# Using grouped variables in regression

In a dataset, income is split up in different groups, e.g. person 1 is in income group 5, person 2 in income group 11 etc. The groups are of unequal size (i.e. group 1: 0 < x < 2500, group 11: 7000 < x < 100000).

I would like to model the edcuation level of a kindergarten child (assuming it depends on parental income). Do I have to use dummy variables for each category or can I, under the assumption of uniform distribution, use midpoints of each group and then use actual income values?

The usual approaches I tend to see are either

(i) to ignore the ordering and treat as nominal categories (thus throwing away a lot of potential information; or

(ii) use scores of some kind, often in a fairly arbitrary fashion (your midpoints would count), in effect imposing a lot of information you don't really have.

Neither is necessarily bad, they both make compromises.

The R package treats ordered factors differently from nominal factors, by fitting orthogonal polynomials to the numbered levels (i.e. it treats their value as 0,1,2 ..), which potentially overcomes the scoring issue; the first few such terms would tend to account for smooth functions of the level, but all possible terms would correspond to the fit of a nominal category (if set up in a somewhat more interpretable way).

Another alternative is to fit some kind of smooth monotonic function (perhaps a monotonic spline, say) to whatever scores you have.

• If I decide to use a dummy variable for each group but the base group (effictively g-1 dummies) and also include a dummy that takes value 1 if the family income lies below the poverty threshold - will this lead to near-multicollinearity? [Correlation between inc1+inc2+inc3 and poor = 0.79, correlation between inc8+inc9+inc10 = -0.54] Commented May 8, 2014 at 15:42

Since the data are structured as levels of a factor, you should use dummy variables. Using midpoints is problematic in that it obscures the structure of the data, since income is not measured as a continuous variable and individuals do not have the midpoint values imputed. Additionally, at the top category there is the problem of what income should be imputed since there is putatively no upper limit.

An unwritten principle of statistics is never to through away information unless absolutely necessary. In this case, by imputing midpoint values, you're throwing away information on the categories of income in the data.

One other option -- implied by the idea of "grouped data" -- is to use a multilevel model (sometimes called a "mixed model" in the social sciences") to fit a regression model in which individuals are nested within income groups. You can then allow the intercepts and slopes to vary across groups. This approach is the best option if you have a relatively large number of income groups (e.g., above 10, although this is only a rough benchmark).