Density variability of COVID-19 face mask litter: A cautionary tale for pandemic PPE waste monitoring

Despite the requirement for data to be normally distributed with variance being independent of the mean, some studies of plastic litter, including COVID-19 face masks, have not tested for these assumptions before embarking on analyses using parametric statistics. Investigation of new data and secondary analyses of published literature data indicate that face masks are not normally distributed and that variances are not independent of mean densities. In consequence, it is necessary to either use nonparametric analyses or to transform data prior to undertaking parametric approaches. For the new data set, spatial and temporal variance functions indicate that according to Taylor's Power Law, the fourth-root transformation will offer most promise for stabilizing variance about the mean.


Introduction
Surveys undertaken to assess the magnitude of abandoned waste, such as plastic pollution, often wish to compare the standing-stock or the accumulation rates of litter between or among different locations and contrasting times. Determining the spatial and temporal uncertainty associated with estimates of mean values of environmental variables is essential to the scientific treatment of sampling design and data interpretation ( France 1988 ;France and Peters 1992 ). Standard sampling theory is based on preliminary examination of data to ensure that they follow a normal distribution, and that the variance of the sample is independent of the mean ( Elliott 1979 ;Green 1979 ). If the distribution of data satisfies the assumption of normality, then parametric statistics can be used; if not, then it becomes necessary to either revert to equivalent nonparametric statistics or to subject the data to some form of transformation to stabilize the variance (i.e., eliminate the correlation between variances and means), prior to parametric statistical analysis ( Taylor 1961 ). Despite this, however, even the most cursory perusal of the published literature on plastic pollution indicates that it is not uncommon for this necessary step to be overlooked. Instead, many researchers immediately proceed to performing parametric analyses such as linear regressions and analysis of variance and the like, even though conclusions reached in such situations may be suspect at best and can be erroneous at worst ( Snedecor and Cochran 1968 ). Unfortunately, these problems have been repeated in some recent surveys of face mask litter associated with the COVID-19 pandemic. Overviews concerning the monitoring of plastic debris in general (e.g., Velander and Mocogni 1999 ;Ryan et al. 2009 ;Smith and Markic 2013 ), and pandemic personal protective equipment (PPE) waste in particular ( Benson et al. 2021 ;Kurtralam-Muniasamy et al. 2022 ;Mvovo and Magagula 2022 ;Roberts et al. 2022 ;Shruti et al. 2022 ;Silva et al. 2021, have, if they explore issues related to sampling programs at all, focused on protocols for the collection of litter rather than on the subsequent analysis of the ensuing data. This, despite that cautionary tales concerning the mathematical treatment of data are known to be important for refining the interpretation of environmental phenomena (e.g., France 2011 ). In terms of pandemic PPE waste, Cueva's (2022) consideration of how temporal variability affects sampling programs is an exception.
This study specifically examines the issue of statistical variability about estimates of the mean density of abandoned face masks, from which a cautionary tale is crafted regarding future analysis protocols. This paper therefore addresses the stated mandate for this special issue in terms of solicitating contributions that specifically focus on "methodological and analytical advancements on pandemic waste monitoring " ( Silva and Walker 2022 ).  France et al. 2019 ). Sampling took place during the overwinter period from November 2021 to April 2022, at a time when the wearing of face masks was compulsory for admittance to all public buildings in the province. Further study details will be provided in a subsequent paper on pandemic waste. For the present methodological contribution, it suffices to note that the entire surface area of each parking lot as well as its immediate surrounding fringe (combined areal extent determined by GIS analysis) was completely scoured of all observable face masks (disposable surgical, damaged, and reusable cloth combined) for each collection period such that the abundance of litter could be expressed on a per areal (m − 2 ) basis for the analyses described below. Fifty parking lots, ranging in size from 1,386 to 54,383 m 2 , of a diverse typology (grocery stores, shopping malls, health centres, schools, recreation centres, varied single shops, automobile service stations, restaurants, offices, and hotels) were surveyed. All buildings associated with the parking lots were open throughout the duration of the study. Four parking lots were sampled weekly ( n = 25 times); 9 lots were sampled monthly ( n = 7 times); and 41 lots were sampled bimonthly ( n = 4 times), in addition to the initial collection period, for the half-yearlong survey. This represents one of the most extensive sampling efforts undertaken to assess the extent of PPE littering in relation to the COVID-19 pandemic. Standard statistical methods were employed for analyses of linear regression equations, Spearman's rank correlation coefficients, and Shapiro-Wilk and Chi-squared distributional tests ( Snedecor and Cochran 1968 ;Elliot 1979 ) and for data transformations ( Taylor 1961 ), all calculated through use of established online programs (socscistatistics.com, calculator.net, calculatorsoup.com).

