# dist: Details of the distributions In MGLM: Multivariate Response Generalized Linear Models

 dist R Documentation

## Details of the distributions

### Description

An object that specifies the distribution to be fitted by the `MGLMfit` function, or the regression model to be fitted by the `MGLMreg` or `MGLMsparsereg` functions. Can be chosen from `"MN"`, `"DM"`, `"NegMN"`, or `"GDM"`.

### Details

#### "MN": Multinomial distribution

A multinomial distribution models the counts of d possible outcomes. The counts of categories are negatively correlated. The density of a d category count vector y with parameter p=(p_1, …, p_d) is

P(y|p) = C_{y_1, …, y_d}^{m} prod_{j=1}^{d} p_j^{y_j},

where m = sum_{j=1}^d y_j, 0 < p_j < 1, and sum_{j=1}^d p_j = 1. Here, C_k^n, often read as "n choose k", refers the number of k combinations from a set of n elements.

The `MGLMreg` function with `dist="MN"` calculates the MLE of regression coefficients β_j of the multinomial logit model, which has link function p_j = exp(Xβ_j) / (1 + sum_{j=1}^{d-1} exp(Xβ_j)), j=1,…,d-1. The `MGLMsparsereg` function with `dist="MN"` fits regularized multinomial logit model.

#### "DM": Dirichlet multinomial distribution

When the multivariate count data exhibits over-dispersion, the traditional multinomial model is insufficient. Dirichlet multinomial distribution models the probabilities of the categories by a Dirichlet distribution. The density of a d category count vector y, with parameter α = (α_1, …, α_d), α_j > 0, is

P(y|α) = C_{y_1, …, y_d}^{m} prod_{j=1}^d {Gamma(α_j+y_j)Gamma(sum_{j'=1}^d α_j')} / {Gamma(α_j)Gamma(sum_{j'=1}^d α_j' + sum_{j'=1}^d y_j')},

where m = sum_{j=1}^d y_j. Here, C_k^n, often read as "n choose k", refers the number of k combinations from a set of n elements.

The `MGLMfit` function with `dist="DM"` calculates the maximum likelihood estimate (MLE) of (α_1, …, α_d). The `MGLMreg` function with `dist="DM"` calculates the MLE of regression coefficients β_j of the Dirichlet multinomial regression model, which has link function α_j = exp(Xβ_j), j=1,…,d. The `MGLMsparsereg` function with `dist="DM"` fits regularized Dirichlet multinomial regression model.

#### "GDM": Generalized Dirichlet multinomial distribution

The more flexible Generalized Dirichlet multinomial model can be used when the counts of categories have both positive and negative correlations. The probability mass of a count vector y over m trials with parameter (α, β)=(α_1, …, α_{d-1}, β_1, …, β_{d-1}), α_j, β_j > 0, is

P(y|α,β) =C_{y_1, …, y_d}^{m} prod_{j=1}^{d-1} {Gamma(α_j+y_j)Gamma(β_j+z_{j+1})Gamma(α_j+β_j)} / {Gamma(α_j)Gamma(β_j)Gamma(α_j+β_j+z_j)},

where z_j = sum_{k=j}^d y_k and m = sum_{j=1}^d y_j. Here, C_k^n, often read as "n choose k", #' refers the number of k combinations from a set of n elements.

The `MGLMfit` with `dist="GDM"` calculates the MLE of (α, β)=(α_1, …, α_{d-1}, β_1, …, β_{d-1}). The `MGLMreg` function with `dist="GDM"` calculates the MLE of regression coefficients α_j, β_j of the generalized Dirichlet multinomial regression model, which has link functions α_j=exp(Xα_j) and β_j=exp(Xβ_j), j=1, …, d-1. The `MGLMsparsereg` function with `dist="GDM"` fits regularized generalized Dirichlet multinomial regression model.

#### "NegMN": Negative multinomial distribution

Both the multinomial distribution and Dirichlet multinomial distribution are good for negatively correlated counts. When the counts of categories are positively correlated, the negative multinomial distribution is preferred. The probability mass function of a d category count vector y with parameter (p_1, …, p_{d+1}, β), sum_{j=1}^{d+1} p_j = 1, p_j > 0, β > 0, is

P(y|p,β) = C_{m}^{β+m-1} C_{y_1, …, y_d}^{m} prod_{j=1}^d p_j^{y_j} p_{d+1}^β = (β_m)/(m!) C_{y_1, …, y_d}^{m} prod_{j=1}^d p_j^{y_j} p_{d+1}^β,

where m = sum_{j=1}^d y_j. Here, C_k^n, often read as "n choose k", refers the number of k combinations from a set of n elements.

The `MGLMfit` function with `dist="NegMN"` calculates the MLE of (p_1, …, p_{d+1}, β). The `MGLMreg` function with `dist="NegMN"` and `regBeta=FALSE` calculates the MLE of regression coefficients (α_1,…,α_d, β) of the negative multinomial regression model, which has link function p_{d+1} = 1/(1 + sum_{j=1}^d exp(Xα_j)), p_j = exp(Xα_j) p_{d+1}, j=1, …, d. When `dist="NegMN"` and `regBeta=TRUE`, the overdispersion parameter is linked to covariates via β=exp(Xα_{d+1}), and the function `MGLMreg` outputs an estimated matrix of (α_1, …, α_{d+1}). The `MGLMsparsereg` function with `dist="NegMN"` fits regularized negative multinomial regression model.

### Author(s)

Yiwen Zhang and Hua Zhou

`MGLMfit`, `MGLMreg`, `MGLMsparsereg`, `dmn`, `ddirmn`, `dgdirmn`, `dnegmn`