The AI Journal

Consider the most common sentence in empirical writing: the effect disappeared once we controlled for X. It is offered as a finding, and it is read as one. The variable under study turns out to have been X in disguise; the coefficient that was doing the work was borrowed. There is something deeply satisfying about this move, because it mimics an experiment. We introduced a cause, we watched the effect go, and we conclude that the effect belonged to the cause. But no cause was introduced, and nothing was removed. A column was added to a matrix. The sentence sounds like a mechanism and is actually a statement about the geometry of a fit.

Two very different situations produce the same output line. In the first, the variable really is unrelated to the outcome; its coefficient is near zero with or without company, and the control merely confirms it. In the second, the variable is related to the outcome but shares so much of its variance with the other predictors that almost nothing is left for it to explain alone. Its unique coefficient, its contribution over and above the rest, collapses toward zero, and the software prints the same b and the same p as the genuine null. Nothing in the output distinguishes the two. The distinction lives in the design and the correlations, not in the table, and the table is what gets quoted.

The second situation is not a curiosity. Lynam, Hoyle and Newman (2006) collected cautionary tales in which partialling produced relations that were not there and dissolved relations that were; the lesson was not that partialling is useless, but that what survives it is not the real effect. It is the effect conditional on a particular, and often arbitrary, set of others. Westfall and Yarkoni (2016) pushed the point from the other side: a control measured with error does not quietly remove confounding, it can distort the estimate in either direction. You can control your way into an effect as easily as out of one. A researcher who treats the partial coefficient as the truth about a variable has mistaken a modelling choice for a measurement.

Suppression makes the trouble sharper still. MacKinnon, Krull and Lockwood (2000) observed that the same arithmetic underlies mediation, confounding and suppression, and that the three are told apart by interpretation, not by the numbers. Suppression is the case where a variable with no marginal relation to the outcome nonetheless carries the effect once another variable is in the model. Watson, Clark, Chmielewski and Kotov (2013) argued that such suppressor effects can be substantively informative rather than noise. The point cuts both ways. A variable can matter while contributing no unique variance, and a variable can show a unique contribution while mattering not at all. The mapping from coefficient to importance is not a function.

This is the asymmetry worth keeping. A zero marginal relation and a zero unique contribution are simply not the same finding. The first is a candidate for unrelated. The second is a stamp that says inseparable in this model, and inseparable is not the same as absent. Merging them under the single word null is where the error enters, because the word then does the argument’s work for it. Two nulls, one word, and the variable is quietly retired.

Why does the elimination reading persist? Because the sentence has the grammar of a cause. Controlling for X, the effect vanished, borrows the authority of a controlled experiment without paying for it. In an experiment you intervene, hold things fixed, and observe. Here you add a predictor and watch the coefficient change, then narrate the change as an explanation. The phrase explained by X performs a counterfactual that was never run. Worse, the control set is usually presented as if it were neutral, as if what remains after partialling is the effect stripped of noise. There is no neutral control set. Every choice of covariates encodes a theory about what should be held constant, and a different defensible set would have produced a different survivor.

What does the data actually license? Not elimination. At most, the weaker sentence: in this model, among these predictors, this variable’s unique share of variance is small. Tools for decomposing that share exist precisely because the unique coefficient is one partition among several and a fragile one when predictors correlate; commonality analysis and relative weights were built to report the shared portion alongside the unique (Kraha, Turner, Nimon, Zientek and Henson, 2012; Nimon and Oswald, 2013). The honest report is the decomposition, not the deletion.

The strongest objection is worth stating plainly. If a variable adds no unique variance, an objector will say, then including it is pointless, and the parsimonious model is the honest one. This is not a foolish position. Parsimony is a genuine virtue, and a model that carries a useless predictor is a worse model. The reply is that parsimony is a claim about a preferred model, not a claim about the world. Drop the variable from your regression and you have made a modelling decision. Drop it from the description of what is going on and you have made a claim about reality that no test you ran can support. The objector also quietly assumes the remaining variables are the right ones, that partialling has exposed the effect itself. It has exposed the effect conditional on a chosen set. Change the set and the effect changes, which is exactly what suppression and collinearity demonstrate.

So I narrow the claim. I do not say a null coefficient is never informative. Where predictors are nearly orthogonal, where constructs are measured with little error, and above all where the design tests for equivalence rather than merely failing to reject, a null is real information, a test that has been given the power to find an effect and did not (Lakens, 2017). The rule is conditional. A null supports elimination only when you have shown it is the unrelated kind. Short of that, it supports the weaker sentence, and the weaker sentence is not the one usually written.

The cost of dropping the distinction is not merely technical. Effects do not usually disappear from a literature because they were disproven. They disappear because the sentence that retired them was never examined, and it never had to be, because it sounded like the conclusion of an experiment. A variable can be argued out of a field one controlling for at a time, and the whole process will look like rigour from beginning to end. The remedy is small and unpleasant: when a coefficient vanishes, say which null you are looking at, and decline to say more than the design can carry.