A common question I get from other faculty members is when they are conducting a longitudinal analysis, i.e., pre-test and post-test, they wonder if where the participants' score at pre-test, i.e., initial level, is associated with their score at post-test, i.e., ending level. in other words, where the participants ended up is largely due to where they started. As an example, I am conducting a study on mindfulness training where we measure their mindfulness ability before and after the training. Although we found a positive effect from pre-test to post-test, observers were curious if those who already had high mindfulness abilities at pre-test would have high mindfulness abilities at post-test. Therefore the training would not change their mindfulness levels that much because the participants already had high mindfulness abilities when they entered the treatment. 

to answer this question, I thought about using correlations between the pre-test and post-test scores. the higher the score on the pre-test, the higher the score on the post-test. and there is a moderate correlation value. no surprise there.

I guess my question though was more about (a) if you had a high pre-test score, is your growth in mindfulness from pre-test to post-test flat, i.e., not much change because of a ceiling effect, or (b) if you had a low pre-test score, is your growth in mindfulness from pre-test to post-test a linear progression because when you start low, there is no where to go but up.

I thought about using HLM to answer this question. Could I use variance in intercepts, e.g., at pre-test, and use the intercepts to predict mindfulness slope? If so, how would I set up the Level 1 and level 2 models for a growth curve?

I tried setting up this model (four waves (pre, post, follow-up 1, follow-up 2). I added a quadratic effect):

Level-1 Model

          MINDFULN = P0 + P1*(WAVE) + P2*(WAVESQ) + e

 Level-2 Model

         P0 = B00 + r0

         P1 = B10 + r1

         P2 = B20 + r2

Mixed Model

      MINDFULN = B00

      + B10*WAVE

      + B20*WAVESQ  + r0 + r1*WAVE  + r2*WAVESQ + e

I got the tau as correlation matrix:

tau (as correlations)

INTRCPT1,P0  1.000 -0.707  0.680

       WAVE,P1 -0.707  1.000 -0.997

     WAVESQ,P2  0.680 -0.997  1.000

I am getting the impression that my model will not allow me to examine this hypothesis. Is there a way I could set up the model to test the hypothesis that those participants with an already high mindfulness ability at pre-test will likely not experience much increase in their mindfulness ability over time, but those participants with a low mindfulness ability at pre-test will likely experience more increases in their mindfulness ability over time?