nnspat: A package for NN Methods and Their Use in Testing Spatial Patterns

Description

nnspat is a package for computation of spatial pattern tests based on NN relations and generation of various spatial patterns.

Details

The nnspat package contains the functions for segregation/association tests based on nearest neighbor contingency tables (NNCTs), and tests for species correspondence, NN symmetry and reflexivity based on the corresponding contingency tables and functions for generating patterns of segregation, association, uniformity, and various non-random labeling (non-RL) patterns for data in two (or more) dimensions. Applications include plant ecology, disease clustering in epidemiology, and so on. #' See (Dixon (1994); Ceyhan (2010, 2017)).

The nnspat functions

The nnspat functions can be grouped as Auxiliary Functions, NNCT Functions, SCCT Functions, RCT Functions, NN-Symmetry Functions, and the Pattern (Generation) Functions.

Auxiliary Functions

Contains the auxiliary functions used in NN methods, such as indices of NNs, number of shared NNs, Q, R and T values, and so on. In all these functions the data sets are either matrices or data frames.

NNCT Functions

Contains the functions for testing segregation/association using the NNCT. The types of the tests are cell-specific tests, class-specific tests, and overall tests of segregation. See (Ceyhan (2009, 2010)).

SCCT Functions

Contains the functions used for testing species correspondence using the NNCT. The types are NN self and self-sum tests and the overall test of species correspondence. See (Ceyhan (2018)).

RCT Functions

Contains the functions for testing reflexivity using the reflexivity contingency table (RCT). The types are NN self reflexivity and NN mixed-non reflexivity. See (Ceyhan and Bahadir (2017); Bahadir and Ceyhan (2018)).

Symmetry Functions

Contains the functions for testing NN symmetry using the NNCT and Q-symmetry contingency table. The types are NN symmetry and symmetry in shared NN structure. See (Ceyhan (2014)).

Pattern Functions

Contains the functions for generating and visualization of spatial patterns of segregation, association, uniformity clustering and non-RL. See (Ceyhan (2014, 2014)).

References

Bahadir S, Ceyhan E (2018). “On the Number of reflexive and shared nearest neighbor pairs in one-dimensional uniform data.” Probability and Mathematical Statistics, 38(1), 123-137.

Ceyhan E (2009). “Class-Specific Tests of Segregation Based on Nearest Neighbor Contingency Tables.” Statistica Neerlandica, 63(2), 149-182.

Ceyhan E (2010). “On the use of nearest neighbor contingency tables for testing spatial segregation.” Environmental and Ecological Statistics, 17(3), 247-282.

Ceyhan E (2010). “Exact Inference for Testing Spatial Patterns by Nearest Neighbor Contingency Tables.” Journal of Probability and Statistical Science, 8(1), 45-68.

Ceyhan E (2010). “New Tests of Spatial Segregation Based on Nearest Neighbor Contingency Tables.” Scandinavian Journal of Statistics, 37(1), 147-165.

Ceyhan E (2010). “Directional clustering tests based on nearest neighbour contingency tables.” Journal of Nonparametric Statistics, 22(5), 599-616.

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

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

Ceyhan E (2014). “Simulation and characterization of multi-class spatial patterns from stochastic point processes of randomness, clustering and regularity.” Stochastic Environmental Research and Risk Assessment (SERRA), 38(5), 1277-1306.

Ceyhan E (2017). “Cell-Specific and Post-hoc Spatial Clustering Tests Based on Nearest Neighbor Contingency Tables.” Journal of the Korean Statistical Society, 46(2), 219-245.

Ceyhan E (2018). “A contingency table approach based on nearest neighbor relations for testing self and mixed correspondence.” SORT-Statistics and Operations Research Transactions, 42(2), 125-158.

Ceyhan E, Bahadir S (2017). “Nearest Neighbor Methods for Testing Reflexivity.” Environmental and Ecological Statistics, 24(1), 69-108.

Dixon PM (1994). “Testing spatial segregation using a nearest-neighbor contingency table.” Ecology, 75(7), 1940-1948.

.onAttach start message

Description

.onAttach start message

Usage

.onAttach(libname, pkgname)

Arguments

Value

invisible()

.onLoad getOption package settings

Description

.onLoad getOption package settings

Usage

.onLoad(libname, pkgname)

Arguments

Value

invisible()

Examples

getOption("nnspat.name")

Expected Values of the Self Entries in a Species Correspondence Contingency Table (SCCT)

Description

Returns a vector of length k of expected values of the self entries (i.e., first column) in a species correspondence contingency table (SCCT) or the expected values of the diagonal entries N_{ii} in an NNCT. These expected values are valid under RL or CSR.

The argument ct can be either the NNCT or SCCT.

See also (Ceyhan (2018)).

Usage

EV.Nii(ct)

Arguments

Value

A vector of length k whose entries are the expected values of the self entries (i.e., first column) in a species correspondence contingency table (SCCT) or of the diagonal entries in an NNCT.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2018). “A contingency table approach based on nearest neighbor relations for testing self and mixed correspondence.” SORT-Statistics and Operations Research Transactions, 42(2), 125-158.

See Also

scct and EV.nnct

Examples

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)
ct

EV.Nii(ct)
ct<-scct(ipd,cls)
EV.Nii(ct)

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
ct<-nnct(ipd,fcls)

EV.Nii(ct)

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

EV.Nii(ct)
ct<-scct(ipd,cls)
EV.Nii(ct)

Expected Value for Cuzick & Edwards T_{comb} Test Statistic

Description

This function computes the expected value of Cuzick & Edwards T_{comb} test statistic in disease clustering, where T_{comb} is a linear combination of some T_k tests.

The argument, n_1, is the number of cases (denoted as n1 as an argument). The number of cases is denoted as n_1 to match the case-control class labeling, which is just the reverse of the labeling in Cuzick and Edwards (1990).

The argument klist is the vector of integers specifying the indices of the T_k values used in obtaining the T_{comb}.

The argument sig is the covariance matrix of the vector of T_k values used in Tcomb, and can be computed via the the covTcomb function.

See page 87 of (Cuzick and Edwards (1990)) for more details.

Usage

EV.Tcomb(n1, n, klist, sig)

Arguments

Value

Returns the expected value of the T_{comb} test statistic

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

Tcomb, and ZTcomb

Examples

n<-20  #or try sample(1:20,1) #try also n<-50, 100
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)

kl<-sample(1:5,3) #try also sample(1:5,2)
kl
sig<-covTcomb(Y,n1,kl)
EV.Tcomb(n1,n,kl,sig)

Expected Value of Cuzick and Edwards T_k^{inv} Test statistic

Description

This function computes the expected value of Cuzick and Edwards T_k^{inv} test statistic which is based on the sum of number of cases closer to each case than the k-th nearest control to the case.

The number of cases are denoted as n_1 (denoted as n1 as an argument) and number of controls as n_0 for both functions (denoted as n0 as an argument), to match the case-control class labeling, which is just the reverse of the labeling in Cuzick and Edwards (1990).

See the function ceTkinv for the details of the T_k^{inv} test.

See (Cuzick and Edwards (1990)) and references therein.

Usage

EV.Tkinv(n1, n0, k)

Arguments

Value

The expected value of Cuzick and Edwards T_k^{inv} test statistic for disease clustering

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

ceTkinv, ceTrun, and EV.Trun

Examples

n1<-20
n0<-25
k<-2 #try also 2, 3

EV.Tkinv(n1,n0,k)

EV.Tkinv(n1,n0,k=1)
EV.Trun(n1,n0)

Expected Values of the Cell Counts in NNCT

Description

Returns a matrix of same dimension as, ct, whose entries are the expected cell counts of the NNCT under RL or CSR. The class sizes given as the row sums of ct and the row and column names are inherited from ct.

See also (Dixon (1994); Ceyhan (2010)).

Usage

EV.nnct(ct)

Arguments

Value

A matrix of the expected values of cell counts in the NNCT.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2010). “On the use of nearest neighbor contingency tables for testing spatial segregation.” Environmental and Ecological Statistics, 17(3), 247-282.

Dixon PM (1994). “Testing spatial segregation using a nearest-neighbor contingency table.” Ecology, 75(7), 1940-1948.

See Also

nnct and EV.tct

Examples

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)

EV.nnct(ct)

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
ct<-nnct(ipd,fcls)
EV.nnct(ct)

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

EV.nnct(ct)

ct<-matrix(c(0,10,5,5),ncol=2)
EV.nnct(ct)

Expected Values of the Cell Counts in RCT

Description

Returns a matrix of same dimension as the RCT, rfct, whose entries are the expected cell counts of the RCT under RL or CSR.

See also (Ceyhan and Bahadir (2017)).

Usage

EV.rct(rfct, nvec)

Arguments

Value

A matrix of the expected values of cell counts in the RCT.

Author(s)

Elvan Ceyhan

References

Ceyhan E, Bahadir S (2017). “Nearest Neighbor Methods for Testing Reflexivity.” Environmental and Ecological Statistics, 24(1), 69-108.

See Also

rct, EV.nnct and EV.tct

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)

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

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
nvec<-as.numeric(table(fcls))
rfct<-rct(ipd,fcls)
EV.rct(rfct,nvec)

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

rfct<-rct(ipd,cls)
EV.rct(rfct,nvec)

Expected Values of the Types I-IV cell-specific tests

Description

Returns a matrix of same dimension as, ct, whose entries are the expected values of the T_{ij} values which are the Types I-IV cell-specific test statistics (i.e., T^I_{ij}-T^{IV}_{ij}) under RL or CSR. The row and column names are inherited from ct. The type argument specifies the type of the cell-specific test among the types I-IV tests.

See also (Ceyhan (2017)) and the references therein.

Usage

EV.tct(ct, type = "III")

Arguments

Value

A matrix of the expected values of Type I-IV cell-specific tests.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2017). “Cell-Specific and Post-hoc Spatial Clustering Tests Based on Nearest Neighbor Contingency Tables.” Journal of the Korean Statistical Society, 46(2), 219-245.

See Also

EV.tctI, tct, and EV.nnct

Examples

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)

EV.tct(ct,2)
EV.tct(ct,"II")
EV.tctI(ct)

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
ct<-nnct(ipd,fcls)
EV.tct(ct,2)

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

EV.tct(ct,2)

ct<-matrix(c(0,10,5,5),ncol=2)
EV.tct(ct,2)

Expected Values of the Type I cell-specific tests

Description

Returns a matrix of same dimension as, ct, whose entries are the expected values of the Type I cell-specific test statistics, T^I_{ij}. The row and column names are inherited from ct. These expected values are valid under RL or CSR.

See also (Ceyhan (2017)) and the references therein.

Usage

EV.tctI(ct)

Arguments

Value

A matrix of the expected values of Type I cell-specific tests.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2017). “Cell-Specific and Post-hoc Spatial Clustering Tests Based on Nearest Neighbor Contingency Tables.” Journal of the Korean Statistical Society, 46(2), 219-245.

See Also

EV.tct, tct and EV.nnct

Examples

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)

EV.tctI(ct)

Finding the index of the NN of a given point

Description

Returns the index (or indices) of the nearest neighbor(s) of subject i given data set or IPD matrix x. It will yield a vector if there are ties, and subject indices correspond to rows (i.e., rows 1:n ) if x is the data set and to rows or columns if x is the IPD matrix.

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.

Usage

NN(x, i, is.ipd = TRUE, ...)

Arguments

Value

Returns the index (indices) i.e., row number(s) of the NN of subject i

Author(s)

Elvan Ceyhan

See Also

kNN and NNsub

Examples

#3D data points
n<-20  #or try sample(1:20,1)
Y<-matrix(runif(3*n),ncol=3)
ipd<-ipd.mat(Y)
NN(ipd,1)
NN(Y,1,is.ipd = FALSE)
NN(ipd,5)
NN(Y,5,is.ipd = FALSE)
NN(Y,5,is.ipd = FALSE,method="max")

#1D data points
X<-as.matrix(runif(15)) # need to be entered as a matrix with one column
#(i.e., a column vector), hence X<-runif(5) would not work
ipd<-ipd.mat(X)
NN(ipd,1)
NN(ipd,5)

#with possible ties in the data
Y<-matrix(round(runif(30)*10),ncol=3)
ny<-nrow(Y)
ipd<-ipd.mat(Y)
for (i in 1:ny)
  cat(i,":",NN(ipd,i),"|",NN(Y,i,is.ipd = FALSE),"\n")

Distances between subjects and their NNs

Description

Returns the distances between subjects and their NNs. The output is an n \times 2 matrix where n is the data size and first column is the subject index and second column contains the corresponding distances to NN subjects.

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.

Usage

NNdist(x, is.ipd = TRUE, ...)

Arguments

Value

Returns an n \times 2 matrix where n is data size (i.e., number of subjects) and first column is the subject index and second column is the NN distances.

Author(s)

Elvan Ceyhan

See Also

kthNNdist, kNNdist, and NNdist2cl

Examples

#3D data points
n<-20  #or try sample(1:20,1)
Y<-matrix(runif(3*n),ncol=3)
ipd<-ipd.mat(Y)
NNdist(ipd)
NNdist(Y,is.ipd = FALSE)
NNdist(Y,is.ipd = FALSE,method="max")

#1D data points
X<-as.matrix(runif(5)) # need to be entered as a matrix with one column
#(i.e., a column vector), hence X<-runif(5) would not work
ipd<-ipd.mat(X)
NNdist(ipd)
NNdist(X,is.ipd = FALSE)

Distances between subjects from class i and their NNs from class j

Description

Returns the distances between subjects from class i and their nearest neighbors (NNs) from class j. The output is a list with first entry (nndist) being an n_i \times 3 matrix where n_i is the size of class i and first column is the subject index in class i, second column is the subject index in NN class j, and third column contains the corresponding distances of each class i subject to its NN among class j subjects. Class i is labeled as base class and class j is labeled as NN class.

The argument within.class.ind is a logical argument (default=FALSE) to determine the indexing of the class i subjects. If TRUE, index numbering of subjects is within the class, from 1 to class size (i.e., 1:n_i), according to their order in the original data; otherwise, index numbering within class is just the indices in the original data.

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.

Usage

NNdist2cl(x, i, j, lab, within.class.ind = FALSE, is.ipd = TRUE, ...)

Arguments

Value

Returns a list with three elements

Author(s)

Elvan Ceyhan

See Also

kthNNdist, kNNdist, and NNdist2cl

Examples

#3D data points
n<-20  #or try sample(1:20,1)
Y<-matrix(runif(3*n),ncol=3)
ipd<-ipd.mat(Y)
#two class case
clab<-sample(1:2,n,replace=TRUE) #class labels
table(clab)
NNdist2cl(ipd,1,2,clab)
NNdist2cl(Y,1,2,clab,is.ipd = FALSE)

NNdist2cl(ipd,1,2,clab,within = TRUE)

#three class case
clab<-sample(1:3,n,replace=TRUE) #class labels
table(clab)
NNdist2cl(ipd,2,1,clab)

#1D data points
n<-15
X<-as.matrix(runif(n))# need to be entered as a matrix with one column
#(i.e., a column vector), hence X<-runif(n) would not work
ipd<-ipd.mat(X)
#two class case
clab<-sample(1:2,n,replace=TRUE) #class labels
table(clab)
NNdist2cl(ipd,1,2,clab)
NNdist2cl(X,1,2,clab,is.ipd = FALSE)

Finding the index of the NN of a given point among a subset of points

Description

Returns the index (indices) of the nearest neighbor(s) of subject i (other than subject i) among the indices of points provided in the subsample ss using the given data set or IPD matrix x. The indices in ss determine the columns of the IPD matrix to be used in this function. It will yield a vector if there are ties, and subject indices correspond to rows (i.e., rows 1:n ) if x is the data set and to rows or columns if x is the IPD matrix.

