Journal of Healthcare Engineering

Journal of Healthcare Engineering / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 6660160 | https://doi.org/10.1155/2020/6660160

Ray-Ming Chen, "Whether Economic Freedom Is Significantly Related to Death of COVID-19", Journal of Healthcare Engineering, vol. 2020, Article ID 6660160, 9 pages, 2020. https://doi.org/10.1155/2020/6660160

Whether Economic Freedom Is Significantly Related to Death of COVID-19

Academic Editor: Saverio Maietta
Received29 Oct 2020
Revised23 Nov 2020
Accepted26 Nov 2020
Published12 Dec 2020

Abstract

COVID-19 has caused a huge mayhem globally. Different economic freedom leads to different performances of a country’s reaction to the pandemic. We study 164 countries and apply mathematical and statistical approaches to tackle the problem: whether economic freedom has a significant impact on the death of COVID-19. We devise a metric, some norms, and some orderings to construct an absolute reference and the actual relation via binary sequences. Then, we use the theoretical binary sequences to construct a probability distribution which linearises the strength of relation between economic freedom and death of COVID-19. Then, the actual relation from the data analysis provides an evidence to the hypothetical testing. Our analysis and model show that there is no significant relation between economic freedom and death of COVID-19.

1. Introduction

Due to COVID-19 pandemic, there are many fatalities across the world. Many countries are baffled by whether to open the market or impose lockdown [13]. It creates a huge chaos in either economic or social stability [4, 5]. This motivates us to study the relation between economic freedom and the death rate or tolls of COVID-19. We locate 164 countries from some datasets [6, 7]—because some of the countries lack statistics of either the economic freedom or the death information regarding COVID-19. Then, we use a series of mathematical and statistical approaches to reach a conclusion. For the mathematical part, we define a new concept of metric which could measure the difference between the scoring structures—this is hardly the case if one adopts the usual Euclidean metric. For reference purpose, one fixes the referential structure (or scoring system) first. Then, one could compute the distances between all the (sampled) multivalued data points and , i.e., . Based on these distances, we could then create an ordering for with respect to the referential structure .

2. Modelling

2.1. Notations and Symbols

For a vector , we use to denote its length; for any set , we use to denote its size (cardinality). Moreover, we use to denote the -th element in . Let denote a binary vector, i.e., each element in is either 0 or 1. Let denote the set of all the binary vectors with total length . Let be a set of countries. Let be a set of attributes of economic freedom (regarded as independent variables). Let be a set of result (regarded as dependent variables). Each time we fix one to study the relation between the attributes and . In this article, we restrict our attribute values to be numerical numbers. The theoretical table is shown in Table 1, and for the actual forms, one could refer to Sections 3.2.1 and 3.2.2.


Countries


We use the notations ; ; and .

2.2. Binary Subvectors and Norm

Definition 1. (subvectors). Suppose is a binary vector. We use to denote all its truncated subvectors consisting of only 1.

Example 1. Suppose . Then, .

We simply abbreviate it as . Indeed reveals the structure of an independent-dependent variable relation.

Definition 2. (binary norm). For any binary vector with , define a binary norm .

One could, according to real situations, adopt other numbers (for example, replace 2 with other numbers) or other forms other than the one provided here.

Claim 1. .

Proof. It follows immediately from the definition.

Definition 3. (linear ordering on ). if and only if , for all .

Example 2. If , then and . Thus, .

Remark 1. A binary norm indeed serves an important technique in revealing the relation between dependent and independent variables. Given two pairs and of numerical data with , if (i.e., they act proportionally), we associate it with a value 1 to indicate such relation and 0, otherwise (i.e., they act inversely). Such mechanism gives a way to look into the fundamental relation between and variables. This kind of analysis is in particular useful when the precision of the data is questionable or when the actual numbers are unknown or more suitable to be interpreted via ranks.

Example 3. (sign vector). Suppose , a set of ordered vectors. Then, we could associate with a sign vector via Remark 1.

Definition 4. (relational vector). Suppose that is a sign vector; we associate it with a relational vector whose -th element is assigned 1 iff and 0, otherwise.

Example 4. Let us continue with Example 3. We could compute its relational vector , and thus and . The higher the value of the norm is, the closer the relation between the dependent and independent variables is.

Definition 5. (equivalence relation ). For all , iff .

Let . One observes that partitions . If , we use to denote the equivalence class whose elements’ norms are all .

