Clarke et al. reply

Sultan points out that because cerebellum and neocortex differ fundamentally in their basic wiring structure, neocortical white matter should be excluded from our calculations. Doing this increases the effective cerebellar (cbl) volume fraction to Fcbl′=Fcbl/(1−FW), where FW is the neocortical white-matter volume fraction. We made this correction in 44 species3,4 and found that its effect was negligible: Fcbl′ is still relatively constant (0.16±0.04, mean±s.d., 13 taxa), whereas the corrected neocortical grey-matter volume fraction is not. The quantities are uncorrelated (r2=0.07, d.f.=11, P=0.4). The cerebellar correction is largest in cetaceans (Fcbl′=0.25±0.06, 2 species), emphasizing our finding that echolocating mammals have unusually large cerebellar volume fractions. Note that the use of a t-test (not ANOVA) was appropriate here because we wanted to know whether a taxon's mean was outlying.

Barton's allometric approach to quantifying size differences relies on fitting data to a single power law (a linear fit in log–log coordinates). He uses data on neocortical volume (N), cerebellar volume (C), and the rest of the brain (BNC). He fits for log N and log C against log (BNC) by pooling primate and insectivore taxa; the fit slope is 1.62 for neocortex and 1.28 for cerebellum. But subtracting these fits discards the difference between these two baseline relationships, which constitutes the dominant size trend between neocortex and cerebellum. As the underlying reason for the fitted trend is unknown, it is unclear whether the resulting residuals are interpretable.

Calculating residuals creates another problem: pooled data from multiple taxa are generally not well fitted by a single power law1,5. As a result, residuals calculated to a single line contain systematic errors (see Fig. 13 of ref. 1 and Fig. 4 of ref. 5). For instance, fits for different taxa often have similar slopes but different intercepts, a phenomenon known as grade shifting. This is apparent in Barton's analysis, in which nearly all primates fall above the fit lines, causing neocortical and cerebellar residuals to be mostly positive for primates and negative for insectivores (see Barton's Fig. 1), and leading to an artefactual correlation.

To remedy this, we considered the primates separately, dividing them into lemurs and lorises (haplorhines) and tarsiers, monkeys and apes (strepsirhines). After fitting each group to its own power law, we recalculated neocortical and cerebellar residuals and found that one set of residuals can explain only 9% of the variance in the other set (r2=0.09, 39 d.f., P=0.1). Recalculation including insectivores gives similar results, although classification of insectivores poses more difficulties than primates6,7. Therefore, the correlation reported by Barton does not exist within taxa; it largely (although indirectly) reflects differences in brain architecture among taxa. The same arguments also apply to his other analyses.

A third type of error arises when components are combined for analysis. The sum of two power laws of differing exponent can never yield a power law. Therefore any analysis (including Barton's) that combines multiple components (for instance, the rest of the brain, or neocortex equals grey plus white matter) intrinsically contradicts the power-law assumption8.

Underlying our disagreements with Barton is the fact that allometric relationships constitute ad hoc models that often provide only a rough fit to the data5. Where power-law phenomena have an explanation, such as in the circulatory system, the tissue in question is substantially more homogeneous than the whole brain. For this reason, using residuals based on subtracting approximate power-law trends may not be particularly effective at identifying relationships in the architecture of mammalian brains.