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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?

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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 $V_i = Z_i D Z_i^\top + \sigma^2 I_{n_i}$.

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  • 2
    $\begingroup$ i.e., "Wald statistics" $\endgroup$ – Ben Bolker Sep 20 '18 at 13:21
  • $\begingroup$ 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) ? $\endgroup$ – time Sep 20 '18 at 13:58

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