Appendix D: Elaboration modeling

There are different ways to introduce control variables into the analyses of causal relationships. One method is to use elaboration models (also called the elaboration paradigm, but that sounds a bit big-for-its-britches because it’s really a very simple tool). We’ll look at this in the context of bivariate (one independent variable, one dependent variable) statistical analysis. (Another way to introduce control variables is to use multiple regression, and there are still other techniques for specific types of bivariate statistical analysis.) The same logic can be applied to qualitative data analysis as well.

An elaboration model is fairly simple. If we introduce a control variable, we want to measure the effect of the independent variable \(\left ( IV \right )\) on the dependent variable \(\left ( DV \right )\) while controlling for the control variable \(\left ( CV \right )\). Other phrasings are helpful for understanding what we’re after:

What is the effect of the \(IV\) on the \(DV\) variable holding the \(CV\)constant?

What is the effect of the \(IV\) on the \(DV\) variable independent of the influence of the \(CV\)?

What is the effect of the \(IV\) on the \(DV\) variable, regardless of the \(CV\)?

For example, we might observe that men make higher wages than women, and we find this to be a statistically significant relationship using a t-test to compare men’s and women’s average wages. Someone might challenge that finding, saying that there’s a third variable at play: Years in the workforce. Women are more likely to take time off for raising children, so maybe they tend to make less money because they haven’t put in as much time in the workforce. Does the original finding hold up to this challenge? We’d want to see if men make higher wages than women, controlling for years in the workforce. Our \(IV\) is gender, our \(DV\) is wages, and our \(CV\) is years in workforce. We could test the influence of this \(CV\) on our causal relationship of interest by asking: What is the effect of gender on wages, controlling for years in the workforce? Put differently,

What is the effect of gender on wages, holding workers’ years in the workforce constant?

What is the effect of gender on wages, independent of the influence of workers’ years in the workforce?

What is the effect of gender on wages, regardless of workers’ years in the workforce?

An elaboration model applies the “holding the \(CV\) constant” phrasing quite literally. To investigate this question, we could divide our workers into, say, three categories, based on their values for the control variable: <6 years in the workforce, 6 – 10 years in the workforce, and >10 years in the workforce. Then, we could measure the relationship between sex and wages within each of those three levels. That would be three separate t-tests: One t-test for just the <6 years group, one for just the 6 – 10 years group, and one for just the >10 years group. We would be measuring the relationship between our IV and DV three times, while literally holding the \(CV\) constant each time.

What might we learn from applications of elaboration models?

The control variable may have no influence on the causal relationship: If the original wage gap persists throughout the three t-tests, we would conclude that the men make higher wages than women, controlling for years in the workforce.

The control variable may wholly explain away the purported causal relationship, meaning it was a spurious relationship to begin with: If the wage gap disappears throughout the three t-tests, we would conclude that there is no relationship between sex and wages when controlling for years in the workforce and that the simple bivariate relationship between sex and wages is spurious; sex and wages are both related to years in the workforce, but they are not directly related to each other.

The control variable, quite often, will partially explain away the causal relationship under investigation, meaning that some, but not all, of the relationship between the \(IV\) and \(DV\) is really due to both of them being related to the \(CV\): If the three t-tests reveal that men have higher wages than women, but to a lesser degree than in the original t-test conducted with the entire sample at once, we would conclude that there is, indeed, a wage gap, but part of the wage gap is attributed to differences in men’s and women’s years in the workforce.

The control variable may help to better specify the relationship between the IV and DV: If the t-tests reveal no wage gap among the <6 year workers, a moderate wage gap among the 6 – 10 year workers, and a larger wage gap among the >10 year workers, the control variable has helped us describe the relationship between sex and wages with better specificity.

In crazy, uncommon cases, the control variable may have a suppressor effect, revealing a stronger relationship between the \(IV\) and \(CV\) or even changing the direction (direct to inverse) of the relationship between the \(IV\) and \(CV\). If our three t-tests revealed that, within each of the groups, women had higher wages than men, we would conclude that we need to spend more time with our data to figure out the complex causal relationships at work between sex, wages, and years in the workforce!