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.

Usage

NNsub(ss, x, i, is.ipd = TRUE, ...)

Arguments

Value

Returns a list with the elements

Author(s)

Elvan Ceyhan

See Also

NN and kNN

Examples

#3D data points 
n<-20  #or try sample(1:20,1)
Y<-matrix(runif(3*n),ncol=3)
ipd<-ipd.mat(Y)
#indices of the subsample ss
ss<-sample(1:n,floor(n/2),replace=FALSE)
NNsub(ss,ipd,2)
NNsub(ss,Y,2,is.ipd = FALSE)
NNsub(ss,ipd,5)

#1D data points
n<-15
X<-as.matrix(runif(n))# need to be entered as a matrix with one column
#(i.e., a column vector), hence X<-runif(n) would not work
ipd<-ipd.mat(X)
#two class case
clab<-sample(1:2,n,replace=TRUE) #class labels
#indices of the subsample ss
ss<-sample(1:n,floor(n/2),replace=FALSE)
NNsub(ss,ipd,2)
NNsub(ss,ipd,5)

#with possible ties in the data
Y<-matrix(round(runif(60)*10),ncol=3)
ipd<-ipd.mat(Y)
ss<-sample(1:20,10,replace=FALSE) #class labels
NNsub(ss,ipd,2)
NNsub(ss,ipd,5)

Vector of Shared NNs and Number of Reflexive NNs

Description

Returns the Qvec and R where Qvec=(Q_0,Q_1,\ldots) with Q_j is the number of points shared as a NN by j other points i.e., number of points that are NN of i points, for i=0,1,2,\ldots and R is the number of reflexive pairs where points A and B are reflexive iff they are NN to each other.

Usage

Ninv(x, is.ipd = TRUE, ...)

Arguments

Value

Returns a list with two elements

Author(s)

Elvan Ceyhan

See Also

Qval, Qvec, sharedNN, Rval, and QRval

Examples

#3D data points
n<-20  #or try sample(1:20,1)
Y<-matrix(runif(3*n),ncol=3)
ipd<-ipd.mat(Y)
W<-Wmat(ipd)
sharedNN(W)
Qvec(W)
Ninv(ipd)
Ninv(Y,is.ipd = FALSE)
Ninv(Y,is.ipd = FALSE,method="max")

#1D data points
n<-15
X<-as.matrix(runif(n))# need to be entered as a matrix with one column 
#(i.e., a column vector), hence X<-runif(n) would not work
ipd<-ipd.mat(X)
W<-Wmat(ipd)
sharedNN(W)
Qvec(W)
Ninv(ipd)

#with possible ties in the data
Y<-matrix(round(runif(30)*10),ncol=3)
ny<-nrow(Y)
ipd<-ipd.mat(Y)
W<-Wmat(ipd)
sharedNN(W)
Qvec(W)
Ninv(ipd)

N_t Value (found with the definition formula)

Description

This function computes the N_t value which is required in the computation of the asymptotic variance of Cuzick and Edwards T_k test. Nt is defined on page 78 of (Cuzick and Edwards (1990)) as follows. 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).

This function yields the same result as the asyvarTk and varTk functions with $Nt inserted at the end.

See (Cuzick and Edwards (1990)) for more details.

Usage

Nt.def(a)

Arguments

Value

Returns 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

asyvarTk, varTk, and varTkaij

Examples

n<-20  #or try sample(1:20,1)
Y<-matrix(runif(3*n),ncol=3)
k<-2 #try also 2,3
a<-aij.mat(Y,k)
Nt.def(a)

N_{tkl} Value

Description

This function computes the N_{tkl} value which is required in the computation of the exact and asymptotic variance of Cuzick and Edwards T_{comb} test, which is a linear combination of some T_k tests. N_{tkl} is defined on page 80 of (Cuzick and Edwards (1990)) as follows. Let a_{ij}(k) be 1 if j is a k NN of i and zero otherwise and N_t(k,l) = \sum \sum_{i \ne m}\sum a_{ij}(k) a_{mj}(l).

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.

See (Cuzick and Edwards (1990)) for more details.

Usage

Ntkl(dat, k, l, nonzero.mat = TRUE, ...)

Arguments

Value

Returns the N_{tkl} value. 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

asycovTkTl, and covTkTl

Examples

n<-20  #or try sample(1:20,1)
Y<-matrix(runif(3*n),ncol=3)
k<-1 #try also 2,3 or sample(1:5,1)
l<-1 #try also 2,3 or sample(1:5,1)
c(k,l)

Ntkl(Y,k,l)
Ntkl(Y,k,l,nonzero.mat = FALSE)
Ntkl(Y,k,l,method="max")

Number of Shared and Reflexive NNs

Description

Returns the Q and R values where Q is the number of points shared as a NN by other points i.e., number of points that are NN of other points (which occurs when two or more points share a NN, for data in any dimension) and R is the number of reflexive pairs where A and B are reflexive iff they are NN to each other.

These quantities are used, e.g., in computing the variances and covariances of the entries of the nearest neighbor contingency tables used for Dixon's tests and other NNCT tests.

Usage

QRval(njr)

Arguments

Value

A list with two elements

Author(s)

Elvan Ceyhan

See Also

Qval, Qvec, sharedNN, Rval, and Ninv

Examples

#3D data points
n<-20  #or try sample(1:20,1)
Y<-matrix(runif(3*n),ncol=3)
ipd<-ipd.mat(Y)
ninv<-Ninv(ipd)
QRval(ninv)
W<-Wmat(ipd)
Qvec(W)$q

#1D data points
n<-15
X<-as.matrix(runif(n))# need to be entered as a matrix with one column 
#(i.e., a column vector), hence X<-runif(n) would not work
ipd<-ipd.mat(X)
ninv<-Ninv(ipd)
QRval(ninv)
W<-Wmat(ipd)
Qvec(W)$q

#with possible ties in the data
Y<-matrix(round(runif(30)*10),ncol=3)
ny<-nrow(Y)
ipd<-ipd.mat(Y)
ninv<-Ninv(ipd)
QRval(ninv)
W<-Wmat(ipd)
Qvec(W)$q

Q-symmetry Contingency Table (QCT)

Description

Returns the k \times 3 contingency table for Q-symmetry (i.e., Q-symmetry contingency table (QCT)) given the IPD matrix or data set x where k is the number of classes in the data set. Each row in the QCT is the vector of number of points with shared NNs, Q_i=(Q_{i0},Q_{i1},Q_{i2}) where Q_{ij} is the number of class i points that are NN to class j points for j=0,1 and Q_{i2} is the number of class i points that are NN to class j or more points. That is, this function pools the cells 3 or larger together for k classes, 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. Rows the QCT are labeled with the corresponding class labels.

Q-symmetry is also equivalent to Pielou's second type of NN symmetry or the symmetry in the shared NN structure for all classes.

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.

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

Usage

Qsym.ct(x, lab, is.ipd = TRUE, ...)

Arguments

Value

Returns the k \times 3 QCT where k is the number of classes in the data set.

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

sharedNNmc, Qsym.test and scct

Examples

n<-20  #or try sample(1:20,1)
Y<-matrix(runif(n*3),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.ct(Y,cls,is.ipd = FALSE)
Qsym.ct(Y,cls,is.ipd = FALSE,method="max")

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
Qsym.ct(ipd,fcls)

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

Qsym.ct(ipd,cls)

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

Value

A list with the elements

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)

Skewness of Cuzick and Edwards T_k Test statistic

Description

This function estimates the skewness of Cuzick and Edwards T_k test statistic under the RL hypothesis. Skewness of a random variable T is defined as E(T-\mu)^3/(E(T-\mu)^2)^{1.5} where \mu=E T.

Skewness is used for Tango's correction to Cuzick and Edwards kNN test statistic, T_k. Tango's correction is a chi-square approximation, and its degrees of freedom is estimated using the skewness estimate (see page 121 of Tango (2007)).

The argument, n_1, is the number of cases (denoted as n1 as an argument) and k is the number of NNs considered in T_k test statistic. The argument of the function is the A_{ij} matrix, a, which is the output of the function aij.mat. However, inside the function we symmetrize the matrix a as b <- (a+a^t)/2, to facilitate the formulation.

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).

Usage

SkewTk(n1, k, a)

Arguments

Value

The skewness of Cuzick and Edwards T_k test statistic for disease clustering

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.

Tango T (2007). “A class of multiplicity adjusted tests for spatial clustering based on case-control point data.” Biometrics, 63, 119-127.

See Also

ceTk, EV.Tk, and varTk

Examples

n<-20  #or try sample(1:20,1)
Y<-matrix(runif(3*n),ncol=3)
cls<-sample(0:1,n,replace = TRUE)
n1<-sum(cls==1)

k<-sample(1:5,1) # try also 3, 5, sample(1:5,1)
k
a<-aij.mat(Y,k)

SkewTk(n1,k,a)

Cuzick & Edwards Tcomb Test Statistic

Description

This function computes the value of Cuzick & Edwards T_{comb} test statistic in disease clustering, where T_{comb} is a linear combination of some T_k tests.

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.

The argument klist is the vector of integers specifying the indices of the T_k values used in obtaining the T_{comb}.

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.cov (default=FALSE) is for using the asymptotic covariance or the exact (i.e., finite sample) covariance for the vector of T_k values used in Tcomb in the standardization of T_{comb}. If asy.cov=TRUE, the asymptotic covariance is used, otherwise the exact covariance is used.

See page 87 of (Cuzick and Edwards (1990)) for more details.

Usage

Tcomb(
  dat,
  cc.lab,
  klist,
  case.lab = NULL,
  nonzero.mat = TRUE,
  asy.cov = FALSE,
  ...
)

Arguments

Value

Returns the value of the T_{comb} test statistic

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, EV.Tcomb, and ZTcomb

Examples

n<-20  #or try sample(1:20,1) #try also n<-50, 100
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)

kl<-sample(1:5,3) #try also sample(1:5,2)
kl
Tcomb(Y,cls,kl)
Tcomb(Y,cls,kl,method="max")
Tcomb(Y,cls+1,kl,case.lab=2)
Tcomb(Y,cls,kl,nonzero.mat = FALSE)
Tcomb(Y,cls,kl,asy.cov = TRUE)

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
Tcomb(Y,fcls,kl,case.lab="a")

T value in NN structure

Description

Returns the 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. Note that in the NN digraph, T+R is the sum of the indegrees of the points in the reflexive pairs.

This quantity (together with Q and R) is used in computing the variances and covariances of the entries of the reflexivity contingency table. See (Ceyhan and Bahadir (2017)) for further details.

Usage

Tval(W, R)

Arguments

Value

Returns the T value. See the description above for the details of this quantity.

Author(s)

Elvan Ceyhan

See Also

Qval, Qvec, sharedNN, and Rval

Examples

#3D data points
n<-10
Y<-matrix(runif(3*n),ncol=3)
ipd<-ipd.mat(Y)
W<-Wmat(ipd)
R<-Rval(W)
Tval(W,R)

#1D data points
X<-as.matrix(runif(15)) # need to be entered as a matrix with one column
#(i.e., a column vector), hence X<-runif(5) would not work
ipd<-ipd.mat(X)
W<-Wmat(ipd)
R<-Rval(W)
Tval(W,R)

#with ties=TRUE in the data
Y<-matrix(round(runif(30)*10),ncol=3)
ipd<-ipd.mat(Y)
W<-Wmat(ipd,ties=TRUE)
R<-Rval(W)
Tval(W,R)

The incidence matrix W for the NN digraph

Description

Returns the W=(w_ij) matrix which is used to compute Q, R and T values in the NN structure. w_{ij}=I( point j is a NN of point i)) i.e., w_{ij}=1 if point j is a NN of point i and 0 otherwise.

The argument ties is a logical argument (default=FALSE) to take ties into account or not. If TRUE the function takes ties into account by making w_{ij}=1/m if point j is a NN of point i and there are m tied NNs and 0 otherwise. If FALSE, w_{ij}=1 if point j is a NN of point i and 0 otherwise. The matrix W is equivalent to A=(a_{ij}) matrix with k=1, i.e., Wmat(X)=aij.mat(X,k=1).

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.

Usage

Wmat(x, ties = FALSE, is.ipd = TRUE, ...)

Arguments

Value

The incidence matrix W=(w_ij) where w_{ij}=I( point j is a NN of point i)), i.e., w_{ij}=1 if point j is a NN of point i and 0 otherwise.

Author(s)

Elvan Ceyhan

See Also

aij.mat, aij.nonzero, and aij.theta

Examples

n<-3
X<-matrix(runif(3*n),ncol=3)
ipd<-ipd.mat(X)
Wmat(ipd)
Wmat(X,is.ipd = FALSE)

n<-5
Y<-matrix(runif(3*n),ncol=3)
ipd<-ipd.mat(Y)
Wmat(ipd)
Wmat(Y,is.ipd = FALSE)
Wmat(Y,is.ipd = FALSE,method="max")

Wmat(Y,is.ipd = FALSE)
aij.mat(Y,k=1)

#1D data points
X<-as.matrix(runif(5)) # need to be entered as a matrix with one column
#(i.e., a column vector), hence X<-runif(5) would not work
ipd<-ipd.mat(X)
Wmat(ipd)
Wmat(X,is.ipd = FALSE)

#with ties=TRUE in the data
Y<-matrix(round(runif(15)*10),ncol=3)
ipd<-ipd.mat(Y)
Wmat(ipd,ties=TRUE)
Wmat(Y,ties=TRUE,is.ipd = FALSE)

Chi-square Approximation to Cuzick and Edwards T_k Test statistic

Description

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

This approximation is suggested by Tango (2007) since T_k statistic had high skewness rendering the normal approximation less efficient. The chi-square approximation is as follows: \frac{T_k- ET_k}{\sqrt{Var T_k}} \approx \frac{\chi^2_\nu-\nu}{\sqrt{2 \nu}} where \chi^2_\nu is a chi-square random variable with \nu df, and \nu=8/skewnees(T_k) (see SkewTk for the skewness).

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.

The logical argument nonzero.mat (default=FALSE) is for using the A matrix if FALSE or just the matrix of nonzero locations in the A matrix (if TRUE).

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 (Tango (2007)) and the references therein.

Usage

Xsq.ceTk(
  dat,
  cc.lab,
  k,
  case.lab = NULL,
  nonzero.mat = TRUE,
  asy.var = FALSE,
  ...
)

Arguments

Value

A list with the elements

Author(s)

Elvan Ceyhan

References

Tango T (2007). “A class of multiplicity adjusted tests for spatial clustering based on case-control point data.” Biometrics, 63, 119-127.

See Also

ceTk, ZceTk and SkewTk

Examples

set.seed(123)
n<-20
Y<-matrix(runif(3*n),ncol=3)
cls<-sample(0:1,n,replace = TRUE)

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

Xsq.ceTk(Y,cls,k)
Xsq.ceTk(Y,cls,k,nonzero.mat=FALSE)
Xsq.ceTk(Y,cls+1,k,case.lab = 2)
Xsq.ceTk(Y,cls,k,method="max")

Xsq.ceTk(Y,cls,k,asy.var=TRUE)

Overall NN Symmetry Test with Chi-square Approximation

Description

An object of class "Chisqtest" performing the hypothesis test of equality of the expected values of the off-diagonal cell counts (i.e., entries) under RL or CSR in the NNCT for k \ge 2 classes. That is, the test performs Dixon's or Pielou's (first type of) overall NN symmetry test which is appropriate (i.e., have the appropriate asymptotic sampling distribution) for completely mapped data or for sparsely sample data, respectively. (See Pielou (1961); Dixon (1994); Ceyhan (2014) for more detail).

