Collapsing (combining) two variables into one for analysis

Question

I came across a topic I am somewhat confused about: the merging of two variables.

Assume we have two measurements from the same subjects. The two variables ($x_1$ and $x_2$) are measuring something similar, but not exactly the same thing. The variables (or combined variable, called $x_{12}$) will later be used as an explanatory variable ($X$) of some other variable ($Y$).

For example, let's say we want to estimate the IQ of someone, and we only have the IQ of his father and mother (and we don't know the child's gender).

What statistical (and non-statistical) issues are relevant for deciding on whether or not to combine the two measurements into one?

Some issues to consider:

1. Let us say that we would later fit a linear regression of the type $Y$~$X$ (where $X$ is either $x_1$ and $x_2$ or the combination of the two), is there a time we would rather merge the two variables ($x_1$, $x_2$) into one?
2. How does the association of the two variables ($x_1$ and $x_2$) relevant to the decision on whether or not to merge them?
3. Is there any relation between $x_1$/$x_2$ and $Y$ that might influence the merging decision?
4. What if $x_1$ and $x_2$ are ordinal variables, or forced integer variables, does that make a difference on the value of collapsing them in order to be merged?
5. Is there any other issue to consider on this topic that I didn't mention?

Answer 1

Building somewhat on previous answers:

1. Yes, there would. In regression analysis, you suppose that your regressors $X$ are orthogonal to each other. If you consider $IQ$, income, number of years of schooling, race of the parents, you will end up with regressors that are strongly correlated. It will cause your estimates for both $x_1$ and $x_2$ to be imprecisely estimated (large standard errors.) and you may wrongly conclude that they are not significant. For a discussion on the effect of correlated regressors, see this link. Basically, in a choir, it is hard to know who sings what.

2. Possible combinations: sum, average, first principal component of the two (or more) series. You would have to justify your choice here. In the case of years of schooling, you could add both numbers and declare it "parents' education."

3. What exactly do you mean ?

4. No, you can do it with any type of variables. See the example with years of schooling.

5. Be advised that any aggregation of data leads to a loss of information. It is your job as a researcher to weigh the pros and cons of such choice.

Answer 2

1. Yes, see (2)
2. If the predictor variables affect your dependent variable in the same way, and you have a theoretical basis for collapsing the variables, it makes sense to merge them.

An example of an association that would make you want to collapse variables:

If you examine factors predicting individual income, parent's education stands out. Imagine you know the highest level of education of both the father and mother of the individual. It turns out that the key factor is highest level of income of either of the parents. If you do a regression analysis, you find that it doesn't matter which parent had a higher level of education, or whether both had the same level of education. Instead, what predicts almost all of the individual's income out of those two variables is simply the highest level of education between the two parents. So in this case, you would want to combine the two variables and call the variable "highest level of parental education."

Answer 3

2. How does the association of the two variables (x1 and x2) relevant to the decision on whether or not to merge them?

If variables x1 and x2 represent two different methods to measure the same element, combining the two can provide a more accurate description of the element you wish to measure. E.g. if you need a variable that indicates body size, one can combine variables x1 = height and x2 = weight, either by summing x1 + x2 or by peforming a principal compenents analysis and choosing only the first component.

If variables x1 and x2 are repeated measures of the same variable, creating a variable which is the average of the two may give a more accurate measure of the variable in question.

5. Is there any other issue to consider on this topic that I didn't mention?

If the variables are repeated measures, or the variables measure the same element but from two different subjects (mother and father from above) or two different parts (e.g. wrist width from both the left and the right hand of same person):

It is ill-advised to create an average between them if either x1 or x2 includes outliers, or the differences between the two variables (x1 - x2) are very large.