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Take the "Orthodont" dataset from "nlme" for an example. Let's say we want to fit a linear mixed effect model "distance ~ age + random intercept + random slope for age". It seems to me that the following two specifications are equivalent. However, they produce different SE and variance components estimates.

library(nlme) data("Orthodont") m.reml1 <- lmer(distance ~ age + (age | Subject), data = Orthodont, REML = T) m.reml2 <- lmer(distance ~ age + (1 | Subject) + (0 + age | Subject), data = Orthodont, REML = T) summary(m.reml1) summary(m.reml2)

From m.reml1, we get SE 0.77525 and 0.07125 for (Intercept) and age.

From m.reml2, we get SE 0.71380 and 0.06561 for (Intercept) and age.

The variance components are also different.

So why the difference? Are they really specifying different models?

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The first model assumes that the random intercepts and random slopes for age are correlated, whereas the second one assumes that they are independent.

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