In a random-slope model, the addition of Level-2 (L2) covariates implies a x-level interaction between the random Level-1 variable and the L2 covariate. In Stata (and maybe R, SAS), despite it's implicit nature, the x-level interaction needs to be explicitly entered into the model syntax by creating the interaction term(s).
Is a model considered mis-specified if one does not include these x-level interaction terms? For example, if the fixed effect of a L2 covariate is siginificant (and theoretically important) but it's x-level interaction is not significant, must the x-level interaction remain specified in the model?
No. It is perfectly ok to not have a cross level interaction but still specify a random slope.
Conside the following random intercept - random slope model where x is a level 1 predictor and z is level 2 covariate
y_i = a_ij + b_ij *x_i + e_i
a_j= g01 + g02*z +u_j
b_j= g11 + g12*z +w_j
The g02 coefficient is the coefficient for the direct effect of the model, and g12 is the coefficient for the crosslevel interaction. If you do not include the interaction term then your random slope would be estimated from the error term:
b_j= g11 + w_j
This does not affect in any way the a_j equation which specifies the direct effect of z. You can also potentially remove the random slope (b_j) all together (e.g. if it does not improve the model) and still test for the direct effect of z (as long as you keep the random intercept).
right of course, that makes perfect sense.
I have 3 level-2 covariates of interest which have significant main effects, but only one has a significant x-level interaction effect with the random L1 slope variable. I was worried that I would be violating a specification rule if I omitted the non-significant x-level effects.
Many thanks.
I concur with George
You may be interested in this rather old paper on my RGate which tries to put cross-level interactions into an overall graphical framework
https://www.researchgate.net/publication/222639471_Individuals_and_their_ecologies_analysing_the_geography_of_chronic_illness_within_a_multilevel_modelling_framework?ev=prf_pub
You need quite a lot of data to have the statistical power to detect such effects especially if you have several of them - but it is interesting to read this paper to see how Bayesian MCMC analysis copes with even relatively small data sets - but note in this simulation he has a large sample size at level 1 in each group
How many countries for multilevel modeling? A comparison of Frequentist and Bayesian approaches
http://www.daniel-stegmueller.com/files/Stegmueller_AJPS2013.pdf
Kelvyn
Article Individuals and their ecologies: Analysing the geography of ...
- Badges
- Science method
- Similar topics
- Mathematics
- Statistics
- Statistical Techniques
- Structural Equation Modeling