asyvarTk: Asymptotic Variance of Cuzick and Edwards T_k Test statistic

In nnspat: Nearest Neighbor Methods for Spatial Patterns

Asymptotic Variance of Cuzick and Edwards T_k Test statistic

Description

This function computes the asymptotic variance of Cuzick and Edwards T_k test statistic based on the number of cases within kNNs of the cases in the data.

The argument, n_1, is the number of cases (denoted as n1 as an argument). The number of cases are denoted as n_1 and number of controls as n_0 in this function to match the case-control class labeling, which is just the reverse of the labeling in \insertCitecuzick:1990;textualnnspat.

The logical argument nonzero.mat (default=TRUE) is for using the A matrix if FALSE or just the matrix of nonzero locations in the A matrix (if TRUE) for computing N_s and N_t, which are required in the computation of the asymptotic variance. N_s and N_t are defined on page 78 of (\insertCitecuzick:1990;textualnnspat) as follows. N_s=∑_i∑_j a_{ij} a_{ji} (i.e., number of ordered pairs for which kNN relation is symmetric) and N_t= ∑ ∑_{i \ne l}∑ a_{ij} a_{lj} (i.e, number of triplets (i,j,l) i,j, and l distinct so that j is among kNNs of i and j is among kNNs of l). For the A matrix, see the description of the functions aij.mat and aij.nonzero.

See (\insertCitecuzick:1990;textualnnspat) for more details.

Usage

asyvarTk(dat, n1, k, nonzero.mat = TRUE, ...)

Arguments

dat	The data set in one or higher dimensions, each row corresponds to a data point.
n1	Number of cases
k	Integer specifying the number of NNs (of subject i)
nonzero.mat	A logical argument (default is TRUE) to determine whether the A matrix or the matrix of nonzero locations of the A matrix will be used in the computation of N_s and N_t. If TRUE the nonzero location matrix is used, otherwise the A matrix itself is used.
...	are for further arguments, such as method and p, passed to the dist function.

Value

A list with the elements

asy.var	The asymptotic variance of Cuzick and Edwards T_k test statistic for disease clustering
Ns	The N_s value standing for the number of ordered pairs for which kNN relation is symmetric, see the description.
Nt	The N_t value standing for the number of triplets (i,j,l) i,j, and l distinct so that j is among kNNs of i and j is among kNNs of l see the description.

Author(s)

Elvan Ceyhan

References

\insertAllCited

See Also

ceTk, varTk, and varTkaij

Examples

n<-20  #or try sample(1:20,1)
Y<-matrix(runif(3*n),ncol=3)
cls<-sample(0:1,n,replace = TRUE)  #or try cls<-rep(0:1,c(10,10))
n1<-sum(cls==1)
k<-3 #try also 2,3

asyvarTk(Y,n1,k)
asyvarTk(Y,n1,k,nonzero.mat=FALSE)
asyvarTk(Y,n1,k,method="max")

nnspat documentation built on Aug. 30, 2022, 9:06 a.m.