Inference of model coefficients — cap

This function performs inference on the model coefficient $\beta$.

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 $\gamma$. Default is `NULL`. If `gamma = NULL`, an error warning will be returned.
beta	a $q$-dimensional vector, the model coefficient $\beta$. Default is `NULL`. If `beta = NULL`, when `boot = FALSE`, $\beta$ will be estimated using the provided $\gamma$.
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 $\Sigma_{i}$. We assume there exits a $p$-dimensional vector $\gamma$ such that $z_{it}:=\gamma'y_{it}$ satisfies the multiplicative heteroscedasticity: $$\log(\mathrm{Var}(z_{it}))=\log(\gamma'\Sigma_{i}\gamma)=\beta_{0}+x_{i}'\beta_{1},$$ where $x_{i}$ contains explanatory variables of subject $i$, and $\beta_{0}$ and $\beta_{1}$ are model coefficients.

The $\beta$ 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 $\beta$ 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 $\beta$ coefficient, the test statistic, and the $p$-value.

When boot = TRUE,

Inference

point estimate of the $\beta$ 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 $\beta$ coefficient in each iteration.

References

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

Author

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>

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

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 \(\gamma\). Default is `NULL`. If `gamma = NULL`, an error warning will be returned.
beta	a \(q\)-dimensional vector, the model coefficient \(\beta\). Default is `NULL`. If `beta = NULL`, when `boot = FALSE`, \(\beta\) will be estimated using the provided \(\gamma\).
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`.