# How many people are infected?
A case study on SARS-CoV-2 prevalence in Austria111Thanks to Daniel Ladenhauf
for providing me with the information about the data. All errors are mine.

Gabriel Ziegler222The University of Edinburgh, School of Economics; 31
Buccleuch Place, Edinburgh, EH8 9JT, UK<EMAIL_ADDRESS>
###### Abstract

Using recent data from voluntary mass testing, I provide credible bounds on
prevalence of SARS-CoV-2 for Austrian counties in early December 2020. When
estimating prevalence, a natural missing data problem arises: no test results
are generated for non-tested people. In addition, tests are not perfectly
predictive for the underlying infection. This is particularly relevant for
mass SARS-CoV-2 testing as these are conducted with rapid Antigen tests, which
are known to be somewhat imprecise. Using insights from the literature on
partial identification, I propose a framework addressing both issues at once.
I use the framework to study differing selection assumptions for the Austrian
data. Whereas weak monotone selection assumptions provide limited
identification power, reasonably stronger assumptions reduce the uncertainty
on prevalence significantly.

Keywords: Prevalence, Partial Identification, SARS-CoV-2, COVID-19, Austria

##  Introduction

An important measure for (health) policy during a pandemic is prevalence. As a
quantification of current disease status, prevalence is the proportion of
people in a given population having the disease. (Rothman, 2012) Besides
providing an evaluation of how widespread the diseases is, prevalence is
relevant in how accurate diagnostic tests are. (Zhou et al., 2014) However,
prevalence is not directly observable in many situations.

Often prevalence is inferred from diagnostic tests results, which are
indicative of the disease. There are at least two problems arising in such a
situation. First, tests are usually not perfectly indicative for the disease.
A test might produce either false positives, false negatives, or both.
Recently, Ziegler (2020) discusses how this problem worsens when the test
accuracy is evaluated with respect to an imperfect reference test. In such
cases, the test’s information is ambiguous. Second, the tested population is
usually not the whole population and the testing pool’composition matters for
inference of overall prevalence. The latter problem is even more severe in
cases, when then the composition of the testing pool is unknown. For example,
when testing is voluntary it is not obvious whether disease-susceptible people
are more or less likely to take the test. That is, people self-select into the
testing pool. Selection problems as occurring with voluntary testing are
ubiquitous within Economics. Without strong assumption on unobservable data,
Manski (1989) shows that this problem leads to an identification problem and
therefore it might not be possible to assign a unique number to the relevant
statistic.

In this note, I use data of voluntary COVID-19 mass testing in Austria in
December 2020 to provide bounds on SARS-CoV-2 (point) prevalance at that time.
Building on the work of Manski and Molinari (2021), Stoye (2020), and Ziegler
(2020),333Sacks et al. (2020) use a related approach. I address both of the
problems mentioned within one framework. This allows me to illustrate how much
knowledge about prevalence can be obtained just from the data alone (with
minimal assumptions). Furthermore, the framework provides a simple method to
address the identifying power of varying (stronger) assumptions about the
selection problem.

In the first half (4/12–15/12) of December 2020, every Austrian municipality
provided voluntary SARS-CoV-2 tests for their population via rapid Antigen
tests. In many municipalities testing was available only for a few,
consecutive days. The goal of the policy was to identify otherwise undetected
SARS-CoV-2 infected people. For this, people with typical symptoms, people who
were tested regularly before in their workplace, quarantined people, and
children below school starting age444Austrian school starting age is 6 years
with September 1st as cutoff date. were explicitly asked to _not_ attend the
testing. (Sozialministerium, 2020) This and anecdotal evidence suggests
negative selection into testing. That is, tested people are less susceptible
of being infected by SARS-CoV-2. DerStandard (2020) provides data of test
results and participation on the county level. The dataset only covers $7$ out
of the $9$ Austrian states (Bundesländer).555Data on Vorarlberg is missing.
Data for Carinthia does not include the number of positive tests results and
is therefore omitted from the analysis. Three counties are combined due to
local misreporting of data (Amstetten, Scheibbs, Waidhofen a.d. Ybbs). Since
the analysis is county-by-county, it is unaffected by these data issues.

