In my previous post, I showed that the calibration-in-time reconstruction from Seebergsee by Larocque-Tobler et al (2011) had not been cross-validated, and that once it had been cross-validated, the performance was terrible.

Now it is time to look a little closer at the chironomid counts from Seebergsee. The text reports that fossil samples contained between 0 and 60 chironomids, and that adjacent samples with fewer than 30 chironomids were merged.

By comparing figure 4a and 4b from Larocque-Tobler et al (2011), we can work out which samples have been merged, and hence what the count sums are. For example, the oldest sample in figure 4b has 66% *Tanytarsus* sp and 33% *Conynoneura*; this must be the merger of the two oldest samples with chironomids in figure 4a. This merged sample has a count sum of only 12 chironomids, far below the thirty promised and far too small to expect a precise reconstruction. Overall, 69% of the merged samples have count sums lower than 30. Elsewhere, the corrigendum to Larocque-Tobler (2015) admitted that the counts sums in that paper were substantially lower than reported. A pattern emerges.

Look again at figure 4a above. Zoom in. Notice that the lowest count for any species is four chironomid head capsules: there are no bars showing counts of only one or two chironomids. There are also no bars showing five or six chironomids either: all counts seem to be multiples of four. This is a rather curious distribution of counts.

Under very optimistic assumptions, the probability that a given count is divisible by four is 1/4. Hence, the probability that all 186 counts are divisible by four is 10^{-112}.

More realistically, there will be many rare taxa with one or two head capsules. In the Lake Żabińskie fossil chironomid count data from Larocque-Tobler et al (2015), only 15% of the counts are divisible by four (rounding half chironomids up), lowering the probability that all 186 counts are divisible by four to 10^{-153}.

These counts are improbable. Not as improbable as the proverbial monkey typing Hamlet, but if we set our simian thespian the shorter target of

To be, or not to be, that is the question:

Whether ’tis nobler in the mind to suffer

The slings and arrows of outrageous fortune,

Or to take

then, ignoring capitalisation, spacing and punctuation (just as I normally do – together with spelling), success is as improbable as the chironomid counts from Seebergsee.

But how else to explain these counts other than by blind chance?

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