#' WP method
#'
-#' This function implements an R0 estimation due to White and Pagano (Statistics in Medicine, 2008). The method is based on maximum likelihood estimation in a Poisson transmission model. See details for important implementation notes.
+#' This function implements an R0 estimation due to White and Pagano (Statistics
+#' in Medicine, 2008). The method is based on maximum likelihood estimation in a
+#' Poisson transmission model. See details for important implementation notes.
#'
-#' This method is based on a Poisson transmission model, and hence may be most most valid at the beginning of an epidemic. In their model, the serial distribution is assumed to be discrete with a finite number of posible values. In this implementation, if the serial distribution is assumed known, it is taken to be a discretized version of a gamma distribution with mean \code{mu}, shape parameter one, and largest possible value based on parameter \code{tol}. When the serial distribution is unknown, the function implements a grid search algorithm to find the maximum likelihood estimator over all possible gamma distributions with unknown mean and variance, restricting these to a prespecified grid (see \code{search} parameter).
+#' This method is based on a Poisson transmission model, and hence may be most
+#' most valid at the beginning of an epidemic. In their model, the serial
+#' distribution is assumed to be discrete with a finite number of posible
+#' values. In this implementation, if \code{mu} is not {NA}, the serial
+#' distribution is taken to be a discretized version of a gamma distribution
+#' with mean \code{mu}, shape parameter one, and largest possible value based on
+#' parameter \code{tol}. When \code{mu} is \code{NA}, the function implements a
+#' grid search algorithm to find the maximum likelihood estimator over all
+#' possible gamma distributions with unknown mean and variance, restricting
+#' these to a prespecified grid (see \code{search} parameter).
#'
-#' When the serial distribution is taken to be \code{known}, sensitivity testing of the parameter \code{mu} is strongly recommended. If the serial distribution is \code{unknown}, the likelihood function can be flat near the maximum, resulting in numerical instability of the optimizer. When the serial distribution is \code{unkown} the implementation takes considerably longer to run. Users should be careful about units of time (e.g. are counts observed daily or weekly?) when implementing.
+#' When the serial distribution is known (i.e., \code{mu} is not \code{NA}),
+#' sensitivity testing of \code{mu} is strongly recommended. If the serial
+#' distribution is unknown (i.e., \code{mu} is \code{NA}), the likelihood
+#' function can be flat near the maximum, resulting in numerical instability of
+#' the optimizer. When \code{mu} is \code{NA}, the implementation takes
+#' considerably longer to run. Users should be careful about units of time
+#' (e.g., are counts observed daily or weekly?) when implementing.
#'
-#' The model developed in White and Pagano (2008) is discrete, and hence the serial distribution is finite discrete. In our implementation, the input value \code{mu} is that of a continuous distribution. The algorithm when \code{method="known"} disretizes this input, and hence the mean of the serial distribution returned in the list \code{SD} will differ from \code{mu} somewhat. That is to say, if the user notices that the input \code{mu} and out put mean of \code{SD} are different, this is to be expected, and is caused by the discretization.
+#' The model developed in White and Pagano (2008) is discrete, and hence the
+#' serial distribution is finite discrete. In our implementation, the input
+#' value \code{mu} is that of a continuous distribution. The algorithm
+#' discretizes this input when \code{mu} is not \code{NA}, and hence the mean of
+#' the serial distribution returned in the list \code{SD} will differ from
+#' \code{mu} somewhat. That is to say, if the user notices that the input
+#' \code{mu} and output mean of \code{SD} are different, this is to be expected,
+#' and is caused by the discretization.
#'
+#' @param NT Vector of case counts.
+#' @param mu Mean of the serial distribution (needs to match case counts in time
+#' units; for example, if case counts are weekly and the serial
+#' distribution has a mean of seven days, then \code{mu} should be set
+#' to one). The default value of \code{mu} is set to \code{NA}.
