Gradual, autocatalytic and punctuational models of hominid brain evolution: A cautionary tale

Laurie Godfrey

Gradual, Autocatalytic and Punctuatlonal Models of H o m i n i d Brain Evolution: A Cautionary Tale

Department of Anthropolog),, University of Massachusetts, Amherst, MA 01003, U.S.A.

Kenneth H. Jacobs Department of Anthropolo~)~, Universit6t Frankfurt, Siesmayerstr, 70, D-6000 Frankfi~rt/3/fabz, West Germany

Received 9 September 1980 and accepted 4 December 1980 Keywords: evolutionary models,

hominid encephalization, log-log transformations, methodology.

The tempo and pattern of macroevolutionary change have bearing on current controversies concerning the processes of macroevolutionary change. In this paper we reexamine the evidence lbr increase in cranial capacity in light of traditional arguments and interpretations, paying special attention to methodological problems inherent in data transformations and manipulations. Despite a large volume of descriptive material published on the subject, there has been relatively little attention paid to testing competing hypotheses of processes of macroevolutionary change (punctuational, gradual, autocatalytic); the descriptions which have emerged often reflect unstatcd and, perhaps, unrealized a priori assumptions made by researchers. Log log transformations of cranial capacity versus time are misleading and inappropriate. A uniform pattern of change in hominid cranial capacity is less obvious than generally assumed.

1. I n t r o d u c t i o n

O n e of the most thoroughly documented macroevolutionary trends is tile trend toward increasing cranial capacity in the hominid lineage. T h e general consensus is that: (a) fbssil hominids, at least since the advent of Homo, have undergone a steady, gradual increase in brain size (both in absolute measure and relative to b o d y weight) over time (Tobias, 1971; Lestrel & Read, 1973; Holloway, 1975; Lestrel, 1976; Falk, in press); (b) the pace of increasing encephalization accelerated through time (Kurt~n, 1959, 1972; Campbell, 1964; Tobias, 1971 ; Holloway, 1972; Gabow, 1977) until checked by selection against further increase sometime prior to the U p p e r Paleolithic (Campbell, 1964; Bilsborough, 1973; Lestrel & Read; 1973; Olivier, 1973; Lestrel, 1976). This paper examines various procedures used to elucidate the tempo and pattern of encephalic change. Whereas it is clear that hominid brain size has increased dramatically in the past several million years and that little of' this increase can be eplained as an allometric effect of the concomitant (but far less dramatic) increase in b o d y weight (cf. Leutenegger, 1973 ; Pilbeam & Gould, 1974; Holloway, 1975 ; M c H e n r y , 1975 ; Passingham 1975; Fatk, in press; Deacon et al., n.d. ; Jacobs & Godfrey, in prep.), it is our contention that the tempo and pattern of hominid brain evolution are still poorly understood, in part due to the use of certain methodologically unsound operations. 2. D o u b l e L o g a r i t h m i c

Transformations

Perhaps the most thorough attempt to d o c u m e n t the pattern and tempo of evolutionary increment in the hominid brain has been that of Lestrel (1976), which represented a refinement of a similar earlier attempt by Lestrel & R e a d (1973). Using published estimates of cranial capacity and geological age for fossil hominds, Lestrel (1976) performed a series of regressions documenting the changes in hominid cranial capacity over time. Journal of Human Evolution (1981) 10, 255-272

0047-2484/81/030255 + 18 $02.00/0

9 1981 Academic Press Inc. (London) Limited

256

i,. GODFREY AND K. H. JACOBS

Figure 1 reproduces Lestrel's log-log transformation of cranial capacity versus time. With the "present" set at "year" ~ 0, a regression of cranial capacity on time produces a negative slope. As Lestrel notes, the slope changes markedly at roughly 400,000200,000 years ago. H e draws several conclusions t)om these data: first, the plot shows a "generally consistent trend" toward increasing encephalization in fossil hominids prior to 200,000 years ago (p. 212). (Indeed the linear regression fitted to these log-log transformed data explains 88% of the variance; r = 0.94.) Second, "cranial capacity reached a maximum sometime between 0.2 • 106 and 0.4 • 106 m.y.B.P, and then began to fluctuate independently of time until the present" (p. 209). The latter stasis was initiated over a period of "hundreds of thousands of years" (p. 207). Also, according to Lestrel, the slightly decreasing encephalization seen in this period (r =- @0"20) is probably a spurious consequence of male sampling bias in Neandertal populations. Following Leutenegger (1972, 1974), Lestrel tentatively attributes this stasis to increasingly difficult parturition. Presumably, as fetal head size increased in the later Middle Pleistocene, negative selection induced by death in childbirth would have countered positive selection for increasing encephalization. Figure 2 a, b and c shows plots of three log-log transformed artificial data sets which were generated to conform to three distinct hypotheses (Figure 3, Tables I and 2). Figure 1. Plot of cranial capacity versus time, double logarithmically transformed. Redrawn from Lestrel (1976, p. 208, figure 1). These data were based on a sample of 155 points, including 30 fossils exceeding 200,000 B.P., and numerous samples of geologically modern populations. Reprinted with permission from the Journal of Human Evolution. Copyright by Academic Press Inc. (London) Ltd.

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HOMINID BRAIN SIZE INCREASE Figure 2. Plots of double logarithmically transformed artificial data sets (see text). Criteria used to generate these data sets are specified in Table 1, and the entire data set for one of these reproduced in Table 2. (a) Gradual uniform change; (b) autocatalytic acceleration; (c) punctuationat change (three stases).

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Gradual U n i f o r m Change ( N o Stasis) T h e c r a n i a l c a p a c i t y figures a t t w o p o i n t s (4 m . y . a n d 2 0 0 , 0 0 0 B.P.) i n L e s t r e l ' s d a t a set w e r e u s e d to g e n e r a t e a l i n e a r r e l a t i o n s h i p b e t w e e n c r a n i a l c a p a c i t y a n d t i m e . P r e d i c t e d v a l u e s w e r e t h e n c a l c u l a t e d for 67 p o i n t s r a n g i n g i n g e o l o g i c a l a g e f r o m 50 y e a r s to 4 m i l l i o n y e a r s B.P. ( T a b l e 3, c o l u m n s 1 a n d 2), a n d " e r r o r " w a s i n t r o d u c e d b y a l l o w i n g c r a n i a l c a p a c i t y to d e v i a t e ~ = 0 - 1 9 % f r o m t h e p r e d i c t e d v a l u e s ( T a b l e 3, c o l u m n 3).

