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R/stimodels.R
In adespatial: Multivariate Multiscale Spatial Analysis
#' Space-time interaction in ANOVA without replication
#'
#' Function \code{stimodels} performs two-way ANOVA to test space-time
#' interaction (STI) without replicates using one among a set of possible models
#' described in Legendre et al. (2010).
#' Function \code{quicksti} allows performing space-time ANOVA in a simplified
#' way. In many models, degrees of freedom are saved by coding space and/or
#' time parsimoniously using distance-based Moran Eigenvector Maps (dbMEM;
#' Borcard & Legendre 2002; Dray et al. 2006).
#'
#' @details The 'stimodels' and 'quicksti' functions should only be used
#' (1) when each site has been sampled during each survey, with no missing data,
#' and (2) when there are no replicate observations of the space-time points. When
#' there is replication, use a regular permutational Manova function such as adonis2.
#'
#' When the sites form a one-dimensional spatial transect, or a meandering line
#' such as the course of a river, with regular sampling intervals, and the time series
#' has fairly equal time intervals, one can use the S and Ti arguments to indicate
#' the number of sites in space and the number of surveys along time. The order of
#' the sites in each temporal block of the input data file will be taken to correspond
#' to the spatial positions of the sites along the transect (from 1 to S), and the
#' order of the time blocks in the data file will be taken to indicate the temporal
#' order of the surveys (from 1 to Ti). The function will then compute dbMEM
#' eigenfunctions corresponding to the spatial and temporal positions of the data
#' rows in the input data file as if they were on straight lines.
#'
#' When the sites do not form a regular, one-dimensional spatial transect, one must
#' provide a file of spatial coordinates of the sites to argument S. Similarly, when
#' the time series has unequal time intervals, one must provide a file of temporal
#' coordinates of the surveys to argument Ti.
#'
#' Alternatively, one can use arguments COD.S or COD.T to provide a matrix of Helmert
#' contrasts to be used in the analysis in place of dbMEM eigenfunctions. One can do
#' that, for example, when there are only a few surveys along time and it would not
#' be useful to represent these few surveys by dbMEM eigenfunctions. That matrix can
#' have the class "matrix" or "numeric"; they both work in functions stimodels and
#' quicksti. Arguments COD.S and COD.T can also be used to provide matrices
#' containing other types of eigenfunctions, for example AEM eigenfunctions, to be
#' used instead of dbMEM matrices computed by stimodels or quicksti. However, do not
#' code both the space and time factors as Helmert contrasts; that would not leave
#' residual degrees of freedom for the test of the interaction.
#'
#' In \code{stimodels}, tests for space-time interaction and space or time main
#' effects are conducted using one of the following models:
#' \itemize{
#' \item Model 2 - Space and Time are coded using Helmert contrasts for the main
#' effects. The interaction cannot be tested.
#' \item Model 3a - Space is coded using dbMEM variables whereas Time is coded
#' using Helmert contrasts.
#' \item Model 3b - Space is coded using Helmert contrasts whereas Time is coded
#' using dbMEM variables.
#' \item Model 4 - Both Space and Time are coded using dbMEM variables; the
#' interaction is coded as the Hadamard (or elementwise) product of the
#' space-coding by the time-coding dbMEMs.
#' \item Model 5 - Space and Time are coded using Helmert contrasts for the main
#' factor effects; the interaction is coded as the Hadamard product of the
#' space-coding by the time-coding dbMEM variables.
#' \item Model 6a - Nested model. Testing for the existence of spatial
#' structure (common or separate) using dbMEM (or other) variables coding for Space.
#' \item Model 6b - Nested model. Testing for the existence of temporal structure
#' (common or separate) using dbMEM (or other) variables coding for Time.
#' \item Model 7 - Space and Time are coded using dbMEM variables for the main
#' factor effects, but they are coded using Helmert contrasts for the interaction
#' term (not recommended). }
#'
#' With Models 2, 6a and 6b, the interaction test is not available.
#'
#' When using \code{quicksti}, space-time interaction is first tested using
#' Model 5. Depending on the outcome of this test, the main factors are tested
#' using different strategies. If the interaction is not significant then the
#' test of main factors is also done following Model 5. If the interaction is
#' significant, then a nested model (6a) is used to know whether separate
#' spatial structures exist and another (6b) to know whether separate temporal
#' structures exist. In \code{quicksti} function space and time are always
#' considered fixed factors (F ratios are constructed using residual MS in the
#' denominator).
#'
#' For the interaction the permutations are unrestricted, whereas for the main
#' factors the permutations are restricted within time blocks (for the test of
#' factor Space) or space blocks (for the test of factor Time). By default, the
#' function computes dbMEM for space and time coding, but other space and/or
#' time descriptors can be provided by the user, through \code{COD.S} and
#' \code{COD.T}.
#'
#' @param Y Site-by-species response data table. Assumes row blocks
#' corresponding to times, i.e. within each block all sites are provided,
#' always in the same order.
#' @param S Number of spatial points (when they are aligned on a transect or a
#' time series and equispaced) or a matrix of spatial coordinates (when the
#' sites are on a two-dimensional surface or on a line but very irregularly
#' spaced).
#' @param Ti Number of time campaigns (when equispaced) or a matrix (a
#' vector) of temporal coordinates (when the time campaigns are very
#' irregularly spaced).
#' @param model Linear space-time model to be used (can be either "2", "3a",
#' "3b", "4", "5", "6a", "6b", or "7").
#' @param nperm Number of permutations in the significance tests.
#' @param alpha In \code{quicksti}, confidence level for the interaction test.
#' Depending on the decision for the interaction test, the main factors are
#' tested differently.
#' @param nS Number of space dbMEMs to use (by default, -1, all dbMEMs with
#' positive autocorrelation are used).
#' @param nT Number of time dbMEMs to use (by default, -1, all dbMEMs with
#' positive autocorrelation are used).
#' @param Sfixed Logical: is factor Space fixed, or not (if FALSE, it is
#' considered a random factor).
#' @param Tfixed Logical: is factor Time fixed, or not (if FALSE, it is
#' considered a random factor).
#' @param COD.S Spatial coding functions to be used instead of dbMEM. The
#' number of columns must be lower than \code{S} and the number of rows must
#' be equal to the number of rows in \code{Y}. – Do not use full coding of S
#' (binary variables or Helmert contrasts) in COD.S to code for the factor to
#' be tested, space, in Model 6a: there would be no residual degrees of freedom
#' left for the test.
#' @param COD.T Temporal coding functions to be used instead of dbMEM. The
#' number of columns must be lower than \code{Ti} and the number of rows must
#' be equal to the number of rows in \code{Y}. – Do not use full coding of Ti
#' (binary variables or Helmert contrasts) in COD.T to code for the factor to
#' be tested, time, in Model 6b: there would be no residual degrees of freedom
#' left for the test.
#' @param save.X If TRUE, the explanatory bloc-diagonal matrix constructed for
#' model 6a or 6b is saved in the output list with the label X.
#' @param print.res If TRUE, displays the results and additional information
#' onscreen (recommended). If FALSE, only R2, F and P are printed onscreen.
#'
#' @return A list containing the following results:
#' \itemize{
#' \item \code{testS} An object with the result of the space effect test, including
#' the mean squares for the F numerator (\code{MS.num}), the mean squares for
#' the F denominator (\code{MS.den}), the proportion of explained variance
#' (\code{R2}), the adjusted proportion of explained variance (\code{R2.adj}),
#' the F statistics (\code{F}) and its p-value computed from a permutation test
#' (\code{Prob}).
#' \item \code{testT} An object with the result of the time effect test, like \code{testS}.
#' \item \code{teststi} An object with the result of the space-time interaction test,
#' like \code{testS}.
#' \item \code{X.matrix} The bloc-diagonal explanatory matrix used in test of model 6a or 6b
#' }
#'
#' @author Pierre Legendre \email{pierre.legendre@@umontreal.ca}, Miquel De Caceres
#' and Daniel Borcard
#' @seealso \code{\link{trichoptera}}
#'
#' @references
#' Borcard, D. & P. Legendre. 2002. All-scale spatial analysis of ecological data by
#' means of principal coordinates of neighbour matrices. Ecological Modelling 153: 51-68.
#' https://doi.org/10.1016/S0304-3800(01)00501-4.
#'
#' Dray, S., P. Legendre & P. R. Peres-Neto. 2006. Spatial modelling: a
#' comprehensive framework for principal coordinate analysis of neighbour
#' matrices (PCNM). Ecological Modelling 196: 483-493.
#' https://doi.org/10.1016/j.ecolmodel.2006.02.015.
#'
#' Legendre, P. & D. Borcard. 2018. Box-Cox-chord transformations for community
#' composition data prior to beta diversity analysis. Ecography 41: 1820–1824.
#' https://doi.org/10.1111/ecog.03498.
#'
#' Legendre, P., M. De Caceres & D. Borcard. 2010. Community surveys through space
#' and time: testing the space-time interaction in the absence of replication.
#' Ecology 91: 262-272. https://doi.org/10.1890/09-0199.1.
#'
#' @keywords multivariate spatial
#'
#' @examples
#' # The "trichoptera" data set contains the abundances of 56 Trichoptera species captured
#' # during 100 consecutive days in 22 emergence traps along a stream. The 22 traps
#' # (sites) form a regular transect, with geographic positions 1 to 22. The original
#' # daily data collected at each site were pooled into 10 survey periods for the study
#' # of Legendre et al. (2010) in order to reduce the very high proportion of zeros in the
#' # original data matrix. Order of the observations in the data set: the 22 traps (sites)
#' # are nested within the survey weeks, as required by the 'stimodels' and 'quicksti'
#' # functions.
#'
#' data(trichoptera)
#'
#' # log-transform the species data, excluding the Site and Date colums
#'
#' trich.log <- log1p(trichoptera[,-c(1,2)])
#'
#' # A log-chord transformation (Legendre & Borcard 2018) would also be appropriate for
#' # these data: trich.tr <- decostand(log1p(trichoptera[,-c(1,2)]), method="norm")
#'
#' # Example 1. Compute the space-time interaction test using model 5. By specifying the
#' # number of sites (traps), the sofware assumes that they form a regular transect with
#' # equispaced sites. Important note to users – In real analyses, use more than 99
#' # permutations.
#'
#' out.1 <- stimodels(trich.log, S=22, Ti=10, model="5", nperm=99)
#'
#' # The interaction is significant. Because of that, test results for the main effects,
#' # space and time, obtained with model 5, cannot be interpreted. Tests of spatial
#' # variation can be done for individual times using simple RDA against dbMEM.S
#' # variables. Likewise, tests of temporal variation can be done for individual sites
#' # using simple RDA against dbMEM.T variables. A global test of the hypothesis that none
#' # of the times shows a significant spatial structure can be done with model 6a. For a
#' # global test of temporal structure at the various sites, use model 6b.
#'
#' \donttest{ # Code not run during CRAN software tests
#'
#' # Example 2. Run space-time analysis with global tests for main effects after testing
#' # the interaction, which is significant in this example
#'
#' out.2 <- quicksti(trich.log, S=22, Ti=10, nperm=999)
#'
#' # Since the interaction is significant, function 'quicksti' will carry out the
#' # tests of existence of a spatial (at least in one of the time periods) and temporal
#' # (at least at one of the sites) structures using models 6a and 6b, respectively.
#'
#' # 3. Run space-time analysis for two time blocks only, i.e. time periods 6 and 7,
#' # then time periods 8 and 9.
#'
#' # Example 3.1. Time periods 6 and 7. The interaction is not significant. In that case,
#' # function 'quicksti' carries out the tests of the main effects using model 5.
#'
#' out.3 <- quicksti(trich.log[111:154,], S=22, Ti=2, nperm=999)
#'
#' # Example 3.2. Time periods 8 and 9. The interaction is significant. In that case,
#' # 'quicksti' carries out the tests of the spatial effects using model 6a. Model 6b
#' # cannot proceed with the test of the temporal effect because Ti=2. An explanation is
#' # printed in the output list.
#'
#' out.4 <- quicksti(trich.log[155:198,], S=22, Ti=2, nperm=999)
#'
#' # 4. Illustrations of the use of 'COD.S' and 'COD.T' in STI analysis
#'
#' # The following examples illustrate how to use other representations of the spatial or
#' # temporal relationships among observations, through arguments 'COD.S' and
#' # 'COD.T' of functions 'stimodels' and 'quicksti'. The trichoptera data
#' # are used again.
