A SUGGESTED METHOD FOR USING MEANS DATA IN DISCRIMINANT FUNCTIONS ANALYSIS OF ANTHROPOMETRIC VARIABLES Randall R. Skelton Department of Anthropology University of Montana Missoula, MT 59812 an_rrs@selway.umt.edu ABSTRACT Forensic anthropologists often attempt to determine the biological ancestry of individuals from skeletal remains provided by law enforcement agencies. The standard metric technique for determining ancestry, discriminant analysis, is limited in that it conventionally requires data in the form of individual measurements for each group being distinguished. In this study I investigate a way to circumvent this limitation by constructing pooled standard deviations plus correlations matrices using individual measurements data, then appending means and sample sizes from other samples. The resulting files are used as input for an SPSS discriminant analysis procedure. Data from the Boas Anthropometric Data Set are used in this investigation. Classification accuracies are presented for adult and subadult males and females classified using discriminant functions designed to classify adult males by the conventional method and four permutations of the alternative method. The results suggest that the alternative method may be a useful way to develop discriminant functions that include groups for which only means and sample size data are available. Surprisingly, the alternative method appears to give higher accuracy in classifying individuals not used to construct the discriminant functions. I conclude that the alternative method has the potential to be useful in forensic anthropology. INTRODUCTION One definition of forensic anthropology (Haviland, 1994) emphasizes the application of the methods and expertise of physical anthropology to the legal process. In one common situation, the forensic anthropologist takes custody of skeletal material from a law enforcement agency and attempts to determine the age, sex, race, stature, and other individualizing information about any human beings present. These findings may then be matched against lists of missing persons in an attempt to determine the identity of the deceased. One of the important identifying items in a law enforcement agency's description of a person is their race (Rhine 1990a). Unfortunately, non-anthropologists, including law enforcement agents, conceptualize race in a way that often differs greatly from a physical anthropologist's concept of race (Gill 1990), and may include factors that are more properly considered ethnicity (Novotny, _et al_. 1993; St. Hoyme and Iscan 1989). In contrast, what is recorded in a person's skeleton is their biological ancestry. The correspondence between biological ancestry and ethnicity is often inexact. Identification of ancestry is arguably one of the most difficult parts of a forensic anthropological analysis. There are two useful approaches to identifying ancestry (Krogman and Iscan 1986). In the first approach the anthropologist visually inspects the skeleton, looking for characters that have a high frequency in certain populations. These characters are primarily features of the face, such as shape of the nasal aperture and placement of the cheek bones (Angel and Kelley 1990; Brooks _et al_. 1990; Brues 1990; Corruccini 1974; Hinkes 1990; Jones 1930; Napoli and Birkby 1990; Rhine 1990b). In the second approach the anthropologist takes measurements of the skeleton and plugs them into one or more computer-generated formulae, called discriminant functions (Ayers _et al_. 1990; Birkby 1966; Crichton 1966; Fisher and Gill 1990; Giles and Elliot 1962; Gill 1984; Gill _et al_. 1988; Gill and Gilbert 1990; Iscan, 1983, 1990; Iscan and Cotton 1990; Keita 1992; Rightmire 1970). Discriminant functions yield a set of scores for an individual that can be used to classify the individual into a race category. Both of these approaches are useful, and the best analyses use both of them (Novotny _et al_. 