Table 1

T h r e e m o d e l s for i n c r e a s i n g cranial c a p a c i t y u s e d to g e n e r a t e artificial d a t a s e t s , s h o w n w i t h "error' ' i n t r o d u c e d and t h e p o i n t s d o u b l e l o g a r i t h m i c a l l y t r a n s f o r m e d in Figure 2, a n d w i t h o u t " e r r o r " or l o g a r i t h m i c t r a n s f o r m a t i o n in Figure 3

l~odel a. Gradual uniform change Hypothesis: cranial capacity increased at a steady rate t?om four million years ago to the present. The values of cranial capacity and time ibr two points [taken from Lestrel (1976)] were used to define this linear relationship : Cranial capacity (cc) Year (I) 435 4,000,000 B.P. (2) 1375 200,000 B.P. Model b. Autocatalytic acceleration Hypothesis: the tempo of rising encephalization accelerated as cranial capacity itself increased. The values of log cranial capacity and time were taken for the same two points used in constructing model a: Log cranial capacity Year (1) 2-638 4,000,000 B.P. (2) 3.138 200,000 B.P. Model c. Punctuational change Hypothesis: cranial capacity increased in episodes of rapid change punctuated with periods of little or no change (stasis). Data were generated for lhree non-overlapping stases (using a random numbers table) according to the limitations set below'. The ranges, range midpoints and durations of the stases are listed. The coefficients of variation for the resulting three data sets varied from 12-17%, within the range of variation shown for actual fossil hominids, in comparable time periods : Time period Range midpoint Range (1) 4 m.y.-2 m.y. 550 400 699 (2) 2 m.y.-500,000 900 700-1099 (3) 500,000-present 1350 1100-1599

Table 2

C o r r e l a t i o n s a n d s l o p e s of linear r e g r e s s i o n s after d o u b l e l o g a r i t h m i c t r a n s f o r m a t i o n . Slopes a r e for t h e r e g r e s s i o n s of Y (log cc) on X (log time). W h e r e t h e r e g r e s s i o n w a s n o t p r o v i d e d b y L e s t r e l for h i s data, w e r a n t h e c a l c u l a t i o n o u r s e l v e s . (Cranial c a p a c i t y in cm~; t i m e ixi y e a r s )

~<200,000 B.P.

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Lestrel 1976 a Gradual uniform change b Autocatalytic acceleration c Punctuational change

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HOMINID B R A I N SIZE INCREASE

259

T h e g e n e r a t e d values for cranial c a p a c i t y are shown in T a b l e 3 (column 4). T h e s e d a t a were l o g - l o g t r a n s f o r m e d a n d plotted. T h e y were then d i v i d e d into two sets: values for the p e r i o d of time `<<<<200,000B.P. ; values for the p e r i o d of time in excess o f 200,000 B.P. T h e Pearson p r o d u c t m o m e n t c a l c u l a t e d for the l a t t e r set o f 27 points was --0"87 (cf. T a b l e 2), not very different from Lestrel's 0"94 (also based on • = 27). I t should be n o t e d t h a t 10% of tile points in our s a m p l e fall b e t w e e n 200,000 a n d 400,000 B . P . - - t h e r a n g e for w h i c h this d o u b l e l o g a r i t h m i c plot deviates m a r k e d l y from linearity. Lestrel's smaller sample of points in this range p a r t i a l l y accounts for his slightly h i g h e r coefficient of correlation. ( I n d e e d , Lestrel's most recent p o i n t in this range was 360,000 B.P.)* T h e Pearson p r o d u c t m o m e n t c a l c u l a t e d for the p e r i o d o f time `<<<<200,000 for o u r artificial d a t a was +-0.22 (n 40). As w i t h Lestrel's sample, almost none of the v a r i a n c e of t h e points in this r a n g e was e x p l a i n e d b y the l i n e a r regression calculated for the d o u b l e l o g a r i t h m i c a l l y t r a n s f o r m e d data. F o r our smaller sample, the Pearson p r o d u c t m o m e n t o f 0"22 was not significantly different fi'om zero at the 0-05 level o f confidence.

Autocatalytic Acceleration (No Stasis) U s i n g the same two points, b u t log t r a n s f o r m i n g cranial c a p a c i t y , a n e w r e l a t i o n s h i p was g e n e r a t e d to conform to the hypothesis t h a t cranial c a p a c i t y :increased at a n accelera t i n g t e m p o t h r o u g h time, due p r e s u m a b l y to steadily intensifying orthoselective pressures. No p e r i o d o f stasis was i n t r o d u c e d ; therefore, this set of values shows c r a n i a l c a p a c i t y c u r r e n t l y increasing at a faster r a t e t h a n ever before. As in plot a, a n error t e r m was int r o d u c e d ; then time was log t r a n s f o r m e d a n d the values for " l o g cc" a n d " l o g y e a r " plotted. T h e d a t a were then d i v i d e d into two sets as before (less t h a n or e q u a l to, a n d greater t h a n 200,000 B.P.), a n d linear regressions a n d Pearson p r o d u c t m o m e n t s calculated for each. T h e coetficient of correlation c a l c u l a t e d for the set exceeding 200,000 years B.P. was --0.88. T h e coefficient of correlation c a l c u l a t e d for the time p e r i o d between 50 a n d 200,000 B.P. was --0-306. A l t h o u g h t h e r e is a p e r c e p t i b l e difference b e t w e e n the slopes of d a t a sets a a n d b for the l a t t e r time period, n e i t h e r have slopes nor coefficients of correlation significantly different from zero a t the P = 0"05 level.

Punctuationa! Chartge (Three Stases) F i g u r e 2C shows a d o u b l e l o g a r i t h m i c p l o t ofartificiaI d a t a g e n e r a t e d using a p u n c t u a t i o n a l hypothesis of h o m i n i d b r a i n evolution. Three completely non-overlapping stases were constructed a c c o r d i n g to the criteria specified in T a b l e 1. T h e " s a m p l i n g " scheme used for models "a" a n d "b" was a d o p t e d here as well (40 points in the 50 to 200,000 B.P. range, 27 points in the r a n g e exceeding 200,000 B.P.). E r r o r was i n t r o d u c e d using a r a n d o m n u m b e r s table, a n d the d a t a were l o g - l o g transformed. Finally, the d a t a were divid e d into two sets as before, a n d linear regressions a n d coefficients o f correlation c a l c u l a t e d * Lestrel omittcd three of his four samples in the 200,000 400,000 year period from his calculation: Swanscombe (placed by Lestrel at 200,000 B.P.), Steinheim (placed by Lestrel at 250,000 B.P.), and Solo (placed by Lestrel at 310,000 B.P.). The correlation drops slightly if these are included. JFurthermore, some of Lestrel's assigned geological ages were estimated using his regression, thereby improving the "relationship". It is circular to include them in the calculation of the Pearson product moment tbr these data.

