How many people are infected? A case study on SARS-CoV-2 prevalence in Austria
aa r X i v : . [ ec on . GN ] D ec How many people are infected?
A case study on SARS-CoV-2 prevalence in Austria ∗ Gabriel Ziegler † December 23, 2020
Abstract
Using recent data from voluntary mass testing, I provide credible boundson prevalence of SARS-CoV-2 for Austrian counties in early December 2020.When estimating prevalence, a natural missing data problem arises: no testresults are generated for non-tested people. In addition, tests are not perfectlypredictive for the underlying infection. This is particularly relevant for massSARS-CoV-2 testing as these are conducted with rapid Antigen tests, which areknown to be somewhat imprecise. Using insights from the literature on partialidentification, I propose a framework addressing both issues at once. I usethe framework to study differing selection assumptions for the Austrian data.Whereas weak monotone selection assumptions provide limited identificationpower, reasonably stronger assumptions reduce the uncertainty on prevalencesignificantly.
Keywords : Prevalence, Partial Identification, SARS-CoV-2, COVID-19, Aus-tria ∗ Thanks to Daniel Ladenhauf for providing me with the information about the data. All errorsare mine. † The University of Edinburgh, School of Economics; 31 Buccleuch Place, Edinburgh, EH8 9JT,UK; [email protected] . An important measure for (health) policy during a pandemic is prevalence. As a quan-tification of current disease status, prevalence is the proportion of people in a givenpopulation having the disease. (Rothman, 2012) Besides providing an evaluationof how widespread the diseases is, prevalence is relevant in how accurate diagnostictests are. (Zhou et al., 2014) However, prevalence is not directly observable in manysituations.Often prevalence is inferred from diagnostic tests results, which are indicative ofthe disease. There are at least two problems arising in such a situation. First, testsare usually not perfectly indicative for the disease. A test might produce either falsepositives, false negatives, or both. Recently, Ziegler (2020) discusses how this problemworsens when the test accuracy is evaluated with respect to an imperfect referencetest. In such cases, the test’s information is ambiguous. Second, the tested populationis usually not the whole population and the testing pool’composition matters forinference of overall prevalence. The latter problem is even more severe in cases, whenthen the composition of the testing pool is unknown. For example, when testing isvoluntary it is not obvious whether disease-susceptible people are more or less likelyto take the test. That is, people self-select into the testing pool. Selection problemsas occurring with voluntary testing are ubiquitous within Economics. Without strongassumption on unobservable data, Manski (1989) shows that this problem leads toan identification problem and therefore it might not be possible to assign a uniquenumber to the relevant statistic.In this note, I use data of voluntary COVID-19 mass testing in Austria in Decem-ber 2020 to provide bounds on SARS-CoV-2 (point) prevalance at that time. Buildingon the work of Manski and Molinari (2021), Stoye (2020), and Ziegler (2020), I ad-dress both of the problems mentioned within one framework. This allows me toillustrate how much knowledge about prevalence can be obtained just from the dataalone (with minimal assumptions). Furthermore, the framework provides a simplemethod to address the identifying power of varying (stronger) assumptions about theselection problem.In the first half (4/12–15/12) of December 2020, every Austrian municipality pro-vided voluntary SARS-CoV-2 tests for their population via rapid Antigen tests. In Sacks et al. (2020) use a related approach.
