Coronavirus and Inferential Statistics

With the Coronavirus pandemic in 2020, the question has been raised as to whether the virus is seasonal; in other words, does Coronavirus spread more slowly as the temperature increases? I've decided to investigte this question as an example of regression analysis. This example should not be used as a good example of methodology in terms of selecting appropriate data to answer questions (there's quite a large number of methodological flaws in the choice of data); it's more a mathematical demonstration.

Here are the raw data. On March 3, there were 52 countries that had reported at least 2 cases of the virus. These are listed along with the average temperatures of their capital cities in March (except for China, where I used Wuhan's average temperature, and Italy, where I used Milan's). Under the assumption that the number of Coronavirus cases grows exponentially, we'll look at the natural logarithm of the number of cases (specifically, the difference in the logarithms) in order to be able to use linear regression.

Country	Average Temperature in March	Cases, March 3	ln (Cases, March 3)	Cases, March 25	ln (Cases, March 25)	ln (Cases, March 25) − ln (Cases, March 3)
China	6.7	80304	11.2935747118947	81848	11.3126191475588	0.0190444356640604
South Korea	5.7	4812	8.47886807709457	9137	9.12008738299862	0.641219305904052
Australia	17.6	33	3.49650756146648	2252	7.71957398925958	4.2230664277931
Malaysia	27.6	29	3.36729582998647	1624	7.39264752072162	4.02535169073515
Japan	8.7	268	5.59098698051086	1193	7.08422642209792	1.49323944158706
Singapore	28.0	108	4.68213122712422	558	6.32435896238131	1.64222773525709
Philippines	28.7	3	1.09861228866811	552	6.31354804627709	5.21493575760898
Vietnam	20.0	16	2.77258872223978	134	4.89783979995091	2.12525107771113
New Zealand	15.8	2	0.693147180559945	189	5.24174701505964	4.5485998344997
Italy	9.0	2036	7.61874237767041	69176	11.1444092606403	3.52566688296985
Spain	11.2	114	4.7361984483945	39673	10.5884261345462	5.85222768615169
Germany	5.1	157	5.05624580534831	31554	10.3594556428174	5.30320983746909
France	8.8	191	5.25227342804663	22025	9.99993345080438	4.74766002275775
Switzerland	5.3	30	3.40119738166216	8789	9.08125621856465	5.68005883690249
United Kingdom	6.9	39	3.66356164612965	8081	8.99727090623345	5.3337092601038
Netherlands	6.1	18	2.89037175789616	5560	8.62335338724463	5.73298162934846
Austria	5.7	18	2.89037175789616	5282	8.57206009285708	5.68168833496091
Belgium	6.8	8	2.07944154167984	4269	8.35913488675796	6.27969334507813
Norway	3.3	25	3.2188758248682	2566	7.85010354517558	4.63122772030738
Sweden	0.1	15	2.70805020110221	2272	7.72841577984104	5.02036557873883
Portugal	14.9	2	0.693147180559945	2362	7.76726399675731	7.07411681619736
Denmark	3.5	5	1.6094379124341	1591	7.37211802833779	5.76268011590369
Czechia (Czech Republic)	3.6	3	1.09861228866811	1394	7.23993259132047	6.14132030265236
Israel	15.0	10	2.30258509299405	1329	7.19218205871325	4.8895969657192
Finland	-1.3	7	1.94591014905531	792	6.67456139181443	4.72865124275911
Greece	13.2	7	1.94591014905531	743	6.61069604471776	4.66478589566245
Iceland	0.5	9	2.19722457733622	648	6.47389069635227	4.27666611901605
Russia	-1.0	3	1.09861228866811	658	6.48920493132532	5.39059264265721
Romania	5.4	3	1.09861228866811	762	6.63594655568665	5.53733426701854
Croatia	6.4	8	2.07944154167984	382	5.94542060860658	3.86597906692674
San Marino	10.0	8	2.07944154167984	187	5.23110861685459	3.15166707517475
Azerbaijan	7.0	3	1.09861228866811	87	4.46590811865458	3.36729582998647
Georgia	6.6	3	1.09861228866811	73	4.29045944114839	3.19184715248028
Thailand	29.5	43	3.76120011569356	934	6.83947643822884	3.07827632253528
Indonesia	26.4	2	0.693147180559945	686	6.53087762772588	5.83773044716594
India	22.1	5	1.6094379124341	562	6.33150184989369	4.72206393745959
Iran	10.0	1501	7.31388683163346	24811	10.1190423822017	2.8051555505682
Pakistan	17.0	5	1.6094379124341	991	6.89871453432999	5.28927662189589
Qatar	17.0	7	1.94591014905531	526	6.26530121273771	4.3193910636824
Bahrain	21.2	49	3.89182029811063	392	5.97126183979046	2.07944154167984
Egypt	16.9	2	0.693147180559945	402	5.99645208861902	5.30330490805908
Lebanon	16.0	13	2.56494935746154	304	5.71702770140622	3.15207834394469
Iraq	16.6	26	3.25809653802148	316	5.75574221358691	2.49764567556543
Kuwait	19.3	56	4.02535169073515	195	5.27299955856375	1.2476478678286
United Arab Emirates	22.3	21	3.04452243772342	248	5.51342874616498	2.46890630844156
Oman	25.0	6	1.79175946922805	99	4.59511985013459	2.80336038090653
United States	8.3	64	4.15888308335967	51914	10.8573437822964	6.69846069893675
Canada	-2.2	27	3.29583686600433	1739	7.46106551435428	4.16522864834995
Brazil	21.5	2	0.693147180559945	2201	7.69666708152646	7.00351990096652
Ecuador	14.3	6	1.79175946922805	1049	6.9555926083963	5.16383313916824
Mexico	18.1	5	1.6094379124341	370	5.91350300563827	4.30406509320417
Algeria	13.2	5	1.6094379124341	264	5.57594910314632	3.96651119071222