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Table 3

Artificial d a t a set b a s e d on a m o d e l of u n i f o r m , g r a d u a l (i.e. linear) change. The s a m p l i n g schedule s h o w n here w a s selected to c o n f o r m a p p r o x i m a t e l y to Lestrel's data, although only t w o (instead of s o m e 30) data points w e r e generated for the period b e t w e e n 50 a n d 550 B.P. "Error" w a s introduced using a r a n d o m n u m b e r s table, a n d a l l o w i n g cranial capacity to deviate ~-0-19~0 f r o m the predicted values

Year 50 550 1050 1550 2050 2550 3050 3550 4050 4550 5050 5550 6050 6550 7050 7550 8050 8550 9050 9550 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000 100,000 i 10,000 120,000 130,000 140,000 150,000 160,000 170,000 180,000 190,000 200,000 210,000 260,000 310,000 360,000 410,000 460,000 500,000 650,000 800,000 950,000 1,100,000 1,250,000

Predicted cranial capacity (perfect correlation)

R a n d o m error term •

Sample cranial capacity

1424 1424 1424 1424 1424 1424 1424 1424 1423 1423 1423 1423 t 423 1423 1423 1423 1422 1422 1422 1422 1422 1420 1417 1415 1412 1410 1407 1405 1402 1400 1397 1395 1392 1390 1387 1385 1382 1380 1377 1375 1373 I360 1348 1335 1323 1311 1301 1264 1227 1189 1152 1115

-- 10 -- 14 + 12 -- 13 + 13 -- 13 + 08 --08 -- 19 -~ 03 -- 13 -- 05 + 03 + 18 --06 --07 --03 -- 12 + 19 J-03 --06 -- 14 -- 04 --03 + 02 +04 --03 +09 --04 --02 -- O0 --07 +16 + 18 -- O0 + 17 + 01 -- 11 + 12 ~ Ol + 15 --06 -- O0 +03 +03 + 03 + 12 -- 06 + 05 -- 14 + 12 -- 12

1282 1225 1595 1239 1610 1239 1539 1310 1153 1466 1238 1352 1466 1679 1338 1323 1379 1251 1692 1465 1337 1221 1360 I373 1440 1466 1365 1531 1346 1372 1397 1297 1615 1640 1387 1620 1396 1228 1542 1389 1579 1278 1348 t 375 1363 1350 1457 1188 1288 1023 1290 981

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HOMINID BRAIN SIZE INCREASE

T a b l e 3, continued

Year

Predicted cranial capacity (perfect correlation)

R a n d o m error term =1=0-19%

Sample cranial capacity

1078 1041 1004 967 930 880 831 781 732 682 633 583 534 484 435

~ 18 --08 + 18 -- 12 ~ 07 -- 12 - 04 +08 J- 13 + 15 +01 +02 + 03 9- 03 -- 05

1272 958 1185 851 995 774 798 843 827 784 639 595 550 469 413

1,400,000 1,550,000 1,700,000 1,850,000 2,000,000 2,200,000 2,400,000 2,600,000 2,800,000 3,000,000 3,200,000 3,400,000 3,600,000 3,800,000 4,000,000

for each. T h e results were a g a i n quite c o m p a r a b l e to those of Lestrel; c o m p a r e our r - 0"24 to Lestrel's r = 0"20 for the recent time period, a n d our r == 0.92 to Lestrel's r = --0"94 for the period exceeding 200,000 B.P. " T h e relationships" u n d e r l y i n g our three artificial data sets are easily distinguishable w h e n plotted w i t h o u t log t r a n s f o r m a t i o n a n d w i t h o u t error terms (Figure 3). T h e y are only similar in that we required of each a threefold increase in cranial capacity over Figure 3. T h r e e models for increasing cranial capacity, shown without " e r r o r " or data transformation. T h e straight line represents gradual uniform change. T h e dotted line represents autocatalytic change (constant acceleration). T h e dashed lines represent a three-stasis p u n c t u a tional scheme, with no phyletic c o m p o n e n t of size increase added. Needless to say, n u m e r o u s other "plausible" models m i g h t be generated, including some combination of the above, a punetuational model with more stases, or a p u n e t u a t i o n a l model which includes a c o m p o n e n t of phyletic size increase. 1600

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a four million year time period. They are not easily distinguishable when reasonable error terms are introduced and the data are log-log transformed. Two of the conclusions that Lestrel drew from his plot are merely artifacts of the double logarithmic transformation : (1) that there was a "generally consistent" upward trend in cranial capacity prior to roughly 400,000 B.P. ; and (2) that this generally consistent trend came to an end between 200,000 and 400,000 B.P., and was fbllowed by a stasis which has continued into tile current time slice. In fact the "significant" change in slope between 200,000 and 400,000 B.P. reflects nothing but the differences in scale of the original variables "cranial capacity" and "year." While cranial capacity increases threefold (less than one order of magnitude), time increases more thanfive orders of magnitude. Figure 4 shows what happens to model a (the solid line in Figure 3) when cranial capacity and year are both log transformed. We are looking at a distorted picture of a perfectly linear relationship. Note that if the scales of the original variables were equal, double logarithmic transformation would not distort the intrinsic linearity. An obvious example is the simple isometric relationship Y = X. It is linear. Log Y ~- log X is also linear. Figure 4. Gradual uniform change (Figure 3, solid line) double logarithmically transtormed. No error term.

