An Introductory Tutorial Mixed Linear Models An Introductory Tutorial

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

Longitudinal Data Linear The Data Assuming linear

Longitudinal Data No Assumptions The Data Dummy coded

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.

“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

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

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

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

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

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

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

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.

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

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

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

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

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)