Plotting this data, we get:

In the linear regression model, the line of best fit has the form

Where, y is the dependent variable (in this case, the growth in cases), x is the independent variable (in this case, average temperature), a is the y-intercept, and b is the slope. a and b are calculated as follows:

Looking at our data again:

Row #	x	y	xy	x²
1	6.7	0.0190444356640604	0.127597718949205	44.89
2	5.7	0.641219305904052	3.6549500436531	32.49
3	17.6	4.2230664277931	74.3259691291586	309.76
4	27.6	4.02535169073515	111.09970666429	761.76
5	8.7	1.49323944158706	12.9911831418074	75.69
6	28.0	1.64222773525709	45.9823765871985	784
7	28.7	5.21493575760898	149.668656243378	823.69
8	20.0	2.12525107771113	42.5050215542226	400
9	15.8	4.5485998344997	71.8678773850953	249.64
10	9.0	3.52566688296985	31.7310019467287	81
11	11.2	5.85222768615169	65.5449500848989	125.44
12	5.1	5.30320983746909	27.0463701710924	26.01
13	8.8	4.74766002275775	41.7794082002682	77.44
14	5.3	5.68005883690249	30.1043118355832	28.09
15	6.9	5.3337092601038	36.8025938947162	47.61
16	6.1	5.73298162934846	34.9711879390256	37.21
17	5.7	5.68168833496091	32.3856235092772	32.49
18	6.8	6.27969334507813	42.7019147465313	46.24
19	3.3	4.63122772030738	15.2830514770144	10.89
20	0.1	5.02036557873883	0.502036557873883	0.01
21	14.9	7.07411681619736	105.404340561341	222.01
22	3.5	5.76268011590369	20.1693804056629	12.25
23	3.6	6.14132030265236	22.1087530895485	12.96
24	15.0	4.8895969657192	73.343954485788	225
25	−1.3	4.72865124275911	−6.14724661558684	1.69
26	13.2	4.66478589566245	61.5751738227443	174.24
27	0.5	4.27666611901605	2.13833305950802	0.25
28	−1.0	5.39059264265721	−5.39059264265721	1
29	5.4	5.53733426701854	29.9016050419001	29.16
30	6.4	3.86597906692674	24.7422660283311	40.96
31	10.0	3.15166707517475	31.5166707517475	100
32	7.0	3.36729582998647	23.5710708099053	49
33	6.6	3.19184715248028	21.0661912063698	43.56
34	29.5	3.07827632253528	90.8091515147908	870.25
35	26.4	5.83773044716594	154.116083805181	696.96
36	22.1	4.72206393745959	104.357613017857	488.41
37	10.0	2.8051555505682	28.051555505682	100
38	17.0	5.28927662189589	89.9177025722301	289
39	17.0	4.3193910636824	73.4296480826008	289
40	21.2	2.07944154167984	44.0841606836126	449.44
41	16.9	5.30330490805908	89.6258529461984	285.61
42	16.0	3.15207834394469	50.433253503115	256
43	16.6	2.49764567556543	41.4609182143861	275.56
44	19.3	1.2476478678286	24.079603849092	372.49
45	22.3	2.46890630844156	55.0566106782468	497.29
46	25.0	2.80336038090653	70.0840095226633	625
47	8.3	6.69846069893675	55.597223801175	68.89
48	−2.2	4.16522864834995	−9.16350302636989	4.84
49	21.5	7.00351990096652	150.57567787078	462.25
50	14.3	5.16383313916824	73.8428138901058	204.49
51	18.1	4.30406509320417	77.9035781869955	327.61
52	13.2	3.96651119071222	52.3579477174013	174.24
Σ	643.4	220.669855974774	2591.69559117111	11643.76

Plugging all those numbers into the formula above, we get

The line of best fit looks like:

The line of best fit is slightly negative. Note that, even though the slope is only slightly negative, because we've taken the logarithm of the ratio of cases, a slight negative might result in a big difference in the growth in cases if it were statistically significant (and this is a question we'll investigate below). So, a hot country (30°) would show only one-third the growth of that of a cold country (0°). Taking the exponential of the line of best fit, we get:

Now to determine whether the result is statistically significant. In the simple linear regression model, a (1 − α) · 100% confidence interval for the slope parameter is given by:

Typically we would use a 95% confidence interval, so α would be 0.05. If this interval does not contain 0, we would reject the null hypothesis that b (the slope) is zero (meaning there is no relationship). If the interval does contain zero, there is insufficient evidence for rejecting the null hypothesis.

