Covid-19 - (Part 2) A World-Wide Test of the Mask-Law Hypothesis -
Detailed Statistics, Czechia to Kazakhstan
Introduction
(note that if you have previously read the introduction to Part 1, this is essentially the same text, so you can skip right to the graphs and results)
The use of masks and mask-laws to prevent Covid-19
infections is obviously a huge, multi-faceted, complex and controversial
subject. There are a lot of lines of evidence, sometimes contradictory, ranging
from detailed medical studies (e.g. hospital ICU studies), to engineering
studies (e.g. viral aerosol particle hang-time indoors), to ecological
studies (how did mask-laws seem to affect infection rates in various
situation).
So, intrepidly and perhaps foolishly, I thought I would add
to the discussion via looking at how mask-laws seemed to have actually worked
around the world, on a national level. I used publicly available data
sources (as outlined below) and relied on primarily graphical evidence,
supplemented by simple curve-fitting, to examine and judge the evidence as
objectively as I possibly could. I also state the methods that I used and the
overall hypothesis that the data will be used to test.
The evidence and my judgments are shown below, and in three
more blogs, on a case-by-case basis. I broke the data in to 4 blogs, as
there are some 123 countries that had mask-laws in the data, and I didn't want
to overwhelm the reader with that much data all at once.
An accompanying blog will give my analysis of the overall
conclusions, taking into account the evidence of the 123 countries as a
whole. However, by presenting the evidence in this way, it is my
intention that anyone (statistician or interested layperson) can look at the
data and draw their own conclusions.
As for my own qualifications, I am a statistician (or data
scientist, a popular new term) who does operational research for a large
Canadian university. Covid-19 isn't my area of expertise but data
analysis of this sort (practical observational studies of the effect of
targeted interventions on a population) is a fairly broad area, one with which
I am familiar. But, as I say, interested readers should scan through the graphs
and narratives and form their own conclusions.
Statement of Hypothesis and Description of Method of Evaluation
For my purposes the Mask-Law Hypothesis can be stated as:
"A national mask-law will result in the number of new Covid-19 cases
coming down within the one month period after the mask-law date, relative to
the trend in the one month period before that date". It has often
expressed in more informal terms as "a mask-law will help bend the curve
downwards".
Since observational studies of this sort have many potential
confounders, the hypothesis can be restated in a form that attempts to control
for these confounders, by analyzing a large number of cases, on the assumption
that the effects of these confounding variables will tend to cancel out as the
number of cases increases. The restated form of the hypothesis is then: "National
mask-laws will result in new Covid-19 cases coming down within the one month
period after the mask-law date, relative to the trend in the one month period
before that date, in the vast majority of countries in which the mask-law is
implemented".
Below are the detailed, country-by-country statistics for
Covid-19 cases by days before and after that particular country's mask-law, as
indicated on the source Masks4All website (which provided the mask-law date)
and the GitHub repository of Covid-19 statistics (which provided the aggregate
number of cases during the study period, by day).
The graphs show:
- Line
graphs of aggregate case counts in the 4 week periods before and after the
law, to obtain an overall sense of how the numbers evolved during the
study period. Linear and quadratic functions are fit to the data, to
give a visual and mathematical basis for determining whether the general
trend was up, flat or down in that country during the 8 week period.
- Line
graphs with forecasts of cases for the four weeks after the mask-law,
based on the trend during the four weeks before the law, along with the
actual results during that succeeding four weeks. This enables a
comparison of forecasts and actuals, to determine whether or not the
pre-law trend in cases appeared to be affected by the mask-law.
Generally speaking, the two best fitting functions of the linear,
quadratic, exponential and logarithmic cases are shown (determined by
R-square, a conventional measure of model fitness, where the best forms
have R-square closest to 1.00).
- A bar
graph with the number of new cases each week during that time, as well as
the functional form that best fits that data. This gives a visual
and mathematical picture of how new cases varied on a weekly basis.
I use this
data to judge how well the evidence supports the mask-law's apparent
effectiveness for each country. Readers may sometimes dispute my opinion,
though I think the majority of cases are not too difficult to categorize.
However, although the evidence for or against the mask-law hypothesis is
sometimes quite clear-cut, at other times it is only somewhat persuasive and
sometimes it is too inconclusive to render any sort of judgment.
