Which predictor is most important? Predictive utility vs. construct importance
Every so often, I’m asked for my two cents on a correlational study in which the researcher wants to find out which of a set of predictor variables is the most important one. For instance, they may have the results of an intelligence test, of a working memory task and of a questionnaire probing their participants’ motivation for learning French, and they want to find out which of these three is the most important factor in acquiring a nativelike French accent, as measured using a pronunciation task. As I will explain below, research questions such as these can be interpreted in two ways, and whether they can be answered sensibly depends on the interpretation intended.
Interpretation 1: Predictive utility
First, you can interpret questions such as Which of variables A, B and C is the most important factor in X? as follows: If you wanted to guesstimate a person’s result on X (e.g., nativelikeness of accent) and you could only know their score on A, B or C (e.g., intelligence test, working memory task, motivational questionnaire), which one should you pick? Such questions can make sense when you have a battery of tasks and questionnaires that you need to slim down for a future study or evaluation.
Interpreted like this, such questions can be sensibly answered, for instance by comparing the correlation coefficients for AX, BX and CX or by comparing the fit of regression models for each of the three predictor variables.
Interpretation 2: Construct importance
Often, however, it turns out that researchers aren’t interested in the predictive utility of variables A, B and C per se, but rather in the importance of the construct that these variables represent. Concretely, they aren’t so much interested in the participants’ performance on an intelligence test as they are interested in the participants’ intelligence proper. The tests, tasks and questionnaires are merely means for eliciting this information, and imperfect means at that.
For the accent example, then, the intended research question under this interpretation is this: What’s more important for acquiring a nativelike accent in French: your intelligence, your working memory capacity or your motivation?
The difference between this interpretation and the ‘predictive utility’ interpretation may seem small, but whereas I think that such research questions are answerable under the ‘predictive utility’ interpretation, I think they are usually unanswerable when they concern the scientific constructs themselves. The reason for this, as so often, is measurement error.
Due to measurement error, the participants’ scores on the intelligence test, working memory task and motivational questionnaire are but approximations of their true intelligence, working memory capacity and motivation. What is more, the extent to which these scores are affected by measurement error will vary from task to task. This is important because measurement error, on average, attenuates between two variables. As a result, we may find that intelligence scores predict nativelikeness of accent better than working memory scores or motivational questionnaire scores, but this doesn’t have to mean that intelligence itself is more important than working memory capacity or motivation: it may well be the case that motivation is by far the most important factor in accent acquisition, but that this factor is less well captured by the questionnaire than intelligence is by the intelligence test.
When it’s construct importance you’re interested in, the time to worry about measurement error is before running the study. For instance, at the cost of considerably more time and effort on the part of the participants, you may want to use multiple tests and tasks for each of the constructs you’re interested in and conduct a latent variable analysis. Or you can try to find tasks whose reliability in measuring the construct is known so you can correct for measurement error (see Chapter 7 in Faraway’s _Linear models with R). I don’t have much experience with either strategy, though.
When you want to find out which variable is the most important one, think about whether it’s predictive utility or construct importance you’re interested in. If it’s (also) the latter, consider the attenuating effect of measurement error when designing your study.