Unmasking the mask studies: why the effectiveness of surgical masks in preventing respiratory infections has been underestimated
Pratyush K. Kollepara, Alexander F. Siegenfeld, Nassim Nicholas Taleb, Yaneer Bar-Yam
UUnmasking the mask studies: why the effectiveness of surgical masks in preventingrespiratory infections has been underestimated
Pratyush K. Kollepara,
1, 2, ∗ Alexander F. Siegenfeld,
1, 3, ∗ Nassim Nicholas Taleb, and Yaneer Bar-Yam New England Complex Systems Institute, Cambridge, MA Department of Physics, BITS Pilani K K Birla Goa Campus, Goa, India Department of Physics, Massachusetts Institute of Technology, Cambridge, MA Tandon School of Engineering, New York University, Brooklyn, NY
Face masks have been widely used as a protective measure against COVID-19. However, pre-pandemic empirical studies have produced mixed statistical results on the effectiveness of masksagainst respiratory viruses. The implications of the studies’ recognized limitations have not beenquantitatively and statistically analyzed, leading to confusion regarding the effectiveness of masks.Such confusion may have contributed to organizations such as the WHO and CDC initially notrecommending that the general public wear masks. Here we show that when the adherence tomask-usage guidelines is taken into account, the empirical evidence indicates that masks preventdisease transmission: all studies we analyzed that did not find surgical masks to be effective wereunder-powered to such an extent that even if masks were 100% effective, the studies in questionwould still have been unlikely to find a statistically significant effect. We also provide a frameworkfor understanding the effect of masks on the probability of infection for single and repeated expo-sures. The framework demonstrates that more frequently wearing a mask provides super-linearlycompounding protection, as does both the susceptible and infected individual wearing a mask. Thiswork shows (1) that both theoretical and empirical evidence is consistent with masks protectingagainst respiratory infections and (2) that nonlinear effects and statistical considerations regardingthe percentage of exposures for which masks are worn must be taken into account when designingempirical studies and interpreting their results.
In 1910, one of the first western-trained Chinese physicians adapted surgical masks for use against a respiratoryplague that killed more than 60,000 people in four months [1]. The logic behind their function is transparent: a maskcan block some viral or bacterial particles from entering and/or dispersing from the wearer’s respiratory tract. Theyhave been used for prevention in a wide range of disease outbreaks and medical settings, and there is currently ageneral consensus that surgical and cloth masks help prevent infected individuals from spreading COVID-19 [2, 3].Surprisingly, given the logic of their utility, there is less of a consensus that surgical/cloth masks also protect thewearer and many government health organizations did not initially recommend wearing them during the early monthsof the COVID-19 pandemic [4, 5].It is well established that surgical and cloth masks partially block virus-containing airborne droplets of varioussizes [6–15]. Cloth masks, surgical masks, respirator masks (e.g. N95), and powered air-purifying respirators areunderstood to be capable of providing increasing levels of protection. The amount of virus transmitted betweenan infected and a susceptible individual is therefore expected to be reduced if either is wearing a mask, with bothwearing masks giving the best protection. However, this straightforward inference has been difficult to establish inexperimental studies. Here we analyze why some experimental studies find masks to be effective while others donot. We determined that the studies that did not find surgical masks to be effective were under-powered to such anextent that even if the masks were 100% effective, they still would have been unlikely to find a statistically significantresult. Statistical power is the probability that a study will find a statistically significant result if its intervention doesin fact have a certain effect. Our results concerning the statistical power of mask studies are summarized in fig. 1,which shows that all studies that had a large enough sample size and/or adherence for 80% power (above and tothe right of the gray lines) show a statistically significant reduction in infections among mask-wearers. As would beexpected, most studies with less statistical power (towards the lower left) did not find a statistically significant effect.We also provide a framework for understanding the nonlinear effects of mask-wearing on the probability of infection.Experiments that do not take such factors into account provide misleading results unless interpreted carefully. Whilethe precautionary principle [16, 17] would recommend the use of masks during the COVID-19 pandemic in any case(due to the asymmetric risks of using vs. not using masks), the analyses we provide gives consistency to theoreticalanalyses, experimental studies, and epidemiological recommendations. ∗ These two authors contributed equally. a r X i v : . [ q - b i o . Q M ] F e b -2 -1 Effective adherence in the mask group10 N u m b e r o f i n f e c t i o n s i n t h e n o n - m a s k g r o u p
123 45 6 7 89 101112 1314 151617 181920 212223 n No statistically significant reduction in infections in mask group for study n Statistically significant reduction in infections in mask group for study
Power < 80 % Power > 80 %Sample size (shown as the expected number of infections in the non-mask group) necessary for80% statistical power as a function of effective adherence, for various probabilities of infection without a mask:1% 10% 30% 50% 60%
FIG. 1. The effective adherence and sample sizes of studies that found masks to be effective (red triangles) and those that didnot (blue 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, theeffective adherence (i.e. mask effectiveness multiplied by the fraction of exposures for which masks are on average worn in themask group), and probability of being infected without a mask in the setting of the study. Each curve depicts the requiredsample size (expressed as the expected number of infections in the non-mask group) as a function of effective adherence in orderfor the study to have a power of 80%, i.e. an 80% probability of finding a statistically significant result ( p < .
