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No brain expansion in Australopithecus boisei.

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No Brain Expansion in Australopithecus boisei
John Hawks*
Department of Anthropology, University of Wisconsin-Madison, Madison, WI 53706-1393
brain evolution; robust australopithecines; temporal trends
The endocranial volumes of robust australopithecine fossils appear to have increased in size
over time. Most evidence with temporal resolution is concentrated in East African Australopithecus boisei. Including the KNM-WT 17000 cranium, this sample comprises
11 endocranial volume estimates ranging in date from 2.5
million to 1.4 million years ago. But the sample presents
several difficulties to a test of trend, including substantial
estimation error for some specimens and an unusually
low variance. This study reevaluates the evidence, using
randomization methods and a related test using an
explicit model of variability. None of these tests applied to
the A. boisei endocranial volume sample produces significant evidence for a trend in that species, whether or not
the early KNM-WT 17000 specimen is included. Am J
Phys Anthropol 146:155–160, 2011. V 2011 Wiley-Liss, Inc.
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)
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 (Elton et al.,
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
17,000 volume estimate, the coefficient of variation in
the sample examined by Elton et al. (2001) is only
8.2%. Excluding KNM-WT 17000, the within-sample CV
is 6.8%. By comparison, Tobias (1971) reported data on
endocranial volumes of hominoids. Great ape values
include chimpanzees with 9.7%, orangutans at 10.9%,
and gorillas with a CV of 13.1%. 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 (Silverman et al., 2001).
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 (Silverman et al., 2001), 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
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 lowsample variation and small sample size. None of these
tests results in a statistically significant trend in the
Fossil specimens
C 2011
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.
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) estimated the volume as 410 ml.
2. Omo L338-y6 is a juvenile cranium of uncertain age.
Holloway (1981) estimated its volume at 427 ml.
Grant sponsor: Graduate School of the University of WisconsinMadison.
*Correspondence to: John Hawks, Department of Anthropology,
University of Wisconsin–Madison, 5240 Social Science Building,
1180 Observatory Drive, Madison, WI 53706–1393.
Received 22 April 2010; accepted 10 September 2010
DOI 10.1002/ajpa.21420
Published online 16 August 2011 in Wiley Online Library
Elton et al. (2001) 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. Brown et al. (1993) reported an estimate of
‘‘about 490’’ based on similarity with the 491-ml
KNM-ER 23000. Falk et al. (2000) considered it too
fragmentary for an accurate estimate. I concur that
the available estimate cannot be considered independent of other endocasts on which it may have been
4. KNM-WT 17400 preserves only the anterior third of
the endocast, consisting mainly of the frontal lobes.
Brown et al. (1993) gave an estimate of 500 ml by
modeling missing portions after the more complete
KNM-ER 23000, but Holloway (1988b) put the volume
between 390 and 400 ml, and Falk et al. (2000)
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 500ml estimate by Falk et al. (2000) and a high 530-ml
estimate by Tobias (1963). 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), from 450 to 480 ml, again covers roughly a
quarter of the range attributable to the species.
Brown et al. (1993) 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 (Brown et al., 1993) may be the most accurate
assigned to A. boisei.
8. KNM-ER 406 is also well-preserved (Wood, 1991). Its
volume estimate of 525 ml is uncontroversial (Holloway, 1988a).
9. KNM-ER 407 is missing several vault sections including those enclosing the frontal lobe. Holloway (1988a)
estimated the volume at 510 ml; Falk et al. (2000)
prepared a new reconstruction with a volume estimate of 438 ml. The difference between these two
estimates covers nearly 50% of the total range of the
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 (1988b) estimated
the endocast volume at 500 ml; Falk et al. (2000) at
466 ml.
11. KGA 10-525 lacks most of the frontal and anterior
cranial base. Suwa et al. (1997) 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) reconstructed specimens with missing or incomplete frontal
American Journal of Physical Anthropology
Fig. 1. Endocranial volume estimates for specimens of A.
boisei against time. The sample is that used in this study,
excluding Omo 323.
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.
