Response to Naudet et al.

Obviously, correlation doesn't mean causality. However, in sciences and especially in medecine, when a physio-pathological context is added, evidences were reinforced. Legitimate reviews were done on our previous modest study, we will try to reply here.

General considerations

The correlation between vitamin D and sunlight exposure has been broadly documented and known for a long time (Holick, 2016). Our hypothesis was initially based upon COVID-19 outcomes differences between BAME (Black, Asian, Middle East) and White people living at the same latitudes, that could be explained by vitamin D deficiency in the BAME population (Harris, 2006). Links between vitamin D deficiency and COVID-19 fatal outcomes  were widely documented since (Mitchell, 2020), it was not the case at the time of submission date. Furthermore, to assess this, the French National Academy of Medicine has recommended the rapid serum vitamin D (i.e. 25 OHD) testing in people over 60 years of age with Covid-19, and a loading dose of 50,000 to 100,000 IU in case of deficiency, which could help limit respiratory complications (Académie Nationale de Médecine, 2020).

Our manuscript (Lansiaux et al., 2020) had as objective to prospect around this possible link between COVID-19, sunlight exposure and vitamin D. To this aim, in view of our team heterogeneity, we used open public data (because we didn't have access to others) to illustrate this link.

The studied COVID-19 outcome was the «mortality rate». Indeed, it was the number of COVID-19 deaths divided by the number of COVID-19 confirmed cases, related to the prognosis of COVID-19.

Specific considerations

The sunlight exposure average has been measured since 2018 (not the same data as correspondents have used (Naudet et al., 2020)). Weather data are actually unreachable due to a recent website update (Meteo France, 2020). In more, COVID-19 outcomes data were extracted after the daily update so we have different numbers (as presented on our first manuscript), so the 25/05/2020 data were included in our manuscript and not in their letter and they were published online on the 26/05/2020 by Santé Publique France.

Firstly, concerning the so-called mistake of «assumption of binormal distribution» (Naudet et al., 2020), we are sure that it did not escape the authors that we have used a Shapiro–Wilk test, as it is assessed in our previous manuscript and described in Table 1 . As a reminder, it is a test of normality in frequentist statistics which test the null hypothesis that a sample came from a normally distributed population. Monte Carlo simulation has found that Shapiro-Wilk test has the best power for a given significance (Razali and Wah, 2011). As the p-value for the sunlight exposure is 0.3697 (which is higher than the traditional 0,05 p-value and especially higher than the 0,001 p-value that we have fixed as the limit), we can't reject the null hypothesis. In this way, as we have assessed the normal distribution of each variable (including sunlight exposure), we don't violate any Pearson's correlation law because:

Table 1

Shapiro–Wilk test for each variable.

Variable	W	P-value
Inhabitants (nb.)	0.8512	0.03801
Sexe ratio	0.9142	0.2413
Male (%)	0.9172	0.2632
Female (%)	0.9172	0.2632
Age
0–24y (%)	0.9217	0.2999
25–59y (%)	0.7322	0.001754
>60y (%)	0.8481	0.03477
Life expectancy	0.9689	0.8988
Male	0.993	1
Female	0.9381	0.474
Birth rate (‰)	0.8324	0.02243
Death rate (‰)	0.7943	0.008073
Life conditions
Imposed fiscal menages rate (%)	0.7465	0.002456
Health status
Current smokers prevalence	0.9639	0.8377
Diabetic	0.9804	0.9852
Chronic pulmonary insuffisance	0.8436	0.03059
Renal chronic insuffisance	0.9505	0.6439
Obesity prevalence	0.9275	0.3542
Healthcare access
Eldery equipment rate (‰)	0.9769	0.9682
Medical doctor density (‰oo)	0.9404	0.5032
places in short hopistalisation service	0.8676	0.06099
hospitalisation bed density (‰oo)	0.9834	0.9938
Sunlight exposure	0.929	0.3697
Infection
Confirmed cases
Infection rate (%)	0.8311	0.02161
Healed cases
Healed rate(%)	0.9477	0.6034
Deceased cases
Mortality rate (%)	0.9395	0.4916

we have established the link between the sunlight exposure and the COVID-19 mortality (by r computation: R = −0.635688603).

We have obtained the regression line y = −55,459x + 2888,5.

According to our correspondents, if we use their analysis upon the 12 regions as N, we obtain t = 2.604 and a p-value of 0,0263 (the correlation stays significant with this p-value).

In this way, the use of Spearman's correlation would be inappropriate here.