Results and discussion
Examination of the shape and nature of frequency distributions of environmental variables and monitoring parameters through preliminary statistical analysis is an important step toward validating the use of appropriate mathematical procedures ( Heyman et al. 1984 ;Giovanardi et al. 2006 ). In the present case, with both its demonstrated predominance of low counts and skewness or bilateral asymmetry of high counts, the overall frequency distribution of face mask densities in Truro parking lots is clearly different from the traditional bell-shaped curve characteristic of normally distributed data ( Fig. 1 ). A Shapiro-Wilk test confirmed the significant departure from a normality (W = 0.80; p < 0.001). As a result, further analyses of these data require the use of either nonparametric statistics or the need for transformation before parametric tests are undertaken. The lack of concordance with the Poisson distribution (Chi-squared test, p < 0.001), wherein variance is approximately equal to the mean ( Elliott 1979 ), indicates that COVID-19 face masks are dispersed nonrandomly; in other words, the litter is aggregated or follows some form of contagious distribution. A suite of these distributions exists for aggregated data depending on the increasing degree of skewness in the frequency counts; for example, the Thomas, Neyman Type A, Poly-Aeppli, negative binomial, and discrete log-normal distributions ( Elliott 1979 ). In the present case, the absence of decreased frequencies at the lowest densities and lack of concordance with the negative binomial distribution (Chi-squared test, p < 0.01), wherein the variance is much larger than the mean ( Elliott 1979 ), suggests that use of a log transformation of face mask data to remove the dependence of variance on the mean (see below), would be too severe, and that some intermediate transformation to stabilize variance could often be more appropriate ( Downing 1979 ;France 1987 ).
The present finding agrees with some recent surveys of COVID-19 face mask litter. Although frequency plots were not presented, authors likewise noted that their data significantly invalidated the normal distribution Haddad et al. 2021 ;France 2021 ;De-la-Torre et al. 2021 ;Riberio et al. 2022 ;Gunasekaran et al. 2022 ). As a result, further analyses in these studies correctly used nonparametric statistics, such as, for example, the Kruskal-Wallis, Mann-Whitney U, and Dunn's multiple comparison tests, Spearman's rank correlations, and permutational analyses of variance. In contrast, almost twice the number of studies of COVID-19 face mask litter did not explicitly mention testing for normality. Moreover, several of these studies nevertheless proceeded to employ parametric statistics.
Transformation of data to stabilize variance will also alleviate skewness in the distribution as well as increase the probability of variance additivity, both of which are key assumptions for analysis of variance and linear regression ( Snedecor and Cochran 1968 ). The choice of a particular transformation is dependent on the variance to mean ratio, and as such, is a function of the distribution of the data. Because both spatial and temporal variance are almost always proportional to a fractional power of the magnitude of variables ( France 1988 ;France and Peters 1992 ), Taylor's Power Law ( Taylor 1961 ) states that log s 2 is linearly and positively related to log x . The slope of this relationship or "variance function ", b , is an index of aggregation, which, because it incorporates density dependence, allows the relative degree of dispersion of samples of different magnitude to be compared through a single mathematical expression. The larger the value of b , the greater the degree of contagion in the data (i.e., s 2 >> x ). The situation of b = 1 is the special case of the random or Poisson distribution ( s 2 = x ), whereas b = 2 indicates a log-normal distribution ( Elliott 1979 ;Downing 1979 ). The exact variance stabilizing function can be calculated once the index of contagion is known ( Taylor 1961 ;Perry and Woiwod 1992 ): X' = X 1-(b/2) where X is, in this case, the density of face masks per square metre; b is the power function exponent; and X' is the transformed datum. Other transformations that have proven useful in studies of zoobenthos abundance include those of the square-root ( b = 1, 1 -b /2 = 0.5) and the fourth-root ( b = 1.5, 1 -b /2 = 0.25; France 1987 ).