Some authors place KNM-WT 17000 within the A. boisei hypodigm, but many would put it into a different species, often Australopithecus aethiopicus. This earlier species may have been part of a single evolving lineage with
later A. boisei, but need not have been so. Elton et al.
(2001) found significant evidence for a trend in the A.
boisei sample whether or not the sample included KNMWT 17000. Including this early, small specimen tends to
amplify the evidence for a trend. The pattern is likewise
amplified by the assumption of a small volume for the
other early specimen, Omo L338y-6. Holloway (1981)
assessed the Omo L338-y6 juvenile as likely belonging to
A. africanus not A. boisei. This observation gained support due to the lack of an occipital–marginal sinus drainage on the endocast. Thus, both early specimens are
In the following tests, I have retained these specimens
within the A. boisei sample, because including them
tends to stack the deck in favor of a trend. When the
same tests are run without these specimens, the P-values are without exception farther from statistical significance. However, I want to emphasize that these specimens are not included within A. boisei by any consensus,
and their status must be evaluated with evidence beyond
their endocasts.
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 (see Fig. 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.
NO BRAIN EXPANSION IN Australopithecus boisei
Comparing each specimen as a temporal subsample
results in Spearman’s rank-order correlation (q), which
(Elton et al., 2001) reported as significant for their sample of A. boisei EV estimates.
Also, following Leigh (1992) and Konigsberg (1990),
Elton et al. (2001) applied the ‘‘Hubert test’’ (Hubert
et al., 1985), sometimes simply called the ‘‘Gamma’’ (G)
test (Wood et al., 1994; Lockwood et al., 2000). 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 G is estimated to be (M 1 1)/(N 1 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 G 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 G
(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, because 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. Because 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, which
became evident to me when trying to replicate the
results of different studies. As applied by Konigsberg
(1990), following Hubert et al. (1985), the vector of temporal ranks is centered on zero (i.e., the values are . . .
22, 21, 0, 1, 2 . . .). But as applied by Leigh (1992) and
Elton et al. (2001), 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 twotailed test. Elton et al. (2001) and Leigh (1992) did not
report whether their tests are one- or two-tailed; following the procedures they described will result in a onetailed test. Wood (1994) also applied the Hubert test to
test for trends in dental characters of A. boisei, citing
Leigh (1992); these authors also did not specify whether
they performed one-tailed or two-tailed tests. Lockwood
et al. (2000) used the Hubert test (there called the G 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
TABLE 1. Results of tests 1 and 2
Including Omo 323
This study (no Omo 323)
Spearman’s q
Hubert test
Spearman’s q
Hubert test
Model-based test
P value
P [ 0.10
P 5 0.10
P [ 0.05
P 5 0.07
P 5 0.07
to incorrect assessments of statistical significance. This
study applies only two-tailed tests of the null hypothesis
of no trend. For future research, I recommend the zerocentered ranks approach.
Test 1: Lower estimate for KNM-WT 17400
Falk et al. (2000) argued that smaller estimates are
more accurate for several robust australopithecine specimens, and the smaller estimates were generally used by
Elton et al. (2001). One exception is KNM-WT 17400, for
which Elton et al. (2001) used the highest estimate of
500 ml (Brown et al., 1993), even though both Holloway
(1988b) and Falk et al. (2000) 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 q
and the Hubert test statistic G for the sample of Elton et
al. (2001), using the smaller 390-ml estimate for KNMWT 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 nonindependence 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, whereas 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.
American Journal of Physical Anthropology
To attempt to correct for these issues, it is necessary to
use 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 1)/(N 1 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
earlier, including KNM-WT 17000, excluding the
extremely fragmentary Omo 323-1976-896, and using 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 earlier 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%—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 is undersampled by available EV estimates. Because 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% (Tobias,
1971). As a further example, I varied the population CV
parameter of the model-based test, covering the entire
range between 4 and 15%. 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.
