Lee, S., & Wolpoff, M. (2003). The pattern of evolution in Pleistocene human brain size Paleobiology, 29 (2), 186-196 DOI: 10.1666/0094-8373(2003)0292.0.CO;2

There has been a bit of debate around the biology blogoverse recently about the evolution of human brain size. It started off as an “idle speculation” type argument, but then took a satisfying swerve into the evidencey regions of science, which is always satisfying. The upshot of this is that I’ve spent today riffing statistically on some fossil data from a relatively old paper on the evolution of human brain size, and seeing if I can tease some interesting tidbits out of it.

### Some background

It all kicked off when neuroscientist Colin Blakemore made some comments on the evolution of the human brain. He argued that the large increase in brain size we see around 200k years ago may have been a useless “macromutation” that was tolerated due to the abundance of food. The evolutionary implausibility of this was evaluated unfavourably by and Jerry Coyne.

I’ve heard lots of plausible reasons that human brain size may have seen an increase 200k years ago or so; my personal favourite is that it was due to the discovery of fire, which made food easier to eat, giving more calories and allowing children to be born without the fully connected skull required to attach the jaw muscles for chewing tough food. This would loosen up constraints on selection for brain size, allowing brain size to rush ahead.

Either way, none of the hypotheses proposed have had much strong evidence put forward for them yet (maybe when we bring some Evo-Devo to the party?), and this is particularly true of Blakemore’s theory.

More interesting, and more answerable, is the question of whether there was a sudden increase in brain size at all, and if so when exactly it happened. John Hawkes put up a graph of data from a review of human brain evolution by Lee and Wolpoff, and used it to argue that brain size has been increasing gradually for millions of years, with no recent “jump”. In response, Ciarán Brewster did some basic number crunching to establish that, even if there wasn’t a sudden macromutation 200 kya, the human brain seems to have been increasing in size faster over the last 200 kya, compared to before that.

### If and when did brain growth speed up?

I decided to delve a little bit deeper into the data from the paper. The problem with Ciarán’s analysis is that it assumes that if there was a speed-up, it started 200k years ago. This is a slightly problematic assumption, mostly because of the winner’s curse (oh go look it up).

To do a more rigorous test for the existence of a speed-up, and to estimate when the speed-up happened if there was one, I fitted a least-squared break-point model (a model where the slope of a trend line changes on either side of some point). I compared this to to the basic linear trend-line, to see if it explains the data significantly better.

Here is what I get (the points are the fossils, the dashed line is a linear fit, and the coloured lines are the break-point model):

The model shows a definite speed-up of brain size increase recently, and fits the data significantly better than a simple trend line (F(1,90) = 15.8, p < 10^-5). I estimate that the speed-up occured 252kya, and can say with 95% confidence that it lies between 203 and 377 kya. This result is pretty robust to exactly what model we use; I also tried using a model where brain size grew exponentially with time, and this gave a similar break-point: 250kya, with a 95% interval of 167-402 kya (see this graph).

If you prefer non-parametric statistics, here is a loess smoothing of the data, showing a clear kink around 280kya:

Simon Blakemore’s theory of a single, sudden macromutation is, of course, inconsistent with the data; given that each sample is an individual human, if large brain size was a macromutation we’d see each sample either having a massive brain, or a small one. But, there does appear to be a change in the processes driving brain evolution somewhere between 200 and 400 kya.

### Appendix 1: Adding sex into the mix

Examining the literature (and by “examining the literature” I mean “googling the sample names”), of the 94 samples in the dataset, 30 are probably male, 24 are probably female, and 40 have not been sexed, or I couldn’t find out the sex (my analysis was pretty haphazard, so don’t take this as gospel). Both sexes have the same age distribution (see here), so it is unlikely that sex would confound the break-point. However, just to be on the safe side, here is the analysis using just males:

And just females:

Both show a similar break-point. The break-point model explains the data significantly better than the linear fit for males (F(1,26) = 20.4, p < 10^-3) but not for females (F(1,20) = 2.9, p = 0.105). Maybe some hint that the change in brain growth was greater for men than for women, perhaps? [Insert fruitless and subconciously sexist speculation here]. We don't have much data, and you'd want to get an expert to classify the sexes before you drew any conclusions.

### Appendix 2: Lee and Wolpoff’s Slightly Weird Model

In the original paper, Lee and Wolpoff argue that there is in fact no discontinuity anywhere, essentially because if you plot the log of brain size against the log of time, you get a straight line all the way back (Figure 3 in the paper). A log-log relationship corresponds to fitting an exponent model of Y = Ax^{b} (where y is brain size and x is number of years before present), which looks like this:

And indeed, under this model, you cannot find any break points that will significantly improve the fit. The model is pretty weird though; in our case, the model basically corresponds to brain size being inversely proportional to the fifth root of time (b = -0.2). Linear growth makes sense, and exponential growth corresponds to a growth in brain size is proportional to the current brain size, but what does inverse-quintic root growth correspond to? Maybe there is something fundamental going on here, but I expect the goodness of the fit in the graph above is just down to the flexibility of the exponent model, and I’d consider any conclusions drawn from a log-log transformation of the data to be somewhat dodgy (or at least, very underpowered, as the flexibility of the model will tend to obscure true discontinuities).

The authors strange choice of model appears to stem from a slight misunderstanding:

A logarithmic transformation may help avoid the problem of interdependence within the data set because it can be derived from the assumption that the rate of change of cranial capacity at any particular time is proportional to the cranial capacity of the sample at that time

But of course this is an exponential model, which corresponds to a log-linear transformation (the analysis I did above, which still showed a breakpoint); they performed a log-log, which corresponds to a exponent model.

*The data I used, including sex classification, is all here. I’d normally put the code I’d used here as well, but in this case it is in a pretty nasty state. If anyone wants it, I can clean it up.*

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I trust there are no Neanderthal specimens represented in this plotting, as their inclusion would, by definition, invalidate the analysis.

Blakemore’s thesis is concerned with a postulated event in the sapiens lineage, so only specimens from that lineage are relevant.

The data above includes all known Homo skulls, but none of the results change if you exclude the 24 Neandertals. In fact, you see the same results if you exclude Sapiens but keep Neandertals; the trends are pan-Homo, and aren’t confined to a specific lineage (though if you exclude Erectus everything goes skewiff, as you’d expect).

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.

Would we get a similar fit, if we were to ignore all specimens prior to 1000kya? Presumably no one would suggest that changes prior to that date have any relevance to the argument as to what what might have happened around 200-375,000 years ago?

The reason for a geometric (log-linear) model is that it represents a constant proportion of increase per unit time. A linear (non-geometric) model requires the rate of increase relative to the current value to decline with time.

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.

@Cyril

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

@John

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?

Thanks Luke.

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?

I agree with John Hawks that the exponential model is not very weird.

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.

Correction:

“may have stopped sooner in Homo Erectus (as it does in modern humans

who are pygmies) than it does in modern humans.”@Ohwilleke: I think you’ve got a bit mixed up between the exponential and inverse quintic models.

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).