brain

Background

An increase in brain size was one of the major trends of human evolution [1][2]. At the beginning of the Pleistocene, the average endocranial volume of fossil Homo specimens was approximately 750 ml [3]. By 30,000 years ago, this average value had increased to nearly 1500 ml [1][2]. Much of this increase occurred within the period following 800,000 years ago [1][2], during which mean endocranial volume in \emph{Homo} increased by approximately 70 ml per 100,000 years. This trend occurred in all regions of the Old World [2], which may have included either a single [4][2] or multiple species of archaic Homo [5][3].

Less well known is that the terminal Pleistocene and Holocene (ca. 30,000 years ago to present) witnessed a substantial decline in endocranial volume [6][7][1]. This decrease occurred within modern \emph{Homo sapiens}, and has been observed in many parts of the world [6][7][8]. The scope of this decrease is remarkable: for example, within the past 10,000 years the average endocranial volume in European females reduced from a mean of 1502 ml to a recent value of 1241 ml [7]. This decrease of approximately 240 ml in 10,000 years is nearly 36 times the rate of increase during the previous 800,000 years.

Brain size is related to body size both across higher taxa [9] and within humans [10]. This suggests the hypothesis that changes in human brain size may result from changes in body size. For example, the larger brain size in early Homo compared to Australopithecus may reflect the simple expansion in body size from earlier hominids [11]. This explanation cannot explain every change in brain size in humans: for example, the long increase in brain size during the Pleistocene did not coincide with increases in body size [3].

What about the reduction in brain size during the last 10,000 years—can it be explained by a reduction in the size of the body? Human body size, as measured from skeletal dimensions, did reduce during the past 30,000 years, at least in some populations [6][7][1][12]. This reduction influenced both mass and stature [7][1][12]. A reduction in overall body size may have resulted from Late Pleistocene and Holocene subsistence strategies, which replaced close-contact ambush hunting of large mammals with projectile weapons, intensive collection of small animals, fish, and shellfish, and ultimately sedentary pastoralism and agriculture [13]. Nutritional inadequacies and disease during the Holocene also may explain reductions in body size [14]. Within Europe, where the trend has been most closely studied, body size rebounded within the past 1000 years as manifested by increases in stature [7].

Several workers have suggested that recent reductions in brain size may have been caused by reductions in body size [6][7][1][15]. A coincidence of reduction in both these measures would lend some support to that hypothesis. However, for a reduction in body size to be a sufficient explanation for reduction in brain size, it is not enough that the reductions occurred at the same time. Natural selection on one character (like body size) will affect a correlated character (like brain size) only to the extent that the two characters are heritable and are genetically correlated. Therefore, to test the hypothesis that selection on body size accounts for reductions in brain size in recent human evolution, we must consider the relationship and genetics of these characters within human populations.

Here, I apply a quantitative genetic model to test the hypothesis that Holocene evolution of brain size may be explained by reductions in body size. The reasons for reduction in body size are unclear, so I consider both body mass and stature as candidates for the target of selection in recent populations. This is a very limited approach, constrained to published estimates of endocranial volume in archaeological populations and estimates of phenotypic correlations and heritability from samples of living humans. No attempt is made to correlate brain size and body size in the same samples of archaeological specimens, as such data are not available at present. Instead, I estimate the amount of body size change that would be necessary to explain the observed change in endocranial volume. This estimate is then assessed for credibility as applied to archaeological samples.

Results and Discussion

Body mass

Body mass is related to brain size in humans with a phenotypic correlation of r≈0.29. The standard deviation of male body mass within recent human populations ranges around 11 kg, a value near the midpoint of within-sex variation in other primate species [16]. Using these values along with the others listed in Table 1, selection on body mass would be expected to reduce the mean endocranial volume by 4.3 ml for each kilogram of reduction in body mass.

The decline in body mass in human populations during the last 10,000 years has been estimated as less than 5 kg, or less than a 10 percent reduction in mass from a Late Upper Paleolithic mean of some 63 kg [1]. A decline of 5 kg would predict a decrease in endocranial volume only around 22 ml. The observed decline in several regions (including Europe, China, Southern Africa, and Australia) is between 100 and 150 ml during the past 10,000 years. Therefore, the reduction in body mass would be expected to have decreased brain size by only one-fifth to one-seventh the observed decline.

We can look at the inverse question: how much reduction in body mass would be required to cause a 150 ml reduction in endocranial volume? Using the same ratio (4.3 ml per kilogram body mass), the endocranial volume contrast would predict a reduction of 34 kg. This value is implausibly high, by more than a factor of five.

The reduction of endocranial volume in these populations is not well explained by body mass according to equation 1. Selection for smaller mass is insufficient to account for reduction in brain size or vault dimensions.

Stature

Applying equation 1 to the parameters for stature and its correlation to brain size, endocranial volume would be expected to change approximately 9.5 ml per centimeter change in stature. This value is less extreme than the reduction in body mass that would be necessary to achieve the same reduction in brain size. But the skeletal record is inconsistent with any great decrease of mean male stature, particularly during the post-Neolithic time period.

Stature estimates exist for a broad sample of ancient European populations, showing approximate stasis in stature during the last 4000–6000 years. Over the same time period, the estimated endocranial volume declined slightly more than 100 ml in Europe from an estimated 1496 ml to 1391 ml. This decline cannot be explained by decreases in stature, because the stature did not change. Additionally, although these early samples are small, Mesolithic Europeans had larger endocranial volumes than Upper Paleolithic Europeans, across the same interval when they underwent a substantial decline in stature. That Mesolithic change in endocranial volume is in the opposite direction expected from the change in stature.

Likewise, the femur lengths of foragers in Southern Africa showed no net decrease over the last 10,000 years. From 5500 to 2500 years ago, both femur length and femur head diameter declined in this region, but they rebounded within the last 2500 years [17]. Across the same 10,000-year time period, Henneberg and Steyn [8] documented a decline in external and internal cranial module. The sample of LSA foragers (before 2000 years BP) had a mean external cranial module of 154.7, Iron Age (2000--200 years BP) had a mean of 149.6, while recent foragers had a mean of 150.3 --- roughly a standard deviation lower than the pre-2000 BP value. Under the hypothesis that change in endocranial volume is predicted by the change in stature, we should predict no net change in endocranial volume in this population. But the reduction in external module corresponds to a reduction in endocranial volume between 100 and 150 ml [8]. However, the LSA sample in that study is very small (n=12) and temporally dispersed.