+#' @param search List of default values for the grid search algorithm. The list
+#' includes three elements: the first is \code{B}, which is the
+#' length of the grid in one dimension; the second is
+#' \code{scale.max}, which is the largest possible value of the
+#' scale parameter; and the third is \code{shape.max}, which is
+#' the largest possible value of the shape parameter. Defaults to
+#' \code{B = 100, scale.max = 10, shape.max = 10}. For both shape
+#' and scale, the smallest possible value is 1/\code{B}.
+#' @param tol Cutoff value for cumulative distribution function of the
+#' pre-discretization gamma serial distribution. Defaults to 0.999
+#' (i.e. in the discretization, the maximum is chosen such that the
+#' original gamma distribution has cumulative probability of no more
+#' than 0.999 at this maximum).
#'
-#' @param NT Vector of case counts
-#' @param mu Mean of the serial distribution (needs to match case counts in time units; for example, if case counts are weekly and the serial distribution has a mean of seven days, then \code{mu} should be set to one). The default value of \code{mu} is set to \code{NA}.
-#' @param method Variable taking one of two possible values: \code{known} or \code{unknown}. If "known", the serial distribution is assumed to be gamma with rate 1/\code{mu} and shape equal to one, if "unknown" then the serial distribution is gamma with unknown parameters. Defaults to "unknown"
-#' @param search List of default values for the grid search algorithm; the list includes three elements: the first is \code{B} which is the length of the grid in one dimension, the second is \code{scale.max} which is the largest possible value of the scale parameter, and the third is \code{shape.max} which is the largest possible value of the shape parameter; defaults to \code{B=100, scale.max=10, shape.max=10}. For both shape and scale, the smallest possible value is 1/\code{B}.
-#' @param tol Cutoff value for cumulative distribution function of the pre-discretization gamma serial distribution, defaults to 0.999 (i.e. in the discretization, the maximum is chosen such that the original gamma distribution has cumulative probability of no more than 0.999 at this maximum).
-#'
-#' @return WP returns a list containing the following components: \code{Rhat} is the estimate of R0, \code{SD} is either the discretized serial distribution (if \code{method="known"}) or the estimated discretized serial distribution (if \code{method="unknown"}), and \code{inputs} is a list of the original input variables \code{NT, mu, method, search, tol}. The list also returns the variable \code{check}, which is equal to the number of non-unique maximum likelihood estimators. The serial distribution \code{SD} is returned as a list made up of \code{supp} the support of the distribution and \code{pmf} the probability mass function.
+#' @return \code{WP} returns a list containing the following components:
+#' \code{Rhat} is the estimate of R0, and \code{SD} is either the
+#' discretized serial distribution (if \code{mu} is not \code{NA}), or
+#' the estimated discretized serial distribution (if \code{mu} is
+#' \code{NA}). The list also returns the variable \code{check}, which is
+#' equal to the number of non-unique maximum likelihood estimators. The
+#' serial distribution \code{SD} is returned as a list made up of
+#' \code{supp} (the support of the distribution) and \code{pmf} (the
+#' probability mass function).
#'
#' @examples
-#'
-#' ## ===================================================== ##
-#' ## Illustrate on weekly data ##
-#' ## ===================================================== ##
-#'
-#' NT <- c(1, 4, 10, 5, 3, 4, 19, 3, 3, 14, 4)
-#' ## obtain Rhat when serial distribution has mean of five days
-#' res1 <- WP(NT=NT, mu=5/7, method="known")
+#' # Weekly data.
+#' NT <- c(1, 4, 10, 5, 3, 4, 19, 3, 3, 14, 4)
+#'
+#' # Obtain R0 when the serial distribution has a mean of five days.
+#' res1 <- WP(NT, mu = 5 / 7)
#' res1$Rhat
-#' ## obtain Rhat when serial distribution has mean of three days
-#' res2 <- WP(NT=NT, mu=3/7, method="known")
+#'
+#' # Obtain R0 when the serial distribution has a mean of three days.
+#' res2 <- WP(NT, mu = 3 / 7)
#' res2$Rhat
-#' ## obtain Rhat when serial distribution is unknown
-#' res3 <- WP(NT=NT)
+#'
+#' # Obtain R0 when the serial distribution is unknown.
