Assessing intrasample variation Analysis of Rapa Nui (Easter Island) museum cranial collections example.код для вставкиСкачать
AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 124:45–58 (2004) Assessing Intrasample Variation: Analysis of Rapa Nui (Easter Island) Museum Cranial Collections Example Vincent H. Stefan* Department of Anthropology, Lehman College, CUNY, Bronx, New York 10468 KEY WORDS intrasample variation; Rapa Nui; craniometrics; cranial discrete traits ABSTRACT Osteological studies both old and new have utilized various Polynesian cranial samples, individually or in combination, to assess the racial composition of prehistoric Polynesians as a group, with regards to other Paciﬁc populations, or to represent the Polynesian peoples as a whole in various multivariate analyses of worldwide populations. However, few of these studies have assessed the degree of intrasample variation produced when data derived from skeletal samples from different Polynesian islands (populations) are pooled to represent “Polynesians” as a whole. A similar argument can be made when data derived from various museum skeletal samples of the same Polynesian population are pooled to produce a larger sample representing that particular Polynesian population (Murrill  Cranial and postcranial skeletal remains from Easter Island; Minneapolis: University of Minnesota Press; Stefan  Am. J. Phys. Anthropol. [Suppl.] 34:147). This study examined Easter Island crania curated at various museums in North America, South America, and Europe to assess whether signiﬁcant differences exist among the museum collections of Rapa Nui (Easter Island) skeletal material. A NORM statistical program (Schafer and Olsen  NORM, version 1.01; University Park: Pennsylvania State University) for multiple Human biological variation within individual skeletal samples has been assessed with regards to temporal processes (Jantz, 1973; Key and Jantz, 1981; Konigsberg, 1990; Owsley and Jantz, 1978; Owsley et al., 1982; Relethford and Crawford, 1995; Relethford et al., 1997) and between skeletal samples when assessing population relationships (Konigsberg and Blangero, 1993; Relethford, 1998; Steadman, 2001). However, relatively few studies have assessed the degree of intrasample variation produced when data from various museum samples are combined into one sample (Sparks et al., 2002; Stefan, 2002). Osteological studies both old and new have utilized various Polynesian cranial samples individually or in combination to assess the racial composition of prehistoric Polynesians as a group (Howells, 1990; Katayama, 1987; Pietrusewsky, 1996; Tagaya and Katayama, 1988), with regards to other Paciﬁc populations (Brace et al., 1990; Giles, 1973; Howells, 1970; Pietrusewsky, 1970, 1990a,b, 1992, 1994; Wagner, 1937), or to represent the Polynesian peo© 2004 WILEY-LISS, INC. imputation of incomplete multivariate datasets was utilized to estimate missing data. A variance comparison method, which utilizes variance/covariance matrices derived from “hypothesis” and “baseline/reference” samples (Key and Jantz  Hum. Evol. 5:457– 469; Key and Jantz  Am. J. Phys. Anthropol. 82:53–59) was used to compare the Rapa Nui museum samples. This method is designed to test whether variability in a “hypothesis” museum sample exceeds “normal within-group variability” represented by the “baseline/reference” sample. The method was applied to six Rapa Nui museum samples (AANMW, MNHN-KB, MNHN-NAE, NHM, MH, and AMNH). The results indicate that the museum “hypothesis,” male and female samples, exhibited little intrasample variability from the “baseline/reference” sample (MAPSE), though the samples were collected at different times and by different individuals. These results show the ability of multiple imputation and variance comparison methodologies to predict missing variables while maintaining the inherent variance/covariance structure and to discriminate sample variation in artiﬁcially assembled samples. Am J Phys Anthropol 124:45–58, 2004. © 2004 Wiley-Liss, Inc. ples as a whole in various multivariate analyses of worldwide populations (Brace and Hunt, 1990; Howells, 1973, 1989, 1995). With regards to the prehistoric Rapa Nui (Easter Islanders), researchers have Grant sponsor: Easter Island Foundation Research Grants; Grant sponsor: PSC-CUNY Award, City University of New York; Grant sponsor: Research, Projects, and Travel Grant, Ofﬁce of Graduate Studies, University of New Mexico. This article was presented at the 71st Annual Meeting of the American Association of Physical Anthropologists at Buffalo, New York, April 2002. *Correspondence to: Vincent H. Stefan, Department of Anthropology, Lehman College, CUNY, 250 Bedford Park Blvd. West, Bronx, NY 10468. E-mail: email@example.com Received 14 August 2002; accepted 16 April 2003. DOI 10.1002/ajpa.10331 Published online 11 August 2003 in Wiley InterScience (www. interscience.wiley.com). 46 V.H. STEFAN also analyzed various museum collections to assess the racial composition of the islanders and their biological relationship to other Polynesian populations (Baker and Gill, 1997; Brace et al., 1991; Chapman, 1993, 1996, 1998; de Quatrefages and Hamy, 1882; Gill and Owsley, 1993; Gill et al., 1997; Henckel, 1939; Howells, 1989; Imbelloni, 1951; Meier, 1975; Meyer and Jablonowski, 1901; Murrill, 1968; Petri, 1936; Pietrusewsky, 1996; Shapiro, 1940; Stefan, 1998, 2000; Stevenson et al., 1998; Tigner and Gill, 1986; Volz, 1895; von Bonin, 1931; Zimple and Gill, 1986). However, few of these studies assessed the degree of intrasample variation produced when data derived from skeletal samples from different Polynesian islands (populations) are pooled to represent “Polynesians” as a whole. A similar argument can be made when data derived from various museum skeletal samples of the same Polynesian population are pooled to produce a larger sample representing that particular Polynesian population (Murrill, 1968; Stefan, 2002). Rapa Nui craniometric data were collected on a sample of 403 individuals from collections in Europe, North and South America, and Rapa Nui. This skeletal series provides an opportunity to determine whether signiﬁcant differences exist between the various Rapa Nui skeletal samples, and if the differences between these samples can be explained by temporal shifts in skeletal morphology (Konigsberg, 1990). The objectives of this study are the following: 1) to assess the range of variation and differences between various museum Rapa Nui cranial samples; 2) to determine if temporal shifts in Rapa Nui cranial morphology exist; and 3) to determine if pooling Rapa Nui cranial samples is unwarranted due to excessive intrasample variation. RAPA NUI CRANIAL COLLECTIONS The collection of Rapa Nui skeletal remains for museums in Europe and North and South America commenced shortly after the Dutch rediscovery of the island in 1722 (Ayers, 1995; Imbelloni, 1951; La Pérouse, 1797; Lavachery, 1935; Murrill, 1965; Nicoll, 1908; Pinart, 1878a,b; Routledge, 1919; Shapiro, 1935; Thomson, 1891). The majority of Rapa Nui skeletal material has been dated to the Late Prehistoric (AD 1680 –1722) and the Protohistoric (AD 1722–1868) periods (Gill and Owsley, 1993; Gill et al., 1983; Owsley et al., 1994). Skeletal material from earlier periods is not available