13. Multilevel Modeling, I

Question 1

What are multilevel models, and why are they used?

Answer 1

Multilevel models are extensions of regression models used when data is structured in groups. These models allow for varying coefficients depending on the group. They are useful when the data has multiple levels, such as individuals nested within cities or repeated measurements over time.

Steps in multilevel modelling:
1. Run a regression where the coefficients (such as slopes and intercepts) vary across groups.
2. Model how those varying coefficients themselves differ across groups (how the relationship between predictors and outcomes differs across different units, such as schools or regions).

Question 2

What is a varying-intercept model?

Answer 2

A varying-intercept model is a regression that includes group indicators, allowing each group to have a different intercept. This means the baseline relationship between the predictor (x) and the outcome (y) can vary across groups.

Yi = aj[i] + Bxi + ei

Question 3

What is a varying-slope model?

Answer 3

A varying-slope model allows the slope of the regression to vary by group. This is done by creating interactions between the continuous predictor (x) and group indicators.

yi = a + Bj[i]Xi + ei

Question 4

What does a model with both varying intercepts and slopes involve?

Answer 4

A model with both varying intercepts and slopes allows both the intercept and the slope to vary by group. Group indicators affect both the baseline level (intercept) and the relationship (slope) between the predictor (x) and the outcome (y).

Yi = aj[i] + Bj[i]xi + ei

Question 5

What does “complete pooling” and “no pooling” mean in the context of multilevel modeling?

Answer 5

Multilevel regrsssion can be thought of as a method for comprimising between two extremes: complete pooling (excluding a categorical predictor from a model) and no pooling (estimating seperate models within each level of the categorical predictor).

Whereas complete pooling ignores variation between units, the no-pooling analysis overstates it.

Multilevel regression can be thought of as a method for
compromising between the two extremes of excluding a categorical predictorcfrom a model (complete pooling), or estimating separate models within eachclevel of the categorical predictor (no pooling).

Whereas complete pooling ignores variation between counties, the no-pooling
analysis overstates it. To put it another way, the no-pooling analysis overfits the
data within each county. To put it another way, looking at all the counties together: the estimates from
the no-pooling model overstate the variation among counties and tend to make
the individual counties look more different than they actually are.