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), with the exception of a
American Journal of Physical Anthropology
lower estimate of 390 ml for KNM-WT 17400. With this
estimate, the nonparametric Spearman’s correlation q 5
0.52, which is nonsignificant (P [ 0.10, two-tailed). For
the two-tailed Hubert test on the sample, P 5 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, because the endocast is less than 50%
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 5 0.07 (compared to P 5 0.10 when Omo 323 is
included). Spearman’s nonparametric correlation for the
sample was 0.58, again nonsignificant (P [ 0.05, twotailed). The model-based test described in this work
came to a very similar result on this sample, with P 5
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 size 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 appearance 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. Fortysix percent had at least one specimen larger than 545
ml, and 19% of simulated samples had specimens more
extreme than both the largest and smallest of the
observed sample.
Test 3: Arbitrary variation
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 (Fig. 2).
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
were as variable in endocranial volume as extant gorillas, then 15.1% of randomly drawn samples would exhibit an apparent trend as strong as 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
NO BRAIN EXPANSION IN Australopithecus boisei
Fig. 2. 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.
The problem with testing a trend in any early hominid
species is similar in form to the problems discussed by
Holloway (1970). 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 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.
Interestingly, this problem does not present itself as a
lack of precision or repeatability. An estimate based on a
fragmentary specimen may be highly repeatable, even
by different observers, when the missing portions are
well known from model specimens.
Instead, the problem may be noted as a bias affecting
the between-species and within-species variances among
early hominins. Presently, samples assigned to different
early hominid species exhibit some anatomical differences. Such differences may reflect neuroanatomical adaptations in these 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, small samples
will always exhibit some chance differences. If the differences between well-preserved endocasts are mainly
idiosyncratic variations, then when we use only other
A. boisei specimens as models for incomplete A. boisei
reconstructions, we will 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 withinspecies differences minimized. The ambiguity about
specimens like Omo L338-y6 pertains to this issue.
Should a variant be included within a species or
assigned to a different one based on the presence of a
diagnostic trait?
Different estimates of endocranial volumes may result
from different reconstructions, including CT versus more
traditional methods of reconstruction. Some authors (e.g.,
Falk et al., 2000) have provided new estimates for certain
specimens that differ by 10% or more from previous estimates, generally representing a reduction in size compared to earlier estimates. Evaluating the accuracy of
individual volume estimates is beyond the scope of this
analysis, but would be clearly desirable. I believe that the
best way to assess the accuracy of such estimates, including the reconstruction of missing parts from models, is for
multiple workers to perform blind replication studies, providing open access to CT and endocast data.
Even with a CV of 10.3%, the variation in A. boisei is
likely undersampled. The extant sample is apparently
male-biased, with only three 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. Moreover, at 10.3%, the CV may be inflated by the
use of a very small estimate for KNM-WT 17400, and the
inclusion of the similarly small KNM-WT 17000, which
may well belong to a different species. The CV values for
other anatomical measurements do not necessarily bear
on the endocranial volume, but it may be relevant that
10.3% would be near the minimum for large (n [ 8) samples of early hominin molar areas, which range up to 17%
for mandibular M1 and M2 areas in A. africanus.
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.
Naturally, more numerous fossil specimens would be
welcomed as a way to improve our statistical understanding of early hominins. Meanwhile, some statistical
methods may bear some increased scrutiny with reference to large samples of living taxa. Resampling
approaches assume that the observed data are characteristic of the population variability that they sample. The
reconstruction necessary for fossil specimens can in
some cases violate that assumption. Using a model-dependent statistical method in this case has highlighted
an instance where the variation of a fossil sample
presents problems for statistical comparisons. Similar
methods might prove fruitful for other hypotheses tested
with fossil samples.
I thank Aaron Sams and Marc Kissel who investigated
the statistical tests of trend and made many helpful
American Journal of Physical Anthropology
comments. Ralph Holloway deserves many thanks for
his advice and encouragement. I also thank Christopher
Ruff, Sarah Elton, and one anonymous reviewer for their
comments, which greatly helped the manuscript.
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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:
xi yi
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
zxi zyi
where zxi and zyi are standardized values of xi and yi,
respectively. Thus, the product–moment correlation is
the dot product of two standardized vectors divided by
their rank (minus 1).
In a randomization test, the different values of x and y
are scrambled with respect to each other. However, the
means x and y and the standard deviations sx and sy are
constant in all 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:
xi yi x y
ðn 1Þsx sy
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boisen, brain, australopithecus, expansion
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