Ron Heck, Fall 2011
EDEP 768E: Seminar in Multilevel Modeling
1
Nov. 30, 2011
Week 15: Assignment 3 Notes
Growth Model Fitting
I was thinking a bit more about how to decide between models built on the random quadratic
term at level 3 versus a model that you might build on a random linear component at level 3.
First I started by looking at whether we would actually need to include the quadratic component
(linear and quadratic components) or could just say the trajectory was linear (linear component
only).
Model 1 (Random Intercept and Level 2 and Level 3)
Linear component only
(EDEP 768) Week 15: Assignment 3 Notes (Growth Model Fitting)
6 parameters tested
-2LL = 53507.966
Model 2 (Random Intercept and Level 2 and Level 3)
Linear and Quadratic components included
2
(EDEP 768) Week 15: Assignment 3 Notes (Growth Model Fitting)
3
-2 LL = 51,843.804
7 parameters tested
We can see that the model with both linear and quadratic components fits better. If we form a
delta chi-square we have
[-2 LL (M1)] - -2LL(M2)]
[53507.966 - 51,843.804 = 1664.162 for 1 df (3.84 chi sq. required)
So it is obvious that the second model with both fits the data better.
Next I fit Model 3. This model and has a random lincon component at school level (8 parameters
estimated)
Model 4 has a random quadratic component at level 3 (8 parameters)
(EDEP 768) Week 15: Assignment 3 Notes (Growth Model Fitting)
4
We would probably favor Model 4 with the quadratic component random at level 3 based on the
fit indices. For the same number of free parameters (8) the fit is better ( Model 3 = AIC =
51,803.608 and Model 4 = 51,788.224). We favor smaller values of AIC.
Finally, I compare Model 5, which is the final model we ran last night with quadcon (16
parameters estimated)
Model 6 is final model we ran last night but with lincon as random lat level 3 (16 parameters)
(EDEP 768) Week 15: Assignment 3 Notes (Growth Model Fitting)
5
We would still probably favor Model 5 with the random quadratic component at level 3 over
Model 6 with the random linear component, again based on the fit indices (AIC and BIC are
smaller).
So this would give you one way to justify that sequence of models that we built using some
model-fitting criteria that are available.
I think we would have to conclude that the trajectories are quadratic (containing a curvilinear
shape suggesting that growth slows). We would also suggest probably building the model on the
random quadratic component as the better of the two strategies in this case. Hope this helps.