Early Holocene populations in Australia have produced a substantial sample of crania, but postcrania from this time period are rare or poorly preserved [18]. The net change in endocranial volume, roughly 130 ml from the terminal Pleistocene to late Holocene skeletal sample [19] would predict a reduction in stature of 13 cm, if the brain size had changed only because of correlated changes in stature. That degree of stature reduction is not biologically impossible although it would be extreme. Further investigation of the evolution of body size in recent Australian hunter-gatherers may be necessary to answer the question.

Why did brain size reduce during the Holocene?

The evidence suggests substantial reductions in brain size in some recent human populations, more than can be explained by correlated changes in body size. It is worth discussing two related points concerning the distribution and causes of this pattern of brain size evolution.

First, was the change global or local in scope? The samples here cover several far-flung geographic areas, but they do not cover all regions of the world. Beals, Smith and Dodd [6] reviewed the global evidence for endocranial volume and showed a decline in the available terminal Pleistocene to Holocene skeletal sample. The Late Pleistocene skeletal sample was in that case strongly biased toward Europe, an area that in contemporary humans has a relatively large average endocranial volume. Thus, it was not obvious whether geographic differences in sampling might explain the reduction in endocranial volume noted in the study. This problem also characterizes the somewhat more course sampling by Ruff and colleagues [1]. Here, the samples of endocranial volumes and body sizes are matched in region to the extent possible; they do represent probable evolutionary trends within these populations. But there are few other comparable sequences of skeletal samples, so it may not be possible to conclude strongly that the reduction in brain size generalizes outside these regions.

A large series of crania from ancient Nubia covers the period from roughly 3400 years ago to 600 years ago [20][21]. Samples show a slight trend toward decrease in the major length, breadth and height measurements from Iron Age (Meroitic, external cranial module 145.2) to Medieval (Christian, external cranial module 143.9) times, but the intermediate series of crania (X-Group, external cranial module 147.1) is somewhat larger in these dimensions than either of the other groups. In this context it would be misleading to speak of a reduction in cranial vault size in this region. Across the same time interval, these samples show a substantial reduction in facial and dental measurements [21].

Second, given that the pattern is widespread if not global, how can we explain the reduction in brain size? Several hypotheses have been presented that may help to explain recent brain evolution. It is beyond the scope of this paper to test these hypotheses but here I review several of the adaptive and non-adaptive alternatives with some notes relating to the observed pattern.

1. Chance. Genetic drift may be considered a null hypothesis for any slight morphological change. However, in the case of brain size evolution during the last 10,000 years, genetic drift is a markedly unlikely hypothesis. Endocranial volume changed by a standard deviation or more, rapidly and directionally, within some very numerous and growing post-agricultural populations.
2. Plasticity. Somatic development in humans is plastic to some degree, depending on uterine and childhood nutritional and disease environments. This plasticity underlies most of the recent secular trend in body mass and stature. However, the brain size reaches 90 percent of its adult value very early in development and most of the variance in living populations is additive. This suggests that brain size may be less plastic than other components of body size. The pattern of decrease does not match stature or mass across the last several thousand years in these populations, suggesting that environmental effects were probably mediated by genetic factors.
3. Climate. Beals, Smith and Dodd [6] presented correlations between endocranial volumes of populations and their local climate, as reflected by latitude or temperature. Smaller-brained populations live in warmer climates, and this relation cannot be explained entirely in terms of body size of contemporary populations. They proposed that post-glacial climate change may have favored smaller brains. However, if the link between climate and brain volume is not mediated through body mass (following Bergmann's rule), it is not obvious why climate should cause brain size reduction.
4. Nutrition. The diets of early agriculturalists were nutritionally challenging in several ways: low in protein content, sometimes low in essential vitamins, and subject to fluctuating supply. The brain is an energetically expensive organ and nutritionally costly to develop. Smaller brains on balance should be advantageous under energetic or nutritional constraint, if they are functionally equivalent. Larger Holocene populations may have been selected for smaller brians for energetic reasons.
5. Function. Smaller brains may have some functional implications, as white matter tracts are shorter and functional areas of the cortex may be more compact. Given the social and ecological changes of the Holocene, it is possible that a different mix of mental and cognitive functions was the target of selection. Despite the long Pleistocene history of human brain evolution, it would be fallacious to assume that larger brains were always adaptive in the context of cognitive changes.
6. Development. Although adult brain size is attained relatively early in development compared to adult body size, brain development continues during adolescence and early adulthood. It is possible that the life history evolution of recent humans has involved changes in the maturation schedule that would impact the ontogeny of brain maturation. If so, then the schedule of brain development after it attains adult size might have been constrained by earlier events, in such a way that faster development or smaller completed size was advantageous.

These hypotheses are not mutually exclusive. To assess them, it will be necessary to collect systematic data from a large sample of crania representing these and other regions of the world. This study represents only an early step toward understanding the cross-regional record of brain size evolution in the Holocene.

Comparative data may also be useful to resolve these hypotheses. The decline of human endocranial volume during the last 10,000 years is paralleled most obviously by the reductions of brain size in domesticated animal species, including dogs, cattle and sheep, compared to their wild progenitors. Nutritional, developmental, and functional issues are all possible explanations for these parallel cases of brain size reduction. Humans are different in many ways from these domesticated species, but exhibit other parallel trends such as decreased skeletal robusticity.

At present, the literature presents a relative hodge-podge of estimates of endocranial volume, based on different original measurements. Estimates taken from the same method are compatible with each other, but it is not obvious that estimates based on different methods can be reconciled. It would be valuable to replace this mixture of measurements with a standard morphometric profile. The size of the endocranial cavity is interesting because of the developmental and energetic aspects of brains. But size is only one aspect of recent brain evolution. A full accounting of the shape of the cranial vault or endocast will be necessary to test hypotheses about why and how the brain reduced in size in these Holocene populations.

Conclusions

The available skeletal samples show a reduction in endocranial volume or vault dimensions in Europe, southern Africa, China, and Australia during the Holocene. This reduction cannot be explained as an allometric consequence of reductions of body mass or stature in these populations. The large population numbers in these Holocene populations, particularly in post-agricultural Europe and China, rule out genetic drift as an explanation for smaller endocranial volume. This is likely to be true of African and Australian populations also, although the demographic information is less secure. Therefore, smaller endocranial volume was correlated with higher fitness during the recent evolution of these populations. Several hypotheses may explain the reduction of brain size in Holocene populations, and further work will be necessary to uncover the developmental and functional consequences of smaller brains.