1993). The discriminant functions approach has an advantage in that discriminant functions require less osteological expertise to use, and the interpretation of the results is less subjective (St. Hoyme and Iscan 1989). The popularity of this approach has led to the development of at least one computer software application, FORDISC, which is at least partly intended for use by non-experts (Jantz and Ousley 1993). One limitation of conventional discriminant functions analysis is that measurements data for each individual in a data set are required for constructing the discriminant functions. Unfortunately, individual measurements data are not commonly published or otherwise made easily available. In contrast, statistical summaries of data sets, including means and sample sizes, are common. Minimally, discriminant analysis requires group means and sample sizes for each variable in the analysis; standard deviations for each variable, pooled over all the groups; and a matrix of the correlations between the variables, pooled over all the groups (Morrison 1967; SPSS Inc. 1988). If the input data are individual measurements, then these statistics are computed during the analysis. However, if it can be assumed that the standard deviations of the measurements and the correlations between the measurements are similar for all populations of _Homo_ _sapiens_, then it might be possible to utilize a modified method, which allows the use of means and sample sizes from groups for which individual measurements are not available. The modified method would entail using pooled standard deviations and a correlations matrix constructed from the individual measurements data that are available. Then, means and samples sizes for the groups for which individual measurements are not available can be combined with the pooled standard deviations and correlations matrix to produce a file that can be used as input for a discriminant analysis. In this study I develop and conduct preliminary tests of the method proposed above. This is not an applied exercise, and no discriminant functions of utility outside this analysis will be produced. My intention is to show that this method is worthy of further testing. MATERIALS AND METHODS I worked with the Boas anthropometric data set, which was kindly provided by R.L. Jantz and S. Ousley of the University of Tennessee, Knoxville (Jantz _et al_., 1992). This data set contains six useful cranial and six useful postcranial measurements on over 15,000 then living Native Americans of many groups. These data were collected about 100 years ago by colleagues of Franz Boas, a revered figure in American anthropology (Hays, 1964). Although the Boas data set consists of measurements of living people and, therefore, cannot be applied directly to investigations of the skeleton, these measurements should behave similarly to skeletal measurements. The subset of measurements included in the Boas data set, which were used in this study are those listed in Jantz _et al_. (1992), namely: head length, head breadth, face height, face breadth, nose height, nose breadth, standing height, shoulder height, finger height, finger reach, sitting height, and shoulder breadth. These twelve measurements were chosen because values for them are present for the majority of individuals in the data set. All twelve measurements were used in the discriminant analyses and no attempt was made to examine the performance of different classes of variables (i.e. postcranial vs cranial). The Boas data set includes data for individuals of all ages and both sexes. I selected the adult males for use in calculating the discriminant functions, since they are the most commonly represented individuals in the Boas data set. Adult males with missing values were excluded. I identified within the Boas data set 47 groups that contained 30 or more adult males without missing values. I used 90% of the adult males from each of these 47 groups for the analysis data set used to calculate discriminant functions, and reserved 10% for a test series to see how well the discriminant functions worked with males that were not actually used in the construction of the formulas. For example, the Boas data set contained values for all 12 variables for 114 adult male Apache. Of these, 90% (103 individuals) were included in the data set used to construct the discriminant functions, and 10% (11 individuals) were placed in the data set reserved for testing the accuracy of the functions. Hereafter I refer to these data sets as the 'adult males' data set and the 'test' data set, respectively. I extracted three additional data sets from the Boas data set for testing purposes, consisting of: all adult females of the 47 groups, all subadult females of the 47 groups, and all subadult males of the 47 groups. Individuals with missing values were excluded. The number of individuals used in each of the