due to the extensive practice of cremation known to have occurred during the 13th century and most probably through the ﬁnal phases of ahu use (Shaw, 1996; Van Tilburg, 1994). The existing skeletal material was recovered from modiﬁed image ahus (Love, 1993), semipyramidal ahu crypts (Stevenson, 1984), and secondary cave burials (Shaw, 1996). The utilization of image ahus, semipyramidal ahus, and caves as burial sites began around 1700 and contin- ued up until the 1860s, when the ﬁrst missionaries came to the island. The Rapa Nui skeletal samples investigated by this researcher and others, primarily G.W. Gill, D.W. Owsley, and S.J. Baker, include museum collections from North and South America and Europe from which craniometric data were collected. Detailed provenience information was gathered for each sample if available. However, exact provenience information on all Rapa Nui collections was not available due to the unsystematic manner in which early expeditions collected the material (Chapman, 1996). The Rapa Nui skeletal material utilized in this study is curated at the Anthropologische Abteilung, Naturhistorisches Museum Wien, Austria; the Museo Nacional de Historia Natural (MNHN), Santiago, Chile; the Natural History Museum, London, UK; the Musée de l’Homme, Paris, France; the American Museum of Natural History, New York, New York; and the Museo Antropológico Padre Sabastián Englert (MAPSE), Rapa Nui. A summary of available provenience information and circumstances of collection for each sample is provided in Table 1. INTRAGROUP VARIABILITY There are several measures of intragroup variability available on a univariate level. These include the coefﬁcient of variation, comparison of ranges statistic, the variance ratio of Morant (1935), and the sigma ratio of Howells (1936) (see also Key and Jantz, 1990a, p. 458). Van Vark (1984) and Howells (1974) attempted multivariate measures of variability. However, univariate measures suffer from various underlying problems common to univariate statistics (i.e., they do not consider variable intercorrelation and have problems with simultaneous conﬁdence levels in signiﬁcance tests) (Key and Jantz, 1990b, p. 53), and few multivariate methods possess a formal test of the null hypothesis that the variances are equal. Key and Jantz (1990a) previously attempted several multivariate measures of intragroup variability, including intrasample discriminant functions, intrasample clustering techniques, and calculating Mahalanobis D2s between all possible pairs of individuals; however, all these attempts deﬁed easy interpretation. Recently, Key and Jantz (1990a,b) employed a multivariate statistical method, the variance comparison method, which utilizes variance/covariance matrices derived from hypothesis and reference samples. It is an adaptation of certain techniques used in twin research (Bock and Vandenberg, 1968). The method can be most easily understood with reference to ordinary discriminant analysis. The standard discriminant model can be deﬁned as: 共W ⫺1 ⴱ A ⫺ L ⴱ I兲 ⴱ V ⫽ 0, where W is the pooled within-group covariance matrix and A is the covariance matrix among groups. In 47 ASSESSING INTRASAMPLE VARIATION TABLE 1. Male and female museum sample1 Museum Who collected Date collected Rapa Nui provenience Male (n) Female (n) Reference AANMW Paymaster J. Weisser 1877–1879 or 1882 Unknown 6 7 MNHN-KB 1911 and 1918 Unknown 22 18 MNHN-NAE Dr. Walter Knoche Padre Bienvenido de Estella Thor Heyerdahl 1955–1956 9 10 Murrill (1965, 1968) NHM Lord Crawford Ahu Tepeu, Ahu Hekii, and Ahu Vinapu Unknown 28 25 Nicoll (1908) MH Katherine Scoresby Routledge Alphonse Pinart AMNH MAPSE 1903 and 1914–1915 Volz (1895), Petri (1936), Pietrusewsky (1986), Jumeau (1997) Knoche (1925) Imbelloni (1951) Routledge (1919) 1877 and 1934–1935 La Pérouse Bay and Vaihu 40 26 Alfred Métraux Harry L. Shapiro 1935 Ahu Tahai and Ahu Tautira 11 11 George W. Gill 1981–1991 Various 61 47 177 144 Total Pinart (1878b) Lavachery (1935) Chapin (1935), Shapiro (1935) Gill (1986), Gill et al. (1997) 1 AANMW, Anthropologische Abteilung, Naturhistorisches Museum Wien, Vienna, Austria; MNHN-KB, Museo Nacional de Historia Natural (MNHN), Santiago, Chile, Knoche/Bienvenido collection; MNHN-NAE, Museo Nacional de Historia Natural (MNHN), Santiago, Chile, Norwegian Archaeological Expedition collection; NHM, Natural History Museum, London, UK; MH, Musée de l’Homme, Paris, France; AMNH, American Museum of Natural History, New York, NY; MAPSE, Museo Antropológico Padre Sabastián Englert, Rapa Nui. the null case, A can be regarded as an estimate of W. L, the vector of eigenvalues of (W⫺1 * A), provides a test of the null hypothesis. V is the matrix of eigenvectors of (W⫺1 * A), and can be utilized to deﬁne linear combinations of variables which maximize distances among groups. I is the identity matrix. The variance comparison model is deﬁned as: 共W ⫺1 ⴱ H ⫺ L ⴱ I兲 ⴱ V ⫽ 0. With this model, W is a covariance matrix obtained from a sample thought to represent normal withingroup variability, which Key and Jantz (1990a,b) term “baseline variability.” The hypothesis matrix H will contain a component of baseline variability represented by W, plus an unknown component A. The statistics of interest are the eigenvalues and eigenvectors of the asymmetric product matrix W⫺1 * H, which provides a test of the null hypothesis that A ⫽ 0. The results of this test can indicate whether there are signiﬁcant differences between the baseline sample and the hypothesis samples, and to determine whether there are unknown forces producing excessive variation between samples. Key and Jantz (1990a,b) utilized a 2 test statistic for evaluating the statistical signiﬁcance of the test, following Bartlett (1951). This method tests the overall heterogeneity present, as well as residual heterogeneity remaining after successively removing each eigenvalue. It also tests the cumulative contribution of each eigenvalue to the overall heterogeneity. However, Petersen (1998, 2000) recently criticized their methods and suggested that an F-test would be more appropriate when dealing with small sample sizes. He also noted that Key and Jantz (1990a,b) utilized the wrong number of degrees of freedom (df) for the 2 test. These problems would have the result of increasing the probability of committing a type I error. To account for these difﬁculties, this research utilized both the 2 test and the F-test, and the F-test statistic calculated in accordance with (Rao, 1973). MATERIALS AND METHODS Measurements This study incorporates measures of the cranial vault, face, and interorbital region as deﬁned by Bass (1995), Gill et al. (1988), Howells (1973), and Martin and Saller (1957). The list of 49 standardized measurements collected is provided in the Appendix. To take these measurements, ﬁve instruments were used: a pair of sliding calipers, a pair of spreading calipers, a head spanner, a simometer, and a palatometer. The complete dataset of 49 variables was reduced to 30 variables (Table 2) following the univariate ANOVA procedure, discussed below, to identify those variables which were signiﬁcantly different among museum samples and more likely to reﬂect excessive intrasample variation. Museum collections utilized The methodologies of missing data estimation and intrasample variance comparisons could only be applied to the Anthropologische Abteilung, Naturhistorisches Museum Wien, Vienna, Austria, Weisser collection; the Museo Nacional de Historia Natural, Santiago, Chile, Knoche/Bienvenido collection and the Norwegian Archaeological Expedition collection; the Natural History Museum, London, UK, Lord Crawford/Katherine Scoresby Routledge collection; 48 6 7 22 18 9 10 28 25 40 26 11 11 61 47 321 Percentage of sample in parentheses. 