1. An Introductory Tutorial
Mixed Linear Models
An Introductory Tutorial

2. Longitudinal Data Mean Structures
What kind of treatment trajectory do your subjects take?
Mean Structures
Linear
Assumes that subjects improve steadily aX+b
Quadratic
Subjects’ follow a part of or a parabola cX^2+bX+a
Cubic
Subjects’ follow a part of or a cubic dX^3+cX^2+bX+a
Log
Decreases/Increases quickly, then slows
1/x
Decreases/increases to a floor/ceiling
Dummy coding
Assumes no particular treatment progression

3. Longitudinal Data
Linear
The Data Assuming linear

4. Longitudinal Data
No Assumptions
The Data Dummy coded

5. How do we include the within subject correlation in our model?
Remember the random intercept model
Linear time trend and random intercept
This assumes equal correlation between all time points.

6. “Repeated” covariance structures
Instead of specifying a random intercept, we can change the structure of the errors to allow for dependence
We still make the normal assumption, but we drop the independence assumption, and specify a dependence structure

7. Compound symmetry The compound symmetric structure The error variance
Time 1
Time 2
Time 3
Time 1
Time 2
Time 3
The error variance

8. Benefits and problems of CS
1. Simple structure only 2 parameters
2. Same as random intercept
-Problems:
1. Designed for clustered data not
longitudinal
2. Assumes same variance at each time

9. Auto Regressive 1 - AR(1) Simplest longitudinal structure
Assumes that consecutive time points all have the same correlation, and that the correlation between non-consecutive time points is a result only of the consecutive correlations
For example: time 1, and time 2 have a correlation of .5, and so do time 2 and time 3. Because time 1 is related to time 2, which in turn is related to time 3, time 1 is related to time 3 with correlation .25

10. Benefits and problems of AR(1)
1. Simple structure, only 2 parameters
2. Takes into account longitudinal sample
-Problems:
1. Equal correlation between all
consecutive time points:
Needs equally spaced time points
2. No direct relationships allowed between
non-consecutive time points
2. Assumes same variance at each time

11. Auto regressive heterogeneous - ARH(1)
Slightly more complex structure
Assumes that consecutive time points all have the same correlation, and that the correlation between non-consecutive time points is a result only of the consecutive correlations
Allows for different variances at each time point

12. Benefits and problems of ARH(1)
1. Moderately complex structure
# of parameters = # of time points +1
2. Takes into account longitudinal sample
3. Unequal variances allowed
-Problems:
1. Equal correlation between all
consecutive time points:
Needs equally spaced time points
2. No direct relationships allowed between
non-consecutive time points

13. Ante Dependence - ANTE(1)
Fairly complex structure
Assumes that the correlation between non-consecutive time points is a result only of the consecutive correlations
Allows for different variances at each time point
Allows for the correlation of time 1 and time 2 to be different from the correlation between time 2 and 3.

14. Benefits and problems of ARH(1)
1. Does not need equally spaced time points
2. Takes into account longitudinal sample
3. Unequal variances allowed
-Problems:
1. Moderately complex structure
# of parameters = 2 * (# of time points) -1
2. No direct relationships allowed between
non-consecutive time points

15. Toeplitz (heterogeneous)- TOEP and TOEPH
Homogeneous Heterogeneous
Assumes “stationarity” : all consecutive time points have the same correlation, and all time points separated by one other time point have the same correlation, … etc
Can allow for different variances at each time point
Allows for direct relationships between non-consecutive time points

16. Benefits and problems of TOEP/TOEPH
2. Direct relationships allowed between
non-consecutive time points
2. Takes into account longitudinal sample
3. Unequal variances can be allowed
-Problems:
1. Moderately/fairly complex structure
TOEP
# of parameters = # of time points TOEPH
# of parameters = 2 * (# of time points) –1
1. Needs equally spaced time points
2. No direct relationships allowed between

17. Most general correlation structure
Unstructured- UN/UNR
Most general correlation structure
Assumes nothing

18. Benefits and problems of UNR
2. No Assumptions made
-Problems:
1. Very complex structure
# of parameters = (# of time points) (# of time points +1)/2
2. Sometimes SPSS will fail to fit the model (convergence
problems)