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I f it is still not clear that log transforming " t i m e " is inappropriate especially when the scales of " t i m e " and the other variable(s) are grossly different, consider what happens to the --0"94 correlation between cranial capacity and year calculated for the portion of Lestrel's data set exceeding 200,000 B.P. when one moves the zero point. Time(on Earth) is linear and the placement of the point of reference (year "zero") should be arbitrary. Without log transformation, reversing the time scale will reverse the sign of the Pearson product m o m e n t but will not otherwise affect it. With log-log transformation, however, the story changes. Taking the example of Lestrel's data for the period of time exceeding

HOMINID B R A I N SIZE INCREASE

26~

200,000 B.P. and reversing the time scale, the Pearson product moment drops 10% to @0"837.* Now only 70% (not 88%) of the variance is explained. This is because such a reversal "expands" the early part of the record where, according to Lestrel's data, cranial capacity increases slowly, and contracts the latter part of the record, where cranial capacity increases rapidly. 3. G o o d n e s s o f Fit

It is generally believed that one can decipher the "real" biological relationship between any two variables by comparing the variances explained by linear regressions calculated for log-log, semilog, and untransformed data sets. Several years ago, Dr Thomas Hursh (then at the Lawrence Hall of Science, Berkeley) and one of us (LRG) began designing classroom materials to teach hypothesis testing to undergraduates. Our objective was to teach students "how to discover" biological regularities or relationships using computer generated fake data sets which conformed to various known biological relationships. By specifying the equation, the ranges for variables X and Y and their "error" terms, an infinite number of data sets conforming to any given biological relationship could be produced. The students would examine the scattergram generated by the computer, decide whether and how to transform the data, examine the relative strengths of the Pearson product moments calculated for the various data transfbrmations, and, we thought, "discover" the underlying "biological relationship". The trouble was, when any kind of non-trivial error term was introduced, our "goodness of fit" test was simply inadequate. We ourselves could not reproduce the relationships used to generate the original scattergrams; the "wrong" transformation sometimes explained the most variance. Nor is this a rare or trivial problem. Inconsistencies in the descriptions of numerous well-studied relationships are commonplace in the biological literature. Length-length and length-diameter relationships among closely related organisms are not always thought to be linear; length-weight relationships are not universally described as curvilinear. What should the relationship between two body "lengths" look like--in an ontogenetic series --for example ? In adults of a single species ? In adults of a group of closely related taxa ? How should "lengths" relate to "diameters" ? What about the relationship between "weights" and "lengths"? Within the last decade, the study of rules governing the form of organisms has matured dramatically. Researchers in physiology and biomechanics have begun debating alternative hypotheses and constructing a set of concrete expectations (Schmidt-Nielsen, 1972; Alexander & Goldspink, 1977; Pedley, 1977). Here the relationship between hypothesis and mathematical treatment is explicit. For example, if a group of organisms are geometrically similar, their length-length and length-diameter relationship will be isometric (linear). Furthermore, we know that the relationships between their body lengths and weights, surface areas, cross-sectional areas, basal metabolic power, heartbeat, etc., will all be curvilinear. And we can even predict what the exponents should look like. * To avoid a mathematical impossibility (log transforming a zero value), we moved Lestrel's values for Makapansgat from 4,000,000 B.P. to 3,900,000 B.P. before reversing the time scale. The values of the Pearson product moment calculated tbr the time reversed and nonreversed data sets are 0.83 and 0-93 respectively.

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L. G O D F R E Y AND K. H . JACOBS

In other words, we have a set of biological expectations--a reason to make specific data transformations even if our range of points is too restricted and our goodness of fit test too weak to clearly demonstrate linearity or curvilinearity. I f we suspect elastic similarity, we would have a different set of expectations; indeed, we would expect a curvilinear relationship between "lengths" and "diameters" (cf. McMahon, 1975). It is much less clear how geological age should relate to biological variables--be they lengths, diameters, volumes, etc. It is here that we most sorely lack a concrete set of expectations. The goodness of fit test often fails when the differences between the predicted values for the various data trans[brmations--at least over the range considered--are trivial. This is the case for cranial capacity versus time when sample size is limited (Blumenberg, 1978). O f course, log transformation will sometimes seem to allow "better" predictions to be made and therefore to have a practical value. However, even when used merely to predict values, we must recognize that log transformation is appropriate only when the underlying relationship is nonlinear. Also, log transformation may introduce other difficulties that may outweigh the apparent improvement in fit. Steel & Torrie (1960, pp. 332 ft.) for example, caution against assuming that the confidence limits and tests of" significance contructed for transformed data will be valid tbr the untransformed data. Furthermore, data transformations are often used not merely to predict but to describe or discover the form or nature of the biological relationship between two variables, and here is where great care must be exercised in testing alternative hypotheses lest the researcher simply fall into the trap of having his unstated and, perhaps, unrealized a priori assumptions dictate his results. One can calculate the linear regression which best fits any bivariate set of points; it does not follow that the relationship between the variables involved is linear ! Similarly, log transformation of a linear data set is possible, and may seem to "improve" the strength of the correlation, but this does not demonstrate anything that is not patently obvious before examining the data set: log transformation reduces the variance and therefore, perhaps, the messiness of the data. How, then, can we tell which is the appropriate transformation ? It would help to set out the underlying assumptions for each transformation--the inherent connection between hypothesis and mathematical treatment. The point is: each hypothesis requires a particular mathematical treatment and each mathematical treatment assumes a particular hypothesis. We cannot select the appropriate mathematical treatment for a data set without first considering our biological expectations, our alternative hypotheses, and the fit of our raw data set to each of the possible models. Let us take the example of the relationship between cranial capacity and time. When are semilog, log-log and no transformations appropriate? When is linear regression appropriate, and when is it inappropriate even to calculate the linear relationship between two untransformed variables ? Let us examine the three biological models (a, b and c) posed above, and consider their appropriate mathematical descriptions.

Gradual Un~'orm Change (No Stasis) This hypothesis implies roughly constant directional selection (orthoselection) over long periods of time. It requires (demands) no logarithmic transformation. I f constant uniform directional selection were operating, the raw data should look linear. A single linear equation should accurately describe the relationship between year and cranial capacity. Linear regression to predict either variable would be appropriate.

IIOMINID BRAIN SIZE INCREASE

265

Autocatalytic Acceleration Stated briefly, this hypothesis means that selection pressure increases constantly as a function of itself, as part of feedback loop (cf. Mayr, 1963; Tobias, 1971 ; Holloway, 1972). Here, a semilog ~ransformation (with cranial capacity log transformed) is appropriate, since the tempo of change in cranial capacity should increase in a predictable manner, rising as cranial capacity itself increases. Theoretically, if such a process were operating, the raw data should look curvilinear; log transformation of cranial capacity should straighten the curve by "slowing" the rate of increase in cranial capacity as the values of cranial capacity increase, and the correlation between "log cranial capacity" and " y e a r " should be stronger than the correlation between "cranial capacity" and "year". T h e equation which describes the relationship between the original variables predicting time as a function of cranial capacity would be in the formy - - bxa @ c. Log transformation of cranial capacity would be necessary to make prediction possible, since a linear regression can only accurately describe a curvilinear relationship after log transformation. There is no reason to log transform "year", which neither accelerates nor decelerates with changes in cranial capacity. T h e distortions resulting from such data transformation have already been discussed.