2.3. Probability Distribution

Suppose is the sampling population. Define a statistic on by its binary norm. The range for is . Define a counting by . Now, we could define the probability distribution for by by

One observes that

This probability distribution reveals the relation between the independent variables and the dependent variables. This would serve our theoretical distribution for our statistical testing : the economic freedom and the death of COVID-19 has no significant relation, i.e., the economic freedom has no great impact on the death of COVID-19. For a concrete construction of such probability distribution, one could refer to Section 6.1.

2.4. Metric

A metric or a distance function is a non-negative function on satisfying identity, symmetry, and triangle properties. In this article, it suffices to define a metric on a closed interval of real number. Fix , where and . Let be a finite vector whose first element is , last element is , and all the other elements are incrementally increased and lie between and . Let be the set of all such vectors. Let be arbitrary. Let denote the vector whose elements are the projections from and . One observes that is closed under .

Definition 6. (atomic norm)

Definition 7. (metric). Define by

Example 5. Suppose the closed interval and and . Then, . Hence, the norm ; the norm ; and . Thus, .

Claim 2. is a metric on .

Proof. This can be shown by the definitions and some techniques.

This metric will be used in Section 3.3. This metric basically measures the differences between the structures of the attributes in the scoring system. The more similar the structures are, the lower the distances are. Unlike the static Euclidean distance, this metric takes the interval structures into consideration.

2.5. Procedures

Let us summarise the whole procedure of our modelling for the sake of data analysis. Let . Let denote the death rate (or tolls, depending on the context) for the country .(1)Define a metric on a real interval, in particular the transformed interval, an interval for attribute values which lie between 0 and 100, for the range of attribute values of economic freedom, and calculate (one could refer to Section 3.3).(2)Rank via the sorted distances with a rank function in which iff .(3)Rank via the sorted distance with a rank function in which iff .(4)Form the vector .(5)Convert into a sign vector , where if and if .(6)Construct the probability distribution for the quotient space .(7)Perform statistical testing by locating the position of and significant level for the batch of country.(8)Apply the Monte Carlo approach on the sampled batches of countries repeatedly.(9)With the threshold probability 0.5, based on binary distribution for the whole spectrum of statistical testing, perform the overall statistical testing.(10)Draw a conclusion for the relation between and .

3. Data Analysis

Following the procedures in Section 2.5, we start to collect, analyse, and produce a report via data analysis. Since the data are huge and hard to handle by the one-off approach, we resort to the sampling technique and reach a conclusion via statistical testing.

3.1. Sampling

The raw data consist of 164 countries (we use 1 to 164 to name the countries) up to 2020, June 27th. Since the size is too huge, we apply the Monte Carlo approach to sample the 164 countries. We do 20 times (or 20 batches: S1 to S20) sampling with 25 countries over the 164 countries per sampling. The sampled batches are listed in Tables 2 and 3.


OrderS1S2S3S4S5S6S7S8S9S10

1682569102765725664163
2141127939855857515115019
32195556141151591126510
41294130457351301369117
51271715174119571281396952
6991513389201601471002718
789141562612495192711388
82431327335118122835365
9128764484153291291684105
1010828487015015947110150
1111262126671517111813415626
129213161219086918789161
13981046698089391075282
14181198450375071522127
15148128399784138111113257
16124121343455615481529
17118959275911581016620
18321698181134112546127116
19769226875971441135638
20144285142853246709741
2141601374861217614092
2213710314413738758684102128
2385120532132124132119317
2478163117122111122997416322
2552118627766193512388103


OrderS11S12S13S14S15S16S17S18S19S20

176918359641191291541537
2231245363461238823055
3291511262414313763543119
41212612955531081224148163
57365914386541524076155
61578618100597676913318
74954102504061201986
868119508552156938914699
91587104743215270146145122
104887440681112714357162
11104118711536101251342625
129967125791358513670101156
13142591585215636638131147
14136139201593937110613110
15414357699630672020134
16115148151108506650142679
17151565449275153546344
187715273647162269825157
19126169013615412413256704
2010882938713295891518520
21953916198138138441117167
22603744216212076113118153
233661654120691411088497
24142861091213484741148
257410103119157136735258136

3.2. Sampled Data
3.2.1. Economic Freedom

Corresponding to the form listed in Table 1, we associate with S1 and define , where Property Rights, Judicial Effectiveness, Government Integrity, Tax Burden, Government Spending, Fiscal Health, Business Freedom, Labor Freedom, Monetary Freedom, Trade Freedom, Investment Freedom, and Financial Freedom. The attribute values are based on a 100-point scoring system [6]. Due to the limitation of space, we list only the first sampling (or S1) regarding its attributes of economic freedom in Table 4. We omit the other 19 similar tables of this form. serves as the set of our independent variables.


