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##' Estimate probabilities in contingency table
##'
##' @title Estimate probabilities in contingency table
##' @aliases multinomial kappa.multinomial kappa.table gkgamma
##' @param x Formula (or matrix or data.frame with observations, 1 or 2 columns)
##' @param data Optional data.frame
##' @param marginal If TRUE the marginals are estimated
##' @param transform Optional transformation of parameters (e.g., logit)
##' @param vcov Calculate asymptotic variance (default TRUE)
##' @param IC Return ic decomposition (default TRUE)
##' @param ... Additional arguments to lower-level functions
##' @export
##' @examples
##' set.seed(1)
##' breaks <- c(-Inf,-1,0,Inf)
##' m <- lvm(); covariance(m,pairwise=TRUE) <- ~y1+y2+y3+y4
##' d <- transform(sim(m,5e2),
##' z1=cut(y1,breaks=breaks),
##' z2=cut(y2,breaks=breaks),
##' z3=cut(y3,breaks=breaks),
##' z4=cut(y4,breaks=breaks))
##'
##' multinomial(d[,5])
##' (a1 <- multinomial(d[,5:6]))
##' (K1 <- kappa(a1)) ## Cohen's kappa
##'
##' K2 <- kappa(d[,7:8])
##' ## Testing difference K1-K2:
##' estimate(merge(K1,K2,id=TRUE),diff)
##'
##' estimate(merge(K1,K2,id=FALSE),diff) ## Wrong std.err ignoring dependence
##' sqrt(vcov(K1)+vcov(K2))
##'
##' ## Average of the two kappas:
##' estimate(merge(K1,K2,id=TRUE),function(x) mean(x))
##' estimate(merge(K1,K2,id=FALSE),function(x) mean(x)) ## Independence
##' ##'
##' ## Goodman-Kruskal's gamma
##' m2 <- lvm(); covariance(m2) <- y1~y2
##' breaks1 <- c(-Inf,-1,0,Inf)
##' breaks2 <- c(-Inf,0,Inf)
##' d2 <- transform(sim(m2,5e2),
##' z1=cut(y1,breaks=breaks1),
##' z2=cut(y2,breaks=breaks2))
##'
##' (g1 <- gkgamma(d2[,3:4]))
##' ## same as
##' \dontrun{
##' gkgamma(table(d2[,3:4]))
##' gkgamma(multinomial(d2[,3:4]))
##' }
##'
##' ##partial gamma
##' d2$x <- rbinom(nrow(d2),2,0.5)
##' gkgamma(z1~z2|x,data=d2)
##' @author Klaus K. Holst
multinomial <- function(x,data=parent.frame(),marginal=FALSE,transform,vcov=TRUE,IC=TRUE,...) {
formula <- NULL
if (inherits(x,"formula")) {
trm <- terms(x)
if (length(attr(trm,"term.labels"))>1) {
x <- update(x,as.formula(paste0(".~ interaction(",
paste0(attr(trm,"term.labels"),collapse=","),")")))
trm <- terms(x)
}
formula <- x
x <- as.matrix(model.frame(trm,data))
if (ncol(x)>1)
x <- x[,c(seq(ncol(x)-1)+1,1),drop=FALSE]
} else {
trm <- NULL
}
if (!vcov) IC <- FALSE
if (is.table(x) && IC) x <- lava::Expand(x)
if (NCOL(x)==1) {
if (!is.table(x)) {
x <- as.factor(x)
lev <- levels(x)
k <- length(lev)
n <- length(x)
P <- table(x)/n
} else {
n <- sum(x)
P <- x/n
lev <- names(x)
k <- length(lev)
}
if (IC) {
IC <- matrix(0,n,k)
for (i in seq(k)) {
IC[,i] <- (1*(x==lev[i])-P[i])/n
};
varcov <- crossprod(IC)
} else {
IC <- varcov <- NULL
if (vcov) {
varcov <- tcrossprod(cbind(P))/n
diag(varcov) <- P*(1-P)/n
}
}
coefs <- as.vector(P); names(coefs) <- paste0("p",seq(k))
res <- list(call=match.call(), coef=coefs,P=P,
vcov=varcov,IC=IC*NROW(IC),
position=seq(k),levels=list(lev),data=x, terms=trm)
class(res) <- "multinomial"
return(res)
}
if (!is.table(x)) {
if (NCOL(x)!=2L) stop("Matrix or data.frame with one or two columns expected")
x <- as.data.frame(x)
x[,1] <- as.factor(x[,1])
x[,2] <- as.factor(x[,2])
lev1 <- levels(x[,1])
lev2 <- levels(x[,2])
k1 <- length(lev1)
k2 <- length(lev2)
M <- table(x)
n <- sum(M)
} else {
lev1 <- rownames(x)
lev2 <- colnames(x)
k1 <- length(lev1)
k2 <- length(lev2)
M <- x
n <- sum(x)
}
Pos <- P <- M/n
if (IC) {
IC <- matrix(0,n,k1*k2)
for (j in seq(k2)) {
for (i in seq(k1)) {
pos <- (j-1)*k1+i
IC[,pos] <- (x[,1]==lev1[i])*(x[,2]==lev2[j])-P[i,j]
Pos[i,j] <- pos
}
}; IC <- IC/n
} else {
IC <- varcov <- NULL
}
coefs <- as.vector(P);
names(coefs) <- as.vector(outer(seq(k1),seq(k2),function(...) paste0("p",...)))