Theoretic Framework many municipalities testing was available only for a few, consecutive days. The goalof the policy was to identify otherwise undetected SARS-CoV-2 infected people. Forthis, people with typical symptoms, people who were tested regularly before in theirworkplace, quarantined people, and children below school starting age were explic-itly asked to not attend the testing. (Sozialministerium, 2020) This and anecdotalevidence suggests negative selection into testing. That is, tested people are less sus-ceptible of being infected by SARS-CoV-2. DerStandard (2020) provides data of testresults and participation on the county level. The dataset only covers 7 out of the 9Austrian states (Bundesl¨ander). Let c = 1 denote a person who is infected with SARS-CoV-2 and c = 0 otherwise.Participation in the mass testing is indicted with t = 1 (again t = 0 otherwise). Onlyif the person was tested, she can obtain a positive test result denoted with a = 1(and a = 0 otherwise). The population (of a county) is a distribution P ( a, c, t ), butobserved data are just P ( a | t = 1). In particular, note that P ( a = 1) = P ( a = 1 | t =1) P ( t = 1) because a positive test can only be observed for tested people. γ := P ( a = 1 | t = 1) . . . test yield ρ := P ( c = 1) . . . prevalance τ := P ( t = 1) . . . proportion of tested people . Accuracy of the test is given by sensitivity and specificity given by σ := P ( a = 1 | c = 1 , t = 1) = P ( a = 1 , c = 1 | t = 1) P ( c = 1 | t = 1) (1) π := P ( a = 0 | c = 0 , t = 1) = P ( a = 0 , c = 0 | t = 1) P ( c = 0 | t = 1) , (2) Austrian school starting age is 6 years with September 1st as cutoff date. Data on Vorarlberg is missing. Data for Carinthia does not include the number of positivetests results and is therefore omitted from the analysis. Three counties are combined due to localmisreporting of data (Amstetten, Scheibbs, Waidhofen a.d. Ybbs). Since the analysis is county-by-county, it is unaffected by these data issues. respectively. In line with Ziegler (2020), Antigen tests correspond to ambiguousinformation and therefore I make the following assumption the both, sensitivity andspecificity, are only known to lie within an interval.
Assumption 1 (Ambiguous Information) . The test satisfies σ ∈ [ σ, σ ] and π ∈ [ π, π ] . Assumption 1 alone provides sharp bounds on prevalence ρ := P ( c = 1): Proposition 1.
If Assumption 1 holds, then ρ = (cid:20) τ γ + π − σ + π − , τ γ + π − σ + π − − τ ) (cid:21) Proof.
First, consider fixed σ and π . Then by the law of total probability γ = σP ( c = 1 | t = 1) + (1 − π )(1 − P ( c = 1 | t = 1)) ⇐⇒ P ( c = 1 | t = 1) = γ + π − σ + π − P ( c = 1 | t = 0) ∈ [0 , ρ ∈ (cid:20) τ γ + π − σ + π − , τ γ + π − σ + π − − τ ) (cid:21) . The fraction is increasing in π and decreasing in σ . The result follows by evaluatingat the respective extremes. (cid:4) The prevalance bounds in Proposition 1 are pretty wide in applications as willbe seen later. However, they are not completely trivial in the sense of just statingprevalence is bounded by 0 and 1 although they rely on minimal assumptions aboutthe (untested) population. As can be seen in the proof of Proposition 1, P ( c = 1 | t =0) is trivially bounded without stronger assumptions, which leads to wide boundson prevalence. As explained above, there is some indication of negative selectioninto the testing pool in the case of Austrian mass testing. This knowledge can beused to narrow bounds on prevalence. This extraneous information is formalized inAssumption 2. A potentially more satisfying way of modeling selection is bounding the odds ratio. Stoye(2020) uses such bounds in his analysis under the assumption of π = π = 1, i.e. the test does not Empirical Analysis
Assumption 2 (Selection) . The population satisfies P ( c = 1 | t = 0) P ( c = 1 | t = 1) ∈ [ κ, κ ] , with κ ≥ . When κ ≥
1, then P ( c = 1 | t = 0) ≥ P ( c = 1 | t = 1) so that tested people are less likely to be infected than untested people. This corresponds to the negative selectionexplained above. On the other hand, if κ ≤
1, then there is positive selection, whichseems more appropriate in the case of PCR testing. Indeed, Manski and Molinari(2021) use such an assumption in their study on prevalence of SARS-CoV-2. Theirassumption (called test-monotonicity) corresponds to ( κ, κ ) = (0 , Proposition 2.
Suppose Assumption 1 and Assumption 2 hold. If κ ≤ σ + π − γ + π − , then ρ = (cid:20) ( τ + (1 − τ ) κ ) γ + π − σ + π − , ( τ + (1 − τ ) κ ) γ + π − σ + π − (cid:21) . Otherwise the upper bound is given by Proposition 1.Proof.
