A Data Driven Investigation on the Relationship between Tests,

Conﬁrmed Cases and Positive Rate: The Case of Greece

Nikolaos Papadakis

a

and Dimitrios Bakirtzis

Hellenic Mediterranean University, School of Applied Sciences, Department of Mechanical Engineering, Greece

Keywords:

COVID-19, Conﬁrmed Sases, Positive Rate, RT-PCR, Greece.

Abstract:

While the pandemic of COVID-19 has proved a global challenge for humanity, it has also provided an un-

precedented opportunity to study infectious diseases because of the globally coordinated effort to collect daily

data and make it publicly available. As a result, it is possible to access data and try to improve on models and

approaches. It is probably, so far, the largest effort to conduct testing on population at a global scale for an

infectious disease. The aim of this project is to report on the relationship between the daily values of the test

samples, the conﬁrmed cases and the positive rate for COVID-19 in Greece. A discussion on the volatility of

the reported metrics by media coverage is carried out, to highlight the potential pitfalls of using the conﬁrmed

cases or positive rate in isolation. For the case of Greece, a dependence in the number of tests and conﬁrmed

cases on the weekdays is identiﬁed. That dependence seemed to decrease with increasing number of tests.

A comparison takes place between Greece and Austria which is a country with a similar size and also data

regarding the RT-PCR tests and antigen tests are publicly available.

1 INTRODUCTION

During the COVID-19 pandemic, the media cov-

erage usually revolves around the daily number of

conﬁrmed cases and the daily number of casualties.

Those quantities (the conﬁrmed cases and deaths) are

used as indicators of the current state of the pandemic.

Usually there is no mention of the daily number of

tests. However, it is logical that there should be a

relationship between tests and conﬁrmed cases. I.e.

more tests are more likely to produce more conﬁrmed

cases. Therefore quoting in isolation the number of

conﬁrmed cases can be misleading.

For example, 500 conﬁrmed cases carry a totally

different interpretation and signiﬁcance in the context

in case of 1000 or 1000000 performed tests. There-

fore, quoting the number of cases would be more

meaningful if the number of daily performed tests re-

mained constant or if they were somehow included in

the quantity that is reported.

Obviously, controlling the number of tests in a

meaningful manner is not possible in this context, so

the use of Positive Rate conveys useful information

from both conﬁrmed cases and tests, because it is de-

ﬁned as the ratio of conﬁrmed case to the number of

tests. However, the positive rate can also be mislead-

a

https://orcid.org/0000-0001-5909-5559

ing when there is high ﬂuctuation in the number of

tests.

Another common metric is the Rate of Transmis-

sion (R

t

or R

0

) which is depended mainly on the num-

ber of conﬁrmed cases. However, the rate of transmis-

sion is not usually reported by media. This is prob-

ably R

t

values keep changing as new data become

available. Therefore, by the time, the R

t

values stop

changing (usually after a week or so depending on the

algorithm), there is little point in reporting them to the

public.

1.1 Description of Case of Greece

In the particular case of Greece, the media coverage

of the COVID-19 focuses on the number of conﬁrmed

cases. As mentioned before, focusing only on that

quantity can be misleading as an indicator of the pan-

demic’s current state. In Greece, it is more evident

because there is a weekly variation in terms of daily

performed tests, which in turn affects the conﬁrmed

cases.

Figure 1 presents the absolute number of per-

formed RT-PCR tests grouped per weekday for a pe-

riod of 62 weeks.

It can be seen that, while Tuesday to Saturday ap-

pear to have similar max values, Monday and Sun-

Papadakis, N. and Bakirtzis, D.

A Data Driven Investigation on the Relationship between Tests, Conﬁrmed Cases and Positive Rate: The Case of Greece.

DOI: 10.5220/0010975300003123

In Proceedings of the 15th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2022) - Volume 5: HEALTHINF, pages 777-784

ISBN: 978-989-758-552-4; ISSN: 2184-4305

Copyright

c

2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

777

Figure 1: Absolute number of RT-PCR tests grouped per

weekday.

day have comparably lower max values. Also, the

100 percentile for Monday and Sunday is approxi-

mately equal or less than the median of the other

days). So, the reported number of tests in Greece is

usually greater at days following the working week-

days (Tuesday to Saturday), and falls on the days fol-

lowing weekend days (Sunday and Monday).

Figure 2 presents the absolute number of con-

ﬁrmed cases grouped per weekday. It can be seen

that Tuesday has the highest max value. Wednesday

to Saturday appear to have similar max values, while

Monday and Sunday appear to have similar max val-

ues which are lower than the rest.