position1 <- position2 <- NULL
if (marginal) {
p1 <- rowSums(P)
p2 <- colSums(P)
names(p1) <- paste0("p",seq(k1),".")
names(p2) <- paste0("p",".",seq(k2))
coefs <- c(coefs,p1,p2)
position1 <- length(P)+seq(k1)
position2 <- length(P)+k1+seq(k2)
if (!is.null(IC)) {
ic1 <- apply(Pos,1,function(x) rowSums(IC[,x]))
ic2 <- apply(Pos,2,function(x) rowSums(IC[,x]))
IC <- cbind(IC,ic1,ic2)
colnames(IC) <- names(coefs)
}
}
if (!missing(transform) && !is.null(IC)) {
f <- function(p) do.call(transform,list(p))
D <- diag(numDeriv::grad(f,coefs),ncol=length(coefs))
coefs <- f(coefs)
IC <- IC%*%t(D)
}
if (vcov && !is.null(IC)) varcov <- crossprod(IC)
res <- list(call=match.call(),
formula=formula,
coef=coefs,P=P,vcov=varcov,IC=IC*NROW(IC), position=Pos,
call=match.call(), levels=list(lev1,lev2), data=x,
position1=position1,position2=position2, ## Position of marginals)
terms=trm
)
class(res) <- "multinomial"
if (length(list(...))>0) {
res <- structure(estimate(res,...),class=c("multinomial","estimate"))
}
return(res)
}
##' @export
model.frame.multinomial <- function(formula,...) {
formula$data
}
##' @export
IC.multinomial <- function(x,...) {
x$IC
}
##' @export
coef.multinomial <- function(object,...) {
object$coef
}
##' @export
vcov.multinomial <- function(object,...) {
object$vcov
}
##' @export
predict.multinomial <- function(object,newdata,type=c("prob","map"),...) {
if (missing(newdata) || is.null(newdata)) newdata <- object$data
if (!is.null(object$formula) && is.data.frame(newdata)) {
trm <- terms(object$formula)
newdata <- model.frame(trm,newdata)[,-1]
}
px <- rowSums(object$P)
idx <- match(trim(as.character(newdata)),trim(rownames(object$P)))
pcond <- object$P
for (i in seq(nrow(pcond))) pcond[i,] <- pcond[i,]/px[i]
pr <- pcond[idx,,drop=FALSE]
if (tolower(type[1])%in%c("map","class")) {
pr <- colnames(pr)[apply(pr,1,which.max)]
}
return(pr)
}
##' @export
print.multinomial <- function(x, ...) {
cat("Call: "); print(x$call)
cat("\nJoint probabilities:\n")
print(x$P, quote=FALSE)
if (length(dim(x$P))>1) {
cat("\nConditional probabilities:\n")
print(predict(x, newdata=rownames(x$P)), quote=FALSE)
}
cat("\n")
print(estimate(NULL, coef=coef(x), vcov=vcov(x)))
## stderr <- diag(vcov(x))^.5
## StdErr <- x$position
## StdErr[] <- stderr[StdErr]
## cat("\nStd.Err:\n")
## print(StdErr,quote=FALSE)
## cat("\nPosition:\n")
## print(x$position,quote=FALSE)
}
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