Because uncertainties exist with respect to how sampling interval may affect the variability of litter abundance ( Ryan et al. 2009 ;Smith and Markic 2013 ;Cueva 2022 ), data for the three collection schedules were kept separate for variance function analyses. The regressions describing the spatial variance of mean face mask density for the present study ( Fig. 2 ) were as follows: log s 2 = 2.84 + 1.40 (log x ), r 2 = 0.79, n = 4 for lots sampled bi-monthly; log s 2 = 0.75 + 1.68 (log x ), r 2 = 0.97, n = 7 for lots sampled monthly, and log s 2 = -1.40 + 1.72 (log x ), r 2 = 0.60, n = 25 for lots sampled weekly. The regressions describing the temporal variance of mean face mask density were as follows: log s 2 = -0.13 + 1.33 (log x ), r 2 = 0.70, n = 41 for the bi-monthly sampled lots, log s 2 = 1.16 + 1.36 (log x ), r 2 = 0.82, n = 5 for the monthly sampled lots, and log s 2 = -8.89 + 2.75 (log x ), r 2 = 0.91, n = 4 for weekly sampled lots. The average aggregation index, b , was 1.60 among different parking lots sampled at the same time, and 1.81 among the same lots sampled at different times. According to Taylor's Power Law ( Taylor 1961 ;Perry 1987 ), these variance functions indicate that the fourth-root transformation could offer promise for stabilizing variances in relation to mean densities of COVID-19 face masks in this survey.
Studies of zoobenthos aggregation have suggested that the log transformation is on average too extreme in that it tends to over-transform data, whereas the square-root transformation under-transforms data, and that a compromise, the fourth-root transformation, is the most useful to apply routinely ( Downing 1979 ). This is a conclusion that has led a lively discussion in the literature ( Taylor 1980 ;Chang and Winnell 1981 ;Downing 1981 ), much of it concerning perennial differences of opinion concerning the mutually exclusive objectives of precision vs. generality in sampling design ( Downing 1980 ;France 1987 ), as well as problems that can sometimes arise when applying the Power Law in situations characterized by the presence of very low densities ( Riddle 1989 ;Routledge and Swartz 1991 ).
Re-examination of the published literature on densities of littered COVID-19 face masks indicates the generality of the present findings by demonstrating that sample variability is regularly not independent of mean density. The first suggestion of this is shown by an analysis of data from 12 studies in which both means and ranges of densities were presented. Here, a very strong cross-study correlation (Spearman's rank coefficient = 0.91, p = 0.00004) was found between means and the differences between mean and maximum densities. In short, among studies with mean face mask densities between 0.01 and 8.0 × 10 − 3 m − 2 , those data sets having higher mean densities also displayed correspondingly higher variability about those means. Secondary analyses on density variability within individual studies could be performed on 22 sets of data presented in 18 papers ( Akhbarizadeh et al. 2021 ;Ammendolia et al. 2021 ;De-la-Torre et al. 2021 ;France 2021 ;Haddad et al. 2021 ;Rakib et al. 2021 ;Thiel et al. 2021 ;Abedin et al. 2022 ;Amuah et al. 2022 ;Cueva 2022 ;De-la-Torre et al. 2022 ;Gunasekaran et al. 2022 ;Hassan et al. 2022 ;Mghili et al. 2022 ;Ribeiro et al. 2022 ;Sajorne et al. 2022 ;Tesfaldet et al. 2022 ). In all cases, various diverse measures of data variability, such as standard deviation (SD), standard error (SE), SD range, boxplot range, overall range, unspecified error bar range, were found to be positively correlated with mean densities, with significant relationships (Spearman's rank correlation coefficients) existing in fully 16 of the cases ( Fig. 3 ). The number of paired mean and variability metrics in these studies ranged from 4 to 25. In four other studies in which there were only three sets of paired metrics, there was a perfect concordance between higher values of both, whereas in a fifth study the concordance was high, and in a sixth study the concordance was low. Finally, in two further studies in which there were only two pairs of metrics, in both cases higher mean densities corresponded to higher variability about those means. The present sampling program collected face masks from the entire surface of parking lots of variable size, whereas many of the other studies used a form of standardized sampling of small areas -such as line transect counts -deemed to be spatially representative of the larger landscape. The fact that similar codependencies between means and variability exist in all these different sampling scenarios and physical situations suggests the generality of the findings.