As in the proof of Proposition 1, P ( c = 1 | t = 1) = γ + π − σ + π − , but now P ( c = 1 | t =0) ∈ [ κP ( c = 1 | t = 1) , κP ( c = 1 | t = 1)], which is below one because of κ ≤ σ + π − γ + π − .Then ρ ∈ (cid:20) ( τ + (1 − τ ) κ ) γ + π − σ + π − , ( τ + (1 − τ ) κ ) γ + π − σ + π − (cid:21) . The fraction is increasing in π and decreasing in σ . The result follows by evaluatingat the respective extremes. (cid:4) The dataset was already explained in Section 1. It remains to get data on the test’saccuracy as formalized in Assumption 1. Rapid Antigen test were used in the masstesting in Austria. To best of my knowledge, there is no publicaly available data onwhich specific test was used by each municipality. However, I personally obtained produce false-positives. Without this assumption, bounds on the odds ratio seem rather intractable.Furthermore, such an assumption is problematic in the application to Antigen tests. the data for a few municipalities in Graz-Umgebung. In these municipalities theStandard Q COVID-19 Rapid Antigen Test of SD Biosensor/Roche for detection ofSARS-CoV-2 was used. I will use this test as if it was used in all the municipalitiesin the dataset. Kaiser et al. (2020) provide an independent analysis of the Standard Q Antigentest. Relative to a PCR test, they find a (point estimate for) sensitivity of 89 , , .
2% to 97 . . Due to the imperfectness of the referencePCR test, the Antigen test does not have a unique value for sensitivity and specificityas discussed by Ziegler (2020). Using the method proposed by Ziegler (2020) andevaluating across all possible PCR’s sensitivities, Standard Q’s accuracy is boundedby σ ∈ [53 . , . π ∈ [99 . , It remains to specify the selection parameters ( κ, κ ). For this, I will considerseveral cases corresponding to different assumptions about selection and explain theeffects using the city of Graz (Graz-Stadt) as an example. Graz had a particpationrate of slightly more than 20% and of these 0 .
9% obtained a positive test result. Theresults for each county (together with participation τ and test yield γ ) are shown inTable 1 and Table 2. No Assumption.
This case corresponds to Proposition 1. Here any kind of se-lection, negative or positive, is allowed for. Correspondingly, the bounds onprevalence are rather wide. For Graz the bounds are [0 . . No selection.
Next, consider a scenario of no selection, which embodies a verystrong assumption and one which might not be appropriate in the current con-text. No selection means that the testing decision was as if randomly assignedand therefore the results from the tested people is representative for the entire Many Antigen tests currently on the market have very similar quality in terms of observedsensitivity and specificity relative to a PCR test. Therefore, the use of a different test would notchange the results significantly. For all considered PCR tests, their 95% confidence intervals exclude perfect sensitivity. In these calculations, I use the point estimates of Kaiser et al. (2020).
Empirical Analysis population. Mathematically this means P ( c = 1 | t = 0) = P ( c = 1 | t = 1) (orequivalently in the current framework, κ = κ = 1). With this strong (and mostlikely unwarranted assumption), the bounds for Graz reduce to [0 . , . Negative Selection.
As explained before this assumption is credible in the cur-rent context, but it is still very weak as it just imposes P ( c = 1 | t = 0) ≥ P ( c = 1 | t = 1) (with κ = 1 and κ = ∞ ). However, with negative selectiononly it is still possible that every untested person is infected and therefore theupper bound is not reduced relative to the No Assumption scenario. For Grazthe bounds are [0 . . Restricted Negative Selection.
Here, the selection assumption from before is main-tained, but there is also an upper bound on selection. In particular, at most P ( c = 1 | t = 0) = 2 × P ( c = 1 | t = 1), i.e. non-tested people are twice as likelyinfected than tested ones. Formally, this case is obtained with κ = 1 and κ = 2.For Graz this additional assumption gives prevalence bounds [0 . . Small Ambigous Selection.
Finally, consider a case where no knowledge aboutthe direction of selection is warranted, but there is evidence for small selec-tion, meaning P ( c = 1 | t = 0) is close to P ( c = 1 | t = 1). In particular, herethe assumption is that non-tested people have an infection probability between95% and 105% of the tested people’s infection probability (i.e. κ = 0 .