1 MAPSE Total AMNH MH NHM MNHN-NAE MNHN-KB 15 0 0 0 0 0 0 0 0 0 0 0 1 (0.091) 1 (0.016) 2 (0.043) 4 (0.012) 0 0 0 0 0 0 0 1 (0.040) 0 0 0 0 2 (0.033) 1 (0.021) 4 (0.012) 14 13 12 1 (0.167) 0 0 0 0 2 (0.094) 0 1 (0.056) 0 0 0 0 2 (0.071) 1 (0.036) 0 1 (0.040) 0 0 0 0 0 0 0 1 (0.091) 2 (0.033) 4 (0.066) 1 (0.021) 3 (0.064) 6 (0.019) 13 (0.04) 0 0 0 0 0 0 0 1 (0.040) 0 0 0 0 1 (0.016) 0 2 (0.006) 11 10 0 0 1 (0.045) 0 0 0 0 0 0 0 0 1 (0.091) 0 1 (0.021) 3 (0.009) 0 0 0 1 (0.056) 0 0 0 1 (0.040) 1 (0.025) 0 0 0 0 1 (0.021) 4 (0.012) 9 8 0 0 2 (0.091) 0 0 0 0 0 1 (0.025) 0 1 (0.091) 0 2 (0.033) 1 (0.021) 7 (0.022) 0 0 1 (0.045) 0 0 0 3 (0.107) 0 1 (0.025) 0 0 0 3 (0.049) 1 (0.021) 9 (0.028) 7 6 Number of missing variable 0 0 0 1 (0.056) 1 (0.111) 1 (0.100) 2 (0.071) 2 (0.080) 0 1 (0.038) 0 0 4 (0.066) 2 (0.043) 14 (0.044) 0 0 1 (0.045) 0 0 0 3 (0.107) 2 (0.080) 1 (0.025) 2 (0.077) 0 0 0 2 (0.043) 11 (0.034) 0 0 0 0 1 (0.111) 0 0 2 (0.080) 1 (0.025) 1 (0.038) 0 2 (0.182) 2 (0.033) 2 (0.043) 11 (0.034) 1 (0.167) 0 0 1 (0.056) 0 1 (0.100) 1 (0.036) 3 (0.120) 0 2 (0.077) 0 0 3 (0.049) 4 (0.085) 15 (0.047) 0 0 4 (0.182) 2 (0.111) 1 (0.111) 2 (0.200) 3 (0.107) 3 (0.120) 5 (0.125) 1 (0.038) 2 (0.182) 1 (0.091) 4 (0.066) 8 (0.170) 36 (0.112) 0 2 (0.286) 3 (0.136) 3 (0.167) 1 (0.111) 1 (0.100) 1 (0.036) 3 (0.120) 7 (0.175) 3 (0.115) 2 (0.182) 2 (0.182) 18 (0.295) 6 (0.128) 52 (0.162) 4 (0.667) 5 (0.714) 8 (0.364) 10 (0.556) 5 (0.556) 5 (0.500) 12 (0.429) 6 (0.240) 23 (0.575) 16 (0.615) 6 (0.545) 3 (0.273) 15 (0.246) 12 (0.255) 130 (0.405) Male Female Male Female Male Female Male Female Male Female Male Female Male Female 5 TABLE 2. Number of individuals missing variable data1 4 3 2 1 0 Sex Sample AANMW Total V.H. STEFAN the Musée de l’Homme, Paris, France, Alphonse Pinart/Alfred Métraux collection; the American Museum of Natural History, New York, Shapiro collection; and Museo Antropológico Padre Sabastián Englert (MAPSE), Rapa Nui, G.W. Gill collection, due to the limitations imposed by the NORM program. Table 1 contains the sample breakdowns as to number of individuals analyzed and number of males and females in each museum sample. In this study, males and females were analyzed separately. The vast majority of crania in the collections examined were isolated, with no associated postcranial elements. Therefore, sex determination methods which assess pelvic morphology and age determination methods which assess pubic symphysis morphology and/or sternal rib end morphology could not be utilized in this study. An accurate, reliable, and proven method for assessing the age of Polynesian/ Rapa Nui crania was not available. The sex of each cranium was determined utilizing standard anthroposcopic techniques of assessing cranial morphology (Bass, 1995; Buikstra and Ubelaker, 1994), taking into account that Rapa Nui females tended to appear more robust than European females or females of other populations (Baker and Gill, 1997). Any crania with ambiguous or indeterminate sex were not included in the analyses conducted. Inaccuracies in sex assessment, if present, would be a methodological error present in all the museum sample data and would not affect any single pariwise comparison. Cranial metric data were collected only from adult individuals, as evidenced by a fused sphenooccipital sychondrosis (Krogman and Iscan, 1986), and only the most complete crania were selected for data collection. Age determination beyond that of being an “adult” was not assessed or deemed necessary for the analyses that were to be conducted, as justiﬁed below. It was noted that signiﬁcant differences in some cranial measurements occur among different age cohorts of a population (Israel, 1973; MooreJansen, 1989; Ruff, 1980). Ruff (1980) examined 16 craniofacial dimensions of an Indian Knoll, Kentucky sample and found six dimensions (nasion-basion, basion-bregma, glabella-opisthocranion, bizygomatic breadth, basion-prosthion length, and bigonial angle) that were signiﬁcantly larger in a 35–50-year age group than in a 20 –34-year age group. A few of these dimensions (basionbregma, glabella-opisthocranion, and basion-prosthion length) were found to have increased by 2 mm or more. Ruff (1980) argued that if skeletal samples with potentially different demographic structures are compared, the observed differences might be due to the age characteristics of the samples, and not to genetic or environmental differences. Imbedded within the total range of morphological variation observed within a given skeletal sample is variation produced by sex and age effects. In order for a skeletal sample to adequately represent the underlying population from which it was drawn, it ASSESSING INTRASAMPLE VARIATION needs to document all the variation present within that population, including aspects of individual, sex, and age variation. The more comprehensive the sample, the greater the probability that the sample will represent its parent population. A sample biased towards individuals of a particular age cohort may not reﬂect the total range of variation present in the parent population, and analyses utilizing that sample may produce erroneous results. The purpose of this study was to investigate whether independent Rapa Nui hypothesis samples possess greater-than-expected variation as compared to a reference, baseline sample which represents the underlying population. If the baseline sample encompasses the total range of morphological variation of the underlying population, it will include individuals of all age cohorts. Therefore, even if an individual hypothesis sample is biased toward a speciﬁc age cohort, that sample would not be expected to have a variance/covariance structure exceeding that of the baseline sample. If the baseline sample is biased toward a speciﬁc age cohort, then comparisons with the hypothesis samples would produce results indicating “greater than” (not observed in this study; see Results) or “less than/equal to” variance, depending on whether the hypothesis sample is or is not similarly biased. The likelihood of ﬁnding independent skeletal samples, collected by different individuals at different times, each with the same age cohort bias, is in my opinion low. The exact demographic/age cohort structure of a single population’s samples is irrelevant if the results of intrasample variability assessment analyses fail to reject the null hypothesis of no differences between the baseline reference and each hypothesis sample. The collections examined in this study do not appear to be biased towards individuals of any particular age cohort, and appear to have the similar demographic structures. Crania spanning the range of “younger” adults with complete dentition to edentulous “older” adults were observed in each collection. The individuals who personally gathered or commissioned the gathering