When data has a nested structure, like students within schools, random-effects models let you estimate group-specific coefficients while pooling information across groups. The lme4 package in R fits these models using syntax like (1 | group) for random intercepts and (slope | group) for random slopes.
Chapters
So far we've looked at slopes and intercepts. But what if a variable is nested within another distribution? For example, our school data could be illustrated by this figure. We have students nested within classrooms and classrooms nested within schools. More generically, a nested relationship is hierarchical in nature and it creates a multi level model. Mathematically, this is called a mapping from one distribution to another. Practically, it gives us a way to pool or share information across replicates. Thus, outliers from groups with small sample sizes have less impact if we treat them as a random effect. This equation depicts a simple random effect. The equation on top is the relationship to the data given the I th beta. The random effect assumes that beta is drawn from a normal distribution with the mean mu and and the standard deviation sigma. This is the algebraic representation of the multilevel model. In R. We will use the LME4 package. This package is commonly used and is newer than the LMER or LEMUR package, which was created by some of the same authors. Other packages also exist because the numerical methods are still an open research question. With the LME4 notation, we we need to specify a random effect. To specify a random intercept, we use parentheses with a one pipe random effect group. On US keyboards, the pipe key is above the enter key on the right side of the keyboard. To specify a random slope, we use parentheses slope pipe random effect group. Now that we've seen the theory behind random effects, let's dive into some examples with code.
