Face mask and Covid-19 (DANMASK study): a Bayesian interpretation
In the famous study, DANMASK, the authors conclude that wearing surgical mask together with social distancing practice did not reduce the risk of SARS-Cov-2 infection. However, using a Bayesian inference, I show that their result is consistent with a risk reduction of up to 28%.
Rarely a scientific study generates a lot of controversies, but that is the case for the DANMASK study . In this study, the authors found that face mask wearing did not reduce the risk of infection with SARS-Cov-2 (odds ratio, 0.82; 95% CI, 0.54 to 1.23; P = 0.33) . However, most newspapers and health experts were not swayed by the evidence, and they categorically say that “you need to wear them anyway” . Here, I would like to offer my interpretation of the study’s result within context of previous data.
First, about the DANMASK study: it was designed as a randomized controlled trial in Denmark. Ultimately the study included 6024 individuals who were randomly assigned to either the intervention (n = 3030) or control group (n = 2994). The intervention was social distancing recommendation plus mask wearing. The control group was also recommended to follow the social distancing rule but face mask wearing was not recommended. This sample size was calculated based on the assumption that wearing face mask cuts the risk of infection by 50% (eg 2% in the control and 1% in the intervention group). The primary outcome was the incidence of SARS-Cov-2 infection confirmed by antibody testing and PCR. In a per-protocol analysis, the outcome can be summarized as follows:
· 42 / 2392 individuals (1.76%) in the intervention group were infected;
· 53 / 2470 individuals (2.15%) in the control group were infected;
· the risk ratio (RR) = 0.82, with 95% confidence interval [CI] ranging from 0.55 to 1.22; P = 0.33.
As you can see, the result is not consistent with the hypothesis of 50% risk reduction at all. Actually, the result is consistent with a risk reduction of 45%, but it is also consistent with an increased risk of 22%. Some people may tempt to say that the result is inconclusive.
Well, not so soon.
Some critics harshly criticized that the study was underpowered, meaning that the sample size was not adequate to find a real effect. However, this criticism is totally wrong. When a study has been completed, power is no longer relevant . The information of effect is contained in the confidence interval, not the post-hoc power. We should focus on the observed data and its uncertainty, not retrospective power calculation. And, to that end, I would like to offer my Bayesian interpretation of the data.
In the previous article , I say that Bayesian theorem is a way of synthesizing information from different sources in a logical way. In other words, the present data should be considered within the context of our pre-existing knowledge to derive an updated knowledge about the effect of face mask on infection risk.
Several studies on the effect of face mask on acute respiratory infections (ARI) in non-healthcare settings have been carried out, and their results have recently been synthesized by a meta-analysis published in September 2020 . The result of this meta-analysis apparently shows that surgical mask had a modest but statistically non-significant reduction in ARI (odds ratio, 0.96; 95% CI, 0.80 to 1.15). However, I am interested in the result of 5 cluster randomized trials which showed that face mask was associated with 13% reduction in the odds of ARI (odds ratio, 0.87; 95% CI, 0.74 to 1.04) . We can quantify our knowledge of the effect of face mask on infection risk by the Normal distribution with:
· mean = log(0.87) = -0.16; and
· standard deviation = log(1.04 / 0.74) / 3.92 = 0.087
Now, the evidence provided by the study  can also be quantified by a Normal distribution with:
· mean = log(0.82) = -0.20; and
· standard deviation = log(1.22 / 0.55) / 3.92 = 0.203
Using the Bayesian theorem, I can estimate the posterior mean and standard deviation of log risk ratio as follows (you can check my calculation):
· mean = -0.174
· standard deviation = 0.081
In other words, the risk ratio = exp(-0.174) = 0.84, and 95% credible interval ranges from 0.72 to 0.99. The result is visualized as follows:
The above posterior result means that face mask wearing was associated with an average of 16% reduction in the risk of SARS-Cov-2 infection, and that the data are consistent with a 1% to 28% reduction of risk.
So, my analysis suggests that the result is ‘positive’.
My next question is concerned with the magnitude of effect: what is the probability that face mask wearing reduces the risk of infection by at least 50% (as hypothesized by the authors), 40%, 30%, 20% 10%, and 5%? Here are my results:
· Probability of reduction by at least 50%: 0.000
· Probability of reduction by at least 40%: 0.000
· Probability of reduction by at least 30%: 0.009
· Probability of reduction by at least 20%: 0.247
· Probability of reduction by at least 10%: 0.788
· Probability of reduction by at least 5%: 0.930
The above analysis suggests that the effect of face mask wearing on risk reduction is likely to be very modest, ranging from 5 to 10% at best.
Finally, I think that the interpretation should be considered within context. Denmark has so far recorded more than 70,000 cases of Covid-19 and 781 deaths . The Covid-19 situation in Denmark has not been as severe as, say, in the US. Therefore, my interpretation is as follows: in settings with moderate Covid-19 infection (such as Denmark), wearing face mask may modestly protect the wearers from infection with SARS-Cov-2.
We tend to forget the real purpose of surgical mask which is designed to protect the patient from being infected by the surgeon, not to protect the surgeon. So, the critical question is: does face mask wearing protect the people around the wearers? We don’t know yet. This question should be addressed by a cluster randomized trial.
PS: A shorter version of this commentary has been published in the Ann Int Med website under the title “A Bayesian interpretation of the effect of face mask on SARS-Cov-2 infection”. The authors have responded to my comment that the Bayesian analysis was not part of the study's protocol, so they did not conduct such an analysis.