of the crania of their collections were more intent on the collection of intact complete crania (personal opinion). Because this study examined each sample as a “whole” and each sample was a composite of individuals from all age cohorts, further division of the various museum samples into age-speciﬁc cohorts was not performed. Of the 403 total crania for which data were available, only 355 were from the museum samples under investigation and had reliable determination of sex. The new dataset, comprised of 355 individuals and 30 variables, had 15.6% missing data. In order to minimize the amount of missing data, individuals missing data for more than 15 of the 30 variables were eliminated from the dataset. The “pruned” dataset, comprised of 321 individuals and 30 variables, had only 9.5% missing data. A detailed summary of the number of individuals missing a certain number of missing variables for each sample is given 49 in Table 2. As is evident from Table 2, 83.8% (n ⫽ 269) of the individuals of the total sample have data for 24 or more of the 30 variables, while only 16.2% of the individuals of the total sample have data for only 15–23 of the 30 variables. Though any given individual could have 50.0% missing data (4 individuals, 1.2% of total sample), the overall percentage of missing data was signiﬁcantly reduced. It is the “pruned” dataset that will be utilized in the missing data estimation procedures and the intrasample variability assessment. The Museo Antropológico Padre Sabastián Englert (MAPSE) Rapa Nui Gill collection will serve as the baseline sample. The justiﬁcation for this selection is founded on a number of points. First, it is a well-documented sample comprising individuals from various sites from all regions of the island and spans the period of the Late Prehistoric through the Protohistoric (Gill and Owsley, 1993; Gill et al., 1983). Second, the sample is the single largest collection of Rapa Nui crania in the world. Third, it represents a population which existed prior to the island population having been reduced to approximately 111 individuals as a result of civil conﬂict, slavery, exploitation, and disease (Bahn and Flenley, 1992). The Gill collection is the best approximation of the island’s breeding population, against which the other samples will be compared. The Anthropologische Abteilung, Naturhistorisches Museum Wien, Museo Nacional de Historia Natural, Natural History Museum, Musée de l’Homme, and American Museum of Natural History samples will serve as the hypothesis samples to be tested. Data analysis Interobserver error. In order to utilize the Rapa Nui data collected by Gill, interobserver error rates were calculated. The author measured 30 individual crania measured by Gill (28, Marquesas Island; 2, Rapa Nui), with interobserver error rates tested via pairwise t-tests and a method error statistic across all variables. Bonferroni and Dunn-Šidák methods for calculating the experimentwise error rate for multiple comparisons, ␣⬘, was utilized to determine the signiﬁcance levels of each comparison (Šidák, 1967; Ury, 1976). The chances of committing a type I error (rejecting a null hypothesis that is correct) increase with the number of tests performed. In order to minimize the chance of committing a type I error, an experimentwise error rate, ␣⬘ ⫽ 1 ⫺ (1 ⫺ ␣)1/k, was calculated, where ␣ is the desired experimentwise error rate, and k is the number of pairwise comparisons to be made. The experimentwise error rate lowers the probability of making a type I error for each comparison, so that the probability of making any type I error in the entire series of tests does not exceed ␣ (Sokal and Rohlf, 1995). To maintain an experimentwise error rate of ␣ ⫽ 0.05, with 49 variable comparisons being conducted, each comparison was determined to be signiﬁcant only if the P-value for t-tests was less than the ␣⬘ ⫽ 0.001 level. 50 V.H. STEFAN The method error statistic (Solow, 1966; Utermohle and Zegura, 1982), which tests for random error within replicate data, with high values of the statistic indicating greater measurement error, was also calculated to assess methodological differences between myself and other researchers. The method error statistic (S), also known as the technical error measurement, is calculated as follows: S ⫽ 冑 冘 共x N 1i ⫺ x 2i 兲 2 i⫽1 2N where x1 is the original measurement value, x2 is the repeated measurement value, and N is the sample size. The size of S is a function of the scale of the measurements, which in this study were millimeters (mm). Univariate analysis. An analysis of variance (ANOVA) procedure was undertaken, with Scheffé post hoc analysis, to determine if signiﬁcant differences existed for each variable within the male and female museum collections and to determine which of the museum samples were the most different for each signiﬁcant variable. As with interobserver error rate analyses, an experimentwise error rate for multiple comparisons, ␣⬘, was calculated by the Scheffé test to determine the signiﬁcance levels of each pairwise comparison, in order to maintain an overall error rate of ␣ ⫽ 0.05 (SAS Institute, 1990, p. 942–945). The interobserver error and univariate analyses reported here were performed in SAS (version 6) (SAS Institute, 1990). Missing data estimation. Multivariate analysis procedures require that there be no missing data: every observation must have a value for each variable entered into the analysis; otherwise, the observation is eliminated. Deleting observations results in large amounts of information being lost, and the remaining completely observed cases would be unrepresentative of the population which they were intended to reﬂect. In order to optimize the number of observations utilized from each Rapa Nui museum collection, the problem of missing variable values needs to be addressed. Considerable attention has been given to the problems of missing data estimation and of the most appropriate procedures for estimating missing data (e.g., Droessler, 1981). Traditionally, three options have been available to anthropological researchers for estimating missing data: substitution of group means, substitution of grand means, or prediction of missing measurements by means of multiple regression. These alternatives have their associated advantages and disadvantages (Droessler, 1981, p. 80 – 84). Substitution of group means for missing data has the effect of decreasing variance within groups, while concurrently increasing intergroup distances based on comparisons of within-group and between- group variances. Substitution of means over all groups (grand means) increases within-group variance while decreasing between-group differences. When working with a null hypothesis of no difference among groups, the use of the grand mean would be a more conservative procedure than the use of individual group means. However, the use of either the group or grand means has one additional problem. Many multivariate techniques involve the consideration of each observation as a composite of interrelated morphological variables, where proportion as well as size is represented by these measurements (Droessler, 1981). When mean values are substituted for missing values into particularly large or small individuals, there may be inconsistencies in size with