R/intensity.R

In SmoothHazard: Estimation of Smooth Hazard Models for Interval-Censored Data

### intensity.R --- 
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## author: Thomas Alexander Gerds
## created: Feb  6 2016 (08:47) 
## Version: 
## last-updated: Apr  2 2024 (18:04) 
##           By: Thomas Alexander Gerds
##     Update #: 29
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## 
### Commentary: 
## 
### Change Log:
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## 
### Code:
##'  M-spline estimate of the transition intensity function
##' and the cumulative transition intensity function
##' for survival and illness-death models
##'
##' The estimate of the transition intensity function is a linear
##' combination of M-splines and the estimate of the cumulative transition
##' intensity function is a linear combination of I-splines (the integral of a
##' M-spline is called I-spline). The coefficients \code{theta} are the same for
##' the M-splines and I-splines.
##' 
##' Important: the theta parameters returned by \code{idm} and \code{shr} are in fact
##' the square root of the splines coefficients. See examples.
##'
##' This function is a R-translation of a corresponding Fortran function called \code{susp}. \code{susp} is
##' used internally by \code{idm} and \code{shr}.
##' 
##' @title M-spline estimate of the transition intensity function
##' @param times Time points at which to estimate the intensity function
##' @param knots Knots for the M-spline
##' @param number.knots Number of knots for the M-splines (and I-splines see details)
##' @param theta The coefficients for the linear combination of M-splines (and I-splines see details)
##' @param linear.predictor Linear predictor beta*Z. When it is non-zero, 
##' transition and cumulative transition are multiplied by \code{exp(linear.predictor)}. Default is zero. 
##' @return
##' \item{times}{The time points at which the following estimates are evaluated.}
##' \item{intensity}{The transition intensity function evaluated at \code{times}.}
##' \item{cumulative.intensity}{The cumulative transition intensity function evaluated at \code{times}}
##' \item{survival}{The "survival" function, i.e., exp(-cumulative.intensity)}
##' 
##' @seealso \code{\link{shr}}, \code{\link{idm}} 
##' @examples
##' data(testdata)
##' fit.su <- shr(Hist(time=list(l, r), id) ~ cov,
##'               data = testdata,method = "Splines",CV = TRUE)
##' intensity(times = fit.su$time, knots = fit.su$knots,
##'            number.knots = fit.su$nknots, theta = fit.su$theta^2)
##' 
##' \donttest{
##'   data(Paq1000)
##'   fit.idm <-  idm(formula02 = Hist(time = t, event = death, entry = e) ~ certif,
##'                   formula01 = Hist(time = list(l,r), event = dementia) ~ certif,
##'                   formula12 = ~ certif, method = "Splines", data = Paq1000)
##'   # Probability of survival in state 0 at age 80 for a subject with no cep given
##'   # that he is in state 0 at 70
##'   su0 <- (intensity(times = 80, knots = fit.idm$knots01, 
##'                    number.knots = fit.idm$nknots01, 
##'                    theta = fit.idm$theta01^2)$survival
##'          *intensity(times = 80, knots = fit.idm$knots02, 
##'                    number.knots = fit.idm$nknots02, 
##'                    theta = fit.idm$theta02^2)$survival)/
##'         (intensity(times = 70, knots = fit.idm$knots01, 
##'                    number.knots = fit.idm$nknots01, 
##'                    theta = fit.idm$theta01^2)$survival
##'         *intensity(times = 70, knots = fit.idm$knots02, 
##'                    number.knots = fit.idm$nknots02, 
##'                    theta = fit.idm$theta02^2)$survival)
##'   # Same result as:  