Punctuational Change (Three Stases) Briefly stated, the punctuational view which has gained support in recent years a m o n g some paleobiologists (see discussion) is that major macroevolutionary transitions proceed in episodes of rapid change interspersed with long periods of stasis or negligible change. It requires a stepladder picture of macroevolutionary change, although punctuationalists allow for minor adaptational fine-tuning and, notably, phyletic shifts in size during periods of stasis. T h e number of "stases" involved in any macroevolutionary trend should vary markedly ; three was selected here because it represents a reasonable n u m b e r for a m a m m a l in a four million year period. It also represents a strongly punctuational interpretation of hominid brain evolution. No phyletic component of size increase was included in any of our stases. Needless to say, were we to increase the number of "steps" or introduce a phyletic component of size increase our punctuational model would be more difficult to distinguish from gradual uniform change. An increase in the number of steps or a single more dramatic step occurring late in the evolution of hominid cranial capacity would create a picture resembling a pattern of acceleration, but the gradual increase in tempo required by the autocatalytic model should not occur. Furthermore, sampling error itself m a y help produce an apparently curvilinear pattern of evolutionary change. I f the probability of missing (i.e. not sampling) an episode of rapid evolutionary change increases rapidly as we go back in time, it is possible that only the last such episode can be expected to have been sampled. Furthermore, if we do miss a rapid or q u a n t u m evolutionary event (but we take samples on either side of the event), the slope calculated for the change in any morphological trait greatly affected by the event will depend upon the time interval on either side of the quantum event; it will be lower for longer intervals and higher for shorter intervals. This would be true even if the actual magnitude of the change in morphology were the same for all such events. For these reasons, and because variance in cranial capacity m a y be high and sample size poor, a punctuational model m a y be, in practice, very difficult to distinguish ti'om either models a or b. For the punctuational case, linear regression cannot be applied to the whole data set

266

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GODFREY

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K.

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JACOBS

w i t h o u t seriously distorting the " r e a l " biological relationship b e t w e e n the variables. S e p a r a t e linear regressions for each stasis ( u n t r a n s f o r m e d d a t a ) are a p p r o p r i a t e , a n d they should show slopes a n d Pearson p r o d u c t m o m e n t s close t o zero. A linea~i regression a p p l i e d to the whole d a t a set will e x p l a i n a large a m o u n t of the v a r i a n c e simply b e c a u s e of t h e threefold increase in c r a n i a l c a p a c i t y e n c o m p a s s e d in these stases. But it w o u l d not be a p p r o p r i a t e except to d e m o n s t r a t e the overall p a t t e r n of c h a n g e on a g r a n d scale, a n d w o u l d n o t d e m o n s t r a t e t h a t the d a t a are linear. L o g t r a n s f o r m a t i o n of c r a n i a l c a p a c i t y w o u l d also be i n a p p r o p r i a t e , a l t h o u g h it can p e r h a p s b e justified if there is a clear reason to expect the number or magnitude of the steps to increase, a n d the r e s e a r c h e r clearly states an i n t e n t i o n to describe the p a t t e r n of m a c r o e v o l u t i o n a r y c h a n g e over long periods e n c o m p a s s i n g several stases. A g a i n such a p a t t e r n c a n n o t b e used to m a k e predictions within stases, or to describe the biological relationship b e t w e e n c r a n i a l c a p a c i t y a n d y e a r on a fine scale. L o g t r a n s f o r m a t i o n of c r a n i a l c a p a c i t y a n d y e a r will distort a n d h i d e the p u n c t u a t i o n a l n a t u r e of the r a w d a t a even if the d a t a show no phyletic c o m p o n e n t of size increase within stases, a n d a l i n e a r regression a p p l i e d to such l o g - l o g t r a n s t b r m e d d a t a is i n a p p r o p r i a t e . 4. T h e F o s s i l D a t a W e now t u r n to the fossil r e c o r d of h o m i n i d c r a n i a l capacity, a n d ask w h e t h e r it a c t u a l l y describes a g r a d u a l i s t a n d continuous p a t t e r n . C r a n i a l capacities a n d geological ages for 95 fossil h o m i n i d specimens r a n g i n g in age from 3.6 m.y. to 0.012 m.y. were g a t h e r e d from the l i t e r a t u r e [Figure 5 ; see J a c o b s & G o d f r e y (in prep.) for a fuller discussion]. W h e r e m u l t i p l e c r a n i a l c a p a c i t y estimates existed for a single specimen, the most r e c e n t estimate was used. Geological ages used are those in general a c c e p t a n c e a n d are b a s e d w h e r e Figure 5. Linear regressions for unstransformed data, Australopithecus and IIomo. Solid lines represent the linear regression calculated tbr whole generic sets. Dashed lines represent regressions calculated for subsets of Homo. Sample sizes for the regressions (circled) and means (subscripts) are indicated. The value of cranial capacity indicated for Australopithecus afarensis is an estimate (the approximate midpoint of the probable range). Further details are given in tile text and in Table 4. 1600 t

• , 1400

~t,omo/__|

~- 1200

g ~3 moo @

/

Us . . . . . .

/

J%

H4 BOO HQs~~ Q4

Aus/ro/op/lhecos A5 ILi ~'5

@

A6 9 i 3"0

i 2"5

A2*

oA 5

? AI

I

2.0 Time

1'5 (m~,.)