##  Theoretic Framework

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). Only if 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 $P(a,c,t)$, but observed 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.

$\displaystyle\gamma$ $\displaystyle:=P(a=1|t=1)$ $\displaystyle\ldots$ test
yield $\displaystyle\rho$ $\displaystyle:=P(c=1)$ $\displaystyle\ldots$
prevalance $\displaystyle\tau$ $\displaystyle:=P(t=1)$ $\displaystyle\ldots$
$\displaystyle\text{proportion of tested people}.$

Accuracy of the test is given by sensitivity and specificity given by

$\displaystyle\sigma$
$\displaystyle:=P(a=1|c=1,t=1)=\frac{P(a=1,c=1|t=1)}{P(c=1|t=1)}$ (1)
$\displaystyle\pi$
$\displaystyle:=P(a=0|c=0,t=1)=\frac{P(a=0,c=0|t=1)}{P(c=0|t=1)},$ (2)

respectively. In line with Ziegler (2020), Antigen tests correspond to
ambiguous information and therefore I make the following assumption the both,
sensitivity and specificity, are only known to lie within an interval.

######  Assumption 1 (Ambiguous Information).

The test satisfies

$\displaystyle\sigma\in\left[\underline{\sigma},\overline{\sigma}\right]\quad\text{and}\quad\pi\in\left[\underline{\pi},\overline{\pi}\right].$

1 alone provides sharp bounds on prevalence $\rho:=P(c=1)$:

######  Proposition 1.

If 1 holds, then

$\displaystyle\rho=\left[\tau\frac{\gamma+\underline{\pi}-1}{\overline{\sigma}+\underline{\pi}-1},\tau\frac{\gamma+\overline{\pi}-1}{\underline{\sigma}+\overline{\pi}-1}+(1-\tau)\right]$

###### Proof.

First, consider fixed $\sigma$ and $\pi$. Then by the law of total probability

$\displaystyle\gamma$ $\displaystyle=\sigma P(c=1|t=1)+(1-\pi)(1-P(c=1|t=1))$
$\displaystyle\iff\quad$ $\displaystyle
P(c=1|t=1)=\frac{\gamma+\pi-1}{\sigma+\pi-1}$

and $P(c=1|t=0)\in[0,1]$. Then, again by the law of total probability

$\displaystyle\rho\in\left[\tau\frac{\gamma+\pi-1}{\sigma+\pi-1},\tau\frac{\gamma+\pi-1}{\sigma+\pi-1}+(1-\tau)\right].$

The fraction is increasing in $\pi$ and decreasing in $\sigma$. The result
follows by evaluating at the respective extremes. ∎

The prevalance bounds in 1 are pretty wide in applications as will be seen
later. However, they are not completely trivial in the sense of just stating
prevalence is bounded by $0$ and $1$ although they rely on minimal assumptions
about the (untested) population. As can be seen in the proof of 1,
$P(c=1|t=0)$ is trivially bounded without stronger assumptions, which leads to
wide bounds on prevalence. As explained above, there is some indication of
negative selection into the testing pool in the case of Austrian mass testing.
This knowledge can be used to narrow bounds on prevalence. This extraneous
information is formalized in 2.666A potentially more satisfying way of
modeling selection is bounding the odds ratio. Stoye (2020) uses such bounds
in his analysis under the assumption of $\underline{\pi}=\overline{\pi}=1$,
i.e. the test does not 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.

######  Assumption 2 (Selection).

The population satisfies

$\displaystyle\frac{P(c=1|t=0)}{P(c=1|t=1)}\in\left[\underline{\kappa},\overline{\kappa}\right],$

with $\underline{\kappa}\geq 0$.

