One absent condition is enough

Anyone who wants to find a cause by comparison compares cases that differ in the factor in question. For this John Stuart Mill required cases that have “every circumstance in common save one” (Mill 1843, Book III, ch. 8) (Vol. I, §3.3).

Two patients differ in age, genetic make-up, diet, sleep and in a thousand further respects. If causal reasoning required all of this to agree, no clinical study would ever have justified a result.

The diagram shows recovery with two clusters: the treatment together with the absorption of the drug in the body, or a spontaneous remission together with an effective immune response.

What a comparison must prevent is not that background factors vary. It is that a competing cluster is fired in the one case and not in the other. The comparison then reflects the coming and going of this cluster, not the factor under test.

A cluster is a conjunction. If one of its conjuncts is reliably absent, the whole conjunction is false. If the study excludes patients with a tendency to spontaneous remission, the lower cluster is switched off. The immune response may then vary as it will from patient to patient, without any effect on recovery or on the comparison.

For the treatment cluster the reverse holds: its partner, the absorption of the drug, must be present in all cases; otherwise the comparison also measures the varying absorption. The administration protocol and the monitoring of intake ensure this.

The aim, then, is not to keep as much as possible constant. It is to find, for each competing cluster, the one conjunct that is easiest to keep absent. Minimising the homogeneity demand is no concession to practice; it is what makes efficient experimentation possible in the first place.

Not every competing cluster has a conjunct that can be kept absent. For such clusters there is randomisation. It does not switch the cluster off; nothing in a study of human beings could do that. It makes the firing of the cluster independent of the assignment to treatment: the cluster is fired equally often in both arms.

This yields two different homogeneity demands for two different inferences.

• Gate A, for reading off a whole theory from a table: every hidden combination of background conditions has the same truth value throughout the field. Every contradictory configuration refutes this; if none occurs, merely no violation has been found.
• Gate B, for a single comparison of two arms: no competing combination is fired with a different frequency together with the factor under test. Exclusion criteria, protocol and randomisation ensure this.

A randomised study passes Gate B and as a rule fails Gate A. That is no defect. It does not aim to explain every case but to isolate one factor. The statin study shows how far it gets in doing so.