Methods

Endocranial volume

Studies of skeletal samples from different regions of the world are very consistent in finding reductions of endocranial volume during the last 10,000 years [6][22][7] [19] [23] [8][24]. However, there are discrepancies among studies in the both the method of estimation and the time periods for which skeletal samples are available. These are listed in Table 1.

Estimation methods

The literature on brain size in archaeological specimens refers to several different measurements:

1. Endocranial volume: directly measured by mustard seed, shot or water displacement of endocasts, or estimated from tomographic (CT) or magnetic resonance (MRI) methods. These different measurement methods can lead to systematically different results and so should not be combined without accounting for the measurement bias. The endocranial volume is larger than the brain volume (because of the intervening fluid and meningeal membranes).
2. Brain weight: directly measured from cadavers or estimated from CT or MRI based on brain volume and estimated tissue density.

   Some notable large-sample studies of variation within contemporary human populations have examined brain weight [25]. Brain weight and endocranial volume are strongly correlated but not identical. The volume of the skull includes fluid and tissue components that are not included with cadaver brain weights, while different means of preservation of cadaver brains may inflate the variability of some brain weight datasets. The problems of brain weight measurement are not directly relevant to archaeological samples, where there are no brains to weigh. But brain weight remains important because of the present-day samples in which we can estimate the phenotypic correlation of brain and body size. Where possible, I have included present-day samples that include either endocranial volume or cranial measurements, for direct comparability with the archaeological samples.

3. Cranial module: The external cranial module is the arithmetic mean of three external measurements of the skull: maximum length (glabella-opisthocranion), maximum breadth (euryon-euryon) and cranial height (basion-bregma). These external measurements include not only the brain but also the thickness of cranial bones.

   In some populations considered here, the thickness of cranial vault bones declined during the Holocene. This means that a decrease in the external module may be explained in part by a decrease in thickness, and some correction must be made to consider endocranial volume. The effect of thickness can be quite substantial; a decrease of 5 mm of thickness around a skull with an external module of 160 mm would increase its endocranial volume by around 180 ml. Where measurements of thickness are available, one approach is to subtract twice the vault thickness from the external module, resulting in an internal cranial module. This is the approach taken by Henneberg [7], for example, who reports both internal cranial module and resulting estimates of endocranial volume derived from regression on internal module.

The current paper uses the generic term ``brain size'' to refer to any of these estimation methods. Each of the four regions considered here is represented by at least one study that uses consistent estimation methods within the region. Even though different regions may be characterized by different methods of estimation, these differences should not bias the results within each region. But when different regions produce a common result, it remains possible that the magnitude of changes may actually diverge from each other due to differences in estimation methods.

One fundamental problem remains. Estimates of heritability and brain-body phenotypic correlation within human samples typically involve brain weight (for autopsy studies) or brain volume (for MRI or CT studies). Estimates from skeletal samples typically involve endocranial volume or cranial module. We cannot know that the heritability of the skeletal measures is equal to that of the soft-tissue measures.

Regions

The literature includes sufficient data to consider the reduction of brain size in four regions of the world.

The greatest temporal detail is available from Europe, reviewed by Henneberg [7]. Samples of up to several thousand skulls have estimates of endocranial volume. The largest set of these are based on external measurements, corrected for average vault thickness. The literature also includes a substantial number of direct measurements of endocranial volume by seed or water displacement. Henneberg [7] reports a Mesolithic mean endocranial volume for males of 1567 ml (based on internal cranial module of 144.1). This estimate is based on a relatively small sample of 35 individuals. For Neolithic and Eneolithic samples, with 1017 individuals, the mean endocranial volume estimate reduced to 1496 ml (internal cranial module 141.9), Bronze and Iron Age samples had a mean estimate of 1468 ml (internal cranial module 141.0), Roman period mean estimate 1452 ml (internal cranial module 140.5), and Early Middle Ages 1449 (internal cranial module 140.4). Late Middle Ages had a mean estimate 1418 (internal cranial module 139.4), and ``Modern Times'' (which comprises post-Medieval samples) corresponded to a mean estimate of 1391 ml (internal cranial module 138.5). Female samples across this time period exhibited a similar degree of size change; from a Neolithic mean of 1373 ml to 1210 ml in the ``Modern Times'' sample.

Henneberg's study was notable for its discussion of the limitations of these data, which are compiled from many sources. The reliance on external dimensions does tend to increase the interstudy comparability of the values, but necessitates relying on regression predictions of endocranial volume, which necessarily involve some error. The overall change is substantial enough to overcome the plausible methodological inconsistencies, but it is appropriate to be cautious between time intervals (e.g., Early to Late Middle Ages) where the amount of change is minimal.

Endocranial volume in southern Africa was considered by Henneberg and Steyn [8], estimating from measurements of external and internal cranial module. The sample covers the time period after 30,000 radiocarbon years BP, however, the vast majority of specimens date to the last 2000 years. Henneberg and Steyn [8] showed a statistically significant decline in both male and female crania, separated by morphological criteria.

Much of this sample, together with a larger selection of archaeological crania, were included in a later study by Stynder and colleagues [26] using morphometric methods. This study demonstrated an increase in craniofacial size during the last 4000 years, which appears to contradict the findings of Henneberg and Steyn [8]. The resolution between these two results is twofold. Most obviously, Stynder and colleagues [26] did not include landmarks that would indicate cranial breadth across the parietals, as these are not easily digitized. The breadth values are those showing the most consistent decreases in the sample studied by [8]. Secondarily, Stynder et al. [26] included facial measurements in their sample, so that the centroid size of crania was determined by both facial and vault dimensions. The allometric shape analyses in this paper demonstrated that larger centroid size was associated with allometric increase in the face and relative decrease in the vault. The implications of this allometry for the absolute vault dimensions are not clear, although the direct measurements indicate a reduction in vault size for the sample measured by Henneberg and Steyn [8]. It would be valuable to look at these allometric questions comprehensively with both landmark and caliper measurements in the southern African sample.

