# pois.env: Fit the envelope model in poisson regression In Renvlp: Computing Envelope Estimators

 pois.env R Documentation

## Fit the envelope model in poisson regression

### Description

Fit the envelope model in poisson regression with dimension u.

### Usage

```pois.env(X, Y, u, asy = TRUE, init = NULL)
```

### Arguments

 `X` Predictors. An n by p matrix, p is the number of predictors and n is number of observations. The predictors must be continuous variables. `Y` Response. An n by 1 matrix. The univariate response must be counts. `u` Dimension of the envelope. An integer between 0 and p. `asy` Flag for computing the asymptotic variance of the envelope estimator. The default is `TRUE`. When p and r are large, computing the asymptotic variance can take much time and memory. If only the envelope estimators are needed, the flag can be set to `asy = FALSE`. `init` The user-specified value of Gamma for the envelope subspace in poisson regression. An p by u matrix. The default is the one generated by function pois.envMU.

### Details

This function fits the envelope model in poisson regression,

Y = exp(μ + β' X), Σ_{X}=ΓΩΓ'+Γ_{0}Ω_{0}Γ'_{0}

using the maximum likelihood estimation. When the dimension of the envelope is between 1 and p-1, the starting value and blockwise coordinate descent algorithm in Cook et al. (2016) is implemented. This model works the best when X is multivariate normal.

### Value

The output is a list that contains the following components:

 `beta` The envelope estimator of the canonical parameter. `SigmaX` The envelope estimator of the covariance matrix of X. `Gamma` An orthonormal basis of the envelope subspace. `Gamma0` An orthonormal basis of the complement of the envelope subspace. `eta` The estimated beta of the canonical parameter with respect to Gamma. `Omega` The coordinates of SigmaX with respect to Gamma. `Omega0` The coordinates of SigmaX with respect to Gamma0. `mu` The estimated intercept of the canonical parameter. `loglik` The maximized log likelihood function. `covMatrix` The asymptotic covariance of vec(beta). The covariance matrix returned are asymptotic. For the actual standard errors, multiply by 1 / n. `asySE` The asymptotic standard error for elements in beta under the envelope model. The standard errors returned are asymptotic, for actual standard errors, multiply by 1 / sqrt(n). `ratio` The asymptotic standard error ratio of the standard multivariate linear regression estimator over the envelope estimator, for each element in beta. `n` The number of observations in the data.

### References

Cook, R. D., Zhang, X. (2015). Foundations for Envelope Models and Methods. Journal of the American Statistical Association 110, 599 - 611.

Cook, R. D., Forzani, L. and Su, Z. (2016) A Note on Fast Envelope Estimation. Journal of Multivariate Analysis. 150, 42-54.

### Examples

```data(horseshoecrab)
X1 <- as.numeric(horseshoecrab[ , 1] == 2)
X2 <- as.numeric(horseshoecrab[ , 1] == 3)
X3 <- as.numeric(horseshoecrab[ , 1] == 4)
X4 <- as.numeric(horseshoecrab[ , 2] == 2)
X5 <- as.numeric(horseshoecrab[ , 2] == 3)
X6 <- horseshoecrab[ , 3]
X7 <- horseshoecrab[ , 5]
X <- cbind(X1, X2, X3, X4, X5, X6, X7)
Y <- horseshoecrab[ , 4]

## Not run: u <- u.pois.env(X, Y)
## Not run: u

m <- pois.env(X, Y, 1)
m\$beta
```

Renvlp documentation built on Aug. 8, 2022, 1:06 a.m.