The type="dixon" refers to Dixon's overall NN symmetry test and type="pielou" refers to Pielou's first type of overall NN symmetry test. The symmetry test is based on the chi-squared approximation of the corresponding quadratic form and type="dixon" yields an extension of Dixon's NN symmetry test, which is extended by Ceyhan (2014) and type="pielou" yields Pielou's overall NN symmetry test.

The function yields the test statistic, p-value and df which is k(k-1)/2, description of the alternative with the corresponding null values (i.e., expected values) of differences of the off-diagonal entries,(which is 0 for this function) and also the sample estimates (i.e., observed values) of absolute differences of the off-diagonal entries of NNCT (in the upper-triangular form). The functions also provide names of the test statistics, the description of the test and the data set used.

The null hypothesis is that all E(N_{ij})=E(N_{ji}) for i \ne j in the k \times k NNCT (i.e., symmetry in the mixed NN structure) for k \ge 2. In the output, if if type="pielou", the test statistic, p-value and the df are valid only for (properly) sparsely sampled data.

See also (Pielou (1961); Dixon (1994); Ceyhan (2014)) and the references therein.

Usage

Xsq.nnsym(dat, lab, type = "dixon", ...)

Arguments

Value

A list with the elements

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.

Dixon PM (1994). “Testing spatial segregation using a nearest-neighbor contingency table.” Ecology, 75(7), 1940-1948.

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.ss, Znnsym.dx and Znnsym2cl

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))

Xsq.nnsym(Y,cls)
Xsq.nnsym(Y,cls,method="max")
Xsq.nnsym(Y,cls,type="pielou")

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))

Xsq.nnsym(Y,fcls)
Xsq.nnsym(Y,fcls,type="pielou")

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

Xsq.nnsym(Y,cls)
Xsq.nnsym(Y,cls,type="pielou")

Z-test for Cuzick and Edwards T_{comb} statistic

Description

An object of class "htest" performing a z-test for Cuzick and Edwards T_{comb} test statisticin disease clustering, where T_{comb} is a linear combination of some T_k tests.

For disease clustering, Cuzick and Edwards (1990) developed a k-NN test T_k based on number of cases among k NNs of the case points, and also proposed a test combining various T_k tests, denoted as T_{comb}.

See page 87 of (Cuzick and Edwards (1990)) for more details.

Under RL of n_1 cases and n_0 controls to the given locations in the study region, T_{comb} approximately has N(E[T_{comb}],Var[T_{comb}]) 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.

The argument klist is the vector of integers specifying the indices of the T_k values used in obtaining the T_{comb}.

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.cov (default=FALSE) is for using the asymptotic covariance or the exact (i.e., finite sample) covariance for the vector of T_k values used in Tcomb in the standardization of T_{comb}. If asy.cov=TRUE, the asymptotic covariance is used, otherwise the exact covariance is used.

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

Usage

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

Arguments

Value

A list with the elements

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

Tcomb, EV.Tcomb, and covTcomb

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))

kl<-sample(1:5,3) #try also sample(1:5,2)
ZTcomb(Y,cls,kl)
ZTcomb(Y,cls,kl,method="max")

ZTcomb(Y,cls,kl,nonzero.mat=FALSE)
ZTcomb(Y,cls+1,kl,case.lab = 2,alt="l")
ZTcomb(Y,cls,kl,conf=.9,alt="g")
ZTcomb(Y,cls,kl,asy=TRUE,alt="g")

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
ZTcomb(Y,fcls,kl,case.lab="a")

Z-test for Cuzick and Edwards T_{run} statistic

Description

An object of class "htest" performing a z-test for Cuzick and Edwards T_{run} test statistic which is based on the number of consecutive cases from the cases in the data under RL or CSR independence.

Under RL of n_1 cases and n_0 controls to the given locations in the study region, T_{run} approximately has N(E[T_{run}],Var[T_{run}]) distribution for large n.

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.

The logical argument var.sim (default=FALSE) is for using the simulation estimated variance or the exact variance for the variance of T_{run} in its standardization. If var.sim=TRUE, the simulation estimated variance is used for Var[T_{run}] (see varTrun.sim), otherwise the exact variance (see varTrun) is used. Moreover, when var.sim=TRUE, the argument Nvar.sim represents the number of resamplings (without replacement) in the RL scheme, with default being 1000.

The function varTrun might take a very long time when data size is large (even larger than 50); in this case, it is recommended to use var.sim=TRUE in this function.

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

Usage

ZTrun(
  dat,
  cc.lab,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95,
  case.lab = NULL,
  var.sim = FALSE,
  Nvar.sim = 1000,
  ...
)

Arguments

Value

A list with the elements

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

ceTrun, ZceTk, and ZTcomb

Examples

n<-20  #or try sample(1:20,1) #try also 40, 50, 60
set.seed(123)
Y<-matrix(runif(3*n),ncol=3)
cls<-sample(0:1,n,replace = TRUE)  #or try cls<-rep(0:1,c(10,10))

ZTrun(Y,cls)
ZTrun(Y,cls,method="max")
ZTrun(Y,cls,var.sim=TRUE)
ZTrun(Y,cls+1,case.lab = 2,alt="l") #try also ZTrun(Y,cls,conf=.9,alt="g")

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
ZTrun(Y,fcls,case.lab="a")

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

Value

A list with the elements

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")

NN Symmetry Test with Normal Approximation

Description

An object of class "cellhtest" performing hypothesis test of equality of the expected values of the off-diagonal cell counts (i.e., entries) for each pair i,j of classes under RL or CSR in the NNCT for k \ge 2 classes. That is, the test performs Dixon's or Pielou's (first type of) NN symmetry test which is appropriate (i.e., have the appropriate asymptotic sampling distribution) for completely mapped data or for sparsely sample data, respectively. (See Pielou (1961); Dixon (1994); Ceyhan (2014) for more detail).

The type="dixon" refers to Dixon's NN symmetry test and type="pielou" refers to Pielou's first type of NN symmetry test. The symmetry test is based on the normal approximation of the difference of the off-diagonal entries in the NNCT and are due to Pielou (1961); Dixon (1994).

The function yields a contingency table of the test statistics, p-values for the corresponding alternative, expected values (i.e., null value(s)), lower and upper confidence levels and sample estimate for the N_{ij}-N_{ji} values for i \ne j (all in the upper-triangular form except for the null value, which is 0 for all pairs) and also names of the test statistics, estimates, null values, the description of the test, and the data set used.

The null hypothesis is that all E(N_{ij})=E(N_{ji}) for i \ne j in the k \times k NNCT (i.e., symmetry in the mixed NN structure) for k \ge 2. In the output, if if type="pielou", the test statistic, p-value and the lower and upper confidence limits are valid only for (properly) sparsely sampled data.

See also (Pielou (1961); Dixon (1994); Ceyhan (2014)) and the references therein.

Usage

Znnsym(
  dat,
  lab,
  type = "dixon",
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95,
  ...
)

Arguments

Value

A list with the elements

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.

Dixon PM (1994). “Testing spatial segregation using a nearest-neighbor contingency table.” Ecology, 75(7), 1940-1948.

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.ss.ct, Znnsym.ss, Znnsym.dx.ct, Znnsym.dx and Znnsym2cl

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))

Znnsym(Y,cls)
Znnsym(Y,cls,method="max")
Znnsym(Y,cls,type="pielou")
Znnsym(Y,cls,type="pielou",method="max")

Znnsym(Y,cls,alt="g")
Znnsym(Y,cls,type="pielou",alt="g")

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
Znnsym(Y,fcls)

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

Znnsym(Y,cls)
Znnsym(Y,cls,type="pielou")

NN Symmetry Test with Normal Approximation for Two Classes

Description

An object of class "htest" performing hypothesis test of equality of the expected value of the off-diagonal cell counts (i.e., entries) under RL or CSR in the NNCT for k=2 classes. That is, the test performs Dixon's or Pielou's (first type of) NN symmetry test which is appropriate (i.e., have the appropriate asymptotic sampling distribution) for completely mapped data and for sparsely sample data, respectively. (See Ceyhan (2014) for more detail).

The symmetry test is based on the normal approximation of the difference of the off-diagonal entries in the NNCT and are due to Pielou (1961); Dixon (1994).

The type="dixon" refers to Dixon's NN symmetry test and type="pielou" refers to Pielou's first type of NN symmetry test.

The function yields the test statistic, p-value for the corresponding alternative, the confidence interval, estimate and null value for the parameter of interest (which is the difference of the off-diagonal entries in the NNCT), and method and name of the data set used.

The null hypothesis is that all E(N_{12})=E(N_{21}) in the 2 \times 2 NNCT (i.e., symmetry in the mixed NN structure).

See also (Pielou (1961); Dixon (1994); Ceyhan (2014)) and the references therein.

Usage

Znnsym2cl(
  dat,
  lab,
  type = "dixon",
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)

Arguments

Value

A list with the elements

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.

Dixon PM (1994). “Testing spatial segregation using a nearest-neighbor contingency table.” Ecology, 75(7), 1940-1948.

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

Znnsym2cl.ss.ct, Znnsym2cl.ss, Znnsym2cl.dx.ct, Znnsym2cl.dx, Znnsym.ss.ct, Znnsym.ss, Znnsym.dx.ct, Znnsym.dx, Znnsym.dx.ct, Znnsym.dx and Znnsym

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))

Znnsym2cl(Y,cls)
Znnsym2cl(Y,cls,type="pielou")

Znnsym2cl(Y,cls,alt="g")
Znnsym2cl(Y,cls,type="pielou",alt="g")

Closeness or Proximity Matrix for Tango's Spatial Clustering Tests

Description

This function computes the A=a_{ij}(\theta) matrix useful in calculations for Tango's test T(\theta) for spatial (disease) clustering (see Eqn (2) of Tango (2007). Here, A=a_{ij}(\theta) is any matrix of a measure of the closeness between two points i and j with aii = 0 for all i = 1, \ldots,n, and \theta = (\theta_1,\ldots,\theta_p)^t denotes the unknown parameter vector related to cluster size and \delta = (\delta_1,\ldots,\delta_n)^t, where \delta_i=1 if z_i is a case and 0 otherwise. The test is then

T(\theta)=\sum_{i=1}^n\sum_{j=1}^n\delta_i \delta_j a_{ij}(\theta)=\delta^t A(\theta) \delta

where A=a_{ij}(\theta).

T(\theta) becomes Cuzick and Edwards T_k tests statistic (Cuzick and Edwards (1990)), if a_{ij}=1 if z_j is among the kNNs of z_i and 0 otherwise. In this case \theta=k and aij.theta becomes aij.mat (more specifically, aij.mat(dat,k) and aij.theta(dat,k,model="NN").

In Tango's exponential clinal model (Tango (2000)), a_{ij}=\exp\left(-4 \left(\frac{d_{ij}}{\theta}\right)^2\right) if i \ne j and 0 otherwise, where \theta is a predetermined scale of cluster such that any pair of cases far apart beyond the distance \theta cannot be considered as a cluster and d_{ij} denote the Euclidean distance between two points i and j.

In the exponential model (Tango (2007)), a_{ij}=\exp\left(-\frac{d_{ij}}{\theta}\right) if i \ne j and 0 otherwise, where \theta and d_{ij} are as above.

In the hot-spot model (Tango (2007)), a_{ij}=1 if d_{ij} \le \theta and i \ne j and 0 otherwise, where \theta and d_{ij} are as above.

The argument model has four options, NN, exp.clinal, exponential, and hot.spot, with exp.clinal being the default. And the theta argument specifies the scale of clustering or the clustering parameter in the particular spatial disease clustering model.

See also (Tango (2007)) and the references therein.

Usage

aij.theta(dat, theta, model = "exp.clinal", ...)

Arguments

Value

The A=a_{ij}(\theta) matrix useful in calculations for Tango's test T(\theta).

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.

Tango T (2000). “A test for spatial disease clustering adjusted for multiple testing.” Statistics in Medicine, 19, 191-204.

Tango T (2007). “A class of multiplicity adjusted tests for spatial clustering based on case-control point data.” Biometrics, 63, 119-127.

See Also

aij.mat, aij.nonzero and ceTk

Examples

n<-20  #or try sample(1:20,1)
Y<-matrix(runif(3*n),ncol=3)
k<-3#1 #try also 2,3

#aij for CE's Tk
Aij<-aij.theta(Y,k,model = "NN")
Aij2<-aij.mat(Y,k)
sum(abs(Aij-Aij2)) #check equivalence of aij.theta and aij.mat with model="NN"

Aij<-aij.theta(Y,k,method="max")
Aij2<-aij.mat(Y,k)
range(Aij-Aij2)

theta=.2
aij.theta(Y,theta,model = "exp.clinal")
aij.theta(Y,theta,model = "exponential")
aij.theta(Y,theta,model = "hot.spot")

Asymptotic Covariance between T_k and T_l Values

Description

This function computes the asymptotic covariance between T_k and T_l values which is used in the computation of the asymptotic variance of Cuzick and Edwards T_{comb} test, which is a linear combination of some T_k tests. The limit is as n_1 goes to infinity.

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) in the computations.

See page 80 of (Cuzick and Edwards (1990)) for more details.

Usage

asycovTkTl(dat, n1, k, l, nonzero.mat = TRUE, ...)

Arguments

Value

Returns the asymptotic covariance between T_k and T_l values.

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

covTkTl, covTcomb, and Ntkl

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

asycovTkTl(Y,n1,k,l)
asycovTkTl(Y,n1,k,l,nonzero.mat = FALSE)
asycovTkTl(Y,n1,k,l,method="max")

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

Value

A list with the elements

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")

pdf of the Bivariate Normal Distribution

Description

Computes the value of the probability density function (i.e., density) of the bivariate normal distribution at the specified point X, with mean mu and standard deviations of the first and second components being s_1 and s_2 (denoted as s1 and s2 in the arguments of the function, respectively) and correlation between them being rho (i.e., the covariance matrix is \Sigma=S where S_{11}=s_1^2, S_{22}=s_2^2, S_{12}=S_{21}=s_1 s_2 rho).

Usage

bvnorm.pdf(X, mu = c(0, 0), s1 = 1, s2 = 1, rho = 0)

Arguments

Value

The value of the probability density function (i.e., density) of the bivariate normal distribution at the specified point X, with mean mu and standard deviations of the first and second components being s_1 and s_2 and correlation between them being rho.

Author(s)

Elvan Ceyhan

See Also

mvrnorm

Examples

mu<-c(0,0)
s1<-1
s2<-1
rho<-.5

n<-5
Xp<-cbind(runif(n),runif(n))
bvnorm.pdf(Xp,mu,s1,s2,rho)

Cuzick and Edwards T_k Test statistic

Description

This function computes 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 based on number of cases among k NNs of the case points. Let z_i be the i^{th} point and d_i^k be the number cases among k NNs of z_i. Then Cuzick-Edwards' k-NN test is T_k=\sum_{i=1}^n \delta_i d_i^k, where \delta_i=1 if z_i is a case, and 0 if z_i is a control.

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).

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

Usage

ceTk(dat, cc.lab, k = 1, case.lab = NULL, ...)

Arguments

Value

Cuzick and Edwards T_k test statistic for disease clustering

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

Tcomb, seg.ind, Pseg.coeff and ceTkinv

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))

ceTk(Y,cls)
ceTk(Y,cls,method="max")
ceTk(Y,cls,k=3)
ceTk(Y,cls+1,case.lab = 2)

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
ceTk(Y,fcls,case.lab="a") #try also ceTk(Y,fcls)

#############
n<-40
Y<-matrix(runif(3*n),ncol=3)
cls<-sample(1:4,n,replace = TRUE)  # here ceTk(Y,cls) gives an error message

Cuzick and Edwards T_k^{inv} Test statistic

Description

This function computes Cuzick and Edwards T_k^{inv} test statistic based on the sum of number of cases closer to each case than the k-th nearest control to the case.