Brown and Maeda [22][19] reported on diachronic change of skeletal measurements in Holocene north China and Australia. They showed that the endocranial volume of males decreased from a mean of 1510 ml in early Neolithic (5500--6000 year old) samples down to 1400 ml in present-day Chinese. The change is consistent with a trend toward decrease across time intervals, despite relatively small sample sizes (n=10 to n=20 in the archaeological samples). Present-day Chinese people appear to vary in cranial size from north to south, possibly by more than 100 ml [19][6], and it is not obvious which samples of contemporary Chinese make the most relevant comparisons. So a decrease of 100 ml over the last 6000 years may either overstate or understate the actual change in endocranial volume in this population.

Wu and colleagues [27] confirmed the trend toward smaller cranial size from Bronze Age to recent northern Chinese populations. The study included a much larger sample of crania than examined by Brown and Maeda [22], but endocranial volume itself was not measured. The length, breadth and height of the skull all underwent significant reductions from the Bronze Age, roughly 3000 years ago, to the present.

Brown [19] presented a comparison of 19 male Australian crania from the terminal Pleistocene and 23 contemporary crania of Aboriginal Australians. The terminal Pleistocene sample stretches across a substantial range of dates, the earliest specimens possibly older than 30,000 years, to as little as 9000 for the large Coobool Creek sample. The Pleistocene people were larger in body size than recent Australians, and exhibit larger teeth and greater skeletal robusticity. The mean endocranial volume of the terminal Pleistocene males is 1405 ml; the recent mean is 1272 ml, for a decrease of just over 130 ml.

In qualitative terms, the strongest documentation of the decline in endocranial volume is from Europe, due to both sample size and sample preservation. The other three skeletal samples show a comparable magnitude of decrease. In China, this decline occurred over roughly the same time interval as in Europe; in South Africa and Australia the reductions may have unfolded over a longer period of time. In all cases, the estimated reduction of endocranial volume was greater than 100 ml within males, roughly 7 percent of the mean.

Mass and stature

Like brain size, stature and body mass provide challenges in the archaeological record.

Mass is a parameter of fundamental biological interest, but it depends strongly on soft tissue body composition and is therefore estimated only with substantial error from skeletal samples. In a global survey of the Pleistocene human skeletal record, Ruff and colleagues [1] estimated a mean body mass for Late Upper Paleolithic humans as 62.9 kg; this estimate was derived from 71 skeletal specimens, mostly from Europe. The ``living worldwide'' value cited in that study was 58.2 kg, a reduction of less than 5 kg from the Late Upper Paleolithic value, although the samples are geographically inconsistent.

Stature should be a better proxy for body size in the archaeological record, because it exhibits less phenotypic plasticity and because it relates more directly to measurable skeletal quantities such as long bone lengths. This increases the geographic sample available to test hypotheses of temporal change, because either long bone lengths or stature estimates exist for Europe, Southern Africa, and China.

Frayer [13] reported an Upper Paleolithic male mean stature of 174 cm with a standard deviation of 9.4 cm. The Mesolithic male mean stature in that study was 165 cm with a standard deviation of 6.6 cm. The reduction in female stature values was concordant with the male values, with roughly half the number of sampled individuals. Maximum femur length reduced from 466 to 446 mm in male individuals between these time periods, with standard deviations of 38 and 29 mm, respectively.

Henneberg [7] lists a series of stature estimates from rural Poland since the 13th century. Both male and female statures were in approximate stasis over that time period, until the 19th century. Koepke and Baten [28] put together a broader sample of anthropometric measures from across Europe during the last 2000 years, and also concluded that heights had been ``stagnant'' across that interval. Brief excursions of stature in some parts of Europe may nevertheless have occurred. Steckel [29] collated a series of stature estimates from Northern European skeletal samples dating from the 9th to the 19th centuries. Across this region, the mean male stature declined from roughly 173.4 to a low of 166.2 cm during the 18th century, a reduction of 7 cm. That decline may have been presaged by an increase in the post-classical period suggested by the data of Koepke and Baten [28]. Neither trend was noted in the samples considered by Frayer [30] or Henneberg [7].

Sealy and Pfeiffer [31] measured and performed stable isotope composition analysis of femora from the Cape region of South Africa, dating to the last 10,000 years. The male-attributed femora with measurable lengths in this study date to the period between 6000 and 1000 years ago. They show no significant decline in maximal length across this period. Femoral head diameter reduced slightly and significantly between the earlier male sample (before 4000 years ago) and later males (between 1000 and 4000 years ago). Pfeiffer and Sealy [17] revisited this sample and added evidence from more recnet skeletal individuals. The results showed that stature tended to rebound to a larger mean within the last 2000 years, roughly equal to the initial sample before 6000 years ago. Across this entire time period, the stature and mass of the archaeological population was within the range exhibited by present-day Khoisan peoples.

The documentation of stature by long bone lengths is the best available source of data on body size in archaeological samples. Conservatively, we can conclude that the skeletal record documents a modest reduction of stature since the Upper Paleolithic in Europe, most of which had occurred by the Mesolithic. In Europe and China, the skeletal record is consistent with approximate stasis of stature during the last 5000 years, with some geographic and temporal excursions from the broad pattern.

Body mass is unlikely to have changed is a very different pattern from stature. Fatness is poorly documented skeletally and is at present the largest component of variation in within-sex mass in industrial populations, but this varied much less substantially in pre-industrial peoples.

Quantitative genetic model

For both body size parameters, the error of skeletal estimates is substantial. Therefore, here I adopt a very conservative test of the null hypothesis: (1) Determine the amount of change in body size that would minimally be required to explain the observed change in brain size; and (2) Evaluate whether that amount of change in body size is credible given the skeletal record. The skeletal record addresses point (2), but for point (1) we must turn to a quantitative genetic model relating the evolutionary dynamics of correlated characters.

The allometry of brain and body size has been investigated extensively among both living and fossil organisms. From a quantitative genetic perspective, Lande [32] developed mathematical expectations for allometric change in the population mean of a single phenotypic character in response to selection on a correlated character. This change is given by Equation 2b in Lande (1979) [32]:

Equation 1

[note: HTML is difficult to represent bar over letters; these are z-bar in the manuscript]

Δz_i/Δz_b indicates the change in the population mean z_i of one character (here, endocranial volume) with a correlated change in the mean z_b of a selected character (here, body size). The genetic correlation between the two characters is γ_ib, while h_iσ_i is the square root of the additive genetic variance of character i.

For this study, the null hypothesis is that brain volume should be predicted by equation 1, given the parameter estimates and the change in body size. This is equivalent to the hypothesis that brain size has changed entirely due to its genetic correlation with body size. The parameters in equation 1 have all been estimated in one or more contemporary human populations.