the existing measurements. The estimation of missing data through multiple regression is a much more complicated and timeconsuming procedure than the substitution of means. The technique has several advantages. Predicting missing values from existing variables overcomes the problem of maintaining consistency in the size of cranial dimensions within observations. Additionally, within-group variation will be affected to a lesser extent with regression estimates than that which will result from mean substitution. Regression-based estimates will also result in less alteration of within-group variances than results from either group or grand mean estimates (Droessler, 1981). However, imputing predicted values from regression models tends to inﬂate observed correlations (Schafer, 1997). Regression predictions can also compound sampling error when missing data are predicted for small groups and when relatively few complete observations are available for generating the regression equations. Another potential problem when utilizing imputed datasets as if they were real in any standard statistical method is that any standard errors, P-values, and other measures of variability calculated could be misleading because they would fail to reﬂect any uncertainty due to missing data (Schafer, 1997). This research will utilize the NORM statistical program (Schafer and Olsen, 1997) for multiple imputation of incomplete multivariate datasets and techniques described by Schafer (1997) to assess the validity in combining point estimates and covariance matrices to be utilized in the intragroup variability analyses discussed below. These methods will minimize the problems inherent in missing value estimations discussed above. NORM utilizes an expectation-maximization (EM) algorithm, a computational method for ﬁnding maximum-likelihood estimates of parameters from an incomplete dataset, and data augmentation (DA), a Marchov chain Monte Carlo (MCMC) technique that can be used to generate multiple imputations of missing data. Multiple imputation, utilizing EM and DA, solves an incomplete-data problem by repeatedly solving the complete-data version. In multiple imputation, the unknown missing data are replaced 51 ASSESSING INTRASAMPLE VARIATION by simulated values. Each of the completed datasets is analyzed by standard complete-data methods. The variability among the results of analyses provides a measure of the uncertainty due to missing data, which, when combined with measures of ordinary sample variation, leads to a single inferential statement about the parameters of interest (Schafer, 1997). For each individual male and female Rapa Nui museum sample, ﬁve imputed datasets were generated with their associated variance/covariance matrices. Average variance/covariance matrices were then generated from the imputed matrices. A test statistic, D, ⫺ Q 0 兲/k, ⫺ Q 0 兲⬘T̃ ⫺1 共Q D ⫽ 共Q and its associated P-value were calculated to ensure that the averaged variance/covariance matrix was not signiﬁcantly different from the original museum dataset with missing variables (Schafer, 1997, p. 112–114). The test statistic is assessed utilizing an F-distribution. This test assessed the two sources of variance produced by the multiple imputation process: 1) the within-imputation variance, which is the average of the complete-data variance estimates, and 2) the between-imputation variance, which is the variance of the complete-data point estimates. These two sources of imputation variance serve to imitate the normal sample variation (J.W. Graham, personal communication) observed with complete datasets. This statistical procedure will utilize the NORM 2.03 statistical program (Schafer, 1999) for multiple imputation of incomplete multivariate datasets and techniques described by Schafer (1997). Intrasample variability assessment. As discussed above, the variance comparison model was deﬁned as: 共W ⫺1 ⴱ H ⫺ L ⴱ I兲 ⴱ V ⫽ 0, and the statistics of interest were the latent roots and vectors of the asymmetric product matrix W⫺1 * H. However, these values cannot be calculated directly, since some of the eigenvalues may be imaginary. Therefore, the following procedures were used. First, the covariance matrices, W and H, were calculated from the baseline and hypothesis samples, respectively, for each sex. W was inverted, and R found, the Cholesky half-root decomposition of W⫺1. The latent vectors (V) of the symmetric matrix RHR⬘ were then calculated, and the eigenvectors maximizing the variation among individuals in the hypothesis sample were identiﬁed via the equation X ⫽ R⬘ * V. The latent roots (L), the vector of eigenvalues of RHR⬘, were then calculated, which provided a test of the null hypothesis that A ⫽ 0, i.e., no variability above the baseline was present. Both the 2-test and F-test statistics were calculated. The statistical analyses reported here were performed on SAS and SAS IML (version 6) (SAS Institute, 1990). TABLE 3. Interobserver error rates and method error statistics Variable GOL NOL XCB XFB WFB ZYB BBH BNL FRC PAC NPH NAL ASB BPL STB JUB FOL OBB OBD OBH EKB NLB NLH FMB AUB WCB AUR PBH NAR SSR PRR BPH MXB MXS ZOB ZOS ALB ALS WNB ZMB SSS MDH MDB WMH IML XML PAD MAB MAL t-test P-value Mean deviation (in mm) Method error statistic 0.4235 1.0000 0.4146 0.7687 0.1608 0.4238 0.1034 0.0502 0.3259 0.0961 0.1033 0.5365 0.7287 1.0000 0.2339 0.1613 1.0000 0.7450 0.2827 0.2641 0.2703 0.4892 0.3256 0.1070 1.0000 0.3788 0.0033 0.0001* 0.0098 0.0003* 0.0001* 1.0000 0.5732 0.0006* 0.2246 0.0830 0.0881 0.0501 0.0431 0.3757 0.0296 0.8012 1.0000 0.0089 0.1610 0.5309 0.1101 0.1033 0.0431 ⫺0.1290 0.0000 ⫺0.1935 ⫺0.0645 ⫺0.2581 0.1333 ⫺0.2857 ⫺0.5000 0.0667 ⫺0.3333 ⫺0.2667 ⫺0.1333 0.2069 0.0000 0.5161 0.2963 0.0000 ⫺0.0645 0.3226 ⫺0.2000 0.5000 0.1333 ⫺0.0645 ⫺0.5806 0.0000 ⫺0.3448 ⫺1.1333 ⫺1.2667 ⫺0.6207 ⫺0.9333 ⫺1.3571 0.0000 0.1379 0.7143 ⫺0.6897 0.2222 0.5714 0.5185 ⫺0.2857 ⫺0.2143 0.5714 0.0645 0.0000 0.5161 0.8276 0.4828 ⫺0.4444 ⫺0.2581 ⫺0.5714 0.3162 0.3216 0.4655 0.4282 0.3651 0.3216 0.3333 0.4907 0.1313 0.3939 0.3216 0.4152 1.1260 0.2722 0.8563 0.3922 0.3086 0.3873 0.5916 0.3474 0.8498 0.3714 0.1291 0.7188 0.2582 0.7440 0.7987 0.7071 0.4818 0.5571 0.7577 0.3922 0.4629 0.4303 1.0856 0.2402 0.6383 0.5000 0.2722 0.4513 0.5092 0.5000 0.4472 0.4082 1.1339 1.4577 0.5189 0.3162 0.5443 * Signiﬁcant at experimentwise error rate of ␣⬘ ⱕ 0.001. RESULTS Interobserver error The results of interobserver error analyses (Table 3) indicate that four variables were signiﬁcantly different between the data collected by the researcher and those collected by Gill (␣⬘ ⱕ 0.001): porionbregma P ⫽ 0.0001, porion-subspinale P ⫽ 0.0003, porion-prosthion P ⫽ 0.0001, and maxillofrontal subtense P ⫽ 0.0006. Despite their signiﬁcance, however, the mean differences between the data were no greater than ⫾1.5 mm (porion-bregma, ⫺1.2667 mm; porion-subspinale, ⫺0.9333; porionprosthion, ⫺1.3571 mm; and maxillofrontal subtense, 0.7143 mm). The method error statistics for these variables were: porion-bregma, 0.7071; porion- 52 V.H. STEFAN TABLE 4. Male and female museum sample significant variables Variable GOL NOL XCB XFB* ZYB* JUB* AUB BNL BPL FRC* NPH* NAL ASB WCB* OBB* OBD* OBH EKB FMB* AUR* NAR SSR* PRR* MXS* ALB* ALS* SSS WMH IML* MAL* Male P-value Female P-value 0.0250 0.0140 0.2764 0.0295 