T_k^{inv} test statistic is an extension of the run length test allowing a fixed number of controls in the run sequence.

T_k^{inv} test statistic is defined as T_k^{inv}=\sum_{i=1}^n \delta_i \nu_i^k where \delta_i=1 if z_i is a case, and 0 if z_i is a control and \nu_i^k is the number of cases closer to the index case than the k nearest control, i.e., number of cases encountered beginning at z_i until k-th control is encountered.

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.

Usage

ceTkinv(dat, k, cc.lab, case.lab = NULL, ...)

Arguments

Value

A list with two elements

Author(s)

Elvan Ceyhan

References

There are no references for Rd macro ⁠\insertAllCites⁠ on this help page.

See Also

ceTrun, ceTk, and Tcomb

Examples

n<-20
Y<-matrix(runif(3*n),ncol=3)
cls<-sample(0:1,n,replace = TRUE)  #or try cls<-rep(0:1,c(10,10))
cls
k<-2 #also try 3,4

ceTkinv(Y,k,cls)
ceTkinv(Y,k,cls+1,case.lab = 2)
ceTkinv(Y,k,cls,method="max")

ceTrun(Y,cls)
ceTkinv(Y,k=1,cls)

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
ceTkinv(Y,k,fcls,case.lab="a") #try also ceTrun(Y,fcls)

#############
n<-40
Y<-matrix(runif(3*n),ncol=3)
cls<-sample(1:4,n,replace = TRUE)  #here ceTkinv(Y,k,cls) #gives error

Cuzick and Edwards T_{run} Test statistic

Description

This function computes Cuzick and Edwards T_{run} test statistic based on the sum of the number of successive cases from each cases until a control is encountered in the data for detecting rare large clusters.

T_{run} test statistic is defined as T_{run}=\sum_{i=1}^n \delta_i d_i^r where \delta_i=1 if z_i is a case, and 0 if z_i is a control and d_i^r is the number successive cases encountered beginning at z_i until a control is encountered.

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.

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

Usage

ceTrun(dat, cc.lab, case.lab = NULL, ...)

Arguments

Value

A list with two elements

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, Tcomb and ceTkinv

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))

ceTrun(Y,cls)
ceTrun(Y,cls,method="max")
ceTrun(Y,cls+1,case.lab = 2)

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
ceTrun(Y,fcls,case.lab="a") #try also ceTrun(Y,fcls)

#############
n<-40
Y<-matrix(runif(3*n),ncol=3)
cls<-sample(1:4,n,replace = TRUE)  #here ceTrun(Y,cls) #gives an error message

Entries for the Types I-IV cell-specific tests

Description

Returns a matrix of same dimension as, ct, whose entries are the values of the Types I-IV cell-specific test statistics, T^I_{ij}-T^{IV}_{ij}. The row and column names are inherited from ct. The type argument specifies the type of the cell-specific test among the types I-IV tests. Equivalent to the function tct in this package.

See also (Ceyhan (2017)) and the references therein.

Usage

cellsTij(ct, type = "III")

Arguments

Value

A matrix of the values of Type I-IV cell-specific tests

Author(s)

Elvan Ceyhan

References

Ceyhan E (2017). “Cell-Specific and Post-hoc Spatial Clustering Tests Based on Nearest Neighbor Contingency Tables.” Journal of the Korean Statistical Society, 46(2), 219-245.

See Also

tct and nnct

Examples

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)
ct

type.lab<-c("I","II","III","IV")
for (i in 1:4)
{ print(paste("T_ij values for cell specific tests for type",type.lab[i]))
  print(cellsTij(ct,i))
}

cellsTij(ct,"II")

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
ct<-nnct(ipd,fcls)
cellsTij(ct,2)

#############
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)
cellsTij(ct,2)

ct<-matrix(c(0,10,5,5),ncol=2)
cellsTij(ct,2)

Covariance Matrix of the Cell Counts in an NNCT

Description

Returns the covariance matrix of cell counts N_{ij} for i,j=1,\ldots,k in the NNCT, ct. The covariance matrix is of dimension k^2 \times k^2 and its entries are cov(N_{ij},N_{kl}) when N_{ij} values are by default corresponding to the row-wise vectorization of ct. If byrow=FALSE, the column-wise vectorization of ct is used. These covariances are valid under RL or conditional on Q and R under CSR.

See also (Dixon (1994, 2002); Ceyhan (2010, 2017)).

Usage

cov.nnct(ct, varN, Q, R, byrow = TRUE)

Arguments

Value

The k^2 \times k^2 covariance matrix of cell counts N_{ij} for i,j=1,\ldots,k in the NNCT, ct

Author(s)

Elvan Ceyhan

References

Ceyhan E (2010). “On the use of nearest neighbor contingency tables for testing spatial segregation.” Environmental and Ecological Statistics, 17(3), 247-282.

Ceyhan E (2017). “Cell-Specific and Post-hoc Spatial Clustering Tests Based on Nearest Neighbor Contingency Tables.” Journal of the Korean Statistical Society, 46(2), 219-245.

Dixon PM (1994). “Testing spatial segregation using a nearest-neighbor contingency table.” Ecology, 75(7), 1940-1948.

Dixon PM (2002). “Nearest-neighbor contingency table analysis of spatial segregation for several species.” Ecoscience, 9(2), 142-151.

See Also

covNrow2col, cov.tct, and cov.nnsym

Examples

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)

cov.nnct(ct,varN,Qv,Rv)
cov.nnct(ct,varN,Qv,Rv,byrow=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))
ct<-nnct(ipd,cls)

W<-Wmat(ipd)
Qv<-Qvec(W)$q
Rv<-Rval(W)
varN<-var.nnct(ct,Qv,Rv)

cov.nnct(ct,varN,Qv,Rv)
cov.nnct(ct,varN,Qv,Rv,byrow=FALSE)

#1D data points
n<-20  #or try sample(1:20,1)
X<-as.matrix(runif(n))# need to be entered as a matrix with one column
#(i.e., a column vector), hence X<-runif(n) would not work
ipd<-ipd.mat(X)
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)$q
Rv<-Rval(W)
varN<-var.nnct(ct,Qv,Rv)
Qv<-Qvec(W)$q
Rv<-Rval(W)
varN<-var.nnct(ct,Qv,Rv)
cov.nnct(ct,varN,Qv,Rv)

Covariance Matrix of the Differences of the Off-Diagonal Cell Counts in an NNCT

Description

Returns the covariance matrix of the differences of the cell counts, N_{ij}-N_{ji} for i,j=1,\ldots,k and i \ne j, in the NNCT, ct. The covariance matrix is of dimension k(k-1)/2 \times k(k-1)/2 and its entries are cov(N_{ij}-N_{ji}, N_{kl}-N_{lk}) where the order of i,j for N_{ij}-N_{ji} is as in the output of ind.nnsym(k). These covariances are valid under RL or conditional on Q and R under CSR.

The argument covN is the covariance matrix of N_{ij} (concatenated rowwise).

See also (Dixon (1994); Ceyhan (2014)).

Usage

cov.nnsym(covN)

Arguments

Value

The k(k-1)/2 \times k(k-1)/2 covariance matrix of the differences of the off-diagonal cell counts N_{ij}-N_{ji} for i,j=1,\ldots,k and i \ne j in the NNCT, ct

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.

Dixon PM (1994). “Testing spatial segregation using a nearest-neighbor contingency table.” Ecology, 75(7), 1940-1948.

See Also

var.nnsym, cov.tct, cov.nnct and cov.seg.coeff

Examples

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)
ct

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

cov.nnsym(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)

cov.nnsym(covN)

Covariance Matrix of Segregation Coefficients in a Multi-class Case

Description

Returns the covariance matrix of the segregation coefficients in a multi-class case based on the NNCT, ct. The covariance matrix is of dimension k(k+1)/2 \times k(k+1)/2 and its entry i,j correspond to the entries in the rows i and j of the output of ind.seg.coeff(k). The segregation coefficients in the multi-class case are the extension of Pielou's segregation coefficient for the two-class case. These covariances are valid under RL or conditional on Q and R under CSR.

The argument covN is the covariance matrix of N_{ij} (concatenated rowwise).

See also (Ceyhan (2014)).

Usage

cov.seg.coeff(ct, covN)

Arguments

Value

The k(k+1)/2 x k(k+1)/2 covariance matrix of the segregation coefficients for the multi-class case based on the NNCT, ct

Author(s)

Elvan Ceyhan

References

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

See Also

seg.coeff, var.seg.coeff, cov.nnct and cov.nnsym

Examples

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)

cov.seg.coeff(ct,covN)

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
ct<-nnct(ipd,fcls)

cov.seg.coeff(ct,covN)

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

cov.seg.coeff(ct,covN)

Covariance Matrix of the Entries of the Type I-IV TCTs

Description

Returns the covariance matrix of the entries T_{ij} for i,j=1,\ldots,k in the TCT for the types I, III, and IV cell-specific tests. The covariance matrix is of dimension k^2 \times k^2 and its entries are cov(T_{ij},T_{kl}) when T_{ij} values are by default corresponding to the row-wise vectorization of TCT. The argument covN must be the covariance matrix of N_{ij} values which are obtained from the NNCT by row-wise vectorization. The functions cov.tctIII and cov.tct3 are equivalent. These covariances are valid under RL or conditional on Q and R under CSR.

See also (Ceyhan (2017)).

Usage

cov.tct(ct, covN, type = "III")

Arguments

Value

The k^2 \times k^2 covariance matrix of the entries T_{ij} for i,j=1,\ldots,k in the Type I-IV TCTs

Author(s)

Elvan Ceyhan

References

Ceyhan E (2017). “Cell-Specific and Post-hoc Spatial Clustering Tests Based on Nearest Neighbor Contingency Tables.” Journal of the Korean Statistical Society, 46(2), 219-245.

See Also

cov.nnct and cov.nnsym

Examples

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)

cov.tct(ct,covN,type=1)
cov.tct(ct,covN,type="I")
cov.tct(ct,covN,type="II")
cov.tct(ct,covN,type="III")
cov.tct(ct,covN,type="IV")
cov.tctI(ct,covN)

cov.tct(ct,covN)
cov.tctIII(ct,covN)
cov.tct3(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)

cov.tct(ct,covN,type=3)
cov.tct(ct,covN,type="III")

cov.tctIII(ct,covN)
cov.tct3(ct,covN)

Conversion of the Covariance Matrix of the Row-wise Vectorized Cell Counts to Column-wise Vectorized Cell Counts in an NNCT

Description

Converts the k^2 \times k^2 covariance matrix of row-wise vectorized cell counts N_{ij} for i,j=1,\ldots,k in the NNCT, ct to the covariance matrix of column-wise vectorized cell counts. In the output, the covariance matrix entries are cov(N_{ij},N_{kl}) when N_{ij} values are corresponding to the column-wise vectorization of ct. These covariances are valid under RL or conditional on Q and R under CSR.

See also (Dixon (1994, 2002); Ceyhan (2010, 2017)).

Usage

covNrow2col(covN)

Arguments

Value

The k^2 \times k^2 covariance matrix of column-wise vectorized cell counts N_{ij} for i,j=1,\ldots,k in the NNCT, ct.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2010). “On the use of nearest neighbor contingency tables for testing spatial segregation.” Environmental and Ecological Statistics, 17(3), 247-282.

Ceyhan E (2017). “Cell-Specific and Post-hoc Spatial Clustering Tests Based on Nearest Neighbor Contingency Tables.” Journal of the Korean Statistical Society, 46(2), 219-245.

Dixon PM (1994). “Testing spatial segregation using a nearest-neighbor contingency table.” Ecology, 75(7), 1940-1948.

Dixon PM (2002). “Nearest-neighbor contingency table analysis of spatial segregation for several species.” Ecoscience, 9(2), 142-151.

See Also

cov.nnct

Examples

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)

covNrow<-cov.nnct(ct,varN,Qv,Rv)
covNcol1<-cov.nnct(ct,varN,Qv,Rv,byrow=FALSE)
covNcol2<-covNrow2col(covNrow)

covNrow
covNcol1
covNcol2

all.equal(covNcol1,covNcol2)

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

covNrow<-cov.nnct(ct,varN,Qv,Rv)
covNcol1<-cov.nnct(ct,varN,Qv,Rv,byrow=FALSE)
covNcol2<-covNrow2col(covNrow)

covNrow
covNcol1
covNcol2

all.equal(covNcol1,covNcol2)

#1D data points
n<-20  #or try sample(1:20,1)
X<-as.matrix(runif(n))# need to be entered as a matrix with one column
#(i.e., a column vector), hence X<-runif(n) would not work
ipd<-ipd.mat(X)
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)$q
Rv<-Rval(W)
varN<-var.nnct(ct,Qv,Rv)
Qv<-Qvec(W)$q
Rv<-Rval(W)
varN<-var.nnct(ct,Qv,Rv)
cov.nnct(ct,varN,Qv,Rv)

Covariance matrix for T_k values in Tcomb

Description

This function computes the covariance matrix for the T_k values used in the T_{comb} test statistics, which is a linear combination of some T_k tests.

The argument, n_1, is the number of cases (denoted as n1 as an argument). The number of cases is denoted as n_1 to match the case-control class labeling, which is just the reverse of the labeling in Cuzick and Edwards (1990).

The argument klist is the vector of integers specifying the indices of the T_k values used in obtaining the T_{comb}.

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.cov (default=FALSE) is for using the asymptotic covariance or the exact (i.e., finite sample) covariance for the vector of T_k values used in Tcomb. If asy.cov=TRUE, the asymptotic covariance is used, otherwise the exact covariance is used.

See page 87 of (Cuzick and Edwards (1990)) for more details.

Usage

covTcomb(dat, n1, klist, nonzero.mat = TRUE, asy.cov = FALSE, ...)

Arguments

Value

Returns the covariance matrix for the T_k values used in Tcomb.

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

asycovTkTl, covTcomb, and Ntkl

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)

kl<-sample(1:5,3) #try also sample(1:5,2)
kl
covTcomb(Y,n1,kl)
covTcomb(Y,n1,kl,method="max")
covTcomb(Y,n1,kl,nonzero.mat = FALSE)

covTcomb(Y,n1,kl,asy=TRUE)

Finite Sample Covariance between T_k and T_l Values

Description

This function computes the exact (i.e., finite sample) covariance between T_k and T_l values which is used in the computation of the exact variance of Cuzick and Edwards T_{comb} test, which is a linear combination of some T_k tests.

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.

See page 80 of (Cuzick and Edwards (1990)) for more details.

Usage

covTkTl(dat, n1, k, l, nonzero.mat = TRUE, ...)

Arguments

Value

Returns the exact covariance between T_k and T_l values.

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

asycovTkTl, covTcomb, and Ntkl

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

covTkTl(Y,n1,k,l)
covTkTl(Y,n1,k,l,method="max")
asycovTkTl(Y,n1,k,l)

covTkTl(Y,n1,k,l,nonzero.mat = FALSE)
asycovTkTl(Y,n1,k,l,nonzero.mat = FALSE)

Interpoint Distance Matrix for Standardized Data

Description

This function computes and returns the distance matrix computed by using the specified distance measure to compute the distances between the rows of a data matrix which is standardized row or column-wise. That is, the output is the interpoint distance (IPD) matrix of the rows of the given set of points x dist function in the stats package of the standard R distribution. The argument column is the logical argument (default=TRUE) to determine row-wise or column-wise standardization. If TRUE each column is divided by its standard deviation, else each row is divided by its standard deviation. This function is different from the dist function in the stats package. dist returns the distance matrix in a lower triangular form, and dist.std.data returns in a full matrix of distances of standardized data set. ... are for further arguments, such as method and p, passed to the dist function.