The upper left hand corner graphs, giving the functional
forms (visually and mathematically) for the aggregate data are measures of how
well the mask-laws seem to be doing in their expected roles (i.e. bending the
curve downwards).
- Concave
Down -The second order polynomial fit to the data (the yellow line)
tends to bend downwards. This is generally supportive of the
mask-laws (positive evidence), though in some cases this could
just be the continuation of a trend that began before the law was passed.
- Linear
- The second order polynomial and the linear function nearly overlie each
other, indicating that the number of new cases was essentially steady
during the period. This would not be supportive of the
mask-law's effectiveness (neutral evidence), though it could be argued
that the mask-law prevented an exponential increase.
- Concave
Up - The second order polynomial fit to the data (the yellow line)
tends to bend upwards. This is generally not supportive of the
mask-laws (negative evidence). Indeed, it could be argued
that in these cases the mask-law not only didn't work, but actually
contributed to making the situation worse (perhaps by creating a false
sense of security).
The upper right hand corner graphs show the forecast of cases, based on the
four weeks of data before the mask-law date. As noted, some alternative
trend lines are given, those which best fit the pre-law data (based on their
R-square, a statistical measure of goodness-of-fit).
I should note that forecasts of this sort, based on limited data, can be
misleading, so you have to use your common sense as well as the statistical
R-square measure, when deciding whether they are appropriate. For
example, exponential functions don't generally apply over a very long range of
data in the real world (resources eventually run out), so care has to be taken
when utilizing them. Also, a quadratic function can predict a decline in
aggregate cases, which is physically impossible, so care must be taken to not
rely on this form in these instances.
The actual case counts during the four weeks after the mask-law are also
shown. Again, there are three main possibilities:
- Actuals
Below the Trend - If the number of cases are below the forecasts, that
indicates that the mask-law appears to have slowed or reversed the
trend. This is generally supportive of the mask law's
effectiveness (positive evidence for the mask-law hypothesis).
- Actuals
Mirror Trend - If the number of cases are very close to the forecast,
that indicates that the mask-law had no observable effect on the
trend. This would not be supportive of the mask- law's
effectiveness, but wouldn't indicate that the mask-law was
counter-productive (neutral evidence as regards the mask-law hypothesis).
- Actuals
Above the Trend - If the number of cases exceeds the forecasts, that
indicates that cases grew faster than the pre-existing trend and that the
the mask-law did not slow the trend, it may have even sped it up. This
is generally not supportive of the mask-law's effectiveness (negative
evidence against the mask-law hypothesis).
The lower
left hand bar graphs give the number of new cases, on a week-by-week basis,
before and after the mask-law. It also has as best-fit function for the
new cases data, to aid the visual interpretation. There are three main
ideal-type cases, as well as many intermediate possibilities:
- The
graph has a hump-like shape, with new cases rising before the mask-law
date, peaking shortly after the mask-law, then falling. This
is generally supportive of the mask-law's effectiveness (positive evidence
for the mask-law hypothesis).
- The
graph is more or less flat, with no strong pattern of growth or shrinkage
over the course of the eight week period. This would not be
supportive of the mask-law's effectiveness, but wouldn't indicate that the
mask-law was counter-productive (neutral evidence as regards the mask-law
hypothesis).
- The
graph shows steady, often exponential, growth after the mask-law date, compared
to before that date. This is generally not supportive of the
mask-law's effectiveness (negative evidence against the mask-law
hypothesis).
The three
lines of evidence are then considered and an overall evaluation is given, via a
five-level categorization. The categories are given below:
- Category
2: Strong positive evidence. The aggregate case graph is
generally downward, actual cases are below the pre-law forecasts and new
cases clearly fall after the mask-law date.
- Category
1: Weak positive evidence. The majority of the lines of evidence
favour the mask-law, but the evidence is relatively weak and/or has some
contrary indications.
- Category
0: Neutral evidence. None of the lines of evidence show very
persuasive evidence, one way or the other.
- Category
-1: Weak negative evidence. The majority of the lines of
evidence do not favour the mask-law, but the evidence is relatively weak
and/or has some contrary indications.
- Category
-2: Strong negative evidence. The aggregate case graph is
generally upward, actual cases are above the pre-law forecasts and new
cases clearly rise (often exponentially) after the mask-law date.