+#' # NOTE: This implementation will take longer to run.
+#' res3 <- WP(NT)
#' res3$Rhat
-#' ## find mean of estimated serial distribution
-#' serial <- res3$SD
-#' sum(serial$supp*serial$pmf)
-#' TODO - talk to Jane about this example - should we have tested SD in our simulations as well in the paper?
#'
-#' ## ========================================================= ##
-#' ## Compute Rhat using only the first five weeks of data ##
-#' ## ========================================================= ##
+#' # Find the mean of the estimated serial distribution.
+#' serial <- res3$SD
+#' sum(serial$supp * serial$pmf)
#'
-#'
-#' res4 <- WP(NT=NT[1:5], mu=5/7, method="known") # serial distribution has mean of five days
-#' res4$Rhat
+#' @importFrom stats pexp qexp
#'
-#'
#' @export
+WP <- function(NT, mu = NA,
+ search = list(B = 100, shape.max = 10, scale.max = 10),
+ tol = 0.999) {
+ if (is.na(mu)) {
+ print("You have assumed that the serial distribution is unknown.")
+ res <- WP_unknown(NT, B = search$B, shape.max = search$shape.max,
+ scale.max = search$scale.max, tol = tol)
+ Rhat <- res$Rhat
+ p <- res$p
+ range.max <- res$range.max
+ JJ <- res$JJ
+ } else {
+ print("You have assumed that the serial distribution is known.")
+ range.max <- ceiling(qexp(tol, rate = 1 / mu))
+ p <- diff(pexp(0:range.max, 1 / mu))
+ p <- p / sum(p)
+ res <- WP_known(NT = NT, p = p)
+ Rhat <- res
+ JJ <- NA
+ }
+
+ return(list(Rhat = Rhat,
+ check = length(JJ),
+ SD = list(supp = 1:range.max, pmf = p)))
+}
+
+#' WP method background function WP_known
+#'
+#' This is a background/internal function called by \code{WP}. It computes the
+#' maximum likelihood estimator of R0 assuming that the serial distribution is
+#' known and finite discrete.
+#'
+#' @param NT Vector of case counts.
+#' @param p Discretized version of the serial distribution.
+#'
+#' @return The function returns the maximum likelihood estimator of R0.
+#'
+#' @noRd
+WP_known <- function(NT, p) {
+ k <- length(p)
+ TT <- length(NT) - 1
+ mu_t <- rep(0, TT)
+
+ for (i in 1:TT) {
+ Nt <- NT[i:max(1, i - k + 1)]
+ mu_t[i] <- sum(p[1:min(k, i)] * Nt)
+ }
+
+ Rhat <- sum(NT[-1]) / sum(mu_t)
+ return(Rhat)
+}
+
+#' WP method background function WP_unknown
+#'
+#' This is a background/internal function called by \code{WP}. It computes the
+#' maximum likelihood estimator of R0 assuming that the serial distribution is
+#' unknown but comes from a discretized gamma distribution. The function then
+#' implements a simple grid search algorithm to obtain the maximum likelihood
+#' estimator of R0 as well as the gamma parameters.
+#'
+#' @param NT Vector of case counts.
+#' @param B Length of grid for shape and scale (grid search parameter).
+#' @param shape.max Maximum shape value (grid \code{search} parameter).
+#' @param scale.max Maximum scale value (grid \code{search} parameter).
+#' @param tol cutoff value for cumulative distribution function of the serial
+#' distribution (defaults to 0.999).
+#'
+#' @return The function returns \code{Rhat}, the maximum likelihood estimator of
+#' R0, as well as the maximum likelihood estimator of the discretized
+#' serial distribution given by \code{p} (the probability mass function)
+#' and \code{range.max} (the distribution has support on the integers
+#' one to \code{range.max}). The function also returns \code{resLL} (all
+#' values of the log-likelihood) at \code{shape} (grid for shape
+#' parameter) and at \code{scale} (grid for scale parameter), as well as
+#' \code{resR0} (the full vector of maximum likelihood estimators),
+#' \code{JJ} (the locations for the likelihood for these), and \code{J0}
+#' (the location for the maximum likelihood estimator \code{Rhat}). If
+#' \code{JJ} and \code{J0} are not the same, this means that the maximum
+#' likelihood estimator is not unique.
