interpretation of fitting a constant shift model, where the regression relationships for group members and nonmembers are identical, other than being shifted up or down; that is,
for nonmembers and
for members. The
Would two different regression relationships be better still? Say there is only one numerical predictor
for nonmembers (
for members (
corresponds to the two different regression lines
for nonmembers (since
for members (since
The
on
These constructions can be easily generalized to multiple predictors, with different variations of models obtainable. For example, a regression model with unequal slopes for some predictors and equal slopes for others is fit by including products of the indicator and the predictor for the ones with different slopes and not including them for the predictors with equal slopes. Appropriate
A reasonable question to ask at this point is “Why bother to fit the full model? Isn't it just the same as fitting two separate regressions on the two groups?” The answer is no. The full model fit above assumes that the variance of the errors is the same (the constant variance assumption), while fitting two separate regressions allows the variances to be different. The fitted slope coefficients from the full model will, however, be identical to those from two separate fits. What is gained by analyzing the data this way is the comparison of versions of pooled, constant shift, and full models based on group membership, including different slopes for some variables and equal slopes for others, something that is not possible if separate regressions are fit to the two groups.
Another way of saying that the relationship between a predictor and the target is different for members of the two different groups is that there is an interaction effect between the predictor and group membership on the target. Social scientists would say that the grouping has a moderating effect on the relationship between the predictor and the target. The fact that in the case of a grouping variable, the interaction can be fit by multiplying the two variables together has led to a practice that is common in some fields: to try to represent any interaction between variables (that is, any situation where the relationship between a predictor and the target is different for different values of another predictor) by multiplying them together. Unfortunately, this is not a very reasonable way to think about interactions for numerical predictors, since there are many ways that the effect of one variable on the target can differ depending on the value of another that have nothing to do with product functions. See Section 15.6 for further discussion.
2.4.1 EXAMPLE — ELECTRONIC VOTING AND THE 2004 PRESIDENTIAL ELECTION
The 2000 US presidential election matching Republican George W. Bush against Democrat Al Gore attracted worldwide attention because of its close and controversial results, particularly in the state of Florida. The 2004 election, pitting the incumbent Bush against John Kerry, is less discussed, but was also controversial, in part because of the introduction of electronic voting machines in some polling places across the country (such machines were introduced in part because of the irregularities in paper balloting that occurred in Florida in the 2000 election). Some of the manufacturers of electronic voting machines were strong supporters of President Bush, and this, along with the fact that the machines did not produce a paper trail, led to speculation about whether the machines could be manipulated to favor one candidate over the other.