Since there
are 123 countries in the study, I will have to break the data into
several blogs, just to avoid overloading blogger with graphs. This
is Part 2, with countries from Czechia to Kazakhstan (alphabetically ordered).
Note that not all countries passed mandatory mask laws and therefore not all
countries could be included in this study.
Part 1, with countries from Algeria to Cuba is in the link below:
https://dodecahedronbooks.blogspot.com/2021/09/covid-19-1-world-wide-test-of-mask-law.html
Czechia
Mask Law Date: 18/03/2020
Aggregate Cases Functional Form: Concave Up
Forecasts vs Actuals: Actuals higher than forecasts.
New Cases Bar Graph: New cases higher after mask-law date than before.
Categorization: -2 (Strong Negative Evidence)
Comment: The yellow quadratic function in the line graph is trending upwards indicating rising Covid rates. Although there is a hump-like appearance to new case bar graph, it rises throughout most of the post-mask law date period, which is contrary to expectations for the mask law. Actuals are much higher in the post-law period than would be expected by the best-fit pre-law forecast. So, this case is considered weak negative evidence, in terms of the mask-law hypothesis.
Denmark
Mask Law Date: 22/08/2020
Aggregate Cases Functional Form: Concave Up
Forecasts vs Actuals: Actuals about the same as forecasts.
New Cases Bar Graph: New cases higher after mask-law date than before.
Categorization: -2 (Strong Negative Evidence)
Comment: The quadratic has an upward slope and the bar graph of new cases rises in an exponential fashion after the mask law date. Actuals are about the same in the post-law period than would be expected by the best-fit pre-law forecast. Thus, this case is categorized as strong negative evidence against the mask-law hypothesis.
Djibouti
Mask Law Date: 10/05/2020
Aggregate Cases Functional Form: Concave Up
Forecasts vs Actuals: Actuals are higher than forecasts.
New Cases Bar Graph: New cases are higher after mask-law date than before.
Categorization: -2 (Strong Negative Evidence)
Comment: The quadratic function has a strong concave up appearance and the bar graph of new cases rises after the mask law date, rather than falls. Actuals are much higher than the pre-law forecasts. So, this case is categorized as strong negative evidence.
Dominica
Mask Law Date: 10/05/2020
Aggregate Cases Functional Form: Concave Up
Forecasts vs Actuals: Actuals are above forecasts.
New Cases Bar Graph: New cases are higher after mask-law date than before.
Categorization: -1 (Weak Negative Evidence)
Comment: The quadratic function is up and the new case count is up after the mask law. Actuals exceed forecasts. However, there is obviously very little evidence to go on here, so it will be categorized as weak negative evidence.
Dominican Republic
Mask Law Date: 18/04/2020
Aggregate Cases Functional Form: Concave Up
Forecasts vs Actuals: Actuals below forecasts.
New Cases Bar Graph: New cases are higher after mask-law date than before.
Categorization: -1 (Strong Negative Evidence)
Comment: The quadratic function is concave up and the bar graph for the new case count rises fairly steadily after the mask law. Actuals track slightly below the best-fit forecast, however. Thus, this is categorized as weak negative evidence.
Ecuador
Mask Law Date: 08/04/2020
Aggregate Cases Functional Form: Concave Up
Forecasts vs Actuals: Actuals exceed forecasts.
New Cases Bar Graph: New cases are higher after mask-law date than before.
Categorization: -2 (Strong Negative Evidence)
Comment: Again, the quadratic function is concave up and the bar graph of new cases shows a steep rise in the post-mask law period, though with some inconsistency. Post-law actuals are much higher than the pre-law forecasts. Thus this case is also categorized as strong negative evidence against the mask-law hypothesis.
Egypt
Mask Law Date: 26/04/2020
Aggregate Cases Functional Form: Concave Up
Forecasts vs Actuals: Actuals are higher than forecasts.
New Cases Bar Graph: New cases are higher after mask-law date than before.
Categorization: -2 (Strong Negative Evidence)
Comment: The quadratic function is concave up and the bar graph of new cases shows a strong and steady exponential rise. Actuals are higher than forecasts. Given these facts, this case is categorized as strong negative evidence.