+#'
+#' @importFrom stats pgamma qgamma
+#'
+#' @noRd
+WP_unknown <- function(NT, B = 100, shape.max = 10, scale.max = 10,
+ tol = 0.999) {
+ shape <- seq(0, shape.max, length.out = B + 1)
+ scale <- seq(0, scale.max, length.out = B + 1)
+ shape <- shape[-1]
+ scale <- scale[-1]
+
+ resLL <- matrix(0, B, B)
+ resR0 <- matrix(0, B, B)
+
+ for (i in 1:B)
+ for (j in 1:B) {
+ range.max <- ceiling(qgamma(tol, shape = shape[i], scale = scale[j]))
+ p <- diff(pgamma(0:range.max, shape = shape[i], scale = scale[j]))
+ p <- p / sum(p)
+ mle <- WP_known(NT, p)
+ resLL[i, j] <- computeLL(p, NT, mle)
+ resR0[i, j] <- mle
+ }
+
+ J0 <- which.max(resLL)
+ R0hat <- resR0[J0]
+ JJ <- which(resLL == resLL[J0], arr.ind = TRUE)
+ range.max <- ceiling(qgamma(tol, shape = shape[JJ[1]], scale = scale[JJ[2]]))
+ p <- diff(pgamma(0:range.max, shape = shape[JJ[1]], scale = scale[JJ[2]]))
+ p <- p / sum(p)
+
+ return(list(Rhat = R0hat, J0 = J0, ll = resLL, Rs = resR0, scale = scale,
+ shape = shape, JJ = JJ, p = p, range.max = range.max))
+}
+
+#' WP method background function computeLL
+#'
+#' This is a background/internal function called by \code{WP}. It computes the
+#' log-likelihood.
+#'
+#' @param p Discretized version of the serial distribution.
+#' @param NT Vector of case counts.
+#' @param R0 Basic reproductive ratio.
+#'
+#' @return This function returns the log-likelihood at the input variables and
+#' parameters.
+#'
+#' @noRd
+computeLL <- function(p, NT, R0) {
+ k <- length(p)
+ TT <- length(NT) - 1
+ mu_t <- rep(0, TT)
+
+ for (i in 1:TT) {
+ Nt <- NT[i:max(1, i - k + 1)]
+ mu_t[i] <- sum(p[1:min(k, i)] * Nt)
+ }
-WP <- function(NT, mu="NA", method="unknown", search=list(B=100, shape.max=10, scale.max=10), tol=0.999){
-
- if(method=="unknown"){
-
- print("You have assumed that the serial distribution is unknown.")
- res <- WP_unknown(NT=NT, B=search$B, shape.max=search$shape.max, scale.max=search$scale.max, tol=tol)
- Rhat <- res$Rhat
- p <- res$p
- range.max <- res$range.max
- JJ <- res$JJ
-
-
- }
-
- if(method=="known"){
-
- if(mu=="NA"){
+ mu_t <- R0 * mu_t
+ LL <- sum(NT[-1] * log(mu_t)) - sum(mu_t)
- res <- "NA"
- print("For method=known, the mean of the serial distribution must be specified.")
-
- } else {
-
- print("You have assumed that the serial distribution is known.")
-
- range.max <- ceiling(qexp(tol, rate=1/mu))
- p <- diff(pexp(0:range.max, 1/mu))
- p <- p/sum(p)
- res <- WP_known(NT=NT, p=p)
- Rhat <- res$Rhat
- JJ <- NA
- }
-
- }
-
-return(list(Rhat=Rhat, check=length(JJ), SD=list(supp=1:range.max, pmf=p), inputs=list(NT=NT, mu=mu, method=method, search=search, tol=tol)))
-
+ return(LL)
}
git.nmode.ca / Rnaught / blobdiff
