# Kruskal-Wallis Test, Sample Size per Group: Results from a Covid-19 paper looking at viral load by age groups

Hello,
Covid-19 is a hot topic, there are many papers published, and many of them with contradicting results. I’d like to know how you would judge the results from this current study (pre-print) from one of Germany’s most famous virologists: http://bit.ly/SARS-2-load

The study investigates viral load for people infected with Covid-19. Assumption: higher viral load is strongly associated with higher probability of being infectious. Background: School opening and the question, whether kids may increase the pandemic spread or not. Methods: DV is viral load, IV is age category (e.g. Kindergarten, grade school, uni, adult). Results: no significant differences between kids and adults. Conclusion: Kids may spread Covid-19 to the same extent as adults do, thus school opening might increase pandemic spread.

I don’t want to start a discussion about pros and cons of school opening. I have four questions/concerns:

1. I was wondering, whether the very different sample sizes per category (e.g. n=16 for grade school compared to n = 2071 for people aged > 45), and the Kruskal-Wallis test, can give “robust” results.
Edit: The pairwise comparisons are based on three different approaches, Tukey’s HSD (probably with 1-way-anova as input?), t-Test (Bonferroni adjusted) and Dunn’s Test for pairwise comparisons. But the point is the same: very different N’s per group.

2. The absence of “statistical evidence” was interpreted as “there is no difference” (i.e. accepting H0, which is actually not possible in NHST, only in Bayesian framework?)

3. Which viral load is actually infectious?

4. Isn’t the “majority” of viral loads for kids and children rather below the average, while for adults etc. it looks rather uniform distributed, implying that kids are indeed less infectious (though this assumption of course is biased due to small N in groups as well…)

P.S.: The author of the study also linked to a study (https://science.sciencemag.org/content/early/2020/04/28/science.abb8001) that concluded that kids are only about 1/3 as infectious as adults. That’s a good thing that people are not afraid promoting “counterpart-results”.

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I have more questions than answers at this point.

1. Was there any explanation why they simply didn’t estimate the relationship between increasing age and viral load using some regression method (either logistic or a robust parametric technique) vs. formulating the problem as a hypothesis test? The section on the viral load distribution reports there were 3,712 individuals to estimate how a change in age is related to a change in viral load.

2. Taking for granted that this decision was going to be done in the frequentist framework, does anyone think the research hypothesis tested was not correct for the actual decision in question? They appeared to test the H_0 \ne H_1 The only relevant question from a decision viewpoint is: Do the data clearly indicate that younger individuals are less infectious than older ones (H_1 < H_0)?

Given the costs of a wrong decision along with other plausible explanations for the observed result, I don’t think much data is needed. A simple mini-max analysis would lead one to conclude to keep social distancing in place by keeping schools closed.

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FWIW, I just found a “re-analysis” of the study in question: https://twitter.com/eggersnsf/status/1257993188943572993?s=20

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I did a quick scan of the confidence intervals reported when I wrote my initial post. I noticed that the grouping by grade suggested a difference, but the grouping by age was less clear. I didn’t think that was worth noting at the time.

I thought it was obvious that any regression on the reported data would be sensitive to how the data were categorized. I’d much rather see the regression done on the individual data points.

Edit: here is a direct link to the Leonhard Held reanalysis. It was in the twitter post but here is a direct reference for those interested in the topic:

Held, L. (2020) A Discussion and Reanalysis of Results Reported in Jones (2020)
https://osf.io/bkuar/