#'
#' # Example 4.1. Explicitly compute dbMEMs for the spatial structure along the regular
#' # transect, using function 'dbmem' of adespatial, and provide it to 'stimodels'
#' # or 'quicksti' as argument 'COD.S'. The dbMEMs must first be computed on the
#' # transect, then repeated (Ti-1) times to provide Ti repeats in total.
#'
#' dbMEM.S1 <- as.matrix(dbmem(1:22))
#' dbMEM.S10 <- dbMEM.S1
#' for(j in 2:10) dbMEM.S10 <- rbind(dbMEM.S10, dbMEM.S1)
#' out.5 <- stimodels(trich.log, S=22, Ti=10, model="5", COD.S=dbMEM.S10, nperm=999)
#'
#' # Results should be identical to those in output file out.1 of Example 1, except for
#' # P-values which can vary slightly.
#'
#' # Example 4.2. Assume now that the sampling sites have irregular positions, as
#' # described by the following matrix of geographic coordinates 'xy.trich'. Provide
#' # this matrix to argument S of 'stimodels'
#'
#' xy.trich = matrix(c(1:5,11:15,21:25,31:35,41,42,rep(c(1,2),11)),22,2)
#' plot(xy.trich, asp=1) # Plot a quick map of the site positions
#' out.6 <- stimodels(trich.log, S=xy.trich, Ti=10, model="5", nperm=999)
#'
#' # Example 4.3. Compute a matrix of dbMEMs for the sites. The coding matrix provided to
#' # argument 'COD.S' must contain repeated dbMEM.S codes because that matrix must have
#' # the same number of rows as matrix Y. Construct coding matrix dbMEM.S10 containing the
#' # dbMEM.S codes repeated 10 times.
#'
#' dbMEM.S1 <- as.matrix(dbmem(xy.trich))
#' dbMEM.S10 = dbMEM.S1
#' for(i in 1:9) dbMEM.S10 <- rbind(dbMEM.S10, dbMEM.S1)
#' out.7 <- stimodels(trich.log, S=22, Ti=10, model="5", COD.S=dbMEM.S10, nperm=999)
#'
#' # Compare the results with those obtained in the output file out6, example 4.2.
#'
#' # Careful: If an analysis requires a dbMEM coding matrix for 'COD.T', the dbMEM.T
#' # codes must follow the required data arrangement: sites must be nested within times.
#' # The following function can be used to construct a dbMEM.T matrix.
#'
#' MEM.T <- function(s, tt, coord.T=NULL)
#' # Documentation of function MEM.T –
#' # Generate a matrix of temporal eigenfunctions for input into stimodels,
#' # with sites nested within times.
#' # Arguments –
#' # s : number of space points (sites)
#' # tt : number of time points
#' # coord.T : optional matrix or vector giving the time point coordinates
#' {
#' n <- s*tt
#' if(is.null(coord.T)) coord.T <- as.matrix(1:tt)
#' MEM.TT <- as.matrix(dbmem(coord.T))
#' dbMEM.T <- matrix(0,n,ncol(MEM.TT)) # Empty matrix to house dbMEM.T
#' beg.x <- seq(1, n, by=s)
#' for(i in 1:tt) { # Fill tt blocks of rows with identical MEM.TT values
#' for(j in 1:s) dbMEM.T[(beg.x[i]+j-1),] <- MEM.TT[i,]
#' }
#' dbMEM.T
#' }
#'
#' # Example of use of function MEM.T
#'
#' dbMEM.T <- MEM.T(s=6, tt=5)
#' # Check the size of the dbMEM.T output matrix
#' dim(dbMEM.T)
#'
#' } # End of code not run during CRAN software tests
#'
#' @importFrom MASS ginv
#' @importFrom stats model.matrix
#' @aliases stimodels quicksti
#' @export stimodels quicksti
#' @rdname stimodels
#'
'stimodels' <- function(Y, S, Ti, model="5", nperm=999, nS=-1, nT=-1, Sfixed=TRUE,
Tfixed=TRUE, COD.S=NULL, COD.T=NULL, save.X=FALSE, print.res=TRUE)
{
if(!is.logical(print.res)) {
stop("Wrong operator; 'print.res' should be either 'FALSE' or 'TRUE'",
call.=FALSE)
}
if(model!="2" && model!="3a" && model!="3b" && model!="4" && model!="5"
&& model!="6a" && model!="6b" && model!="7") {
stop(paste("Unrecognized model ",model,
"; 'model' should be '2', '3a', '3b', '4', '5', '6a','6b' or '7'.",
sep=""), call.=FALSE)
}
# Checking that the data represent abundances or abundance-like data
if (any(Y < 0))
stop("Data contain negative values\n", call.=FALSE)
if (any(is.na(Y)))
stop("Data contain 'NA' values\n", call.=FALSE)
skip.Manova <- FALSE
res <- NA
aa <- system.time( {
# Sets the number and location of spatial and temporal points
S <- as.matrix(S)
if(dim(S)[1]==1 && dim(S)[2]==1) {
s <- S[1,1]
sitesX <- c(1:s)
} else {
s <- dim(S)[1]
sitesX <- S
}
Ti <- as.matrix(Ti)
if(dim(Ti)[1]==1 && dim(Ti)[2]==1) {
tt <- Ti[1,1]
timesX <- c(1:tt)
} else {
tt <- dim(Ti)[1]
timesX <- Ti
}
# Total number of rows in matrix
n <- s*tt
p <- dim(Y)[2]
# Check response data file containing species data
Y <- as.matrix(Y)
p <- dim(Y)[2]
if(dim(Y)[1] != n) stop("The number of rows in species file is not (S x Ti)",
call.=FALSE)
# Center response data
Y <- scale(Y, center=TRUE, scale=FALSE)
if(print.res) {
cat("=======================================================\n")
cat(" Space-time ANOVA without replicates\n")
cat(" \n")
cat(" Pierre Legendre, Miquel De Caceres, Daniel Borcard\n")
cat("=======================================================\n\n")
cat(" Number of space points (s) =", s,'\n')
cat(" Number of time points (tt) =", tt,'\n')
cat(" Number of observations (n = s*tt) =", n,'\n')
cat(" Number of response variables (p) =", p,'\n','\n')
}
# Generates space and time helmert contrasts
A <- as.factor(rep(1:s,tt))
B <- rep(1,s)
for(i in 2:tt) B <- c(B,rep(i,s))
B <- as.factor(B)
HM <- model.matrix(~ A+B, contrasts = list(A="contr.helmert", B="contr.helmert"))
HM.S <- as.matrix(HM[,2:s]) # Helmert contrasts for Space, no intercept
HM.T <- as.matrix(HM[,(s+1):(s+tt-1)]) # Helmert contrasts for Time, no intercept
# Generate dbMEM variables for Space (if necessary)
if(model=="3a"|| model=="6a" || model=="7"|| model=="4"|| model=="5") {
if(is.null(COD.S)) { # Generate spatial dbMEMs if not given by user
if(print.res) cat(" Computing dbMEMs to code for space\n")
if(s==2) {
dbMEM.S <- as.matrix(rep(c(-1,1),tt))
#
if(model=="6a") {
cat("Model 6a requires that 's' be larger than 2. When s=2, full",
"coding of the site locations by a binary variable or Helmert",
"contrast does not leave any degree of freedom for the residuals",
"in the test of the Space factor.\n\n")
skip.Manova <- TRUE
}
}
else {
dbMEM.S.tmp <- dbmem(sitesX, MEM.autocor="positive")
SS <- as.matrix(dbMEM.S.tmp)
dbMEM.S.thresh <- give.thresh(dist(sitesX))
if(print.res) cat(" Truncation level for space dbMEMs =",
dbMEM.S.thresh, "\n")
dbMEM.S <- SS
for(j in 2:tt) dbMEM.S <- rbind(dbMEM.S,SS)
if(nS==-1) nS <- ncol(SS)
else {
if(nS > ncol(SS)) {
cat("\n Number of requested spatial variables nS =", nS,
"larger than available\n")
cat(" The", ncol(SS),
"available spatial dbMEMs will be used\n\n")
nS <- ncol(SS)
}
else dbMEM.S <- dbMEM.S[, 1:nS, drop = FALSE]
}
}
} else {
dbMEM.S <- apply(as.matrix(COD.S),2,scale,center=TRUE,scale=TRUE)
nS <- dim(dbMEM.S)[2]
if(nS>=s)
stop("The number of spatial coding functions must be lower than S",
call.=FALSE)
if(nrow(as.matrix(COD.S))!=dim(Y)[1])
stop("The number of rows in COD.S must be equal to the number",
"of observations", call.=FALSE)
}
}
# Generate dbMEM variables for Time (if necessary)
if(model=="3b" || model=="6b"|| model=="7"||model=="4" || model=="5") {
if(is.null(COD.T)) { # Generate temporal dbMEM variables if not given by user
if(tt==2) { # Two points in time: construct 2 Helmert contrast variables
dbMEM.T=as.matrix(c(rep(-1, s), rep(1,s))) #DB
#
if(model=="6b") {
cat("Model 6b requires that 'tt' be larger than 2. When tt=2, full",
"coding of the times by a binary variable or Helmert",
"contrast does not leave any degree of freedom for the residuals",
"in the test of the Time factor.\n\n")
skip.Manova <- TRUE
}
} else { # COD.T is NULL, tt>2
if(print.res) cat(" Computing dbMEMs to code for time\n")
TT <- as.matrix(dbmem(timesX))
dbMEM.T.thresh <- give.thresh(dist(timesX))
if(print.res) cat(" Truncation level for time dbMEMs =",
dbMEM.T.thresh, "\n\n")
beg <- seq(1,n,by=s) # 1 7 13 19 25
dbMEM.T <- matrix(0,n,ncol(TT)) # Empty matrix 30x2 to house dbMEM.T
for(i in 1:tt) { # Fill tt blocks of rows with identical MEM.T values
for(j in 1:s) dbMEM.T[(beg[i]+j-1),] <- TT[i,]
}
if(nT==-1) { nT <- ncol(TT) # nT: number of dbMEM variables in TT
} else { # If the arguments specify an imposed value to nT
if(nT > ncol(TT)) {
cat("\n Number of requested temporal variables nT =", nT,
"larger than available\n")
cat(" The", ncol(TT), "available temporal dbMEM will be used\n\n")
nT <- ncol(TT)
} else {
dbMEM.T <- dbMEM.T[, 1:nT, drop = FALSE]
}
}
} # End of if(is.null(COD.T))
} else { # COD.T is present
dbMEM.T=apply(as.matrix(COD.T),2,scale,center=TRUE,scale=TRUE)
nT <- dim(dbMEM.T)[2]
if(nT>=tt) stop("The number of temporal coding functions",
"must be lower than Ti", call.=FALSE)
if(nrow(as.matrix(COD.T))!=dim(Y)[1])
stop("The number of rows in COD.T must be equal to the number",
"of observations", call.=FALSE)
}
}
if(!skip.Manova) { # Begin of "(!skip.Manova)"
if(print.res) {
if(model!="2" && model!="3b" && model!="6b") {
if(s==2) #DB To avoid number = -1
cat(" Number of space coding functions = 1 \n\n")
else
cat(" Number of space coding functions =", nS,'\n\n')
}
if(model!="2" && model!="3a" && model!="6a") {
if(tt==2) #DB To avoid number = -1
cat(" Number of time coding functions = 1 \n\n")
else
cat(" Number of time coding functions =", nT, "\n\n")}
}
test.STI <- TRUE