five data sets is shown in table 1 by group. Note that the group names are those used in the Boas anthropometric data set and may not be the currently preferred name for the people in question. Since most applications of discriminant functions seek to classify individuals into a relatively small number of groups, I chose to work with 50 subsamples of four of the 47 groups, rather than analyzing all 47 groups simultaneously. Dividing the sample into subsamples also enabled me to generate a relatively large number of combinations of groups. I selected groups at random by drawing four numbered pieces of paper from a well-shaken box. I substituted one group for another in two of the subsamples to make sure that each group was included in at least three, but not more than six, of the 50 subsamples. The number of times each group was used in a subsample is shown in table 1. The group composition of each of the 50 subsamples is shown in table 2. Using the correlation program from the SPSS statistical package, I constructed four files consisting of the pooled standard deviations plus a correlations matrix (hereafter SD&C files), using samples of individuals from the Boas data set. The first sample consisted of 1215 adult males of those groups in the Boas data set containing fewer than 30 adult males, and produced a SD&C file which I refer to as the 'other males' file. This file does not include any of the males in the 'all males' data set or the 'test' data set. The second sample consisted of all 4649 adult males from the 47 large Native American groups (the individuals from the 'adult males' and 'test' data sets), which produced a SD&C file I refer to as the 'all males' file. The third consisted of 7,629 adults of both sexes and all Native American groups, which produced a file I refer to as the 'all adults' file. The fourth consisted of 15,149 individuals of both sexes and all ages from all Native American groups, and produced a SD&C file which I refer to as the 'all people' file. Individuals with missing values were not excluded from this procedure with the exception of some of the adult males. I used the discriminant program from the SPSS statistical package for the discriminant analyses. For each of the 50 subsamples I provided the individual measurements for the adult males in the subsample as input data, performed a discriminant analysis, and requested classification of the individuals in the four test sets. I saved the means and samples sizes generated by this procedure, added them to each of the four SD&C files, and repeated the analysis using each of the data files thus generated. I used the ttest program from the SPSS statistical package to compare mean classification accuracies resulting from specific data set and SD&C file combinations. In summary, the procedure was as follows. For each of the 50 subsamples of the 47 large groups I performed a conventional discriminant analysis using the individual measurements data for adult males of the four groups in the subsample. Then for each of the 50 subsamples I performed four variations of the alternative method by appending group means and sample sizes for the adult males in the subsampled groups to each of the four previously prepared SD&C files. The resulting five discriminant analyses were used to classify the adult males, test set males, adult females, subadult females, and subadult males of the four groups in the subsample. This procedure produced classification accuracies for each combination of data set and discriminant analysis methodology, a total of 25 separate classification accuracies per subsample. The classification accuracies were then averaged over the 50 subsamples. RESULTS The results of these procedures, presented as average classification accuracies over the 50 subsamples, are shown in table 3. The classification accuracies using the traditional method of providing individual measurements data as input are shown in the column labelled 'conventional analysis'. This method gives the highest accuracies for the individuals whose measurements were provided as input, but gives relatively poor accuracy with the four test sets. Using the 'all males' SD&C file resulted in a non- significant decrease of 1.71% in the mean classification accuracy for the adult males (p=.175). Using any of the remaining three SD&C files produced a significant decrease in the mean accuracy of classification of the adult