Usage

dist.std.data(x, column = TRUE, ...)

Arguments

Value

A distance matrix whose i,j-th entry is the distance between rows i and j of x, which is standardized row-wise or column-wise.

Author(s)

Elvan Ceyhan

See Also

dist, ipd.mat, and ipd.mat.euc

Examples

n<-20  #or try sample(1:20,1)
Y<-matrix(runif(3*n),ncol=3)
ipd<-ipd.mat(Y)
range(ipd)

ipd2<-dist.std.data(Y) #distance of standardized data
range(ipd2)

ipd2<-dist.std.data(Y,method="max") #distance of standardized data
range(ipd2)

#############
Y<-matrix(runif(60,0,100),ncol=3)
ipd<-ipd.mat(Y)
range(ipd)

ipd2<-dist.std.data(Y) #distance of standardized data
range(ipd2)

Converts a lower triangular distance matrix to a full distance matrix

Description

Converts a lower triangular distance matrix to a full distance matrix with zeroes in the diagonal. The input is usually the result of the dist function in the stats package. This function is adapted from Everitt's book (Everitt (2004))

Usage

dist2full(dis)

Arguments

Value

A square (symmetric) distance matrix with zeroes in the diagonal.

Author(s)

Elvan Ceyhan

References

Everitt BS (2004). An R and S-Plus Companion to Multivariate Analysis. Springer-Verlag, London, UK.

See Also

dist

Examples

#3D data points
n<-3
X<-matrix(runif(3*n),ncol=3)
dst<-dist(X)
dist2full(dst)

The Euclidean distance between two vectors, matrices, or data frames

Description

Returns the Euclidean distance between x and y which can be vectors #' or matrices or data frames of any dimension (x and y should be of same dimension).

This function is equivalent to Dist function in the pcds package but is different from the dist function in the stats package of the standard R distribution. dist requires its argument to be a data matrix and dist computes and returns he distance matrix computed by using the specified distance measure to compute the distances between the rows of a data matrix (Becker et al. (1988)), while euc.dist needs two arguments to find the distances between. For two data matrices A and B, dist(rbind(as.vector(A),as.vector(B))) and euc.dist(A,B) yield the same result.

Usage

euc.dist(x, y)

Arguments

Value

Euclidean distance between x and y

Author(s)

Elvan Ceyhan

References

Becker RA, Chambers JM, Wilks AR (1988). The New S Language. Wadsworth &amp Brooks/Cole.

See Also

dist from the base package stats and Dist from the package pcds

Examples

B<-c(1,0); C<-c(1/2,sqrt(3)/2);
euc.dist(B,C);
euc.dist(B,B);

x<-runif(10)
y<-runif(10)
euc.dist(x,y)

xm<-matrix(x,ncol=2)
ym<-matrix(y,ncol=2)
euc.dist(xm,ym)

euc.dist(xm,xm)

dat.fr<-data.frame(b=B,c=C)
euc.dist(dat.fr,dat.fr)
euc.dist(dat.fr,cbind(B,C))

Exact version of Pearson's chi-square test on NNCTs

Description

An object of class "htest" performing exact version of Pearson's chi-square test on nearest neighbor contingency tables (NNCTs) for the RL or CSR independence for 2 classes. Pearson's \chi^2 test is based on the test statistic \mathcal X^2=\sum_{j=1}^2\sum_{i=1}^2 (N_{ij}-\mu_{ij})^2/\mu_{ij}, which has \chi^2_1 distribution in the limit provided that the contingency table is constructed under the independence null hypothesis. The exact version of Pearson's test uses the exact distribution of \mathcal X^2 rather than large sample \chi^2 approximation. That is, for the one-sided alternative, we calculate the p-values as in the function exact.pval1s; and for the two-sided alternative, we calculate the p-values as in the function exact.pval2s with double argument determining the type of the correction.

This test would be equivalent to Fisher's exact test fisher.test if the odds ratio=1 (which can not be specified in the current version), and the odds ratio for the RL or CSR independence null hypothesis is \theta_0=(n_1-1)(n_2-1)/(n_1 n_2) which is used in the function and the p-value and confidence interval computations are are adapted from fisher.test.

See Ceyhan (2014) for more details.

Usage

exact.nnct(
  ct,
  alternative = "two.sided",
  conf.level = 0.95,
  pval.type = "inc",
  double = FALSE
)

Arguments

Value

A list with the elements

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.

See Also

fisher.test, exact.pval1s, and exact.pval2s

Examples

n<-20
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)
ct

exact.nnct(ct)
fisher.test(ct)

exact.nnct(ct,alt="g")
fisher.test(ct,alt="g")

exact.nnct(ct,alt="l",pval.type = "mid")

#############
ct<-matrix(sample(10:20,9),ncol=3)
fisher.test(ct) #here exact.nnct(ct) gives error message, since number of classes > 2

p-value correction to the one-sided version of exact NNCT test

Description

In using Fisher's exact test on the 2 \times 2 nearest neighbor contingency tables (NNCTs) a correction may be needed for the p-value. For the one-sided alternatives, the probabilities of more extreme tables are summed up, including or excluding the probability of the table itself (or some middle way). Let the probability of the contingency table itself be p_t=f(n_{11}|n_1,n_2,c_1;\theta_0) where \theta_0=(n_1-1)(n_2-1)/(n_1 n_2) which is the odds ratio under RL or CSR independence and f is the probability mass function of the hypergeometric distribution. For testing the one-sided alternative H_o:\,\theta=\theta_0 versus H_a:\,\theta>\theta_0, we consider the following four methods in calculating the p-value:

• [(i)] with S=\{t:\,t \geq n_{11}\}, we get the table-inclusive version which is denoted as p^>_{inc},

• [(ii)] with S=\{t:\,t> n_{11}\}, we get the table-exclusive version, denoted as p^>_{exc}.

• [(iii)] Using p=p^>_{exc}+p_t/2, we get the mid-p version, denoted as p^>_{mid}.

• [(iv)] We can also use Tocher corrected version which is denoted as p^>_{Toc} (see tocher.cor for details).

See (Ceyhan (2010)) for more details.

Usage

exact.pval1s(ptable, pval, type = "inc")

Arguments

Value

A modified p-value based on the correction specified in type.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2010). “Exact Inference for Testing Spatial Patterns by Nearest Neighbor Contingency Tables.” Journal of Probability and Statistical Science, 8(1), 45-68.

See Also

exact.pval2s and tocher.cor

Examples

ct<-matrix(sample(20:40,4),ncol=2)
ptab<-prob.nnct(ct)
pv<-.3
exact.pval1s(ptab,pv)
exact.pval1s(ptab,pv,type="exc")
exact.pval1s(ptab,pv,type="mid")

p-value correction to the two-sided version of exact NNCT test

Description

In using Fisher's exact test on the 2 \times 2 nearest neighbor contingency tables (NNCTs) a correction may be needed for the p-value. For the one-sided alternatives, the probabilities of more extreme tables are summed up, including or excluding the probability of the table itself (or some middle way).

There is additional complexity in p-values for the two-sided alternatives. A recommended method is adding up probabilities of the same size and smaller than the probability associated with the current table. Alternatively, one can double the one-sided p-value (see (Agresti (1992)).

Let the probability of the contingency table itself be p_t=f(n_{11}|n_1,n_2,c_1;\theta_0) where \theta_0=(n_1-1)(n_2-1)/(n_1 n_2) which is the odds ratio under RL or CSR independence and f is the probability mass function of the hypergeometric distribution.

**Type (I):** For double the one-sided p-value, we propose the following four variants:

• [(i)] twice the minimum of p_{inc} for the one-sided tests, which is table-inclusive version for this type of two-sided test, and denoted as p^I_{inc},

• [(ii)] twice the minimum of p_{inc} minus twice the table probability p_t, which is table-exclusive version of this type of two-sided test, and denoted as p^I_{exc},

• [(iii)] table-exclusive version of this type of two-sided test plus p_t, which is mid-p-value for this test, and denoted as p^I_{midd},

• [(iv)]Tocher corrected version (see tocher.cor for details).

**Type (II):** For summing the p-values of more extreme —than that of the table— cases in both directions, the following variants are obtained. The p-value is p=\sum_S f(t|n_1,n_2,c_1;\theta=1) with

• [(i)] S=\{t:\,f(t|n_1,n_2,c_1;\theta=1) \leq p_t\}, which is called table-inclusive version, p^{II}_{inc},

• [(ii)] the probability of the observed table is included twice, once for each side; that is p=p^{II}_{inc}+p_t, which is called twice-table-inclusive version, p^{II}_{tinc},

• [(iii)] table-inclusive minus p_t, which is referred as table-exclusive version, p^{II}_{exc},

• [(iv)] table-exclusive plus one-half the p_t, which is called mid-p version, p^{II}_{mid} and,

• [(v)]Tocher corrected version, p^{II}_{Toc}, is obtained as before.

See (Ceyhan (2010)) for more details.

Usage

exact.pval2s(ptable, pval, type = "inc", double = FALSE)

Arguments

Value

A modified p-value based on the correction specified in type.

Author(s)

Elvan Ceyhan

References

Agresti A (1992). “A Survey of Exact Inference for Contingency Tables.” Statistical Science, 7(1), 131-153.

Ceyhan E (2010). “Exact Inference for Testing Spatial Patterns by Nearest Neighbor Contingency Tables.” Journal of Probability and Statistical Science, 8(1), 45-68.

See Also

exact.pval1s and tocher.cor

Examples

ct<-matrix(sample(20:40,4),ncol=2)
ptab<-prob.nnct(ct)
pv<-.23
exact.pval2s(ptab,pv)
exact.pval2s(ptab,pv,type="exc")
exact.pval2s(ptab,pv,type="mid")

Auxiliary Functions for Computing Covariances Between Cell Counts in the TCT

Description

Five functions: cov.2cells, cov.cell.col, covNijCk, cov2cols and covCiCj

These are auxiliary functions for computing covariances between entries in the TCT for the types I-IV cell-specific tests. The covariances between T_{ij} values for i,j=1,\ldots,k in the TCT require covariances between two cells in the NNCT, between a cell and column sum, and between two column sums in the NNCT. cov.2cells computes the covariance between two cell counts N_{ij} and N_{kl} in an NNCT, cov.cell.col and covNijCk are equivalent and they compute the covariance between cell count N_{ij} and sum of column k, C_k, cov2cols and covCiCj are equivalent and they compute the covariance between sums of two columns, C_i and C_j. The index arguments refer to which entry or column sum is intended in the NNCT. The argument covN must be the covariance between N_{ij} values which are obtained from NNCT by row-wise vectorization. These covariances are valid under RL or conditional on Q and R under CSR.

Usage

cov.2cells(i, j, k, l, ct, covN)

cov.cell.col(i, j, k, ct, covN)

covNijCk(i, j, k, ct, covN)

cov.2cols(i, j, ct, covN)

covCiCj(i, j, ct, covN)

Arguments

Value

cov.2cells returns the covariance between two cell counts N_{ij} and N_{kl} in an NNCT, cov.cell.col and covNijCk return the covariance between cell count N_{ij} and sum of column k, C_k, cov2cols and covCiCj return the covariance between sums of two columns, C_i and C_j.

Author(s)

Elvan Ceyhan

See Also

cov.tct and cov.nnct

Examples

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)

cov.2cells(1,1,1,2,ct,covN)

cov.cell.col(2,2,1,ct,covN)
covNijCk(2,2,1,ct,covN)

cov.2cols(2,1,ct,covN)
covCiCj(2,1,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)

cov.2cells(2,3,1,2,ct,covN)

cov.cell.col(1,1,2,ct,covN)
covNijCk(1,1,2,ct,covN)

cov.2cols(3,4,ct,covN)
covCiCj(3,4,ct,covN)

Base Class-specific Chi-square Tests based on NNCTs

Description

Two functions: base.class.spec.ct and base.class.spec.

Both functions are objects of class "classhtest" but with different arguments (see the parameter list below). Each one performs class specific segregation tests due to Dixon for k \ge 2 classes. That is, each one performs hypothesis tests of deviations of entries in each row of NNCT from the expected values under RL or CSR for each row. Recall that row labels in the NNCT are base class labels. The test for each row i is based on the chi-squared approximation of the corresponding quadratic form and are due to Dixon (2002).

Each function yields the test statistic, p-value and df for each base class i, description of the alternative with the corresponding null values (i.e., expected values) for the row i, estimates for the entries in row i for i=1,\ldots,k. The functions also provide names of the test statistics, the description of the test and the data set used.

The null hypothesis for each row is that the corresponding N_{ij} entries in row i are equal to their expected values under RL or CSR.

See also (Dixon (2002); Ceyhan (2009)) and the references therein.

Usage

base.class.spec.ct(ct, covN)

base.class.spec(dat, lab, ...)

Arguments

Value

A list with the elements

Author(s)

Elvan Ceyhan

References

Ceyhan E (2009). “Class-Specific Tests of Segregation Based on Nearest Neighbor Contingency Tables.” Statistica Neerlandica, 63(2), 149-182.

Dixon PM (2002). “Nearest-neighbor contingency table analysis of spatial segregation for several species.” Ecoscience, 9(2), 142-151.

See Also

NN.class.spec.ct, NN.class.spec, class.spec.ct and class.spec

Examples

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)

base.class.spec(Y,cls)
base.class.spec.ct(ct,covN)
base.class.spec(Y,cls,method="max")

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
ct<-nnct(ipd,fcls)

base.class.spec(Y,fcls)
base.class.spec.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)

base.class.spec(Y,cls)
base.class.spec.ct(ct,covN)

Pielou's Cell-specific Segregation Test with Normal Approximation (for Sparse Sampling)

Description

Two functions: cell.spec.ss.ct and cell.spec.ss.

Both functions are objects of class "cellhtest" but with different arguments (see the parameter list below). Each one performs hypothesis tests of equality of the expected values of the cell counts (i.e., entries) in the NNCT for k \ge 2 classes. Each test is appropriate (i.e., have the appropriate asymptotic sampling distribution) when that data is obtained by sparse sampling.

Each cell-specific segregation test is based on the normal approximation of the entries in the NNCT and are due to Pielou (1961).

Each function yields a contingency table of the test statistics, p-values for the corresponding alternative, expected values, lower and upper confidence levels, sample estimates (i.e., observed values) and null value(s) (i.e., expected values) for the N_{ij} values for i,j=1,2,\ldots,k and also names of the test statistics, estimates, null values, the description of the test, and the data set used.

The null hypothesis is that all E(N_{ij})=n_i c_j /n where n_i is the sum of row i (i.e., size of class i) c_j is the sum of column j in the k \times k NNCT for k \ge 2. In the output, the test statistic, p-value and the lower and upper confidence limits are valid only for (properly) sparsely sampled data.

See also (Pielou (1961); Ceyhan (2010)) and the references therein.

Usage

cell.spec.ss.ct(
  ct,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)

cell.spec.ss(
  dat,
  lab,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95,
  ...
)

Arguments

Value

A list with the elements

Author(s)

Elvan Ceyhan

References

Ceyhan E (2010). “On the use of nearest neighbor contingency tables for testing spatial segregation.” Environmental and Ecological Statistics, 17(3), 247-282.

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

cell.spec.ct and cell.spec

Examples

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)

cell.spec.ss(Y,cls)
cell.spec.ss.ct(ct)
cell.spec.ss.ct(ct,alt="g")

cell.spec.ss(Y,cls,method="max")

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
ct<-nnct(ipd,fcls)

cell.spec.ss(Y,fcls)
cell.spec.ss.ct(ct)

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

cell.spec.ss(Y,cls,alt="l")
cell.spec.ss.ct(ct)
cell.spec.ss.ct(ct,alt="l")

Class-specific Chi-square Tests based on NNCTs

Description

Two functions: class.spec.ct and class.spec.