El Salvador
Mask Law Date: 08/04/2020
Aggregate Cases Functional Form: Concave Up
Forecasts vs Actuals: Actuals are about the same as forecasts.
New Cases Bar Graph: New cases are up after mask-law date compared with before.
Categorization: -2 (Strong Negative Evidence)
Comment: Again, the quadratic function is concave up and the new cases bar graph shows a steady exponential rise in cases. Actuals track forecasts closely, though in the last week actuals are higher than forecast. Thus, this is categorized as strong negative evidence.
Equatorial Guinea
Mask Law Date: 02/05/2020
Aggregate Cases Functional Form: Concave Up
Forecasts vs Actuals: Actuals are about the same as forecasts.
New Cases Bar Graph: New cases are higher after mask-law date than before.
Categorization: -2 (Strong Negative Evidence)
Comment: The quadratic function is concave up and the new cases bar graph shows a considerable increase in cases after the mask law compared to before. Actuals track forecasts closely, though in the last week actuals are slightly higher than forecast. Therefore this is categorized as strong negative evidence against the mask-law hypothesis.
Ethiopia
Mask Law Date: 02/05/2020
Aggregate Cases Functional Form: Concave Up
Forecasts vs Actuals: Actuals are below forecasts.
New Cases Bar Graph: New cases are higher after mask-law date than before.
Categorization: -1 (Weak Negative Evidence)
Comment: The quadratic function is concave up, though that is mainly due to the influence of the final data point. The bar graph does show a drop in cases after the mask law, but then turns around sharply to show a substantial increase. However, actuals are below forecasts. So, overall this is categorized as weak positive evidence.
France
Mask Law Date: 11/05/2020
Aggregate Cases Functional Form: Concave Down
Forecasts vs Actuals: Actuals about the same as forecasts.
New Cases Bar Graph: New cases are lower after mask-law date than before.
Categorization: 0 (Neutral Evidence)
Comment: The quadratic function is concave down in this instance, but the new cases bar graph shows that cases after the mask law were about the same as in the two week period previous to the law (i.e. a continuation of an ongoing trend). This is reinforced by the actuals vs forecast graph, which shows actuals tracking the pre-law best-fit function very closely. So, this is categorized as neutral evidence, neither for or against the mask law hypothesis.
Gabon
Mask Law Date: 11/05/2020
Aggregate Cases Functional Form: Concave Up
Forecasts vs Actuals: Actuals are greater than forecasts.
New Cases Bar Graph: New cases are higher after mask-law date than before.
Categorization: -2 (Strong Negative Evidence)
Comment: The quadratic function is concave up and new cases show a large exponential rise after the mask law date. Actuals after the mask-law are much higher than forecast. Thus, this instance is considered strong negative evidence against the mask-law hypothesis.
Gambia
Mask Law Date: 11/05/2020
Aggregate Cases Functional Form: Concave Up
Forecasts vs Actuals: Actuals are higher than forecasts.
New Cases Bar Graph: New cases are higher after mask-law date than before.
Categorization: -2 (Strong Negative Evidence)
Comment: This is another case where the quadratic function is concave up and the new case graph shows a very large increase in cases after the mask law date. Also, actuals are much higher than forecasts. So it is categorized as strong negative evidence against the mask-law hypothesis.
Georgia
Mask Law Date: 20/04/2020
Aggregate Cases Functional Form: Close to Linear
Forecasts vs Actuals: Actuals are lower than forecasts.
New Cases Bar Graph: New cases are lower after mask-law date than before.
Categorization: 2 (Strong Positive Evidence)
Comment: Although the quadratic and Linear functions overlay each other, there is a noticeable drop in new cases after the mask law date, resulting in the familiar hump-like shape (though there is an uptick a the end). Actuals are well below both the quadratic and exponential forecasts. This will therefore be categorized as strong evidence in favour of the mask law hypothesis.
Germany
Mask Law Date: 27/04/2020
Aggregate Cases Functional Form: Concave Down
Forecasts vs Actuals: Actuals are below forecasts.
New Cases Bar Graph: New cases a fewer after mask-law date than before.
Categorization: 0 (Neutral Evidence)
Comment: Although the quadratic is concave down, that trend obviously began before the mask law date. In fact, the new cases bar graph shows an exponential decline. Actuals are somewhat below the forecast curve, however. So, since the post-law decline can be seen as a continuation of an existing trend, this will be considered neutral evidence.
