nll_nb: Negative log-likelihood for NB

In depower: Power Analysis for Differential Expression Studies

Negative log-likelihood for NB

Description

The negative log-likelihood for two independent samples of negative binomial distributions.

Usage

nll_nb_null(param, value1, value2, equal_dispersion, ratio_null)

nll_nb_alt(param, value1, value2, equal_dispersion)

Arguments

param	(numeric: ⁠(0, Inf)⁠) A vector of NB parameters. Must be in the following order for each scenario: Null and unequal dispersion: c(mean, dispersion1, dispersion2) Alternative and unequal dispersion: c(mean1, mean2, dispersion1, dispersion2) Null and equal dispersion: c(mean, dispersion) Alternative and equal dispersion: c(mean1, mean2, dispersion) for groups 1 and 2.
value1	(integer: ⁠(0, Inf)⁠) The vector of NB values from group 1. Must not contain NAs.
value2	(integer: ⁠(0, Inf)⁠) The vector of NB values from group 2. Must not contain NAs.
equal_dispersion	(Scalar logical) If TRUE, the log-likelihood is calculated assuming both groups have the same population dispersion parameter. If FALSE (default), the log-likelihood is calculated assuming different dispersions.
ratio_null	(Scalar numeric: ⁠(0, Inf)⁠) The ratio of means assumed under the null hypothesis (group 2 / group 1). Typically ratio_null = 1 (no difference).

Details

These functions are primarily designed for speed in simulation. Limited argument validation is performed.

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.

Unequal dispersion parameters

When the dispersion parameters are not equal, the likelihood is

\begin{aligned} L(r, \mu, \theta_1, \theta_2 \mid X_1, X_2) = & \left( \frac{\theta_1^{\theta_1}}{\Gamma(\theta_1)} \right)^{n_1} \frac{\mu^{\sum{x_{1i}}}}{(\mu + \theta_1)^{\sum{x_{1i} + n_1 \theta_1}}} \times \\ & \left( \frac{\theta_2^{\theta_2}}{\Gamma(\theta_2)} \right)^{n_2} \frac{(r \mu)^{\sum{x_{2j}}}}{(r \mu + \theta_2)^{\sum{x_{2j} + n_2 \theta_2}}} \times \\ & \prod_{i = 1}^{n_1}{\frac{\Gamma(x_{1i} + \theta_1)}{x_{1i}!}} \prod_{j = 1}^{n_2}{\frac{\Gamma(x_{2j} + \theta_2)}{x_{2j}!}} \end{aligned}

and the parameter space is \Theta = \left\{ (r, \mu, \theta_1, \theta_2) : r, \mu, \theta_1, \theta_2 > 0 \right\}. The log-likelihood is

\begin{aligned} l(r, \mu, \theta_1, \theta_2) = \ &n_1 \left[ \theta_1 \ln \theta_1 - \ln \Gamma(\theta_1) \right] + \\ &n_2 \left[ \theta_2 \ln \theta_2 - \ln \Gamma(\theta_2) \right] + \\ &(n_1 \bar{x}_1 + n_2 \bar{x}_2) \ln(\mu) - n_1 (\bar{x}_1 + \theta_1) \ln(\mu + \theta_1) + \\ &n_2 \bar{x}_2 \ln(r) - n_2 (\bar{x}_2 + \theta_2) \ln(r \mu + \theta_2) + \\ &\sum_{i = 1}^{n_1}{\left( \ln \Gamma(x_{1i} + \theta_1) - \ln(x_{1i}!) \right)} + \\ &\sum_{j = 1}^{n_2}{\left( \ln \Gamma(x_{2j} + \theta_2) - \ln(x_{2j}!) \right)} \end{aligned}

Equal dispersion parameters

When the dispersion parameters are equal, the likelihood is

\begin{aligned} L(r, \mu, \theta \mid X_1, X_2) = & \left( \frac{\theta^{\theta}}{\Gamma(\theta)} \right)^{n_1 + n_2} \times \\ & \frac{\mu^{\sum{x_{1i}}}}{(\mu + \theta)^{\sum{x_{1i} + n_1 \theta}}} \frac{(r \mu)^{\sum{x_{2j}}}}{(r \mu + \theta)^{\sum{x_{2j} + n_2 \theta}}} \times \\ & \prod_{i = 1}^{n_1}{\frac{\Gamma(x_{1i} + \theta)}{x_{1i}!}} \prod_{j = 1}^{n_2}{\frac{\Gamma(x_{2j} + \theta)}{x_{2j}!}} \end{aligned}

and the parameter space is \Theta = \left\{ (r, \mu, \theta) : r, \mu, \theta > 0 \right\}. The log-likelihood is

\begin{aligned} l(r, \mu, \theta) = \ &(n_1 + n_2) \left[ \theta \ln \theta - \ln \Gamma(\theta) \right] + \\ &(n_1 \bar{x}_1 + n_2 \bar{x}_2) \ln(\mu) - n_1 (\bar{x}_1 + \theta) \ln(\mu + \theta) + \\ &n_2 \bar{x}_2 \ln(r) - n_2 (\bar{x}_2 + \theta) \ln(r \mu + \theta) + \\ &\sum_{i = 1}^{n_1}{\left( \ln \Gamma(x_{1i} + \theta) - \ln(x_{1i}!) \right)} + \\ &\sum_{j = 1}^{n_2}{\left( \ln \Gamma(x_{2j} + \theta) - \ln(x_{2j}!) \right)} \end{aligned}

Value

Scalar numeric negative log-likelihood.

References

\insertRef

rettiganti_2012depower

\insertRef

aban_2009depower

See Also

mle_nb

Examples

#----------------------------------------------------------------------------
# nll_nb_*() examples
#----------------------------------------------------------------------------
library(depower)

set.seed(1234)
d <- sim_nb(
  n1 = 60,
  n2 = 40,
  mean1 = 10,
  ratio = 1.5,
  dispersion1 = 2,
  dispersion2 = 8
)

nll_nb_alt(
  param = c(mean1 = 10, mean2 = 15, dispersion1 = 2, dispersion2 = 8),
  value1 = d[[1L]],
  value2 = d[[2L]],
  equal_dispersion = FALSE
)

nll_nb_null(
  param = c(mean = 10, dispersion1 = 2, dispersion2 = 8),
  value1 = d[[1L]],
  value2 = d[[2L]],
  equal_dispersion = FALSE,
  ratio_null = 1
)

depower documentation built on Nov. 5, 2025, 5:21 p.m.