Description

For a broad class of testing problems, such as the test of single-parameter null hypotheses using t-statistics, a proper, asymptotically valid test statistics joint null distribution is the multivariate Gaussian distribution with mean vector zero and covariance matrix equal to the correlation matrix of the vector influence curve for the estimator of the parameter of interest. The function corr.null estimates the correlation matrix of the vector influence curve for such parameters and returns samples from the corresponding normal distribution. Arguments to the function allow for refinements in calculating the resulting null distribution estimate.

Usage

Arguments

Details

This function is called internally when the argument nulldist='ic' is evaluated in the main user-level functions MTP or EBMTP. Formatting of the data objects X, W, Y, and especially Z occurs at execution begin of the main user-level functions.

Based on the value of test, the appropriate correlation matrix of the vector influence curve is calculated. Once the correlation matrix is obtained, one may sample vectors of null test statistics directly from a multivariate normal distribution rather than relying on permutation-based or bootstrap-based resampling. Because the Gaussian distribution is continuous, we expect this choice of null distribution to suffer less from discreteness than either the permutation or the bootstrap distribution. Additionally, in large-scale settings, use of null distributions derived from the vector influence function typically reduce computational bottlenecks associated with resampling methods.

Because the influence curve null distributions have been implemented for parametric, standardized t-statistics, the options robust and standardize are not allowed. Influence curve null distributions are available for the following values of test: 't.onesamp', 't.pair', 't.twosamp.equalvar', 't.twosamp.unequalvar', 'lm.XvsZ', 'lm.YvsXZ', 't.cor', and 'z.cor'.

In the simpler cases involving one-sample and two-sample tests of means, the correlation matrices are obtained via calls to cor. For two-sample tests, the correlation matrix corresponds to the following transformation of the group-specific covariance matrices: cov(X(group1))/n1 + cov(X(group2))/n2, where n1 and n2 are sample sizes of each group. When weights are present, the internal function IC.CorXW.NA is called to calculate weighted estimates of the (group) covariance matrices from each subject's estimated vector influence curve. The calculations are similar in spirit to those in cov.wt, but they are done in a way which allows for handling NA elements in the estimated vector influence curve IC_n. The correlation matrix corresponding to IC_n * (IC_n)^t is calculated.

For linear regression models, corr.null calculates the vector influence curve associated with each subject/sample. The vector has length equal to the number of hypotheses. The internal function IC.Cor.NA is used to calculate IC_n * (IC_n)^t in a manner which allows for NA-handling when the influence curve may contain missing elements. For linear regression models of the form E[Y|X], IC_n takes the form (E[((X^t)X)^(-1)] (X^t)_i Y_i) - Y_i-hat. Influence curves for correlation parameters are more complicated, and the user is referred to the references below.

Once the correlation matrix sigma' corresponding to the variance covariance matrix of the vector influence curve sigma =IC_n * (IC_n)^t is obtained, one may sample from N(0,sigma') to obtain null test statistics.

If ic.quant.trans=TRUE, the matrix of null test statistics can be quantile transformed to produce a matrix which accounts for the joint dependencies between test statistics (down columns), but which has marginal t-distributions (across rows). If marg.null and marg.par are not specified (=NULL), the following default t-distributions are applied:

t.onesamp

df=n-1;

t.pair

df=n-1, where n is the number of unique samples, i.e., the number of observed differences between paired samples;

t.twosamp.equalvar

df=n-2;

t.twosamp.unequalvar

df=n-1; N.B., this is not recommended, since the effective degrees of freedom are unknown. With sufficiently large n, a normal approximation should yield similar results.

lm.XvsZ

df=n-p, where p is the number of variables in the regression equation;

lm.YvsXZ

df=n-p, where p is the number of variables in the regression equation;

t.cor

df=n-2;

z.cor

N.B., also not recommended. Fisher's z-statistics are already normally distributed. Marginal transformation to a t-distribution makes little sense.

Value

A matrix of null test statistics with dimension the number of hypotheses (typically nrow(X)) by the number of desired samples (B).

Author(s)

Houston N. Gilbert

References

K.S. Pollard and Mark J. van der Laan, "Resampling-based Multiple Testing: Asymptotic Control of Type I Error and Applications to Gene Expression Data" (June 24, 2003). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 121. http://www.bepress.com/ucbbiostat/paper121

S. Dudoit and M.J. van der Laan. Multiple Testing Procedures and Applications to Genomics. Springer Series in Statistics. Springer, New York, 2008.

H.N. Gilbert, M.J. van der Laan, and S. Dudoit, "Joint Multiple Testing Procedures for Inferring Genetic Networks from Lower-Order Conditional Independence Graphs" (2009). In preparation.

See Also

boot.null,MTP, MTP-class, EBMTP, EBMTP-class, get.Tn, ss.maxT, mt.sample.teststat,get.Tn, wapply, boot.resample

Examples