Figure 2: Absolute number of conﬁrmed cases grouped per

weekday.

Similarly to the pattern observed in tests, the ab-

solute number of conﬁrmed cases is usually greater

in days following working weekdays, and falls during

the days following weekend days.

Figure 3 presents a representative part of the time

series for conﬁrmed cases, RT-PCR tests and posi-

tive rate which corresponds to July of 2021. The red

shaded areas correspond to weekend, where the num-

ber of tests is smaller, and it can be seen that the pos-

itive rate increases signiﬁcantly on Sundays.

1.2 Normalising Conﬁrmed Cases and

Tests with Respect to Weekly

Average

When examining a quantity (e.g. the conﬁrmed cases

or the number of tests) to determine the inﬂuence of

the weekdays, the absolute numbers contain two dif-

ferent sources of variability. I.e. any variation within

Figure 3: Detail of time series for conﬁrmed cases, RT-PCR

tests and Positive rate, the red areas center around a Sunday.

the week (short term) and also the current state of the

pandemic which has a greater time scale (long term).

For example, if the conﬁrmed cases are consid-

ered, then a value of 10 in a week when there are only

a few cases (e.g. 10) can have a totally different mean-

ing within the context of a week with a rash of cases

(e.g. 1000).

Therefore, in order to isolate the effect of the

weekday on the quantity, a normalisation is consid-

ered. Each daily value within a week is normalised

with respect to the week’s average. The aim of this

normalisation is to report the number of conﬁrmed

cases relative to the week’s average. Reporting the

daily quantity as a percentage of the weeks average al-

lows a more meaningful intra-week comparison (i.e.

independently of the disease state). The duration of

the week is selected because it is considered as a time

unit.

The fact that the week can be considered a good

time unit can be evidenced by applying an Auto Cor-

relation Function (ACF) (Park, 2018), (Papoulis and

Pillai, 2002), (Box et al., 2015), (Percival, 1993), ,

or measure the Allan Variance (Allan, 1966),(Scharf

et al., 1995), (Percival and Mondal, 2012),(Lu et al.,

2013), (Malkin, 2012). In the ACF case, there is a

higher correlation value every 7th day (see Figure 4

top). The Allan variance shows that the minimum

standard deviation is when the samples are 6 days

apart (so every 7th day) — see Figure 4 bottom.

Figure 5 presents the tests per weekday nor-

malised with respect to the average tests for the cor-

responding week. On Monday and Sunday the cases

are signiﬁcantly lower (less than 50%).

Figure 6 presents the conﬁrmed cases per week-

day normalised with respect to the average of the cor-

responding week. It can be seen that on Monday

and Sunday the cases are signiﬁcantly lower (close

to 50%)

From Figures 5 and 6 it can be seen that although

the normalised values of Sundays and Mondays are

lower than the other days, the median of the nor-

malised values are in general the same between tests

and conﬁrmed cases. E.g. although the normalised

COVID-ex 2022 - Special Session on "COVID-19 epidemic data mining and EXploration"

778

Figure 4: ACF (top) and Allan variance (bottom) for the

conﬁrmed cases.

Figure 5: tests cases normalised with respect to week aver-

age vs. weekday.

conﬁrmed cases for Tuesday are approximately about

30% of the week average, the number of tests are also

increased by approximately the same. The only ex-

ception is Monday, in which the median of the nor-

malised conﬁrmed cases are about 45% of the week’s

average compared to the normalised tests which are

about the 70% of the weeks average.

This is in agreement with Figure 3, which shows

a more smooth progression of the positive rate, fol-

lowed by a brief peak of the positive rate on Mondays.

This is attributed to the smaller number of tests that

are reported on Mondays (close to 50% of the weeks

Figure 6: Conﬁrmed cases normalised with respect to week

average vs. weekday.

average). Additionally the tests that are reported on

Monday are carried out on Sunday, usually on peo-

ple with more grave symptomatology (so they are less

random).

Therefore, reporting the positive rate —i.e. con-

ﬁrmed cases over number of tests (

conﬁrmed cases

number of tests

)—

can also have huge ﬂuctuations due to the dependence

between tests and conﬁrmed cases mentioned above

(this is not a case of random sampling, usually people

submitted to a RT-PCR test have either a symptom or

an exposure).

The aim of this paper is to investigate the relation-

ship between the conﬁrmed cases and the number of

tests. To make the comparison feasible the ratio of the

daily reported value is normalised with respect to the

smoothed value, and the respective ratio for the tests.