test.S <- TRUE
test.T <- TRUE
# Generate matrices X (variables for the factor of interest)
# and W (covariables) for each test to be done
if(model=="5") {
XSTI <- dbMEM.S*dbMEM.T[,1]
if(dim(dbMEM.T)[2]>1)
for(j in 2:dim(dbMEM.T)[2]) XSTI <- cbind(XSTI,dbMEM.S*dbMEM.T[,j])
XS <- HM.S
XT <- HM.T
if(print.res) {
cat(" MODEL V: HELMERT CONTRAST FOR TESTING MAIN FACTORS.\n")
cat(" SPACE AND TIME dbMEMs FOR TESTING INTERACTION.\n")
}
} else if(model=="4") {
XSTI <- dbMEM.S*dbMEM.T[,1]
if(dim(dbMEM.T)[2]>1)
for(j in 2:dim(dbMEM.T)[2]) XSTI = cbind(XSTI,dbMEM.S*dbMEM.T[,j])
XS <- dbMEM.S
XT <- dbMEM.T
if(print.res) {
cat(" MODEL IV: dbMEMs FOR BOTH SPACE AND TIME.\n")
}
} else if(model=="3a") {
XSTI <- dbMEM.S*HM.T[,1]
if(tt>2) for(j in 2:(tt-1)) XSTI <- cbind(XSTI,dbMEM.S*HM.T[,j])
XS <- dbMEM.S
XT <- HM.T
if(print.res) {
cat(" MODEL IIIa: dbMEMs FOR SPACE AND HELMERT CONTRASTS FOR TIME.\n")
}
} else if(model=="3b") {
XSTI <- dbMEM.T*HM.S[,1]
if(dim(HM.S)[2]>1)
for(j in 2:(s-1)) XSTI <- cbind(XSTI,dbMEM.T*HM.S[,j])
XS <- HM.S
XT <- dbMEM.T
if(print.res) {
cat(" MODEL IIIb: HELMERT CONTRASTS FOR SPACE AND dbMEMs FOR TIME.\n")
}
} else if(model=="7") {
XSTI <- HM.S*HM.T[,1]
if(dim(HM.T)[2]>1) for(j in 2:dim(HM.T)[2]) XSTI <- cbind(XSTI,HM.S*HM.T[,j])
XS <- dbMEM.S
XT <- dbMEM.T
if(print.res) {
cat(" MODEL VII: dbMEMs FOR BOTH SPACE AND TIME,",
"BUT HELMERT CONTRAST FOR INTERACTION.\n")
}
} else if(model=="6a") {
# BEGIN: New code (PL) to assemble a block diagonal matrix for XS
XS <- matrix(0,n,tt*nS) # Empty matrix
beg.x <- seq(1,n,by=s) # Beginning rows, vertical
beg.y <- seq(1,tt*nS,by=nS) # Beginning columns, horizontal
tmp <- dbMEM.S[1:s,]
for(j in 1:tt)
XS[(beg.x[j]):(beg.x[j]+s-1),(beg.y[j]):(beg.y[j]+nS-1)] <- tmp
XT <- HM.T
XSTI = NULL
if(print.res) {
cat(" MODEL VIa: NESTED MODEL.\n")
cat(" TESTING FOR THE EXISTENCE",
"OF SPATIAL STRUCTURE (COMMON OR SEPARATE)\n")
}
test.STI <- FALSE
test.T <- FALSE
} else if(model=="6b") {
# BEGIN: New code (PL)) to assemble a block diagonal matrix for XT
beg.x <- seq(1,n,by=s) # Rows to fill at the beginning
beg.y <- seq(1,s*nT,by=nT) # Columns to fill at the beginning
TT = as.matrix(dbMEM.T[beg.x,]) # dbMEM for times, only once
XT <- matrix(0,n,s*nT) # Empty matrix to house the diagonal blocks
for(j in 1:s) { # Write groups of s columns in XT
for(i in 1:tt) { # Fill tt rows with T1 to T12 in each group of 3 columns
XT[(beg.x[i]+j-1), (beg.y[j]):(beg.y[j]+nT-1)] <- TT[i,1:nT]
}
} ## End of New code
XS <- HM.S
XSTI = NULL
if(print.res) {
cat(" MODEL VIb: NESTED MODEL.\n")
cat(" TESTING FOR THE EXISTENCE",
"OF TEMPORAL STRUCTURE (COMMON OR SEPARATE).\n")
}
test.STI <- FALSE
test.S <- FALSE
} else if(model=="2") {
XS <- HM.S
XT <- HM.T
XSTI <- NULL
if(print.res) {
cat(" MODEL II: HELMERT CONTRAST FOR SPACE AND TIME.",
"NO INTERACTION TERM.\n")
}
test.STI <- FALSE
}
if(print.res) {
cat(" Number of space variables =", dim(XS)[2],'\n')
cat(" Number of time variables =", dim(XT)[2],'\n')
if(test.STI) {
cat(" Number of interaction variables =", dim(XSTI)[2],'\n')
cat(" Number of residual degrees of freedom =",
(s*tt-dim(XS)[2]-dim(XT)[2]-dim(XSTI)[2]-1),"\n\n")
if((s*tt-dim(XS)[2]-dim(XT)[2]-dim(XSTI)[2]-1) <= 0)
cat("Not enough residual degrees of freedom for testing.\n")
} else {
cat(" Number of residual degrees of freedom =",
(s*tt-dim(XS)[2]-dim(XT)[2]-1),"\n\n")
if((s*tt-dim(XS)[2]-dim(XT)[2]-1) <= 0)
cat("Not enough residual degrees of freedom for testing.\n")
}
}
# Call Manova by RDA, function manovRDa.R
res <- manovRDa(Y=Y,s=s,tt=tt,S.mat=XS,T.mat=XT,STI.mat=XSTI, Sfixed=Sfixed,
Tfixed=Tfixed, S.test=test.S, T.test=test.T, STI.test=test.STI,
model = model, nperm=nperm, save.X=save.X)
if(test.STI==TRUE) {
cat(' Interaction test: R2 =', round(res$testSTI$R2, 4),
' F =',round(res$testSTI$F, 4),' P(',nperm,'perm) =',res$testSTI$Prob,'\n')
}
if(test.S==TRUE) {
cat(' Space test: R2 =', round(res$testS$R2, 4),
' F =',round(res$testS$F, 4),' P(',nperm,'perm) =',res$testS$Prob,'\n')
}
if(test.T==TRUE) {
cat(' Time test: R2 =', round(res$testT$R2, 4),
' F =',round(res$testT$F, 4),' P(',nperm,'perm) =',res$testT$Prob,'\n\n')
}
} # End of "if(!skip.Manova)"
})
aa[3] <- sprintf("%2f",aa[3])
if(print.res) {
cat("-------------------------------------------------------\n")
cat(" Time for computation =",aa[3]," sec",'\n')
cat("=======================================================\n\n")
}
class(res) <- "sti"
invisible(res)
}
#' @rdname stimodels
'quicksti' <- function(Y, S, Ti, nperm=999, alpha = 0.05, COD.S=NULL, COD.T=NULL,
save.X=FALSE, print.res=TRUE)
{
if(!is.logical(print.res)) {
stop("Wrong operator; 'print.res' should be either 'FALSE' or 'TRUE'.",
call.=FALSE)
}
aa <- system.time( {
# Sets the number and location of spatial and temporal points
S <- as.matrix(S)
if(dim(S)[1]==1 && dim(S)[2]==1) {
s <- S[1,1]
sitesX <- c(1:s)
} else {
s <- dim(S)[1]
sitesX <- S
}
Ti <- as.matrix(Ti)
if(dim(Ti)[1]==1 && dim(Ti)[2]==1) {
tt <- Ti[1,1]
timesX <- c(1:tt)
} else {
tt <- dim(Ti)[1]
timesX <- Ti
}
# Total number of rows in matrix
n <- s*tt
# Check response data file containing species data
Y <- as.matrix(Y)
p <- dim(Y)[2]
if(dim(Y)[1] != n) stop("The number of rows in species file is not (S x Ti).",
call.=FALSE)
# Center response data
Y <- scale(Y, center=TRUE, scale=FALSE)
if(print.res) {
cat("=========================================================\n")
cat(" Space-time ANOVA without replicates\n")
cat(" \n")
cat(" Pierre Legendre, Miquel De Caceres, Daniel Borcard\n")
cat("---------------------------------------------------------\n\n")
cat(" Number of space points (s) =", s,'\n')
cat(" Number of time points (tt) =", tt,'\n')
cat(" Number of observations (n = s*tt) =", n,'\n')
cat(" Number of response variables (p) =", p,'\n')
cat(" Significance level for the interaction test (alpha) =", alpha,'\n','\n')
}
# Generates spatial dbMEMs if not provided by user
if(is.null(COD.S)) {
if(print.res) cat(" Computing dbMEMs to code for space\n")
if(s==2) {
dbMEM.S <- as.matrix(rep(c(-1,1),tt))
nS <- dim(dbMEM.S)[2]
if(print.res) cat("\n There are only two sites. A vector of Helmert",
"contrasts will be used to represent them\n\n")
}
else {
SS <- as.matrix(dbmem(sitesX))
nS <- ncol(SS)
dbMEM.S.thresh <- give.thresh(dist(sitesX))
if(print.res) cat(" Truncation level for space dbMEMs =", dbMEM.S.thresh,"\n")
dbMEM.S <- SS
for(j in 2:tt) dbMEM.S = rbind(dbMEM.S,SS)
}
} else {
dbMEM.S <- apply(as.matrix(COD.S),2,scale,center=TRUE,scale=TRUE)
nS <- dim(dbMEM.S)[2]
if(nS >= s) stop("The number of spatial coding functions must be smaller",
"than S.", call.=FALSE)
if(nrow(as.matrix(COD.S))!=dim(Y)[1]) stop("The number of rows in COD.S must",
"be equal to the number of observations.", call.=FALSE)
}
# Generate dbMEM variables for time if not provided by user
if(is.null(COD.T)) {
nT <- trunc(tt/2)
if(print.res) cat(" Computing dbMEMs to code for time\n")
if(tt==2) {
dbMEM.T <- as.matrix(c(rep(-1, s), rep(1,s)))
if(print.res) cat("\n There are only two time points. A vector of",
"Helmert contrasts will be used to represent them\n\n")
} else {
TT <- as.matrix(dbmem(timesX))
nT <- ncol(TT)
dbMEM.T.thresh <- give.thresh(dist(timesX))
if(print.res) cat(" Truncation level for time dbMEMs =",dbMEM.T.thresh,"\n\n")
beg.x <- seq(1,n,by=s) # New code (PL)
nT = ncol(TT)
dbMEM.T <- matrix(0,n,nT) # Empty matrix 30x2 to house dbMEM.T
for(i in 1:tt) { # Fill tt blocks of rows with identical MEM.T values
for(j in 1:s) dbMEM.T[(beg.x[i]+j-1),] <- TT[i,]
}
}
} else {
dbMEM.T <- apply(as.matrix(COD.T),2,scale,center=TRUE,scale=TRUE)
nT <- dim(dbMEM.T)[2]
if(nT >= tt) stop("The number of temporal coding functions must be",
"lower than Ti.", call.=FALSE)
if(nrow(as.matrix(COD.T))!=dim(Y)[1]) stop("The number of rows in",
"COD.T must be equal to the number of observations.", call.=FALSE)
}
if(s*tt-s-tt-nT*nS-1<=0) stop("Not enough degrees of freedom for testing",
"interaction.", call.=FALSE)
if(print.res) {
cat(" Number of space coding functions =", nS,'\n')
cat(" Number of time coding functions =", nT, "\n\n")
}
# Generates space and time helmert contrasts
A <- as.factor(rep(1:s,tt))
B <- rep(1,s)
for(i in 2:tt) B <- c(B,rep(i,s))
B <- as.factor(B)
HM <- model.matrix(~ A + B, contrasts =list(A="contr.helmert", B="contr.helmert"))
HM.S <- as.matrix(HM[,2:s])
HM.T <- as.matrix(HM[,(s+1):(s+tt-1)])
res2 <- NULL
res3 <- NULL
# Test significance for interaction effect
#
# Defines X (variables for the factor of interest)
# and W (covariables) for the space-time test
#
XSTI <- dbMEM.S*dbMEM.T[,1]
if(dim(dbMEM.T)[2]>1) for(j in 2:dim(dbMEM.T)[2])
XSTI <- cbind(XSTI,dbMEM.S*dbMEM.T[,j])
if(print.res) {
cat("------------------------------------------\n")
cat(" Testing space-time interaction (model 5)\n")
cat("------------------------------------------\n\n")
cat(" Number of space variables =", dim(HM.S)[2],'\n')
cat(" Number of time variables =", dim(HM.T)[2],'\n')
cat(" Number of interaction variables =", dim(XSTI)[2],'\n')
