R/robust.mmm.test.R

In lawstat: Tools for Biostatistics, Public Policy, and Law

#' Robust Mudholkar--McDermott--Mudholkar Test for Ordered Variances
#' 
#' A test for a monotonic trend in variances \insertCite{Mudholkar_etal_1995}{lawstat}. 
#' The test statistic is based on
#' a combination of the finite intersection approach and the two-sample \eqn{t}-test 
#' using Miller's transformation. By default, \code{NA}s are omitted.
#' 
#'
#' @inheritParams lnested.test
#'
#'
#' @return A list with the following elements:
#' \item{T}{the statistic and \eqn{p}-value of the test based on the Tippett \eqn{p}-value combination.}
#' \item{F}{the statistic and \eqn{p}-value of the test based on the Fisher \eqn{p}-value combination.}
#' \item{N}{the statistic and \eqn{p}-value of the test based on the Liptak \eqn{p}-value combination.}
#' \item{L}{the statistic and \eqn{p}-value of the test based on the Mudholkar-George \eqn{p}-value combination.}
#' 
#' Each of the list elements is a list of class \code{"htest"} with the following elements:
#' \item{statistic}{the value of the test statistic.}
#' \item{p.value}{the \eqn{p}-value of the test.}
#' \item{method}{type of test performed.}
#' \item{data.name}{a character string giving the name of the data.}
#' 
#' @references
#' \insertAllCited{}
#' 
#' @seealso \code{\link{neuhauser.hothorn.test}}, \code{\link{levene.test}}, 
#' \code{\link{lnested.test}}, \code{\link{ltrend.test}}, \code{\link{mma.test}}
#' 
#' @keywords htest robust variability
#' 
#' @author Kimihiro Noguchi, Yulia R. Gel
#' 
#' @export
#' @examples
#' data(pot)
#' robust.mmm.test(pot[, "obs"], pot[, "type"], tail = "left")$N
#' 
robust.mmm.test <-
    function(y, group, tail = c("right", "left", "both"))
    {
        ### stop the code if the length of y does not match the length of group ###
        
        if (length(y) != length(group))
        {
            stop("the length of the data (y) does not match the length of the group")
        }
        
        ### assign tail and name ###
        
        tail <- match.arg(tail)
        DNAME <- deparse(substitute(y))
        y <- y[!is.na(y)]
        group <- group[!is.na(y)]
        METHOD <- "Mudholkar et al. (1995) test"
        
        ### sort the order just in case the input is not sorted by group ###
        
        reorder <- order(group)
        group <- group[reorder]
        y <- y[reorder]
        
        ### initialize variables ###
        
        N <- length(y)
        n <- tapply(y, group, length)
        k <- length(n)
        m <- k - 1
        s <- tapply(y, group, var)
        v <- double()
        start.pos <- c(1)
        
        ### calculate statistic ###
        
        for (i in 1:m)
        {
            j <- sum(n[1:i]) + 1
            start.pos <- c(start.pos, j)
        }
        
        for (i in 1:k)
        {
            start <- start.pos[i]
            end <- start + n[i] - 1
            suby <- y[start:end]
            suby.length <- length(suby)
            
            for (j in 1:suby.length)
            {
                subsuby <- suby[-j]
                jackvar <- var(subsuby)
                v <- c(v, jackvar)
            }
        }
        n.group <- n[group]
        s.group <- s[group]
        
        w <- n.group * log(s.group) - (n.group - 1) * log(v)
        w.mean <- tapply(w, group, mean)
        w.mean.group <- w.mean[group]
        q.sq <- sum((w - w.mean.group) ^ 2) / (N - k)
        yi <- double(m)
        
        for (i in 1:m)
        {
            ni <- n[1:i]
            j <- i + 1
            ni2 <- n[1:j]
            wi.mean <- w.mean[1:i]
            yi[i] <-
                (sum(ni) * w.mean[i + 1] - sum(ni * wi.mean)) / sqrt((n[i + 1]) ^ (-1) *
                                                                         sum(ni) * sum(ni2))
        }
        
        t <- double(m)
        p <- double(m)
        q <- double(m)
        
        for (i in 1:m)
        {
            subyi <- 0
            j <- i - 1
            
            if (i > 1)
                subyi <- yi[1:j]
            
            subyi.sq <- (subyi) ^ 2
            t[i] <- yi[i] / sqrt(((N - k) * q.sq + sum(subyi.sq)) / (N - k + i - 1))
            statistic <- t[i]
            