When $\underline{\kappa}\geq 1$, then $P(c=1|t=0)\geq P(c=1|t=1)$ so that
tested people are _less_ likely to be infected than untested people. This
corresponds to the negative selection explained above. On the other hand, if
$\overline{\kappa}\leq 1$, then there is positive selection, which seems 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. Their
assumption (called test-monotonicity) corresponds to
$(\underline{\kappa},\overline{\kappa})=(0,1)$.

######  Proposition 2.

Suppose 1 and 2 hold. If
$\overline{\kappa}\leq\frac{\sigma+\pi-1}{\gamma+\pi-1}$, then

$\displaystyle\rho=\left[(\tau+(1-\tau)\underline{\kappa})\frac{\gamma+\underline{\pi}-1}{\overline{\sigma}+\underline{\pi}-1},(\tau+(1-\tau)\overline{\kappa})\frac{\gamma+\overline{\pi}-1}{\underline{\sigma}+\overline{\pi}-1}\right].$

Otherwise the upper bound is given by 1.

###### Proof.

As in the proof of 1, $P(c=1|t=1)=\frac{\gamma+\pi-1}{\sigma+\pi-1}$, but now
$P(c=1|t=0)\in[\underline{\kappa}P(c=1|t=1),\overline{\kappa}P(c=1|t=1)]$,
which is below one because of
$\overline{\kappa}\leq\frac{\sigma+\pi-1}{\gamma+\pi-1}$. Then

$\displaystyle\rho\in\left[(\tau+(1-\tau)\underline{\kappa})\frac{\gamma+\pi-1}{\sigma+\pi-1},(\tau+(1-\tau)\overline{\kappa})\frac{\gamma+\pi-1}{\sigma+\pi-1}\right].$

The fraction is increasing in $\pi$ and decreasing in $\sigma$. The result
follows by evaluating at the respective extremes. ∎

##  Empirical Analysis

The dataset was already explained in Section 1. It remains to get data on the
test’s accuracy as formalized in 1. Rapid Antigen test were used in the mass
testing in Austria. To best of my knowledge, there is no publicaly available
data on which specific test was used by each municipality. However, I
personally obtained the data for a few municipalities in Graz-Umgebung. In
these municipalities the Standard Q COVID-19 Rapid Antigen Test of SD
Biosensor/Roche for detection of SARS-CoV-2 was used. I will use this test _as
if_ it was used in all the municipalities in the dataset.777Many Antigen tests
currently on the market have very similar quality in terms of observed
sensitivity and specificity relative to a PCR test. Therefore, the use of a
different test would not change the results significantly.

Kaiser et al. (2020) provide an independent analysis of the Standard Q Antigen
test. Relative to a PCR test, they find a (point estimate for) sensitivity of
$89,0\%$ and specificity of $99,7\%$. Although a PCR test is (close to)
perfect when it comes to specificity, it is known that it might not have
perfect sensitivity. Several PCR tests were evaluated for sensitivity by
Alcoba-Florez et al. (2020). They find point estimates ranging from $60.2\%$
to $97.9\%$.888For all considered PCR tests, their $95\%$ confidence intervals
exclude perfect sensitivity.. Due to the imperfectness of the reference PCR
test, the Antigen test does not have a unique value for sensitivity and
specificity as discussed by Ziegler (2020). Using the method proposed by
Ziegler (2020) and evaluating across all possible PCR’s sensitivities,
Standard Q’s accuracy is bounded by $\sigma\in[53.58\%,87.65\%]$ and
$\pi\in[99.53\%,100\%]$.999In these calculations, I use the point estimates of
Kaiser et al. (2020).

It remains to specify the selection parameters
$(\underline{\kappa},\overline{\kappa})$. For this, I will consider several
cases corresponding to different assumptions about selection and explain the
effects using the city of Graz (Graz-Stadt) as an example. Graz had a
particpation rate of slightly more than $20\%$ and of these $0.9\%$ obtained a
positive test result. The results for each county (together with participation
$\tau$ and test yield $\gamma$) are shown in LABEL:tab:1 and LABEL:tab:2.