Additionally, differences between the total num-

ber of tests and RT-PCR tests are reported and con-

trasted with respect to their effect on the conﬁrmed

cases. The dataset of Greece is selected because apart

from data on conﬁrmed cases there are available data

for RT-PCR tests and rapid tests.

2 METHODOLOGY

2.1 Raw Data Collection

The data is collected through the publicly available

dataset at covid-19-greece herokuapp. The data sets

that are downloaded are:

• conﬁrmed cases

• rapid tests

• RT-PCR tests

The data that were collected cover a period be-

tween the 1st of January 2020 until the 6th of October

2021.

The daily data have been checked against the Our

World in Data (OWID) dataset, and they were found

A Data Driven Investigation on the Relationship between Tests, Conﬁrmed Cases and Positive Rate: The Case of Greece

779

in good agreement. The Greek Heroku dataset was

selected because it offers more granular data for the

number and type of tests.

2.2 Preprocessing

2.2.1 Data Cleaning and Outliers

In the dataset some values were detected as abnormal

and they were removed from the dataset. More specif-

ically:

• negative number of tests were recorded

• on 2021-05-02 about minus 3.5 million rapid tests

and minus 4.6 million RT-PCR tests were re-

ported, and on 2021-05-03 about the same number

was added.

In the above cases, the relevant data were dropped

from the dataset.

Additionally, on the 4th, 6th, 7th and 8th of June

2020 the number of conﬁrmed cases is non zero,

while the number of tests is zero. On those dates the

tests were set equal to the number of conﬁrmed cases.

2.2.2 Smoothing Algorithm

The data exhibited a repeated pattern, i.e. 5 days a

week the number of RT-PCR and rapid (antigen or

Lateral Flow Device) tests were considerably greater

than the other two. Figure 7 presents the time series of

the conﬁrmed cases and the RT-PCR tests (although

not presented here the variability of the total tests is

even greater).

Figure 7: Time series of conﬁrmed cases and PCR tests dur-

ing the entire data collection period.

This is attributed to the fact that usually the cases

and tests carried out on a day are reported on the next

day. Therefore the tests and conﬁrmed cases of the

weekend are reported on Sunday and Monday. Lower

number of tests are more likely to yield a lower abso-

lute number of conﬁrmed cases.

The smoothing methodology uses a weighted

moving average with a centred Gaussian window (4

stds), with a window period of 15 days. The dates at

the ends of the time series are removed.

The OWID dataset uses a similar methodology

with a moving average of 7 days, which is not cen-

tred.

2.2.3 Handling of Tests

The Greek Heroku dataset includes data for both rapid

tests and RT-PCR tests. Figure 8 presents the num-

ber of RT-PCR and rapid tests in log scale. For the

rapid tests, a signiﬁcant variation is evident as time

progresses.

Figure 8: Comparison of RT-PCR and rapid tests in log

scale.

Apart from the fact that the RT-PCR tests ap-

pear to be stable in numbers, there are other signif-

icant differences in RT-PCR and rapid tests (at least

in Greece). Rapid tests are performed either a) in

a free testing location that everyone can have them-

selves tested, b) at pharmacies or c) as part of the

self testing programme (only a and b are considered

valid rapid tests). On the other hand, RT-PCR tests

are more expensive and their use is resource-limited.

As a result, there is signiﬁcant variation in the number

of rapid tests, while RT-PCR tests have a more or less

steady count.

As a predictive variable, RT-PCR tests have the

following advantages:

• relatively constant number of tests throughout the

time series

• selective nature

• better sensitivity and speciﬁcity than the corre-

sponding rapid tests.

According to (Brihn et al., 2021) among 1732

paired samples from asymptomatic patients, the (anti-

gen based) rapid test sensitivity was 60.5%, and

speciﬁcity was 99.5% when compared with RT-PCR.

Among 307 symptomatic cases, sensitivity and speci-

ﬁcity were 72.1% and 98.7%, respectively. Others

(Kortela and et. al, 2021), (Mistry et al., 2021) in

a metastudy also report that rapid tests exhibit great

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780

variation in sensitivity (between 37.7% and 99.2%)

and speciﬁcity (92.4% and 100.0%).

Because of the above reasons, only RT-PCR tests

are considered in the following analysis.

2.3 Normalised Positive Rate

The daily positive rate (PR) is deﬁned as the ratio of

conﬁrmed cases to the number of tests:

PR =

conﬁrmed

tests

There are many ways this can be expressed, e.g.

as a percentage, or as a value of conﬁrmed cases per

a speciﬁc number of tests. In this work the Positive

Rate (and all other similar quantities) are expressed

as conﬁrmed cases per 1000 tests.

