output of lmer function

I have a question regarding understanding of the output of lmer function under lme4 package in R.

fit <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy)

res <- summary(fit)

se <- coefficients(res)[,2]

> se
(Intercept)        Days
6.824556    1.545789


My question is: is the standard error based on Wald statistics or profile likelihood?

The standard errors are based on the asymptotic covariance matrix of the maximum likelihood estimates. In this case, the model is of the form $$y_i = X_i \beta + Z_i b_i + \varepsilon_i,$$

where $y_i$ is the outcome vector for subject $i$ ($i = 1, \ldots, n$), $X_i$ and $Z_i$ are the design matrices for the fixed effects $\beta$ and the random effects $b_i$, respectively, and $\varepsilon_i \sim N(0, \sigma^2 I_{n_i})$ and $b_i \sim N(0, D)$. Then the covariance matrix for $\hat \beta$ is $$\mbox{var}(\hat\beta) = \Bigl ( \sum_{i = 1}^n X_i^\top V_i^{-1} X_i \Bigr )^{-1},$$ where $V_i = Z_i D Z_i^\top + \sigma^2 I_{n_i}$.

• i.e., "Wald statistics" – Ben Bolker Sep 20 '18 at 13:21
• Is it possible to get profile likelihood interval of the estimates, like the nuisance parameter have by this codes: pp <- profile(fit,"theta_")#profiling of the deviance ci = confint(pp) ? – time Sep 20 '18 at 13:58