males (p=.04 or less in all cases), although the largest decrease in accuracy was a comparatively minor 4.02%. All four SD&C files produced a significant increase in the mean classification accuracy of the test set males, the adult females, the subadult males, and the subadult females (p<.001 in all cases). In many cases the classification accuracies are drastically better. Using the all people SD&C file, there was a significant difference in the classification accuracies of adult males and females (p<.001), but the sex distinction was not significant when using any of the remaining three SD&C files (p=.596 or greater in all cases). DISCUSSION The traditional method for constructing discriminant functions appears to give the highest accuracies for the individuals actually used in the construction of the functions, but the alternative method appears to give higher classification accuracies with individuals not used to construct the functions. The implication of these results is that the more generalized the data used to calculate the pooled standard deviations and the correlations matrix, the more closely the standard deviations and correlations approach their 'true' values for _H_. _sapiens_, and the better the discriminant functions perform over a wide range of individuals. There must, however, be a limitation to the useful degree of generality of the data used to calculate the pooled standard deviations and the correlations matrix, since the 'all people' SD&C file gave slightly poorer results than the 'all males' and 'all adults' SD&C files. The addition of subadults to the data set used to construct such a SD&C file appears to detract from that file's overall accuracy of classification. This suggests that subadults may exhibit different standard deviations for the measurement or different correlations between measurements than adults. CONCLUSIONS The results of this study suggest that discriminant functions constructed using an alternative method wherein group means and sample sizes for variables are appended to SD&C files constructed using data from other individuals and groups may be useful for classifying individuals. The results also suggest that, surprisingly, the alternative method may be more effective than the conventional method for classifying individuals not used in the actual construction of the discriminant functions. These observations suggest that forensic anthropologists might consider widespread sharing of measurements data from many skeletal populations, which would allow construction of a SD&C file that would be useful over a wide range of populations. This would allow customization of discriminant functions for specific applications by simply adding appropriate means and sample sizes to the SD&C file and performing a discriminant analysis using the resulting file as input. ACKNOWLEDGEMENTS I would like to thank R.L. Jantz, S.D. Ousley, and all the other people involved with compiling, maintaining, and distributing the Boas Anthropometric Data Set for their efforts. University of Montana's Computing and Information Services Department provided the hardware and software. Tom Foor, John Price, and Scott Catey contributed expertise and inspiration at various stages of this project. TABLE 1: GROUPS AND SAMPLE SIZES |GROUP NAME | ADULT | TEST | ADULT |SUBADULT|SUBADULT| TIMES | | | MALES | SAMPLE |FEMALES |FEMALES | MALES | USED IN | | | N | N | N | N | N |SUBSAMPLE| |------------+--------+--------+--------+--------+--------+---------| |Apache | 103 | 11 | 35 | 76 | 71 | 3 | |Arapaho | 53 | 6 | 8 | 5 | 15 | 4 | |Cherokee-OK | 108 | 12 | 38 | 14 | 59 | 4 | |Cherokee | 159 | 18 | 111 | 64 | 89 | 5 | |Cheyenne | 28 | 3 | 3 | 6 | 11 | 4 | |Chickasaw | 95 | 11 | 35 | 28 | 41 | 3 | |Chilcotin | 27 | 3 | 23 | 6 | 9 | 6 | |Chippewa | 318 | 35 | 169 | 157 | 170 | 4 | |Choctaw | 303 | 34 | 72 | 39 | 43 | 6 | |Coahuilla | 32 | 4 | 19 | 50 | 94 | 6 | |Comanche | 76 | 9 | 36 | 10 | 22 | 5 | |Concow | 28 | 3 | 12 | 6 | 8 | 3 | |Cree | 74 | 8 | 62 | 26 | 37 | 5 | |Creek | 71 | 8 | 6 | 2 | 7 | 5 | |Crow | 208 | 23 | 104 | 64 | 65 | 6 | |Eskimo | 46 | 5 | 26 | 6 | 10 | 5 | |Goajira | 29 | 3 | 14 | 8 | 5 | 3 | |Haida | 40 | 4 | 9 | 6 | 5 | 4 | |Hoopa | 36 | 4 | 13 | 7 | 14 | 3 | |Kiowa | 72 | 8 | 36 | 20 | 20 | 4 | |Klamath | 60 | 7 | 32 | 50 | 42 | 6 | |Kutenai | 47 | 5 | 4 | 17 | 34 | 3 | |Kwakiutl | 47 | 5 | 47 | 7 | 10 | 3 | |Lillooet | 74 | 8 | 70 | 