I noted at the time that this paragraph:

“To explore the issue, O’Brien and her colleagues looked at children in grades 2 and 5 (mean age 9)…”

was a perfect example of egregious misuse of statistics in the sciences. I happened to think back on it, and searched on the phrase “camel distribution” (the obvious name for the shape of the age distribution of 2nd and 5th graders) and found this post, which came out a month after the sleep disorders article.

]]>First, we could calibrate Be-10 and Al-26 production rates separately, determine the uncertainties in each calibrated production rate, and then propagate these uncertainties into the ratio. Unfortunately, as pointed out very clearly in the Borchers paper about production rate calibration, the scaling models do not fit the calibration data at acceptable confidence, so it is not possible to make a justifiable uncertainty estimate for the production rate estimates, which in turn means that this approach would not yield a justifiable uncertainty estimate for the ratio either.

Alternatively, we could use the LSDn scaling model to predict the 26/10 ratio for each of the samples where the ratios were measured, and then compare that to the measured ratios. This comparison yields a reduced chi-squared value of 2.8 for, I think, 86 DOF, which has effectively zero probability-of-fit. Thus we have the same problem as in the Borchers paper, which is not surprising because it is basically the same calculation, only using slightly different data.

The standard deviation of the measured ratios with respect to the LSDn predicted ratios is 9% (which is basically the same as the SD of the ratios considered by themselves at 9.5%). If we argue that there is still some unquantified analytical error in the measurements, then 9% is probably a useful upper bound for the uncertainty in the ratio. However, 9% uncertainty means that the ratio is only constrained to be between 6.4 and 7.6 at 68% confidence, and, as discussed briefly above, that large an uncertainty range is incompatible with many burial-dating results. I don’t think anyone thinks the ratio is 6.4 or 7.6.

Basically, the problem is that if we are making Be-10 measurements at 3% and Al-26 measurements at 6% precision, the standard deviation of measured 26/10 ratios from simply exposed samples should be only 6.5%. The fact that it is 9% indicates that there is some unquantified analytical uncertainty present, which makes it hard to estimate the true uncertainty in the ratio.

If we just ignore the reported measurement uncertainties and the question of whether or not the measurements are consistent with the predicted ratios at measurement uncertainty, then we could compute the standard error of the mean (9% / sqrt(90)) to be 1%. So that is probably a good lower bound on the uncertainty in the ratio. To summarize, I think it’s clear that the uncertainty in the ratio is less than 9% and more than 1%, but it is not clear (at least to me) how to make a better uncertainty estimate. Unfortunately, this is not super helpful.

I think the most sensible strategy to improve this situation would be to make (or compile existing) measurements on higher-elevation and/or older samples where there is more Al-26 to measure. Most of the measurements in the existing calibration data set are from low-elevation sites with late-glacial ages where Al-26 concentrations are relatively low. Alternatively, we could try to estimate the ratio from near-saturated surfaces in low-erosion-rate environments such at Antarctica, where it is possible to make very precise measurements. A handful of such measurements have been done. That might be a good subject for a future blog post.

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