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I am confused about a result in my OLS regression.

I am regressing health on both crime level and ubanization and a couple of commonly encountered covariates in the literature such as, for example, age, gender, education, cultural background, income etc.

Health is a variable at an individual level. Crime level is a variable at a regional (German state) level aggregated. Urbanization is a variable at an individual level. I have around 50,000 observations.

I first find that crime is insignificant in my OLS regression. However, if I take urbanization out of the specification, crime becomes suddenly significant.

Therefore I believe that urbanization is an apparent confounder that is correlated with crime. However, I have difficulties to interpret this concisely, or in other words, figure out the causality.

I can hardly imagine that crime 'predicts' urbanization, which in turn 'predicts' health. I would have expected the opposite to happen, perhaps that urbanization at the individual level becomes insignificant when adding crime to the specification.

Is there a rationale for crime to become insignificant when urbanization was added, or in other words, why did crime become insignificant and not urbanization? With my limited understanding I can't make sense of it. What is the rule for which variable to become insignificant when adding a control variable that confounds the relationship?

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A very important thing to realize here is that, as you describe it, "urbanization" is at the individual level while "crime" is at the state level. To be honest, I'm not 100% clear on what you mean by that, but it does certainly sound that "urbanization" is a more informative measure for an individual.

Since "urbanization" is probably highly correlated with "crime", but still more individually related to the outcome of each measure, it is very natural that the regression model would determine that "crime" has very little effect given "urbanization".

Think of it this way: given you know an individual's "urbanization" score (which is at the individual level), knowing their "crime" score (which is only at the state level) will not add much useful information about predicting that individual's health.

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  • $\begingroup$ Thanks, in fact, that's what I am thinking too. Unless I can verify this mathematically, it will remain speculation though. In other words, that this model has a cluster hierarchy, could reverse the causality. $\endgroup$
    – CodeTrek
    May 25 '15 at 1:11
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The first statistical reason that I can think about is (multi) collinearity, which means that several of your predictors are correlated among themselves. What you may inspect is if there is a significant correlation between the two indicated predictors, i.e., urbanization and crime. It may be that urbanization cancels out the effect for crime.

Theorizing on the relation between crime and health, I would say that this relation is not entirely unexpected. The crime rate in a region may be related with the economic wealth of that region, and this may in turn be related to health. There may also be other "untested" factors that mediate such a relation. Please note that we may theorize about this but with correlation/regression analysis we do not have evidence for a causal effect.

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  • $\begingroup$ I have used income/education as well as control variable. But it makes sense. It is just striking, so you say that there is possibly another confounder hidden in that? I problem, essentially is, that of course they are correlated, but why is urbanization cancelling out crime and not vice versa? $\endgroup$
    – CodeTrek
    May 24 '15 at 21:02
  • $\begingroup$ indeed, I forgot to tell that in addition to correlation measures among the predictors the "variance inflation factor" is an interesting indicator if there is substantial collinearity. $\endgroup$ May 24 '15 at 21:07

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