05, two-tailed).The curves are calculated assuming equal-sized non-mask and mask groups, with each curve representing a different probabilityof infection in the non-mask group; the total sample size is thus the expected number of infected individuals in the non-maskgroup (which is what is plotted) divided by the probability of infection and then multiplied by two. We assume that theprobability of infection is reduced linearly with increasing effective adherence (see fig. 2); this assumption will underestimatethe true necessary sample size. The scattered data points depict the size and effective adherence of studies taken from arecent systematic review [18]; the study numbers correspond to those in the first column of fig. 5. The effective adherence forthe studies are overestimated by assuming that masks are 100% effective; even with this assumption, the studies numbered 1through 14 were found to have less than 80% statistical power. Note that the power of the studies (reported in fig. 5) dependon the probability of infection in the non-mask group and the size of the mask group in addition to the information present inthis figure; the location of the studies in this figure relative to the curves should therefore be considered only approximately. Inparticular, some of the studies have smaller overall sample sizes than implied by this figure, due to their non-mask and maskgroups not being equal in size. Mathematical details can be found in the Appendix.
STATISTICAL POWER
Some empirical studies find masks to be effective in preventing disease transmission while others do not [16, 18–23].However, due to poor statistics, even the studies with negative results are not inconsistent with masks being highlyeffective. While some of the studies conducted a power analysis to estimate the sample size required to obtain astatistically significant result with 80% probability (i.e. to achieve 80% power, the standard level by convention),these power analyses did not take in to account the possibility of low adherence (i.e. masks being worn for a lowpercentage of exposure events) and/or the possibility of a very low probability of infection even without a mask.When we consider such factors, none of the studies we analyze that did not find masks to be effective had sufficientstatistical power.The sample of studies we consider is taken from a recent systematic review [18]; see table II for a list of studiesthat were excluded and why. Most of the studies we examine measure whether surgical masks protect the wearer; P r o b a b ili t y o f i n f e c t i o n ˜ v T . . . . % o f p o p u l a t i o n i n f e c t e d HeterogeneousHomogeneous
FIG. 2.
Left:
A susceptible individual’s probability of infection as a function of effective adherence αγ (mask effectiveness γ multiplied by the fraction of exposures for which the mask is worn α ) for various values of that individual’s total effective expo-sure ˜ v T (the total effective exposure is proportional to the number of exposure events). Note that if a mask were 100% effective( γ = 1), then the effective adherence would simply equal the adherence α . The curves denoting the infection probabilities aregiven by 1 − e − (1 − αγ )˜ v T (eq. (5)). For high values of ˜ v T , the infection probability is nonlinear in the adherence, while for lowvalues of ˜ v T , 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 averagethe total effective exposure is relatively low, it may be high for the individuals who make up the bulk of those being infected.Thus, while using the average effective exposure would predict an approximately linear decrease of infection probability withincreasing adherence, such an approach may overestimate the expected effect of partial mask usage. The dashed curve depictsthe expected percentage of infected individuals for the homogeneous case in which everyone experiences the same total effectiveexposure, whereas the solid curve depicts a case in which the exposure is heterogeneous; in both cases, the percentages ofindividuals that would be infected without masks (e.g. in a control group) are identical (approximately 10%). the exceptions are studies nos. 8, 12 and 15, which measure whether masks prevent the wearer from infecting others,and studies nos. 1, 7, 13, and 18, in which both the susceptible and infected individuals sometimes wore masks (seetable I).In order to account for adherence, we make two conservative assumptions that will result in our overestimating thestudies’ statistical powers. First, we assume that the degree to which a mask reduces the probability of infection isproportional to the fraction of exposures for which it is worn (e.g. we assume wearing a mask half as often provideshalf as much protection); in fact, wearing a mask half as often will reduce the probability of infection by less thanhalf as much (see fig. 2), meaning that we overestimate the statistical power of these studies. Second, the numberswe calculate represent the power the studies would have had were masks 100% effective (i.e. were it impossible tobecome infected while wearing a mask). To the extent that masks are less than 100% effective, even larger samplesizes would be needed.For example, a randomized control trial (RCT) at the Hajj pilgrimage [24] assumed a reduction in infection ratefrom 12% to 6% in order to determine the sample size necessary for a statistical power of 80%. After taking intoaccount that the randomization was done by cluster (i.e. tent) rather than individual, the required sample size was ∼ . .
2% reduction in the probability of infection.However, the adherence in the control group was 1.8%, meaning that the maximum possible expected reduction ininfection between the two groups would be . − . − . = 0 .
014 (eq. (10)). Thus the effective adherence value used forthis study is 0.014. In addition, the probability of infection without masks is reported to be quite low (2%). Underthese conditions, the required sample size to achieve the desired statistical power of 80% would be ∼ Viral dose v
01 Probability of infection p ( v ) Effective exposure f ( v ) Convex region Number of exposure events01 Total effective exposure ˜ v T Probability of infection p inf FIG. 3.
