## One More for the Bad Statistics in Astronomy File…

It’s been a while since I last posted anything in the file marked Bad Statistics, but I can remedy that this morning with a comment or two on the following paper by Robertson et al. which I found on the arXiv via the Astrostatistics Facebook page. It’s called *Stellar activity mimics a habitable-zone planet around Kapteyn’s star* and it the abstract is as follows:

Kapteyn’s star is an old M subdwarf believed to be a member of the Galactic halo population of stars. A recent study has claimed the existence of two super-Earth planets around the star based on radial velocity (RV) observations. The innermost of these candidate planets–Kapteyn b (P = 48 days)–resides within the circumstellar habitable zone. Given recent progress in understanding the impact of stellar activity in detecting planetary signals, we have analyzed the observed HARPS data for signatures of stellar activity. We find that while Kapteyn’s star is photometrically very stable, a suite of spectral activity indices reveals a large-amplitude rotation signal, and we determine the stellar rotation period to be 143 days. The spectral activity tracers are strongly correlated with the purported RV signal of “planet b,” and the 48-day period is an integer fraction (1/3) of the stellar rotation period. We conclude that Kapteyn b is not a planet in the Habitable Zone, but an artifact of stellar activity.

It’s not really my area of specialism but it seemed an interesting conclusions so I had a skim through the rest of the paper. Here’s the pertinent figure, Figure 3,

It looks like difficult data to do a correlation analysis on and there are lots of questions to be asked about the form of the errors and how the bunching of the data is handled, to give just two examples.I’d like to have seen a much more comprehensive discussion of this in the paper. In particular the statistic chosen to measure the correlation between variates is the Pearson product-moment correlation coefficient, which is intended to measure *linear* association between variables. There may indeed be correlations in the plots shown above, but it doesn’t look to me that a straight line fit characterizes it very well. It looks to me in some of the cases that there are simply two groups of data points…

However, that’s not the real reason for flagging this one up. The real reason is the following statement in the text:

Aargh!

No matter how the p-value is arrived at (see comments above), it says *nothing* about the “probability of no correlation”. This is an error which is sadly commonplace throughout the scientific literature, not just astronomy. The point is that the p-value relates to the probability that the given value of the test statistic (in this case the Pearson product-moment correlation coefficient, *r*) would arise by chace in the sample if the null hypothesis H (in this case that the two variates are uncorrelated) were true. In other words it relates to P(*r*|H). It does *not *tells us anything directly about the probability of H. That would require the use of *Bayes’ Theorem*. If you want to say anything at all about the probability of a hypothesis being true or not you should use a Bayesian approach. And if you don’t want to say anything about the probability of a hypothesis being true or not then what are you trying to do anyway?

If I had my way I would ban *p*-values altogether, but it people are going to use them I do wish they would be more careful about the statements make about them.

May 21, 2015 at 8:58 am

It’s even worse than that! The p-value isn’t Pr(statistic = r | H), which would at least be a likelihood of sorts; instead it is Pr(statistic > r | H), implying an integral over unobserved data that takes it an extra step further away from something that would be useful for evaluating Pr(H | statistic = r).

May 21, 2015 at 10:40 am

Yes, I should have been clearer about the p-value relating to the probability P(r|H) rather than being it.

Moreover, I strongly suspect that the p-value itself comes from the standard calculation of this type which assumes Gaussian errors with the same variance…