Unmasking the mask studies: why the effectiveness of surgical masks in preventing respiratory infections has been underestimated

Abstract

Background: Pre-pandemic empirical studies have produced mixed statistical results on the effectiveness of masks against respiratory viruses, leading to confusion that may have contributed to organizations such as the WHO and CDC initially not recommending that the general public wear masks during the coronavirus disease 2019 pandemic.

Methods: A threshold-based dose-response curve framework is used to analyse the effects of interventions on infection probabilities for both single and repeated exposure events. Empirical studies on mask effectiveness are evaluated with a statistical power analysis that includes the effect of adherence to mask usage protocols.

Results: When the adherence to mask usage guidelines is taken into account, the empirical evidence indicates that masks prevent disease transmission: all studies we analysed that did not find surgical masks to be effective were under-powered to such an extent that even if masks were 100% effective, the studies in question would still have been unlikely to find a statistically significant effect. We also provide a framework for understanding the effect of masks on the probability of infection for single and repeated exposures. The framework demonstrates that masks can have a disproportionately large protective effect and that more frequently wearing a mask provides super-linearly compounding protection.

Conclusions: This work shows (1) that both theoretical and empirical evidence is consistent with masks protecting against respiratory infections and (2) that non-linear effects and statistical considerations regarding the percentage of exposures for which masks are worn must be taken into account when designing empirical studies and interpreting their results.

Keywords: Surgical mask; dose–response curve; face mask; non-linear effects; statistical power.

Figure 1

The effective adherence and sample sizes of studies that found masks to be effective (triangles) and those that did not (squares). Empirical studies with higher levels of statistical power consistently show that masks protect the wearer; studies with lower statistical power are mixed, as would be expected. The statistical power depends on the sample size, the effective adherence (i.e. mask effectiveness multiplied by the fraction of exposures for which masks are on average worn in the mask group) and probability of being infected without a mask in the setting of the study. Each curve depicts the required sample size (expressed as the expected number of infections in the non-mask group) as a function of effective adherence in order for the study to have a power of 80%. The scattered data points depict the size and effective adherence of studies taken from a recent systematic review; the numbers on the data points correspond to row numbers of the studies in Supplementary Data and Methods: Supplementary Table S1. (Note that they do NOT correspond to the numbers in References.) The effective adherence for the studies are overestimated by assuming that masks are 100% effective; even with this assumption, the studies numbered 1 through 14 were found to have <80% statistical power. See Supplementary Data and Methods for data and mathematical details.

Figure 2

Mask usage frequency and its non-linear effects on infection probability. Left: A susceptible individual’s probability of infection as a function of effective adherence (mask effectiveness multiplied by the fraction of exposures for which the mask is worn ) for various values of that individual’s total effective exposure (the total effective exposure is proportional to the number of exposure events). For high values of , the infection probability is non-linear in the adherence, whereas for low values of , the infection probability decreases approximately linearly with adherence. Right: For a group of individuals (e.g. in an arm of a study), the total effective exposure will in general vary from individual to individual such that even if on average the total effective exposure is relatively low, it may be high for the individuals who make up the bulk of those being infected. The dashed curve depicts the expected percentage of infected individuals for the homogeneous case in which everyone experiences the same total effective exposure, whereas the solid curve depicts a case in which the exposure is heterogeneous; in both cases, the percentages of individuals that would be infected without masks (e.g. in a control group) are identical (~10%).

Figure 3

Non-linearities in the dose response curve and probability of infection. Left: A representative function for a susceptible individual’s probability of infection p as a function of viral dose v for a single exposure event, together with the effective exposure , where f(v) is convex for all v, whereas p(v) is convex for sufficiently small v. The convexity of f(v) (which is demonstrated in Methods) yields an S-curve for p(v). Note that for any particular viral dose v, the effective exposure can vary from individual to individual. Right: A depiction of how the total effective exposure and the probability of eventually becoming infected scale with the number of exposure events. The total effective exposure is the sum of the effective exposures from each exposure event; see Methods for details.

Figure 4

Effect of one vs both individuals wearing a mask. The effect of both the susceptible and infected individual wearing a mask can be much larger than the effect of only one of them wearing a mask. In the depicted example, the total effective exposure if neither the infected nor susceptible individual are wearing masks is such that the probability of infection is very close to one. If each mask reduces the effective exposure by a factor of four, then the probability of infection if only one of the two individuals is wearing a mask is , i.e. a reduction in risk by a factor of 1.08. If both individuals are wearing a mask, however, the probability of infection is = 0.46, corresponding to a reduction in risk by a factor of 2.17, which is greater than the product of the effects of each mask individually (shown by the red dotted curve). For illustrative purposes, we have assumed that the infectious individual wearing a mask has the same effect as the susceptible individual wearing a mask, but relaxing this assumption will not qualitatively change the results; see Supplementary Data and Methods for details.