0.0059* 0.0103* 0.0216 0.0137 0.0374 0.0052* 0.0132* 0.0235 0.0276 0.0002* 0.0019* 0.0013* 0.0120 0.0496 0.0046* 0.0086* 0.0258 0.0002* 0.0163* ⬍0.0001* 0.0851 0.7085 0.0217 0.0316 0.0020* 0.0198* 0.5542 0.4364 0.0278 ⬍0.0001* 0.2626 0.0234 0.3194 0.1748 0.5463 0.9894 0.6796 0.7200 0.0206 0.0007* 0.1133 0.3126 0.3389 0.3451 0.2249 0.2910 0.0358 0.4164 0.6052 ⬍0.0001* ⬍0.0001* 0.0005* 0.8860 0.7765 0.7348 0.0892 * Signiﬁcant pairwise differences found with Scheffé’s post hoc comparisons. subspinale, 0.5571; porion-prosthion, 0.7577; and maxillofrontal subtense, 0.4303. Due to this fact, it is not unreasonable to pool the data collected by this researcher and those collected by Gill and associates. Univariate analysis The results of ANOVA analyses are presented in Table 4. There were 27 variables that were statistically signiﬁcant for males and nine variables for females, with six variables in common for males and females at the ␣ ⫽ 0.05 level. The results of Scheffé post hoc comparisons are presented in Table 5. Of the 30 variables that were signiﬁcant at the ␣ ⫽ 0.05 level, 17 variables had signiﬁcant pairwise differences, and only two had signiﬁcant comparisons for both the male and female samples (WCB and MXS). Though not all 30 variables had signiﬁcant pairwise comparisons in Scheffé post hoc comparisons, all will be utilized in further analyses. The ANOVA analysis noted some degree of difference between the male samples and female samples, though these differences could not be identiﬁed through Scheffé post hoc comparisons. Missing data estimation For each museum sample utilized, ﬁve imputed datasets were created for males and females separately. An average imputed dataset and variance/ covariance matrix were then calculated. Each aver- aged imputed dataset was then compared to the original dataset without missing value estimates, to determine if the two datasets were signiﬁcantly different. The D-statistic and its associated P-value for each museum/sex sample comparison are listed in Table 6. The D-statistic was assessed utilizing an F-distribution. As is evident, no average imputed dataset was signiﬁcantly different from its original dataset with missing values. Intrasample variability assessment Tables 7 and 8 contain the 2-test and F-test statistics and their associated P-values for comparisons between the Museo Antropológico Padre Sabastián Englert male and female samples with the Anthropologische Abteilung, Naturhistorisches Museum Wien, Museo Nacional de Historia Natural, Natural History Museum, Musée de l’Homme, and American Museum of Natural History male and female samples, respectively. With the exception of the male and female Anthropologische Abteilung, Naturhistorisches Museum Wien samples and the male American Museum of Natural History sample, analyses of the latent roots/eigenvalues of the collections indicate that there is no excessive heterogeneity within the remaining collections, and indicate that there are no signiﬁcant differences between the hypothesis samples and the baseline sample. It is also evident that the F-test provides a slightly more conservative test of signiﬁcance as compared to the 2-test, though this does not signiﬁcantly change the interpretation of results. Examination of the 2-test and F-test statistics for the male and female Anthropologische Abteilung, Naturhistorisches Museum Wien (AANMW) samples (Tables 7 and 8) indicated greater levels of variability than the baseline male and female samples within these samples. For the male AANMW sample, after the ﬁrst three latent roots/eigenvalues were removed for the 2-test and the ﬁrst two latent roots/eigenvalues for the F-test, no signiﬁcant variation remained. For the female AANMW sample, after the ﬁrst two latent roots/eigenvalues were removed for the 2-test and the ﬁrst latent root/eigenvalue for the F-test, no signiﬁcant variation remained. The cause of this excessive variability may due to temporal changes (the AANMW sample was collected in 1882; the baseline sample was collected from 1981–1991) or to the small sizes of these hypothesis samples (male, n ⫽ 6; female, n ⫽ 7). The investigation of the cause will be discussed below. Upon examination of the 2-test and F-test statistics for the male American Museum of Natural History sample (Table 7), it is evident that there is at least one signiﬁcant eigenvector. These results indicate that variability signiﬁcantly greater than baseline exists in this sample, though not to the extent seen in the AANMW samples. No signiﬁcant variability remains after removing the ﬁrst latent root/ eigenvalue. Again, the possible cause of the excessive variability may be temporal changes, but is ASSESSING INTRASAMPLE VARIATION 53 TABLE 5. Male and female museum sample Scheffé post hoc comparisons significant variables1 Variable XFB ZYB JUB FRC NPH WCB OBB OBD FMB AUR SSR PRR MXS ALB ALS IML MAL Male pairwise comparison Female pairwise comparison MAPSE—MNHN-KB, MH, NHM MAPSE—MNHN-KB MAPSE—MNHN-KB MAPSE—MNHN-KB MNHN-KB—MH MAPSE—MNHN-KB MNHN-KB—NHM MNHN-KB—MH, NHM MAPSE—MNHN-KB MAPSE—MNHN-KB MAPSE—MNHN-KB, MH, NHM MNHN-KB—NHM MH—MNHN-NAE MAPSE—MNHN-KB, MH MH—MAPSE, MNHN-KB, MNHN-NAE; MNHN-NAE—NHM, AANMW MAPSE—NHM MAPSE—NHM MAPSE—MNHN-KB MAPSE—MNHN-KB 1 AANMW, Anthropologische Abteilung, Naturhistorisches Museum Wien, Vienna, Austria; MNHN-KB, Museo Nacional de Historia Natural (MNHN), Santiago, Chile, Knoche/Bienvenido collection; MNHN-NAE, Museo Nacional de Historia Natural (MNHN), Santiago, Chile, Norwegian Archaeological Expedition collection; NHM, Natural History Museum, London, UK; MH, Musée de l’Homme, Paris, France, MAPSE, Museo Antropológico Padre Sabastián Englert, Rapa Nui. TABLE 6. Average imputed dataset significance1 Museum sample AANMW MNHN-KB MNHN-NAE NHM MH AMNH MAPSE Sex D-statistic df Male Female Male Female Male Female Male Female Male Female Male Female Male Female 0.041 0.003 0.047 0.013 0.035 0.029 0.032 0.054 0.020 0.015 0.017 0.096 0.010 0.017 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 2 ⬁ ⬁ ⬁ ⬁ ⬁ ⬁ ⬁ ⬁ ⬁ ⬁ ⬁ ⬁ ⬁ ⬁ P-value ⬎0.9999 ⬎0.9999 ⬎0.9999 ⬎0.9999 ⬎0.9999 ⬎0.9999 ⬎0.9999 ⬎0.9999 ⬎0.9999 ⬎0.9999 ⬎0.9999 ⬎0.9999 ⬎0.9999 ⬎0.9999 1 AANMW, Anthropologische Abteilung, Naturhistorisches Museum Wien, Vienna, Austria; MNHN-KB, Museo Nacional de Historia Natural (MNHN), Santiago, Chile, Knoche/Bienvenido collection; MNHN-NAE, Museo Nacional de Historia Natural (MNHN), Santiago, Chile, Norwegian Archaeological Expedition collection; NHM, Natural History Museum, London, UK; MH, Musée de l’Homme, Paris, France; AMNH, American Museum of Natural History, New York, NY; MAPSE, Museo Antropológico Padre Sabastián Englert, Rapa Nui. 2 Critical value: 1.46. more likely a result of sample size (n ⫽ 11) or of an individual outlier. ANTHROPOLOGISCHE ABTEILUNG, NATURHISTORISCHES MUSEUM WIEN VARIABILITY ASSESSMENT In order to determine if the excessive variability observed in the Anthropologische Abteilung, Naturhistorisches Museum Wien (AANMW) male and female samples was due to temporal changes or sample size, I added to the samples data from Rapa Nui crania curated in the Bioanthropology Collection at Indiana University (IU, Bloomington, IN; male, n ⫽ 5; female, n ⫽ 1). During data collection, the word “Cooke” was observed written on each cranium, a possible reference