In a second hand, we formally disagree on the follow statement «at least for sunlight exposure, since two neighboring regions have more similar climates than two distant regions» (Naudet et al., 2020). Indeed, for example in the «Provence-Alpes-Côte d'Azur» region, we can observe a diversity of climate (mountain, mediterranean); in more, «Auvergne-Rhônes-Alpes» region has mountain, but borderline regions have different climates (for instance, semi-oceanic and oceanic for «Nouvelle-Aquitaine» region) (Météo France, 2020). To prove this statement, firstly, we invite our opponents to use of basic knowledges which assess that regions, with an average size of 44,605.083 km² (Table 2 ), have climate more independent than neighboring towns. This is all the more reinforced as we reason on sunlight exposure average by region. Then, to use the Shapiro–Wilk test (as we have done in our manuscript (Lansiaux et al., 2020)), we have to prove the pseudo-independence (indeed, the perfect independence can't be prove) of a same variable different values between them. In order to do this, we have used the «turning point test» (published by Irénée-Jules Bienaymé in 1874 (Seneta, 2001)), the time on the abscissa axis was replaced by French regions (therefore it's not a time-series data as usual but a geographical series data). This method «is reasonable for a test against cyclicity but poor as a test against trend». As we wanted to study the inner-independence of variables (and especially for the sunlight exposure), we had to break free from cyclicity thanks to the turning points test. Our null hypothesis was that they are independently, identically distributed random variables. As we have 128 (or 140 if Corsica is included) turning points (Table 3 ), we obtained a z of 0.205 with 12 regions (without Corsica) and z of 0.393 with Corsica inclusion (Table 4 ). As the observed number of turning points is in the 95% trust interval, we can't reject the null hypothesis; we may have used, in our manuscript, inner-independent identically distributed random variables.

Table 2

Area of French regions (Remy, 2006).

	Area (km²)
Auvergne-Rhône-Alpes	69,711
Bourgogne-Franche-Comté	47,784
bretagne	27,208
centre-val de loire	39,151
Grand Est	57,441
Hauts-de-France	31,806
Île-de-France	12,011
Normandie	29,907
Nouvelle-Aquitaine	84,036
Occitanie	72,724
Pays de la Loire	32,082
Provence-Alpes-Côte d'Azur	31,400
Total	535,261
Average	44,605.083

Table 3

Turning points for each variable.

	Turning points with Corsica	Turning points without Corsica
Sexe ratio	7	6
Male (%)	Dependence between them
Female (%)
Age
0–24y (%)	6	6
25–59y (%)	7	7
>60y (%)	6	5
Life expectancy	7	6
Male	Dependence between them
Female
Birth rate (‰)	3	3
Death rate (‰)	5	4
Life conditions
Imposed fiscal menages rate (%)	6	5
Health status
Current smokers prevalence	7	6
Diabetic	6	6
Chronic pulmonary insuffisance prevalence	9	9
Severe renal chronic insuffisance prevalence	7	7
Obesity prevalence	7	7
Healthcare access
Eldery equipment rate (‰)	7	7
Medical doctor density (‰oo)	10	9
Hospitalization bed density	7	6
Sunlight exposure	9	8
Infection
Confirmed cases
Infection rate (%)	8	7
Healed cases
Healed rate(%)	10	9
Deceased cases
Mortality rate (%)	6	5
Turning points sum	140	128

Table 4

Results of Turning point test.

Number of			z	Expected number of turning points	95% Trust Interval
Regions	Points	Turning points			Minor range	Major range
12	192	128	0,20,489,932	128	116,5,489,471	139,4,510,529
13	208	140	0,393,606,616	138,6,666,667	126,7,480,329	150,5,853,004

To respond to the reductio ad absurdum, we are extremely disappointed thatour correspondents  can't read and can't interpret correctly a Pearson/Spearman coefficient. In fact, a negative coefficient indicates a negative correlation and not a positive one as our correspondents assess (Naudet et al., 2020): «sunlight exposure makes people building nursing homes». With a Pearson/Spearman coefficient of −0,69, the correct statement would be «sunlight exposure makes people building less nursing homes», if we interpret correctly the negative coefficient.

Concerning the exclusion of the Corsican region, we have done it after the data extraction from the different public health already quoted in our manuscript (Lansiaux et al., 2020). In order to justify, we used two indicators: the eldery equipment rate (in view of the large aged people infected by the COVID-19) and the hospitalization bed density (indeed, infected seniors require an hospitalization due to theirs comorbidities and not only a lockdown). Two of three trackers (the last one, the Corsican medical doctor density, had no significant difference with others regions) had significant differences with the others French regions (Tables 5 and 6 ) (Lansiaux et al., 2020). Therefore, they have conducted us to exclude the Corsican region before the sunlight exposure data extraction (so it was an a priori choice), and not in order to «hack» the p-value what we have been accused of.

Table 5

Healthcare access data in Corse.

Healthcare access	Corse
Eldery equipment rate (‰)	72
Medical doctor density (‰oo)	306
Places in short hopistalisation service	1.163
Hospitalization bed density (‰oo)	321

Table 6

Healthcare access statistical parameters with the inclusion of the Corse region.

Healthcare access	Average	Variance	SD	CI 95%
Eldery equipment rate (‰)	144,5	707,103	26,591	130,045	147,303
Medical doctor density (‰oo)	326,7	1769,244	42,062	303,835	330,226
Hospitalization bed density (‰oo)	387,5	933,923	30,56	370,887	390,505

Finally, if our correspondents persist in their thesis of p-hacking, we will oppose them that their computed p-value 0,03 (Naudet et al., 2020). Using a ratione absurdum, this one stays above the usual medical significance threshold of 5%. In fact, thanks to this «negators» (according to the antique definition), even if we are wrong we are right. Therefore, we thank them for the fact that they, thus,ensure the validity of our hypothesis, despite the mistakes they think have discovered.

Conclusion

We are extremely enthusiastic about debating in a relaxing manner. Although the heart has its reasons that reason ignores, we must show restraint and rigor, especially in these sanitary hard times. This formal correspondence was very instructive for us (on modesty, respect…), we hope that it was and will for our correspondents.