Likelihood ratio test for BNB ratio of means

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

Usage

lrt_bnb(data, 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().

ratio_null

(Scalar numeric: 1; (0, Inf))
The 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 \(\chi^2\) 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_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 (sample 2 / sample 1).
5		alternative	Point estimates under the alternative hypothesis.
5	1	mean1	Estimated mean of sample 1.
5	2	mean2	Estimated mean of sample 2.
5	3	dispersion	Estimated dispersion.
6		null	Point estimates under the null hypothesis.
6	1	mean1	Estimated mean of sample 1.
6	2	mean2	Estimated mean of sample 2.
6	3	dispersion	Estimated dispersion.
7		n1	The sample size of sample 1.
8		n2	The sample size of sample 2.
9		method	Method used for the results.
10		ratio_null	Assumed population ratio of means.
11		mle_code	Integer indicating why the optimization process terminated.
12		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 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 sample 2 with respect to sample 1 and \(r_{null}\) is a constant for the assumed null population ratio of means (typically \(r_{null} = 1\)).

The LRT statistic is

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

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().

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_bnb()

Examples

#----------------------------------------------------------------------------
# lrt_bnb() examples
#----------------------------------------------------------------------------
library(depower)

set.seed(1234)
sim_bnb(
  n = 40,
  mean1 = 10,
  ratio = 1.2,
  dispersion = 2
) |>
  lrt_bnb()
#> $chisq
#> [1] 9.020431
#> 
#> $df
#> [1] 1
#> 
#> $p
#> [1] 0.002669784
#> 
#> $ratio
#> [1] 1.227978
#> 
#> $alternative
#> $alternative$mean1
#> [1] 9.650008
#> 
#> $alternative$mean2
#> [1] 11.84999
#> 
#> $alternative$dispersion
#> [1] 1.532668
#> 
#> 
#> $null
#> $null$mean1
#> [1] 10.75
#> 
#> $null$mean2
#> [1] 10.75
#> 
#> $null$dispersion
#> [1] 1.532671
#> 
#> 
#> $n1
#> [1] 40
#> 
#> $n2
#> [1] 40
#> 
#> $method
#> [1] "Asymptotic LRT for bivariate negative binomial ratio of means"
#> 
#> $ratio_null
#> [1] 1
#> 
#> $mle_code
#> [1] 0
#> 
#> $mle_message
#> [1] "relative convergence (4)"
#>