It is important to note that parameter estimates may be conservative or nonconservative in their effects under the null hypothesis. The genetic correlation of the two traits must be less than 1. So measuring change in units of standard deviations, the null hypothesis predicts that brain size should change relatively less than body size. However, the absolute change must be considered relative to heritability and variance of the two phenotypic traits. Brain size should change more relative to a given change in body size if:

1. the genetic correlation of brain and body sizes is higher,
2. the heritability of brain size is higher,
3. the phenotypic variation of brain size is higher,
4. the heritability of body size is lower, or
5. the phenotypic variation of body size is lower.

If the parameter estimates are in error in these directions, the test of the null hypothesis will be conservative to some degree—that is, the null hypothesis will be accepted in cases where the true parameter values would lead to rejection.

Estimates of heritability and variances are available for humans and for some other species of primates, both for brain volume and for body mass and stature. The availability of different estimates makes it possible to consider their consistency with each other and the likely effects of error.

Mass and stature are considered separately as independent variables in the analysis.

Brain size variation

The skeletal samples above allow estimates of standard deviations for each sample. However, because of the limited sizes of archaeological samples, these estimates of variability may either overstate or understate the variation of ancient populations.

There is substantial sexual dimorphism of both brain and body size in humans. The simplest way to correct for variation due to sex is to consider males and females separately. All estimates of parameter values in living humans are reported from male- or female-specific samples. Archaeological samples often permit assessment of individual sex, although there is necessarily some error in these assessments. Where possible, this study reports values for males, and assumes that variation is distributed like that of males in living human populations.

Additionally, phenotypic estimates in humans may include confounding age effects. A few cited studies use age-controlled samples, but many rely on postmortem measures in samples with a broad range of age-at-death. Archaeological samples always include age-related variability, although this is likely distributed differently than in many surveys of living humans.

Peper and colleagues [33] reviewed heritability estimates for total and regional brain volume based on MRI studies of twins. Most studies have yielded high estimates for the heritability of total brain volume, ranging from 0.97 [34], 0.94 [35], 0.90 [36] and 0.89 [37]. One outlier study reported a lower estimate of heritability (0.66), but this came from a sample of only 10 MZ and 10 DZ twin pairs [38]. In the current study, the use of a high estimate of heritability will tend to bias the result toward accepting the null hypothesis, since a more heritable character will be expected to change more under the effect of correlation with body size.

Brain-body genetic correlation

The genetic correlation between brain size and body size is not known for humans. However, the phenotypic correlations between brain volume or mass and body mass or stature have been extensively studied. The largest sample of these metrics was published from Danish autopsies by Pakkenberg and Voight [25]. Holloway [10] computed correlations between brain mass, stature and body mass in this dataset; these are reported in Table 1.

Ankney [39], using the data from Ho et al. [40], reports phenotypic correlations between brain mass and stature as r=0.20 for white males and r=0.24 for white females, r=0.20 for black males and r=0.15 for black females. These values are lower than those computed from the Danish data. Both sets of estimates should be regarded as underestimates because of the confounding effect of age variation in the sample. On the other hand, these are phenotypic correlations, and the genetic correlation may be lower than the phenotpic values due to effects induced by the environment or gene-environment interactions. Here, I employ the higher reported estimates of correlations because they have a conservative effect on the hypothesis test: A higher correlation predicts a more substantial change in brain size.

Parameter values in nonhuman primates

Estimates of brain-body correlations and heritabilities in humans have mostly been taken in European or American population samples. These estimates may therefore be biased dietary Westernization and concomitant changes in body mass index. To address this possibility, we can consider these relationships in non-human primates.

Rogers and colleagues [43] measured brain volume and body mass in captive free-ranging baboons (Papio hamadryas) with known pedigrees. They found brain-body phenotypic correlation of r=0.29 (r²=0.086) for males and r=0.16 (r²=0.026) for females. The heritability of brain volume was estimated as 0.52. The heritability of body mass in this captive population was previously estimated as 0.50 [44].

Falk and colleagues [45] found phenotypic correlations in rhesus macaques (Macaca mulatta) between brain volume and body mass to be r=0.54 for males and r=0.40 for females.

Stature is not strictly comparable between humans and other primates, because of the obvious difference in locomotor anatomy.

These comparisons allow several conclusions:

1. The heritability of body mass is approximately the same in humans as in other primates.
2. Heritability of brain size in humans is substantially higher than reported in other primates. Using a high estimate should bias against rejection of the null hypothesis.
3. The phenotypic correlations between brain size and mass in these primates are within the range reported for humans.

Thus, as near as possible, using the human values for these parameter estimates will provide an appropriate test of the null hypothesis, that changes in brain size were caused by changes in body size in recent human populations.

References

The endocranial volumes estimated for late Australopithecus boisei specimens (e.g., after 1.8 Ma) are larger than those of earlier specimens. Elton et al (2001) [1] found that this trend is statistically significant, arguing for the evolution of larger brains over time. Such a trend bears on the ecology and social behavior of A. boisei, and lends some doubt to the idea that brain size evolution in early Homo was exceptional [1].

But the A. boisei sample has some unusual aspects that may complicate the test of a trend. One question is whether the early KNM-WT 17000 specimen represents A. boisei or another species (possibly, Australopithecus aethiopicus). Another question arises from the very small variation of estimated endocranial volumes in the A. boisei sample. Even including the small KNM-WT 17000 volume estimate, the coefficient of variation in the sample examined by Elton et al (2001) [1] is only 8.2 percent. Excluding KNM-WT 17000, the within-sample CV is 6.8 percent. By comparison, Tobias (1971) [2] reported data on endocranial volumes of hominoids. Great ape values include chimpanzees with 9.7 percent, orangutans at 10.9 percent, and gorillas with a CV of 13.1 percent. According to these estimates A. boisei had less variation than any living hominoids, even though its craniodental variation was as great as gorillas or orangutans [3].

There are several possible interpretations for the low variation of the A. boisei sample: (1) A. boisei actually had very low size dimorphism; (2) its endocranial variation has been greatly undersampled, or (3) the sample has been biased by estimation error. Other characters of the A. boisei sample show extensive variability compared to extant hominoids [3], so that monomorphism for this species seems unlikely. Low sample variance is a special concern because estimation error might lead to false positive results in a test of trend.