22 | 26 | 3 | |Makah | 40 | 5 | 30 | 32 | 37 | 4 | |Malecite | 43 | 5 | 15 | 22 | 29 | 3 | |Menomini | 105 | 12 | 26 | 57 | 62 | 4 | |Micmac | 108 | 12 | 82 | 65 | 88 | 4 | |Mississauga | 74 | 8 | 48 | 57 | 94 | 4 | |Munsee | 55 | 6 | 17 | 12 | 13 | 5 | |Navajo | 60 | 7 | 10 | 13 | 68 | 4 | |Ojibwa | 142 | 16 | 60 | 47 | 49 | 4 | |Okanagan | 35 | 4 | 23 | 15 | 21 | 5 | |Oneida | 58 | 6 | 44 | 64 | 62 | 5 | |Paiute | 75 | 8 | 30 | 40 | 58 | 4 | |Pawnee | 40 | 4 | 7 | 12 | 17 | 5 | |Piegan | 56 | 6 | 14 | 13 | 31 | 3 | |San Luis Rey| 62 | 7 | 45 | 32 | 29 | 4 | |Seneca | 31 | 4 | 26 | 20 | 25 | 4 | |Shoshoni | 32 | 4 | 3 | 43 | 18 | 5 | |Shuswap | 131 | 15 | 93 | 104 | 104 | 3 | |Sioux | 565 | 63 | 203 | 92 | 71 | 5 | |Stalo | 49 | 5 | 21 | 48 | 49 | 4 | |Thompson | 106 | 12 | 139 | 64 | 53 | 3 | |Tsimshian | 58 | 7 | 43 | 27 | 28 | 5 | |Ute | 62 | 7 | 20 | 8 | 19 | 6 | |Zuni | 61 | 7 | 10 | 13 | 11 | 4 | |------------+--------+--------+--------+--------+--------+---------| |TOTALS: | 4182 | 467 | 1993 | 596 | 1943 | | |------------+--------+--------+--------+--------+--------+---------| TABLE 2: SUBSAMPLE COMPOSITIONS | SUBSAMPLE | GROUPS INCLUDED | |-----------+----------------------------------------------| | 1 | Cherokee-OK, Cheyenne, Haida, Hoopa | | 2 | Coahuilla, Haida, Mississauga, Navajo | | 3 | Chickasaw, Choctaw, Piegan, Sioux | | 4 | Chippewa, Comanche, Malecite, Ute | | 5 | Cherokee-OK, Cherokee, Chilcotin, Sioux | | 6 | Cherokee, Concow, Munsee, Tsimshian | | 7 | Coahuilla, Crow, Okanagan, Tsimshian | | 8 | Chippewa, Coahuilla, Menomini, Piegan | | 9 | Chippewa, Cree, Paiute, Ute | | 10 | Creek, Micmac, Mississauga, Stalo | | 11 | Arapaho, Comanche, Mississauga, Paiute | | 12 | Cree, Kiowa, Klamath, Oneida | | 13 | Eskimo, Pawnee, San Luis Rey, Zuni | | 14 | Chilcotin, Crow, Okanagan, Paiute | | 15 | Choctaw, Goajira, Menomini, Ojibwa | | 16 | Cheyenne, Crow, Klamath, Ute | | 17 | Klamath, Micmac, Pawnee, Thompson | | 18 | Chickasaw, Lillooet, San Luis Rey, Thompson | | 19 | Cheyenne, Makah, Munsee, Shuswap | | 20 | Arapaho, Comanche, Oneida, Tsimshian | | 21 | Creek, Makah, Munsee, Okanagan | | 22 | Cree, Navajo, Stalo, Ute | | 23 | Klamath, Kutenai, Pawnee, Seneca | | 24 | Haida, Menomini, Shoshoni, Zuni | | 25 | Choctaw, Creek, Seneca, Sioux | | 26 | Cherokee-OK, Coahuilla, Stalo, Tsimshian | | 27 | Apache, Crow, Eskimo, Okanagan | | 28 | Comanche, Eskimo, Kiowa, Kwakiutl | | 29 | Cherokee, Choctaw, Kutenai, Sioux | | 30 | Apache, Chilcotin, Oneida, Shoshoni | | 31 | Arapaho, Comanche, Concow, Piegan | | 32 | Haida, Paiute, San Luis Rey, Shoshoni | | 33 | Chippewa, Cree, Hoopa, Malecite | | 34 | Cherokee-OK, Kwakiutl, Thompson, Ute | | 35 | Creek, Crow, Makah, Navajo | | 36 | Apache, Cherokee, Micmac, Ojibwa | | 37 | Cheyenne, Malecite, Pawnee, Zuni | | 38 | Chilcotin, Eskimo, Oneida, Seneca | | 39 | Creek, Kutenai, Munsee, Sioux | | 40 | Coahuilla, Seneca, Shoshoni, Tsimshian | | 41 | Crow, Kwakiutl, Micmac, Ute | | 42 | Cherokee, Hoopa, Lillooet, Oneida | | 43 | Chilcotin, Choctaw, Ojibwa, Okanagan | | 44 | Eskimo, Klamath, Makah, Menomini | | 45 | Coahuilla, Navajo, Pawnee, Shuswap | | 46 | Arapaho, Concow, Cree, San Luis Rey | | 47 | Chilcotin, Goajira, Kiowa, Shoshoni | | 48 | Chickasaw, Munsee, Ojibwa, Zuni | | 49 | Goajira, Kiowa, Klamath, Lillooet | | 50 | Choctaw, Mississauga, Shuswap, Stalo | |-----------+----------------------------------------------| TABLE 3: AVERAGE ACCURACIES |DATA SET |CONVENTIONAL| OTHER | ALL | ALL | ALL |OVERALL | | | ANALYSIS | MALES | MALES | ADULTS | PEOPLE |ACCURACY| | | | FILE | FILE | FILE | FILE | | |----------------+------------+--------+--------+--------+--------+--------| |Adult Males | 78.49% | 75.60% | 76.78% | 75.93% | 74.47% | 76.25% | |Test Set | 67.53% | 85.96% | 87.46% | 86.49% | 86.36% | 82.76% | |Adult Females | 55.62% | 76.26% | 76.98% | 76.32% | 64.16% | 69.87% | |Subadult Males | 40.32% | 52.91% | 55.39% | 60.72& | 57.08% | 53.29% | |Subadult Females| 34.70% | 59.20% | 61.40% | 66.18% | 53.17% | 54.93% | |----------------+------------+--------+--------+--------+--------+--------| |OVERALL | 55.33% | 69.98% | 71.60% | 73.13% | 67.05% | 67.42% | |----------------+------------+--------+--------+--------+--------+--------| REFERENCES CITED Angel, J.L and Kelley, J.O. 1990 Inversion of the posterior edge of the jaw ramus: new race trait. 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