S1

6864.845.747.579.934.583.660.264.679.986.48070
14133.720.825.586.194.512.453.5607752520
2157.346.745.670.454.64.660.549.577.267.86050
12964.672.349.699.861.819.366.663.38175.44550
12769.862.641.876.980.683.876.766.48273.26540
9958.334.736.776.179.387.56758.470.987.67560
8986.474.490.364.145.49966.845.376.486.49580
257.13338.885.974.686.365.752.181.288.47070
1284126.537.488.377.17364.843.468.164.26030
10893.379.193.97157.898.390.486.78792.28080
11262.542.742.291.57187.780.66777.786.26560
9243.142.124.87971.723.641.962.869.175.45050
875.862.653.391.582.179.382.560.273.888.28070
1820.111.223.186.354.214.258.352.969.967.81540
14859.54843.480.785.996.48363.774835560
12472.556.155.190.370.485.658.66378.186.47050
148.33024.891.479.299.954.761.681661010
3269.961.173.476.480.890.57564.785.2898570
7662.150.546.88074.880.277.77477.768.48050
1484.562.580.246.717.27775.261.180.586.48570
4186.384.6934219.797.788.786.284.686.49080
13758.43846.663.767.4646258.875.975.84550
8544.630.82590.872.5045.647.875.677.46050
7864.854.649.691.873.555.960.152.577.681.27060
5236.545.129.477.490.879.248.657.662.760.83520

3.2.2. COVID-19

Now we start to introduce the dependent variables. Indeed we tackle an individual dependent variable each time. Since the data are huge, we only extract the data [7] for sampling one (or S1) as shown in Table 5. Corresponding to the form listed in Table 1, we associate with S1 and define Total Confirmed COVID-19 Cases, Death Toll of COVID-19, Total Recovered COVID-19 Cases, and Population of the Countries. Due to the limitation of space, we list only the first sampling (or S1) regarding its dependent variables. We omit the other 19 similar tables of this form. Moreover, in the later analysis, we only take and fix as our dependent variable. If the readers are interested in other dependent variables (or , , or other mixed forms), they could simply follow the same approach provided in this article.


S1

68Hungary412757826639660521
141Sudan9257572401443828543
21Brazil128005456109697526212541690
129Senegal635498419316734279
127Sao Tome and Principe71213219219087
99Mexico20839225779120562128914507
89Luxembourg41731103968625810
2Albania22695112982877821
128Saudi Arabia174577147412047134805142
108New Zealand15202214845002100
112North Macedonia575826822062083375
92Malawi10051326019119281
98Mauritius341103261271749
18Bolivia28503913733811670618
148Thailand316258304069798329
124Romania2569715791818119238321
1Afghanistan304516831030638910996
32Chile263360506822343119114153
76Jamaica686105392961033
14Belgium6110697311691811589102
41Denmark12675604115085792000
137South Africa12459023406411159297807
85Lebanon16973311446825627
78Jordan1104983010201800
52Ethiopia5425891688114903773

3.3. Metric

Since we have defined an interval metric in Section 2.4, we could apply it over here. Here we measure the distance between every sampled data and the fixed reference vector . We construct the distances for the 164 countries based on economic freedom (for example, the data of sample one could be referred from Table 4) in Tables 6 and 7. Since all the data are presented in the form of 100-point score for the attribute values in Table 4, we need to transform the values in the table to the interval . For example, the reference vector (we still use to represent to newly transformed vector) will be . Each country sampled in S1 will be transformed into , for example, are the converted data for the first country sampled in the first sampling or country 68. The economic freedom vector for each sampled country is converted by the same way. The converted data are not tabulated. Then, we apply in Section 2.4 on the converted data and repeat the whole processes for other samplings. The complete results regarding the distance for the 20 sampled countries are presented in Tables 6 and 7. The cell in the tables means the value , where denotes the -th country sampled in -th sampling and denotes the converted data for .


S1S2S3S4S5S6S7S8S9S10

61.080.182.274.278.373.253.959.266.872.2
76.276.875.472.473.684.072.883.480.471.3
70.772.471.159.285.168.666.562.373.274.2
80.882.171.482.364.977.274.385.770.265.0
76.871.883.478.572.867.176.978.982.280.7
75.083.469.765.584.449.267.777.478.375.8
65.585.157.371.371.172.471.378.374.461.8
80.470.474.868.777.265.067.867.357.573.2
76.969.581.574.873.171.880.873.174.868.1
73.361.360.171.480.483.270.272.372.780.4
62.373.871.976.583.464.265.077.657.371.3
63.074.949.977.871.980.182.662.765.582.3
72.474.274.082.278.465.566.045.780.772.5
75.872.874.879.954.179.964.280.770.778.3
68.876.966.065.174.870.474.469.274.867.1
71.177.877.371.482.359.283.260.183.271.8
71.375.872.463.072.882.671.675.433.684.4
74.373.172.475.874.482.168.874.076.874.5
69.563.071.362.782.664.959.474.459.274.0
85.177.784.059.784.074.374.071.465.182.1
82.166.071.354.160.172.070.769.564.263.0
67.782.759.467.774.072.880.174.882.376.9
84.073.357.580.474.871.174.872.860.964.9
67.772.265.067.869.267.875.078.572.268.2
80.765.073.878.633.671.377.272.361.882.7


