funsXsq.nnref: Reflexivity Test with Chi-square Approximation

Description

Two functions: Xsq.nnref.ct and Xsq.nnref.

Both functions are objects of class "Chisqtest" but with different arguments (see the parameter list below). Each one performs hypothesis tests of equality of the expected values of the diagonal cell counts (i.e., entries) under RL or CSR in the RCT for \(k \ge 2\) classes. That is, each test performs an overall NN reflexivity test (for the vector of entries \((1,1)\) and \((2,2)\), respectively, in the RCT) which is appropriate (i.e., have the appropriate asymptotic sampling distribution) for completely mapped data. (See ceyhan:NNreflexivity2017;textualnnspat for more detail).

Each reflexivity test is based on the chi-squared approximation of the corresponding quadratic form for the vector of diagonal entries in the RCT and are due to ceyhan:NNreflexivity2017;textualnnspat.

Each function yields the test statistic, \(p\)-value and df which is 2, description of the alternative with the corresponding null values (i.e., expected values) of the diagonal entries and also the sample estimates (i.e., observed values) of the diagonal entries of RCT (as a vector). The functions also provide names of the test statistics, the description of the test and the data set used.

The null hypothesis is that \(E(N_{11},N_{22})=(R P_{aa},R P_{ab})\) in the RCT, where \(R\) is the number of reflexive NNs and \(P_{aa}\) is the probability of any two points selected are being from the same class and \(P_{ab}\) is the probability of any two points selected are being from two different classes.

Usage

Xsq.nnref.ct(rfct, nvec, Qv, Tv)
Xsq.nnref(dat, lab, ...)

Value

A list with the elements

statistic: The chi-squared test statistic for overall NN reflexivity test
p.value: The \(p\)-value for the hypothesis test
df: Degrees of freedom for the chi-squared test, which is 2 for this function.
estimate: Estimates of the parameters, i.e., the observed diagonal entries \((1,1)\) and \((2,2)\) in the RCT, rfct.
est.name,est.name2: Names of the estimates, they are identical for this function.
null.value: Hypothesized null values for the diagonal entries \((1,1)\) and \((2,2)\) in the RCT, which are \(E(N_{11})=R P_{aa}\) and \(E(N_{22})=R P_{ab}\), respectively).
method: Description of the hypothesis test
ct.name: Name of the contingency table, rfct, returned by Xsq.nnref.ct only
data.name: Name of the data set, dat, returned by Xsq.nnref only

Arguments

rfct: An RCT, used in Xsq.nnref.ct only
nvec: The vector of class sizes, used in Xsq.nnref.ct only
Qv: The number of shared NNs, used in Xsq.nnref.ct only
Tv: \(T\) value, which is the number of triplets \((z_i, z_j, z_k)\) with \(NN(z_i) = NN(z_j) = z_k\) and \(NN(z_k) = z_j\) where \(NN(\cdot)\) is the nearest neighbor function, used in Xsq.nnref.ct only.
dat: The data set in one or higher dimensions, each row corresponds to a data point, used in Xsq.nnref only
lab: The vector of class labels (numerical or categorical), used in Xsq.nnref only
...: are for further arguments, such as method and p, passed to the dist function, used in Xsq.nnref only

Author

Elvan Ceyhan

References

Examples

Run this code

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

ipd<-ipd.mat(Y)
W<-Wmat(ipd)
Qv<-Qvec(W)$q
R<-Rval(W)
Tv<-Tval(W,R)

nvec<-as.numeric(table(cls))
rfct<-rct(ipd,cls)

Xsq.nnref(Y,cls)
Xsq.nnref.ct(rfct,nvec,Qv,Tv)

Xsq.nnref(Y,cls,method="max")

#############
n<-40
Y<-matrix(runif(3*n),ncol=3)
cls<-sample(1:4,n,replace = TRUE)  #or try cls<-rep(1:2,c(10,10))

ipd<-ipd.mat(Y)
W<-Wmat(ipd)
Qv<-Qvec(W)$q
R<-Rval(W)
Tv<-Tval(W,R)

nvec<-as.numeric(table(cls))
rfct<-rct(ipd,cls)

Xsq.nnref(Y,cls)
Xsq.nnref.ct(rfct,nvec,Qv,Tv)

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