Left:
A representative function for a susceptible individual’s probability of infection p as a function of viral dose v fora single exposure event, together with the effective exposure ˜ v ≡ f ( v ) ≡ − ln(1 − p ( v )). f ( v ) is convex for all v , while p ( v ) isconvex for sufficiently small v . The convexity of f ( v ) (which is demonstrated in the Appendix) yields an S-curve for p ( v ). Notethat for any particular viral dose v , the effective exposure ˜ v = f ( v ) can vary from individual to individual. Right:
A depictionof how the total effective exposure ˜ v T and the probability of eventually becoming infected scale with the number of exposureevents. The total effective exposure is the sum of the effective exposures from each exposure event; see Appendix for details. and false positive rates (approximately 2% and 0 . p < . NONLINEAR EFFECTS
We now describe a framework to account for the nonlinear aspects of mask effectiveness. Given that there is athreshold for the viral dose (the amount of the virus inhaled) below which the probability of infection is very small dueto the innate immune system [27, 28], and given that the probability of infection p will converge to one (for susceptibleindividuals) as the viral dose v is increased without limit, the probability of infection as a function of viral dose p ( v )is described by a sigmoid function or S-curve (fig. 3, see Appendix for details). (Concave curves have also been usedto model dose response curves, but such an approach ignores threshold effects [29, 30].) For a single exposure event,we can define the dimensionless effective exposure ˜ v ≡ − ln(1 − p ( v )) such that the probability of infection is 1 − e − ˜ v .Conveniently, the effective exposure is additive for independent exposure events, i.e. the total probability of infectionis given by 1 − e − ˜ v T where the total effective exposure ˜ v T is simply equal to the sum of the effective exposures foreach exposure event.Because the probability of infection is a concave function of the total effective exposure (fig. 3), the protectionafforded by a mask is super-linear in the percentage of exposures for which it is worn (e.g. wearing a mask twiceas often is more than twice as effective; see fig. 2). These nonlinear effects can be substantial for high cumulativeexposures. Under such conditions, a mask may need to be worn for most or nearly all of the exposure events inorder to provide significant protection; otherwise the individual is likely to be infected during the exposures for whichthe mask is not worn. In the limit of an extremely high total exposure, a mask will of course not have an effect onthe probability of infection since a susceptible individual will be infected with nearly 100% probability regardless ofwhether or not the mask is worn.On the other hand, for low total exposures, the protection masks provide will be approximately proportional tothe fraction of exposures for which they are worn. It should be noted that the total exposure of individuals canvary within any given study, such that even if the overall probability of infection is low, most of those who wereinfected may have been subjected to high cumulative exposures. Studies with low overall probabilities of infectionalso have an additional difficulty, which is that large sample sizes will be necessary in order that there may be enoughinfections in the non-mask group to produce a statistically meaningful comparison. In other words, for sufficientlylow total exposure, the probability of infection will be quite low even without a mask, and so further reductions tothis probability, even if proportionally large, will be small in absolute terms.We can also analyze certain compound effects that are not considered in most empirical studies. For instance, ˜ v /
16 ˜ v / v Total effective exposure ˜ v T . . . P r o b a b ili t y o f i n f e c t i o n One maskTwo masks No maskOne MaskTwo MasksTwo Masks (independent)
FIG. 4. The effect of both the susceptible and infected individual wearing a mask can be much larger than the effect of onlyone of them wearing a mask. In the depicted example, the total effective exposure ˜ v if neither the infected nor susceptibleindividual are wearing masks is such that the probability of infection p inf (˜ v ) is very close to 1. If each mask reduces theeffective exposure by a factor of 4, then the probability of infection if only one of the two individuals is wearing a mask is p inf (˜ v /
4) = 0 .
92, i.e. a reduction in risk by a factor of 1 .
08. If both individuals are wearing a mask, however, the probability ofinfection is p inf (˜ v /
16) = 0 .
46, corresponding to a reduction in risk by a factor of 2 .
17, which is greater than the product of theeffects of each mask individually (shown by the red dotted curve). For illustrative purposes we have assumed that the infectiousindividual wearing a mask has the same effect as the susceptible individual wearing a mask, but relaxing this assumption willnot qualitatively change the results; see Appendix for details. masks worn on both the infected and susceptible individuals may prevent a transmission event even if neither maskindividually would have. Furthermore, this compound effect may be super-linear: if the effect of only an infectedindividual wearing a mask is to reduce the infection probability by a factor of p and the effect of only a susceptibleindividual wearing a mask is to reduce the infection probability by a factor of p , both individuals wearing a maskcould reduce the infection probability by far greater than a factor of p p , especially for large total effective exposures.In the example shown in fig. 4, the probability of transmission is reduced by only a factor of 1 .
08 (a 7% reduction)due to one or the other individuals wearing a mask, while if both wear a mask, the probability of transmission willbe reduced by a factor of 2 .
