Convert RE variances from cov to cor/sd #536

library(rstanarm)
library(lme4)
data(cake)
set.seed(123)

m1 <- stan_lmer(angle ~ temp + (temp | replicate),
                data = cake,
                chains = 1)

bayestestR::describe_posterior(m1, effects = "all")
#> ...
#> 
#> # Random effects SD/Cor: replicate
#> 
#> Parameter          |   Median |         95% CI |     pd |          ROPE | % in ROPE |  Rhat |    ESS
#> ----------------------------------------------------------------------------------------------------
#> (Intercept)        | 2.90e-03 | [ 0.00,  0.70] |   100% | [-0.82, 0.82] |      100% | 0.999 | 612.00
#> temp ~ (Intercept) | 1.86e-04 | [-0.01,  0.01] | 59.20% | [-0.82, 0.82] |      100% | 0.999 | 395.00
#> temp               | 1.12e-03 | [ 0.00,  0.00] |   100% | [-0.82, 0.82] |      100% | 1.000 | 325.00
#> 
#> ...


posteriors <- as.data.frame(m1)

rho <- with(posteriors, {
  `Sigma[replicate:temp,(Intercept)]` / 
    (sqrt(`Sigma[replicate:temp,temp]`) * sqrt(`Sigma[replicate:(Intercept),(Intercept)]`))
})

# Cor
bayestestR::describe_posterior(rho)
#> Summary of Posterior Distribution
#> 
#> Parameter | Median |        95% CI |     pd |          ROPE | % in ROPE
#> -----------------------------------------------------------------------
#> Posterior |   0.19 | [-0.92, 0.95] | 59.20% | [-0.10, 0.10] |     9.05%


# Slopes
a <- with(posteriors, sqrt(`Sigma[replicate:temp,temp]`))

bayestestR::describe_posterior(a)
#> Summary of Posterior Distribution
#> 
#> Parameter | Median |       95% CI |   pd |          ROPE | % in ROPE
#> --------------------------------------------------------------------
#> Posterior |   0.03 | [0.02, 0.05] | 100% | [-0.10, 0.10] |      100%


# Compare to:
lme4::lmer(angle ~ temp + (temp | replicate),
           data = cake) |> 
  lme4::VarCorr()
#>  Groups    Name        Std.Dev. Corr 
#>  replicate (Intercept) 2.604527      
#>            temp        0.027073 0.134
#>  Residual              4.825927