cat(" Number of residual degrees of freedom =",
(s*tt-dim(XSTI)[2]-dim(HM.S)[2]-dim(HM.T)[2]-1),"\n\n")
if((s*tt-dim(XSTI)[2]-dim(HM.S)[2]-dim(HM.T)[2]-1) == 0)
stop("Not enough residual degrees of freedom for testing.",
"Try with a smaller number of space or time variables.", call.=FALSE)
}
res <- manovRDa(Y=Y,s=s,tt=tt,S.mat=HM.S,T.mat=HM.T,STI.mat=XSTI, S.test=TRUE,
T.test=TRUE, STI.test=TRUE, model="5", nperm=nperm, save.X=save.X)
cat(' Interaction test: R2 =', round(res$testSTI$R2, 4),
' F =',round(res$testSTI$F,4),' P(',nperm,'perm) =',res$testSTI$Prob,"\n")
# Begin models 6a and 6b if Prob<=alpha
if(res$testSTI$Prob<=alpha) {
# Model="6a"
XS <- dbMEM.S
for(j in 1:(tt-1)) XS = cbind(XS,dbMEM.S*HM.T[,j])
XT <- HM.T
if(print.res) {
cat("----------------------------------------------------\n")
cat(" Testing for separate spatial structures (model 6a)\n")
cat("----------------------------------------------------\n\n")
cat(" Number of space variables =", dim(XS)[2],'\n')
cat(" Number of time variables =", dim(XT)[2],'\n')
cat(" Number of residual degrees of freedom =",
(s*tt-dim(XS)[2]-dim(XT)[2]-1),"\n\n")
}
if(s==2) { # skip.manova
if(!print.res) cat("\n")
cat("Model 6a requires that 's' be larger than 2. When s=2, full",
"coding of the site locations by a binary variable or Helmert",
"contrast does not leave any degree of freedom for the residuals",
"in the test of the Space factor.\n\n")
res2$X.matrix <- NA
} else {
res2 <- manovRDa(Y=Y,s=s,tt=tt,S.mat=XS,T.mat=XT,STI.mat=NULL,
S.test=TRUE, T.test=FALSE, STI.test=FALSE, model="6a",
nperm=nperm, save.X=save.X)
res$testS <- res2$testS
cat(' Space test: R2 =', round(res$testS$R2,4), ' F =',
round(res$testS$F,4),' P(',nperm,'perm) =',res$testS$Prob,"\n")
} # End of "if(s==2)"
# Model="6b"
XT <- dbMEM.T
for(j in 1:(s-1)) XT <- cbind(XT,dbMEM.T*HM.S[,j])
XS <- HM.S
if(print.res) {
cat("-----------------------------------------------------\n")
cat(" Testing for separate temporal structures (model 6b)\n")
cat("-----------------------------------------------------\n\n")
cat(" Number of space variables =", dim(XS)[2],'\n')
cat(" Number of time variables =", dim(XT)[2],'\n')
cat(" Number of residual degrees of freedom =",
(s*tt-dim(XS)[2]-dim(XT)[2]-1),"\n\n")
}
if(tt==2) { # skip.manova
if(!print.res) cat("\n")
cat("Model 6b requires that 'tt' be larger than 2. When tt=2, full",
"coding of the times by a binary variable or Helmert",
"contrast does not leave any degree of freedom for the residuals",
"in the test of the Time factor.\n\n")
res3$X.matrix <- NA
} else {
res3 <- manovRDa(Y=Y,s=s,tt=tt,S.mat=XS,T.mat=XT,STI.mat=NULL,
S.test=FALSE, T.test=TRUE, STI.test=FALSE, model="6b",
nperm=nperm, save.X=save.X)
res$testT <- res3$testT
cat(' Time test: R2 =', round(res$testT$R2,4),' F =',
round(res$testT$F,4),' P(',nperm,'perm) =',res$testT$Prob,"\n\n")
} # End of "if(tt==2)"
} else { # End of "if(res$testSTI$Prob<=alpha)", # End of Models 6a and 6b
save.X = FALSE
# Tests of Space and Time from the results of model="5"
if(print.res) {
cat(
"---------------------------------------------------------------------\n")
cat(
" Testing for common spatial and common temporal structures (model 5)\n")
cat(
"---------------------------------------------------------------------\n\n")
}
cat(' Space test: R2 =', round(res$testS$R2,4),
' F =',round(res$testS$F,4),' P(',nperm,'perm) =',res$testS$Prob,'\n')
cat(' Time test: R2 =', round(res$testT$R2,4),
' F =',round(res$testT$F,4),' P(',nperm,'perm) =',res$testT$Prob,'\n\n')
}
})
aa[3] <- sprintf("%2f",aa[3])
if(print.res) {
cat("---------------------------------------------------------\n")
cat(" Time for computation =",aa[3]," sec",'\n')
cat("=========================================================\n\n")
}
if(save.X) {
out <- list(res, res2$X.matrix, res3$X.matrix)
names(out) <- c("tests", "space.blocks", "time.blocks")
invisible(out)
}
else {
invisible(res)
}
}
'manovRDa' <- function(Y, s, tt, S.mat=NULL, T.mat=NULL, STI.mat=NULL, Sfixed=TRUE,
Tfixed=TRUE, S.test=TRUE, T.test=TRUE, STI.test=TRUE, model="5",
nperm=999, save.X)
{
n <- nrow(Y)
p <- ncol(Y)
a <- ncol(S.mat)
b <- ncol(T.mat)
if(!is.null(STI.mat)) {cc <- ncol(STI.mat)}
else {cc = 0}
A <- S.mat
B <- T.mat
if(!is.null(STI.mat)) AxB <- STI.mat
# Compute projector of A and Yfit.A
invA <- ginv(t(A) %*% A)
projA <- A %*% invA %*% t(A)
Yfit.A <- projA %*% Y
# Compute projector of B and Yfit.B
invB <- ginv(t(B) %*% B)
projB <- B %*% invB %*% t(B)
Yfit.B <- projB %*% Y
# Compute projector of AxB and Yfit.AxB
if(!is.null(STI.mat)) {
invAxB <- ginv(t(AxB) %*% AxB)
projAxB <- AxB %*% invAxB %*% t(AxB)
Yfit.AxB <- projAxB %*% Y
} else {
projAxB=NULL
}
# Create a "compound matrix" to obtain R-square and adjusted R-square
if(!is.null(STI.mat)) {
ABAxB <- cbind(A,B,AxB)
} else {
ABAxB <- cbind(A,B)
}
# Compute projector of ABAxB and Yfit.ABAxB
invABAxB <- ginv(t(ABAxB) %*% ABAxB)
projABAxB <- ABAxB %*% invABAxB %*% t(ABAxB)
Yfit.ABAxB <- projABAxB %*% Y
# If save.X is TRUE, save the block-diagonal matrix X used in test of model 6a or 6b
X <- NA
if(save.X && model=="6a") X <- S.mat
if(save.X && model=="6b") X <- T.mat
# Compute Sums of squares (SS) and Mean squares (MS)
SS.Y <- sum(Y*Y)
SS.Yfit.ABAxB <- sum(Yfit.ABAxB*Yfit.ABAxB)
SS.Yfit.A <- sum(Yfit.A*Yfit.A)
SS.Yfit.B <- sum(Yfit.B*Yfit.B)
if(!is.null(STI.mat)) SS.Yfit.AxB <- sum(Yfit.AxB*Yfit.AxB)
MS.A <- SS.Yfit.A/a
MS.B <- SS.Yfit.B/b
if(!is.null(STI.mat)) MS.AxB <- SS.Yfit.AxB/cc
MS.Res <- (SS.Y-SS.Yfit.ABAxB)/(n-(a+b+cc)-1)
# Test interaction (unrestricted permutations)
if(STI.test==TRUE) {
nPGE.AxB = 1
Fref.AxB <- MS.AxB/MS.Res
nPGE.AxB=
.Call("sti_loop",nperm,Y,s,tt,a,b,cc,SS.Y,Fref.AxB,projAxB,projABAxB) ### NM
P.AxB <- nPGE.AxB/(nperm+1)
R2 <- SS.Yfit.AxB/SS.Y
R2a <- 1-((n-1)/(n-dim(STI.mat)[2]-1))*(1-R2)
testSTI <-
list(MS.num=MS.AxB, MS.den=MS.Res, R2=R2, R2.adj=R2a, F=Fref.AxB, Prob=P.AxB)
} else {
testSTI <- NULL
}
# Test factor A (space) using restricted permutations within time blocks
if(S.test==TRUE) {
#########################
nPGE.A=1
if(Tfixed==FALSE && !is.null(STI.mat)) {
# Time random factor in crossed design with interaction
Fref.A <- MS.A/MS.AxB
MS.den <- MS.AxB
T_fixed <- 1
} else if(Tfixed==FALSE && model=="6b") {
# Time random factor in nested design
Fref.A <- MS.A/MS.B
MS.den <- MS.B
T_fixed <- 2
} else {
Fref.A <- MS.A/MS.Res
MS.den <- MS.Res
T_fixed <- 3
}
nPGE.A=.Call("s_loop",nperm,Y,s,tt,a,b,cc,SS.Y,Fref.A,projA,projB,projAxB,
projABAxB,T_fixed) ### NM
P.A <- nPGE.A/(nperm+1)
R2 <- SS.Yfit.A/SS.Y
R2a <- 1-((n-1)/(n-a-1))*(1-R2)
testS <- list(MS.num=MS.A, MS.den=MS.den, R2=R2, R2.adj=R2a, F=Fref.A, Prob=P.A)
} else {
testS <- NULL
}
# Test factor B (time) using restricted permutations within time blocks
if(T.test==TRUE) {
nPGE.B = 1
if(Sfixed==FALSE && !is.null(STI.mat)) {
# Space random factor in crossed design with interaction
Fref.B <- MS.B/MS.AxB
MS.den <- MS.AxB
T_fixed=1
} else if(Sfixed==FALSE && model=="6a") { # Space random factor in nested design
Fref.B <- MS.B/MS.A
MS.den <- MS.A
T_fixed=2
} else {
Fref.B <- MS.B/MS.Res
MS.den <- MS.Res
T_fixed=3
}
nPGE.B <- .Call("t_loop",nperm,Y,s,tt,a,b,cc,SS.Y,Fref.B,projA,projB,projAxB,
projABAxB,T_fixed) # NM
P.B <- nPGE.B/(nperm+1)
R2 <- SS.Yfit.B/SS.Y
R2a <- 1-((n-1)/(n-b-1))*(1-R2)
testT <- list(MS.num=MS.B, MS.den=MS.den, R2=R2, R2.adj=R2a, F=Fref.B, Prob=P.B)
} else {
testT <- NULL
}
return(list(testSTI = testSTI, testS = testS, testT=testT, X.matrix=X))
}
'restrictedPerm' <- function(nobs.block, nblock, n, restPerm, vec)
#
# restPerm == 0: Unrestricted permutation.
#
# restPerm == 1: Restricted permutation of the observations within each block.
# The data are arranged by blocks, with all observations forming a block
# placed in adjacent positions in the file. Example:
# BLOCK-1: obs1, obs2, ... obs-nobs.block; BLOCK-2: obs1, obs2, ... obs-nobs.block; etc.
#
# restPerm == 2: Restricted permutation of the observations across blocks (within the
# blocks formed by each nth observation).
#
# Vector 'vec' contains the initial order of the objects, e.g. vec=c(1:n).
# At the end of the function, it gives the permuted order of the objects.
#
# Examples: toto0 <- restrictedPerm(6,4,24,0,c(1:24))
# toto1 <- restrictedPerm(6,4,24,1,c(1:24))
#
# Pierre Legendre, January 2006
# Miquel De Caceres, February 2009
{
if(restPerm == 0) {
vec <- sample(vec[1:n],n)
} else if(restPerm==1) {
for(j in 1:nblock) {
i1 <- nobs.block*(j-1)+1
i2 <- nobs.block*j
vec[i1:i2] <- sample(vec[i1:i2],nobs.block)
}
} else {
vecT <- vec
for(j in 1:nobs.block) {
ind <- sample(1:nblock)
for(i in 1:nblock) {
vec[nobs.block*(i-1)+j] <- vecT[nobs.block*(ind[i]-1)+j]
}
}
}
return(vec)
}
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#' Space-time interaction in ANOVA without replication
#'
#' Function \code{stimodels} performs two-way ANOVA to test space-time
#' interaction (STI) without replicates using one among a set of possible models
#' described in Legendre et al. (2010).