Both functions are objects of class "classhtest" but with different arguments (see the parameter list below). Each one performs class specific segregation tests for the rows if type="base" and columns if type="NN" for k \ge 2 classes. That is, each one performs hypothesis tests of deviations of entries in each row (column) of NNCT from the expected values under RL or CSR for each row (column) if type="base" ("NN"). Recall that row labels of the NNCT are base class labels and column labels in the NNCT are NN class labels. The test for each row (column) i is based on the chi-squared approximation of the corresponding quadratic form and are due to Dixon (2002) (Ceyhan (2009)).

The argument covN must be covariance of row-wise (column-wise) vectorization of NNCT if type="base" (type="NN").

Each function yields the test statistic, p-value and df for each base class i, description of the alternative with the corresponding null values (i.e., expected values) for the row (column) i, estimates for the entries in row (column) i for i=1,\ldots,k if type="base" (type="NN"). The functions also provide names of the test statistics, the description of the test and the data set used.

The null hypothesis for each row (column) is that the corresponding N_{ij} entries in row (column) i are equal to their expected values under RL or CSR.

See also (Dixon (2002); Ceyhan (2009)) and the references therein.

Usage

class.spec.ct(ct, covN, type = "base")

class.spec(dat, lab, type = "base", ...)

Arguments

Value

A list with the elements

References

Ceyhan E (2009). “Class-Specific Tests of Segregation Based on Nearest Neighbor Contingency Tables.” Statistica Neerlandica, 63(2), 149-182.

Dixon PM (2002). “Nearest-neighbor contingency table analysis of spatial segregation for several species.” Ecoscience, 9(2), 142-151.

See Also

base.class.spec.ct, base.class.spec, NN.class.spec.ct and NN.class.spec

Examples

n<-20
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

class.spec(Y,cls)
class.spec(Y,cls,type="NN")

class.spec.ct(ct,covN)
class.spec.ct(ct,covN,type="NN")

class.spec(Y,cls,method="max")

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
ct<-nnct(ipd,fcls)

class.spec(Y,fcls)
class.spec(Y,fcls,type="NN")

class.spec.ct(ct,covN)
class.spec.ct(ct,covN,type="NN")

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

class.spec(Y,cls)
class.spec(Y,cls,type="NN")

class.spec.ct(ct,covN)
class.spec.ct(ct,covN,type="NN")

Covariance Matrix of the Self Entries in a Species Correspondence Contingency Table (SCCT)

Description

Two functions: covNii.ct and covNii.

Both functions return the covariance matrix of the self entries (i.e., first column entries) in a species correspondence contingency table (SCCT) but have different arguments (see the parameter list below). The covariance matrix is of dimension k \times k and its entries are cov(S_i,S_j) where S_i values are the entries in the first column of SCCT (recall that S_i equals diagonal entry N_{ii} in the NNCT). These covariances are valid under RL or conditional on Q and R under CSR.

The argument ct which is used in covNii.ct only, can be either the NNCT or SCCT. And the argument Vsq is the vector of variances of the diagonal entries N_{ii} in the NNCT or the self entries (i.e., the first column) in the SCCT.

See also (Ceyhan (2018)).

Usage

covNii.ct(ct, Vsq, Q, R)

covNii(dat, lab, ...)

Arguments

Value

A vector of length k whose entries are the variances of the self entries (i.e., first column) in a species correspondence contingency table (SCCT).

The k \times k covariance matrix of cell counts S_i in the self (i.e., first) column of the SCCT or of the diagonal cell counts N_{ii} for i=1,\ldots,k in the NNCT.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2018). “A contingency table approach based on nearest neighbor relations for testing self and mixed correspondence.” SORT-Statistics and Operations Research Transactions, 42(2), 125-158.

See Also

scct, cov.nnct, cov.tct and cov.nnsym

Examples

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)

vsq<-varNii.ct(ct,Qv,Rv)
covNii(Y,cls)
covNii.ct(ct,vsq,Qv,Rv)

covNii(Y,cls,method="max")

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
ct<-nnct(ipd,fcls)

covNii(Y,fcls)
covNii.ct(ct,vsq,Qv,Rv)

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

vsq<-varNii.ct(ct,Qv,Rv)
covNii(Y,cls)
covNii.ct(ct,vsq,Qv,Rv)

Functions for Covariances of the Entries of the Types I, III and IV TCTs

Description

Four functions: cov.tctI, cov.tctIII, cov.tct3, and cov.tctIV.

These functions return the covariances between entries in the TCT for the types I, III, and IV cell-specific tests in matrix form which is of dimension k^2 \times k^2. The covariance matrix entries are cov(T_{ij},T_{kl}) when T_{ij} values are by default corresponding to the row-wise vectorization of TCT. The argument CovN must be the covariance between N_{ij} values which are obtained from the NNCT by row-wise vectorization. The functions cov.tctIII and cov.tct3 are equivalent. These covariances are valid under RL or conditional on Q and R under CSR.

See also (Ceyhan (2017)).

Usage

cov.tctI(ct, CovN)

cov.tctIII(ct, CovN)

cov.tct3(ct, CovN)

cov.tctIV(ct, CovN)

Arguments

Value

Each of these functions returns a k^2 \times k^2 covariance matrix, whose entries are the covariances of the entries in the TCTs for the corresponding type I-IV cell-specific test. The row and column names are inherited from ct.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2017). “Cell-Specific and Post-hoc Spatial Clustering Tests Based on Nearest Neighbor Contingency Tables.” Journal of the Korean Statistical Society, 46(2), 219-245.

See Also

cov.tct and cov.nnct

Examples

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)

cov.tctI(ct,covN)
cov.tctIII(ct,covN)
cov.tctIV(ct,covN)

Functions for the k^{th} and k NN distances

Description

Two functions: kthNNdist and kNNdist.

kthNNdist returns the distances between subjects and their k^{th} NNs. The output is an n \times 2 matrix where n is the data size and first column is the subject index and second column contains the corresponding distances to k^{th} NN subjects.

kNNdist returns the distances between subjects and their k NNs. The output is an n \times (k+1) matrix where n is the data size and first column is the subject index and the remaining k columns contain the corresponding distances to k NN subjects.

Usage

kthNNdist(x, k, is.ipd = TRUE, ...)

kNNdist(x, k, is.ipd = TRUE, ...)

Arguments

Value

kthNNdist returns an n \times 2 matrix where n is data size (i.e., number of subjects) and first column is the subject index and second column is the k^{th} NN distances.

kNNdist returns an n \times (k+1) matrix where n is data size (i.e., number of subjects) and first column is the subject index and the remaining k columns contain the corresponding distances to k NN subjects.

Author(s)

Elvan Ceyhan

See Also

NNdist and NNdist2cl

Examples

#Examples for kthNNdist
#3D data points, gives NAs when n<=k
n<-20  #or try sample(1:20,1)
Y<-matrix(runif(3*n),ncol=3)
ipd<-ipd.mat(Y)
kthNNdist(ipd,3)
kthNNdist(Y,3,is.ipd = FALSE)
kthNNdist(ipd,5)
kthNNdist(Y,5,is.ipd = FALSE)
kthNNdist(Y,3,is.ipd = FALSE,method="max")

#1D data points
X<-as.matrix(runif(5)) # need to be entered as a matrix with one column
#(i.e., a column vector), hence X<-runif(5) would not work
ipd<-ipd.mat(X)
kthNNdist(ipd,3)

#Examples for kNNdist
#3D data points, gives NAs if n<=k for n,n+1,...,kNNs
n<-20  #or try sample(1:20,1)
Y<-matrix(runif(3*n),ncol=3)
ipd<-ipd.mat(Y)
kNNdist(ipd,3)
kNNdist(ipd,5)
kNNdist(Y,5,is.ipd = FALSE)

kNNdist(Y,5,is.ipd = FALSE,method="max")

kNNdist(ipd,1)
kthNNdist(ipd,1)

#1D data points
X<-as.matrix(runif(5)) # need to be entered as a matrix with one column
#(i.e., a column vector), hence X<-runif(5) would not work
ipd<-ipd.mat(X)
kNNdist(ipd,3)

Functions for the k^{th} and k NN distances

Description

Two functions: kthNNdist2cl and kNNdist2cl.

kthNNdist2cl returns the distances between subjects from class i and their k^{th} NNs from class j. The output is a list with first entry (kth.nndist) is an n_i \times 3 matrix where n_i is the size of class i and first column is the subject index for class i, second column is the index of the k^{th} NN of class i subjects among class j subjects and third column contains the corresponding k^{th} NN distances. The other entries in the list are labels of base class and NN class and the value of k, respectively.

kNNdist2cl returns the distances between subjects from class i and their k NNs from class j. The output is a list with first entry (ind.knndist) is an n_i \times (k+1) matrix where n_i is the size of class i, first column is the indices of class i subjects, second to (k+1)-st columns are the indices of k NNs of class i subjects among class j subjects. The second list entry (knndist) is an n_i \times k matrix where n_i is the size of class i and the columns are the kNN distances of class i subjects to class j subjects. The other entries in the list are labels of base class and NN class and the value of k, respectively.

The argument within.class.ind is a logical argument (default=FALSE) to determine the indexing of the class i subjects. If TRUE, index numbering of subjects is within the class, from 1 to class size (i.e., 1:n_i), according to their order in the original data; otherwise, index numbering within class is just the indices in the original data.

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.

Usage

kthNNdist2cl(x, k, i, j, lab, within.class.ind = FALSE, is.ipd = TRUE, ...)

kNNdist2cl(x, k, i, j, lab, within.class.ind = FALSE, is.ipd = TRUE, ...)

Arguments

Value

kthNNdist2cl returns the list of elements

kNNdist2cl returns the list of elements

Author(s)

Elvan Ceyhan

See Also

NNdist2cl, kthNNdist, and kNNdist

Examples

#Examples for kthNNdist2cl
#3D data points
n<-20  #or try sample(1:20,1)
Y<-matrix(runif(3*n),ncol=3)
ipd<-ipd.mat(Y)
#two class case
clab<-sample(1:2,n,replace=TRUE) #class labels
table(clab)
kthNNdist2cl(ipd,3,1,2,clab)
kthNNdist2cl(Y,3,1,2,clab,is.ipd = FALSE)
kthNNdist2cl(ipd,3,1,2,clab,within = TRUE)

#three class case
clab<-sample(1:3,n,replace=TRUE) #class labels
table(clab)
kthNNdist2cl(ipd,3,2,3,clab)

#1D data points
n<-15
X<-as.matrix(runif(n))# need to be entered as a matrix with one column
#(i.e., a column vector), hence X<-runif(n) would not work
ipd<-ipd.mat(X)
#two class case
clab<-sample(1:2,n,replace=TRUE) #class labels
table(clab)
kthNNdist2cl(ipd,3,1,2,clab) # here kthNNdist2cl(ipd,3,1,12,clab) 
#gives an error message

kthNNdist2cl(ipd,3,"1",2,clab)

#Examples for kNNdist2cl
#3D data points
n<-20  #or try sample(1:20,1)
Y<-matrix(runif(3*n),ncol=3)
ipd<-ipd.mat(Y)
#two class case
clab<-sample(1:2,n,replace=TRUE) #class labels
table(clab)
kNNdist2cl(ipd,3,1,2,clab)
kNNdist2cl(Y,3,1,2,clab,is.ipd = FALSE)

kNNdist2cl(ipd,3,1,2,clab,within = TRUE)

#three class case
clab<-sample(1:3,n,replace=TRUE) #class labels
table(clab)
kNNdist2cl(ipd,3,1,2,clab)

#1D data points
n<-15
X<-as.matrix(runif(n))# need to be entered as a matrix with one column
#(i.e., a column vector), hence X<-runif(n) would not work
ipd<-ipd.mat(X)
#two class case
clab<-sample(1:2,n,replace=TRUE) #class labels
table(clab)

kNNdist2cl(ipd,3,1,2,clab)
kNNdist2cl(ipd,3,"1",2,clab) #here kNNdist2cl(ipd,3,"a",2,clab) 
#gives an error message

Dixon's Overall Test of Segregation for NNCT

Description

Two functions: overall.nnct.ct and overall.nnct.

Both 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 NNCT. That is, each test is Dixon's overall test of segregation based on NNCTs for k \ge 2 classes. This overall test is based on the chi-squared approximation of the corresponding quadratic form and are due to Dixon (1994, 2002). Both functions exclude the last column of the NNCT (in fact any column will do and last column is chosen without loss of generality), to avoid ill-conditioning of the covariance matrix (for its inversion in the quadratic form).

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

The null hypothesis is that all N_{ij} entries are equal to their expected values under RL or CSR.

See also (Dixon (1994, 2002); Ceyhan (2010, 2017)) and the references therein.

Usage

overall.nnct.ct(ct, covN)

overall.nnct(dat, lab, ...)

Arguments

Value

A list with the elements

Author(s)

Elvan Ceyhan

References

Ceyhan E (2010). “On the use of nearest neighbor contingency tables for testing spatial segregation.” Environmental and Ecological Statistics, 17(3), 247-282.

Ceyhan E (2017). “Cell-Specific and Post-hoc Spatial Clustering Tests Based on Nearest Neighbor Contingency Tables.” Journal of the Korean Statistical Society, 46(2), 219-245.

Dixon PM (1994). “Testing spatial segregation using a nearest-neighbor contingency table.” Ecology, 75(7), 1940-1948.

Dixon PM (2002). “Nearest-neighbor contingency table analysis of spatial segregation for several species.” Ecoscience, 9(2), 142-151.

See Also

overall.seg.ct, overall.seg, overall.tct.ct and overall.tct

Examples

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.nnct(Y,cls)
overall.nnct.ct(ct,covN)

overall.nnct(Y,cls,method="max")

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
ct<-nnct(ipd,fcls)

overall.nnct(Y,fcls)
overall.nnct.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.nnct(Y,cls)
overall.nnct.ct(ct,covN)

Overall Segregation Tests for NNCTs

Description

Two functions: overall.seg.ct and overall.seg.

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 NNCT or TCT. That is, each test is one of Dixon's or Types I-IV overall test of segregation based on NNCTs or TCTs for k \ge 2 classes. Each overall test is based on the chi-squared approximation of the corresponding quadratic form and are due to Dixon (1994, 2002) and to Ceyhan (2010, 2017), respectively. All 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), see the relevant functions under See also section below.

The type="dixon" or "nnct" refers to Dixon's overall test of segregation, and type="I"-"IV" refers to types I-IV overall tests, respectively.

Each function yields the test statistic, p-value and df which is k(k-1) for type II and Dixon's 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 description of the test and the data set used.

The null hypothesis is that all N_{ij} or T_{ij} entries for the specified type are equal to their expected values under RL or CSR, respectively.

See also (Dixon (1994, 2002); Ceyhan (2010, 2010)) and the references therein.

Usage

overall.seg.ct(ct, covN, type)

overall.seg(dat, lab, type, ...)

Arguments

Value

A list with the elements

Author(s)

Elvan Ceyhan

References

Ceyhan E (2010). “New Tests of Spatial Segregation Based on Nearest Neighbor Contingency Tables.” Scandinavian Journal of Statistics, 37(1), 147-165.

Ceyhan E (2010). “On the use of nearest neighbor contingency tables for testing spatial segregation.” Environmental and Ecological Statistics, 17(3), 247-282.

