# funs.overall.tct: Types I-IV Overall Tests of Segregation for NNCT In nnspat: Nearest Neighbor Methods for Spatial Patterns

## Description

Two functions: overall.tct.ct and overall.tct.

All functions are objects of class "Chisqtest" but with different arguments (see the parameter list below). Each one performs hypothesis tests of deviations of cell counts from the expected values under RL or CSR for all cells (i.e., entries) combined in the TCT. That is, each test is one of Types I-IV overall test of segregation based on TCTs for k ≥ 2 classes. This overall test is based on the chi-squared approximation of the corresponding quadratic form and are due to \insertCiteceyhan:SJScorrected2010,ceyhan:jkss-posthoc-2017;textualnnspat. Both functions exclude some row and/or column of the TCT, to avoid ill-conditioning of the covariance matrix of the NNCT (for its inversion in the quadratic form). In particular, type-II removes the last column, and all other types remove the last row and column.

Each function yields the test statistic, p-value and df which is k(k-1) for type II test and (k-1)^2 for the other types, description of the alternative with the corresponding null values (i.e. expected values) of TCT entries, sample estimates (i.e. observed values) of the entries in TCT. The functions also provide names of the test statistics, the method and the data set used.

The null hypothesis is that all Tij entries for the specified type are equal to their expected values under RL or CSR.

## Usage

 1 2 3 overall.tct.ct(ct, covN, type = "III") overall.tct(dat, lab, type = "III", ...) 

## Arguments

 ct A nearest neighbor contingency table, used in overall.tct.ct only covN The k^2 \times k^2 covariance matrix of row-wise vectorized entries of NNCT, ct ; used in overall.tct.ct only. type The type of the overall segregation test, default="III". Takes on values "I"-"IV" (or equivalently 1-4, respectively. dat The data set in one or higher dimensions, each row corresponds to a data point, used in overall.tct only lab The vector of class labels (numerical or categorical), used in overall.tct only ... are for further arguments, such as method and p, passed to the dist function. used in overall.tct only

## Value

A list with the elements

 statistic The overall chi-squared statistic for the specified type stat.names Name of the test statistic p.value The p-value for the hypothesis test df Degrees of freedom for the chi-squared test, which is k(k-1) for type="II" and (k-1)^2 for others. estimate Estimates of the parameters, TCT, i.e., matrix of the observed T_{ij} values which is the TCT. est.name,est.name2 Names of the estimates, former is a longer description of the estimates than the latter. null.value Matrix of hypothesized null values for the parameters which are expected values of the the T_{ij} values in the TCT. null.name Name of the null values method Description of the hypothesis test ct.name Name of the contingency table, ct, returned by overall.tct.ct only data.name Name of the data set, dat, returned by overall.tct only

Elvan Ceyhan

## References

\insertAllCited

overall.seg.ct, overall.seg, overall.nnct.ct and overall.nnct
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 n<-20 #or try sample(1:20,1) Y<-matrix(runif(3*n),ncol=3) ipd<-ipd.mat(Y) cls<-sample(1:2,n,replace = TRUE) #or try cls<-rep(1:2,c(10,10)) ct<-nnct(ipd,cls) W<-Wmat(ipd) Qv<-Qvec(W)$q Rv<-Rval(W) varN<-var.nnct(ct,Qv,Rv) covN<-cov.nnct(ct,varN,Qv,Rv) #default is byrow overall.tct(Y,cls) overall.tct(Y,cls,type="I") overall.tct(Y,cls,type="II") overall.tct(Y,cls,type="III") overall.tct(Y,cls,type="IV") overall.tct(Y,cls,method="max") overall.tct.ct(ct,covN) overall.tct.ct(ct,covN,type="I") #cls as a factor na<-floor(n/2); nb<-n-na fcls<-rep(c("a","b"),c(na,nb)) ct<-nnct(ipd,fcls) overall.tct(Y,fcls) overall.tct.ct(ct,covN) ############# n<-40 Y<-matrix(runif(3*n),ncol=3) ipd<-ipd.mat(Y) cls<-sample(1:4,n,replace = TRUE) #or try cls<-rep(1:2,c(10,10)) ct<-nnct(ipd,cls) W<-Wmat(ipd) Qv<-Qvec(W)$q Rv<-Rval(W) varN<-var.nnct(ct,Qv,Rv) covN<-cov.nnct(ct,varN,Qv,Rv) overall.tct(Y,cls) overall.tct.ct(ct,covN)