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17.4 Negative-binomial-2-log generalized linear model (negative binomial regression)

Stan also supplies a single function for a generalized linear model with negative binomial likelihood and log link function, i.e. a function for a negative binomial regression. This provides a more efficient implementation of negative binomial regression than a manually written regression in terms of a negative binomial likelihood and matrix multiplication.

17.4.1 Probability mass function

If $x\in \mathbb{R}^{n\cdot m}, \alpha \in \mathbb{R}^n, \beta\in \mathbb{R}^m, \phi\in \mathbb{R}^+$, then for $y \in \mathbb{N}^n$, $$\text{NegBinomial2LogGLM}(y~|~x, \alpha, \beta, \phi) = \prod_{1\leq i \leq n}\text{NegBinomial2}(y_i~|~\exp(\alpha_i + x_i\cdot \beta), \phi).$$

17.4.2 Sampling statement

y ~ neg_binomial_2_log_glm(x, alpha, beta, phi)

Increment target log probability density with neg_binomial_2_log_glm_lupmf(y | x, alpha, beta, phi).
Available since 2.19

17.4.3 Stan functions

real neg_binomial_2_log_glm_lpmf(int y | matrix x, real alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi.
Available since 2.23

real neg_binomial_2_log_glm_lupmf(int y | matrix x, real alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi dropping constant additive terms.
Available since 2.25

real neg_binomial_2_log_glm_lpmf(int y | matrix x, vector alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi.
Available since 2.23

real neg_binomial_2_log_glm_lupmf(int y | matrix x, vector alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi dropping constant additive terms.
Available since 2.25

real neg_binomial_2_log_glm_lpmf(array[] int y | row_vector x, real alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi.
Available since 2.23

real neg_binomial_2_log_glm_lupmf(array[] int y | row_vector x, real alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi dropping constant additive terms.
Available since 2.25

real neg_binomial_2_log_glm_lpmf(array[] int y | row_vector x, vector alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi.
Available since 2.23

real neg_binomial_2_log_glm_lupmf(array[] int y | row_vector x, vector alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi dropping constant additive terms.
Available since 2.25

real neg_binomial_2_log_glm_lpmf(array[] int y | matrix x, real alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi.
Available since 2.18

real neg_binomial_2_log_glm_lupmf(array[] int y | matrix x, real alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi dropping constant additive terms.
Available since 2.25

real neg_binomial_2_log_glm_lpmf(array[] int y | matrix x, vector alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi.
Available since 2.18

real neg_binomial_2_log_glm_lupmf(array[] int y | matrix x, vector alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi dropping constant additive terms.
Available since 2.25