Solved – Difference between using a dumthe variable in a multiple regression and clustering on that dumthe

In a multiple regression, the coefficients become insignificant using vce(cluster year) in Stata. Will including year dummies solve the problem? How are these two approaches different?

"Insignificant" coefficients are not a "problem" unless your method is wrong (but significant effects would be a problem in that case too, however nary a statistician comes asking for feedback here on a method when they "achieve statistical significance"). I can propose a 1,000 methods, one of which is guaranteed to give you $p < 0.05$ if that's what your study needs. Alternately, I can just randomly generate p-values for you and you can keep rolling through until you find $p < 0.05$.

I think that what you're asking is about the difference between using random effects and fixed effects in a mixed model. VCE takes variable arguments that indicate clusters, whether they're "teeth within a mouth" or "residents in a community". One's decision about how to handle correlation in the data using either approach should be driven by the scientific question. Not the significance of the estimated effects.

Let's contrast the mixed modeling approaches to handling cluster variables:

Fixed effects: explicitly estimate an intercept indicating a difference in intercepts for each cluster. The working probability model does indicate that the slopes in each cluster are the same, but this can be avoided by fitting interactions. This approach is only sane when the number of clusters is small and there are many observations within a cluster.

Random effects: the clustering in Stata corresponds to a random intercept model, or repeated measures ANOVA (for linear models using the identity link). This approach imposes the same assumptions of the working probability model for the association between the exposure and outcome of interest within each cluster (homogeneous effects), but random intercepts is the analogous approach to the fixed effects setting to handle heterogeneous effects. This approach is only sane when the number of clusters is large and there are relatively few observations per cluster. The underlying assumption is that the distribution of random effects is normal.

Fixed effects have the advantage over random effects because they handle small numbers of clusters very well and can estimate effects from any type of assumed probability distribution. Random effects on the other hand make a lot of sense when data are highly correlated and clusters are too large in number to estimate their effects exactly.

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