Here, I conduct three new tests of the null hypothesis of stasis of endocranial volume in A. boisei. These tests explore the effect of estimation error on the appearance of a trend in the sample, as well as the effect of low sample variation and small sample size. None of these tests find a statistically significant trend in the sample.

Materials and methods

Fossil specimens

Estimating endocranial volume can be challenging even for relatively complete specimens, considering the subtle distortion exhibited by many fossils. For more fragmentary cranial remains, the estimation of endocranial volume requires not only the correction of distortions but also the reconstruction of missing portions.

Endocranial volume estimates for specimens of A. boisei against time. The sample is that used in this study, excluding Omo 323.

The eleven cranial specimens of Australopithecus boisei listed below vary in their completeness and preservation of relevant anatomy. There is no explicit way of statistically controlling for error in the estimation of endocranial volume, considering the diversity of methods of reconstruction. In several cases, different workers have provided competing estimates. For less complete specimens, choosing one estimate above another must involve a close critique of anatomical details. The following list reviews the anatomical condition of each of these specimens. It is not an exhaustive list of volume estimates, but focuses on the range between credible extremes for the more disputed specimens. This gives an impression of the boundary conditions for measurement accuracy for each specimen.

1. KNM-WT 17000 is a well preserved skull with relatively small vault fragments missing. Walker et al. (1986) [4] estimated the volume as 410 ml.
2. Omo L338-y6 is a juvenile cranium of uncertain age. Holloway (1981) [5] estimated its volume at 427 ml. Elton et al (2001) [1] estimated an adult volume 4% higher, or 444 ml.
3. The Omo 323-1976-896 cranial remains are exceedingly fragmentary. One side of the posterior cranial base is preserved, allowing a relatively good estimate of the posterior endocast breadth. The preserved frontal and parietal elements do not join with each other or the temporal; their small size and unknown positions do not allow an accurate estimate of endocast volume. [6] reported an estimate of ``about 490'' based on similarity with the 491 ml KNM-ER 23000. Falk et al (2000) [7] considered it too fragmentary for an accurate estimate. I concur; the available estimate cannot be considered independent of other endocasts on which it may have been based.
4. KNM-WT 17400 preserves only the anterior third of the endocast, consisting mainly of the frontal lobes. Brown et al. (1993) [6] gave an estimate of 500 ml by modeling missing portions after the more complete KNM-ER 23000, but Holloway (1988) [8] put the volume between 390 and 400 ml, and Falk et al. (2000) [7] adopted an estimate of 390 ml.
5. OH 5 has good preservation of the endocast, but an uncertain join between the anterior and posterior portions of the vault. This discontinuity has caused a disparity in estimates of its volume, including a low 500 ml estimate by Falk et al (2000) [7] and a high 530 ml estimate by Tobias (1963) [9]. The range of estimates on this well-preserved specimen covers nearly a quarter of the range of variation cited for A. boisei as a whole.
6. KNM-ER 13750 preserves only the superior vault, accounting for under half of the total endocranial contour. The range of estimates provided by Falk et al (2000) [7], from 450 to 480 ml, again covers roughly a quarter of the range attributable to the species. Brown (1993) [6] reported a higher estimate of 500 ml.
7. KNM-ER 23000 is a nearly complete vault missing the midline cranial base. Its endocranial volume of 491 ml [6] may be the most accurate assigned to A. boisei.
8. KNM-ER 406 is also well-preserved [10]. Its volume estimate of 525 ml is uncontroversial [11].
9. KNM-ER 407 is missing several vault sections including those enclosing the frontal lobe. Holloway (1988) [11] estimated the volume at 510 ml; Falk et al (2000) [7] prepared a new reconstruction with a volume estimate of 438 ml. The difference between these two estimates covers nearly 50 percent of the total range of the sample.
10. KNM-ER 732 has good preservation of the left side of the vault, but is not complete across the rear of the cranium or basicranium, making a mirror reconstruction problematic. Holloway (1988) [8] estimated the endocast volume at 500 ml; Falk et al (2000) [7] at 466 ml.
11. KGA 10-525 lacks most of the frontal and anterior cranial base. Suwa et al. (1997) [12] estimated its volume at 545 ml.

The damaged or missing frontals of many specimens have added to ambiguity about their reconstructed volume. Robust endocasts that preserve this region, such as KNM-WT 17400, differ in their anatomy from other taxa, especially early Homo. Falk et al (2000) [7] reconstructed specimens with missing or incomplete frontal endocasts using more complete robust australopithecine endocasts as models; this resulted in substantially smaller endocranial estimates for OH 5, KNM-ER 732 and KNM-ER 407.

Tests of temporal trends

Most A. boisei specimens with EV estimates date to the approximate center of the species' temporal span. The reason for the appearance of a trend is quite clear: there is little variation in the center of the species' temporal range; the latest two specimens are also the two largest; the earliest two specimens include two of the three smallest (Figure 1).

A test of a temporal trend might be conducted in several ways. A simple linear regression of endocranial volumes against time will test for a trend, but may be confounded by small numbers of specimens at early and late temporal extremes. Testing for a difference in means among temporal subsamples may address this problem. Comparing each specimen as a temporal subsample results in Spearman's rank-order correlation (ρ), which [1] reported as significant for their sample of A. boisei EV estimates.

Also, following Leigh (1992) [13] and Konigsberg (1990) [14], Elton et al (2001) [1] applied the "Hubert test" [15], sometimes simply called the "Gamma" (Γ) test [16] [17]. This test is a randomization test of association of one continuous and one ranked variable, involving four steps:

1. The age of each specimen is converted to a rank within the sample. For a two-tailed significance test, ranks are standardized with a mean of zero.
2. The endocranial volume of each specimen is multiplied by its temporal rank, and all the values thus obtained are summed. This is equivalent to calculating the dot product of a vector of endocranial volumes with a vector of ranks.
3. The sample is reordered at random an arbitrarily large number of times, each time obtaining the dot product of endocranial volume and rank vectors.
4. The statistic Γ is estimated to be (M+1)/(N+1), where M is the number of permutations with dot products greater than or equal to that of the observed sample, and N is the number of permutations examined. A Γ ≤ 0.05 is taken as a significant rejection of the null hypothesis of no trend.

