Package 'nnspat'

Description

.onAttach start message

Usage

.onAttach(libname, pkgname)

Arguments

Value

invisible()

Description

.onLoad getOption package settings

Usage

.onLoad(libname, pkgname)

Arguments

Value

invisible()

Examples

getOption("nnspat.name")

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")

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")

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")

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)

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)

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

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

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

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)

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)

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)

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)

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)

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)

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)

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)

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)

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 the 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))

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)

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)

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)

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)

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)

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)

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)

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