#' Function \code{quicksti} allows performing space-time ANOVA in a simplified
#' way. In many models, degrees of freedom are saved by coding space and/or
#' time parsimoniously using distance-based Moran Eigenvector Maps (dbMEM;
#' Borcard & Legendre 2002; Dray et al. 2006).
#'
#' @details The 'stimodels' and 'quicksti' functions should only be used
#' (1) when each site has been sampled during each survey, with no missing data,
#' and (2) when there are no replicate observations of the space-time points. When
#' there is replication, use a regular permutational Manova function such as adonis2.
#'
#' When the sites form a one-dimensional spatial transect, or a meandering line
#' such as the course of a river, with regular sampling intervals, and the time series
#' has fairly equal time intervals, one can use the S and Ti arguments to indicate
#' the number of sites in space and the number of surveys along time. The order of
#' the sites in each temporal block of the input data file will be taken to correspond
#' to the spatial positions of the sites along the transect (from 1 to S), and the
#' order of the time blocks in the data file will be taken to indicate the temporal
#' order of the surveys (from 1 to Ti). The function will then compute dbMEM
#' eigenfunctions corresponding to the spatial and temporal positions of the data
#' rows in the input data file as if they were on straight lines.
#'
#' When the sites do not form a regular, one-dimensional spatial transect, one must
#' provide a file of spatial coordinates of the sites to argument S. Similarly, when
#' the time series has unequal time intervals, one must provide a file of temporal
#' coordinates of the surveys to argument Ti.
#'
#' Alternatively, one can use arguments COD.S or COD.T to provide a matrix of Helmert
#' contrasts to be used in the analysis in place of dbMEM eigenfunctions. One can do
#' that, for example, when there are only a few surveys along time and it would not
#' be useful to represent these few surveys by dbMEM eigenfunctions. That matrix can
#' have the class "matrix" or "numeric"; they both work in functions stimodels and
#' quicksti. Arguments COD.S and COD.T can also be used to provide matrices
#' containing other types of eigenfunctions, for example AEM eigenfunctions, to be
#' used instead of dbMEM matrices computed by stimodels or quicksti. However, do not
#' code both the space and time factors as Helmert contrasts; that would not leave
#' residual degrees of freedom for the test of the interaction.
#'
#' In \code{stimodels}, tests for space-time interaction and space or time main
#' effects are conducted using one of the following models:
#' \itemize{
#' \item Model 2 - Space and Time are coded using Helmert contrasts for the main
#' effects. The interaction cannot be tested.
#' \item Model 3a - Space is coded using dbMEM variables whereas Time is coded
#' using Helmert contrasts.
#' \item Model 3b - Space is coded using Helmert contrasts whereas Time is coded
#' using dbMEM variables.
#' \item Model 4 - Both Space and Time are coded using dbMEM variables; the
#' interaction is coded as the Hadamard (or elementwise) product of the
#' space-coding by the time-coding dbMEMs.
#' \item Model 5 - Space and Time are coded using Helmert contrasts for the main
#' factor effects; the interaction is coded as the Hadamard product of the
#' space-coding by the time-coding dbMEM variables.
#' \item Model 6a - Nested model. Testing for the existence of spatial
#' structure (common or separate) using dbMEM (or other) variables coding for Space.
#' \item Model 6b - Nested model. Testing for the existence of temporal structure
#' (common or separate) using dbMEM (or other) variables coding for Time.
#' \item Model 7 - Space and Time are coded using dbMEM variables for the main
#' factor effects, but they are coded using Helmert contrasts for the interaction
#' term (not recommended). }
#'
#' With Models 2, 6a and 6b, the interaction test is not available.
#'
#' When using \code{quicksti}, space-time interaction is first tested using
#' Model 5. Depending on the outcome of this test, the main factors are tested
#' using different strategies. If the interaction is not significant then the
#' test of main factors is also done following Model 5. If the interaction is
#' significant, then a nested model (6a) is used to know whether separate
#' spatial structures exist and another (6b) to know whether separate temporal
#' structures exist. In \code{quicksti} function space and time are always
#' considered fixed factors (F ratios are constructed using residual MS in the
#' denominator).
#'
#' For the interaction the permutations are unrestricted, whereas for the main
#' factors the permutations are restricted within time blocks (for the test of
#' factor Space) or space blocks (for the test of factor Time). By default, the
#' function computes dbMEM for space and time coding, but other space and/or
#' time descriptors can be provided by the user, through \code{COD.S} and
#' \code{COD.T}.
#'
#' @param Y Site-by-species response data table. Assumes row blocks
#' corresponding to times, i.e. within each block all sites are provided,
#' always in the same order.
#' @param S Number of spatial points (when they are aligned on a transect or a
#' time series and equispaced) or a matrix of spatial coordinates (when the
#' sites are on a two-dimensional surface or on a line but very irregularly
#' spaced).
#' @param Ti Number of time campaigns (when equispaced) or a matrix (a
#' vector) of temporal coordinates (when the time campaigns are very
#' irregularly spaced).
#' @param model Linear space-time model to be used (can be either "2", "3a",
#' "3b", "4", "5", "6a", "6b", or "7").
#' @param nperm Number of permutations in the significance tests.
#' @param alpha In \code{quicksti}, confidence level for the interaction test.
#' Depending on the decision for the interaction test, the main factors are
#' tested differently.
#' @param nS Number of space dbMEMs to use (by default, -1, all dbMEMs with
#' positive autocorrelation are used).
#' @param nT Number of time dbMEMs to use (by default, -1, all dbMEMs with
#' positive autocorrelation are used).
#' @param Sfixed Logical: is factor Space fixed, or not (if FALSE, it is
#' considered a random factor).
#' @param Tfixed Logical: is factor Time fixed, or not (if FALSE, it is
#' considered a random factor).
#' @param COD.S Spatial coding functions to be used instead of dbMEM. The
#' number of columns must be lower than \code{S} and the number of rows must
#' be equal to the number of rows in \code{Y}. – Do not use full coding of S
#' (binary variables or Helmert contrasts) in COD.S to code for the factor to
#' be tested, space, in Model 6a: there would be no residual degrees of freedom
#' left for the test.
#' @param COD.T Temporal coding functions to be used instead of dbMEM. The
#' number of columns must be lower than \code{Ti} and the number of rows must
#' be equal to the number of rows in \code{Y}. – Do not use full coding of Ti
#' (binary variables or Helmert contrasts) in COD.T to code for the factor to
#' be tested, time, in Model 6b: there would be no residual degrees of freedom
#' left for the test.
#' @param save.X If TRUE, the explanatory bloc-diagonal matrix constructed for
#' model 6a or 6b is saved in the output list with the label X.
#' @param print.res If TRUE, displays the results and additional information
#' onscreen (recommended). If FALSE, only R2, F and P are printed onscreen.
#'
#' @return A list containing the following results:
#' \itemize{
#' \item \code{testS} An object with the result of the space effect test, including
#' the mean squares for the F numerator (\code{MS.num}), the mean squares for
#' the F denominator (\code{MS.den}), the proportion of explained variance
#' (\code{R2}), the adjusted proportion of explained variance (\code{R2.adj}),
#' the F statistics (\code{F}) and its p-value computed from a permutation test
#' (\code{Prob}).
#' \item \code{testT} An object with the result of the time effect test, like \code{testS}.
#' \item \code{teststi} An object with the result of the space-time interaction test,
#' like \code{testS}.
#' \item \code{X.matrix} The bloc-diagonal explanatory matrix used in test of model 6a or 6b
#' }
#'
#' @author Pierre Legendre \email{pierre.legendre@@umontreal.ca}, Miquel De Caceres
#' and Daniel Borcard
#' @seealso \code{\link{trichoptera}}
#'
#' @references
#' Borcard, D. & P. Legendre. 2002. All-scale spatial analysis of ecological data by
#' means of principal coordinates of neighbour matrices. Ecological Modelling 153: 51-68.
#' https://doi.org/10.1016/S0304-3800(01)00501-4.
#'
#' Dray, S., P. Legendre & P. R. Peres-Neto. 2006. Spatial modelling: a
#' comprehensive framework for principal coordinate analysis of neighbour
#' matrices (PCNM). Ecological Modelling 196: 483-493.
#' https://doi.org/10.1016/j.ecolmodel.2006.02.015.
#'
#' Legendre, P. & D. Borcard. 2018. Box-Cox-chord transformations for community
#' composition data prior to beta diversity analysis. Ecography 41: 1820–1824.
#' https://doi.org/10.1111/ecog.03498.
#'
#' Legendre, P., M. De Caceres & D. Borcard. 2010. Community surveys through space
#' and time: testing the space-time interaction in the absence of replication.
#' Ecology 91: 262-272. https://doi.org/10.1890/09-0199.1.
#'
#' @keywords multivariate spatial
#'
#' @examples
#' # The "trichoptera" data set contains the abundances of 56 Trichoptera species captured
#' # during 100 consecutive days in 22 emergence traps along a stream. The 22 traps
#' # (sites) form a regular transect, with geographic positions 1 to 22. The original
#' # daily data collected at each site were pooled into 10 survey periods for the study
#' # of Legendre et al. (2010) in order to reduce the very high proportion of zeros in the
#' # original data matrix. Order of the observations in the data set: the 22 traps (sites)
#' # are nested within the survey weeks, as required by the 'stimodels' and 'quicksti'
#' # functions.
#'
#' data(trichoptera)
#'
#' # log-transform the species data, excluding the Site and Date colums
#'
#' trich.log <- log1p(trichoptera[,-c(1,2)])
#'
#' # A log-chord transformation (Legendre & Borcard 2018) would also be appropriate for
#' # these data: trich.tr <- decostand(log1p(trichoptera[,-c(1,2)]), method="norm")
#'
#' # Example 1. Compute the space-time interaction test using model 5. By specifying the
#' # number of sites (traps), the sofware assumes that they form a regular transect with
#' # equispaced sites. Important note to users – In real analyses, use more than 99
#' # permutations.
#'
#' out.1 <- stimodels(trich.log, S=22, Ti=10, model="5", nperm=99)
#'
#' # The interaction is significant. Because of that, test results for the main effects,
#' # space and time, obtained with model 5, cannot be interpreted. Tests of spatial
#' # variation can be done for individual times using simple RDA against dbMEM.S
#' # variables. Likewise, tests of temporal variation can be done for individual sites
#' # using simple RDA against dbMEM.T variables. A global test of the hypothesis that none
#' # of the times shows a significant spatial structure can be done with model 6a. For a
#' # global test of temporal structure at the various sites, use model 6b.
#'
#' \donttest{ # Code not run during CRAN software tests
#'
#' # Example 2. Run space-time analysis with global tests for main effects after testing
#' # the interaction, which is significant in this example
#'
#' out.2 <- quicksti(trich.log, S=22, Ti=10, nperm=999)
#'
#' # Since the interaction is significant, function 'quicksti' will carry out the
#' # tests of existence of a spatial (at least in one of the time periods) and temporal
#' # (at least at one of the sites) structures using models 6a and 6b, respectively.
#'
#' # 3. Run space-time analysis for two time blocks only, i.e. time periods 6 and 7,
#' # then time periods 8 and 9.
#'
#' # Example 3.1. Time periods 6 and 7. The interaction is not significant. In that case,
#' # function 'quicksti' carries out the tests of the main effects using model 5.
#'
#' out.3 <- quicksti(trich.log[111:154,], S=22, Ti=2, nperm=999)
#'
#' # Example 3.2. Time periods 8 and 9. The interaction is significant. In that case,
#' # 'quicksti' carries out the tests of the spatial effects using model 6a. Model 6b
#' # cannot proceed with the test of the temporal effect because Ti=2. An explanation is
#' # printed in the output list.
#'
#' out.4 <- quicksti(trich.log[155:198,], S=22, Ti=2, nperm=999)
#'
#' # 4. Illustrations of the use of 'COD.S' and 'COD.T' in STI analysis
#'
#' # The following examples illustrate how to use other representations of the spatial or
#' # temporal relationships among observations, through arguments 'COD.S' and
#' # 'COD.T' of functions 'stimodels' and 'quicksti'. The trichoptera data
#' # are used again.