##'   predict(fit.idm, s = 70, t = 80, conf.int = FALSE) # see first element               
##' }
##' @export  
#' @author R: Celia Touraine <Celia.Touraine@@isped.u-bordeaux2.fr> and Thomas Alexander Gerds <tag@@biostat.ku.dk>
#' Fortran: Pierre Joly <Pierre.Joly@@isped.u-bordeaux2.fr> 
intensity <- function(times,knots,number.knots,theta,linear.predictor=0) {
    if (length(theta)!= number.knots+2) stop(paste0("For ",
                  number.knots,
                  " knots we need ",
                  number.knots+2,
                  " coefficients. But, length of argument theta as provided is ",
                  length(theta)))
    cumulative.intensity=rep(0,length(times))   # risque cumule
    intensity=rep(0,length(times))  # risque
    survival=rep(0,length(times))   # survie
    TF=rep(0,length(times)) # T si z[i-1]<=times[.]<z[i], F sinon
    som=0
    for (i in 5:(number.knots+3)) {
        TF = ( (knots[i-1]<=times) & (times<knots[i]) )
        if (sum(TF) != 0) { 
            ind = which(TF) 
            mm3=rep(0,length(ind))
            mm2=rep(0,length(ind))
            mm1=rep(0,length(ind))
            mm=rep(0,length(ind))
            im3=rep(0,length(ind))
            im2=rep(0,length(ind))
            im1=rep(0,length(ind))
            im=rep(0,length(ind))
            j = i-1
            if (j>4) { 
                som = sum(theta[1:(j-4)])
            }
            ht = times[ind]-knots[j] #
            htm = times[ind]-knots[j-1] #
            h2t = times[ind]-knots[j+2] #
            ht2 = knots[j+1]-times[ind] #
            ht3 = knots[j+3]-times[ind] #
            hht = times[ind]-knots[j-2] #
            h = knots[j+1]-knots[j]
            hh = knots[j+1]-knots[j-1]
            h2 = knots[j+2]-knots[j]
            h3 = knots[j+3]-knots[j]
            h4 = knots[j+4]-knots[j]
            h3m = knots[j+3]-knots[j-1]
            h2n = knots[j+2]-knots[j-1]
            hn= knots[j+1]-knots[j-2]
            hh3 = knots[j+1]-knots[j-3]
            hh2 = knots[j+2]-knots[j-2]
            mm3[ind] = ((4*ht2*ht2*ht2)/(h*hh*hn*hh3))
            mm2[ind] = ((4*hht*ht2*ht2)/(hh2*hh*h*hn))+((-4*h2t*htm*ht2)/(hh2*h2n*hh*h))+((4*h2t*h2t*ht)/(hh2*h2*h*h2n))
            mm1[ind] = (4*(htm*htm*ht2)/(h3m*h2n*hh*h))+((-4*htm*ht*h2t)/(h3m*h2*h*h2n))+((4*ht3*ht*ht)/(h3m*h3*h2*h))
            mm[ind] = 4*(ht*ht*ht)/(h4*h3*h2*h)
            im3[ind] = (0.25*(times[ind]-knots[j-3])*mm3[ind])+(0.25*hh2*mm2[ind])+(0.25*h3m*mm1[ind])+(0.25*h4*mm[ind])
            im2[ind] = (0.25*hht*mm2[ind])+(h3m*mm1[ind]*0.25)+(h4*mm[ind]*0.25)
            im1[ind] = (htm*mm1[ind]*0.25)+(h4*mm[ind]*0.25)
            im[ind] = ht*mm[ind]*0.25
            cumulative.intensity[ind] = som +(theta[j-3]*im3[ind])+(theta[j-2]*im2[ind])+(theta[j-1]*im1[ind])+(theta[j]*im[ind])
            intensity[ind] = (theta[j-3]*mm3[ind])+(theta[j-2]*mm2[ind])+(theta[j-1]*mm1[ind])+(theta[j]*mm[ind])
            ## if (any(is.na(intensity))) browser()
        } # fin if (sum(TF) != 0)
    } # fin for
    TF = (times>=knots[number.knots+3])
    if (sum(TF) != 0) {
        ind = which(TF)
        som = sum(theta[1:(number.knots+2)])
        cumulative.intensity[ind] = som
        intensity[ind] = 4*theta[number.knots+2]/(knots[number.knots+3]-knots[number.knots+2])
    }
    TF = (times<knots[4])
    if (sum(TF) != 0) {
        ind = which(TF)
        cumulative.intensity[ind] = 0
        intensity[ind] = 0
    }
    e = exp(linear.predictor)
    intensity=intensity*e
    cumulative.intensity=cumulative.intensity*e
    survival = exp(-cumulative.intensity)
    return(list(times=times,intensity=intensity,cumulative.intensity=cumulative.intensity,survival=survival))
}

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### intensity.R ends here