combine(npc)

R Documentation

Combine

Description

Provides the Nonparametric combination of dependent variables. It allows to combine partial tests with respect to variables or groups (contrasts). It also allows to obtain a global test by repeatedly applying this function together with the function t2p.

Usage

combine(P,fun=c("Direct","Fisher","Liptak","Max_T","Tippett"),which=c(2,3),W=NULL) 

Arguments

P	an array containing the permutation distribution of the partial tests. It can be a two or a three dimension array. The first dimension is always equal to B+1, where B is the number of considered Monte Carlo permutations. It can also be an array of p-values related to the partial test to be combined in order to obtain a global test.
fun	the combining function to be applied in the nonparametric combination of partial tests. See Details for the description of the combining functions.
which	the dimension with respect to which the nonparametric combination has to be done. Set which = 2 if you want to combine partial tests with respect to p variables, set which = 3 if the combination of partial test has to be done with respect to C samples (or contrasts)
W	An (optional) vector of weights, with the length equal to p if which = 2 or equal to C if which = 3.

Details

This function is very flexible and represents the core point of the Nonparametric combination methodology. It allows to combine arrays with either the permutation distribution of the partial tests or their related p-values. If fun = "Direct" or "Max_T", then the input matrix P must be the matrix with the permutation values of the test statistic. Otherwise, if fun = "Fisher", "Liptak" or "Tippett" the input matrix must be made of the p-values of the partial tests. The argument "which" specifies the dimension with respect to the combination has to be done.

The "Direct" function is the (weighted) sum of the partial statistics f(w,x) = t(w)%*%x, where w is the weigth vector and x is the considered vector of the partial statistics. The "Max_T" function corresponds to the maximum of the (weighted) partial statistics, f(w,x) = max(t(w)%*%x), where the maximum is referred to the dimension specified by the which argument.

The "Fisher" function is f(w,p) = -2*t(w)%*%log(p), where w is the weigth vector and p is the considered vector of the p-values of the partial statistics. The "Liptack" function is the (weighted) sum of the inverse standard normal C.d.f. f(w,p) = t(w)*%qnorm(1-p). The "Tippett" function is f(w,p) = min(t(w)%*%p).

Value

A matrix T1 with dimensions [B+1, p] if which = 3 or with dimensions [B+1, C] if which = 2 containing the values of the combined partial tests.

See Also

t2p,permute

Examples

 
## Example on producing plastic film from Krzanowski (1998, p. 381) 
## Two sample problem with three (dependent) variables. Partial tests are required with one-sided alternatives. 
 
n1<-9
n2<-11
n<-n1+n2
 
tear <- c(6.5, 6.2, 5.8, 6.5, 6.5, 6.9, 7.2, 6.9, 6.1, 6.3,
 6.7, 6.6, 7.2, 7.1, 6.8, 7.1, 7.0, 7.2, 7.5, 7.6)
gloss <- c(9.5, 9.9, 9.6, 9.6, 9.2, 9.1, 10.0, 9.9, 9.5, 9.4,
 9.1, 9.3, 8.3, 8.4, 8.5, 9.2, 8.8, 9.7, 10.1, 9.2)
opacity <- c(4.4, 6.4, 3.0, 4.1, 0.8, 5.7, 2.0, 3.9, 1.9, 5.7,
 2.8, 4.1, 3.8, 1.6, 3.4, 8.4, 5.2, 6.9, 2.7, 1.9)
Y <- cbind(tear, gloss, opacity)
 
X <- c(rep(1/n1,n1),rep(-1/n2,n2)) 
# X: vector of contrasts: the test statistics is the difference of sampling means for all variables
 
A= c(1,1,-1) 
# array specifying the alternatives (“greater” for ‘tear’ and ‘gloss’, “less” for ‘opacity’
 
source("permute.r")
source("combine.r")
source("t2p.r")
 
set.seed(3)
T<-permute(X,Y,A,B=1000)
 
dim(T)
 
# [1] 1001 3 
 
P<-t2p(T)
 
P[1,]
 
# [1] 0.994 0.014 0.252 # partial p-values;
 
par(mfrow=c(1,p)) # data representation;
for(j in 1:p){boxplot(Y[,j]~X,main=colnames(Y)[j],names=c("Sample2","Sample1"))}
 
T1 <- combine(P,fun="Fisher",which=2,W=c(0.5,0.2,0.3))
dim(T1)
 
#[1] 1001 1
 
global.p <- t2p(T1)[1]
global.p
 
# [1] 0.248 # global p-value.
 
######### ONE WAY ANOVA: 
######### WAIS score example, from Mack & Wolfe (1981). 
 
n = 3 ; C = 5;
Y = c(8.62,9.94,10.06,9.85,10.43,11.31,9.98,10.69,11.4,9.12,9.89,10.57,4.8,9.18,9.27)
X = array(0,dim=c(n*C,C))
 
for(k in 1:C){
X[(n*(k-1)+1):(n*k),k] = 1/n 
}
 
# X: matrix whose columns are indicator vectors of each group; the partial tests are squared sample means
 
x11() ; boxplot(Y~rep(1:5,each=3))
 
set.seed(9)
T = permute(X,Y,A = 0,B=1000)
dim(T)
 
# [1] 1001 5
 
T[1,]
 
# [1] 91.0116 110.8809 114.2761 97.2196 60.0625 # squared group means 
 
T1 = combine(T,fun="Direct",which=2,W=rep(n,C)) # T1 is equivalent to the between-group deviance
p.glob = t2p(T1)[1]
p.glob
 
# [1] 0.052 # global p-value
 
T1[1] - n*C*mean(Y)^2 # between-groups deviance
 
#[1] 16.55796

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