(EDEP 768) Week 5: Notes 1

Week 5: Notes

Let’s review for a few moments.

What are some of the advantages of multilevel models?

  • Study relationships between individuals and groups (e.g., interactions)
  • Takes advantage of natural nesting of individual within higher order social groupings
  • Avoid the aggregation vs. disaggregation problem
  • Adjust estimates at the group level for similarities among individuals within groups (i.e., grand-mean centering)
  • More accurate hypothesis tests because standard errors are adjusted for similarities of individuals nested in groups
  • Define variables at their correct levels (e.g., organizational resources as a function of the number of organizations in the study, as opposed to the number of individuals)
  • Examine a wider range of outcomes (intercepts, slopes)
  • Can focus hypothesis tests on variance in effects at the group level, as well as their level or the size of the effect.

Today we first want to look a little more closely at the meaning of “slopes as outcomes.”

  • Models with intercepts as the outcome (βo) answer questions such as:
    Why are math outcome higher in some schools than in other schools?
  • Slope models answer questions such as:
    Why do some schools have greater SES effects on math achievement than other schools?

We printed out estimates of the relationship between student SES and their math achievement for each school. We can see them in the following 10 schools.
Notably, we can see that some are positive. This indicates schools where low SES students actually score a bit higher in math than their higher SES peers. Six of the 10, however, are negative, with estimates ranging from about -7 to -14. / / -6.842
1.936
3.032
4.440
-11.621
-12.028
3.647
0.280
-14.554
-7.812

In the Week 4 Follow Up Handout, I made the point that the pattern of results from the multiple regression analysis (which is only at the school level) is very similar to the MIXED (two-level) analysis. The estimates are not the same size but we can see which ones are significant and which ones are not in both models (school composition and school quality composite). In the first output the intercept is the SES effect when the predictors were zero.

We can see below that the mean slope effect is 0, but that there is considerable variability (i.e., the lowest gap is -21.14 points and the largest advantage is 16.69 points).

I then grand-mean centered the regression model so the intercept would be the SES-achievement value would be 0 (i.e., when all were at their grand means also). You can see the coefficients are all the same but the intercept is now basically 0 as it is in the descriptive table above. .

Comments

Of course, the regression model estimates are not “optimal” because they are not conducted in a multilevel modeling context.

Another difference you will notice is that in the two-level analysis the school variables that explain variation in the SES-math achievement slope are shown as cross-level interactions. They are actually formed as interactions in the model (Zenroll_y*lowSES).

A couple of notes about estimation.

  1. Power of tests for individual regression depends on total sample size. The power of tests of higher level effects depends on number of groups more than total sample size.
  2. Intercepts are usually more reliably estimated since they do not depend on other variables that people are measured on within groups. Slopes are typically not as reliably estimated because they depend on other variables.