95 and κ = 1 . . . County Positive Tests Participation No Assumption No SelectionAmstetten, Scheibbs, Waidhofen adY 0.21% 30.98% [0.00%, 69.14%] [0.00%, 0.40%]Bruck an der Leitha 0.10% 39.21% [0.00%, 60.87%] [0.00%, 0.20%]Baden 0.10% 31.65% [0.00%, 68.41%] [0.00%, 0.19%]Gmuend 0.29% 28.95% [0.00%, 71.20%] [0.00%, 0.54%]Gaenserndorf 0.11% 34.99% [0.00%, 65.08%] [0.00%, 0.21%]Hollabrunn 0.15% 37.91% [0.00%, 62.20%] [0.00%, 0.27%]Horn 0.05% 37.26% [0.00%, 62.78%] [0.00%, 0.09%]Korneuburg 0.15% 47.08% [0.00%, 53.05%] [0.00%, 0.27%]Krems (Land) 0.09% 35.67% [0.00%, 64.39%] [0.00%, 0.18%]Krems (Stadt) 0.12% 27.79% [0.00%, 72.27%] [0.00%, 0.23%]Lilienfeld 0.12% 33.07% [0.00%, 67.00%] [0.00%, 0.23%]Moedling 0.13% 37.98% [0.00%, 62.11%] [0.00%, 0.23%]Melk 0.20% 31.11% [0.00%, 69.01%] [0.00%, 0.38%]Mistelbach 0.13% 46.67% [0.00%, 53.45%] [0.00%, 0.25%]Neunkirchen 0.19% 29.14% [0.00%, 70.97%] [0.00%, 0.36%]St. Poelten (Stadt) 0.19% 22.46% [0.00%, 77.62%] [0.00%, 0.35%]St. Poelten (Land) 0.16% 39.78% [0.00%, 60.33%] [0.00%, 0.30%]Tulln 0.09% 30.23% [0.00%, 69.83%] [0.00%, 0.18%]Wiener Neustadt (Stadt+Land) 0.13% 29.43% [0.00%, 70.64%] [0.00%, 0.23%]Waidhofen an der Thaya 0.13% 26.18% [0.00%, 73.88%] [0.00%, 0.23%]Zwettl 0.17% 31.98% [0.00%, 68.12%] [0.00%, 0.32%]Innsbruck-Stadt 0.19% 29.21% [0.00%, 70.89%] [0.00%, 0.35%]Imst 0.32% 28.80% [0.00%, 71.37%] [0.00%, 0.60%]Innsbruck-Land 0.22% 34.75% [0.00%, 65.39%] [0.00%, 0.41%]Kitzbuehel 0.28% 31.42% [0.00%, 68.74%] [0.00%, 0.52%]Kufstein 0.30% 32.43% [0.00%, 67.75%] [0.00%, 0.56%]Landeck 0.36% 32.44% [0.00%, 67.78%] [0.00%, 0.68%]Lienz 0.37% 25.25% [0.00%, 74.93%] [0.00%, 0.69%]Reutte 0.18% 35.48% [0.00%, 64.64%] [0.00%, 0.34%]Schwaz 0.48% 28.26% [0.00%, 71.99%] [0.00%, 0.89%]Bruck-Muerzzuschlag 0.49% 22.98% [0.00%, 77.22%] [0.02%, 0.91%]Deutschlandsberg 0.37% 21.13% [0.00%, 79.01%] [0.00%, 0.69%]Graz-Stadt 0.90% 20.65% [0.10%, 79.70%] [0.49%, 1.68%]Graz-Umgebung 0.19% 26.89% [0.00%, 73.20%] [0.00%, 0.36%]Hartberg-Fuerstenfeld 0.19% 19.97% [0.00%, 80.10%] [0.00%, 0.36%]Leibnitz 0.19% 19.47% [0.00%, 80.60%] [0.00%, 0.36%]
Empirical Analysis
