eppls {Renvlp}

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 = \mu + \Gamma\eta X + \varepsilon, \Sigma=\Gamma\Omega\Gamma' + \Gamma_{0}\Omega_{0}\Gamma'_{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