Confidence interval of parameters of mixed-model in R

[Cynderella]

Hi , can anyone please explain me the following output produces in `R` when I ran the command `confint(lmer())` .

Here is the R code :

    library(lme4)
    fm1 <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy)
    confint(fm1)

And the output is :

    Computing profile confidence intervals ...
                     2.5 %     97.5 %
    .sig01       14.3815760  37.715996
    .sig02       -0.4815007   0.684986
    .sig03        3.8011641   8.753383
    .sigma       22.8982669  28.857997
    (Intercept) 237.6806955 265.129515
    Days          7.3586533  13.575919

I have understood the last two row which are confidence interval for Intercept and Days , respectively . But what are those `sig01` , `sig02` , `sig03` , `sigma` ?

If I write down the model `Reaction ~ Days + (Days | Subject)` :

\(\text{Reaction}_{ij}=\beta_{0j}+\beta_{1j}\text{Days}_{ij}+e_{ij}\)
\(\beta_{0j}=\gamma_{00}+u_{0j}\)
\(\beta_{1j}=\gamma_{10}+u_{1j}\)

Combining the last two equations into first one , that is , by substituting the level-2 equation to level-1 equation, we have ;

\(\text{Reaction}_{ij}=\gamma_{00}+\gamma_{10}\text{Days}_{ij}+u_{0j}+u_{1j}\text{Days}_{ij}+e_{ij}\)

* Is `sig01` variance of \(u_{0j}\) ?

* Is `sig02` variance of \(u_{1j}\) ?

* Is `sig03` variance of \(e_{ij}\) ?

* What is `sigma` in the output of `confint` ?

Many Thanks! Regards .

 

[TheEcologist]

Yes, it's not very intuitively named. But If you conduct

confint.merMod(fm1,oldNames=FALSE)

You'll see that the output makes more sense, as the first three parameter simply refer to estimates of the variances for each random effect and their correlation/covariance. The forth gives confidence intervals for the residual variance.

                                   2.5 %     97.5 %
sd_(Intercept)|Subject        14.3814761  37.715996
cor_Days.(Intercept)|Subject  -0.4815007   0.684986
sd_Days|Subject                3.8011641   8.753383
sigma                         22.8982669  28.857997
(Intercept)                  237.6806955 265.129515
Days                           7.3586533  13.575919

Good luck,

 

[Cynderella]

I have not understood the second one . What is

                                   2.5 %     97.5 %
cor_Days.(Intercept)|Subject  -0.4815007   0.684986

?

What does "cor_Days.(Intercept)|Subject" mean ?

Thanks again . Regards .

 

[TheEcologist]

the variances for each random effect and their correlation/covariance.

It's the correlation between random effects.

 

[Cynderella]

the first three parameter simply refer to estimates of the variances for each random effect and their correlation/covariance. The forth gives confidence intervals for the residual variance.

.

Are the first four gives confidence interval of **Variance** or confidence interval of **Standard Deviation** ?
From your Quote , it seems confidence interval of **Variance** . But if I tried empirically it matches with confidence interval of **Standard Deviation** .

library(lme4)
model <- lmer(Reaction ~ Days + (1|Subject), sleepstudy)
confint.merMod(model,oldName=FALSE)

s2 <- VarCorr(model)$Subject[1]
n <- nrow(ranef(model)$Subject)
CI <- (n - 1) * s2 / qchisq(c(0.975, 0.025), df = n - 1)
sqrt(CI) #matches with first row of confint

Would you please tell me am I doing any mistake ?

Thanks & Regards .