It is perhaps of interest that although the Hubert test uses the dot product of the two vectors, the use of the product-moment correlation yields precisely the same Γ (shown in Appendix). Samples for which the dot product shows a significant trend are samples that have significant correlations between EV and temporal ranks. This suggests a weakness of the test, since a correlation is a measure not of change over time, but of fit to a linear model. A sample may have a significant correlation with very little change, if its variance is also very low. Hence, the interpretation of the test depends on whether the variance is biologically realistic. Since A. boisei appears to be relatively invariant in endocranial volume compared to sexually dimorphic hominoids, the test might be confounded by error in the sample of EV estimates.

The Hubert test has been applied in the anthropological literature in two partially incompatible ways. As applied by Konigsberg (1990) [14], following Hubert (1985) [15], the vector of temporal ranks is centered on zero (i.e., the values are ... -2, -1, 0, 1, 2 ...). But as applied by Leigh (1992) [13] and Elton et al (2001) [1], the temporal ranks are simple ordinal ranks (i.e., 1, 2, 3, ...). These two alternatives are mathematically equivalent for performing a one-tailed test. But while the first alternative (zero-centered ranks) readily admits a two-tailed test, the second alternative requires a bit more algorithmic complexity for a two-tailed test. Elton et al (2001) [1] and Leigh (1992) [13] did not report whether their tests are one- or two-tailed; following the procedures they described will result in a one-tailed test. Wood et al. (1994) [17] also applied the Hubert test to test for trends in dental characters of A. boisei, citing Leigh (1992) [13]; these authors also did not specify whether they performed one-tailed or two-tailed tests. Lockwood (2000) [16] employed the Hubert test (there called the Γ statistic), and explicitly described a two-tailed approach. One-tailed tests ignore the strength of any negative associations in the permuted samples, and therefore lead to incorrect assessments of statistical significance. The current study applies only two-tailed tests of the null hypothesis of no trend.

Test 1: Lower estimate for KNM-WT 17400

Falk et al (2000) [7] argued that smaller estimates are more accurate for several robust australopithecine specimens, and the smaller estimates were generally used by Elton et al (2001) [1]. One exception is KNM-WT 17400, for which Elton et al (2001) [1] used the highest estimate of 500 ml [6], even though both Holloway (1988) [8] and Falk et al (2000) [7] adopted much lower estimates, between 390 and 400 ml. This smaller estimate would make KNM-WT 17400 the smallest member of the sample. A small size for this specimen at the center of the species' time range increases overall sample variability and decreases the relative contribution of early specimens to that variability. This makes KNM-WT 17400 very important to any test of a trend.

As a preliminary step, I recalculated Spearman's ρ and the Hubert test statistic Γ for the sample of Elton et al (2001) [1], using the smaller 390 ml estimate for KNM-WT 17400. This replicates the methods of that study, except for the change in size of the single KNM-WT 17400 specimen.

Test 2: Model-based simulation values

A difficulty of the A. boisei sample is the non-independence of estimates. Less complete specimens have been reconstructed using explicit information from more complete endocasts, chiefly Sts 5 and OH 5. The sample should therefore have reduced variation compared to a sample of intact crania. A reduced variance may increase the chance that a null hypothesis of stasis will be falsely rejected. This is a context in which randomization tests are potentially invalid: they do not assume a statistical distribution, but they do assume independence.

An additional aspect of the problem is that the state of preservation of fossils may be autocorrelated with time. In the present sample, the early and late specimens are relatively complete, while the middle of the time range is dominated by incomplete specimens. This situation arises frequently in paleontology, because species abundance is often highest at the center of a species' temporal range. Early and late specimens will be more likely attributed to a species if their anatomy is unambiguous --- which is more likely if they are more complete. Early or late specimens may be represented at different fossil localities than the majority of specimens, again requiring more complete specimens for confident assignment. In a Holocene context, specimens are likely to be more fragmentary and rarer earlier in time. These situations present the possibility of finding spurious trends due to differential preservation.

To attempt to correct for these issues, it is necessary to employ tests that rely on an explicit model of sample variability, instead of randomization of the sample values themselves. A simple model-based test replaces the sample EV estimates with new random deviates from a normal distribution. A normal distribution takes two parameters: the population mean and standard deviation. Deviates drawn from this distribution are independent; an arbitrary number of simulated samples may be obtained by repeatedly drawing new values to replace the sample values.

Here, the model-based sampling technique was used to generate samples with the same temporal ranks as the observed data, but with new EV values. In cases where the observed sample has two specimens of the same date, two specimens in all simulated samples were assigned the same temporal rank. The observed A. boisei sample has two such pairs of specimens. As in the Hubert test, the computer generated an arbitrarily large number of simulated samples (in this study, 100,000). The dot product of EV and temporal rank vectors in each simulated sample is compared to the dot product of the observed sample. The significance measure is taken as (M+1)/(N+1), where N is the number of simulated samples, and M is the number of those samples in which the absolute value of the dot product is more extreme than the observed value. This is a two-tailed test of the null hypothesis of no trend. I refer to the test below as the ``model-based Hubert test.''

This test was applied to the A. boisei sample described above, including KNM-WT 17000, excluding the extremely fragmentary Omo 323-1976-896, and employing an estimate of 390 ml for KNM-WT 17400. Simulated samples were generated using the observed sample mean (468 ml) and standard deviation (49.1).

Test 3: Arbitrary variation

The model-based Hubert test described above is not limited to the observed sample variation. It can also be applied using a different value for the population standard deviation.

This option is relevant to the A. boisei endocranial volume sample, because the sample of estimates may have lower variation than the population from which the specimens were drawn. Even with the lower estimate of 390 ml for KNM-WT 17400, the CV of the observed A. boisei sample is still only 10.3 percent --- between chimpanzees (9.7) and orangutans (10.9). This value might be uncharacteristic of the A. boisei population, if its sexual dimorphism or temporal variability are undersampled by available EV estimates. Since the test described here derives its simulated EV estimates from a model distribution, it is easy to apply a more variable model --- for example, matching the CV of gorillas at 13.1 percent [2]. As a further example, I varied the population CV parameter of the model-based test, covering the entire range between 4 percent to 15 percent This range encompasses the CVs of all extant hominoids. In all cases I assumed a mean equal to the A. boisei sample mean (468 ml). Using this procedure, it is possible to evaluate whether possible underestimation of variability in the observed sample may affect the significance of the test of no trend.

