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) The vector of initial values for NB parameters. Must be in the following order for each scenario: Null and unequal dispersion: c(mean1, dispersion1, dispersion2) Alternative and unequal dispersion: c(mean1, mean2, dispersion1, dispersion2) Null and equal dispersion: c(mean1, dispersion) Alternative and equal dispersion: c(mean1, mean2, dispersion) for groups 1 and 2.
value1	(numeric) The vector of NB values from group 1. Must not contain NAs.
value2	(numeric) 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. Arguments are not checked.

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(y_{1i} + \theta_1) - \ln(y_{1i}!) \right)} + \\ &\sum_{j = 1}^{n_2}{\left( \ln \Gamma(y_{2j} + \theta_2) - \ln(y_{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(y_{1i} + \theta) - \ln(y_{1i}!) \right)} + \\ &\sum_{j = 1}^{n_2}{\left( \ln \Gamma(y_{2j} + \theta) - \ln(y_{2j}!) \right)} \end{aligned}

Value

Scalar numeric negative log-likelihood.

References

\insertRef

rettiganti_2012depower

\insertRef

aban_2009depower

See Also

sim_nb(), stats::nlminb(), stats::nlm(), stats::optim()

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, dispersion = 2, dispersion2 = 8),
  value1 = d[[1L]],
  value2 = d[[2L]],
  equal_dispersion = FALSE
)

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

depower documentation built on April 3, 2025, 9:23 p.m.