prototest.univariate {prototest}

Perform Prototype or F Tests for Significance of Groups of Predictors in the Univariate Model

Description

Perform prototype or F tests for significance of groups of predictors in the univariate model. Choose either exact or approximate likelihood ratio prototype tests (ELR) or (ALR) or F test or marginal screening prototype test. Options for selective or non-selective tests. Further options for non-sampling or hit-and-run null reference distributions for selective tests.

Usage

prototest.univariate(x, y, type = c("ALR", "ELR", "MS", "F"), 
selected.col = NULL, lambda, mu = NULL, sigma = 1, hr.iter = 50000, 
hr.burn.in = 5000, verbose = FALSE, tol = 10^-8)

Arguments

x

input matrix of dimension n-by-p, where p is the number of predictors in a single predetermined group of predictors. Will be mean centered and standardised before tests are performed.

y

response variable. Vector of length emphn, assumed to be quantitative.

type

type of test to be performed. Can only select one at a time. Options include the exact and approximate likelihood ratio prototype tests of Reid et al (2015) (ELR, ALR), the F test and the marginal screening prototype test of Reid and Tibshirani (2015) (MS). Default is ELR.

selected.col

preselected columns specified by user. Vector of indices in the set {1, 2, ..., p}. If specified, a non-selective (classical) version of the chosen test it performed. In particular, this means the classicial \chi^2_1 reference distribution for the likelihood ratio tests and the F reference for the F test. Default is NULL, which directs the function to estimate the selected set with the lasso or the marginal screening procedure, depending on the test.

lambda

regularisation parameter for the lasso fit. Must be supplied when selected.col is NULL. Will be supplied to glmnet. This is the unstandardised version, equivalent to lambda/n supplied to glmnet.

mu

mean parameter for the response. See Details below. If supplied, it is first subtracted from the response to yield a mean-zero (at the population level) vector for which we proceed with testing. If NULL (the default), this parameter is treated as nuisance parameter and accounted for as such in testing.

sigma

error standard deviation for the response. See Details below. Must be supplied. If not, it is assumed to be 1. Required for the computation of some of the test statistics.

hr.iter

number of hit-and-run samples required in the reference distrbution of a selective test. Applies only if selected.col is NULL. Default is 50000. Since dependent samples are generated, large values are required to generate good reference distributions. If set to 0, the function tries to apply a non-sampling selective test (provided selected.col is NULL), if possible. If non-sampling test is not possible, the function exits with a message.

hr.burn.in

number of burn-in hit-and-run samples. These are generated first so as to make subsequent hit-and-run realisations less dependent on the observed response. Samples are then discarded and do not inform the null reference distribution.

verbose

should progress be printed?

tol

convergence threshold for iterative optimisation procedures.

Details

The model underpinning each of the tests is

y = \mu + \theta\cdot\hat{y} + \epsilon

where \epsilon \sim N(0, \sigma^2I) and \hat{y} depends on the particular test considered.

In particular, for the ELR, ALR and F tests, we have \hat{y} = P_M\left(y - \mu\right), where P_M = X_M\left(X_M^\top X_M\right)^{-1}X_M^\top. X_M is the input matrix reduced to the columns in the set M, which, in turn, is either provided by the user (via selected.col) or selected by the lasso (if selected.col is NULL). If the former, a non-selective test is performed; if the latter, a selective test is performed, with the restrictions Ay\leq b, as set out in Lee et al (2015).

For the marginal screening prototype (MS) test, \hat{y} = x_{j^*} where x_j is the j^{th} column of x and j^* = {\rm argmax}_j |x_j^\top y|.

All tests test the null hypothesis H_0: \theta = 0. Details of each are described in Reid et al (2015).

Value

A list with the following four components:

ts

The value of the test statistic on the observed data.

p.val

Valid p-value of the test.

selected.col

Vector with columns selected. If initially NULL, this will now contain indices of columns selected by the automatic column selection procedures of the test.

y.hr

Matrix with hit-and-run replications of the response. If sampled selective test was not performed, this will be NULL.

Author(s)

Stephen Reid

References

Reid, S. and Tibshirani, R. (2015) Sparse regression and marginal testing using cluster prototypes. http://arxiv.org/pdf/1503.00334v2.pdf. Biostatistics doi: 10.1093/biostatistics/kxv049
Reid, S., Taylor, J. and Tibshirani, R. (2015) A general framework for estimation and inference from clusters of features. Available online: http://arxiv.org/abs/1511.07839.

See Also

prototest.multivariate

Examples

require (prototest)

### generate data
set.seed (12345)
n = 100
p = 80

X = matrix (rnorm(n*p, 0, 1), ncol=p)


beta = rep(0, p)
beta[1:3] = 0.1 # three signal variables: number 1, 2, 3
signal = apply(X, 1, function(col){sum(beta*col)})
intercept = 3

y = intercept + signal + rnorm (n, 0, 1)

### treat all columns as if in same group and test for signal

# non-selective ELR test with nuisance intercept
elr = prototest.univariate (X, y, "ELR", selected.col=1:5)
# selective F test with nuisance intercept; non-sampling
f.test = prototest.univariate (X, y, "F", lambda=0.01, hr.iter=0) 
print (elr)
print (f.test)

### assume variables occur in 4 equally sized groups
num.groups = 4
groups = rep (1:num.groups, each=p/num.groups)

# selective ALR test -- select columns 21-25 in 2nd group; test for signal in 1st; hit-and-run
alr = prototest.multivariate(X, y, groups, 1, "ALR", 21:25, lambda=0.005, hr.iter=20000)
# non-selective MS test -- specify first column in each group; test for signal in 1st
ms = prototest.multivariate(X, y, groups, 1, "MS", c(1,21,41,61)) 
print (alr)
print (ms)