~A11 1.0

I I

I

r 0"5

J ~ I

I ] 0

HOMINID BRAIN SIZE INCREASE

267

possible on tile most recent geochronological, t~unal and paleomagnetic evidence. Disagreement on the temporal placement of certain specimens obviously exists, but the ages used here represent at least the best current guess. Figure 5 presents a plot of mean cranial capacity for temporally lumped subsets on the genera Australopithecus (A) and Homo (H). Specimens which have been sometimes considered Australopithecus, sometimes Homo (i.e. habilines from Olduvai Gorge), tbrm a distinct subset represented by the letter Q . Sample sizes for subsets of Homo vary from 3 to 14, and the groups were selected because of temporal rather than morphological homogeneity. '~HQ" represents the rnean for one subset when the "habilines" are included in the genus Homo. T h e regression lines in this figure were derived from analyses of individual specimen cranial capacities (untransformed) and geological ages in million years B.P. T h e y represent the hypothesis of gradual linear change. The solid lines were calculated tor whole generic samples; the broken lines show regressions calculated for subsets of Homo. T h e n's are indicated for each mean, and each regression. As can be seen in the figure, Australopithecus changed little in cranial capacity over a several million year period. In contrast, the Homo sample exhibits a slope that is fully 15 times greater than that of the Australopithecus sample. Here the habilines are excluded ti~om both samples. When the habilines are included in either group and excluded from the other, the picture changes little. Including the habilines in the Homo sample does not change its linear regression significantly. When included in Australopithecus and excluded from Homo, the slope of the linear regression for Australopithecus is still negligible (the slight increment attributable to body size increase) and the slope for Homo is still dramatically greater than that of Australopithecus. The correlation between cranial capacity and geological age for Homo even when the habilines are excluded, is highly significant (0.79; n -- 74). This is not likely attributable to a concomitant body size increase, there being little discernible trend in Homo body size from the Early Pleistocene of Lake T u r k a n a to the Late Pleistocene European specimens of our sample. This trend is certainly not surprising, the progressive encephalization of Homo having long been recognized. On the other hand, the actual tbssil data exhibit intriguing deviations from the pattern predicted by the regression. As can be seen in Figure 5, the regression seriously underestimates cranial capacity for the period from two million to one and a half million years B.P. (even with habilines included), equally seriously overestimates brain size between one million and 250 thousand B.P., and then again underestimates in the 100 to 12 thousand year period. Such a pattern is typical o f a curvilinear relationship. A curvilinear pattern is not inconsistent with a strict gradualist interpretation of the overall phyletic trend. It merely implies a steadily accelerating pace of encephalization, due presumably to steadily intensifying orthoselective pressures and a population structure conducive to rapid evolutionary change in the Middle Pleistocene (Gabow, 1977). This is indeed the gist of the traditional gradualist explanation, in which a rapidly increasing adaptive value of learning and intelligence is postulated. It is inferred from the archaeological record, which seemingly evinces an accelerating trajectory of reliance on tool use and manufacture, morphological and technical complexity of the tools themselves, and variability and complexity of sociocultural systems. These and other features of hominid evolution have been repeatedly suggested as the driving forces behind intensifying orthoselective pressures for large brains. While this m a y indeed have been the case, the precipitous cessation of encephalization

268

L. GODFREY AND K. H. JACOBS

in the U p p e r Pleistocene, after n e a r l y two million years of acceleration, has yet to be e x p l a i n e d . I t is usually s i m p l y asserted t h a t e n c e p h a l i z a t i o n "just s t o p p e d " , A s s u m i n g for the m o m e n t t h a t this true, it raises the i n t r i g u i n g possibility t h a t d u r i n g o t h e r phases o f h o m i n i d evolution, e n c e p h a l i z a t i o n m i g h t have a g a i n "just s t o p p e d " . T o e v a l u a t e this possibility, the entire fossil s a m p l e was d i v i d e d into five non-discrete t i m e intervals (in no case r e p r e s e n t i n g fewer t h a n 10 i n d i v i d u a l s ) : 2 m i l l i o n to 800 t h o u s a n d years, 900 to 400 t h o u s a n d years, 400 to 200 t h o u s a n d years, 200 to 45 t h o u s a n d years, a n d 45 to 12 t h o u s a n d years. Regressions of c r a n i a l c a p a c i t y on age expressed in millions of years were d e r i v e d for e a c h interval a n d are presented (dashed lines) in F i g u r e 5 a n d in T a b l e 4. Subset regressions: (a) correlations a n d slopes of linear reg r e s s i o n s after s e m i l o g t r a n s f o r m a t i o n . R e g r e s s i o n s of log cc on year (Homo only); (b) correlations and slopes of linear regressions after no t r a n s f o r m a t i o n . R e g r e s s i o n s of cranial capacity on year (Homo only)

Table 4

f

Data set

n

LE*

Time (m.y.) --~ GE'~ r

r2

F-test ~ significance Slope of slope

(a) Semilog transformed

15 15 10 25 25

2'0 0.9 0.4 0-2 0.045

0-8 0.4 0.2 0-045 0.012

--0.42 --0-62 --0.02 --0.61 --0.004

0.18 0-38 0.00 0"37 0.00

--0.055 --0'224 --0.015 --0.990 --0.012

NS P < 0.01 NS P < 0.0I NS

(b) Untransformed

15 15 I0 25 25

2.0 0'9 0-4 0.2 0.045

0.8 0.4 0'2 0.045 0.012

--0.42 --0'63 +0.006 --0.62 --0.01

0'18 0.39 0.00 0.39 0.00

--109"1 --525'6 + 1t-7 --3225-3 --130.8

NS P < 0.05 NS P < 0.01 NS

*LE = less than or equal to. ~'GE -- greater than or equal to. (Cranial capacity in cm3; time in million year units.) Interestingly, the first interval, despite a length o f 1-2 million years, exhibits a slope t h a t is n o t significantly g r e a t e r t h a n t h a t of the a u s t r a l o p i t h e c i n e s a m p l e discussed earlier. T h i s is in stark contrast to the h a l f million y e a r second interval, for w h i c h the slope is fully five times as g r e a t as t h a t of the previous period. T h e t h i r d interval, from 400 to 200 t h o u s a n d years, once a g a i n exhibits a negligible slope. F r o m 200 to 45 t h o u s a n d years, however, the slope is over six times as g r e a t as t h a t of the r e l a t i v e l y longer p r e c e d i n g p e r i o d of r a p i d increase. A n d , as expected, the p e r i o d from 45 to 12 t h o u s a n d years shows n o c r a n i a l c a p a c i t y increase. T h u s it w o u l d seem possible to a r g u e either a p u n c t u a t i o n a l or a n a u t o c a t a l y t i c m o d e l o f e v o l u t i o n a r y c h a n g e in h u m a n c r a n i a l capacity. T h e a c t u a l d a t a show some curvilinearity, a n d a n i n t r i g u i n g s u p e r i m p o s e d staircase p a t t e r n , b u t the s a m p l e sizes a r e small a n d the staircase p a t t e r n , p e r h a p s , unconvincing. T h e hypothesis of intensifying selection w o u l d seem to be s u p p o r t e d b y the a p p a r e n t c u r v i l i n e a r i t y of the d a t a . B u t semilog t r a n s f o r m a t i o n does n o t s t r a i g h t e n t h e curve v e r y well, a n d w e still face slopes for l i n e a r regressions a p p l i e d to subsets w h i c h differ m a n y f o l d . I n fact, the p a t t e r n o f significance o f subset slopes is e x a c t l y the s a m e for u n t r a n s f o r m e d a n d semilog t r a n s f o r m e d d a t a ,

