Qsym.test {nnspat}

Pielou's Second Type of NN Symmetry Test with Chi-square Approximation

Description

An object of class "Chisqtest" performing the hypothesis test of equality of the probabilities for the rows in the Q-symmetry contingency table (QCT). Each row of the QCT is the vector of Qij values where Q_{ij} is the number of class i points that are NN to j points. That is, the test performs Pielou's second type of NN symmetry test which is also equivalent to Pearson's test on the QCT (Pielou (1961)). Pielou's second type of NN symmetry is the symmetry in the shared NN structure for all classes, which is also called Q-symmetry. The test is appropriate (i.e., have the appropriate asymptotic sampling distribution) provided that data is obtained by sparse sampling, although simulations suggest it seems to work for completely mapped data as well. (See Ceyhan (2014) for more detail).

The argument is.ipd is a logical argument (default=TRUE) to determine the structure of the argument x. If TRUE, x is taken to be the inter-point distance (IPD) matrix, and if FALSE, x is taken to be the data set with rows representing the data points.

The argument combine is a logical argument (default=TRUE) to determine whether to combine the 3rd column and the columns to the left. If TRUE, this function pools the cells 3 or larger together for k classes in the QCT, so, Q_2, Q_3 etc. are pooled, so, the column labels are Q_0, Q_1 and Q_2 with the last one is actually sum of Q_j for j \ge 2 in the QCT. If FALSE, the function does not perform the pooling of the cells.

The function yields the test statistic, p-value and df which is (k-1)(n_c-1) where n_c is the number of columns in QCT (which reduces to 2(k-1), if combine=TRUE). It also provides the description of the alternative with the corresponding null values (i.e., expected values) of the entries of the QCT and also the sample estimates of the entries of QCT (i.e., the observed QCT). The function also provides names of the test statistics, the description of the test and the data set used.

The null hypothesis is the symmetry in the shared NN structure for each class, that is, all E(Q_{ij})=n_i Q_j/n where n_i the size of class i and Q_j is the sum of column j in the QCT (i.e., the total number of points serving as NN to class j other points). (i.e., symmetry in the mixed NN structure).

See also (Pielou (1961); Ceyhan (2014)) and the references therein.

Usage

Qsym.test(x, lab, is.ipd = TRUE, combine = TRUE, ...)

Arguments

x

The IPD matrix (if is.ipd=TRUE) or a data set of points in matrix or data frame form where points correspond to the rows (if is.ipd = FALSE).

lab

The vector of class labels (numerical or categorical)

is.ipd

A logical parameter (default=TRUE). If TRUE, x is taken as the inter-point distance matrix (IPD matrix), otherwise, x is taken as the data set with rows representing the data points.

combine

A logical parameter (default=TRUE). If TRUE, the cells in column 3 or columns to the left are merged in the QCT, so, Q_2, Q_3 etc. are pooled, so, the column labels are Q_0, Q_1 and Q_2 with the last one is actually sum of Q_j for j \ge 2 in the QCT. If FALSE, the function does not perform the pooling of the cells.

...

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

Value

A list with the elements

statistic

The chi-squared test statistic for Pielou's second type of NN symmetry test (i.e., Q-symmetry which is equivalent to symmetry in the shared NN structure)

p.value

The p-value for the hypothesis test

df

Degrees of freedom for the chi-squared test, which is (k-1)(n_c-1) where n_c is the number of columns in QCT (which reduces to 2(k-1) if combine=TRUE).

estimate

Estimates, i.e., the observed QCT.

est.name, est.name2

Names of the estimates, they are identical for this function.

null.value

Hypothesized null values for the entries of the QCT, i.e., the matrix with entries E(Q_{ij})=n_i Q_j/n where n_i the size of class i and Q_j is the sum of column j in the QCT (i.e., the total number of points serving as NN to class j other points).

method

Description of the hypothesis test

data.name

Name of the data set, x

Author(s)

Elvan Ceyhan

References

Ceyhan E (2014). “Testing Spatial Symmetry Using Contingency Tables Based on Nearest Neighbor Relations.” The Scientific World Journal, Volume 2014, Article ID 698296.

Pielou EC (1961). “Segregation and symmetry in two-species populations as studied by nearest-neighbor relationships.” Journal of Ecology, 49(2), 255-269.

See Also

Znnsym and Xsq.nnsym

Examples

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)
Qsym.ct(ipd,cls)

Qsym.test(ipd,cls)
Qsym.test(Y,cls,is.ipd = FALSE)
Qsym.test(Y,cls,is.ipd = FALSE,method="max")

Qsym.test(ipd,cls,combine = FALSE)

#cls as a faqctor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
Qsym.test(ipd,fcls)
Qsym.test(Y,fcls,is.ipd = FALSE)

#############
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))

Qsym.test(ipd,cls)
Qsym.test(Y,cls,is.ipd = FALSE)