Results

Test 1: Lower estimate for KNM-WT 17400

The first tests performed were on the A. boisei sensu lato sample of Elton et al (2001) [1], with the exception of a lower estimate of 390 ml for KNM-WT 17400. With this estimate, the nonparametric Spearman's correlation ρ = 0.52, which is nonsignificant (p>0.10, two-tailed). For the two-tailed Hubert test on the sample, p=0.10. For both tests, the lower estimate for KNM-WT 17400 causes the significance of a temporal trend in A. boisei to completely disappear. This low estimate currently appears to be a consensus for the specimen, although it must be treated cautiously since the endocast is less than 50 percent complete. This single specimen illustrates well the importance of accurate estimates.

Test 2: Model-based simulated values

The removal from the sample of the 490 ml estimate for Omo 323-1976-896 actually enhances the appearance of a trend. This is reflected by the Hubert test result, with p=0.07 (compared to p=0.10 when Omo 323 is included). Spearman's nonparametric correlation for the sample was 0.58, again nonsignificant (p>0.05, two-tailed). The model-based test described in this paper came to a very similar result on this sample, with p=0.07. Both these tests failed to reject the null hypothesis of no trend for the A. boisei sample.

Further examination of the simulated samples gave some indication of the relationship between sample variability and the appearance of a trend. One hypothesis might be that the sizes of early KNM-WT 17000 specimen is actually relatively extremely small, and the late KGA 10-525 specimen is actually relatively extremely big, resulting in the apperance of a steady expansion from smallest to biggest through the sample. The simulated samples, in which specimens are drawn from a population with equal standard deviation (49.1) to the A. boisei sample, rejected this hypothesis. Forty-four percent of the simulated samples had at least one specimen smaller than 390 ml, the smallest in the observed sample. Forty-six percent had at least one specimen larger than 545 ml, and 19 percent of simulated samples had specimens more extreme than both the largest and smallest of the observed sample.

Test 3: Arbitrary variation

Result of Test 3, testing the significance of a trend in A. boisei with a range of models for population CV. Each point represents 100,000 simulated samples of equal mean to the A. boisei sample and CV given as on the x-axis. The greater the assumed variation in the underlying population, the greater the chance that an increase over time equal or greater than that in the A. boisei sample will be observed. There is no significant trend for any model of variation within the range of living great apes and humans.

An alternative hypothesis is that the appearance of a trend is due to low sample variability, increasing the correlation of EV and temporal rank. The result of the model-based test applied to a range of model CV between 4% and 15% shows the close relationship of significance of the A. boisei trend and population variation. Briefly, the greater the variation in the population, the more likely each simulated sample will present a trend at least as great as that in the observed sample. If the A. boisei sample was drawn from a population with greater EV variability, then the level of correlation of EV with time is less surprising. If the A. boisei population was as variable in endocranial volume as extant gorillas, then 15.1% of randomly drawn samples would exhibit an apparent trend as strong or stronger than the observed sample. With the extant sample, it is not possible to confirm this hypothesis of underrepresentation --- in particular, body size dimorphism does not necessarily follow from variability in cranial and masticatory variability.

Discussion

The problem with testing a trend in any early hominid species is similar in form to the problems discussed by Holloway (1970) [18]. All reconstructions are based on relevant knowledge of the anatomy of other specimens. Whether reconstructions are done on crania, endocasts, or CT data, they all rely on knowledge of more complete specimens — for A. boisei endocasts, these models include OH 5 and KNM-ER 23000, and the well-known \emph{A. africanus} endocast Sts 5. When we test hypotheses using samples of reconstructions, we are to some extent including multiple instances of these well-known specimens, spread through many semi-independent reconstructions. There is no ready statistical model to incorporate the effects of estimation error from fragmentary specimens. These estimates are likely to be biased by the use of more complete specimens as models, the more frequent preservation of some parts of the cranial surface as opposed to others, or unrecognized sex differences in fossil individuals. In other words, one effect of estimation error is to reduce the variation within the fossil sample.

Estimation error may also tend to elevate the between-species differences among early hominins. Presently, samples assigned to different early hominid species exhibit some anatomical differences. For example, These differences may result from differing neuroanatomical adaptations in these different species. If so, then it would be anatomically misleading to use a specimen of A. africanus like Sts 5 as a model for the reconstruction of an incomplete A. boisei specimen. On the other hand, differences are observed between very small samples, and may be idiosyncratic rather than systematic. Instead of distinctive adaptations, they may represent only chance differences between small samples. In this case, the use of only other A. boisei specimens as models for incomplete A. boisei reconstructions would tend to artificially inflate the differences between A.boisei and A. africanus, as well as artificially reducing variation within A. boisei. The smaller the sample, the more likely that between-species differences will be inflated by reconstruction and within-species differences minimized.

Even with a CV of 10.3%, the variation in A. boisei is likely undersampled. The extant sample is apparently male-biased, with only 3 presumed females (KNM-ER 732, KNM-WT 17400, and KNM-ER 407). Incomplete specimens have been reconstructed by modeling after more complete crania, reducing variation from anatomical differences. Beyond this, temporal fluctuations should tend to inflate variability with or without a directional trend.

All of these factors also must affect the samples currently assigned to Homo habilis (including KNM-ER 1470), which taken together have an endocranial volume CV of 12.6%. Endocranial volume has a disproportionately important role in differentiating between smaller and larger Plio-Pleistocene Homo morphs, and this may bias the consideration of evolutionary trends in early Homo.

The only solution for these problems is the discovery of more specimens. But in the meantime, it would be appropriate to exercise caution in the interpretation of variability within and among species. Significant differences among species are tested with reference to within-species variation. For estimated characters like endocast volume, within-species variation is potentially biased by estimation error. This bias may often tend to inflate between-species differences and reduce within-species variation attributed to fossil samples.

Appendix

The dot product is commonly used in vector transformations, but interpreting it in the context of a temporal trend may not be intuitive. The dot product of two vectors is the sum of the products of their respective elements:

This product is a measure of the projection of one vector onto the other; it increases as the angle between the vectors (taken from the origin) decreases. The dot product of two perpendicular vectors is zero.

The product-moment correlation between two vectors is:

where z_xi and z_yi are standardized values of x_i and y_i, respectively. Thus, the product-moment correlation is the dot product of two standardized vectors divided by their rank ( - 1).

In a randomization test, the different values of x and y are scrambled with respect to each other. However, the sample means and and the sample standard deviations s_x and s_y are constant in all of these randomized samples, because each includes exactly the same specimens. Thus, within any random set of permutations of a sample, the product-moment correlation can be obtained by a simple linear transformation from the dot product:

References