Ceyhan E (2017). “Cell-Specific and Post-hoc Spatial Clustering Tests Based on Nearest Neighbor Contingency Tables.” Journal of the Korean Statistical Society, 46(2), 219-245.

Dixon PM (1994). “Testing spatial segregation using a nearest-neighbor contingency table.” Ecology, 75(7), 1940-1948.

Dixon PM (2002). “Nearest-neighbor contingency table analysis of spatial segregation for several species.” Ecoscience, 9(2), 142-151.

See Also

overall.nnct.ct, overall.nnct, overall.tct.ct and overall.tct

Examples

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

type<-"dixon" #try also "nnct", I", "II", "III", and "IV"
overall.seg(Y,cls,type)
overall.seg(Y,cls,type,method="max")
overall.seg(Y,cls,type="I")

overall.seg.ct(ct,covN,type)
overall.seg.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.seg(Y,fcls,type="I")
overall.seg.ct(ct,covN,type)

#############
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.seg(Y,cls,type="I")
overall.seg.ct(ct,covN,type)

Types I-IV Overall Tests of Segregation for NNCT

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 \ge 2 classes. This overall test is based on the chi-squared approximation of the corresponding quadratic form and are due to Ceyhan (2010, 2017). 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 description of the test 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.

See also (Ceyhan (2010, 2017)) and the references therein.

Usage

overall.tct.ct(ct, covN, type = "III")

overall.tct(dat, lab, type = "III", ...)

Arguments

Value

A list with the elements

Author(s)

Elvan Ceyhan

References

Ceyhan E (2010). “New Tests of Spatial Segregation Based on Nearest Neighbor Contingency Tables.” Scandinavian Journal of Statistics, 37(1), 147-165.

Ceyhan E (2017). “Cell-Specific and Post-hoc Spatial Clustering Tests Based on Nearest Neighbor Contingency Tables.” Journal of the Korean Statistical Society, 46(2), 219-245.

See Also

overall.seg.ct, overall.seg, overall.nnct.ct and overall.nnct

Examples

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)

The functions for probability of selecting a number of points from respective classes

Description

The ancillary probability functions used in computation of the variance-covariance matrices of various NN spatial tests such as NNCT tests and tests based on other contingency tables. These functions can be classified as pij and Pij type functions. The pij functions are for individual probabilities and the corresponding Pij functions are the summed pij values. For example p_{iijk} is the probability of any 4 points with 2 from class i, and others are from classes j and k. These probabilities are for data from RL or CSR.

Usage

p11(k, n)

P11(nvec)

p12(k, l, n)

P12(nvec)

p111(k, n)

P111(nvec)

p1111(k, n)

P1111(nvec)

p112(k, l, n)

P112(nvec)

p122(k, l, n)

p123(k, l, m, n)

P123(nvec)

p1234(k, l, m, p, n)

P1234(nvec)

p1223(k, l, m, n)

p1123(k, l, m, n)

P1123(nvec)

p1122(k, l, n)

P1122(nvec)

p1112(k, l, n)

P1112(nvec)

Arguments

Value

Probability values for the selected points being from the indicated classes.

See Also

pk

Species Correspondence Contingency Table (SCCT)

Description

Two functions: scct.ct and scct.

Both functions return the k \times 2 species correspondence contingency table (SCCT) but have different arguments (see the parameter list below).

SCCT is constructed by categorizing the NN pairs according to pair type as self or mixed. A base-NN pair is called a self pair, if the elements of the pair are from the same class; a mixed pair, if the elements of the pair are from different classes. Row labels in the RCT are the class labels and the column labels are "self" and "mixed". The k \times 2 SCCT (whose first column is self column with entries S_i and second column is mixed with entries M_i) is closely related to the k \times k nearest neighbor contingency table (NNCT) whose entries are N_{ij}, where S_i=N_{ii} and M_i=n_i-N_{ii} with n_i is the size of class i.

The function scct.ct returns the SCCT given the inter-point distance (IPD) matrix or data set x, and the function scct returns the SCCT given the IPD matrix. SCCT is a k \times 2 matrix where k is number of classes in the data set. (See Ceyhan (2018) for more detail, where SCCT is labeled as CCT for correspondence contingency table).

The argument ties is a logical argument (default=FALSE for both functions) to take ties into account or not. If TRUE a NN contributes 1/m to the NN count if it is one of the m tied NNs of a subject.

The argument nnct is a logical argument for scct.ct only (default=FALSE) to determine the structure of the argument x. If TRUE, x is taken to be the k \times k NNCT, and if FALSE, x is taken to be the IPD matrix.

The argument lab is the vector of class labels (default=NULL when nnct=TRUE in the function scct.ct and no default specified for scct).

Usage

scct.ct(x, lab = NULL, ties = FALSE, nnct = FALSE)

scct(dat, lab, ties = FALSE, ...)

Arguments

Value

Returns the k \times 2 SCCT where k is the number of classes in the data set.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2018). “A contingency table approach based on nearest neighbor relations for testing self and mixed correspondence.” SORT-Statistics and Operations Research Transactions, 42(2), 125-158.

See Also

nnct, tct, rct and Qsym.ct

Examples

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))
NNCT<-nnct(ipd,cls)
NNCT

scct(Y,cls)
scct(Y,cls,method="max")

scct.ct(ipd,cls)
scct.ct(ipd,cls,ties = TRUE)
scct.ct(NNCT,nnct=TRUE)

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
scct.ct(ipd,fcls)

#############
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))
NNCT<-nnct(ipd,cls)
NNCT

scct(Y,cls)

scct.ct(ipd,cls)
scct.ct(NNCT,nnct=TRUE)

Pielou's Segregation Coefficients for NNCTs

Description

Two functions: Pseg.coeff and seg.coeff.

Each function computes segregation coefficients based on NNCTs. The function Pseg.coeff computes Pielou's segregation coefficient (Pielou (1961)) for the two-class case (i.e., based on 2 \times 2 NNCTs) and seg.coeff is the extension of Pseg.coeff to the multi-class case (i.e., for k \times k NNCTs with k \ge 2) and provides a k \times k matrix of segregation coefficients (Ceyhan (2014)). Both functions use the same argument, ct, for NNCT.

Pielou's segregation coefficient (for two classes) is S_P = 1-(N_{12} + N_{21})/(E[N_{12}] + E[N_{21}]) and the extended segregation coefficents (for k \ge 2 classes) are S_c = 1 -(N_{ii})/(E[N_{ii}]) for the diagonal cells in the NNCT and S_c = 1 -(N_{ij} + N_{ji})/(E[N_{ij}] + E[N_{ji}]) for the off-diagonal cells in the NNCT.

Usage

Pseg.coeff(ct)

seg.coeff(ct)

Arguments

Value

Pseg.coeff returns Pielou's segregation coefficient for 2 \times 2 NNCT seg.coeff returns a k \times k matrix of segregation coefficients (which are extended versions of Pielou's segregation coefficient)

Author(s)

Elvan Ceyhan

References

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

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

seg.ind, Zseg.coeff.ct and Zseg.coeff

Examples

#Examples for Pseg.coeff
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)
ct
Pseg.coeff(ct)

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
ct<-nnct(ipd,fcls)

Pseg.coeff(ct)

#############
ct<-matrix(sample(1:25,9),ncol=3)
#Pseg.coeff(ct)

#Examples for seg.coeff
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)
ct
seg.coeff(ct)

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
ct<-nnct(ipd,fcls)

seg.coeff(ct)

#############
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)
ct<-nnct(ipd,cls)

seg.coeff(ct)

Variances of the Self Entries in a Species Correspondence Contingency Table (SCCT)

Description

Two functions: varNii.ct and varNii.

Both functions return a vector of length k of variances of the self entries (i.e., first column) in a species correspondence contingency table (SCCT) or the variances of the diagonal entries N_{ii} in an NNCT, but have different arguments (see the parameter list below). These variances are valid under RL or conditional on Q and R under CSR.

The argument ct which is used in varNii.ct only, can be either the NNCT or SCCT.

See also (Ceyhan (2018)).

Usage

varNii.ct(ct, Q, R)

varNii(dat, lab, ...)

Arguments

Value

A vector of length k whose entries are the variances of the self entries (i.e., first column) in a species correspondence contingency table (SCCT) or of the diagonal entries in an NNCT.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2018). “A contingency table approach based on nearest neighbor relations for testing self and mixed correspondence.” SORT-Statistics and Operations Research Transactions, 42(2), 125-158.

See Also

scct, var.nnct, var.tct, var.nnsym and covNii

Examples

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)

varNii(Y,cls)
varNii.ct(ct,Qv,Rv)

varNii(Y,cls,method="max")

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
ct<-nnct(ipd,fcls)

varNii(Y,fcls)
varNii.ct(ct,Qv,Rv)

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

varNii(Y,cls)
varNii.ct(ct,Qv,Rv)

Functions for Variances of Cell Counts in the Types I, III and IV TCTs

Description

Three functions: var.tctI, var.tctIII, and var.tctIV.

These functions return the variances of T_{ij} values for i,j=1,\ldots,k in the TCT in matrix form which is of the same dimension as TCT for types I, III and IV tests. The argument covN must be the covariance between N_{ij} values which are obtained from the NNCT by row-wise vectorization. These variances are valid under RL or conditional on Q and R under CSR.

See also (Ceyhan (2017)).

Usage

var.tctI(ct, covN)

var.tctIII(ct, covN)

var.tctIV(ct, covN)

Arguments

Value

Each of these functions returns a matrix of same dimension as, ct, whose entries are the variances of the entries in the TCT for the corresponding type of cell-specific test. The row and column names are inherited from ct.

References

Ceyhan E (2017). “Cell-Specific and Post-hoc Spatial Clustering Tests Based on Nearest Neighbor Contingency Tables.” Journal of the Korean Statistical Society, 46(2), 219-245.

See Also

var.tct and var.nnct

Aij matrices for computation of Moments of Cuzick and Edwards T_k Test statistic

Description

Two functions: aij.mat and aij.nonzero.

The function aij.mat yields the A=(a_{ij}(k)) matrix where a_{ij}(k) = 1 if z_j is among the kNNs of z_i and 0 otherwise due to Tango (2007). This matrix is useful in calculation of the moments of Cuzick-Edwards T_k tests.

The function aij.nonzero keeps only nonzero entries, i.e., row and column entries where in each row, for the entry (r_1,c_1) r_1 is the row entry and c_1 is the column entry. Rows are from 1 to n, which stands for the data point or observation, and column entries are from 1 to k, where k is specifying the number of kNNs (of each observation) considered. This function saves in storage memory, but needs to be carefully unfolded in the functions to represent the actual the A matrix.

See also (Tango (2007)).

Usage

aij.mat(dat, k, ...)

aij.nonzero(dat, k, ...)

Arguments

Value

The function aij.mat returns the A_{ij} matrix for computation of moments of Cuzick and Edwards T_k Test statistic while the function aij.nonzero returns the (locations of the) non-zero entries in the A_{ij} matrix

Author(s)

Elvan Ceyhan

References

Tango T (2007). “A class of multiplicity adjusted tests for spatial clustering based on case-control point data.” Biometrics, 63, 119-127.

See Also

aij.theta and EV.Tkaij

Examples

n<-20  #or try sample(1:20,1)
Y<-matrix(runif(3*n),ncol=3)
k<-3 #try also 2,3

Aij<-aij.mat(Y,k)
Aij
Aij2<-aij.mat(Y,k,method="max")
range(Aij,Aij2)

apply(Aij,2,sum) #row sums of Aij

aij.nonzero(Y,k)
aij.nonzero(Y,k,method="max")

Correction Matrices for the Covariance Matrix of NNCT entries

Description

Two functions: correct.cf1 and correct.cf1.

Each function yields matrices which are used in obtaining covariance matrices of T_{ij} values for types I and II tests from the usual Chi-Square test of contingency tables (i.e., Pielou's test) applied on NNCTs. The output matrices are to be term-by-term multiplied with the covariance matrix of the entries of NNCT. See Sections 3.1 and 3.2 in (Ceyhan (2010)) or Sections 3.5.1 and 3.5.2 in (Ceyhan (2008)) for more details.

Usage

correct.cf1(ct)

correct.cf2(ct)

Arguments

Value

Both functions return a correction matrix which is to be multiplied with the covariance matrix of entries of the NNCT so as to obtain types I and II overall tests from Pielou's test of segregation. See the description above for further detail.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2008). “New Tests for Spatial Segregation Based on Nearest Neighbor Contingency Tables.” https://arxiv.org/abs/0808.1409v3 [stat.ME]. Technical Report # KU-EC-08-6, Koç University, Istanbul, Turkey.

Ceyhan E (2010). “New Tests of Spatial Segregation Based on Nearest Neighbor Contingency Tables.” Scandinavian Journal of Statistics, 37(1), 147-165.

See Also

nnct.cr1 and nnct.cr2

Examples

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)

#correction type 1
CM1<-correct.cf1(ct)
CovN.cf1<-covN*CM1

#correction type 2
CM2<-correct.cf2(ct)
CovN.cf2<-covN*CM2

covN
CovN.cf1
CovN.cf2

Expected Value for Cuzick and Edwards T_k Test statistic

Description

Two functions: EV.Tk and EV.Tkaij.

Both functions compute the expected value of Cuzick and Edwards T_k test statistic based on the number of cases within kNNs of the cases in the data under RL or CSR independence.

The number of cases are denoted as n_1 (denoted as n1 as an argument) for both functions and number of controls as n_0 (denoted as n0 as an argument) in EV.Tk, to match the case-control class labeling, which is just the reverse of the labeling in Cuzick and Edwards (1990).

The function EV.Tkaij uses Toshiro Tango's moments formulas based on the A=(a_{ij}) matrix (and is equivalent to the function EV.Tk, see Tango (2007), where a_{ij}(k) = 1 if z_j is among the kNNs of z_i and 0 otherwise.

See also (Ceyhan (2014)).

Usage

EV.Tk(k, n1, n0)

EV.Tkaij(k, n1, a)

Arguments

Value

The expected value of Cuzick and Edwards T_k test statistic for disease clustering

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.

Tango T (2007). “A class of multiplicity adjusted tests for spatial clustering based on case-control point data.” Biometrics, 63, 119-127.

See Also

ceTk and EV.Tcomb

Examples

n1<-20
n0<-25
k<-1 #try also 3, 5, sample(1:5,1)

EV.Tk(k,n1,n0)

###
n<-20  #or try sample(1:20,1)
Y<-matrix(runif(3*n),ncol=3)
cls<-sample(0:1,n,replace = TRUE)
n1<-sum(cls==1)
n0<-sum(cls==0)
a<-aij.mat(Y,k)

EV.Tk(k,n1,n0)
EV.Tkaij(k,n1,a)

Expected Value for Cuzick and Edwards T_{run} Test statistic

Description

Two functions: EV.Trun and EV.Trun.alt.

Both functions compute the expected value of Cuzick and Edwards T_{run} test statistic based on the number of consecutive cases from the cases in the data under RL or CSR independence.

The number of cases are denoted as n_1 (denoted as n1 as an argument) and number of controls as n_0 for both functions (denoted as n0 as an argument), to match the case-control class labeling, which is just the reverse of the labeling in Cuzick and Edwards (1990).

The function EV.Trun.alt uses a loop and takes slightly longer than the function EV.Trun, hence EV.Trun is used in other functions.

See also (Cuzick and Edwards (1990)).

Usage

EV.Trun(n1, n0)

EV.Trun.alt(n1, n0)

Arguments

Value

The expected value of Cuzick and Edwards T_{run} test statistic for disease clustering

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

ceTrun and EV.Tk

Examples

n1<-20
n0<-25

EV.Trun(n1,n0)

NN Class-specific Chi-square Tests based on NNCTs

Description

Two functions: NN.class.spec.ct and NN.class.spec.

