10.2 Indirect Correlation
Figure 10.3: What happens to the regression coefficient for the effect of political interest if we add a confounder to the model? Numbers represent correlations (lines) or standardized regression coefficients (arrows).
When is a variable a confounder and when does it change the effect of another predictor a lot if it is added to the regression model? The answer to the first part of this question is easy: A confounder is a variable that is correlated with both the predictor and dependent variable but is not (yet) included in the regression model. Because of the correlations, a confounder establishes an indirect correlation between the predictor and dependent variable.
The size of the indirect correlation equals the product of the correlation between confounder and predictor and the correlation between confounder and dependent variable. In Figure 10.3, the correlation between age and political interest is .12 and the correlation between age and newspaper reading time is .88, the indirect correlation between interest in politics and reading time established by age is .12 * .88 = .11.
10.2.1 Indirect correlation and size of confounding
In Figure 10.3, we start with a simple regression model with political interest as the only predictor of newspaper reading time. Respondent’s age, however, is correlated both with political interest (\(r\) = 0.12) and with newspaper reading time (\(r\) = 0.88). Age creates a positive indirect correlation between political interest and reading time.
As long as age is not included in the regression model, the model believes that the indirect correlation due to age is part of the effect of political interest. It assigns the indirect correlation due to age to the effect of political interest, that is, it includes the indirect correlation in the regression coefficient of political interest. In this situation, the regression coefficient for political interest expresses both the effect of political interest itself and the effect of age (the confounder).
Once we add age as a new predictor to the regression model, the indirect correlation due to age is removed from the effect of political interest. The effect of age on newspaper reading time is now correctly assigned to age. As a result, the value of the regression coefficient for political interest changes if we add age as a new predictor.
The size of the change is related to the size of the indirect correlation. The larger the indirect correlation, the more the regression coefficient of political interest changes if age (the former confounder) is included as a new predictor. This answers the second part of the question with which we started Section 10.2: When is a variable a stronger confounder?
If you love the details: The size of the change in the standardized regression coefficient is not exactly the same as the size of the indirect correlation. It is equal to the correlation between the confounder (age) and the predictor (political interest) times the standardized regression coefficient of the effect of the confounder (age) on the dependent variable (newspaper reading time) that controls for the effect of the predictor (political interest).
10.2.2 Confounders are not included in the regression model
Finally, it is important to remember that a confounder, such as age in the present example, is a variable that is not included in our regression model. As long as it is not included, the indirect correlation between predictor (political interest) and outcome (newspaper reading time) due to the confounder (age) is not controlled for when the effect of the predictor is estimated. The estimated effect is confused (confounded) with the effect of the confounder.
Once the confounder (age) is added to the regression model, however, the estimated effects are controlling for the variable formerly known as a confounder. The effects no longer partly represent the effect of the former confounder. In other words, they are no longer confounded by the effect of that variable. The former confounding variable now is a predictor or, if we are not interested in its effects, a covariate or control variable in the regression model.
10.2.3 Randomization for avoiding confounders
There is a very important way to minimize the chance of having any confounders at all, namely, randomization in an experiment. Remember the example of Chapter 7, where participants saw a video clip with Angelina Jolie, George Clooney, or no celebrity endorsing a charity. The video clip with or without a celebrity endorser is the experimental treatment here. If we let chance decide which video clip a participant sees, we randomize the experimental treatment.
How does this help us to avoid having confounders? The example research aimed to find out whether the celebrity endorser affects the willingness to donate to a charity. The experimental treatment (celebrity endorser video) is the independent variable or predictor variable in the model. If participants’ scores on this variable — in the example, seeing Jolie, Clooney, or no celebrity endorser — are random, the variable is expected not to correlate with any other characteristic of the participants when the experiment starts.
For example, female and male participants would have the same chance to see Jolie, Clooney, or no endorser. We expect one third of all females and one third of all males to see Jolie, to see Clooney, and to see no endorser. If there is no systematic difference between females and males in this respect, participant’s experimental treatment is not correlated with sex of the participant. The same reasoning applies to every other characteristic of the participant at the start of the experiment: age, hair colour, favourite movie star, and so on. We expect that all of these participant characteristics are not correlated with the experimental treatment variable.
Let us now turn to the definition of a confounder: A confounder is a variable that is correlated with both the predictor and dependent variable but is not included in the regression model. A confounder must be correlated with the predictor, which is the experimental treatment here. Thanks to experimental randomization, we expect that all participant characteristics that are not included in the experiment do not meet this criterion. There should not be any confounders!
We have learned about probabilities and expectations in previous chapters. These principles also apply to experimental randomization. Even if we may expect to have equal numbers of females and males seeing Angelina Jolie in our example experiment, we can end up with more females than males seeing Jolie in our experiment due to chance. In this situation, the experimental treatment variable is correlated with the sex of the participant, so participant sex is a confounder if it is also correlated with the dependent variable (willingness to donate) and not included in the analysis. Experimental randomization does not guarantee that there are no confounders but it is our best instrument to minimize the chance of having confounders.