S11S12S13S14S15S16S17S18S19S20

69.582.667.382.666.872.880.881.183.254.1
82.371.157.583.274.072.371.672.571.473.6
71.883.471.974.452.967.749.977.270.472.8
68.771.980.873.657.573.367.890.368.872.2
68.773.270.252.980.171.680.649.169.558.9
62.980.175.877.482.669.564.982.232.375.8
59.671.682.379.949.149.973.371.371.649.9
61.072.879.984.080.757.375.465.573.575.0
83.262.778.878.574.380.671.473.580.167.8
60.171.678.549.161.069.278.352.967.182.0
78.884.062.768.680.775.480.177.671.380.1
75.076.568.883.974.584.085.771.475.457.3
59.782.671.680.757.371.433.674.074.967.7
85.778.984.466.566.054.172.772.060.974.2
82.170.467.182.272.771.476.584.484.477.6
68.668.883.473.379.933.679.959.749.983.9
83.459.271.659.678.374.157.571.683.281.5
78.683.278.380.772.382.071.372.480.162.9
71.973.171.985.781.171.174.859.271.490.3
73.372.575.462.774.872.465.583.484.084.4
72.466.082.372.478.278.281.569.264.276.5
66.054.181.570.773.873.369.574.465.073.1
80.749.973.171.673.382.276.273.374.865.1
71.377.780.184.768.777.674.864.982.168.8
78.574.282.772.862.985.768.780.782.985.7

4. Absolute Reference

By the derived distances presented in Tables 6 and 7, we could construct absolute references. The absolute references would server as the benchmarks for other internal structures. Let us use to denote the set of sampled countries in -th sampling. Let be arbitrary.

Definition 8. (ordering of the sampled countries). iff .

Based on this ordering, we could generate the absolute references (Tables 8 and 9). Let us take S1 for example: . From these absolute references (or ordering for the samplings), we could view the structure (or interval) difference between the ideal scoring (or ) and real scoring results. Indeed, an absolute reference is a reference acting like ordering without specific scales. Such reference is useful when the precise values are unknown or when the precision of the data is questionable. In this article, we use relative distances between a country’s economic freedom and others to create such ordering.


S1S2S3S4S5S6S7S8S9S10

682863766160721076688
112921565637561445615692
921185314248717148537
8960144877711811256117
7876489211111839873157
1374311797124891598388105
148173989905714711114022
761633313775122122769726
21955122119115125708919
1241191731531391236429
11626265512421479163
9812030703819191192182
108621263113291581616365
3210989813261754611038
9913195108495461136510
18186284357513084113116
141127461827658110113218
12712813267803213210084128
1284284121150359913412727
2121937445501282727150
522534775986357415052
12941445015141861395241
4110369285911295269161
8515115169201591151102103
141485451485151361520


S11S12S13S14S15S16S17S18S19S20

4965340406666401336
142378714314366143637
4856574915637535631156
6887838753156714271155
1573912515968137897118157
601489115157111122895797
1154315821397673111148147
73124545464124761976122
128126981230267043148
765490119473705426163
112616108965486170119
298293551201231109830153
1261191824629512082855
9516271004611913210814610
10865747432120841468499
991010450135108933813118
746750521325114111310167
774286361381016735145134
1041391296927134271342525
368644595013850524144
419110263861522515458162
23591617952162152696379
15151038536691291511520
15115115110915485442085136
1361120136591361364204

5. Ordering for COVID-19 Fatalities

Based on Table 5 and other omitted tables, we start to construct the ordering (or ranking) based on the fatalities of COVID-19.

Definition 9. (ordering on fatalities). iff , where is the death toll for -th country sampled in -th sampling.

Based on this ordering, we have the results presented in Tables 10 and 11. Let us take the cells in S1 for example: .


