Beta shape parameters from the parameters of each scheme

Converts the pair (mu, phi) of one of three parameterisations into the shapes $(a, b)$ of the beta density $f(y) = y^{a - 1}(1 - y)^{b - 1}/B(a, b)$, $0 < y < 1$.

Usage

brs_repar(mu, phi, repar = 2L)

Arguments

mu: Numeric vector: the first parameter, i.e. the mean in $(0, 1)$ for repar = 1, 2 and the shape $p > 0$ for repar = 0.
phi: Numeric vector (or scalar): the second parameter, i.e. the precision $\phi > 0$, the dispersion $\phi \in (0, 1)$ or the shape $q > 0$.
repar: Integer (0, 1 or 2) selecting the scheme (default 2).

Value

A data.frame with columns shape1 ($a$) and shape2 ($b$), one row per element of the recycled inputs.

Details

`repar`	`mu`, `phi`	$(a, b)$	$E[Y]$	$\mathrm{Var}[Y]$
0	shapes $p, q > 0$	$(p, q)$	$p/(p + q)$	$pq/\{(p + q)^2 (p + q + 1)\}$
1	mean $\mu \in (0, 1)$, precision $\phi > 0$	$(\mu\phi, (1 - \mu)\phi)$	$\mu$	$\mu(1 - \mu)/(1 + \phi)$
2	mean $\mu \in (0, 1)$, dispersion $\phi \in (0, 1)$	$(\mu\tau, (1 - \mu)\tau)$, $\tau = (1 - \phi)/\phi$	$\mu$	$\phi\,\mu(1 - \mu)$

The three describe the same family: $a + b$ is the precision, and the dispersion of repar = 2 is $$\phi = \frac{\mathrm{Var}[Y]}{\mu(1 - \mu)} = \frac{1}{1 + a + b},$$ the share of the largest possible variance $\mu(1 - \mu)$, which is approached as $a + b \to 0$. It is not a coefficient of variation.

In Lopes (2023) these are eq. eqn_beta_p1 (shapes $p, q$), "parametrizacao 1" (Ferrari and Cribari-Neto, 2004; eq. eqn_beta_p2) and "parametrizacao 2" (Bayer, 2011; eq. eqn_beta_p3), where the dispersion is written $\sigma$. The package names the second parameter phi in every scheme. Regression on the shapes (repar = 0) is a package extension. Admissible links per scheme: section 'Reparameterizations and links' of brs.

References

Lopes, J. E. (2023). Modelos de regressao beta para dados de escala. Master's dissertation, Universidade Federal do Parana, Curitiba. URI: https://hdl.handle.net/1884/86624.

Ferrari, S. L. P., and Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31(7), 799–815. doi:10.1080/0266476042000214501

Bayer, F. M. (2011). Modelagem e inferencia em regressao beta. PhD thesis, Universidade Federal de Pernambuco.

Examples

# One beta distribution in the three parameterisations: shapes (6, 14)
brs_repar(mu = 0.3, phi = 20, repar = 1)      # mean 0.3, precision 20
#>   shape1 shape2
#> 1      6     14
brs_repar(mu = 0.3, phi = 1 / 21, repar = 2)  # mean 0.3, dispersion 1/(1 + 20)
#>   shape1 shape2
#> 1      6     14
brs_repar(mu = 6, phi = 14, repar = 0)        # shapes p = 6, q = 14
#>   shape1 shape2
#> 1      6     14

# Back from shapes: mean, precision a + b, dispersion 1 / (1 + a + b)
sh <- brs_repar(mu = 0.3, phi = 20, repar = 1)
c(mean = sh$shape1 / (sh$shape1 + sh$shape2),
  precision = sh$shape1 + sh$shape2,
  dispersion = 1 / (1 + sh$shape1 + sh$shape2))
#>        mean   precision  dispersion 
#>  0.30000000 20.00000000  0.04761905 

# E[Y] and Var[Y] agree across parameterisations: 0.3 and 0.21 / 21 = 0.01
mu <- 0.3
c(var_repar1 = mu * (1 - mu) / (1 + 20),
  var_repar2 = mu * (1 - mu) * (1 / 21),
  var_shapes = 6 * 14 / ((6 + 14)^2 * (6 + 14 + 1)))
#> var_repar1 var_repar2 var_shapes 
#>       0.01       0.01       0.01 

# Vectorised: one row per observation
brs_repar(mu = c(0.2, 0.5, 0.8), phi = 0.1, repar = 2)
#>   shape1 shape2
#> 1    1.8    7.2
#> 2    4.5    4.5
#> 3    7.2    1.8

`repar`	`mu`, `phi`	\((a, b)\)	\(E[Y]\)	\(\mathrm{Var}[Y]\)
0	shapes \(p, q > 0\)	\((p, q)\)	\(p/(p + q)\)	\(pq/\{(p + q)^2 (p + q + 1)\}\)
1	mean \(\mu \in (0, 1)\), precision \(\phi > 0\)	\((\mu\phi, (1 - \mu)\phi)\)	\(\mu\)	\(\mu(1 - \mu)/(1 + \phi)\)
2	mean \(\mu \in (0, 1)\), dispersion \(\phi \in (0, 1)\)	\((\mu\tau, (1 - \mu)\tau)\), \(\tau = (1 - \phi)/\phi\)	\(\mu\)	\(\phi\,\mu(1 - \mu)\)