Sign of coefficient changes when contextual variables are added

Question

I have an interpretation question.

I am running binary multilevel models on whether or not households have bank accounts. Apart from relevant economic, social and demographical household level variables I also include two contextual variables for the primary sampling units (PSU), which are either villages or urban neighborhoods. These two measure the wealth level (assets) and the educational level of the PSU’s. Because a relevant explanatory variable is whether a bank branch is within the vicinity of the household and because households in urban areas should be expected to be closer to the nearest bank branch I expect the dummy variable for URBAN to be positively associated with having a bank account. The proportion of households with bank accounts are higher in urban (53.1) than in rural (30.9) PSU’s and average wealth and educational levels are higher in urban than in rural PSU’s.

However, the dummy for URBAN has a positive sign when the two contextual variables for PSU wealth and education are not included and a negative sign when they are included. How should I interpret this?

KML

Answer 1

This sort of problem is almost exactly what Andrew Gelman found in voting patterns in different states in the USA; his Rich State, poor state, red state, blue state gives a great explanation (and is interesting if you are into politics). He found that wealthier states are more likely to vote Democratic, but wealthier people are more likely to vote Republican. Fascinating.

What's going on in your case?

the dummy for URBAN has a positive sign when the two contextual variables for PSU wealth and education are not included

So, when not controlling for PSU wealth and education, urban people are more likely to have bank accounts.

and a negative sign when they are included. How should I interpret this?

but if you control for wealth and education, urban people are less likely to have bank accounts.

The single best way to see this is probably to plot the predicted probability of having a bank account against wealth and education, with separate lines for urban and rural people.