
#> [1] 14.375
#> 
#> $alternative$dispersion1
#> [1] 1.545421
#> 
#> $alternative$dispersion2
#> [1] 11.08002
#> 
#> 
#> $null
#> $null$mean1
#> [1] 13.17212
#> 
#> $null$mean2
#> [1] 13.17212
#> 
#> $null$dispersion1
#> [1] 1.31389
#> 
#> $null$dispersion2
#> [1] 10.18563
#> 
#> 
#> $n1
#> [1] 60
#> 
#> $n2
#> [1] 40
#> 
#> $method
#> [1] "Asymptotic LRT for independent negative binomial ratio of means"
#> 
#> $equal_dispersion
#> [1] FALSE
#> 
#> $ratio_null
#> [1] 1
#> 
#> $mle_code
#> [1] 0
#> 
#> $mle_message
#> [1] "relative convergence (4)"
#>