            ### calculate log of the component probabilities ###
            
            df <- N - k + i - 1
            
            if (tail == "left")
            {
                if (i == 1)
                    METHOD <- paste(METHOD, "(left-tailed)")
                p.value = pt(statistic, df, lower.tail = TRUE)
                log.p.value = pt(statistic,
                                 df,
                                 lower.tail = TRUE,
                                 log.p = TRUE)
                log.q.value = pt(statistic,
                                 df,
                                 lower.tail = FALSE,
                                 log.p = TRUE)
            }
            
            else if (tail == "right")
            {
                if (i == 1)
                    METHOD <- paste(METHOD, "(right-tailed)")
                p.value = pt(statistic, df, lower.tail = FALSE)
                log.p.value = pt(statistic,
                                 df,
                                 lower.tail = FALSE,
                                 log.p = TRUE)
                log.q.value = pt(statistic,
                                 df,
                                 lower.tail = TRUE,
                                 log.p = TRUE)
            }
            
            else
            {
                tail = "both"
                
                if (i == 1)
                    METHOD <- paste(METHOD, "(two-tailed)")
                p.value = pt(statistic, df, lower.tail = TRUE)
                log.p.value = pt(statistic,
                                 df,
                                 lower.tail = TRUE,
                                 log.p = TRUE)
                log.q.value = pt(statistic,
                                 df,
                                 lower.tail = FALSE,
                                 log.p = TRUE)
                
                if (p.value >= 0.5)
                {
                    p.value = pt(statistic, df, lower.tail = FALSE)
                    log.p.value = pt(statistic,
                                     df,
                                     lower.tail = FALSE,
                                     log.p = TRUE)
                    log.q.value = pt(statistic,
                                     df,
                                     lower.tail = TRUE,
                                     log.p = TRUE)
                }
                
                p.value = p.value * 2
                log.p.value = log.p.value + log(2)
                log.q.value = log(1 - p.value)
            }
            
            p[i] <- log.p.value
            q[i] <- log.q.value
        }
        
        ### combine p-values using four p-value combining methods (T,F,N,L) ###
        
        psiT <- exp(min(p))
        psiF <- -2 * sum(p)
        psiN <- -sum(qnorm(p, lower.tail = TRUE, log.p = TRUE))
        A <- pi ^ 2 * m * (5 * m + 2) / (15 * m + 12)
        psiL <- -A ^ (-1 / 2) * sum(p - q)
        
        psiT.pvalue <- 1 - (1 - psiT) ^ m
        psiF.pvalue <- pchisq(psiF, df = 2 * m, lower.tail = FALSE)
        psiN.pvalue <- pnorm(psiN,
                             mean = 0,
                             sd = sqrt(m),
                             lower.tail = FALSE)
        psiL.pvalue <- pt(psiL, df = 5 * m + 4, lower.tail = FALSE)
        
        ### display output ###
        
        PSIT = psiT
        PSIF = psiF
        PSIN = psiN
        PSIL = psiL
        
        names(PSIT) = "Test Statistic (T)"
        names(PSIF) = "Test Statistic (F)"
        names(PSIN) = "Test Statistic (N)"
        names(PSIL) = "Test Statistic (L)"
        
        list(
            T = structure(
                list(
                    statistic = PSIT,
                    p.value = psiT.pvalue,
                    data.name = DNAME,
                    method = METHOD
                ),
                class = "htest"
            ),
            F = structure(
                list(
                    statistic = PSIF,
                    p.value = psiF.pvalue,
                    data.name = DNAME,
                    method = METHOD
                ),
                class = "htest"
            ),
            N = structure(
                list(
                    statistic = PSIN,
                    p.value = psiN.pvalue,
                    data.name = DNAME,
                    method = METHOD
                ),
                class = "htest"
            ),
            L = structure(
                list(
                    statistic = PSIL,
                    p.value = psiL.pvalue,
                    data.name = DNAME,
                    method = METHOD
                ),
                class = "htest"
            )
        )
    }