# Standard deviation of outcome in gamlss model with random intercepts in mean

I'm interested in a simple random-intercepts model: $$Y_{ij} = \alpha_0 + \gamma_i + \epsilon_{ij}$$ where $$\gamma_i \sim N(0, \sigma_{\gamma}^2)$$ independently of $$\epsilon_{ij} \sim N(0, \sigma_{\epsilon}^2)$$.

In nlme:

library(nlme)
data(ergoStool)
l1 <- lme(effort ~ 1, data = ergoStool, random = ~ 1|Subject, method="ML")


and gives estimates of (what I assume are) $$\sigma_\gamma$$ and $$\sigma_\epsilon$$:

summary(l1)

Random effects:
Formula: ~1 | Subject
(Intercept) Residual
StdDev:   0.9090556 2.020727


In gamlss, the same model can be fit, with the random intercepts included in a couple of ways:

library(gamlss)
t1 <- gamlss(effort ~ 1 + random(Subject), data = ergoStool)
t2 <- gamlss(effort ~ 1 + re(random = ~ 1|Subject), data = ergoStool)


The gamlss models give the same fit, and each has three variance 'components', which agree with one another:

c(sigb = getSmo(t1)$$sigb, sige = getSmo(t1)$$sige, exp(t1$sigma.coefficients)) sigb sige (Intercept) 0.9090593 1.0610962 1.9043752 summary(t2) Random effects: Formula: ~1 | Subject (Intercept) Residual StdDev: 0.9090594 1.061096 exp(t2$sigma.coefficients)

(Intercept)
1.904375


The first is obviously the same $$\hat{\sigma}_\gamma$$, but I can't work out how the other two (the residual standard deviation from the random intercept part, and the estimate of the standard deviation of the outcome) relate to $$\hat{\sigma}_\epsilon$$. Is there a relationship between these estimates?