Dear Joop and others,

Precisely this point (test a hypothesized interaction when this follows from theory, also if the corresponding random slope is non-significant) is stressed on p. 106 of Snijders & Bosker (2nd edition 2012). By the way, the same reasoning implies that a hypothesized main effect of a level-2 variable can and should be tested also if there is no significant intercept variance.

A somewhat related point is mentioned on p. 104 of the same amazing book:
(The following quote reads a bit funny, but it is preceded on p. 102 by the introductory sentence, "Model specification is a process guided by the following principles", of which the 8th includes, e.g., "We are constantly making type I and type II errors" which I cannot repeat often enough; ... and here comes the last-mentioned principle:)

"9. Providing tested fixed effects with an appropriate error term in the model (whether or not it is significant). For level-two variables, this is the random intercept term (the residual term in (5.7)). For cross-level interactions, it is the random slope of the level-one variable involved in the interaction (the residual term in (5.8)). For level-one variables, it is the regular level-one residual that one would not dream of omitting. This guideline is supported by Berkhof and Kampen (2004)."
Berkhof, J., and Kampen, J.K. (2004) ‘Asymptotic effect of misspecification in the random part of the multilevel model’. Journal of Educational and Behavioral Statistics, 29, 201–218.

Best wishes,
Tom Snijders

I'm working on a study that seeks to compare the strength of all the the extant theories of story content and news play, so the relative strength of the explanatory variables is crucial.

Question:

For a three-level model, has anyone developed a method to calculate the proportion of explained variance combining all levels simultaneously, perhaps by pooling and discounting the explained variance among the levels?

The question may seem naive or foolish to a statistician.  I know the conventional answer in the field is, "No, it can't be done." Yet as a media sociologist, the question is front and center.  I cannot avoid trying to find an answer regardless of the state of knowledge within the discipline.  To a non-specialist like myself, the difficulty seems to revolve around finding a valid method to assign or incorporate the explained variance of cross-level interactive terms in the regression coefficients for the Level-1 variables.


Briefly, here’s my data set. I am working on a content analyze of news stories from a nationally representative sample.The nested data set has 6,090 news stories, published by 114 newspapers, which are in turn owned by 59 individuals or chains. At Level-1, news stories were coded by graduate students and given dichotomous scores for characteristics that were either present (1) or absent (0).I created additive indexes for each Level-1 variable.At Level-2, I used continuous scale measures of circulation and demographic variables. At Level-3, I had dichotomous ownership charactertics. The dependent variable was a continueous scale variable based on six components of Story Prominence.All the variables were standardized.

Again, please excuse me for not being willing to accept what everyone else seems to have concluded.