A similar quantity (Smoothed positive rate - SPR)

can be calculated using the smoothed conﬁrmed and

the smoothed number of tests:

SPR =

smoothed conﬁrmed

smoothed tests

The normalised positive rate (NPR) for a speciﬁc

day can be deﬁned as:

NPR =

PR

SPR

=

conﬁrmed

tests

smoothed conﬁrmed

smoothed tests

NPR =

conﬁrmed

smoothed conﬁrmed

·

smoothed tests

tests

The NPR indicates the relationship between the

actual value of the positive rate compared to the

smoothed positive rate. Values greater than one indi-

cate that the positive rate calculated only by the daily

data is overestimating the positive rate, and vice versa.

Regarding the units of NPR, because it’s a ratio of

similar quantities, it is a unitless quantity.

2.4 Tests for Comparing Distributions

The 2 sample Kolmogorov-Smirnov test is used to

test whether the distribution of a quantity on different

weekdays exhibits the same empirical cumulative dis-

tribution (Hodges, 1958). The considered quantities

are the conﬁrmed cases, no. of tests and the positive

rate.

To perform the test, every data point (e.g. con-

ﬁrmed cases)—which is associated to a date—is nor-

malised with respect to the average of the relative

week. Then all the dates that correspond to a cer-

tain weekday are grouped together, (there are approx-

imately 60 weeks in the dataset, so there are about

60 data points in each group). Finally, the 2 sam-

ple Kolmogorov-Smirnov test is performed on a pair

of weekday groups. The results are presented in a

heatmap. Due to multiple comparisons between data

the Bonferroni correction is used(Dunn and Dunn,

1961), (G. Miller, 2012). The results are tested

against a Bonferroni-adjusted alpha level of 0.000476

(0.01/21).

2.5 Model Fitting

A very basic model is ﬁtted using a reciprocal func-

tion (

a

0

x

+ a

1

). The reason this model is selected is be-

cause of its simplicity, and also because it represents

the basic relationship between the two quantities (i.e.

the more tests are carried out the more the process

resembles a SRS scheme, while for lower tests num-

bers it is expected that the process will be less random

therefore there will be a higher positive rate).

Conﬁdence intervals are computed and plotted

based on the covariance matrix and an α value of 0.01.

The model parameters are reported on the graph.

3 RESULTS AND DISCUSSION

3.1 Testing of Weekday Distributions

for Conﬁrmed Cases and Tests

Figure 9 presents the results from a 2 sample

Kolmogorov-Smirnov test (Hodges, 1958) which was

used to determine the proximity of the empirical dis-

tributions per weekday for the conﬁrmed cases and

tests.

Figure 9: Kolmogorov Smirnov 2 sample tests p-value

heatmap for the conﬁrmed cases (left) and tests (right) per

weekday. (Smaller values support stronger statistically sig-

niﬁcant rejection of the hypothesis that the distributions

come from the same distribution).

When tested against a Bonferroni adjusted alpha

level 0.000476 (0.01/21), it can be seen that there is

strong statistical evidence that for the conﬁrmed cases

the distribution of Sunday is signiﬁcantly different to

all other days—with the exception of Monday. Sim-

ilarly, Monday’s distribution is signiﬁcantly different

to Tuesday and Wednesday.

A Data Driven Investigation on the Relationship between Tests, Conﬁrmed Cases and Positive Rate: The Case of Greece

781

Similarly, regarding the tests, it appears that Sun-

day’s and Monday’s distributions differ signiﬁcantly

from all the other weekdays (p-value is smaller than

5e-7 in all cases).

3.2 Smoothing

Figure 10 shows the results of the smoothing on the

daily conﬁrmed cases and the daily tests.

Figure 10: Smoothing results for new cases and RT-PCR

tests.

3.3 Positive Rate and Smoothed Positive

rate

Figure 11 presents the daily (raw) positive rate (PR)

and the smoothed positive rate (SPR). The extreme

values can be seen and they generally correspond to

days following weekends (i.e. Monday and Sunday).

Figure 11: Comparison between Actual and Smoothed Pos-

itive rate.

Figure 12 presents the distribution of the positive

rate grouped by day and the results from a 2 sample

Kolmogorov-Smirnov test (Hodges, 1958) which was

used determine the similarity of the positive rate’s em-

pirical distributions per weekday. It is obvious that the

days following the weekend days (Sunday and Mon-

day) exhibit high variations of the positive rate com-

pared to the average of the week.