17 (a 54% reduction). Similarly, just as there can be a super-linear compound effect fromboth individuals wearing masks, there can also be super-linear compound effects when mask-wearing is combinedwith other behaviors that reduce exposure, such as social distancing. Nonlinear effects continue to accumulate whenmultiple individuals perform multiple behavioral changes that reduce exposure. Recognizing these nonlinear effects iskey to appreciating the effectiveness of transmission prevention policies.It should also be noted that the proportional risk reduction from masks is expected to be large because—due to theconvexity of the S-curve (fig. 3) when exposure is low (as is likely for many exposure events)—the probability of themask-wearer being infected is decreased by a greater factor than the decrease in the viral dose [31] (see Appendix).For these low exposure events, although the probability of infection may be small for any given potential transmissionevent, given multiple events, the large factors by which the probabilities of infection decrease due to this convexitycan significantly reduce both the spread of the virus and the probability that the wearer eventually is infected. Inother words, wearing a mask may not only prevent the wearer from spreading viruses to others but may also have asurprisingly large protective effect for the mask-wearer. Indeed, studies that analyze population-level data show thatmasks significantly reduce transmission [17, 32–35].In addition to the probability of infection, the implications of a nonlinear dose-response curve apply to several otheroutcomes as well. In all of the above analyses, the probability of infection can be replaced with the probability ofdeath or the probability of a particular degree of severity of symptoms, each of which can have a unique S-curve (thatcan also vary from individual to individual). Thus, even when a mask does not prevent infection, it may reduce theseverity of symptoms and the chance of long-term health damage or death. It has been observed for the influenzavirus that increasing the viral dose may lead to more adverse symptoms [36–38], an effect that may also apply toSARS-CoV-2 [38–42].
CONCLUSIONS
Masks block some fraction of viral particles from dispersing from those who are infected and from infecting thosewho are susceptible and are understood to prevent disease transmission through this mechanism. However, this simpleunderstanding has been questioned based upon mixed empirical evidence. Here we have shown that studies that didnot find masks to be effective had limited statistical power and therefore do not imply that masks are ineffective. Theempirical evidence as a whole is thus consistent with current epidemiological recommendations to use masks duringthe COVID-19 pandemic.We note that psycho-social effects can reinforce the effect of prevention. The more individuals who wear masks,the less stigma that is associated with wearing them, which may make it more likely for others—including those whoare infectious (whether symptomatic, pre-symptomatic, or asymptomatic)—to wear masks. Masks may also serve asa reminder to take precautions that compound super-linearly with mask-wearing such as social distancing, althoughrisk compensation, in which mask-wearing provides a sense of security that leads to higher risk taking, is also possible.Effective public communication should emphasize that masks should be used in addition to and not as a substitutefor other precautionary measures.We have also shown that for many exposure events, masks will reduce the probability of infection by a greater factorthan the factor by which they filter viral particles. This effect is also compounded non-linearly when both infected andsusceptible individuals wear masks. When interpreted in light of this a priori reasoning and the other considerationsdiscussed above, the evidence indicates that, in addition to preventing the wearer from spreading respiratory infections,masks also protect the wearer from contracting them. The studies that did not find statistically significant effectsprove only that masks cannot offer protection if they are not worn.
Acknowledgements
This material is based upon work supported by the National Science Foundation Graduate Research FellowshipProgram under Grant No. 1122374 and by the Hertz Foundation. We thank Jeremy Rossman for helpful comments.
APPENDIX1. Accounting for non-linearities in the effectiveness of masks
In this section we develop a framework with which to understand the effect of masks. We show that even if maskswere to reduce the viral dose by only a modest factor, they may have a significantly larger impact on the probabilityof infection. We demonstrate that wearing a mask more frequently can super-linearly reduce one’s chance of infection(e.g. wearing a mask 80% of the time reduces one’s probability of infection by more than twice as much as wearinga mask 40% of the time). We also show that when both infected and susceptible individuals wear masks, there canbe a super-linear compound effect (e.g. if only infected individuals wearing masks reduces the probability of infectingsusceptible individuals by a factor of 3 and only susceptible individuals wearing masks reduces the probability ofbeing infected by a factor of 2, then if both wear masks the probability of infection will be reduced by a factor thatis greater than 2 × a. General Framework Although there is insufficient data to precisely describe the probability of infection as a function of the viral doseinhaled in a single exposure event, we can nonetheless derive some constraints on its shape. For a susceptible individual,the probability of infection (or any other outcome such as hospitalization or death) p is a function of the viral dose v ,i.e. the quantity of virus to which the individual is exposed. (This function p ( v ) will vary from individual to individualbased on biological factors, but should retain the general properties described below.) For small v the probability ofa susceptible individual becoming infected will approach zero, and for large v this probability will approach one, so p (0) = 0 and p ( ∞ ) = 1. Since receiving two viral doses at once should not result in a lower probability of infectionthan the hypothetical in which the exposure to each viral dose could be modeled as an independent event, we havethat p ( v + v ) ≥ p ( v ) + p ( v ) − p ( v ) p ( v ) (1)Equality will hold only in the absence of threshold effects; given that such effects are well established, we expect theinequality to be strict for small v and v . In order to characterize the set of functions satisfying eq. (1), we transform p ( v ) using p ( v ) ≡ − e − f ( v ) , or equivalently, f ( v ) ≡ − ln(1 − p ( v )). Eq. 1 is then equivalent to f ( v + v ) ≥ f ( v ) + f ( v ) (2)Thus eq. (1) is equivalent to f ( v ) being convex. Choosing a convex f ( v ) and then transforming back to p ( v ) yieldsan S-curve (fig. 3), also known as a sigmoid function or sigmoid curve.When f ( v ) (cid:28)
