COVID‐19 encephalopathy, Bayes rule, and a plea for case–control studies

We are interested in investigating encephalopathy rates in COVID‐19 with the Bayes rule, a more intuitive way to understand and communicate risks and associations than P‐values. Like in classical statistics, Bayes cannot help much with low numbers from individual case series. Liotta and coauthors provided a sample size large enough to make more robust estimations than any other research we have seen. ¹ They found a substantial association between encephalopathy and greater morbidity in COVID‐19. This association does not prove causality: the most likely cause of encephalopathy in COVID‐19 is multifactorial, and the press has accurately interpreted this information. ²

We replicated selected outcomes from their study using the Bayesian A/B test for summary statistics in Jeffrey's Amazing Software Package (JASP). ³ The test can monitor and update the evidence for or against an association between two variables of interest. Beyond p‐values, Bayes factors (BF) quantitatively measure the evidence after seeing the data. Even better, by estimating the median with a 95% credible interval (95% CrI) of the posterior probability distribution, we can communicate results in real probability terms, something classical statistics cannot do.

We specified a weakly informative prior (N ~0,1) based on level VI evidence on COVID‐19‐related encephalopathy. All the computations with notes are accessible at the Open Science Framework portal (osf.io/p94u8/). A summary of the results is in Table 1, and Figure 1 will help readers to visualize the effect of disease severity on encephalopathy within a Bayesian framework.

Table 1.

Selected outcomes and association with COVID‐19 encephalopathy.

	COVID‐19 (n = 509)		BF 1	Posterior probability median, (95% CrI) 2	P 3
	Encephalopathy (n = 162)	No encephalopathy (n = 347)
Male 4	101	180	4	0.6(0.5–0.7)	0.034
History Neurological disorder	55	79	12	0.6(0.5–0.7)	0.01
Cancer	32	29	236	0.7(0.6–0.8)	<0.001
CVD	21	18	39.5	0.7(0.6–0.8)	0.004
CKD	27	29	19.3	0.7(0.5–0.8)	0.008
COVID‐19 severity	49	113	109	0.97(0.94–0.98)	<0.001
30‐day mortality	35	11	18.7	0.86(0.78–0.92)	<0.001

COVID‐19, Coronavirus disease 2019; BF, Bayes factor; 95% CrI, 95% credible interval; CVD, cerebrovascular disease; CKD, Chronic kidney disease.

¹Bayes factors quantify evidence on a continuous scale around 1, the point of no difference between two groups, ideas, or hypotheses. BF between 0.3 and 3 provide no evidence at all, whereas BF > 3 or < 0.3 provide increasingly strong evidence ⁴ . For very high BF such as in disease severity and mortality, we used the logarithmic (log) BF.

²The result of combining the prior probability (see text) and the BF after seeing the data from Liotta and coauthors, in direct probability terms after rolling back from log‐odds ratios. We can use this probability to quantify the risk of encephalopathy instead of statistical significance, which conveys little practical information to clinicians and patients. For example, a man with COVID‐19 has a probability of 0.6 (60%) of developing encephalopathy at any point during the acute infection; this risk could be as low as 0.5 (indicating the same risk as women) or as high as 0.7, with 95% certainty that the true probability is in the interval.

³ P‐value from the article by Liotta and coauthors.

⁴By comparing men with encephalopathy (101/281) and women (61/228).

Figure 1.

Upper panel: the left probability wheel shows the distribution of probabilities of no difference P(H₀), and a difference P(H +‐) in encephalopathy by COVID‐19 severity. The right probability wheel is the change of these probabilities after the Bayes rule (posterior). Lower panel: the sequential analysis plots how the probabilities increase with more observations.

We believe that these probabilities are overestimates because we worked with descriptive data from a single study. As an example, in our model, the probability that a patient with severe COVID‐19 will be encephalopathic was 97%, which tallies well into the odds ratio of 131 by Liotta and coauthors in their adjusted regression model. A case–control design will recalibrate these results to a less impressive but more accurate figure. COVID‐19 suspect cases admitted during the same period with a negative test result are acceptable as controls, understanding the limits of test accuracy. We should view rates, risks, and effect sizes from case series of COVID‐19 with plenty of caution.