MLE for NB

Maximum likelihood estimates (MLE) for two independent negative binomial outcomes.

Usage

mle_nb_null(
  data,
  equal_dispersion = FALSE,
  ratio_null = 1,
  method = "nlm_constrained",
  ...
)

mle_nb_alt(data, equal_dispersion = FALSE, method = "nlm_constrained", ...)

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 MLEs are calculated assuming both groups have the same population dispersion parameter. If FALSE (default), the MLEs are 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).
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_nb_alt(), a list with the following elements:

Slot	Name	Description
1	`mean1`	MLE for mean of group 1.
2	`mean2`	MLE for mean of group 2.
3	`ratio`	MLE for ratio of means.
4	`dispersion1`	MLE for dispersion of group 1.
5	`dispersion2`	MLE for dispersion of group 2.
6	`equal_dispersion`	Were equal dispersions assumed.
7	`n1`	Sample size of group 1.
8	`n2`	Sample size of group 2.
9	`nll`	Minimum of negative log-likelihood.
10	`nparams`	Number of estimated parameters.
11	`method`	Method used for the results.
12	`mle_method`	Method used for optimization.
13	`mle_code`	Integer indicating why the optimization process terminated.
14	`mle_message`	Additional information from the optimizer.

For mle_nb_null(), a list with the following elements:

Slot	Name	Description
1	`mean1`	MLE for mean of group 1.
2	`mean2`	MLE for mean of group 2.
3	`ratio_null`	Population ratio of means assumed for null hypothesis. `mean2 = mean1 * ratio_null`.
4	`dispersion1`	MLE for dispersion of group 1.
5	`dispersion2`	MLE for dispersion of group 2.
6	`equal_dispersion`	Were equal dispersions assumed.
7	`n1`	Sample size of group 1.
8	`n2`	Sample size of group 2.
9	`nll`	Minimum of negative log-likelihood.
10	`nparams`	Number of estimated parameters.
11	`method`	Method used for the results.
12	`mle_method`	Method used for optimization.
13	`mle_code`	Integer indicating why the optimization process terminated.
14	`mle_message`	Additional information from the optimizer.

Details

These functions are primarily designed for speed in simulation. Missing values are silently excluded.

Suppose \(X_1 \sim \text{NB}(\mu, \theta_1)\) and \(X_2 \sim \text{NB}(r\mu, \theta_2)\), 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 MLEs of \(r\) and \(\mu\) are \(\hat{r} = \frac{\bar{x}_2}{\bar{x}_1}\) and \(\hat{\mu} = \bar{x}_1\). The MLEs of \(\theta_1\) and \(\theta_2\) are found by maximizing the profile log-likelihood \(l(\hat{r}, \hat{\mu}, \theta_1, \theta_2)\) with respect to \(\theta_1\) and \(\theta_2\). When \(r = r_{null}\) is known, the MLE of \(\mu\) is \(\tilde{\mu} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}\) and \(\tilde{\theta}_1\) and \(\tilde{\theta}_2\) are obtained by maximizing the profile log-likelihood \(l(r_{null}, \tilde{\mu}, \theta_1, \theta_2)\).

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_nb() examples
#----------------------------------------------------------------------------
library(depower)

d <- sim_nb(
  n1 = 60,
  n2 = 40,
  mean1 = 10,
  ratio = 1.5,
  dispersion1 = 2,
  dispersion2 = 8
)

mle_alt <- d |>
  mle_nb_alt()

mle_null <- d |>
  mle_nb_null()

mle_alt
#> $mean1
#> [1] 10.4
#> 
#> $mean2
#> [1] 15.6
#> 
#> $ratio
#> [1] 1.5
#> 
#> $dispersion1
#> [1] 2.096105
#> 
#> $dispersion2
#> [1] 5.692378
#> 
#> $equal_dispersion
#> [1] FALSE
#> 
#> $n1
#> [1] 60
#> 
#> $n2
#> [1] 40
#> 
#> $nll
#> [1] 333.6223
#> 
#> $nparams
#> [1] 4
#> 
#> $method
#> [1] "Alternative hypothesis MLEs for independent negative binomial data"
#> 
#> $mle_method
#> [1] "nlm_constrained"
#> 
#> $mle_code
#> [1] 0
#> 
#> $mle_message
#> [1] "relative convergence (4)"
#> 
mle_null
#> $mean1
#> [1] 13.55451
#> 
#> $mean2
#> [1] 13.55451
#> 
#> $ratio_null
#> [1] 1
#> 
#> $dispersion1
#> [1] 1.821658
#> 
#> $dispersion2
#> [1] 5.117131
#> 
#> $equal_dispersion
#> [1] FALSE
#> 
#> $n1
#> [1] 60
#> 
#> $n2
#> [1] 40
#> 
#> $nll
#> [1] 338.4952
#> 
#> $nparams
#> [1] 3
#> 
#> $method
#> [1] "Null hypothesis MLEs for independent negative binomial data"
#> 
#> $mle_method
#> [1] "nlm_constrained"
#> 
#> $mle_code
#> [1] 0
#> 
#> $mle_message
#> [1] "relative convergence (4)"
#>