# eppls: Fit the Envelope-based Partial Partial Least Squares model In Renvlp: Computing Envelope Estimators

 eppls R Documentation

## Fit the Envelope-based Partial Partial Least Squares model

### Description

Fit the Envelope-based Partial Partial Least Squares model for multivariate linear regression with dimension u.

### Usage

eppls(X1, X2, Y, u, asy = TRUE, init = NULL)


### Arguments

 X1 An n by p1 matrix of continuous predictors, where p1 is the number of continuous predictors with p1 < n. X2 An n by p2 matrix of categorical predictors, where p2 is the number of categorical predictors with p2 < n. Y An n by r matrix of multivariate responses, where r is the number of responses. u A given dimension of the Envelope-based Partial Partial Least Squares. It should be an interger between 0 and p1. 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. An r by u matrix. The default is the one generated by function envMU.

### Details

This function the Envelope-based Partial Partial Least Squares model for multivariate linear regression with dimension u,

Y = μ + Γη X + \varepsilon, Σ=ΓΩΓ' + Γ_{0}Ω_{0}Γ'_{0}

using the maximum likelihood estimation. When the dimension of the envelope is between 1 and p1-1, the starting value and blockwise coordinate descent algorithm in Cook et al. (2016) is implemented. When the dimension is p1, then the envelope model degenerates to the standard multivariate linear regression. When the dimension is 0, it means that X and Y are uncorrelated, and the fitting is different.

### Value

The output is a list that contains the following components:

 muY The estimator of mean of Y. mu1 The estimator of mean of X1. mu2 The estimator of mean of X2. beta1 A p1 by r matrix for the estimator of regression coefficients for continuous predictors. beta2 A p2 by r matrix for the estimator of regression coefficients for categorical predictors. Gamma An p1 by d matrix for the orthogonal basis of the Envelope-based Partial Partial Least Squares. Gamma0 An p1 by (p1-d) matrix for the orthogonal basis of the complement of the Envelope-based Partial Partial Least Squares. gamma A p2 by p1 matrix for the estimator of regression coefficients based on the regression of X1 on X2. eta A d by p1 matrix for the coordinates of beta1 with respect to Gamma. Omega A d by d matrix for the coordinates of SigmaX1 with respect to Gamma. Omega0 A (p1-d) by (p1-d) matrix for the coordinates of SigmaX1 with respect to Gamma0. SigmaX1 The estimator of error covariance matrix Sigma[1|2]. SigmaYcX The estimator of error covariance matrix Sigma[Y|X]. loglik The maximized log likelihood function. n The number of observations in the data. covMatrix1 The asymptotic covariance of vec(beta1). The covariance matrix returned are asymptotic. For the actual standard errors, multiply by 1 / n. covMatrix2 The asymptotic covariance of vec(beta2). The covariance matrix returned are asymptotic. For the actual standard errors, multiply by 1 / n. asySE1 The asymptotic standard error matrix for elements in beta1. The multiplication by the reciprocal of square root of n returns actual standard errors. asySE2 The asymptotic standard error matrix for elements in beta2. The multiplication by the reciprocal of square root of n returns actual standard errors.

### References

Park, Y., Su, Z. and Chung, D. (2022+) Envelope-based Partial Partial Least Squares with Application to Cytokine-based Biomarker Analysis for COVID-19.

### Examples

  data(amitriptyline)

Y <- amitriptyline[ , 1:2]
X1 <- amitriptyline[ , 4:7]
X2 <- amitriptyline[ , 3]
u <- u.eppls(X1, X2, Y)

u

m <- eppls(X1, X2, Y, 2)
m


Renvlp documentation built on Jan. 8, 2023, 1:08 a.m.