Stephen T’s Blog Spot

Open almost any study built on regression and you will find a table, often the second one, that lists the outcome regressed on a main variable of interest plus a row of controls. Each row carries a coefficient, an interval, and a p-value, lined up in a neat column. The natural thing to do is read down that column and take each number as the effect of that variable. It is one of the most common mistakes in applied statistics, and it has a name.

The trouble starts with what the model was actually for. A multivariable regression is built to estimate the effect of one thing, the exposure or treatment you care about, and the other variables are included to remove confounding of that particular relationship. They are there to clean up the estimate for your main variable, not to have their own effects measured. Their coefficients are adjusted associations, conditional on everything else in the model, and that adjustment was chosen for your question, not for theirs.

Here is why that matters more than it sounds. Confounding is not a property of a variable in isolation; it is a property of a specific pairing of a cause and an effect. A variable that is a proper confounder for your exposure can play an entirely different role for some other variable in the same model. It might sit on the causal path from that other variable to the outcome, a mediator, so adjusting for it strips out part of that variable’s real effect. Or it might be a common effect of two things, a collider, so adjusting for it opens a spurious association that was not there before. The very same regression that gives you a clean estimate for your exposure can hand you a badly biased estimate for a covariate, because the set of controls that is right for one is wrong for the other.

Daniel Westreich and Sander Greenland named this the Table 2 fallacy in 2013, after the table where it usually happens. Their point was not that the model is broken. The model does its job for the exposure it was designed around. The error is in the reading: treating a table built to answer one causal question as if it answered a dozen at once.

An example makes it concrete. Suppose you estimate the effect of a training program on earnings, adjusting for age, prior schooling, and motivation. For the training question, that may be a sensible set of controls. But if your eye drifts to the coefficient on prior schooling and you read it as the effect of schooling on earnings, you should not. Motivation may be a mediator of schooling, part of how schooling raises earnings, so adjusting for it removes some of schooling’s real effect. And the confounders you would need to estimate schooling’s effect properly, family background and the like, were never in the model, because the training question did not require them. The schooling coefficient is not the effect of schooling; it is a number shaped by a model built for a different purpose.

The discipline is simple to state. Decide, before you model, which single effect you are estimating, and choose the controls for that effect. If you truly need the causal effects of several variables, that is several distinct questions, each deserving its own adjustment set and often its own model, not one table read greedily. When you must show the full output, label the secondary coefficients honestly as adjusted associations rather than effects, and resist interpreting them. And as a reader, do the same: find the one effect the model was built to estimate, and treat the rest as machinery.

For those of us reading and writing evaluation reports, the temptation to mine a regression table for a dozen findings is strong, and it quietly misleads decision-makers about which factors matter. Estimating an effect is a question you answer one at a time.

So here is my question. When you read a regression table, do you interpret only the effect the model was designed to estimate, or do you read down the column as though every coefficient were a finding?