Multiple Random coefficient modell
Message/Author
 Maren Herrmann posted on Wednesday, January 01, 2020 - 12:24 pm
Dear Bengt and Linda,

I wish to run this 3-multiple random coefficient modell.

Level-1
Y = P0 + P1*(v1) + P2*(v2) + P3*(v1*v2) + E
Level-2
P0 = B00 + R0
P1 = B10 + R1
P2 = B20 + R2
P3 = B30 + R3
Level-3
B00 = G000 + U00
B10 = G100 + U10
B20 = G200 + U20
B30 = G300 + U30
-
ANALYSIS: TYPE = THREELEVEL RANDOM;
MODEL:
%WITHIN%
P1 | Y on v1; ! Rand slpe L2
P2 | Y on v2; ! Rand slpe L2
P3 | Y on v1xv2; ! Random slope for interaction L2
%BETWEEN team%; ! Level 2
Y with P1 P2 P3; !corr betw random slope & random intecept is allowed on L2.
%BETWEEN WS%; ! Level 3
B10 | Y on v1; ! Rand slpe L3
B20 | Y on v2; ! Rand slpe L3
B30 | Y on v1xv2; ! Rand slpe for interaction L3
Y with B10 B 20 B30; !corr between random slope & random intecept is allowed on L3
OUTPUT: sampstat tech1 tech2;

Is it correct?
Are also intercepts random on level 2 and level 3?

Sincererly
 Bengt O. Muthen posted on Thursday, January 02, 2020 - 12:19 pm
Have a look at UG ex 9.20 and in particular how Y appears on all 3 levels. This shows that you don't specify random slopes on your highest level ("WS"). Instead, P1-P3 also vary on this highest level - that variation corresponds to your B10-B30.
 Maren Herrmann posted on Friday, January 03, 2020 - 1:53 pm
Thank you very much!

Is this revision correct?

Variable: Names = WS team y v1 v2;
cluster = WS team; ! Level 3: WS; Level 2: team
WITHIN = v1 v2
DEFINE: v1xv2 = v1*v2;
ANALYSIS: TYPE = THREELEVEL RANDOM
MODEL:
%WITHIN%
P1 | y on v1; ! Rand slpe L2
P2 | y on v2; ! Rand slpe L2
P3 | y on v1xv2; ! Rand slpe L2

%BETWEEN team% ! Level 2
P1; ! Rand slpe L3
P2; ! Rand slpe L3
P3; ! Rand slpe L3
y with P1 P2 P3; ! covariance betw. random intercept and random slope L2

%BETWEEN WS% ! Level 3
y with P1 P2 P3; ! covariance betw. random intercept and random slope L3

OUTPUT: sampstat tech1 tech2;

Sincerely
 Maren Herrmann posted on Friday, January 03, 2020 - 4:31 pm
I think a saw a mistake. Here is a another revision, is it correct?

Variable: Names = WS team y v1 v2;
cluster = WS team; ! Level 3: WS; Level 2: team
WITHIN = v1 v2
DEFINE: v1xv2 = v1*v2;
ANALYSIS: TYPE = THREELEVEL RANDOM
MODEL:
%WITHIN%
P1 | y on v1; ! Rand slpe
P2 | y on v2; ! Rand slpe
P3 | y on v1xv2; ! Rand slpe

%BETWEEN team% ! Level 2
P1; ! Rand slpe L2
P2; ! Rand slpe L2
P3; ! Rand slpe L2
y with P1 P2 P3; ! covariance betw. random intercept and random slope L2

%BETWEEN WS% ! Level 3
P1; ! Rand slpe L3
P2; ! Rand slpe L3
P3; ! Rand slpe L3
y with P1 P2 P3; ! covariance betw. random intercept and random slope L3

OUTPUT: sampstat tech1 tech2;

Sincerely
 Bengt O. Muthen posted on Friday, January 03, 2020 - 5:27 pm
Looks about right except you need 2 cluster variables. Run it and see what you get.
 Maren Herrmann posted on Monday, January 06, 2020 - 7:11 am
Thank you. I also ran the model.

But I still have a question: Where are 2 cluster variables needed/is this command not enough (see also above)?:

cluster = WS team; ! Level 3: WS; Level 2: team
 Bengt O. Muthen posted on Monday, January 06, 2020 - 4:01 pm
You are right - that is enough.
 Maren Herrmann posted on Wednesday, January 08, 2020 - 4:57 am
Thank you!
 Maren Herrmann posted on Sunday, January 19, 2020 - 6:17 pm
I still have a question about centering:
According to Ryu (2015), the centering depends on the Level. „...when the interaction involves two Level 1 variables …grand mean centering is recommended to obtain unbiased estimates for Level 2 model. Cluster mean centering is recommended to obtain unbiased estimates for the Level 1 model.“

a) Can you do different centerings with Mplus (within the same mplus-input) using the command Define: center ?
b) Do you know (or do you know a reference) how to center for the level 3 model (grand mean or cluster mean?) if level 1 predictors and their interaction is involved?

Ehri Ryu (2015) The Role of Centering for Interaction of Level 1 Variables
in Multilevel Structural Equation Models, Structural Equation Modeling: A Multidisciplinary
Journal, 22:4, 617-630, DOI: 10.1080/10705511.2014.936491
Sincerely and thank you
 Maren Herrmann posted on Monday, January 20, 2020 - 2:21 am
I would revise my question b)
Of course for the model on level 3 (highest Level) you can only use grand mean centering.
If interactions involves two Level-1-variables (see my model above), Ryu´s (2015) recommendation (for 2-level-models) on how to center can therefore be transferred to 3-level-models, right?
 Bengt O. Muthen posted on Monday, January 20, 2020 - 9:32 am
a) Yes. You can use Define in the same run to grand-mean center some variables and group-mean center others.

Revised b) That's right.
 Maren Herrmann posted on Monday, January 20, 2020 - 5:56 pm
Thank you!

My three-level-model includes two Level-1-variables (v1, v2) an their interaction. I have another questions. Could you please help me with this or give a reference?

a) Interactions (between Level-1-variables) can also have contextual effects, right?

b) Is it possible to test contextual effects for all three variables v1, v2, and their interaction v1Xv2 in one an the same model? So you can test contextual effects between L1 and L2, L2 and L3, and L1 and L3?

c) Would you consider this approach to be correct?
„» Level 1-model: y as function of grand-mean-centered (deviation of level 1 score from the grand mean) v1, v2, and the product of these centered variables.
„» Level 2-model: intercept as function of group -mean-centered (deviation of level-2 team mean from the level-3 workshop mean) v1, v2, and the product of these centered variables.
„» Level 3-model: intercept as function of grand-mean-centered (deviation of level 3-workshop mean from the grand mean) v1, v2 and the product of these centered variables.

Sincerely
 Bengt O. Muthen posted on Wednesday, January 22, 2020 - 1:16 pm
a) Have a look at

Ryu, E. (2015). The role of centering for interaction of level 1 variables in multilevel structural equation models. Structural Equation Modeling: A Multidisciplinary Journal, 22:4, 617-630, DOI: 10.1080/10705511.2014.936491
view abstract contact first author

b) Yes.

c) Check the article.