Leoben 0.19% 20.68% [0.00%, 79.39%] [0.00%, 0.35%]Liezen 0.32% 16.91% [0.00%, 83.19%] [0.00%, 0.60%]Murau 0.31% 18.28% [0.00%, 81.83%] [0.00%, 0.58%]Murtal 0.32% 18.42% [0.00%, 81.69%] [0.00%, 0.59%]Suedoststeiermark 0.21% 22.17% [0.00%, 77.92%] [0.00%, 0.39%]Voitsberg 0.16% 20.27% [0.00%, 79.79%] [0.00%, 0.30%]Weiz 0.16% 19.11% [0.00%, 80.95%] [0.00%, 0.30%]Salzburg (Stadt) 0.27% 19.08% [0.00%, 81.02%] [0.00%, 0.50%]Hallein (Tennengau) 0.73% 23.87% [0.07%, 76.45%] [0.30%, 1.36%]Salzburg-Umgebung (Flachgau) 0.37% 29.24% [0.00%, 70.96%] [0.00%, 0.68%]St. Johann im Pongau (Pongau) 0.42% 22.41% [0.00%, 77.76%] [0.00%, 0.78%]Tamsweg (Lungau) 0.71% 24.17% [0.07%, 76.15%] [0.28%, 1.33%]Zell am See (Pinzgau) 0.50% 24.02% [0.01%, 76.20%] [0.03%, 0.94%]Linz (Stadt) 0.34% 20.90% [0.00%, 79.23%] [0.00%, 0.63%]Steyr (Stadt) 0.33% 22.09% [0.00%, 78.05%] [0.00%, 0.61%]Wels (Stadt) 0.38% 14.79% [0.00%, 85.32%] [0.00%, 0.71%]Braunau am Inn 0.45% 18.39% [0.00%, 81.76%] [0.00%, 0.84%]Eferding 0.26% 26.31% [0.00%, 73.82%] [0.00%, 0.48%]Freistadt 0.30% 26.85% [0.00%, 73.30%] [0.00%, 0.56%]Gmunden 0.45% 24.10% [0.00%, 76.10%] [0.00%, 0.83%]Grieskirchen 0.35% 24.83% [0.00%, 75.33%] [0.00%, 0.65%]Kirchdorf 0.33% 25.15% [0.00%, 75.01%] [0.00%, 0.62%]Linz-Land 0.41% 24.74% [0.00%, 75.45%] [0.00%, 0.77%]Perg 0.25% 24.03% [0.00%, 76.09%] [0.00%, 0.47%]Ried im Innkreis 0.31% 20.35% [0.00%, 79.77%] [0.00%, 0.57%]Rohrbach 0.54% 27.41% [0.02%, 72.87%] [0.08%, 1.00%]Schaerding 0.86% 23.81% [0.10%, 76.57%] [0.44%, 1.60%]Steyr-Land 0.57% 20.14% [0.02%, 80.08%] [0.12%, 1.07%]Urfahr-Umgebung 0.27% 26.56% [0.00%, 73.57%] [0.00%, 0.51%]Voecklabruck 0.48% 25.37% [0.00%, 74.86%] [0.01%, 0.90%]Wels-Land 0.36% 21.45% [0.00%, 78.69%] [0.00%, 0.67%]Eisenstadt (Stadt+Umgebung) / Rust 0.12% 28.12% [0.00%, 71.95%] [0.00%, 0.22%]Guessing 0.27% 21.22% [0.00%, 78.89%] [0.00%, 0.50%]Jennersdorf 0.22% 24.50% [0.00%, 75.60%] [0.00%, 0.42%]Mattersburg 0.07% 25.01% [0.00%, 75.03%] [0.00%, 0.14%]Neusiedl am See 0.11% 29.95% [0.00%, 70.12%] [0.00%, 0.21%]Oberpullendorf 0.17% 28.77% [0.00%, 71.32%] [0.00%, 0.31%]Oberwart 0.26% 22.04% [0.00%, 78.07%] [0.00%, 0.49%]Wien 0.32% 13.10% [0.00%, 86.98%] [0.00%, 0.60%]
Table 2: County-level prevalence with selection assumptions
County ↓ (cid:31) ( κ, κ ) → (1 , ∞ ) (1 ,
2) (0 . , . Empirical Analysis
