In logitnorm: Functions for the Logitnormal Distribution

library(knitr)
opts_chunk$set(out.extra='style="display:block; margin: auto"'
    #, fig.align="center"
    #, fig.width=4.6, fig.height=3.2
    , fig.width=6, fig.height=3.75 #goldener Schnitt 1.6
    , dev.args=list(pointsize=10)
    , dev=c('png','pdf')
    )
knit_hooks$set(spar = function(before, options, envir) {
    if (before){
        par( las=1 )                   #also y axis labels horizontal
        par(mar=c(2.0,3.3,0,0)+0.3 )  #margins
        par(tck=0.02 )                          #axe-tick length inside plots             
        par(mgp=c(1.1,0.2,0) )  #positioning of axis title, axis labels, axis
     }
})
library(logitnorm) 
if( !require(ggplot2) ){
    print("To generate this vignette, ggplo2 is required.")
    exit(0)
}
#library(reshape2)
# genVigs("logitnorm")

Basic usage

Distribution

The logitnormal distribution is useful as a prior density for variables that are bounded between 0 and 1, such as proportions. The following figure displays its density for various combinations of parameters mu (panels) and sigma (lines).

xGrid =c(0+.Machine$double.eps,seq(0,1, length.out=81)[-c(1,81)],1-.Machine$double.eps)
#explore chanign sigma at mu=0 
theta0 <- expand.grid(mu=seq(0,2,length.out=9),
sigma=10^seq(-0.5,0.5,length.out=5)) 
n <- nrow(theta0)

.calcDensityGrid <- function(
        ### Calculate logitnormal density for given combinations
        theta0  ##<< matrix with columns mu and sigma
        ,xGrid = seq(0,1, length.out=81)[-c(1,81)]
){ 
    dx <- apply( theta0, 1, function(theta0i){
                dx <- dlogitnorm(xGrid, mu=theta0i[1], sigma=theta0i[2])
            })
    # dimnames(dx) <- list(iX=NULL,iTheta=NULL)
    # ds <- melt(dx)    # two variables over time
    # ds <- gather(as.data.frame(dx))
    # ds[1:10,]
    ds <- data.frame(
      mu = rep(as.factor(round(theta0[,1],2)), each=length(xGrid)),
      sigma = rep(as.factor(round(theta0[,2],2)), each=length(xGrid)),
      x = rep(as.vector(xGrid), nrow(theta0)),
      value = as.vector(dx)
    )
    ### data frame with columns value,x,mu and sigma, value (density at x) 
}

ds <- .calcDensityGrid(theta0,xGrid=xGrid)
p1 <- ggplot(ds, aes(x=x, y=value, color=sigma, linetype=sigma)) + geom_line(size=1) +
 facet_wrap(~mu,scales="free")+
 theme_bw() +
 theme(axis.title.x = element_blank()) +  ylab("density") +
 theme()
print(p1)

Example: Plot the cumulative distribution

x <- seq(0,1, length.out=81) 
d <- plogitnorm(x, mu=0.5, sigma=0.5)
plot(d~x,type="l")

Mean and Variance

The moments have no analytical solution. This package estimates them by numerical integration:

Example: estimate mean and standard deviation.

(theta <- momentsLogitnorm(mu=0.6,sigma=0.5))

Mode

The mode is found by setting derivatives to zero and optimizing the resulting equation: $logit(x) = \sigma^2(2x-1)+\mu$.

Example: estimate the mode

(mle <- modeLogitnorm(mu=0.6,sigma=0.5))

Parameter Estimation

from upper quantile and - mode (Maximum Likelihood Estimate) - mean (Expected value) - median

Example: estimate the parameters, with mode 0.7 and upper quantile 0.9

(theta <- twCoefLogitnormMLE(0.7,0.9))
x <- seq(0,1, length.out=81) 
d <- dlogitnorm(x, mu=theta[1,"mu"], sigma=theta[1,"sigma"])
plot(d~x,type="l")
abline(v=c(0.7,0.9), col="grey")

When increasing the $\sigma$ parameter, the distribution becomes eventually becomes bi-model, i.e. has two maxima. The unimodal distribution for a given mode with widest confidence intervals is obtained by function twCoefLogitnormMLEFlat.

(theta <- twCoefLogitnormMLEFlat(0.7))
x <- seq(0,1, length.out=81) 
d <- dlogitnorm(x, mu=theta[1,"mu"], sigma=theta[1,"sigma"])
plot(d~x,type="l")
abline(v=c(0.7), col="grey")