#'
#' # Example 4.1. Explicitly compute dbMEMs for the spatial structure along the regular
#' # transect, using function 'dbmem' of adespatial, and provide it to 'stimodels'
#' # or 'quicksti' as argument 'COD.S'. The dbMEMs must first be computed on the
#' # transect, then repeated (Ti-1) times to provide Ti repeats in total.
#'
#' dbMEM.S1 <- as.matrix(dbmem(1:22))
#' dbMEM.S10 <- dbMEM.S1
#' for(j in 2:10) dbMEM.S10 <- rbind(dbMEM.S10, dbMEM.S1)
#' out.5 <- stimodels(trich.log, S=22, Ti=10, model="5", COD.S=dbMEM.S10, nperm=999)
#'
#' # Results should be identical to those in output file out.1 of Example 1, except for
#' # P-values which can vary slightly.
#'
#' # Example 4.2. Assume now that the sampling sites have irregular positions, as
#' # described by the following matrix of geographic coordinates 'xy.trich'. Provide
#' # this matrix to argument S of 'stimodels'
#'
#' xy.trich = matrix(c(1:5,11:15,21:25,31:35,41,42,rep(c(1,2),11)),22,2)
#' plot(xy.trich, asp=1) # Plot a quick map of the site positions
#' out.6 <- stimodels(trich.log, S=xy.trich, Ti=10, model="5", nperm=999)
#'
#' # Example 4.3. Compute a matrix of dbMEMs for the sites. The coding matrix provided to
#' # argument 'COD.S' must contain repeated dbMEM.S codes because that matrix must have
#' # the same number of rows as matrix Y. Construct coding matrix dbMEM.S10 containing the
#' # dbMEM.S codes repeated 10 times.
#'
#' dbMEM.S1 <- as.matrix(dbmem(xy.trich))
#' dbMEM.S10 = dbMEM.S1
#' for(i in 1:9) dbMEM.S10 <- rbind(dbMEM.S10, dbMEM.S1)
#' out.7 <- stimodels(trich.log, S=22, Ti=10, model="5", COD.S=dbMEM.S10, nperm=999)
#'
#' # Compare the results with those obtained in the output file out6, example 4.2.
#'
#' # Careful: If an analysis requires a dbMEM coding matrix for 'COD.T', the dbMEM.T
#' # codes must follow the required data arrangement: sites must be nested within times.
#' # The following function can be used to construct a dbMEM.T matrix.
#'
#' MEM.T <- function(s, tt, coord.T=NULL)
#' # Documentation of function MEM.T –
#' # Generate a matrix of temporal eigenfunctions for input into stimodels,
#' # with sites nested within times.
#' # Arguments –
#' # s : number of space points (sites)
#' # tt : number of time points
#' # coord.T : optional matrix or vector giving the time point coordinates
#' {
#' n <- s*tt
#' if(is.null(coord.T)) coord.T <- as.matrix(1:tt)
#' MEM.TT <- as.matrix(dbmem(coord.T))
#' dbMEM.T <- matrix(0,n,ncol(MEM.TT)) # Empty matrix to house dbMEM.T
#' beg.x <- seq(1, n, by=s)
#' for(i in 1:tt) { # Fill tt blocks of rows with identical MEM.TT values
#' for(j in 1:s) dbMEM.T[(beg.x[i]+j-1),] <- MEM.TT[i,]
#' }
#' dbMEM.T
#' }
#'
#' # Example of use of function MEM.T
#'
#' dbMEM.T <- MEM.T(s=6, tt=5)
#' # Check the size of the dbMEM.T output matrix
#' dim(dbMEM.T)
#'
#' } # End of code not run during CRAN software tests
#'
#' @importFrom MASS ginv
#' @importFrom stats model.matrix
#' @aliases stimodels quicksti
#' @export stimodels quicksti
#' @rdname stimodels
#'
'stimodels' <- function(Y, S, Ti, model="5", nperm=999, nS=-1, nT=-1, Sfixed=TRUE,
Tfixed=TRUE, COD.S=NULL, COD.T=NULL, save.X=FALSE, print.res=TRUE)
{
if(!is.logical(print.res)) {
stop("Wrong operator; 'print.res' should be either 'FALSE' or 'TRUE'",
call.=FALSE)
}
if(model!="2" && model!="3a" && model!="3b" && model!="4" && model!="5"
&& model!="6a" && model!="6b" && model!="7") {
stop(paste("Unrecognized model ",model,
"; 'model' should be '2', '3a', '3b', '4', '5', '6a','6b' or '7'.",
sep=""), call.=FALSE)
}
# Checking that the data represent abundances or abundance-like data
if (any(Y < 0))
stop("Data contain negative values\n", call.=FALSE)
if (any(is.na(Y)))
stop("Data contain 'NA' values\n", call.=FALSE)
skip.Manova <- FALSE
res <- NA
aa <- system.time( {
# Sets the number and location of spatial and temporal points
S <- as.matrix(S)
if(dim(S)[1]==1 && dim(S)[2]==1) {
s <- S[1,1]
sitesX <- c(1:s)
} else {
s <- dim(S)[1]
sitesX <- S
}
Ti <- as.matrix(Ti)
if(dim(Ti)[1]==1 && dim(Ti)[2]==1) {
tt <- Ti[1,1]
timesX <- c(1:tt)
} else {
tt <- dim(Ti)[1]
timesX <- Ti
}
# Total number of rows in matrix
n <- s*tt
p <- dim(Y)[2]
# Check response data file containing species data
Y <- as.matrix(Y)
p <- dim(Y)[2]
if(dim(Y)[1] != n) stop("The number of rows in species file is not (S x Ti)",
call.=FALSE)
# Center response data
Y <- scale(Y, center=TRUE, scale=FALSE)
if(print.res) {
cat("=======================================================\n")
cat(" Space-time ANOVA without replicates\n")
cat(" \n")
cat(" Pierre Legendre, Miquel De Caceres, Daniel Borcard\n")
cat("=======================================================\n\n")
cat(" Number of space points (s) =", s,'\n')
cat(" Number of time points (tt) =", tt,'\n')
cat(" Number of observations (n = s*tt) =", n,'\n')
cat(" Number of response variables (p) =", p,'\n','\n')
}
# Generates space and time helmert contrasts
A <- as.factor(rep(1:s,tt))
B <- rep(1,s)
for(i in 2:tt) B <- c(B,rep(i,s))
B <- as.factor(B)
HM <- model.matrix(~ A+B, contrasts = list(A="contr.helmert", B="contr.helmert"))
HM.S <- as.matrix(HM[,2:s]) # Helmert contrasts for Space, no intercept
HM.T <- as.matrix(HM[,(s+1):(s+tt-1)]) # Helmert contrasts for Time, no intercept
# Generate dbMEM variables for Space (if necessary)
if(model=="3a"|| model=="6a" || model=="7"|| model=="4"|| model=="5") {
if(is.null(COD.S)) { # Generate spatial dbMEMs if not given by user
if(print.res) cat(" Computing dbMEMs to code for space\n")
if(s==2) {
dbMEM.S <- as.matrix(rep(c(-1,1),tt))
#
if(model=="6a") {
cat("Model 6a requires that 's' be larger than 2. When s=2, full",
"coding of the site locations by a binary variable or Helmert",
"contrast does not leave any degree of freedom for the residuals",
"in the test of the Space factor.\n\n")
skip.Manova <- TRUE
}
}
else {
dbMEM.S.tmp <- dbmem(sitesX, MEM.autocor="positive")
SS <- as.matrix(dbMEM.S.tmp)
dbMEM.S.thresh <- give.thresh(dist(sitesX))
if(print.res) cat(" Truncation level for space dbMEMs =",
dbMEM.S.thresh, "\n")
dbMEM.S <- SS
for(j in 2:tt) dbMEM.S <- rbind(dbMEM.S,SS)
if(nS==-1) nS <- ncol(SS)
else {
if(nS > ncol(SS)) {
cat("\n Number of requested spatial variables nS =", nS,
"larger than available\n")
cat(" The", ncol(SS),
"available spatial dbMEMs will be used\n\n")
nS <- ncol(SS)
}
else dbMEM.S <- dbMEM.S[, 1:nS, drop = FALSE]
}
}
} else {
dbMEM.S <- apply(as.matrix(COD.S),2,scale,center=TRUE,scale=TRUE)
nS <- dim(dbMEM.S)[2]
if(nS>=s)
stop("The number of spatial coding functions must be lower than S",
call.=FALSE)
if(nrow(as.matrix(COD.S))!=dim(Y)[1])
stop("The number of rows in COD.S must be equal to the number",
"of observations", call.=FALSE)
}
}
# Generate dbMEM variables for Time (if necessary)
if(model=="3b" || model=="6b"|| model=="7"||model=="4" || model=="5") {
if(is.null(COD.T)) { # Generate temporal dbMEM variables if not given by user
if(tt==2) { # Two points in time: construct 2 Helmert contrast variables
dbMEM.T=as.matrix(c(rep(-1, s), rep(1,s))) #DB
#
if(model=="6b") {
cat("Model 6b requires that 'tt' be larger than 2. When tt=2, full",
"coding of the times by a binary variable or Helmert",
"contrast does not leave any degree of freedom for the residuals",
"in the test of the Time factor.\n\n")
skip.Manova <- TRUE
}
} else { # COD.T is NULL, tt>2
if(print.res) cat(" Computing dbMEMs to code for time\n")
TT <- as.matrix(dbmem(timesX))
dbMEM.T.thresh <- give.thresh(dist(timesX))
if(print.res) cat(" Truncation level for time dbMEMs =",
dbMEM.T.thresh, "\n\n")
beg <- seq(1,n,by=s) # 1 7 13 19 25
dbMEM.T <- matrix(0,n,ncol(TT)) # Empty matrix 30x2 to house dbMEM.T
for(i in 1:tt) { # Fill tt blocks of rows with identical MEM.T values
for(j in 1:s) dbMEM.T[(beg[i]+j-1),] <- TT[i,]
}
if(nT==-1) { nT <- ncol(TT) # nT: number of dbMEM variables in TT
} else { # If the arguments specify an imposed value to nT
if(nT > ncol(TT)) {
cat("\n Number of requested temporal variables nT =", nT,
"larger than available\n")
cat(" The", ncol(TT), "available temporal dbMEM will be used\n\n")
nT <- ncol(TT)
} else {
dbMEM.T <- dbMEM.T[, 1:nT, drop = FALSE]
}
}
} # End of if(is.null(COD.T))
} else { # COD.T is present
dbMEM.T=apply(as.matrix(COD.T),2,scale,center=TRUE,scale=TRUE)
nT <- dim(dbMEM.T)[2]
if(nT>=tt) stop("The number of temporal coding functions",
"must be lower than Ti", call.=FALSE)
if(nrow(as.matrix(COD.T))!=dim(Y)[1])
stop("The number of rows in COD.T must be equal to the number",
"of observations", call.=FALSE)
}
}
if(!skip.Manova) { # Begin of "(!skip.Manova)"
if(print.res) {
if(model!="2" && model!="3b" && model!="6b") {
if(s==2) #DB To avoid number = -1
cat(" Number of space coding functions = 1 \n\n")
else
cat(" Number of space coding functions =", nS,'\n\n')
}
if(model!="2" && model!="3a" && model!="6a") {
if(tt==2) #DB To avoid number = -1
cat(" Number of time coding functions = 1 \n\n")
