MLE for BNB

Maximum likelihood estimates (MLE) for bivariate negative binomial outcomes.

Usage

mle_bnb_null(data, ratio_null = 1, method = "nlm_constrained", ...)

mle_bnb_alt(data, method = "nlm_constrained", ...)

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).
method: (string: "nlm_constrained")
The optimization method. Must choose one of "nlm", "nlm_constrained", "optim", or "optim_constrained". The default bounds for constrained optimization are [1e-03, 1e06].
...: Optional arguments passed to the optimization method.

Value

For mle_bnb_alt, a list with the following elements:

Slot	Name	Description
1	`mean1`	MLE for mean of sample 1.
2	`mean2`	MLE for mean of sample 2.
3	`ratio`	MLE for ratio of means.
4	`dispersion`	MLE for BNB dispersion.
5	`nll`	Minimum of negative log-likelihood.
6	`nparams`	Number of estimated parameters.
7	`n1`	Sample size of sample 1.
8	`n2`	Sample size of sample 2.
9	`method`	Method used for the results.
10	`mle_method`	Method used for optimization.
11	`mle_code`	Integer indicating why the optimization process terminated.
12	`mle_message`	Additional information from the optimizer.

For mle_bnb_null, a list with the following elements:

Slot	Name	Description
1	`mean1`	MLE for mean of sample 1.
2	`mean2`	MLE for mean of sample 2.
3	`ratio_null`	Population ratio of means assumed for null hypothesis. `mean2 = mean1 * ratio_null`.
4	`dispersion`	MLE for BNB dispersion.
5	`nll`	Minimum of negative log-likelihood.
6	`nparams`	Number of estimated parameters.
7	`n1`	Sample size of sample 1.
8	`n2`	Sample size of sample 2.
9	`method`	Method used for the results.
10	`mle_method`	Method used for optimization.
11	`mle_code`	Integer indicating why the optimization process terminated.
12	`mle_message`	Additional information from the optimizer.

Details

These functions are 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 MLEs of \(r\) and \(\mu\) are \(\hat{r} = \frac{\bar{x}_2}{\bar{x}_1}\) and \(\hat{\mu} = \bar{x}_1\). The MLE of \(\theta\) is found by maximizing the profile log-likelihood \(l(\hat{r}, \hat{\mu}, \theta)\) with respect to \(\theta\). When \(r = r_{null}\) is known, the MLE of \(\mu\) is \(\tilde{\mu} = \frac{\bar{x}_1 + \bar{x}_2}{1 + r_{null}}\) and \(\tilde{\theta}\) is obtained by maximizing the profile log-likelihood \(l(r_{null}, \tilde{\mu}, \theta)\) with respect to \(\theta\).

The backend method for numerical optimization is controlled by argument method which refers to stats::nlm(), stats::nlminb(), or stats::optim(). If you would like to see warnings from the optimizer, include argument warnings = TRUE.

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 .

Examples

#----------------------------------------------------------------------------
# mle_bnb() examples
#----------------------------------------------------------------------------
library(depower)

set.seed(1234)
d <- sim_bnb(
  n = 40,
  mean1 = 10,
  ratio = 1.2,
  dispersion = 2
)

mle_alt <- d |>
  mle_bnb_alt()

mle_null <- d |>
  mle_bnb_null()

mle_alt
#> $mean1
#> [1] 9.650008
#> 
#> $mean2
#> [1] 11.84999
#> 
#> $ratio
#> [1] 1.227978
#> 
#> $dispersion
#> [1] 1.532668
#> 
#> $nll
#> [1] 241.691
#> 
#> $nparams
#> [1] 3
#> 
#> $n1
#> [1] 40
#> 
#> $n2
#> [1] 40
#> 
#> $method
#> [1] "Alternative hypothesis MLEs for bivariate negative binomial data"
#> 
#> $mle_method
#> [1] "nlm_constrained"
#> 
#> $mle_code
#> [1] 0
#> 
#> $mle_message
#> [1] "relative convergence (4)"
#> 
mle_null
#> $mean1
#> [1] 10.75
#> 
#> $mean2
#> [1] 10.75
#> 
#> $ratio_null
#> [1] 1
#> 
#> $dispersion
#> [1] 1.532671
#> 
#> $n1
#> [1] 40
#> 
#> $n2
#> [1] 40
#> 
#> $nll
#> [1] 246.2012
#> 
#> $nparams
#> [1] 2
#> 
#> $method
#> [1] "Null hypothesis MLEs for bivariate negative binomial data"
#> 
#> $mle_method
#> [1] "nlm_constrained"
#> 
#> $mle_code
#> [1] 0
#> 
#> $mle_message
#> [1] "relative convergence (4)"
#>