cap_beta {cap} R Documentation

## Inference of model coefficients

### Description

This function performs inference on the model coefficient β.

### Usage

cap_beta(Y, X, gamma = NULL, beta = NULL, method = c("asmp", "LLR"),
boot = FALSE, sims = 1000, boot.ci.type = c("bca", "perc"),
conf.level = 0.95, verbose = TRUE)


### Arguments

 Y a data list of length n. Each list element is a T\times p matrix, the data matrix of T observations from p features. X a n\times q data matrix, the covariate matrix of n subjects with q-1 predictors. The first column is all ones. gamma a p-dimensional vector, the projecting direction γ. Default is NULL. If gamma = NULL, an error warning will be returned. beta a q-dimensional vector, the model coefficient β. Default is NULL. If beta = NULL, when boot = FALSE, β will be estimated using the provided γ. method a character of inference method. If method = "asmp", the inference is made based on the asymptotic variance; if method = "LLR", the likelihood ratio test is conducted. When boot = TRUE, this argument is ignored. boot a logic variable, whether bootstrap inference is performed. sims a numeric value, the number of bootstrap iterations will be performed. boot.ci.type a character of the way of calculating bootstrap confidence interval. If boot.ci.type = "bca", the bias corrected confidence interval is returned; if boot.ci.type = "perc", the percentile confidence interval is returned. conf.level a numeric value, the designated significance level. Default is 0.95. verbose a logic variable, whether the bootstrap procedure is printed. Default is TRUE.

### Details

Considering y_{it} are p-dimensional independent and identically distributed random samples from a multivariate normal distribution with mean zero and covariance matrix Σ_{i}. We assume there exits a p-dimensional vector γ such that z_{it}:=γ'y_{it} satisfies the multiplicative heteroscedasticity:

\log(\mathrm{Var}(z_{it}))=\log(γ'Σ_{i}γ)=β_{0}+x_{i}'β_{1},

where x_{i} contains explanatory variables of subject i, and β_{0} and β_{1} are model coefficients.

The β coefficient is estimated by maximizing the likelihood function. The asymptotic variance is obtained based on maximum likelihood estimator theory.

### Value

When method = "asmp", the output is a q \times 6 data frame containing the estimate of β coefficient, the asymptotic standard error, the test statistic, the p-value, and the lower and upper bound of the confidence interval.

When method = "LLR", the output is a q \times 3 data frame containing the estimate of β coefficient, the test statistic, and the p-value.

When boot = TRUE,

 Inference point estimate of the β coefficient, as well as the corresponding standard error, test statistic, p-value, and the lower and upper bound of the confidence interval. beta.boot the estimate of the β coefficient in each iteration.

### Author(s)

Yi Zhao, Johns Hopkins University, <zhaoyi1026@gmail.com>

Bingkai Wang, Johns Hopkins University, <bwang51@jhmi.edu>

Stewart Mostofsky, Johns Hopkins University, <mostofsky@kennedykrieger.org>

Brian Caffo, Johns Hopkins University, <bcaffo@gmail.com>

Xi Luo, Brown University, <xi.rossi.luo@gmail.com>

### References

Zhao et al. (2018) Covariate Assisted Principal Regression for Covariance Matrix Outcomes <doi:10.1101/425033>

### Examples


#############################################
data(env.example)
X<-get("X",env.example)
Y<-get("Y",env.example)
Phi<-get("Phi",env.example)

# asymptotic variance
re1<-cap_beta(Y,X,gamma=Phi[,2],method=c("asmp"),boot=FALSE)

# likelihood ratio test
re2<-cap_beta(Y,X,gamma=Phi[,2],method=c("LLR"),boot=FALSE)

# bootstrap confidence interval

re3<-cap_beta(Y,X,gamma=Phi[,2],boot=TRUE,sims=500,verbose=FALSE)

#############################################


[Package cap version 1.0 Index]