else
cat(" Number of time coding functions =", nT, "\n\n")}
}
test.STI <- TRUE
test.S <- TRUE
test.T <- TRUE
# Generate matrices X (variables for the factor of interest)
# and W (covariables) for each test to be done
if(model=="5") {
XSTI <- dbMEM.S*dbMEM.T[,1]
if(dim(dbMEM.T)[2]>1)
for(j in 2:dim(dbMEM.T)[2]) XSTI <- cbind(XSTI,dbMEM.S*dbMEM.T[,j])
XS <- HM.S
XT <- HM.T
if(print.res) {
cat(" MODEL V: HELMERT CONTRAST FOR TESTING MAIN FACTORS.\n")
cat(" SPACE AND TIME dbMEMs FOR TESTING INTERACTION.\n")
}
} else if(model=="4") {
XSTI <- dbMEM.S*dbMEM.T[,1]
if(dim(dbMEM.T)[2]>1)
for(j in 2:dim(dbMEM.T)[2]) XSTI = cbind(XSTI,dbMEM.S*dbMEM.T[,j])
XS <- dbMEM.S
XT <- dbMEM.T
if(print.res) {
cat(" MODEL IV: dbMEMs FOR BOTH SPACE AND TIME.\n")
}
} else if(model=="3a") {
XSTI <- dbMEM.S*HM.T[,1]
if(tt>2) for(j in 2:(tt-1)) XSTI <- cbind(XSTI,dbMEM.S*HM.T[,j])
XS <- dbMEM.S
XT <- HM.T
if(print.res) {
cat(" MODEL IIIa: dbMEMs FOR SPACE AND HELMERT CONTRASTS FOR TIME.\n")
}
} else if(model=="3b") {
XSTI <- dbMEM.T*HM.S[,1]
if(dim(HM.S)[2]>1)
for(j in 2:(s-1)) XSTI <- cbind(XSTI,dbMEM.T*HM.S[,j])
XS <- HM.S
XT <- dbMEM.T
if(print.res) {
cat(" MODEL IIIb: HELMERT CONTRASTS FOR SPACE AND dbMEMs FOR TIME.\n")
}
} else if(model=="7") {
XSTI <- HM.S*HM.T[,1]
if(dim(HM.T)[2]>1) for(j in 2:dim(HM.T)[2]) XSTI <- cbind(XSTI,HM.S*HM.T[,j])
XS <- dbMEM.S
XT <- dbMEM.T
if(print.res) {
cat(" MODEL VII: dbMEMs FOR BOTH SPACE AND TIME,",
"BUT HELMERT CONTRAST FOR INTERACTION.\n")
}
} else if(model=="6a") {
# BEGIN: New code (PL) to assemble a block diagonal matrix for XS
XS <- matrix(0,n,tt*nS) # Empty matrix
beg.x <- seq(1,n,by=s) # Beginning rows, vertical
beg.y <- seq(1,tt*nS,by=nS) # Beginning columns, horizontal
tmp <- dbMEM.S[1:s,]
for(j in 1:tt)
XS[(beg.x[j]):(beg.x[j]+s-1),(beg.y[j]):(beg.y[j]+nS-1)] <- tmp
XT <- HM.T
XSTI = NULL
if(print.res) {
cat(" MODEL VIa: NESTED MODEL.\n")
cat(" TESTING FOR THE EXISTENCE",
"OF SPATIAL STRUCTURE (COMMON OR SEPARATE)\n")
}
test.STI <- FALSE
test.T <- FALSE
} else if(model=="6b") {
# BEGIN: New code (PL)) to assemble a block diagonal matrix for XT
beg.x <- seq(1,n,by=s) # Rows to fill at the beginning
beg.y <- seq(1,s*nT,by=nT) # Columns to fill at the beginning
TT = as.matrix(dbMEM.T[beg.x,]) # dbMEM for times, only once
XT <- matrix(0,n,s*nT) # Empty matrix to house the diagonal blocks
for(j in 1:s) { # Write groups of s columns in XT
for(i in 1:tt) { # Fill tt rows with T1 to T12 in each group of 3 columns
XT[(beg.x[i]+j-1), (beg.y[j]):(beg.y[j]+nT-1)] <- TT[i,1:nT]
}
} ## End of New code
XS <- HM.S
XSTI = NULL
if(print.res) {
cat(" MODEL VIb: NESTED MODEL.\n")
cat(" TESTING FOR THE EXISTENCE",
"OF TEMPORAL STRUCTURE (COMMON OR SEPARATE).\n")
}
test.STI <- FALSE
test.S <- FALSE
} else if(model=="2") {
XS <- HM.S
XT <- HM.T
XSTI <- NULL
if(print.res) {
cat(" MODEL II: HELMERT CONTRAST FOR SPACE AND TIME.",
"NO INTERACTION TERM.\n")
}
test.STI <- FALSE
}
if(print.res) {
cat(" Number of space variables =", dim(XS)[2],'\n')
cat(" Number of time variables =", dim(XT)[2],'\n')
if(test.STI) {
cat(" Number of interaction variables =", dim(XSTI)[2],'\n')
cat(" Number of residual degrees of freedom =",
(s*tt-dim(XS)[2]-dim(XT)[2]-dim(XSTI)[2]-1),"\n\n")
if((s*tt-dim(XS)[2]-dim(XT)[2]-dim(XSTI)[2]-1) <= 0)
cat("Not enough residual degrees of freedom for testing.\n")
} else {
cat(" Number of residual degrees of freedom =",
(s*tt-dim(XS)[2]-dim(XT)[2]-1),"\n\n")
if((s*tt-dim(XS)[2]-dim(XT)[2]-1) <= 0)
cat("Not enough residual degrees of freedom for testing.\n")
}
}
# Call Manova by RDA, function manovRDa.R
res <- manovRDa(Y=Y,s=s,tt=tt,S.mat=XS,T.mat=XT,STI.mat=XSTI, Sfixed=Sfixed,
Tfixed=Tfixed, S.test=test.S, T.test=test.T, STI.test=test.STI,
model = model, nperm=nperm, save.X=save.X)
if(test.STI==TRUE) {
cat(' Interaction test: R2 =', round(res$testSTI$R2, 4),
' F =',round(res$testSTI$F, 4),' P(',nperm,'perm) =',res$testSTI$Prob,'\n')
}
if(test.S==TRUE) {
cat(' Space test: R2 =', round(res$testS$R2, 4),
' F =',round(res$testS$F, 4),' P(',nperm,'perm) =',res$testS$Prob,'\n')
}
if(test.T==TRUE) {
cat(' Time test: R2 =', round(res$testT$R2, 4),
' F =',round(res$testT$F, 4),' P(',nperm,'perm) =',res$testT$Prob,'\n\n')
}
} # End of "if(!skip.Manova)"
})
aa[3] <- sprintf("%2f",aa[3])
if(print.res) {
cat("-------------------------------------------------------\n")
cat(" Time for computation =",aa[3]," sec",'\n')
cat("=======================================================\n\n")
}
class(res) <- "sti"
invisible(res)
}
#' @rdname stimodels
'quicksti' <- function(Y, S, Ti, nperm=999, alpha = 0.05, COD.S=NULL, COD.T=NULL,
save.X=FALSE, print.res=TRUE)
{
if(!is.logical(print.res)) {
stop("Wrong operator; 'print.res' should be either 'FALSE' or 'TRUE'.",
call.=FALSE)
}
aa <- system.time( {
# Sets the number and location of spatial and temporal points
S <- as.matrix(S)
if(dim(S)[1]==1 && dim(S)[2]==1) {
s <- S[1,1]
sitesX <- c(1:s)
} else {
s <- dim(S)[1]
sitesX <- S
}
Ti <- as.matrix(Ti)
if(dim(Ti)[1]==1 && dim(Ti)[2]==1) {
tt <- Ti[1,1]
timesX <- c(1:tt)
} else {
tt <- dim(Ti)[1]
timesX <- Ti
}
# Total number of rows in matrix
n <- s*tt
# Check response data file containing species data
Y <- as.matrix(Y)
p <- dim(Y)[2]
if(dim(Y)[1] != n) stop("The number of rows in species file is not (S x Ti).",
call.=FALSE)
# Center response data
Y <- scale(Y, center=TRUE, scale=FALSE)
if(print.res) {
cat("=========================================================\n")
cat(" Space-time ANOVA without replicates\n")
cat(" \n")
cat(" Pierre Legendre, Miquel De Caceres, Daniel Borcard\n")
cat("---------------------------------------------------------\n\n")
cat(" Number of space points (s) =", s,'\n')
cat(" Number of time points (tt) =", tt,'\n')
cat(" Number of observations (n = s*tt) =", n,'\n')
cat(" Number of response variables (p) =", p,'\n')
cat(" Significance level for the interaction test (alpha) =", alpha,'\n','\n')
}
# Generates spatial dbMEMs if not provided by user
if(is.null(COD.S)) {
if(print.res) cat(" Computing dbMEMs to code for space\n")
if(s==2) {
dbMEM.S <- as.matrix(rep(c(-1,1),tt))
nS <- dim(dbMEM.S)[2]
if(print.res) cat("\n There are only two sites. A vector of Helmert",
"contrasts will be used to represent them\n\n")
}
else {
SS <- as.matrix(dbmem(sitesX))
nS <- ncol(SS)
dbMEM.S.thresh <- give.thresh(dist(sitesX))
if(print.res) cat(" Truncation level for space dbMEMs =", dbMEM.S.thresh,"\n")
dbMEM.S <- SS
for(j in 2:tt) dbMEM.S = rbind(dbMEM.S,SS)
}
} else {
dbMEM.S <- apply(as.matrix(COD.S),2,scale,center=TRUE,scale=TRUE)
nS <- dim(dbMEM.S)[2]
if(nS >= s) stop("The number of spatial coding functions must be smaller",
"than S.", call.=FALSE)
if(nrow(as.matrix(COD.S))!=dim(Y)[1]) stop("The number of rows in COD.S must",
"be equal to the number of observations.", call.=FALSE)
}
# Generate dbMEM variables for time if not provided by user
if(is.null(COD.T)) {
nT <- trunc(tt/2)
if(print.res) cat(" Computing dbMEMs to code for time\n")
if(tt==2) {
dbMEM.T <- as.matrix(c(rep(-1, s), rep(1,s)))
if(print.res) cat("\n There are only two time points. A vector of",
"Helmert contrasts will be used to represent them\n\n")
} else {
TT <- as.matrix(dbmem(timesX))
nT <- ncol(TT)
dbMEM.T.thresh <- give.thresh(dist(timesX))
if(print.res) cat(" Truncation level for time dbMEMs =",dbMEM.T.thresh,"\n\n")
beg.x <- seq(1,n,by=s) # New code (PL)
nT = ncol(TT)
dbMEM.T <- matrix(0,n,nT) # Empty matrix 30x2 to house dbMEM.T
for(i in 1:tt) { # Fill tt blocks of rows with identical MEM.T values
for(j in 1:s) dbMEM.T[(beg.x[i]+j-1),] <- TT[i,]
}
}
} else {
dbMEM.T <- apply(as.matrix(COD.T),2,scale,center=TRUE,scale=TRUE)
nT <- dim(dbMEM.T)[2]
if(nT >= tt) stop("The number of temporal coding functions must be",
"lower than Ti.", call.=FALSE)
if(nrow(as.matrix(COD.T))!=dim(Y)[1]) stop("The number of rows in",
"COD.T must be equal to the number of observations.", call.=FALSE)
}
if(s*tt-s-tt-nT*nS-1<=0) stop("Not enough degrees of freedom for testing",
"interaction.", call.=FALSE)
if(print.res) {
cat(" Number of space coding functions =", nS,'\n')
cat(" Number of time coding functions =", nT, "\n\n")
}
# Generates space and time helmert contrasts
A <- as.factor(rep(1:s,tt))
B <- rep(1,s)
for(i in 2:tt) B <- c(B,rep(i,s))
B <- as.factor(B)
HM <- model.matrix(~ A + B, contrasts =list(A="contr.helmert", B="contr.helmert"))
HM.S <- as.matrix(HM[,2:s])
HM.T <- as.matrix(HM[,(s+1):(s+tt-1)])
res2 <- NULL
res3 <- NULL
# Test significance for interaction effect
#
# Defines X (variables for the factor of interest)
# and W (covariables) for the space-time test
#
XSTI <- dbMEM.S*dbMEM.T[,1]
if(dim(dbMEM.T)[2]>1) for(j in 2:dim(dbMEM.T)[2])
XSTI <- cbind(XSTI,dbMEM.S*dbMEM.T[,j])
if(print.res) {
cat("------------------------------------------\n")
cat(" Testing space-time interaction (model 5)\n")
cat("------------------------------------------\n\n")
