Cardiologists and statistics

Here is the conclusion of a paper in a professional cardiology journal that is inconsistent with its results.  Keep in mind that MDs are seldom statisticians.

In this AF cohort, the authors demonstrated that the CHA2DS2-VASc score was not static, and that most patients with AF developed ≥1 new stroke risk factor before presentation with ischemic stroke. The Delta CHA2DS2-VASc score, reflecting the change in score between baseline and follow-up, was strongly predictive of ischemic stroke, reflecting how stroke risk in AF is a dynamic process due to increasing age and incident comorbidities.

Here are the paper's results that led to the authors' conclusion.

The mean baseline CHA2DS2-VASc score was 1.29, which increased to 2.31 during the follow-up, with a mean Delta CHA2DS2-VASc score of 1.02. The CHA2DS2-VASc score remained unchanged in only 40.8% of patients. Among 4,103 patients who experienced ischemic stroke, 89.4% had a Delta CHA2DS2-VASc score ≥1 compared with only 54.6% in patients without ischemic stroke, and 2,643 (64.4%) patients had ≥1 new-onset comorbidity, the most common being hypertension. The Delta CHA2DS2-VASc score was a significant predictor of ischemic stroke that performed better than baseline or follow-up CHA2DS2-VASc scores, as assessed by the C-index and the net reclassification index.

Here is the problem.

The authors' conclusion is of the form Probability of Stroke given High Delta exceeds Probability of Stroke given Low Delta.  These are the probabilities of interest.

The authors' results are of the form Probability of High Delta given Stroke exceeds Probability of Low Delta given Stroke.

These two probability sets are different.  The latter does not imply the former.

Here is a hypothetical example where the Probability of a Stroke given a High Delta is lower than the Probability of a Stroke given a Low Delta, yet the Probability of a High Delta given a Stroke is higher than the Probability of a Low Delta given a Stroke.

Suppose a sample of 100 people where 80 have a high delta and 20 have a low delta.  Suppose the probability of a stroke given a high delta is 30% and the probability of a stroke give a low delta is 80%.  Then 24 of the high delta people have strokes and 16 of the low delta people have strokes, for a total of 40 strokes.  Of the people suffering a stroke, 60% had a high delta and 40% had a low delta.