HOMINID B R A I N SIZE INCREASE

969

alternating non-significant with highly significant change (Table 4). The increases, when they occur, are far more dramatic than would be anticipated by a hypothesis of gradually intensifying selection. A punctuational model might account for these data, but it would have to somehow account for the apparent curvilinearity of the data. The data do demonstrate at least one extremely rapid episode of evolutionary change, as only punctuationalists have maintained possible. Several conclusions can be drawn from these data: first, contrary to Lestrel (1976), 200,000 B.P. does not mark the end of increasing encephalization in the hominid lineage. In double logarithmically transforming his data, Lestrel missed the most dramatic period of increase in encephalization in the history of humankind. Secondly, if we can seriously maintain that cranial capacity just stopped evolving in the Upper Paleolithic (or earlier), we can hardly deny the high probability of similar cessations of much longer duration. The data clearly do not refute long periods ofstasis or very slow change in cranial capacity :for the genus Homo. 5. D i s c u s s i o n

T h e tempo and pattern of macroevolutionary change have bearing on current controversies concerning the processes of macroevolutionary change. In recent years paleobiology has witnessed the consolidation and divergence of two distinct perspectives on the causation of macroevolutionary trends. The traditional gradualist approach asserts the prid macy of what is called "phyletic" evolution--gradual adaptive change in established populations. Macroevolutionary trends are ascribed to the long-term continuous operation of relatively constant and unidirectional selection pressures acting on individual variation at the population level and spreading to neighbouring populations by gene flow. The competing perspective on macroevolution is variously termed the neocatastrophist or punctuational approach (Eldredge & Gould, 1972; Stanley, 1975, Gould & Eldredge, 1977; Stanley, 1979; Gould, 1980). Proponents of this view argue that evolution is a hierarchical process involving different modes of change at three major levels: populational, speciational, and macroevolutionary. 5/[acroevolutionary trends are attributed to the operation of a high order selection process called "species selection" (Stanley 1975, 1979). Gould (1980) calls this the "Wright break" because Sewall Wright (1967) distinguished between microevolutionary trends or shifts attributable to natural selection operating on individual variation within populations and evolutionary trends produced by the differential success of species. To Wright (i 967), speciation and mutation are analogous mechanisms which produce variation upon which "selection" levels act. Although Sewall Wright's macroevolutionary model differs from that of Eldredge, Gould, and Stanley in some very basic ways, Wright also rejects constant directional selection as an explanation of macroevolutionary trends. The punctuationalists are primarily concerned w i t h processes driving evolutionary change and not the rate or temporal pattern of change; however, their theory implies that major macroevolutionary transformations should proceed in episodes of rapid change interspersed with long periods of stasis or negligible change (that is, minor adaptational finetuning and phyletic shifts in size). The macroevolutionary trend toward increase in hominid brain size is widely recognized as one of the more dramatic and well documented macroevolutionary transformations evinced by the mammalian fossil record (Haldane,

270

L. G O D F R E Y A N D K. H . J A C O B S

1949; Tobias, 1971; Bilsborough, 1973; Jerison, 1973; Sacher, 1975; Gabow, 1977). I t should be make an excellent test case for the various theories of macroevolutionary change. Lestrel & Read (1973) and Lestrel (1976) used linear regression to argue a hypothesis of regular change in h u m a n cranial capacity (although a linear plot of log transformed data cannot be interpreted in this way). Others have supported an autocatalytic model of hominid brain expansion. The least favored view has been the punctuational interpretation (but see Zindler, 1978). Yet the sketchy data for fossil hominids would seem not to preclude such an interpretation. As the tbssil record improves it will be easier to choose among theoretical alternatives. However, acceptance or rejection of a punctuational interpretation of h u m a n brain evolution will probably depend more upon data from other fields than on a necessarily poor fossil record. Indeed, the debate over punctuated equilibria encompasses m a n y fields of inquiry. Research on epigenetics (Lovtrup, 1974), chromosomal reshuffling and regulator gene processes (King & Wilson, 1975; Wilson, Carlson & White, 1977), heterochrony (Gould, 1977) and biostratigraphy (Hallam, 1977; Stanley, 1979) all bears on this controversy. Replacement versus phyletic models of h u m a n evolution have been widely discussed in the past, and our perspective on these m a y change as new analytical techniques become available. For example, using mitochondrial D N A (inherited only through females and therefore not subject to sexual recombination) Altan Wilson and his co-workers are trying to isolate the time of the last "bottleneck" t)om whence all living humans are descended. Work on Y-chromosomal D N A should give us a similar handle on this problem. In the interim, those that are inclined to reject the punctuational hypothesis for h u m a n evolution should not design mathematical "tests" ofpunctuationalism that have gradualist conclusions predicated by gradualist a priori assumptions ! At the very least they should realize that this is what they are doing. In examining fossil data, it is imperative that the theoretical implications of various interpretations of hominid brain evolution be carefully and thoroughly considered. Any mathematical manipulation or transformation done should be done for a reason and shouId be explicitly related to a set of hypotheses.

6. Summary An examination of methods used to establish a widely accepted picture of hominid encephalization reveals that the conclusion that brain size increased in a regular fashion and reached its current "stasis" approximately 200,000 years ago is flawed. It is an artifact of an inappropriate double logarithmic transformation of data on cranial capacity versus time. When such data are double logarithmically transformed, it is impossible to distinguish gradualism from alternate models of evolutionary change, including extreme punctuationalism. Artificial data sets designed to illustrate clear cases of gradualism, autocatalysis and punctuationalism were generated to demonstrate some of the problems inherent in double logarithmic data transformation and linear regression. In order to demonstrate any particular evolutionary model, data manipulations should be explicitly theoretical, taking into account both their justification and possible consequences.

We gratefully acknowledge Dr Pete Lestrel for his encouragement and permission, along with that of Academic Press, to print Figure 1 redrawn from his original publication.

