Hi Atta Muhammad Asif,

I must apologise for not being sure that I understand your question correctly. Are you interested in the interpretation of a level-1 variable being random across a level-2 grouping?

If that is the case, your interpretation is actually quite straightforward. Let's assume we have students (lvl-1) in classes (lvl-2). In our example we are interested in the effect of gender on reading skills.

Now, let's assume we estimate a multilevel model with students nested in classes and the reading skills regressed on gender - we find that girls are better at reading than boys. This is a level-1 effect, but it is easy to imagine that the difference between girls and boys is not the same in all classes. To test this, we set the lvl-1 variable 'gender' random across the lvl-2 grouping factor 'class'. The estimate (expressed as variance or standard deviation) indicates the degree to which the 'reading advantage of girls' varies across different classes.

I hope this is an answer that is useful to you.

Yours,

Rense

One of the reason for using multilevel model is when you are interested in the variance (random-effects model is equivalent to ANOVA). By using the information from levle-2 estiamtes, you can partition total variance into within- and between-cluster variances. You can also calculate ICC using these measures and evaluate how similar each observations are within each cluster. In general, you are not interested in estimating the mean of the random-effects. If you would like to esimate mean values for a specific cluster, consider using cluster as fixed-effects (dummy code cluster ID and include in your regression model) rather than using it as a random-effect (i.e. multi-level model).

You write

“If you would like to estimate mean values for a specific cluster, consider using cluster as fixed-effects (dummy code cluster ID and include in your regression model) rather than using it as a random-effect (i.e. multi-level model)”

There are other views on this! The ANOVA dummy variable approach is equivalent to a ‘select if’ belongs to cluster 1, then cluster 2 and fitting a series of separate models so that the estimation of one cluster takes no account of the information contained in the other clusters, so that it lacks efficiency.

The other approach is to estimate or predict the random effects as coming from a distribution with a common (estimated) variance. The advantage of this second approach is that the cluster estimates are precision weighted that is they are shrunk back towards the general overall relation across all clusters when the number of level 1 units is small. This protects you from over-interpretation. This of course requires making the assumption that the information is exchangeable across the level 2 units, but you have made this assumption by using a random –effects model!

As argued by Robinson, G.K. (1991). "That BLUP is a Good Thing: The Estimation of Random Effects". Statistical Science 6 (1): 15–32.

The following papers (that are all on Research Gate) spell out and illustrate this random- effects perspective: apologies for the shameless self-citation:

K JONES, N BULLEN

CONTEXTUAL MODELS OF URBAN HOUSE PRICES - A COMPARISON OF FIXED-COEFFICIENT AND RANDOM-COEFFICIENT MODELS DEVELOPED BY EXPANSION

Andrew Bell, Kelvyn Jones

Explaining Fixed Effects: Random Effects modelling of Time-Series Cross-Sectional and Panel Data

Kelvyn Jones, Caroline Wright, Andrew Bell

Do multilevel models ever give different results?.