To find the mean squared error (denoted as MSE in the equation above), we'll find the sum of the squares of (y_i − ^y_i and divide by the number of observations less 2 (= 50).

i	xi	yi	^yi	(yi − ^yi)²
1	6.7	0.0190444356640604	4.457253950441	19.6977036970566
2	5.7	0.641219305904052	4.494905983211	14.8509010068531
3	17.6	4.2230664277931	4.046846793248	0.0310533595992085
4	27.6	4.02535169073515	3.670326465548	0.126042910519187
5	8.7	1.49323944158706	4.381949884901	8.34464802531102
6	28.0	1.64222773525709	3.65526565244	4.05232165601611
7	28.7	5.21493575760898	3.628909229501	2.51548014786225
8	20.0	2.12525107771113	3.9564819146	3.35340637797271
9	15.8	4.5485998344997	4.114620452234	0.188338104231718
10	9.0	3.52566688296985	4.37065427507	0.714003692808212
11	11.2	5.85222768615169	4.287819802976	2.44737202494224
12	5.1	5.30320983746909	4.517497202873	0.61734434416393
13	8.8	4.74766002275775	4.378184681624	0.136512027705901
14	5.3	5.68005883690249	4.509966796319	1.36911538343684
15	6.9	5.3337092601038	4.449723543887	0.781430746475328
16	6.1	5.73298162934846	4.479845170103	1.57035098549025
17	5.7	5.68168833496091	4.494905983211	1.40845235042505
18	6.8	6.27969334507813	4.453488747164	3.33502323344271
19	3.3	4.63122772030738	4.585270861859	0.00211203283844446
20	0.1	5.02036557873883	4.705757366723	0.0989783270677977
21	14.9	7.07411681619736	4.148507281727	8.55919114818388
22	3.5	5.76268011590369	4.577740455305	1.40408199925974
23	3.6	6.14132030265236	4.573975252028	2.45657050771668
24	15.0	4.8895969657192	4.14474207845	0.554808803088813
25	−1.3	4.72865124275911	4.758470212601	0.000889170962431547
26	13.2	4.66478589566245	4.212515737436	0.204548296022178
27	0.5	4.27666611901605	4.690696553615	0.171421200774195
28	−1.0	5.39059264265721	4.74717460277	0.4139867740523
29	5.4	5.53733426701854	4.506201593042	1.06323459134201
30	6.4	3.86597906692674	4.468549560272	0.36309119945035
31	10.0	3.15166707517475	4.3330022423	1.39555277708684
32	7.0	3.36729582998647	4.44595834061	1.16351281182466
33	6.6	3.19184715248028	4.461019153718	1.61079756872576
34	29.5	3.07827632253528	3.598787603285	0.270931993387713
35	26.4	5.83773044716594	3.715508904872	4.50382427457647
36	22.1	4.72206393745959	3.877412645783	0.713435804530933
37	10.0	2.8051555505682	4.3330022423	2.33431551343581
38	17.0	5.28927662189589	4.06943801291	1.48800623197263
39	17.0	4.3193910636824	4.06943801291	0.06247652759043
40	21.2	2.07944154167984	3.911299475276	3.35570348887919
41	16.9	5.30330490805908	4.073203216187	1.51315017234655
42	16.0	3.15207834394469	4.10709004568	0.912047350451372
43	16.6	2.49764567556543	4.084498826018	2.51810292110125
44	19.3	1.2476478678286	3.982838337539	7.4812669055946
45	22.3	2.46890630844156	3.869882239229	1.96273355864573
46	25.0	2.80336038090653	3.76822175075	0.930957463016217
47	8.3	6.69846069893675	4.397010698009	5.29667210677034
48	−2.2	4.16522864834995	4.792357042094	0.393290022239993
49	21.5	7.00351990096652	3.900003865445	9.63181178273921
50	14.3	5.16383313916824	4.171098501389	0.98552206104668
51	18.1	4.30406509320417	4.028020776863	0.0762004645842644
52	13.2	3.96651119071222	4.212515737436	0.0605182370087725
Total				129.493244162627

Dividing 129.493244162627 by 50, that result is 2.58986488325253; this is MSE in the formula above. Now, we had found Σ(x_i − x)² above; it was 3682.9223. Finally, we need t_0.025,50. Consulting a table, t_0.025,40 = 2.021 and t_0.025,60 = 2.000. Since t_0.025,50 would fall between those two values, we'll use 2.01. So, a 95% confidence interval for the slope parameter is

Since this range includes zero, we cannot conclude, at the 95% confidence interval, that the slope is not zero, and so the nonzero slope that we found is not statistically significant. That doesn't necessarily mean that, in the real world, there is no relationship whatsoever between temperature and spread of the virus, but it does mean that, if there is one, our data don't provide statistically significant evidence of such a relationship.

I had collected these statistics for Coronavirus statistics by country but ended up not using them (for this, at any rate):