GhanaMask Law Date: 23/04/2020
Aggregate Cases Functional Form: Concave Up
Forecasts vs Actuals: Actuals are below forecasts.
New Cases Bar Graph: New cases are up after mask-law date than before.
Categorization: -1 (Weak Negative Evidence)
Comment: Although the quadratic function is concave up, the new case count grows exponentially for most of the time interval, though it does drop off in the final week. The exponential and quadratic forecasts fit the data equally well, so the actuals are either following the trend or are somewhat above it. So, this case is somewhat mixed, though the drop-off in the final week causes it to be categorized as weak negative evidence rather than strong negative evidence.
GreeceMask Law Date: 23/04/2020
Aggregate Cases Functional Form: Concave Down
Forecasts vs Actuals: Actuals above forecasts.
New Cases Bar Graph: New cases are lower after mask-law date than before.
Categorization: 1 (Weak Positive Evidence)
Comment: The quadratic function is concave down, and the new case counts in the bar graph are much lower in the post-law phase than earlier. That said, the actuals are somewhat higher than the best-fit forecast lines. The evidence is mixed, so it is categorized as weak positive evidence.
Grenada
Mask Law Date: 23/04/2020
Aggregate Cases Functional Form: Concave Down
Forecasts vs Actuals: Actuals are below forecasts.
New Cases Bar Graph: New cases are lower after mask-law date than before.
Categorization: 0 (Neutral Evidence)
Comment: The quadratic function is concave down and cases fall and remain low after the mask law date, though the data is sparse and inconsistent. The actuals follow the best-fit forecast trend, the quadratic. Also, the Ns are small, so this will be categorized as neutral evidence.
Guatemala
Mask Law Date: 23/04/2020
Aggregate Cases Functional Form: Concave Up
Forecasts vs Actuals: Actuals higher than forecasts.
New Cases Bar Graph: New cases higher after mask-law date than before.
Categorization: -2 (Strong Negative Evidence)
Comment: The quadratic has a concave up form and cases increase exponentially throughout the period. Post-law actuals are well above the best-fit pre-law trend. All in all, strong negative evidence against the mask-law hypothesis.
Guinea
Mask Law Date: 18/04/2020
Aggregate Cases Functional Form: Concave Up
Forecasts vs Actuals: Actuals exceed forecasts.
New Cases Bar Graph: New cases are up after mask-law date compared to before.
Categorization: -2 (Strong Negative Evidence)
Comment: Again, the quadratic has a strong concave up appearance, the new case counts increase throughout the study period, and actuals are well above forecasts. Thus, this is also strong negative evidence.
Guinea-BissauMask Law Date: 15/05/2020
Aggregate Cases Functional Form: Concave Down
Forecasts vs Actuals: Actuals are lower than forecasts.
New Cases Bar Graph: New cases are lower after mask-law date than before.
Categorization: 1 (Weak Positive Evidence)
Comment: The quadratic form is slightly concave down and the new case count is lower after the law date than before. Actuals are well below the best-fit forecast line. However, the weekly case count graph shows that the trend to lower counts arguably started well before the mask law date, making the characteristic hump shift a bit and the peak occur before the mask-law date. Thus this will be considered weak positive evidence at best.
Haiti
Mask Law Date: 15/05/2020
Aggregate Cases Functional Form: Concave Up
Forecasts vs Actuals: Actuals are above forecasts.
New Cases Bar Graph: New cases rise after mask-law date.
Categorization: -2 (Strong Negative Evidence)
Comment: The quadratic is strongly concave up and the new case count in the bar graph is a very well defined rising exponential form. Also, actuals are well above the forecasts. Thus, this is strong negative evidence against the mask-law hypothesis.
Honduras
Mask Law Date: 13/04/2020
Aggregate Cases Functional Form: Concave Down
Forecasts vs Actuals: Actuals about the same as best-fit forecast.
New Cases Bar Graph: New cases drop after mask-law date.
Categorization: 0 (Neutral Evidence)
Comment: The quadratic is concave down, but the bar graph of new cases shows that the low case count after the mask law is the continuation of an existing trend (as does the actual vs forecast graph). Also the case counts are very small, so this will be considered to be neutral evidence.
