significance.test {CovRegRF}

Significance test

Description

This function runs a permutation test to evaluate the effect of a subset of covariates on the covariance matrix estimates. Returns an estimated p-value.

Usage

significance.test(
  formula,
  data,
  params.rfsrc = list(ntree = 1000, mtry = ceiling(px/3), nsplit = max(round(n/50),
    10)),
  nodesize.set = round(0.5^(1:100) * round(0.632 * n))[round(0.5^(1:100) * round(0.632
    * n)) > py],
  nperm = 500,
  test.vars = NULL
)

Arguments

formula

Object of class formula or character describing the model to fit. Interaction terms are not supported.

data

The multivariate data set which has n observations and px+py variables where px and py are the number of covariates (X) and response variables (Y), respectively. Should be a data.frame.

params.rfsrc

List of parameters that should be passed to randomForestSRC. In the default parameter set, ntree = 1000, mtry = px/3 (rounded up), nsplit = max(round(n/50), 10). See randomForestSRC for possible parameters.

nodesize.set

The set of nodesize levels for tuning. Default set includes the power of two times the sub-sample size (.632n) greater than the number of response variables (py).

nperm

Number of permutations.

test.vars

Subset of covariates whose effect on the covariance matrix estimates will be evaluated. A character vector defining the names of the covariates. The default is NULL, which tests for the global effect of the whole set of covariates.

Value

An object of class (covregrf, significancetest) which is a list with the following components:

pvalue

Estimated *p*-value, see below for details.

best.nodesize

Best nodesize value selected with the proposed tuning method using all covariates including the test.vars.

best.nodesize.control

Best nodesize value selected with the proposed tuning method using only the set of controlling covariates. If test.vars is NULL, returns NULL.

test.vars

Covariates whose effect on the covariance matrix estimates is evaluated.

control.vars

Controlling set of covariates.

predicted.oob

OOB predicted covariance matrices for training observations using all covariates including the test.vars.

predicted.perm

Predicted covariance matrices for the permutations using all covariates including the test.vars. A list of predictions for each permutation.

predicted.oob.control

OOB predicted covariance matrices for training observations using only the set of controlling covariates. If test.vars is NULL, returns NULL.

predicted.perm.control

Predicted covariance matrices for the permutations using only the set of controlling covariates. If test.vars is NULL, returns NULL.

Details

We perform a hypothesis test to evaluate the effect of a subset of covariates on the covariance matrix estimates, while controlling for the rest of the covariates. Define the conditional covariance matrix of Y given all X variables as \Sigma_{X}, and the conditional covariance matrix of Y given only the set of controlling X variables as \Sigma_{X}^{c}. If a subset of covariates has an effect on the covariance matrix estimates obtained with the proposed method, then \Sigma_{X} should be significantly different from \Sigma_{X}^{c}. We conduct a permutation test for the null hypothesis

H_0 : \Sigma_{X} = \Sigma_{X}^{c}

We estimate a p-value with the permutation test. If the p-value is less than the pre-specified significance level \alpha, we reject the null hypothesis.

Testing the global effect of the covariates on the conditional covariance estimates is a particular case of the proposed significance test. Define the unconditional covariance matrix estimate of Y as \Sigma_{root} which is computed as the sample covariance matrix of Y, and the conditional covariance matrix of Y given X as \Sigma_{X} which is obtained with covregrf(). If there is a global effect of X on the covariance matrix estimates, the \Sigma_{X} should be significantly different from \Sigma_{root}. The null hypothesis for this particular case is

H_0 : \Sigma_{X} = \Sigma_{root}

See Also

covregrf predict.covregrf print.covregrf