depower (version 2025.1.20)

mle_nb: MLE for NB

Description

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", ...)

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.

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.

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_2012depower

aban_2009depower

See Also

sim_nb(), nll_nb

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
mle_null