1, it can be shown by Taylor expansion that p ( v ) ≈ f ( v ). Thus, for small viral doses, p ( v ) will beconvex as well. If a mask reduces the viral dose v by a factor b [6, 7], then the mask will reduce the probability ofinfection (or of some other outcome denoted by p ( v ) such as the probability of severe infection or death) by a factorof p ( v ) p ( v/b ) , which depends on v . When p ( v ) is convex, the factor by which the mask reduces the probability of infectionwill be greater than b (since convexity implies that p ( v/b ) < b p ( v ) + (1 − b ) p (0) = b p ( v )). Thus, for small exposures,masks can result in a surprisingly large reduction in the probability of infection. We treat the impact of masks inmore generality below, after introducing a framework for considering multiple exposure events.The S-curve describes the probability of infection for a single exposure event. For N independent exposure events,the probability of getting infected is p inf = 1 − (cid:81) Ni =1 (1 − p ( v i )). Using the form p ( v ) = 1 − e − f ( v ) as discussed above, p inf = 1 − e − (cid:80) Ni =1 f ( v i ) (3)Defining the effective exposure ˜ v ≡ f ( v ) and defining ˜ p (˜ v ) ≡ − e − ˜ v , p inf = ˜ p (˜ v T ) (4)where ˜ v T = (cid:80) i ˜ v i is the total effective exposure. Considering the effective exposure ˜ v rather than the actual dose v is convenient since the effective exposure for repeated independent exposures is simply the sum of the individualeffective exposures. Note that for small effective exposures, the probability of being infected is approximately equalto the effective exposure, i.e. ˜ p (˜ v ) ≈ ˜ v for ˜ v << b. One mask Let γ be the typical amount by which a mask reduces the effective exposure from a single exposure event—i.e.˜ v → (1 − γ )˜ v .Since f (0) = 0 and f ( v ) is convex, the simplest possible expression for f ( v ) is the scale-free form f ( v ) = ( v/v ) β ,for some v > β >
1. In this case, if a mask reduces the viral dose v to v/b , γ can be calculated exactly as γ = 1 − b − β , regardless of v . For small exposures, the infection probability is roughly equal to the effective exposure,which is reduced by a factor greater than b (i.e. − γ > b ) due to convexity ( β > β = 4, a mask filtering half of the viral particles ( b = 2)corresponds to a sixteen-fold reduction in effective exposure ( γ ≈ . f , then γ becomes an effective parameter that may depend on the distribution of viral doses to which an individualis exposed. Regardless of the precise form of f ( v ), however, − γ > b will always hold due to the convexity of f ( v ),i.e. masks will always have a disproportionately large effect on the effective exposure (and thus also on the infectionprobability when the effective exposure is small).Then, if a mask is worn for a fraction α of all exposures, the total effective exposure will be reduced from ˜ v T to(1 − αγ )˜ v T . The probability of infection is thus˜ p ((1 − αγ )˜ v T ) = 1 − e − (1 − αγ )˜ v T (5)(see fig. 2).Thus, we see that for any fixed γ (mask effectiveness) and ˜ v T (total effective exposure without a mask), the benefitof wearing a mask is a convex function of the fraction α of the exposure events for which it is worn. In other words,wearing a mask x times as often will reduce the infection probability by more than a factor of x . Thus, even if maskswere 100% effective ( γ = 1), a study in which participants wear masks 10% of the time would need to have sufficientpower to detect less than a 10% reduction in the probability of infection. Our analysis therefore overestimates thetrue power of the studies. c. Two masks To the extent that two masks together have an approximately linear effect on the effective exposure (e.g. if oneperson wearing a mask reduces effective exposure by 1 − γ and the second person wearing a mask reduces effectiveexposure by 1 − γ , then both wearing masks reduces effective exposure by 1 − γ ≈ (1 − γ )(1 − γ )), the effect onthe probability of transmission will be super-linear, since the probability of infection ˜ p (˜ v ) is concave in the effectiveexposure ˜ v . In other words, especially for individuals who would have received a large total effective exposure withoutmasks, both the susceptible and infectious individuals wearing masks will have a larger effect than would be calculatedif each mask had an independent effect on the probability of transmission.If the effect of the two masks on the effective exposure is super-linear (i.e. 1 − γ < (1 − γ )(1 − γ )), then the effecton the probability of transmission will be super-linear to an even greater extent. If the effect of the two masks on theeffective exposure is sub-linear (i.e. 1 − γ > (1 − γ )(1 − γ )), then whether or not they still have a super-lineareffect on the probability of transmission will depend on the total effective exposure.(Note: Under the simplest possible form for ˜ v = f ( v ), i.e. f ( v ) = ( v/v ) β , if the mask on the infected individualreduces v by a factor of b , the mask on the susceptible individual reduces v by a factor of b , and together the masksreduce v by a factor of b b , then the masks will have a linear effect on effective exposure, i.e. 1 − γ = (1 − γ )(1 − γ ).Under other forms for f ( v ) or assumptions about how the masks affect v , other behavior is possible.)