Indonesia
Mask Law Date: 25/04/2020
Aggregate Cases Functional Form: Concave Down
Forecasts vs Actuals: Actuals are higher than forecasts.
New Cases Bar Graph: New cases are higher after mask-law date than before.
Categorization: -2 (Strong Negative Evidence)
Comment: The quadratic is concave up and the case count rises up to the mask-law date, then plateaus. Though cases don't rise exponentially at that point, neither do they fall. In addition, the actuals exceed the best-fit forecast, so this constitutes strong evidence against the mask-law hypothesis.
Iran
Mask Law Date: 05/07/2020
Aggregate Cases Functional Form: Close to Linear
Forecasts vs Actuals: Actuals are about the same as forecasts.
New Cases Bar Graph: New cases after the mask-law date as before.
Categorization: 0 (Neutral Evidence)
Comment: The quadratic function essentially overlies the linear function. New case counts drop a bit after the mask law, but quickly return to their pre-law levels and somewhat more. Actuals mirror the forecast function. So, at best this is neutral evidence for the mask law hypothesis.
Iraq
Mask Law Date: 20/04/2020
Aggregate Cases Functional Form: Concave Up
Forecasts vs Actuals: Actuals are higher than forecasts.
New Cases Bar Graph: New cases rise after mask-law date than before.
Categorization: -2 (Strong Negative Evidence)
Comment: The quadratic is concave up and the new cases bar graph shows that cases were actually on the downtrend just prior to the mask law, then rose strongly after that (a saddle-like shape). Actuals are much higher than the best-fit forecast. So, this represents strong evidence against the mask-law hypothesis.
Israel
Mask Law Date: 01/04/2020
Aggregate Cases Functional Form: Concave Up
Forecasts vs Actuals: Actuals lower than forecasts.
New Cases Bar Graph: New cases lower after mask-law date than before.
Categorization: 2 (Strong Positive Evidence)
Comment: The quadratic is slightly concave up, but the new cases bar graph shows the hump-like form characteristic of strong positive evidence. Also, actuals are well below the forecasts. Thus, this is categorized as strong positive evidence for the mask-law hypothesis.
Mask Law Date: 04/05/2020
Aggregate Cases Functional Form: Concave Down
Forecasts vs Actuals: Actuals are about the same as forecasts.
New Cases Bar Graph: New cases are lower after mask-law date than before.
Categorization: 0 (Neutral Evidence)
Comment: The quadratic is slightly concave down, but it is readily apparent in the new cases bar graph that cases were falling throughout the period. This is supported by the actuals versus forecasts graph. Therefore the reductions after the mask law are a continuation of an existing trend, and the data is considered to be neutral evidence.
Jamaica
Mask Law Date: 22/04/2020
Aggregate Cases Functional Form: Close to Linear
Forecasts vs Actuals: Actuals are below forecasts.
New Cases Bar Graph: New cases are lower after mask-law date than before.
Categorization: 2 (Strong Positive Evidence)
Comment: The quadratic and linear functions overlie each other. New cases rise, peak at about the mask law date, then fall off sharply, producing a hump-like shape. Actuals are generally lower than forecast. Therefore this is strong evidence for the mask law hypothesis.
Kazakhstan
Mask Law Date: 22/04/2020
Aggregate Cases Functional Form: Concave Up
Forecasts vs Actuals: Actuals are below forecasts.
New Cases Bar Graph: New cases are higher after mask-law date than before.
Categorization: 0 (Neutral Evidence)
Comment: The quadratic is slightly concave up and case counts actually rise steadily after the mask law date (they had fallen just before that date). Actuals are below forecasts, however. So, this is mixed, inconclusive evidence, and can be considered neutral evidence as regards the mask-law hypothesis.
A Quick Summary of these 30 Cases
(+2) Strong Positive Evidence in favour of the mask-law hypothesis: 3 countries
(+1) Weak Positive Evidence in favour of the mask-law hypothesis: 2 countries
(0) Neutral Evidence, neither for or against the mask-law hypothesis: 7 countries
(-1) Weak Negative Evidence against the mask-law hypothesis: 4 countries
(-2) Strong Negative Evidence against the mask-law hypothesis: 14 countries
No comments:
Post a Comment