ZceTk {nnspat}

Z-test for Cuzick and Edwards T_k statistic

Description

An object of class "htest" performing a z-test for Cuzick and Edwards T_k test statistic based on the number of cases within kNNs of the cases in the data.

For disease clustering, Cuzick and Edwards (1990) suggested a k-NN test T_k based on number of cases among k NNs of the case points. Under RL of n_1 cases and n_0 controls to the given locations in the study region, T_k approximately has N(E[T_k],Var[T_k]/n_1) distribution for large n_1.

The argument cc.lab is case-control label, 1 for case, 0 for control, if the argument case.lab is NULL, then cc.lab should be provided in this fashion, if case.lab is provided, the labels are converted to 0's and 1's accordingly. Also, T_1 is identical to the count for cell (1,1) in the nearest neighbor contingency table (NNCT) (See the function nnct for more detail on NNCTs). Thus, the z-test for T_k is same as the cell-specific z-test for cell (1,1) in the NNCT (see cell.spec).

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) in the computations.

The logical argument asy.var (default=FALSE) is for using the asymptotic variance or the exact (i.e., finite sample) variance for the variance of T_k in its standardization. If asy.var=TRUE, the asymptotic variance is used for Var[T_k] (see asyvarTk), otherwise the exact variance (see varTk) is used.

See also (Ceyhan (2014); Cuzick and Edwards (1990)) and the references therein.

Usage

ZceTk(
  dat,
  cc.lab,
  k,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95,
  case.lab = NULL,
  nonzero.mat = TRUE,
  asy.var = FALSE,
  ...
)

Arguments

dat

The data set in one or higher dimensions, each row corresponds to a data point.

cc.lab

Case-control labels, 1 for case, 0 for control

k

Integer specifying the number of NNs (of subject i).

alternative

Type of the alternative hypothesis in the test, one of "two.sided", "less" or "greater".

conf.level

Level of the upper and lower confidence limits, default is 0.95, for Cuzick and Edwards T_k statistic

case.lab

The label used for cases in the cc.lab (if cc.lab is not provided then the labels are converted such that cases are 1 and controls are 0), default is NULL

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 (argument is passed on to asyvarTk). If TRUE the nonzero location matrix is used, otherwise the A matrix itself is used.

asy.var

A logical argument (default is FALSE) to determine whether the asymptotic variance or the exact (i.e., finite sample) variance for the variance of T_k in its standardization. If TRUE, the asymptotic variance is used for Var[T_k], otherwise the exact variance is used.

...

are for further arguments, such as method and p, passed to the dist function.

Value

A list with the elements

statistic

The Z test statistic for the Cuzick and Edwards T_k test

p.value

The p-value for the hypothesis test for the corresponding alternative

conf.int

Confidence interval for the Cuzick and Edwards T_k value at the given confidence level conf.level and depends on the type of alternative.

estimate

Estimate of the parameter, i.e., the Cuzick and Edwards T_k value.

null.value

Hypothesized null value for the Cuzick and Edwards T_k value which is k n_1 (n_1-1)/(n-1) for this function.

alternative

Type of the alternative hypothesis in the test, one of "two.sided", "less", "greater"

method

Description of the hypothesis test

data.name

Name of the data set, dat

Author(s)

Elvan Ceyhan

References

Ceyhan E (2014). “Segregation indices for disease clustering.” Statistics in Medicine, 33(10), 1662-1684.

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, cell.spec, and Xsq.ceTk

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))
k<-1 #try also 2,3, sample(1:5,1)

ZceTk(Y,cls,k)
ZceTk(Y,cls,k,nonzero.mat=FALSE)
ZceTk(Y,cls,k,method="max")

ZceTk(Y,cls+1,k,case.lab = 2,alt="l")
ZceTk(Y,cls,k,asy.var=TRUE,alt="g")