asyvarTk {nnspat}

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 Cuzick and Edwards (1990).

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 (Cuzick and Edwards (1990)) as follows. N_s=\sum_i\sum_j a_{ij} a_{ji} (i.e., number of ordered pairs for which kNN relation is symmetric) and N_t= \sum \sum_{i \ne l}\sum 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 (Cuzick and Edwards (1990)) 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

Cuzick J, Edwards R (1990). “Spatial clustering for inhomogeneous populations (with discussion).” Journal of the Royal Statistical Society, Series B, 52, 73-104.

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")