2. Power analyses
Let p and p be the probabilities of getting infected in the non-mask (size N ) and mask group (size N ), respec-tively. Defining (cid:15) = p − p , the null hypothesis is H : (cid:15) = 0 and the alternate hypothesis is H : (cid:15) (cid:54) = 0. A teststatistic is W = ˆ p − ˆ p (cid:112) ˆ p (1 − ˆ p ) /N + ˆ p (1 − ˆ p ) /N = ˆ p − ˆ p ˆ s (6)where ˆ p and ˆ p refer to the observed fraction of infections, assumed to be normally distributed random variableswhose means are p and p (this approximation is asymptotically exact). We use the shorthand ˆ s for the denominatorof W ; note that ˆ s is an estimator for s , where s = p (1 − p ) /N + p (1 − p ) /N is the sum of the asymptoticvariances of ˆ p and ˆ p . Asymptotically, W − (cid:15)/s follows a standard normal distribution. Using the standard notationΦ( z − α/ ) = 1 − α/ x ) is the standard normal cumulative distribution function, the rejection region under H for a significance level of α is given by the union of W < − z − α/ and W > z − α/ (7)The various studies may use slightly different statistical tests, but the differences between tests should be small and willasymptotically disappear entirely. For any particular values of (cid:15) and s , the probability W < − z − α/ is asymptoticallygiven by Φ( − z − α/ − (cid:15)/s ) and the probability W > z − α/ is asymptotically given by 1 − Φ( z − α/ − (cid:15)/s ) =Φ( − z − α/ + (cid:15)/s ). Thus, given (cid:15) and s , the power, denoted by 1 − β and equal to the probability that the nullhypothesis is rejected if it is indeed false, is asymptotically given by1 − β = Φ( − z − α/ − (cid:15)/s ) + Φ( − z − α/ + (cid:15)/s ) (8)Under the assumptions that masks are fully effective ( γ = 1) and that the probability of infection p inf decreaseslinearly with the adherence, the effect of mask usage is p inf → p inf (1 − a ) (9)where the adherence a is the average fraction of exposure events for which the masks were used (see section 1 of theAppendix; here we use a instead of α for the adherence to avoid confusion with the significance level). Thus, for aninfection probability p inf = p in the non-mask group (size N ), the infection probability in the mask group (size N )will be p = p (1 − a ). Thus, by estimating p and a for each study, we can use eq. (8) to find power of each studygiven the sizes of their non-mask and mask groups, as well as the sample size (i.e. total number of participants) thatwould have been required for 80% power. For the latter estimate, we assume a study design in which the participantsare evenly divided between the non-mask and mask groups (i.e. N + N = 2 N = 2 N ) and rounded up the necessarysample size to the nearest even integer. No. Name Year Mask Use Adhe-rence Size of non-mask group Fraction of non-mask group infected Size of face mask group Fraction of face mask group infected Statisti-cal Power Required sample size for a power of 0.8 Actual sample size Primary Outcome Significant reduction in infections in mask group?(I) (II) (III) (IV) (V) (VI) (VII) (VIII) (IX) (X) (XI) (XII) (XIII) (XIV)1 Cowling (ITTA) [45] 2008 0.37 205 0.06 61 0.07 0.118 2910 266 Antibody Test No2 MacIntyre (ITTA) [19] 2009 45% people used masks 0.36 100 0.030 94 0.064 0.078 6444 194 ILI No3 MacIntyre (PPA) [19] 2009 Less than 2/5 days 0.32 170 0.053 19 0.211 0.066 4636 189 ILI No4
MacIntyre (PPA) [19]
Barasheed (ITTA) [50]
Sung (pre-post) [51]
Choudhry (survey) [52]
Al-Jasser (survey) [53]
Suess (ITTA) [54]
Wu (survey) [55]
Kim (survey) [56]
Lau (survey) [57]
Lau (survey) [58]
Wu (survey) [59]
FIG. 5. Summary of statistical power analysis. Given the adherence levels reported in the studies the sample size necessary fora statistical power of 80% for a two-tailed test and significance level of 0 .
05 (assuming participants are equally divided betweenthe non-mask and mask groups) is presented in column XI. The statistical power given the actual sizes of the non-mask andmask groups is presented in column X. These calculations were made for the case in which masks are 100% effective; if masksare effective but not perfectly so, the necessary sample sizes for 80% power (column XI) will be larger, while the statisticalpowers given the actual sample sizes (column X) will be lower. Studies found to have greater than 80% power are in bold(nos. 15-23), and studies that found a statistically significant reduction in infections in the mask group are italicized (nos. 4,14-23). Adherence is defined as the fraction of exposure for which masks were used; calculations of adherence for each study arepresented in table I. For studies that reported multiple analyses, each analysis is listed as its own entry (e.g. Aiello (2012) [43]performed one analysis in which infection is defined by influenza-like illness (no. 10) and one analysis in which infection isdefined by a positive PCR test result (no. 11)). Note that only in the intention-to-treat analyses are participants randomlydivided between the non-mask and mask groups; in survey and per-protocol analyses, which group a participant belongs todepends on whether or not that individual reported wearing a mask with a frequency above a threshold decided by the study.
