The Random Intercept Model

By the end of this module, you will be able to

- Explain the components of a random intercept model
- Interpret intraclass correlations
- Use the design effect to decide whether MLM is needed
- Explain why ignoring clustering (e.g., regression) leads to inflated chances of Type I errors
- Describe how MLM pools information to obtain more stable inferences of groups

- Review the resources (lecture videos and slides)
- Complete the assigned readings
- Snijders & Bosker ch 3.1–3.4, 4.1–4.5, 4.8

- Attend the Thursday session and participate in the class exercise
- Complete Homework 3

You can view and download the slides here: PDF

Check your learning: Here’s a snapshot of the `sleepstudy`

data:

```
Reaction Days Subject
1 249.6 0 308
2 258.7 1 308
3 250.8 2 308
11 222.7 0 309
12 205.3 1 309
13 203.0 2 309
21 199.1 0 310
22 194.3 1 310
23 234.3 2 310
```

where `Subject`

is the cluster ID. Is `Days`

a
level-1 or a level-2 variable?

Check your learning: \(u_{0j}\) is the new term in a multilevel model (compared to regression). Is it a level-1 or a level-2 deviation variable?

Check your learning: For the diagram in the video, which one is an actual variable in the data?

Check your learning: For the unconditional model, which of the following is a fixed effect?

Note: On the slide around the 9 minute mark, the numbers labeled the “Std.Dev.” is just the square root of the variance components. That is, the standard deviation of the school means and the within-school standard deviation.

Check your learning: For a study, if \(\tau^2_0 = 5\), \(\sigma^2 = 10\), what is the ICC?

Check your learning: The graph below shows the distribution of the
`Reaction`

variable in the `sleepstudy`

data. What
do you think is a good guess for the its ICC?

Note: OLS = ordinary least squares, the estimation method commonly used in regular regression.

Thinking exercise: When \(\sigma^2 / n_j = 0\), \(\lambda_j = 1\), and the empirical Bayes estimate will be the same as the sample school mean, meaning that there is no borrowing of information. Why is there no need to borrow information in this situation?

Note that the `ses`

was standardized in the data set,
meaning that `ses`

= 0 is at the sample mean, and
`ses`

= 1 means one standard deviation above the mean.

Check your learning: In regression, the independent observation assumption means that

Practice yourself: Compute the design effect for `mathach`

for the HSB data. Which of the following is the closest to your
computation?

Bonus Challenge: What is the design effect for a longitudinal study of 5 waves with 30 individuals, and with an ICC for the outcome of 0.5?

While disaggregation yields results with standard errors being too small, aggregation generally results in standard errors that are slightly larger. The main problem of aggregation, however, is that it removes all the information in the lower level, so level-1 predictors cannot be studied. MLM avoids problems of both disaggregation and aggregation.

This part is optional but gives a mathematical explanation of why OLS underestimates the standard error

Check your learning: In the level-2 equation with
`meanses`

as the predictor, what is the outcome variable?

Check your learning: How do you interpret the coefficient for
`meanses`

?

Note: If the 95% CI exlcudes zero, there is evidence that the predictor has a nonzero relation with the outcome.

Check your learning: By default, what type of confidence interval is
computed by the `lme4`

package?