Leoben [0.00%, 79.39%] [0.00%, 0.62%] [0.00%, 0.36%]Liezen [0.00%, 83.19%] [0.00%, 1.10%] [0.00%, 0.62%]Murau [0.00%, 81.83%] [0.00%, 1.06%] [0.00%, 0.61%]Murtal [0.00%, 81.69%] [0.00%, 1.08%] [0.00%, 0.62%]Suedoststeiermark [0.00%, 77.92%] [0.00%, 0.70%] [0.00%, 0.41%]Voitsberg [0.00%, 79.79%] [0.00%, 0.54%] [0.00%, 0.31%]Weiz [0.00%, 80.95%] [0.00%, 0.54%] [0.00%, 0.31%]Salzburg (Stadt) [0.00%, 81.02%] [0.00%, 0.90%] [0.00%, 0.52%]Hallein (Tennengau) [0.30%, 76.45%] [0.30%, 2.40%] [0.28%, 1.42%]Salzburg-Umgebung (Flachgau) [0.00%, 70.96%] [0.00%, 1.17%] [0.00%, 0.71%]St. Johann im Pongau (Pongau) [0.00%, 77.76%] [0.00%, 1.38%] [0.00%, 0.81%]Tamsweg (Lungau) [0.28%, 76.15%] [0.28%, 2.34%] [0.27%, 1.38%]Zell am See (Pinzgau) [0.03%, 76.20%] [0.03%, 1.65%] [0.03%, 0.97%]Linz (Stadt) [0.00%, 79.23%] [0.00%, 1.13%] [0.00%, 0.65%]Steyr (Stadt) [0.00%, 78.05%] [0.00%, 1.08%] [0.00%, 0.63%]Wels (Stadt) [0.00%, 85.32%] [0.00%, 1.32%] [0.00%, 0.74%]Braunau am Inn [0.00%, 81.76%] [0.00%, 1.53%] [0.00%, 0.88%]Eferding [0.00%, 73.82%] [0.00%, 0.83%] [0.00%, 0.50%]Freistadt [0.00%, 73.30%] [0.00%, 0.96%] [0.00%, 0.58%]Gmunden [0.00%, 76.10%] [0.00%, 1.46%] [0.00%, 0.86%]Grieskirchen [0.00%, 75.33%] [0.00%, 1.14%] [0.00%, 0.68%]Kirchdorf [0.00%, 75.01%] [0.00%, 1.09%] [0.00%, 0.65%]Linz-Land [0.00%, 75.45%] [0.00%, 1.35%] [0.00%, 0.80%]Perg [0.00%, 76.09%] [0.00%, 0.83%] [0.00%, 0.49%]Ried im Innkreis [0.00%, 79.77%] [0.00%, 1.02%] [0.00%, 0.59%]Rohrbach [0.08%, 72.87%] [0.08%, 1.73%] [0.07%, 1.04%]Schaerding [0.44%, 76.57%] [0.44%, 2.81%] [0.42%, 1.66%]Steyr-Land [0.12%, 80.08%] [0.12%, 1.93%] [0.11%, 1.12%]Urfahr-Umgebung [0.00%, 73.57%] [0.00%, 0.88%] [0.00%, 0.52%]Voecklabruck [0.01%, 74.86%] [0.01%, 1.57%] [0.01%, 0.93%]Wels-Land [0.00%, 78.69%] [0.00%, 1.20%] [0.00%, 0.70%]Eisenstadt (Stadt+Umgebung) / Rust [0.00%, 71.95%] [0.00%, 0.38%] [0.00%, 0.23%]Guessing [0.00%, 78.89%] [0.00%, 0.90%] [0.00%, 0.52%]Jennersdorf [0.00%, 75.60%] [0.00%, 0.73%] [0.00%, 0.43%]Mattersburg [0.00%, 75.03%] [0.00%, 0.24%] [0.00%, 0.14%]Neusiedl am See [0.00%, 70.12%] [0.00%, 0.36%] [0.00%, 0.22%]Oberpullendorf [0.00%, 71.32%] [0.00%, 0.53%] [0.00%, 0.32%]Oberwart [0.00%, 78.07%] [0.00%, 0.88%] [0.00%, 0.51%]Wien [0.00%, 86.98%] [0.00%, 1.11%] [0.00%, 0.62%] eferences References
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