No Assumption.

This case corresponds to 1. Here any kind of selection, negative or positive,
is allowed for. Correspondingly, the bounds on prevalence are rather wide. For
Graz the bounds are [$0.10\%$, $79.70\%$], which excludes the possibility of
$0\%$ prevalence in contrast to many other counties in LABEL:tab:1.

No selection.

Next, consider a scenario of no selection, which embodies a very strong
assumption and one which might not be appropriate in the current context. No
selection means that the testing decision was as if randomly assigned and
therefore the results from the tested people is representative for the entire
population. Mathematically this means ${P(c=1|t=0)}={P(c=1|t=1)}$ (or
equivalently in the current framework,
$\underline{\kappa}=\overline{\kappa}=1$). With this strong (and most likely
unwarranted assumption), the bounds for Graz reduce to $[0.49\%,1.68\%]$,
which still leaves a interval width of more than $1\%$. This uncertainty stems
from the imprecise testing technology.

Negative Selection.

As explained before this assumption is credible in the current context, but it
is still very weak as it just imposes ${P(c=1|t=0)}\geq{P(c=1|t=1)}$ (with
$\underline{\kappa}=1$ and $\overline{\kappa}=\infty$). However, with negative
selection only it is still possible that _every_ untested person is infected
and therefore the upper bound is not reduced relative to the No Assumption
scenario. For Graz the bounds are [$0.49\%$, $79.70\%$]. Note that the lower
bound is almost five times higher than in the No Assumption scenario, meaning
that this assumption—although weak—has quite some power for the ’best-case’
prevalence.

Restricted Negative Selection.

Here, the selection assumption from before is maintained, but there is also an
upper bound on selection. In particular, at most
${P(c=1|t=0)}=2\times{P(c=1|t=1)}$, i.e. non-tested people are twice as likely
infected than tested ones. Formally, this case is obtained with
$\underline{\kappa}=1$ and $\overline{\kappa}=2$. For Graz this additional
assumption gives prevalence bounds [$0.49\%$, $3.01\%$]. Here, and for all the
other counties as well, the upper bound is reduced by a large amount relative
to the Negative Selection scenario. Thus, this assumption of restricted
negative selection has quite some identification power, although “twice as
likely” does not appear to be unrealistic.

Small Ambigous Selection.

Finally, consider a case where no knowledge about the direction of selection
is warranted, but there is evidence for small selection, meaning
${P(c=1|t=0)}$ is close to ${P(c=1|t=1)}$. In particular, here the assumption
is that non-tested people have an infection probability between $95\%$ and
$105\%$ of the tested people’s infection probability (i.e.
$\underline{\kappa}=0.95$ and $\overline{\kappa}=1.05$). In this scenario,
bounds are pretty tight and for Graz they are [$0.47\%$, $1.75\%$]. Since
there is the possibility of positive selection, the lower bound here is lower
than in the previous cases.