to G.H. Cooke. The cura- tors of the Bioanthropology Collection were unaware of the identity and provenience of the crania, and had no knowledge of the meaning of the word “Cooke” inscribed on them. The collection containing the Rapa Nui crania also contains skeletal material from Peruvian individuals, collected in 1887 during a voyage of the U.S.S. Mohican. Upon further investigation by the collection curators, it was discovered that the entire collection had been donated to the university by the great, great grandson of Dr. G.H. Cooke. Additionally, three Rapa Nui crania (male, n ⫽ 2; female, n ⫽ 1) curated by the Smithsonian Institution (SI, Washington, DC), collected by either paymaster William J. Thomson or ship’s surgeon G.H. Cooke of the U.S.S. Mohican in December 1886 (Cooke, 1899; Thomson, 1891), will be added to the AANMW sample. Though collected by different individuals, the justiﬁcation for adding these two collections to the AANMW samples is the similar time period in which crania were collected. The results from the 2-test and F-test statistics and their associated P-values for the comparisons between the combined AANMW/IU/SI male and female samples and the Museo Antropológico Padre Sabastián Englert male and female samples are presented in Table 9. As is evident, the combined male sample (n ⫽ 13) does not have any variability signiﬁcantly greater than the baseline male sample. For the combined female sample (n ⫽ 9), only the 2-test indicated variability signiﬁcantly greater than the baseline female sample, and no signiﬁcantly greater variability after removing the ﬁrst latent root/eigenvalue. Clearly these statistical procedures are highly sensitive to sample size, and the results of the ﬁrst analyses of the AANMW revealed signiﬁcantly greater levels of variability due to small sample size, not due to temporal changes. If there had been temporal changes in the Rapa Nui 54 V.H. STEFAN TABLE 7. Male intrasample variability vs. baseline sample results1 Museum collection AANMW MNHN-KB MNHN-NAE NHM MH AMNH Root/eigenvalue no. Chi square df P-value F-statistic df P-value Overall 12 23 34 Overall Overall Overall Overall Overall 12 329.18 270.41 225.05 187.90 651.90 303.65 822.00 1,087.76 414.14 349.91 180 180 180 180 660 270 840 1200 330 330 7.8279E-11 1.4874E-05 0.0127 0.3279 0.5814 0.0778 ⬎0.9999 0.9907 0.0011 0.2160 3.4217 1.9179 1.2034 180, 31.1 180, 31.1 180, 31.1 7.4789E-05 0.0170 0.2771 0.9073 1.1060 0.8910 0.7875 1.3154 1.0085 660, 572.8 270, 151.8 840, 717.8 1,200, 919.5 330, 224.2 330, 224.2 0.8863 0.2471 0.9461 ⬎0.9999 0.0137 0.4757 1 Baseline sample: MAPSE, Museo Antropológico Padre Sabastián Englert, Rapa Nui; AANMW, Anthropologische Abteilung, Naturhistorisches Museum Wien, Vienna, Austria; MNHN-KB, Museo Nacional de Historia Natural (MNHN), Santiago, Chile, Knoche/Bienvenido collection; MNHN-NAE, Museo Nacional de Historia Natural (MNHN), Santiago, Chile, Norwegian Archaeological Expedition collection; NHM, Natural History Museum, London, UK; MH, Musée de l’Homme, Paris, France; AMNH, American Museum of Natural History, New York, NY. 2 First latent root/eigenvalue removed. 3 Second latent root/eigenvalue removed. 4 Third latent root/eigenvalue removed. E (exponent), value raised to the power of the given negative number. TABLE 8. Female intrasample variability vs. baseline sample results1 Museum collection Root/eigenvalue no. Chi square df P-value F-statistic df P-value Overall 12 23 Overall Overall Overall Overall Overall 340.93 271.37 227.45 534.31 331.51 689.99 746.48 318.45 210 210 210 540 300 750 780 330 2.7854E-08 0.0028 0.1945 0.5609 0.1019 0.9424 0.8007 0.6660 2.0059 1.0724 210, 26.7 210, 26.7 0.0187 0.4371 0.8345 0.9552 0.7499 0.7957 0.7664 540, 269.5 300, 96.2 750, 399.6 780, 416.0 330, 118.6 0.9592 0.6212 0.9996 0.9965 0.9649 AANMW MNHN-KB MNHN-NAE NHM MH AMNH 1 Baseline sample: MAPSE, Museo Antropológico Padre Sabastián Englert, Rapa Nui; AANMW, Anthropologische Abteilung, Naturhistoisches Museum Wien, Vienna, Austria; MNHN-KB, Museo Nacional de Historia Natural (MNHN), Santiago, Chile, Knoche/Bienvenido collection; MNHN-NAE, Museo Nacional de Historia Natural (MNHN), Santiago, Chile, Norwegian Archaeological Expedition collection; NHM, Natural History Museum, London, UK; MH, Musée de l’Homme, Paris, France; AMNH, American Museum of Natural History, New York, NY. 2 First latent root/eigenvalue removed. 3 Second latent root/eigenvalue removed. E (exponent), value raised to the power of the given negative number. TABLE 9. AANMW/IU/SI sample variability vs. baseline sample results Museum collection Male Female Root/eigenvalue no. Chi square df P-value F-statistic df P-value Overall Overall 12 249.92 342.42 278.60 390 270 270 ⬎0.9999 0.0018 0.3464 0.5055 1.2253 390, 294.2 270, 73.6 ⬎0.9999 0.1515 1 Baseline sample; MAPSE, Museo Antropológico Padre Sabastián Englert, Rapa Nui; AANMW, Anthropologische Abteilung, Naturhistorisches Museum Wien, Vienna, Austria; IU, Indiana University, Bloomington, IN; SI, Smithsonian Institution, Washington, DC. 2 First latent root/eigenvalue removed. crania, by adding data to the AANMW samples from additional crania collected during the sample period (1880s), one would expect the level of excess variability to have remained the same or to have become greater. This was not the case. These results provide support for the conclusions reached regarding the male American Museum of Natural History sample discussed above. The excessive variability observed was likely an artifact of sample size, and not temporal changes or the true existence of excess variability. The results of any 2-test and F-test of intrasample variability must be interpreted with caution and with consideration of the size of the hypothesis sample. DISCUSSION AND CONCLUSIONS The results of this study show the ability of the multiple imputation and variance comparison methodologies to predict missing variables while maintaining the inherent variance/covariance structure and to discriminate among groups in artiﬁcially assembled samples. It is evident that the Anthropologische Abteilung, Naturhistorisches Museum Wien, Austria, Weisser collection; the Museo Nacional de ASSESSING INTRASAMPLE VARIATION Historia Natural, Santiago, Chile, Knoche/Bienvenido and Norwegian Archaeological Expedition collections; the Natural History Museum, London, Lord Crawford/Katherine Scoresby Routledge collection; the Musée de l’Homme, Paris, Alphonse Pinart/ Alfred Métraux collection; and the American Museum of Natural History, New York, Shapiro collection male and female samples exhibit little excess intrasample variability due to the different time periods from which the various museum samples were collected. It is therefore evident that the hypothesis samples represent subsamples of the underlying Rapa Nui population represented by the baseline sample, and are from the same chronological periods. The baseline sample (Museo Antropológico Padre Sabastián Englert, Rapa Nui Gill collection) appears to encompass the range of variability expressed by the hypothesis samples. A critical consideration for any study assessing museum sample variation remains the selection of W, the baseline sample (Key and Jantz, 1990a,b). The sample utilized to estimate baseline variability would have a profound effect on all