Both functions are objects of class "classhtest" but with different arguments (see the parameter list below). Each one performs class specific segregation tests for the columns, i.e., NN categories for k \ge 2 classes. That is, each one performs hypothesis tests of deviations of entries in each column of NNCT from the expected values under RL or CSR for each column. Recall that column labels in the NNCT are NN class labels. The test for each column i is based on the chi-squared approximation of the corresponding quadratic form and are due to Ceyhan (2009).

The argument covN must be covariance of column-wise vectorization of NNCT if the logical argument byrow=FALSE otherwise the function converts covN (which is done row-wise) to columnwise version with covNrow2col function.

Each function yields the test statistic, p-value and df for each base class i, description of the alternative with the corresponding null values (i.e., expected values) for the column i, estimates for the entries in column i for i=1,\ldots,k. The functions also provide names of the test statistics, the description of the test and the data set used.

The null hypothesis for each column is that the corresponding N_{ij} entries in column i are equal to their expected values under RL or CSR.

See also (Dixon (2002); Ceyhan (2009)) and the references therein.

Usage

NN.class.spec.ct(ct, covN, byrow = TRUE)

NN.class.spec(dat, lab, ...)

Arguments

Value

A list with the elements

Author(s)

Elvan Ceyhan

References

Ceyhan E (2009). “Class-Specific Tests of Segregation Based on Nearest Neighbor Contingency Tables.” Statistica Neerlandica, 63(2), 149-182.

Dixon PM (2002). “Nearest-neighbor contingency table analysis of spatial segregation for several species.” Ecoscience, 9(2), 142-151.

See Also

base.class.spec.ct, base.class.spec, class.spec.ct and class.spec

Examples

n<-20
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)
covNrow<-cov.nnct(ct,varN,Qv,Rv)
covNcol<-covNrow2col(covNrow)

NN.class.spec(Y,cls)
NN.class.spec(Y,cls,method="max")

NN.class.spec.ct(ct,covNrow)
NN.class.spec.ct(ct,covNcol,byrow = FALSE)

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
ct<-nnct(ipd,fcls)

NN.class.spec(Y,fcls)
NN.class.spec.ct(ct,covNrow)

#############
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)
covNrow<-cov.nnct(ct,varN,Qv,Rv)
covNcol<-covNrow2col(covNrow)

NN.class.spec(Y,cls)

NN.class.spec.ct(ct,covNrow)
NN.class.spec.ct(ct,covNcol,byrow = FALSE)

Correction Matrices for the NNCT entries

Description

Two functions: nnct.cr1 and nnct.cr1.

Each function yields matrices which are used in obtaining the correction term to be added to the usual Chi-Square test of contingency tables (i.e., Pielou's test) applied on NNCTs to obtain types I and II overall tests. The output contingency tables are to be row-wise vectorized to obtain N_I and N_{II} vectors. See Sections 3.1 and 3.2 in (Ceyhan (2010)) or Sections 3.5.1 and 3.5.2 in (Ceyhan (2008)) for more details.

Usage

nnct.cr1(ct)

nnct.cr2(ct)

Arguments

Value

Both functions return a k \times k contingency table which is to be row-wise vectorized to obtain N_I and N_{II} vectors which are used in the correction summands to obtain types I and II overall tests from Pielou's test of segregation. See the description above for further detail.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2008). “New Tests for Spatial Segregation Based on Nearest Neighbor Contingency Tables.” https://arxiv.org/abs/0808.1409v3 [stat.ME]. Technical Report # KU-EC-08-6, Koç University, Istanbul, Turkey.

Ceyhan E (2010). “New Tests of Spatial Segregation Based on Nearest Neighbor Contingency Tables.” Scandinavian Journal of Statistics, 37(1), 147-165.

See Also

correct.cf1 and correct.cf2

Examples

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)

#correction type 1
ct1<-nnct.cr1(ct)

#correction type 2
ct2<-nnct.cr2(ct)

ct
ct1
ct2

Functions for one versus rest type labeling

Description

Two functions: lab.onevsrest and classirest.

Both functions relabel the points, keeping class i label as is and relabeling the other classes as "rest". Used in the one-vs-rest type comparisons after the overall segregation test is found to be significant.

See also (Ceyhan (2017)).

Usage

lab.onevsrest(i, lab)

classirest(i, lab)

Arguments

Value

Both functions return the data relabeled as class i label is retained and the remaining is relabeled as "rest".

Author(s)

Elvan Ceyhan

References

Ceyhan E (2017). “Cell-Specific and Post-hoc Spatial Clustering Tests Based on Nearest Neighbor Contingency Tables.” Journal of the Korean Statistical Society, 46(2), 219-245.

See Also

pairwise.lab

Examples

n<-20  #or try sample(1:20,1)
cls<-sample(1:2,n,replace = TRUE)  #or try cls<-rep(1:2,c(10,10))
lab.onevsrest(1,cls)
classirest(2,cls)

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
lab.onevsrest("a",fcls)
lab.onevsrest("b",fcls)
classirest("b",fcls)

#cls as a factor
fcls<-rep(letters[1:4],rep(10,4))
lab.onevsrest("b",fcls)
classirest("b",fcls)

Pielou's Overall Test of Segregation for NNCT (for Sparse Sampling)

Description

Two functions: Pseg.ss.ct and Pseg.ss.

Both 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 independence for all cells (i.e., entries) combined in the NNCT. That is, each test is Pielou's overall test of segregation based on NNCTs for k \ge 2 classes. This overall test is based on the chi-squared approximation, is equivalent to Pearson's chi-squared test on NNCT and is due to Pielou (1961). Each test is appropriate (i.e., have the appropriate asymptotic sampling distribution) when that data is obtained by sparse sampling.

Each function yields the test statistic, p-value and df which is (k-1)^2, description of the alternative with the corresponding null values (i.e., expected values) of NNCT entries, sample estimates (i.e., observed values) of the entries in NNCT. 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_{ij})=n_i c_j /n for all entries in the NNCT where n_i is the sum of row i (i.e., size of class i), c_j is the sum of column j in the k \times k NNCT for k \ge 2. In the output, the test statistic and the p-value are valid only for (properly) sparsely sampled data.

See also (Pielou (1961); Ceyhan (2010)) and the references therein.

Usage

Pseg.ss.ct(ct, yates = TRUE, sim = FALSE, Nsim = 2000)

Pseg.ss(dat, lab, yates = TRUE, sim = FALSE, Nsim = 2000, ...)

Arguments

Value

A list with the elements

Author(s)

Elvan Ceyhan

References

Ceyhan E (2010). “On the use of nearest neighbor contingency tables for testing spatial segregation.” Environmental and Ecological Statistics, 17(3), 247-282.

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

overall.nnct.ct, overall.nnct, overall.seg.ct, overall.seg and chisq.test

Examples

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)
ct

Pseg.ss(Y,cls)

Pseg.ss.ct(ct)
Pseg.ss.ct(ct,yates=FALSE)

Pseg.ss.ct(ct,yates=FALSE,sim=TRUE)
Pseg.ss.ct(ct,yates=FALSE,sim=TRUE,Nsim=10000)

Pseg.ss(Y,cls,method="max")
Pseg.ss(Y,cls,yates=FALSE,sim=TRUE,Nsim=10000,method="max")

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
ct<-nnct(ipd,fcls)

Pseg.ss(Y,fcls)
Pseg.ss.ct(ct)

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

Pseg.ss(Y,cls)
Pseg.ss.ct(ct,yates=FALSE)

Pseg.ss(Y,cls, sim = TRUE, Nsim = 2000)
Pseg.ss.ct(ct,yates=FALSE) 

Functions for the number of shared NNs, shared NN vector, and the number of reflexive NNs

Description

Four functions: Qval, Qvec, sharedNN and Rval.

Qval returns the Q value, the number of points with shared nearest neighbors (NNs), which occurs when two or more points share a NN, for data in any dimension.

Qvec returns the Q-value and also yields the Qv vector Qv=(Q_0,Q_1,\ldots) as well for data in any dimension, where Q_j is the number of points shared as a NN by j other points.

sharedNN returns the vector of number of points with shared NNs, Q=(Q_0,Q_1,\ldots) where Q_i is the number of points that are NN to i points, and if a point is a NN of i points, then there are i(i-1) points that share a NN. So, Q=\sum_{i>1} i(i-1)Q_i.

Rval returns the number of reflexive NNs, R (i.e., twice the number of reflexive NN pairs).

These quantities are used, e.g., in computing the variances and covariances of the entries of the nearest neighbor contingency tables used for Dixon's tests and other NNCT tests. The input must be the incidence matrix, W, of the NN digraph.

Usage

Qval(W)

Qvec(W)

sharedNN(W)

Rval(W)

Arguments

Value

The function Qval returns the Q value

The function Qvec returns a list with two elements

The function sharedNN returns a matrix with 2 rows, where first row is the j values and second row is the corresponding vector of Q_j values

The function Rval returns the R value, the number of reflexive NNs.

See the description above for the details of these quantities.

Author(s)

Elvan Ceyhan

See Also

Tval, QRval, sharedNNmc, and Ninv

Examples

#Examples for Qval
#3D data points
n<-10
Y<-matrix(runif(3*n),ncol=3)
ipd<-ipd.mat(Y)
W<-Wmat(ipd)
Qval(W)

#1D data points
X<-as.matrix(runif(10)) # need to be entered as a matrix with one column
#(i.e., a column vector), hence X<-runif(10) would not work
ipd<-ipd.mat(X)
W<-Wmat(ipd)
Qval(W)

#with ties=TRUE in the data
Y<-matrix(round(runif(15)*10),ncol=3)
ipd<-ipd.mat(Y)
W<-Wmat(ipd,ties=TRUE)
Qval(W)

#with ties=TRUE in the data
Y<-matrix(round(runif(15)*10),ncol=3)
ipd<-ipd.mat(Y)
W<-Wmat(ipd,ties=TRUE)
Qval(W)

#Examples for Qvec
#3D data points
n<-10
Y<-matrix(runif(3*n),ncol=3)
ipd<-ipd.mat(Y)
W<-Wmat(ipd)
Qvec(W)

#2D data points
n<-15
Y<-matrix(runif(2*n),ncol=2)
ipd<-ipd.mat(Y)
W<-Wmat(ipd)
Qvec(W)

#1D data points
X<-as.matrix(runif(15)) # need to be entered as a matrix with one column
#(i.e., a column vector), hence X<-runif(15) would not work
ipd<-ipd.mat(X)
W<-Wmat(ipd)
Qvec(W)

#with ties=TRUE in the data
Y<-matrix(round(runif(15)*10),ncol=3)
ipd<-ipd.mat(Y)
W<-Wmat(ipd,ties=TRUE)
Qvec(W)

#Examples for sharedNN
#3D data points
n<-10
Y<-matrix(runif(3*n),ncol=3)
ipd<-ipd.mat(Y)
W<-Wmat(ipd)
sharedNN(W)
Qvec(W)

#1D data points
X<-as.matrix(runif(15)) # need to be entered as a matrix with one column
#(i.e., a column vector), hence X<-runif(5) would not work
ipd<-ipd.mat(X)
W<-Wmat(ipd)
sharedNN(W)
Qvec(W)

#2D data points
n<-15
Y<-matrix(runif(2*n),ncol=2)
ipd<-ipd.mat(Y)
W<-Wmat(ipd)
sharedNN(W)
Qvec(W)

#with ties=TRUE in the data
Y<-matrix(round(runif(30)*10),ncol=3)
ipd<-ipd.mat(Y)
W<-Wmat(ipd,ties=TRUE)
sharedNN(W)

#Examples for Rval
#3D data points
n<-10
Y<-matrix(runif(3*n),ncol=3)
ipd<-ipd.mat(Y)
W<-Wmat(ipd)
Rval(W)

#1D data points
X<-as.matrix(runif(15)) # need to be entered as a matrix with one column
#(i.e., a column vector), hence X<-runif(5) would not work
ipd<-ipd.mat(X)
W<-Wmat(ipd)
Rval(W)

#with ties=TRUE in the data
Y<-matrix(round(runif(30)*10),ncol=3)
ipd<-ipd.mat(Y)
W<-Wmat(ipd,ties=TRUE)
Rval(W)

Functions for row and column sums of a matrix

Description

Two functions: row.sum and col.sum.

row.sum returns the row sums of a given matrix (in particular a contingency table) as a vector and col.sum returns the column sums of a given matrix as a vector. row.sum is equivalent to rowSums function and col.sum is equivalent to colSums function in the base package.

Usage

row.sum(ct)

col.sum(ct)

Arguments

Value

row.sum returns the row sums of ct as a vector col.sum returns the column sums of ct as a vector

Author(s)

Elvan Ceyhan

See Also

rowSums and colSums

Examples

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)

row.sum(ct)
rowSums(ct)

col.sum(ct)
colSums(ct)

#cls as a factor
na<-floor(n/2); nb<-n-na
fcls<-rep(c("a","b"),c(na,nb))
ct<-nnct(ipd,fcls)

row.sum(ct)
rowSums(ct)

col.sum(ct)
colSums(ct)

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

row.sum(ct)
rowSums(ct)

col.sum(ct)
colSums(ct)

Variance of Cuzick and Edwards T_k Test statistic

Description

Two functions: VarTk and VarTkaij.

Both functions compute the (finite sample) variance of Cuzick and Edwards T_k test statistic based on the number of cases within kNNs of the cases in the data under RL or CSR independence.

The common arguments for both functions are n1, representing the number of cases and k. 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 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).

The function VarTkaij uses Toshiro Tango's moments formulas based on the A=(a_{ij}) matrix (and is equivalent to the function VarTk, see Tango (2007), where a_{ij}(k) = 1 if z_j is among the kNNs of z_i and 0 otherwise.

The function varTkaij is equivalent to varTk (with $var extension).

See (Cuzick and Edwards (1990); Tango (2007)).

Usage

varTk(dat, n1, k, nonzero.mat = TRUE, ...)

varTkaij(n1, k, a)

Arguments

Value

The function VarTk returns a list with the elements

The function VarTkaij returns only var.Tk as above.

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.

Tango T (2007). “A class of multiplicity adjusted tests for spatial clustering based on case-control point data.” Biometrics, 63, 119-127.

See Also

asyvarTk

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<-2 #try also 2,3

a<-aij.mat(Y,k)

varTk(Y,n1,k)
varTk(Y,n1,k,nonzero.mat=FALSE)
varTk(Y,n1,k,method="max")

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

a<-aij.mat(Y,k)

varTkaij(n1,k,a)
varTk(Y,n1,k)$var

Variance of Cuzick and Edwards T_{run} Test statistic

Description

Two functions: varTrun and varTrun.sim.

The function varTrun computes the (finite sample) variance of Cuzick and Edwards T_{run} test statistic which is based on the number of consecutive cases from the cases in the data under RL or CSR independence. And the function varTrun.sim estimates this variance based on simulations under the RL hypothesis.

The only common argument for both functions is dat, the data set used in the functions.

n_1 is an argument for varTrun and 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 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. The argument Nsim represents the number of resamplings (without replacement) in the RL scheme, with default being 1000. cc.lab, case.lab and Nsim are arguments for varTrun.sim only.

The function varTrun might take a very long time when data size is large (even larger than 50), hence the need for the varTrun.sim function.

See (Cuzick and Edwards (1990)).

Usage

varTrun(dat, n1, ...)

varTrun.sim(dat, cc.lab, Nsim = 1000, case.lab = NULL)

Arguments

Value

The function varTrun returns the variance of Cuzick and Edwards T_{run} test statistic under RL or CSR independence. And the function varTrun.sim estimates the same variance based on simulations under the RL hypothesis.

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.