Abbreviations: ITTA : Intention-to-treat analysis;
PPA : Per-protocol analysis;
ILI : Influenza-like illness;
ARI : Acuterespiratory infection;
URTI : Upper respiratory tract infection;
PCR : Polymerase chain reaction test (nasopharyngeal swabtest);
SARS : Severe Acute Respiratory Syndrome p is related to the probabilityof infection without masks p by p = p (1 − γa ) where a is the adherence in the non-mask group and γ is maskeffectiveness. Then the probability of infection in the mask group will be p = p (1 − γa ) where a is the adherencein the mask group. The net adherence a is defined by p = (1 − γa ) p , which yields a = a − a − γa . In our analyses weassume γ = 1, which leads to an overestimate for the net adherence a of a = a − a − a (10)We estimate p using the observed fraction of infections in the non-mask group ˆ p . To check the robustness ofour conclusions, we did a sensitivity analysis and found that if ˆ p differs from p by a standard deviation (i.e. if weincrease our estimate of p by (cid:113) N ), all studies that were under-powered ( < (cid:113) N as the standard deviation, which is the maximum possible value ofthe true standard deviation (cid:112) p (1 − p ) /N .) If ˆ p underestimates p by two standard deviations, another study [44]would have greater than 80% power under our assumptions. It should be noted, however, that these assumptionsoverestimate the power in multiple ways (fully effective masks, overestimated adherence values, assuming a linearrelationship between adherence and effectiveness, and the fact that individuals whose infections were not detecteduntil after the start of the study could have actually been infected before they start of the study, i.e. before the maskintervention was implemented).A more significant limitation of our analysis is in the difficulty in estimating adherence. Adherence is often reportedqualitatively, and even when quantitative, it is reported as the amount of time for which one wears a mask, whichmay differ from the fraction of exposures for which masks were worn. To account for this difficulty, our strategy hasbeen to consistently overestimate statistical power; to this end, we have erred on the side of overestimating adherence(see table I), and have also used other overestimating assumptions described in the previous paragraph. TABLE I: Adherence calculations for each study.
No. Name Year Masks used by Description and calculation1 Cowling(ITTA) [45] 2008 Infected pa-tients and theircontacts Household study: 45% of 21 index cases used masks and 21% of 61 contacts wore masks. To overestimateadherence, we assume no transmission occurs while either the index patient or contact is wearing a mask.Neglecting correlations between whether or not the index patient wore a mask and the number of contactsof that index patient, an upper bound for the probability that either a contact or the index patient corre-sponding to that contact used a mask is 45%+21% = 66% (this is likely an overestimate since householdsin which index patients wear masks and households in which contacts wear masks are almost certainly notmutually exclusive). In the control group, 30 % of index patients and 1 % of contacts used masks. Thoseclassified as using masks used them often or always; therefore we assume that they used masks for 80%of all exposures, a likely overestimate since the participants were asked to use masks only when they arenot sleeping or eating. Therefore, the adherence in the mask group is estimated as 0 . × . .
53, andadherence in the control group is estimated as 0 . × . .
25. This leads to a net adherence of 0.37according to eq. (10).2 MacIntyre(ITTA) [19] 2009 Contacts of in-fected patients Household study over 5 days: Contacts were told to use masks when in the same room as the index patient.We consider only the surgical mask group (the other group was using P2 masks). On day 3, maximumadherence was reported: 45% of contacts used masks for most of the time. We assume that those whoused masks used them for 80% of exposures, a likely overestimate since contacts did not use masks whilesleeping, even if the child (index patient) was next to them in bed, and because the contacts could havebeen infected even if they were not in the same room as the index patient. The adherence is estimated as0 . × . .
363 MacIntyre(PPA) [19] 2009 Contacts of in-fected patients Household study over 5 days (see row no. 2): Participants in this arm of the per-protocol analysis usedmasks for < / × . . / × . . . /
24 = 0 . . /
24 = 0 . .
014 (eq. (10)). Notethat the systematic review [18] uses an older pre-print version of this study.6 Alfelali(PPA) [24] 2020 Susceptibleindividuals Hajj study (see row no. 5): Those who wore masks were compared to those who did not. The averagemask use among those who wore masks was 1.637 hours; thus adherence = 1 . /
24 = 0 . . × . = 0 .
23 (see row no. 1 for why the index andcontact mask usages were added together), a likely overestimate, given that the majority of the householdsresided in small one-bedroom apartments and thus were likely in contact for significantly greater than 10.4hours per day on average. Furthermore, contacts could have been infected outside of their homes. Also, itwas reported that 17.6% of individuals in the control group used masks, meaning it was likely that thosein the hand-hygiene-only group did as well (which would further reduce the net adherence).8 Canini(ITTA) [48] 2010 Infectedpatients Household study: Average mask use was 3 . . . = 0 .
38, a likely overestimate given that contacts couldhave been infected outside their homes, or in their homes while not in contact with the index patient.9 Aiello(ITTA) [44] 2010 Susceptibleindividuals University residence hall: Mask usage was recorded inside the residence hall and they were used for 3 . . − = 0 .
33, a likely overestimate because participants were only encouragedbut not required to use masks outside the residence halls, where they may be infected. In addition, theparticipants had left the residence halls for spring break, during which they were not required to wearmasks.10 Aiello(ITTA) [43] 2012 Susceptibleindividuals University residence hall: Masks were used for 5 .
08 hours per day. Adherence = . − = 0 .