HOMINID BRAIN SIZE INCREASE

271

Dr PaulJ. Godfrey drew the figures and advised on analytical procedures; Dr Thomas M. Hursh provided invaluable assistance in working on related materials with one of us (LRG). We also thank Dr John R. Cole for his advice on the format and clarity of the arguments presented here, and C. J. Masse for her assistance in preparing the manuscript for publication. References

Alexander, R. McN. & Goldspink, G. (Eds) (1977). Mechanics and Energetics of Animal Locomotion. London : Chapman and Hail. Bilsborough, A. (1973). A multivariate study of evolutionary change in the hominid cranial vault and some evolution rates. Journal of Human Evolution 2~ 387-404. Blumenberg, B. (1978). Hominid ECV: available data does not permit a choice of model. Journal of Human Evolution 7, 425436. Campbell, B. (1964). Quantitative taxonomy and human evolution. In (S. L. Washburn, Ed.) Classification and Human Evolution, pp. 50-74. Chicago : Aldine. Deacon, T. W., Filler, A. G., Cronin, J. E., & Boaz, N. T. (n.d.). The rate and nature of brain evolution in the Hominidae. Paperpresented at the 49th Annual Meeting of the American Association of Physical Anthropologists, Niagara Falls, April 1980. Eldredge, N. & Gould, S . J . (1972). Punctuated equilibria: an alternative to phyletic gradualism. In (T. J. M. Schopf, Ed.) Models in Paleobiolog)', pp. 82-115. San Francisco: Freeman, Cooper. Endler, J. A. (1977). Geographic Variation, Speciation, and Clines. Princeton: Princeton University Press. Falk, D. (in press). Hominid brain evolution: the approach from paleoneurology. Yearbook of Physical Anthropology. Gabow, S. L. (1977). Population structure and the rate of hominid brain evolution. Journal of Human Evolution 6, 643-665. Gould, S.J. (1975). Allometry in primates, with emphasis on scaling and the evolution of the brain. Contributions to Primatology 5, 244-292. Gould, S.J. (1977). Ontogeny and Phylogeny. Cambridge, Ma.: Belknap. Gould, S.J. (1980). Is a new and general theory of evolution emerging ? Paleobiology 6, 119-130. Gould, S.J. & Eldredge, N. (1977). Punctuated equilibria: the tempo and mode of evolution reconsidered. Paleobiology 3~ 115-15I. Haldane, J. B. S. (1949). Suggestions as to quantitative measurement of rates of evolution. Evolution 3~ 51-56. Hallam, A. (Ed.) (1977), Patterns of Evolution, as Illustrated by the Fossil Record. Amsterdam: Elsevier. Holloway, R. L. (1972). Australopithecine endocasts, brain evolution in the Hominoidea, and a model of hominid evolution. In (R. H. Tuttle, Ed.) 77zeFunctional and Evolutionary Biology of Primates, pp. 185-203. Chicago : Aldine-Atherton. Holloway, R. L. (1975). Early hominid endocasts. In (R. H. Tuttle, Ed.) Primate Functional Morphology and Evolution, pp. 393Md5. The Hague: Mouton. Jacobs, K. H. & Godfrey, L. R. (in prep.). Orthoselection and the evolution of hominid cranial capacity. Jerison, H . J . (1973). Evolution of the Brain and Intelligence. New York: Academic Press. King, M. C. & Wilson, A. C. (1975). Evolution on two levels in humans and chimpanzees. Science 188~ 107-116. Kurtdn, B. (1959). Rates of evolution in fossil mammals. Cold Spring Harbor Symposia on Quantitative Biology 24, 205-215. Kurtdn, B. (1972). Not from the Apes: A History of Man's Origins and Evolution. New York: Vintage Books. Lestrel, P. E. (1976). Hominid brain size versus time: revised regression estimates. Journal of Human Evolution 5~ 207-212. Lestrel, P. E. & Read, D. W. (1973). Hominid cranial capacity versus time; a regression approach. Journal of Human Evolution 2, 405-411. Leutenegger, W. (1972). Newborn size and pelvic dimensions of Australopithecus. Nature 240, 568-569. Leutenegger, W. (1973). Encephalization in Australopithecines: a new estimate. Foliaprimalologica 19, 9-17. Leutenegger, W. (1974). Functional aspects of pelvic morphology in simian primates. Journal of Human Evolution 3, 207-222. Lovtrup, S. (1974). Ep@enetics: A Treatise on Theoretical Biology. New York: Wiley. Mayr, E. (1963). Animal Species and Evolution. Cambridge, Ma. : Belknap. McHenry, H. M. (1975). Fossil hominid body weight and brain size. Nature 254, 686 688. McMahon, T. A. (1975). Using body size to understand the structural design of animals: quadrupedal locomotion. Journal of Applied Physiology 39, 619-627. Olivier, G. (1973). Hominization and cranial capacity. In (M. H. Day, Ed.) Human Evolution, pp. 87-101. London: Taylor & Francis.

272

L, G O D F R E Y

AND

K.

H. JACOBS

Passingham, R. E. (1975). Changes in the size and organization of the brain in man and his ancestors. Brain, Behavior and Evolution 11, 73-90. Pedley, T. (Ed.) (1977). Scale Effects in Animal Locomotion. London: Academic Press. Pilbeam, D. & Gould, S.J. (1974). Size and scaling in human evolution. Science 186, 892-901. Saeher, G. A. (1975). Maturation and longevity in relation to cranial capacity in hominid evolution. In (R. Tuttle, Ed.) Primate Functional Morphology and Evolution, pp. 417441. The Hague: Mouton. Schmidt-Nielsen, K. (1972). How Animals Work. Cambridge: Cambridge University Press. Stanley, S. M. (1975). A theory of evolution above the species level. Proceedingsof the National Academy of Science, U.S.A. 72, 646-650. Stanley, S. M. (1979). Macroevolution : Pattern and Process. San Francisco: W. H. Freeman. Steel, R. G. D. & Torrie, J. H. (1960). Principles and Procedures of Statistics. New York: McGraw-Hill. Tobias, P. V. (1971 ). The Brain in Hominid Evolution. New York: Columbia University Press. Wilson, A. C., Carlson, S. S. & White, T . J . (1977). Biochemical evolution. Annual Review of Biochemistry 46, 573-639. Wright, S. (1967). Comments on the preliminary working papers of Eden and Waddington. In (P. S. Moorehead & M. M. Kaplan, Eds) Mathematical Challenges" to the Neo-Darwinian Theory of Evolution, pp 117-120. Wistar Institute Symposium 5. Zindler, R. E. (1978). On the increase of cranial capacity in mankind's lineage: augments and elaborations. Journal of Human Evolution 7, 295-305.