Figure 12 shows that there is strong statistical ev-

idence that for the positive rate the distribution of

Monday is different to all other days.

Figure 12: Dispersion of positive rate per weekday (left),

Kolmogorov Smirnov 2 sample tests p-value heatmap for

the positive rate per weekday (right).

It is noteworthy that the higher numbers for pos-

itive rate which are observed on the last data points

of the time-series in ﬁgure 11, are probably due to

the signiﬁcantly higher number of rapid tests. Fig-

ure 13 compares the smoothed positive rate expressed

in cases per 1000 tests using only the RT-PCR tests

and the total number of tests. It can be seen, that al-

though at the beginning the curves are identical, as

time progresses the trends become completely differ-

ent. This can be attributed to the changing number of

rapid tests as evidenced in Figure 8, which since July

2021 are performed in greater numbers (close or over

one order of magnitude).

Figure 13: Positive Rate and Smoothed Positive Rate time

series (expressed in conﬁrmed cases per 1000 tests) using

the total number of tests.

3.4 NPR Model Fitting

Figure 14 presents data and the ﬁtted model for the

normalised positive rate with respect to the number

of RT-PCR tests. Compared to the total tests, the

RT-PCR tests plot shows less variability above 10000

tests. So it can be seen that the daily value of the posi-

tive rate tended to be greater when the number of tests

was smaller in number.

Figure 15 presents the ﬁtted model for the Nor-

malised Positive Rate vs. the normalised no of Tests

(Normalised no of tests is deﬁned as the number of

tests with respect to the smoothed number of tests).

COVID-ex 2022 - Special Session on "COVID-19 epidemic data mining and EXploration"

782

Figure 14: Normalised positive rate vs. the RT-PCR tests.

Figure 15: Normalised positive rate vs. the normalised RT-

PCR tests (normalisation is with respect to the smoothed

quantity).

3.5 Model Comparison with Austria

A similar procedure was undertaken for Austria. The

data were obtained from the ofﬁcial Austrian govern-

ment site. During clean-up the only difference was

that negative values were removed for testing, and

also the data on 27th of June 2021 was considered

an outlier and was removed.

Figure 16 presents the ﬁtted reciprocal model for

the normalised positive rate vs the RT-PCR tests for

Greece and Austria.

It can be seen that although the shape is similar,

the number of tests is signiﬁcantly higher and there-

fore there can not be a direct comparison. Presum-

ably this can also be tracked to the positive rate, in

Figure 16: Comparison between the ﬁtted reciprocal models

for NPR vs. the total number of RT-PCR tests for Greece

and Austria.

the sense that since Austria has a signiﬁcantly higher

daily number of RT-PCR tests compared to Greece

(Austria: 78722 compared to Greece: 14155), the

positive rate is expected to be lower for Austria.

Figure 17 presents the ﬁtted reciprocal model for

the normalised positive rate vs the test ratio for Greece

and Austria. The overlapping of the models can be

seen, and it seems that the bulk of the data coincides.

In this case comparisons can me made more readily.

Figure 17: Comparison between the ﬁtted reciprocal models

for NPR vs the test ratio for Greece and Austria.

4 CONCLUSIONS

The relationship between the number of tests and the

conﬁrmed cases for the case of Greece has been inves-

tigated. The variability of the test cases between dif-

ferent weekdays seemed to correlate with the variabil-

ity per weekday of conﬁrmed cases and (ultimately)

the positive rate.

The only consistent exception was the behaviour

on Sundays. A statistically signiﬁcant difference was

identiﬁed in the number of RT-PCR tests that were

performed on Sunday. Similarly, the number of con-

ﬁrmed cases was signiﬁcantly reduced on Sunday.

Additionally, the distribution on the results of the pos-

itive rate on Sunday appear to be statistically different

from the other days. Those results led to the conclu-

sion that the number of conﬁrmed cases and tests did

not change proportionally on Sunday (while it appear

to do so on average on the other days).

Additional ways were considered to try and de-

termine the relationship between tests and conﬁrmed

cases. It was shown that, generally for lower num-

ber of tests the positive rate is expected to be higher

that the week’s average. For higher test numbers the

positive rate appeared to be close to the week’s aver-

age. However, the variation was too great to provide

a useful predictive tool.

A comparison with Austria indicated that even

countries with the same level of population can have

a signiﬁcantly different behaviour.

A Data Driven Investigation on the Relationship between Tests, Conﬁrmed Cases and Positive Rate: The Case of Greece

783

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