42 (see rowno. 9).11 Aiello(ITTA) [43] 2012 Susceptibleindividuals University residence hall: Masks were used for 5 .
08 hours per day. Adherence = . − = 0 .
42 (see rowno. 9).12 MacIntyre(ITTA) [47] 2016 Infectedpatients Household study: In the mask group, index patients were in contact with contacts for an average of 10.4hours, and used masks for an average of 4.4 hours. The adherence in the mask group is thus estimated as . . = 0 .
42. In the control group, average mask usage was 1.4 hours; adherence in the control group isthus estimated as . . = 0 .
13. Net adherence is thus 0 .
33 (eq. (10)).13 Cowling(ITTA) [49] 2009 Infected pa-tients and theircontacts Household study: We compare the hand-hygiene group with the hand-hygiene + mask group. In the hand-hygiene + mask group, 49% of index cases and 26 % of contacts used a mask often or always. We thereforecalculate adherence in the hand-hygiene + mask group as (0 .
49 + 0 . × . .
60 (see row no. 1). Inthe hand-hygiene group, 5 % of contacts and 31 % of index cases used masks, which leads to an adherence= (0 .
31 + 0 . × . .
29 in the hand-hygiene group. Net adherence is thus 0 .
44 (eq. (10)).14 Barasheed(ITTA) [50] 2014 Susceptibleindividuals Hajj pilgramage: 36 people were in the face mask group: 8 people never used a mask; 11 people used masksfor < > > (8 × /
24 + 11 × /
24 + 8 × /
24 + 9 × /
24) = 0 . .
816 Choudhry(Survey)[52] 2006 Susceptibleindividuals Survey study for Hajj pilgrims: We consider the group of male pilgrims who reported using masks most ofthe time, compared to a group who did not use masks. We assume that masks were not used while sleepingor eating, and note that the pilgrims remain susceptible to infection during such activities since they sleptin shared tents. Allotting 10 hours per day for sleeping and eating and other activities during which maskswere not worn, we estimate the adherence as 14 /
24 = 0 . × . . . TABLE II. Studies not included in power analysis.
Name Year Reason for exclusion from power analysisShin [60] 2018 Study was randomized for testing a common cold drug rather than mask usage, and mask usage wascomparable in both of the groups.Zhang [61] 2013 Unknown adherence and incomplete data.Jolie [62] 1998 Animal to human transmission: We consider only human to human transmission for our analysis.Tahir [63] 2019 Animal to human transmission: We consider only human to human transmission for our analysis.Larson [64] 2010 Mask adherence was reported to be ‘poor’ but neither the percentage of participants using masks nor theduration of mask usage was reported, so we could not make an estimate for the adherence.Emamian [65] 2013 Survey study for Hajj pilgrims: Adherence for mask usage was reported only as ‘Yes’ or ‘No’. Evenoccasional use of mask was considered as ‘Yes’. Since adherence data stratified by frequency and/orduration was not reported, we could not make an estimate for the adherence.Deris [66] 2010 Survey study for Hajj pilgrims: Adherence for mask usage was reported only as ‘Yes’ or ‘No’. Sinceadherence data stratified by frequency and/or duration was not reported, we could not make an estimatefor the adherence.Uchida [67] 2017 Survey study for children. Mask usage was reported as ‘using masks at any time or place’. Since adherencedata stratified by frequency and/or duration was not reported, we could not make an estimate for theadherence.Balaban [68] 2012 Survey study for Hajj pilgrims: Adherence for mask usage was reported only as ‘Yes’ or ‘No’. Sinceadherence data stratified by frequency and/or duration was not reported, we could not make an estimatefor the adherence.Zein 2002 Study not available. [1] Goodman, B. The forgotten science behind face masks (2020). URL . webmd . com/lung/news/20200826/the-forgotten-science-behind-face-masks . Accessed 2021-02-04.[2] Considerations for wearing masks. CDC (2020). URL . cdc . gov/coronavirus/2019-ncov/prevent-getting-sick/cloth-face-cover-guidance . html .[3] Coronavirus disease (covid-19) advice for the public. WHO (2020). URL . who . int/emergencies/diseases/novel-coronavirus-2019/advice-for-public .[4] Milne, R. & Khan, M. Coronavirus outlier sweden chooses its own path on face masks. Financial Times (2020). URL . ft . com/content/3148de6c-3b33-42d3-8cf6-d0e4263cea82 .[5] What countries require public mask usage to help contain covid-19? (2020). URL https://masks4all . co/what-countries-require-masks-in-public/ . Accessed 2021-02-04.[6] Booth, C. M., Clayton, M., Crook, B. & Gawn, J. Effectiveness of surgical masks against influenza bioaerosols. Journalof Hospital Infection , 22–26 (2013). URL https://doi . org/10 . . jhin . . . .[7] Gawn, J., Clayton, M., Makison, C. & Crook, B. Evaluating the protection afforded by surgical masks against influenzabioaerosols: gross protection of surgical masks compared to filtering facepiece respirators. Health Safety Exec (2008). URL https://europepmc . org/article/ctx/c3304 .[8] Lindsley, W. G. et al. Dispersion and exposure to a cough-generated aerosol in a simulated medical examination room.
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