The exponential (log-linear) relationship is discussed in the first section, is indeed a sensible model, and also shows a breakpoint (a change in the rate parameter) in a similar manner to the linear relationship. The ‘inverse quintic’ (log-log) relationship is discussed in the last section, is the model that Lee and Wolpoff use (almost certainly by mistake) to show no break point, and is completely unbiological (and is zero-point dependent, making it pretty much useless).

“may have stopped sooner in Homo Erectus (as it does in modern humans who are pygmies) than it does in modern humans.”

The negative sign of the constant “b” is simply a product of the unconventional way that the units are shown on the graph (with larger positive numbers moving into the past rather than into the future). If the dates had the same order as CE/BCE dates, regardless of the dividing point chosen, the constant “b” would be a positive (i.e. non-weird) number.

The magnitude of the constant (e.g. quintic) in absolute terms is also a function of the choice of units. It would be smaller if one chose the time unit of a day, and longer if one chose the time unit of a decade. The natural unit, incidentally, is probably the generation, rather than an absolute number of years. Since generations are probably longer at the very end of the data set and shorter at the start of the data set, the information is probably slightly off. But, because the expected change in generation size comes so late in the data set, it probably doesn’t matter much.

Adjusted for time unit length, the constant “b” in the equation is basically equivalent to growth rate, and is more usefully conceived as akin to the interest rate in a compound interest equation, than it is to the root of a particular number. In the same vein, one rarely thinks of one’s credit card interest rate as “reverse quintic.” Adjusted for time unit length, the constant “B” is essentially a number the measures the strength of selection for a proportionately larger brain size.

The unit A, of course, simply connects the data to whatever unit you are measuring volume with, and could be replaced with a unit neutral measure that is equal to average volume at a given date in time if one wished.

Since every exponential model looks like a breakpoint linear model in one tries to use a linear model, the fact that one can produce a breakpoint linear model from the data is hardly surprising. The absence of a gap for a macromutation at the break point noted, however, makes a breakpoint linear model seem less parsimonious.

The existing of a wide spread in the data also makes slight selection within a random range of something with a lot of little subcomponents seem plausible.

The biggest issue with using a brain size model like this one to make inferences about intelligence, however, is that we don’t have data on total body mass. Brain size tends to predict intelligence only after controlling for body size. A blue whale’s intelligence is not accurately predicted from its brain size alone. If total body size is increasing at roughly the same rate as brain size over this time period, for example, then intelligence may be roughly constant. Likewise, if male and female body size are changing at different rates, that may mute the apparent sex dependent brain size change rates.

John Hawks noted recently that Homo Erectus was probably shorter than some standard references suggest, because the total height inferred in the largest samples is based on grossing up a child to an assumed adult height when growth may have stopped sooner in Homo Erectus (as it does in modern humans) than it does in modern humans. If Homo Erectus started out short and gradually got bigger, then Homo Erectus may have been much smarter than brain size data alone would suggest. The seeming sophistication of Homo Florensis activity points towards the strength of this inferrence.

If the shape of the curve is not affected by the inclusion of the Neanderthals, doesn’t that logically imply that the Neanderthal lineage displays a similar pattern? If so, on the Blakemore thesis, shouldn’t we be postulating a matching mutation in that lineage also?

Thanks for the comments.

Unless I am very confused, a log-linear geometric model (a constant proportional increase per unit time, (1/Y)*dY/dt = a) is just an exponential model (dY/dt = aY), right?

Just re-ran the analysis excluding all Neandertals and all fossils older than 1000ky. No change to the results.

On the inverse-quintic issue, keep in mind that they derived the regression with time in the past as a positive value, hence the log-regression needs to be negative and decreasing with time, instead of positive and increasing.

That brain size increases gradually in all lineages is another pretty strong argument against brain size being a macromutation.

I’ve put the species in the data file, if you want to play around with it yourself.