S 9. Supplement

S 9.1. Limits of detection and quantification

The chemical analysis procedure established the detection and quantification limits to create a min-imum signal-to-noise ratio of 3 and 10, respectively. This resulted in the limits presented in Table S.1.The amount of censoring for each municipality-drug combination is in Table S.2.

Meth MDMA Benzoylecgonine Methadone Hydrocodone Oxycodone

LOD 1.5 1.0 1.0 2.0 2.0 2.0LOQ 10.0 2.5 10.0 2.5 2.5 2.5

Note: Meth = methamphetamine

Table S.1: Detection and quantification limits (ng/L)

S 9.2. Sampled WWTPs

The 19 treatment plants contributing data for this analysis represent a convenience sample of diversesystems in the Northwest United States, specifically in the states of Washington and Oregon. The givenWWTP names are a mix of neighborhood names, city names, or collective names or otherwise convenientlabels that may or may not accurately reflect the catchment area served. We use the term “municipality”in general to indicate the WWTP and the area served. While most municipalities are along the mainnorth-south transportation corridor (US Interstate 5), several locations represent smaller cities or townsalong the coast or inland in the drier regions east of the Cascade mountain range. The relative populationsizes covered by each WWTP and their representative level of precipitation for 2009 are presented inFigure S.1. The final (average for Seattle and Renton; see Section 3.1) population estimates and thesource of those estimates, per the WWTP surveys, are in the second and third columns of Table S.2.

S 9.3. Sample data setup

Calculating a given sample’s index load requires four parameters: the measured concentration (column1 in Table S.3), the titration correction factor to account for titrating the sample to facilitate measurement(column 2), the day’s measured flow through the WWTP (column 3), and the day’s population estimate(column 6, constant in this example; see Section 3.1 for discussion of how this value might vary). The“Final Concentration” and “Final Load” columns (4 and 5) in the sample data setup are intermediatecalculations that represent the titration-corrected concentration and the product of this concentrationand the day’s volume.

For the third and fourth rows, the initial concentrations are censored. For estimating the mean indexload in the presence of censoring via a survival analysis method, the censoring must be represented asintervals to indicate the range the values could take. The final two columns, here labeled “Left” and“Right” (your software may prefer “Begin” and “End”, etc.), are what are inputted to the estimatingsoftware. For measurable observations, the two sides of this interval are the calculated index load. Forvalues below the level of detection (<LOD, where the LOD in the example is 1.5), the left side of theinterval is 0 and the right side is the LOD×titration factor×flow÷population÷(10002), to reflect thatthe initial concentration must be in the range of 0 to 1.5. For values below the level of quantification(<LOQ, where LOQ here is 10.0), we only know the initial concentration is between the LOD and theLOQ, and so the two sides of the interval reflect these values multiplied through: E.g., LOQ×titrationfactor×flow÷population÷(10002).

S 9.4. Robust retransformation

After maximum likelihood estimation, implemented with 50–80% censoring, we apply a robust retrans-formation to avoid retransformation bias as described by Helsel (2012), Kroll and Stedinger (1996), andShumway, Azari, and Kayhanian (2002). In order to create the distribution graphs in Figures 1, S.2, andS.3, we also apply Steps 1–4 below for cases with less than 50% censoring. Given data as in the exampleabove (Table S.3), the primary goal is to have an unbiased estimate of the mean of the data, and the

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Step 1 Take the estimated distribution of the transformed data and place the censored data in appro-priate places on the lower part of the distribution. These plotting positions or percentiles areessentially evenly spaced in the lower end of the distribution, on the transformed scale.

• Specifically, for each censored observation i among all c censored observations within the Nobservations, the plotting position p is given by

p =c

N×

i− 38

c + 14

(4)

where the i for observations below LOD come before the i for observations above LOD butbelow LOQ.

Step 2 Translate the percentiles into values on the transformed scale via the normal distribution quantilefunction.

Step 3 Individually re-transform these predicted values on the transformed scale to the original scale.

Step 4 Combine these predicted values with the original observed values (i.e. the uncensored values)into a new set of data.

Step 5 Calculate the mean of this hypothetical data.

S 9.5. Distributions and estimates for other analytes

Following the discussion of censoring, distribution, error components, and estimates for metham-phetamine and MDMA in the main text, here we present the graphical results for the other analytesconsidered: benzoylecgonine, methadone, hydrocodone, and oxycodone. In the boxplot graphs, the shad-ing of the box represents the amount of censoring for each municipality-drug combination, with darkerboxes indicating more complete data. The WWTPs are ordered by the completeness of the data andthen alphabetically. The background shading and label (Report censoring, MLE, KM, Complete data)indicate for groups of municipality-drugs what method was used to create the estimate in the corre-sponding estimates graphs. In the latter, the shading of the estimate indicator (the diamond) representsthe amount of censoring, and the dotted line the unweighted average estimate across the 19 WWTPs.WWTPs are ordered by the mean annual estimate.

S 9.6. Sensitivity of confidence intervals to changes in US

As described in the Discussion, our annual sampling error estimate of US = 10% came from a study ofsampling (or monitoring) uncertainty in a small European city (Ort et al., 2014a). That exercise involvedrepeated samples of N = 56 taken from over 1000 consecutive daily samples of benzoylecgonine (cocainemetabolite). With fewer than 56 samples in the current analysis, WWTP catchment area sizes of moreor less than 7000 users, and substances that may have more or less variation in use than cocaine, one mayquestion whether a different estimate of annual sampling error might change the results substantially.In Figure S.6, we present the estimates and confidence intervals for a single substance, MDMA, in whichwe have both doubled our sampling uncertainty RSD (top panel) and halved it (bottom panel). (Similargraphs for the other analytes are available upon request.) Compared to the bottom panel of Figure 2,we see that changing one of our four uncertainty parameters has small effects on the resulting CIs. Witha 5% uncertainty, Port Angeles MDMA becomes more clearly significantly higher than average, but weremain uncertain that Renton MDMA could not be essentially 0. With less certainty about the abilityof our 43 to 55 samples to represent the whole year—US = 20%—Port Angeles MDMA levels are clearlynot significantly different than the average, and Tacoma’s CI is more likely to overlap with any otherWWTP CI, but all except the Renton index load remain significantly greater than 0. Compared withboth the size of the CIs and the differences in estimated means, these changes in CI due to differentannual sampling uncertainties are relatively small.

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S 9.7. Observation variability and population size

Figure S.7: Variability of daily drug loads [as coefficient of variation (CV)] vs. population size (P, in 1000s) in five differentcatchments for different substances (duration of studies in days: 1©=1369, 2©=311, 3©=28, 4©=239, 5©=28/35). Reasonsfor CVs exceeding 0.8 in theses studies are: i) observations below limit of quantification (heroin) and ii) pronounced intra-week variability (MDMA). Such a regular weekend effect causes a high CV but does not imply that more samples wouldbe needed for the same acceptable uncertainty. Adapted from European Monitoring Centre for Drugs and Drug Addiction(2016).

Reference

European Monitoring Centre for Drugs and Drug Addiction, Assessing Illicit Drugs in

Wastewater: Advances in Wastewater-based Drug Epidemiology (Insights 22), 2016,

Publications Office of the European Union; Luxembourg, http://dx.doi.org/10.2810/6622.