Table 1: County-level prevalence with no or very strong assumptions County | Positive Tests | Participation | No Assumption | No Selection
---|---|---|---|---
Amstetten, 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%]
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 $\downarrow\;\diagdown\;(\underline{\kappa},\overline{\kappa})\rightarrow$ | $(1,\infty)$ | $(1,2)$ | $(0.95,1.05)$
---|---|---|---
Amstetten, Scheibbs, Waidhofen adY | [0.00%, 69.14%] | [0.00%, 0.67%] | [0.00%, 0.41%]
Bruck an der Leitha | [0.00%, 60.87%] | [0.00%, 0.31%] | [0.00%, 0.20%]
Baden | [0.00%, 68.41%] | [0.00%, 0.32%] | [0.00%, 0.19%]
Gmuend | [0.00%, 71.20%] | [0.00%, 0.92%] | [0.00%, 0.56%]
Gaenserndorf | [0.00%, 65.08%] | [0.00%, 0.35%] | [0.00%, 0.22%]
Hollabrunn | [0.00%, 62.20%] | [0.00%, 0.44%] | [0.00%, 0.28%]
Horn | [0.00%, 62.78%] | [0.00%, 0.14%] | [0.00%, 0.09%]
Korneuburg | [0.00%, 53.05%] | [0.00%, 0.41%] | [0.00%, 0.28%]
Krems (Land) | [0.00%, 64.39%] | [0.00%, 0.29%] | [0.00%, 0.18%]
Krems (Stadt) | [0.00%, 72.27%] | [0.00%, 0.39%] | [0.00%, 0.23%]
Lilienfeld | [0.00%, 67.00%] | [0.00%, 0.39%] | [0.00%, 0.24%]
Moedling | [0.00%, 62.11%] | [0.00%, 0.38%] | [0.00%, 0.24%]
Melk | [0.00%, 69.01%] | [0.00%, 0.63%] | [0.00%, 0.39%]
Mistelbach | [0.00%, 53.45%] | [0.00%, 0.38%] | [0.00%, 0.25%]
Neunkirchen | [0.00%, 70.97%] | [0.00%, 0.62%] | [0.00%, 0.37%]
St. Poelten (Stadt) | [0.00%, 77.62%] | [0.00%, 0.62%] | [0.00%, 0.36%]
St. Poelten (Land) | [0.00%, 60.33%] | [0.00%, 0.47%] | [0.00%, 0.30%]
Tulln | [0.00%, 69.83%] | [0.00%, 0.30%] | [0.00%, 0.18%]
Wiener Neustadt (Stadt+Land) | [0.00%, 70.64%] | [0.00%, 0.40%] | [0.00%, 0.24%]
Waidhofen an der Thaya | [0.00%, 73.88%] | [0.00%, 0.41%] | [0.00%, 0.24%]
Zwettl | [0.00%, 68.12%] | [0.00%, 0.54%] | [0.00%, 0.33%]
Innsbruck-Stadt | [0.00%, 70.89%] | [0.00%, 0.59%] | [0.00%, 0.36%]
Imst | [0.00%, 71.37%] | [0.00%, 1.02%] | [0.00%, 0.62%]
Innsbruck-Land | [0.00%, 65.39%] | [0.00%, 0.68%] | [0.00%, 0.43%]
Kitzbuehel | [0.00%, 68.74%] | [0.00%, 0.87%] | [0.00%, 0.54%]
Kufstein | [0.00%, 67.75%] | [0.00%, 0.93%] | [0.00%, 0.58%]
Landeck | [0.00%, 67.78%] | [0.00%, 1.13%] | [0.00%, 0.70%]
Lienz | [0.00%, 74.93%] | [0.00%, 1.21%] | [0.00%, 0.72%]
Reutte | [0.00%, 64.64%] | [0.00%, 0.56%] | [0.00%, 0.35%]
Schwaz | [0.00%, 71.99%] | [0.00%, 1.52%] | [0.00%, 0.92%]
Bruck-Muerzzuschlag | [0.02%, 77.22%] | [0.02%, 1.61%] | [0.02%, 0.94%]
Deutschlandsberg | [0.00%, 79.01%] | [0.00%, 1.23%] | [0.00%, 0.71%]
Graz-Stadt | [0.49%, 79.70%] | [0.49%, 3.01%] | [0.47%, 1.75%]
Graz-Umgebung | [0.00%, 73.20%] | [0.00%, 0.62%] | [0.00%, 0.37%]
Hartberg-Fuerstenfeld | [0.00%, 80.10%] | [0.00%, 0.65%] | [0.00%, 0.37%]
Leibnitz | [0.00%, 80.60%] | [0.00%, 0.65%] | [0.00%, 0.37%]
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%]

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