These functions permit to use alternate parametrisations for Gamma distributions, from 'shape and scale' to 'mean (mu) and coefficient of variation (cv), and back. gamma_shapescale2mucv does the first conversion, while gamma_mucv2shapescale does the second. The function gamma_log_likelihood is a shortcut for computing Gamma log-likelihood with the alternative parametrisation (mean, cv). See 'details' for a guide of which parametrisation to use.

gamma_shapescale2mucv(shape, scale)

gamma_mucv2shapescale(mu, cv)

gamma_log_likelihood(x, mu, cv, discrete = TRUE, interval = 1, w = 0,
anchor = 0.5)

## Arguments

shape The shape parameter of the Gamma distribution. The scale parameter of the Gamma distribution. The mean of the Gamma distribution. The coefficient of variation of the Gamma distribution, i.e. the standard deviation divided by the mean. A vector of data treated as observations drawn from a Gamma distribution, for which the likelihood is to be computed. A logical indicating if the distribution should be discretised; TRUE by default. The interval used for discretisation; see distcrete. The centering of the interval used for discretisation, defaulting to 0; see distcrete. The anchor used for discretisation, i.e. starting point of the discretisation process; defaults to 0; see distcrete.

## Value

A named list containing 'shape' and 'scale', or mean ('mean') and coefficient of variation ('cv').

## Details

The gamma distribution is described in ?dgamma is parametrised using shape and scale (or rate). However, these parameters are naturally correlated, which make them poor choices whenever trying to fit data to a Gamma distribution. Their interpretation is also less clear than the traditional mean and variance. When fitting the data, or reporting results, it is best to use the alternative parametrisation using the mean (mu) and the coefficient of variation (cv), i.e. the standard deviation divided by the mean.

## Examples


## set up some parameters

mu <- 10
cv <- 1

## transform into shape scale

tmp <- gamma_mucv2shapescale (mu, cv)
shape <- tmp$shape scale <- tmp$scale

## recover original parameters when applying the revert function

gamma_shapescale2mucv(shape, scale) # compare with mu, cv#> $mu #>  10 #> #>$cv
#>  1
#>

## empirical validation:
## check mean / cv of a sample derived using rgamma with
## shape and scale computed from mu and cv

gamma_sample <- rgamma(n = 10000, shape = shape, scale = scale)
mean(gamma_sample) # compare to mu#>  9.835163sd(gamma_sample) / mean(gamma_sample) # compare to cv#>  1.002399