Narrative · Experimental data
The statin trial
A randomised trial with 400 patients. What the risk difference says about causal relevance, and why its sign fixes the direction of the claim.
Chapter 1
Four hundred patients
A study assigns 400 patients by lot to two arms of 200 each: one group receives a statin, the other does not. What is counted is who suffers a cardiovascular event, for instance a heart attack. In each arm half of the patients have a history of high blood pressure (Vol. III, §8.1).
STATIN statin therapy · HBP history of high blood pressure · CVE cardiovascular event · ¬CVE no such event. The history precedes the therapy, the therapy precedes the event.
In the statin arm 5 of 100 patients with high blood pressure and 19 of 100 without high blood pressure suffer an event, together 24 of 200. The event rate is p_1 = 0.12 (Vol. III, §8.1).
In the control arm the figures are 30 of 100 with and 18 of 100 without high blood pressure, together 48 of 200. The event rate is p_0 = 0.24. Fisher's exact test gives p = 0.003 for this difference (Vol. III, §8.1).
Chapter 2
What the lot achieves
Whether a patient suffers an event depends on many things that the study does not record. Such competing conditions cannot be switched off. The lot ensures that they occur equally often in both arms: their firing is independent of the assignment. That is Gate B, and the study passes it.
It does not pass Gate A, and it need not. In each of the four groups there are patients with and without an event, with the same values of all recorded factors. These are contradictory configurations. They show that the three recorded factors do not explain every case. That does not disturb a single comparison.
Chapter 3
The difference and its sign
The risk difference is p_1 − p_0 = 0.12 − 0.24 = −0.12. Volume II shows what this number means under Gate B. One divides the clusters of the complete, unknown theory of the event into three groups: those that contain the statin, those that contain its absence, and all the rest. The difference is then exactly the share of patients in whom a cluster with the statin is fired and no other, minus the share in whom a cluster with absent statin is fired and no other (Vol. II, §5.2).
For the non-occurrence of the event, ¬CVE, the same decomposition holds, and its difference is the negative of the first: +0.12. Being untreated is not part of a sufficient condition for remaining event-free; under this assumption the difference for ¬CVE is exactly the share of patients in whom a cluster with the statin for ¬CVE is fired and no other. A positive difference therefore justifies the conclusion that the statin is an indispensable part of a cluster for ¬CVE (Vol. II, §5.2).
The sign determines the direction of the claim. A two-sided p-value has no sign: it says only that the difference is not zero.
Chapter 4
High blood pressure as partner
Separated by history, it becomes clear where the difference lies. Among patients with high blood pressure, 5% with a statin and 30% without a statin suffer an event (Vol. III, §8.1).
Among patients without high blood pressure the figures are 19% and 18%: practically no difference. Here the statin does not act alone but together with a partner (Vol. III, §8.1).
The program finds the cluster
It is satisfied in 95 of the 100 cases in which it is present (consistency 0.95), and it covers 95 of the 328 patients without an event (coverage 0.29). The remaining patients without an event owe this to other conditions: there are further clusters for ¬CVE (Vol. III, §8.1).
Chapter 5
What is justified and what is not
The following conclusion is justified: the statin is an indispensable part of a cluster for the absence of a cardiovascular event, with high blood pressure as partner, relative to this field of patients and this factor set.
The complete minimal theory is not justified without further data: for that, the remaining clusters for ¬CVE would have to be known. Nor is the transfer to another field of patients justified.
The usual formulation that the statin “lowers the risk” of an event says the same in a weaker form. The theory requires the polarity to be named: the statin is causally relevant to event-free survival. And it requires the sign, for only the sign of the difference, not the significance, determines to which of the two effects the conclusion applies.