In summary, the present investigation of new data and secondary analyses of published literature data indicates that COVID-19 face masks are frequently distributed in a non-random manner throughout many different landscapes around the world, and that variances are dependent upon mean densities. Therefore, it is necessary, should further analyses be undertaken for comparative purposes, to use either nonparametric statistics or to undertake transformations of the data -such as that of the fourth-root -prior to undertaking parametric procedures. The fact that almost a dozen recently published papers on pandemic PPE waste discuss and compare differences in densities of littered face masks between or among different collection sites or times without first examining the data for normality, and sometimes doing so through the employment of statistics that may be inappropriate for that particular data set, is a practise that should be discouraged. This is not the first paper to flag what Mahmoudnia et al. (2022) refer to "common mistakes, " which have been repeated in recent COVID-19 articles.
Our results also have relevance to monitoring surveys of non-COVID-19 related plastic litter. Although this published literature contains many examples in which researchers did not undertake appropriate data screening and transformation procedures, it is important to flag a handful of representative studies that did do so, thereby providing a wider context in which to situate the present findings. For example, due to high variability about the means, Detrot et al. (2013) and Ryan et al. (2020) determined their respective data on litter density to be either log-normally or quasi-Poisson distributed, thereby necessitating the use of nonparametric statistics. Other studies ( Poeta et al. 2014 ;de Ramos et al. 2021 ;Compa et al. 2022 ) also found that their data were nonrandomly distributed, such that nonparametric tests needed to be used.
Concerning the use of transformation to stabilize variance, Rech et al. (2015) and Goncalves et al. (2020) respectively employed the square-root and the fourth-root transformations on their data. In general sampling theory, the closer the coefficients of determination, r 2 , between means and variances are to zero, the more efficient the variance stabilizing function. For the present 6 analyses (three of which are shown in Fig. 2 ), as well as for another 5 secondary analyses performed on similar data presented but not examined in like fashion in previously published studies, although the fourth-root transformation did reduce r 2 in all but one case, it did so by only by 10-20%. In some cases, this was enough to stabilize variances about the means (i.e., by making the slopes between them nonsignificant); in other cases, however, the fourth-root transformation was insufficient to accomplish this. The important point is that each data set is idiosyncratic and needs to be examined individually. For example, in their survey of beach debris, Becherucci et al. (2017) also found several cases where data transformation failed to remove the dependence of variability on the means. This is not surprising, given that there are many examples in zoobenthos surveys where the theoretically optimal transformations failed to uncorrelate means and variances ( France 1987 ). Nevertheless, any reduction in the coefficients of determination between means and variances, even if minor such as going from r 2 values of 0.80 to 0.60 (i.e., a 20% reduction), is an improvement that goes toward increasing assurance in conclusions derived from subsequent parametric analyses ( Snedecor and Cochran 1968 ). It is important to recognize that Taylor's Power Law is not infallible and that sometimes other equations may work better (see exchange in Routledge and Swartz 1991and Perry and Woiwod 1992.
Until more data become available on the spatial and temporal variability of PPE waste - Cueva's (2022) and the present studies are first steps in this direction -our advice is to first assess if the data being investigated are normally distributed. If, as is likely to be the case, that they are not, then the safest approach will be to use nonparametric procedures in subsequent analyses. If it is desirable that parametric tests be employed, then the data should be transformed, either by defaulting to the general fourth-root transformation used here or by calculating the bespoke variance function for the specific data set. Importantly, the results should be, as Becherucci et al. (2017) suggest, "interpreted with caution " (our italicized emphasis). This gives credence to the subtitle of the present contribution. Nor, when it comes to considerations of variance about the mean in sampling theory, is the presence of the word "caution " or its derivatives in the subtitle of a paper to be either the first ( Riddle 1989: Downing 1989, nor the last time that this is done.

Funding
None.