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It does not indicate how likely the alternative hypothesis, that there is a relationship, is.
To know that we have to have some idea of how plausible the alternative hypothesis is.
Even if there were no other problems, this would be enough to be fairly certain that the correlations in Wing et al are spurious. Hypothesis tests are only fully valid if they are designed before the data are observed.
According to the press release, it was the observation of a 10-year cycle in the incidence data that inspired the study.
But since this paper is picking up some media attention, I thought it might be worth pointing out why solar activity is unlikely to become a tool for diagnosing arthritis.
Wing et al analyse the incidence of giant cell arthritis (GCA) and rheumatoid arthritis (RA) with in Olmsted County, Minnesota over five decades.
The lagged correlations between (yearly average and 3-year moving average RA incidence rates) and (F10.7 and AL) are plotted as dotted and solid lines in panels B and D, respectively.
They correlate the 3-year smoothed incident data with the F10.7 index (solar radiation at 10.7 cm wavelength) and the AL index (a proxy for the westward auroral electrojet), allowing for lags of up to 14 years. Rheumatoid arthritis (RA) incidence (1955–2007) and lagged correlation with F10.7 and AL .
Yearly average and 3-year moving average of RA incidence rates are plotted as dotted and solid lines, respectively, in panels (A and C, left scale).
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From the price of wheat (Hersche 1801) to childhood mortality (Skjærvø et al 2015), there seems to be no end to papers reporting spurious correlations with solar variability.