cat(" Number of space variables =", dim(HM.S)[2],'\n')
cat(" Number of time variables =", dim(HM.T)[2],'\n')
cat(" Number of interaction variables =", dim(XSTI)[2],'\n')
cat(" Number of residual degrees of freedom =",
(s*tt-dim(XSTI)[2]-dim(HM.S)[2]-dim(HM.T)[2]-1),"\n\n")
if((s*tt-dim(XSTI)[2]-dim(HM.S)[2]-dim(HM.T)[2]-1) == 0)
stop("Not enough residual degrees of freedom for testing.",
"Try with a smaller number of space or time variables.", call.=FALSE)
}
res <- manovRDa(Y=Y,s=s,tt=tt,S.mat=HM.S,T.mat=HM.T,STI.mat=XSTI, S.test=TRUE,
T.test=TRUE, STI.test=TRUE, model="5", nperm=nperm, save.X=save.X)
cat(' Interaction test: R2 =', round(res$testSTI$R2, 4),
' F =',round(res$testSTI$F,4),' P(',nperm,'perm) =',res$testSTI$Prob,"\n")
# Begin models 6a and 6b if Prob<=alpha
if(res$testSTI$Prob<=alpha) {
# Model="6a"
XS <- dbMEM.S
for(j in 1:(tt-1)) XS = cbind(XS,dbMEM.S*HM.T[,j])
XT <- HM.T
if(print.res) {
cat("----------------------------------------------------\n")
cat(" Testing for separate spatial structures (model 6a)\n")
cat("----------------------------------------------------\n\n")
cat(" Number of space variables =", dim(XS)[2],'\n')
cat(" Number of time variables =", dim(XT)[2],'\n')
cat(" Number of residual degrees of freedom =",
(s*tt-dim(XS)[2]-dim(XT)[2]-1),"\n\n")
}
if(s==2) { # skip.manova
if(!print.res) cat("\n")
cat("Model 6a requires that 's' be larger than 2. When s=2, full",
"coding of the site locations by a binary variable or Helmert",
"contrast does not leave any degree of freedom for the residuals",
"in the test of the Space factor.\n\n")
res2$X.matrix <- NA
} else {
res2 <- manovRDa(Y=Y,s=s,tt=tt,S.mat=XS,T.mat=XT,STI.mat=NULL,
S.test=TRUE, T.test=FALSE, STI.test=FALSE, model="6a",
nperm=nperm, save.X=save.X)
res$testS <- res2$testS
cat(' Space test: R2 =', round(res$testS$R2,4), ' F =',
round(res$testS$F,4),' P(',nperm,'perm) =',res$testS$Prob,"\n")
} # End of "if(s==2)"
# Model="6b"
XT <- dbMEM.T
for(j in 1:(s-1)) XT <- cbind(XT,dbMEM.T*HM.S[,j])
XS <- HM.S
if(print.res) {
cat("-----------------------------------------------------\n")
cat(" Testing for separate temporal structures (model 6b)\n")
cat("-----------------------------------------------------\n\n")
cat(" Number of space variables =", dim(XS)[2],'\n')
cat(" Number of time variables =", dim(XT)[2],'\n')
cat(" Number of residual degrees of freedom =",
(s*tt-dim(XS)[2]-dim(XT)[2]-1),"\n\n")
}
if(tt==2) { # skip.manova
if(!print.res) cat("\n")
cat("Model 6b requires that 'tt' be larger than 2. When tt=2, full",
"coding of the times by a binary variable or Helmert",
"contrast does not leave any degree of freedom for the residuals",
"in the test of the Time factor.\n\n")
res3$X.matrix <- NA
} else {
res3 <- manovRDa(Y=Y,s=s,tt=tt,S.mat=XS,T.mat=XT,STI.mat=NULL,
S.test=FALSE, T.test=TRUE, STI.test=FALSE, model="6b",
nperm=nperm, save.X=save.X)
res$testT <- res3$testT
cat(' Time test: R2 =', round(res$testT$R2,4),' F =',
round(res$testT$F,4),' P(',nperm,'perm) =',res$testT$Prob,"\n\n")
} # End of "if(tt==2)"
} else { # End of "if(res$testSTI$Prob<=alpha)", # End of Models 6a and 6b
save.X = FALSE
# Tests of Space and Time from the results of model="5"
if(print.res) {
cat(
"---------------------------------------------------------------------\n")
cat(
" Testing for common spatial and common temporal structures (model 5)\n")
cat(
"---------------------------------------------------------------------\n\n")
}
cat(' Space test: R2 =', round(res$testS$R2,4),
' F =',round(res$testS$F,4),' P(',nperm,'perm) =',res$testS$Prob,'\n')
cat(' Time test: R2 =', round(res$testT$R2,4),
' F =',round(res$testT$F,4),' P(',nperm,'perm) =',res$testT$Prob,'\n\n')
}
})
aa[3] <- sprintf("%2f",aa[3])
if(print.res) {
cat("---------------------------------------------------------\n")
cat(" Time for computation =",aa[3]," sec",'\n')
cat("=========================================================\n\n")
}
if(save.X) {
out <- list(res, res2$X.matrix, res3$X.matrix)
names(out) <- c("tests", "space.blocks", "time.blocks")
invisible(out)
}
else {
invisible(res)
}
}
'manovRDa' <- function(Y, s, tt, S.mat=NULL, T.mat=NULL, STI.mat=NULL, Sfixed=TRUE,
Tfixed=TRUE, S.test=TRUE, T.test=TRUE, STI.test=TRUE, model="5",
nperm=999, save.X)
{
n <- nrow(Y)
p <- ncol(Y)
a <- ncol(S.mat)
b <- ncol(T.mat)
if(!is.null(STI.mat)) {cc <- ncol(STI.mat)}
else {cc = 0}
A <- S.mat
B <- T.mat
if(!is.null(STI.mat)) AxB <- STI.mat
# Compute projector of A and Yfit.A
invA <- ginv(t(A) %*% A)
projA <- A %*% invA %*% t(A)
Yfit.A <- projA %*% Y
# Compute projector of B and Yfit.B
invB <- ginv(t(B) %*% B)
projB <- B %*% invB %*% t(B)
Yfit.B <- projB %*% Y
# Compute projector of AxB and Yfit.AxB
if(!is.null(STI.mat)) {
invAxB <- ginv(t(AxB) %*% AxB)
projAxB <- AxB %*% invAxB %*% t(AxB)
Yfit.AxB <- projAxB %*% Y
} else {
projAxB=NULL
}
# Create a "compound matrix" to obtain R-square and adjusted R-square
if(!is.null(STI.mat)) {
ABAxB <- cbind(A,B,AxB)
} else {
ABAxB <- cbind(A,B)
}
# Compute projector of ABAxB and Yfit.ABAxB
invABAxB <- ginv(t(ABAxB) %*% ABAxB)
projABAxB <- ABAxB %*% invABAxB %*% t(ABAxB)
Yfit.ABAxB <- projABAxB %*% Y
# If save.X is TRUE, save the block-diagonal matrix X used in test of model 6a or 6b
X <- NA
if(save.X && model=="6a") X <- S.mat
if(save.X && model=="6b") X <- T.mat
# Compute Sums of squares (SS) and Mean squares (MS)
SS.Y <- sum(Y*Y)
SS.Yfit.ABAxB <- sum(Yfit.ABAxB*Yfit.ABAxB)
SS.Yfit.A <- sum(Yfit.A*Yfit.A)
SS.Yfit.B <- sum(Yfit.B*Yfit.B)
if(!is.null(STI.mat)) SS.Yfit.AxB <- sum(Yfit.AxB*Yfit.AxB)
MS.A <- SS.Yfit.A/a
MS.B <- SS.Yfit.B/b
if(!is.null(STI.mat)) MS.AxB <- SS.Yfit.AxB/cc
MS.Res <- (SS.Y-SS.Yfit.ABAxB)/(n-(a+b+cc)-1)
# Test interaction (unrestricted permutations)
if(STI.test==TRUE) {
nPGE.AxB = 1
Fref.AxB <- MS.AxB/MS.Res
nPGE.AxB=
.Call("sti_loop",nperm,Y,s,tt,a,b,cc,SS.Y,Fref.AxB,projAxB,projABAxB) ### NM
P.AxB <- nPGE.AxB/(nperm+1)
R2 <- SS.Yfit.AxB/SS.Y
R2a <- 1-((n-1)/(n-dim(STI.mat)[2]-1))*(1-R2)
testSTI <-
list(MS.num=MS.AxB, MS.den=MS.Res, R2=R2, R2.adj=R2a, F=Fref.AxB, Prob=P.AxB)
} else {
testSTI <- NULL
}
# Test factor A (space) using restricted permutations within time blocks
if(S.test==TRUE) {
#########################
nPGE.A=1
if(Tfixed==FALSE && !is.null(STI.mat)) {
# Time random factor in crossed design with interaction
Fref.A <- MS.A/MS.AxB
MS.den <- MS.AxB
T_fixed <- 1
} else if(Tfixed==FALSE && model=="6b") {
# Time random factor in nested design
Fref.A <- MS.A/MS.B
MS.den <- MS.B
T_fixed <- 2
} else {
Fref.A <- MS.A/MS.Res
MS.den <- MS.Res
T_fixed <- 3
}
nPGE.A=.Call("s_loop",nperm,Y,s,tt,a,b,cc,SS.Y,Fref.A,projA,projB,projAxB,
projABAxB,T_fixed) ### NM
P.A <- nPGE.A/(nperm+1)
R2 <- SS.Yfit.A/SS.Y
R2a <- 1-((n-1)/(n-a-1))*(1-R2)
testS <- list(MS.num=MS.A, MS.den=MS.den, R2=R2, R2.adj=R2a, F=Fref.A, Prob=P.A)
} else {
testS <- NULL
}
# Test factor B (time) using restricted permutations within time blocks
if(T.test==TRUE) {
nPGE.B = 1
if(Sfixed==FALSE && !is.null(STI.mat)) {
# Space random factor in crossed design with interaction
Fref.B <- MS.B/MS.AxB
MS.den <- MS.AxB
T_fixed=1
} else if(Sfixed==FALSE && model=="6a") { # Space random factor in nested design
Fref.B <- MS.B/MS.A
MS.den <- MS.A
T_fixed=2
} else {
Fref.B <- MS.B/MS.Res
MS.den <- MS.Res
T_fixed=3
}
nPGE.B <- .Call("t_loop",nperm,Y,s,tt,a,b,cc,SS.Y,Fref.B,projA,projB,projAxB,
projABAxB,T_fixed) # NM
P.B <- nPGE.B/(nperm+1)
R2 <- SS.Yfit.B/SS.Y
R2a <- 1-((n-1)/(n-b-1))*(1-R2)
testT <- list(MS.num=MS.B, MS.den=MS.den, R2=R2, R2.adj=R2a, F=Fref.B, Prob=P.B)
} else {
testT <- NULL
}
return(list(testSTI = testSTI, testS = testS, testT=testT, X.matrix=X))
}
'restrictedPerm' <- function(nobs.block, nblock, n, restPerm, vec)
#
# restPerm == 0: Unrestricted permutation.
#
# restPerm == 1: Restricted permutation of the observations within each block.
# The data are arranged by blocks, with all observations forming a block
# placed in adjacent positions in the file. Example:
# BLOCK-1: obs1, obs2, ... obs-nobs.block; BLOCK-2: obs1, obs2, ... obs-nobs.block; etc.
#
# restPerm == 2: Restricted permutation of the observations across blocks (within the
# blocks formed by each nth observation).
#
# Vector 'vec' contains the initial order of the objects, e.g. vec=c(1:n).
# At the end of the function, it gives the permuted order of the objects.
#
# Examples: toto0 <- restrictedPerm(6,4,24,0,c(1:24))
# toto1 <- restrictedPerm(6,4,24,1,c(1:24))
#
# Pierre Legendre, January 2006
# Miquel De Caceres, February 2009
{
if(restPerm == 0) {
vec <- sample(vec[1:n],n)
} else if(restPerm==1) {
for(j in 1:nblock) {
i1 <- nobs.block*(j-1)+1
i2 <- nobs.block*j
vec[i1:i2] <- sample(vec[i1:i2],nobs.block)
}
} else {
vecT <- vec
for(j in 1:nobs.block) {
ind <- sample(1:nblock)
for(i in 1:nblock) {
vec[nobs.block*(i-1)+j] <- vecT[nobs.block*(ind[i]-1)+j]
}
}
}
return(vec)
}
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