# Basic usage

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# 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).

    ## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
    ## ℹ Please use `linewidth` instead.
    ## This warning is displayed once every 8 hours.
    ## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
    ## generated.

plot of chunk densityPlots

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")

plot of chunk cumDensityPlot

## 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))

    ##       mean        var
    ## 0.63812093 0.01208171

## 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))

    ## [1] 0.6641416

## 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))

    ##             mu    sigma
    ## [1,] 0.7608886 0.464783

    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")

plot of chunk twCoefLogitnormMLE

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))

    ##              mu    sigma
    ## [1,] 0.01213214 1.444962

    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")

plot of chunk twCoefLogitnormMLEFlat