S1S2S3S4S5S6S7S8S9S10

784353679015158353117
98131117982035351011520
7625126263511811815110222
127151151691515791766926
92176914215091391614092
1087698928416014713464150
85127268448851598412757
2923987855613248150163
14811862561325012987163161
5216842386519568482
12916348505979525610
8962851078913013613288
11242303737958112311038
14110132898019128473152
68959597113131221138865
416093746641144112527
110315631116146746519
18286181241227110089103
128411774512412511197116
1241844121153717246941
13712051221191159910711318
3212114473143275119156128
1412846137552921706629
991193445757515813921105
21143370278686272727


S11S12S13S14S15S16S17S18S19S20

4943834953101532043162
15126536912162253510125
12615909896516715113120
10467203664762642067
121511266336697669254
15116104108391348414215134
769115115913510815298145163
1423910285628513213476147
368757871323011010826148
10856162450152505611810
488287505237129825737
1364216110959123714663136
951485052471361363813397
6010129591389589528479
23659313615613893785156
745997940156689306
6837103746854661914644
41547454120614111314818
165440466685431155
778441003211144406122
7312418115143312011141153
11511125143157120122618119
2911915811986124731437199
9913927552713770705855
1578686211541192715470157

6. Norm and Probability

In this experiment, we only consider and construct its distribution accordingly. Hence, the domain is and the range lies between 0 and (indeed some of the values’ probability is 0). This section generalises Example 3. The higher the value is, the higher the impact of independent variables on dependent variable is.

6.1. Probability Distribution

We have already constructed the theoretical setting of probability distribution for our testing in Section 2.3. Based on that framework and the data given, we could create the theoretical probability distribution in Figure 1.

The (one-tailed) critical values for 5 and 10 percentages are 138 and 78 (via numerical computation), respectively; that is, if the sampled value is larger than the critical values, we should reject : there is no significant relation between the economic freedom and death of COVID-19.

6.2. Real Results

In comparison with the absolute reference, we could generate the binary sign vectors for the real data from each sampling (or simply ) in Table 12—for the formula and explanation of sign vectors, one could refer to Section 2.2. However, in these 0 and 1 representations, it separates the proportional and inversely proportional relation between the economic freedom and death of COVID-19. To take all the factors into consideration, one further analyses the alternative behaviour of 0 and 1. If there are too many alternations between 0 and 1, it would indicate that there is a less relation between those two. On the other hand, if the alternative times are few, then it leads to the longer length of subvector consisting of pure 1. The alternative results are shown in Table 13.


1234567891011121314151617181920

11101001111101111010
01100001000110100101
11111110011011000110
00000101000101110001
10010011111010001010
11010100100110101100
00001010110001010100
01100111011000100011
11110000100011111001
10001001010101101110
01100111101111110111
01000110010000011100
10111011010111100111
00000001101001011010
10110100100100000100
01001011010010110111
10011000011100101100
11111110000000100000
00100001111111111001
00000110100000010111
11011101010100110100
10100000001011101001
00110111101111001110
01001110110110111100


1234567891011121314151617181920

01110111000100100000
01100000100010011100
00000100100001001000
01101001000000000100
10111000100011011001
00100001101000000111
10010010010110001000
01101000000100100101
10000110001001101000
00010001000101100110
11011110000000010100
00000010111000000100
01000101000001000010
01001010110010100001
00000000001001001100
00101100110001100100
10011001100011110011
00100000000000100110
11011000100000010001
00100100001011011100
10000010100000100010
01101000011011011000
10000110100110001101

Based on Table 13 and definitions in Section 2.2, we could compute the binary norm for each sampling batch (or ) as shown in Table 14.


1234567891011121314151617181920

8111581189614566915178111778

7. Conclusion and Future Work

The contribution of death in COVID-19 is very complicated. We use economic freedom to capture a potential factor in such contribution. To verify the truth of great impact from economic freedom, we devise a metric, two norms, absolute ordering, binary ordering, and probability distribution for the statistical testing population. Based on our research, we find out that the economic freedom has no significant relation to the death of COVID-19. This might provide some reference for the decision makers of the countries. In the future research, one could further study the relation between economic freedom and other ratios related to COVID-19. One could also use other nonparametric approaches to enrich the statistical testing. There is another related paper on the same topic [8]. In that paper, the authors use two-step estimators: negative binomial regression and nonlinear least squares, and find out there is a close relation between economic freedom and fatalities of COVID-19. In essence, their approach focuses more on statistical techniques, while ours focuses more on mathematical approaches. For the future researcher, he could compare or combine these methods to yield a comprehensive or generalised theory that could accommodate and single out the factors that cause the discrepancies.

Data Availability

The data supporting the findings of this study are included within the article.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

This study was supported by the Humanities and Social Science Research Planning Fund Project under the Ministry of Education of China (grant no. 20XJAGAT001).

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Copyright © 2020 Ray-Ming Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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