subsequent analyses. The use of the Museo Antropológico Padre Sabastián Englert, Rapa Nui Gill collection to estimate W is justiﬁable, as it represents individuals from all regions of the island and spans the time periods during which ahu and cave burials were practiced (the Late Prehistoric (AD 1680 –1722) through the Protohistoric (AD 1722–1868) periods; Gill and Owsley, 1993; Gill et al., 1983; Owsley et al., 1994; Shaw, 1996; Stevenson, 1984). The results of this research indicate that it is not unwarranted to pool the various Rapa Nui museum cranial samples for further analyses. The methodologies utilized for this research will be incorporated to investigate intrasample variability among other prehistoric Polynesian cranial collections, in order to ensure the accurate assessment of Polynesian prehistory through the use of craniometrics. Though this study utilized Rapa Nui cranial samples from around the world, a number of methodological issues were addressed which are important for all research using skeletal samples. Among these issues are intra- and interobserver error, sampling bias, missing data estimation, and intrasample variation. One ultimate goal of many physical anthropology studies is the assessment of morphological/ phenotypic variation. However, embedded in all morphological data is environmental variation, creating bias and error within the data. One recognizable source of environmental variance is intra- and interobserver variation (error) in the measuring and recording of morphological dimensions. Differences in measuring techniques and equipment can lead to excessive variation when data from multiple sources are pooled together for an analysis. Standardization in techniques, equipment, and periodic assessment of intra- and interobserver error could signiﬁcantly minimize a source of environmental variation (Utermohle and Zegura, 1982). 55 Sampling bias is a source of environmental variation that may or may not be within the control of the individual researcher. Collection and sampling bias with regards to sample size, physical condition of skeletal elements, accessibility of elements, biological age, sex, and race of specimens can lead to bias in the morphological variation of the sample. The demographic/age cohort structure of samples representing a single population might not be a concern when assessing intrasample variation, as was the case for this study. However, knowledge of the demographic/age cohort structure of a sample might be of vital concern when comparing samples from different populations. Researchers need to assess the composition of their samples to insure they are appropriate for the research question under investigation. The estimation of missing data can also contribute to the environmental variation present within sample data. It was discussed above how various missing data estimation techniques produce different types of bias within sample morphological variation. Each technique has its advantages and disadvantages, and the most appropriate method for the research question under investigation needs to be utilized. In many case this is a judgment call by the individual researcher, but that decision needs to be addressed and justiﬁed. Additionally, interpretation of results obtained from estimated data have to take into account the method utilized and the biases that the technique embeds in the data. All the factors discussed above can greatly inﬂuence intrasample variation. This study should encourage other physical anthropologists to assess the degree of intrasample variability produced when: 1) data from various museum skeletal samples are pooled to represent a single population (i.e., Rapa Nui), or a related group of populations (i.e., Polynesians); and 2) when independent data from two or more researchers are pooled. This type of assessment is far too little practiced. ACKNOWLEDGMENTS This research project would not have been possible without the well-documented Rapa Nui skeletal sample established by Dr. George W. Gill, Scientiﬁc Leader of the 1981 Easter Island Anthropological Expedition, and the efforts of his University of Wyoming students, and Chilean and American colleagues. Dr. Douglas W. Owsley, of the Smithsonian Institution, computerized the craniometric data from that project. I thank Dr. Gill and Dr. Owsley for providing me the opportunity to conduct research on these Rapa Nui craniometrics. Their ﬁeld efforts were supported by the National Geographic Society, the Center for Field Research-Earthwatch, the Government of Chile, the University of Chile, the University of Wyoming, the Kon-Tiki Museum, the Smithsonian Institution, and the World Monuments Fund. This particular research project would not have been possible without the additional assistance 56 V.H. STEFAN Porion-nasion3 H-NAR*‡ Porion-subnasale4 H-SSR*‡ Porion-prosthion5 H-PRR*‡ Basion-porion height B-BPH† Maxillofrontal breadth GH-MXB† Maxillofrontal subtense GH-MXS† Zygoorbital breadth GH-ZOB† Zygoorbital subtense GH-ZOS† Alpha chord GH-ALB† Alpha subtense GH-ALS† Simotic cord H-WNB* Bimaxillary breadth H-ZMB* Bimaxillary subtense H-SSS* Mastoid length H-MDH* Mastoid width H-MDB* Cheek height H-WMH* Malar length, inferior H-IML* Malar length, maximum H-XML* Palatal depth M-PAD Maxilloalveolar breadth6 H-MAB* Maxilloalveolar length B-MAL provided by Dra. Silvia Quevedo K. and Dra. Marı́a Eliana Ramı́rez of the Museo Nacional de Historia Natural (Santiago, Chile), Professeur André Langaney, Philippe Mennecier, and Simone Jousse of the Laboratoire d’Anthropologie Biologique, Musée de l’Homme (Paris, France), Professeur Henry de Lumley of the Muséum d’Histoire Naturelle (Paris, France), Domimique Grimaud-Hervé of the Institut de Paléontologie Humaine (Paris, France), Dr. Robert Kruszynski of the Human Origins Group, Natural History Museum (London, UK), Dr. Ian Tattersall and Dr. Kenneth M. Mowbray of the American Museum of Natural History (New York, NY), Dr. Della C. Cook, Indiana University (Bloomington, IN), and Dr. Douglas W. Owsley, Smithsonian Institution (Washington, DC) during my data collection in the United States, Chile, and Europe. I thank Dr. Richard L. Jantz, Dr. Hans C. Petersen, and Dr. John W. Graham for their helpful discussions regarding the methodologies employed in the research. APPENDIX Standardized craniofacial measurements1 Maximum cranial length H-GOL* Nasion-occipital length H-NOL* Maximum cranial breadth H-XCB* Maximum frontal breadth H-XFB* Minimum frontal breadth B-WFB Bizygomatic breadth H-ZYB* Basion-bregma height H-BBH* Basion-nasion length H-BNL* Nasion-bregma chord H-FRC* Bregma-lambda chord H-PAC* Nasion-prosthion height H-NPH* Nasion-alveolare2 B-NAL Biasterionic breadth H-ASB* Basion-prosthion length H-BPL* Bistephanic breadth H-STB* Bijugal breadth H-JUB* Foramen magnum length H-FOL* L. orbital height H-OBH* L. orbital breadth, dacrion B-OBD L. orbital breadth, max-f H-OBB* Biorbital breadth H-EKB* Nasal height H-NLH* Nasal breadth H-NLB* Bifrontal breadth H-FMB* Biauricular breadth H-AUB* Minimum cranial breadth H-WCB* Auricular height B-AUR‡ Porion-bregma height B-PBH‡ 1 B, Bass (1995); GH, Gill et al. (1988); H, Howells (1973); M, Martin and Saller (1957). *Measurement abbreviations from Howells (1989). 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