Methods, Log sem statistics for crania, Procrustes Distances for mandibles & Z-scores

The use of Z-scores to facilitate morphometric comparisons between African Plio-Pleistocene hominin fossils: An example of method
Australopithecus CRISPREvolutionary Biology hominin Homo habilis Morphometrics Paleobiology Plio-Pleistocene

In a morphometric approach based on pairwise comparisons, Thackeray and Odes29 calculated ‘log sem’ statistics associated with regression analyses to compare OH 24 with other crania, based on measurements from original specimens published by Wood.23 The log sem results reflect variability in skull shape. In the case of the comparison between OH 24 and Sts 5 (almost complete skulls), 54 measurements are in common. In the instance of OH 24 and Sts 71 (partial skull), 44 measurements are common to both specimens.

A log sem value was obtained from a comparison between our measurements of OH 7 and Sts 52b mandibles, using landmarks based mainly on points associated with mesiodistal and bucco-lingual diameters (excluding third lower molars because the specimens do not represent fully adult individuals). Forty measurements per specimen were taken from virtual reconstructions of OH 7 and Sts 52b by Spoor et al8 and Benazzi et al.30 respectively. Sts 52b was selected for comparison in this study because of an apparent degree of morphological similarity with OH 7, as discussed by Tattersall.31

Using Procrustes Distances (PD), Spoor et al.8 compared Plio-Pleistocene hominin specimens with a focus on OH 7, based on more than 50 landmarks. For purposes of our study, we used PD data made available through Spoor and Gunz (personal communication to JFT, 2020). Here we integrate the two kinds of shape-related statistics (PD and log sem) by expressing them as standardised Z-scores in relation to data obtained from humans and extant great apes.

The AL 400-1 mandible of A. afarensis has been compared with OH 7 on the basis of the PD method. AL 400-1 has also been compared with OH 7 using the log sem statistic. The difference between Z-scores is expected to be relatively small if the approaches yield consistent results.

Log sem statistics for crania

The log sem statistic has been previously used in analyses of linear measurements obtained from crania of modern specimens in natural history museums32-34 and Plio-Pleistocene hominins.29,32,34 In this method, measurements are subjected to pairwise comparisons, using least squares linear regression to generate an equation of the form y= mx + c, where m is the slope and c is the intercept. In an initial study of pairs of specimens of the same (extant) species in 1997, Thackeray et al.32 reported central tendency of the log-transformed standard error of the m-coefficient, known as ‘log sem’ which is a measure of the degree of scatter around the regression line, reflecting variability in shape. Central tendency of log sem was also discovered using larger samples, associated with a mean log sem value of -1.61 reported in 2007 by Thackeray.33 At least for hominoids, the mean log sem value of -1.61±0.1 was recognised in 2016 as a typical degree of intraspecific morphological variation in extant species.

In response to views expressed by Gordon and Wood,35 Thackeray and Dykes34 emphasised the need to make pairwise comparisons with specimen A on the x-axis and specimen B on the y-axis, and vice versa. Two log sem values are obtained. The absolute difference between these values is termed ‘delta log sem’. The mean delta log sem is small (generally ≤ 0.03) when pairs of specimens of the same species are compared. By contrast, delta log sem values are large (generally >> 0.03) when specimens of different species are compared.34 Thackeray and Dykes34 stated that the number of measurable dimensions (k) obtained from pairs of specimens should be maximised as far as possible to ensure robusticity of the log sem statistic. When this is done, with the number of measurements for pairwise comparisons being greater than 20, there is a tendency for the mean log sem to stabilise around a value of circa -1.6.

In their analyses of cranial measurements of Pan troglodytes, P. paniscus, Gorilla gorilla and H. sapiens (using more than 20 measurable dimensions as published by Gordon and Wood35), Thackeray and Dykes34 obtained the following results from pairwise comparisons:

Mean log sem = -1.612±0.129 (n=8072 pairwise regressions) reflects what is considered to be a typical degree of intraspecific variation within hominoids.

Mean log sem = -1.063±٠.126 (n=26 780 regressions), reflecting the degree of variability from interspecific comparisons.

These results, based on a very large number of regressions, clearly show that there is indeed a significant difference between the log sem values calculated for intraspecific and interspecific comparisons, which is related to similarity (or dissimilarity) in shape.

A criticism that has been levelled against the log sem approach relates to which variables are being measured. Remarkable as it may seem, the degree of intraspecific variability reflected by a mean log sem value of circa -1.61±0.1 has been obtained, not only from cranial variables, but also from measurements from teeth.34

Procrustes Distances for mandibles

Bookstein36 and Duta37 have described the method whereby PD are calculated, reflecting differences in shape between objects (in this case, mandibles). PD statistics serve to quantify the difference between landmarks by using the square root of the sum of squared differences in positions of those landmarks.

Spoor et al.8 calculated PD values from pairwise comparisons of OH 7 and other hominins, using landmarks indicated in their Fig. 2f and 2g. PD were also calculated for purposes of comparisons with H. sapiens, P. troglodytes and G. gorilla. The mean PD for the extant hominoids, based on data obtained by Spoor and Gunz (personal communication to JFT, 2020), provides a ‘within group’ (conspecific) frame of reference.


The mean and standard deviation for PD for pairwise comparisons of extant hominoids (0.089±0.021) are analogous to the mean and standard deviation for log sem values for extant hominoids (-1.61±0.1). These two means can both be related to probabilistic Z-scores, where the Z value of 0 corresponds to the mean value of 0.089 (PD) and also to the mean log sem value of -1.61. One standard deviation above or below the mean is circa 1 and -1, respectively. Likewise, two standard deviations above or below a mean Z-value of 0 correspond to Z-score values of circa 2 and -2, respectively. In this way, log sem statistics (including those calculated for a comparison between OH 24 and skulls of other hominins) and PD values (including those calculated for comparisons between OH 7 and other hominin mandibles) can be expressed on a common (Z) scale.

Article TitleThe use of Z-scores to facilitate morphometric comparisons between African Plio-Pleistocene hominin fossils: An example of method

January 27, 2022


South Africa and East Africa each have a rich palaeoanthropological heritage, but the taxonomy of fossil hominins from these regions is controversial. In this study, two morphometric methods related to the quantification of variability in morphology have been applied to pairwise comparisons of linear measurements of hominoid crania and mandibles. The log-transformed standard error of the m-coefficient (‘log sem’) is calculated from linear regressions. Like Procrustes Distances (PD), log sem statistics can serve to quantify variation in the shape of a cranium or mandible in the context of a constellation of landmarks. In this study, PD and log sem statistics are integrated and standardised using Z-scores, and applied probabilistically to Plio-Pleistocene hominins. As a test case, OH 7 and OH 24 as reference specimens of Homo habilis _are compared to fossils representing other taxa. There is a wide spectrum of variation in Z-scores for specimens attributed to early Homo dated within the period between circa 1.8 Ma and 2 Ma. In terms of morphometric variation predating 1.8 Ma, Z-scores (Z<2) for _Australopithecus afarensis, A. africanus and Homo habilis display a small range of variability. This study serves as a demonstration of a method whereby log sem and PD can be used together to facilitate an objective assessment of morphological variability, applicable in palaeontological contexts.

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