# Estimating the length of incomplete long bones Forensic standards from Guatemala.

код для вставкиСкачатьAMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 120:233–251 (2003) Estimating the Length of Incomplete Long Bones: Forensic Standards From Guatemala Lori E. Wright1* and Mario A. Vásquez2 1 2 Department of Anthropology, Texas A&M University, College Station, Texas 77843-4352 Escuela de Historia, Universidad de San Carlos de Guatemala, Guatemala City, Guatemala KEY WORDS forensic anthropology; stature; long bone fragments; regression; Maya ABSTRACT We report on new standards for estimating long bone length from incomplete bones for use in forensic and archaeological contexts in Central America. The measurements we use closely follow those defined by Steele ([1970] Personal Identification in Mass Disasters; Washington, DC: Smithsonian Institution), but we add several new landmarks. We measured the femur, humerus, tibia, and fibula of 100 Maya skeletons (68 males, 32 females) recovered from forensic exhumations. We derived the equations by regressing bone segment length on bone length, and solved for bone length to maximize the utility of the equations for taller populations. We generated equations for all segments that were significantly correlated with bone length for males, for females, and for both sexes combined, but accepted only regressions with The estimation of standing stature during life is a key goal in the forensic analysis of human skeletal material, because height is a straightforward biological characteristic that may aid the investigator in deducing the identity of a skeleton. Likewise, for archaeological materials, stature provides an important means to address health and adaptation in the past, as it is a sensitive measure of the growth status of a population. Hence, the development of techniques for estimating stature from skeletal remains has been a core focus of both forensic anthropology and bioarchaeology since their inception. In both fields, stature estimation is often problematic because of deterioration of the long bones during burial. Methods to circumvent this problem by estimating stature from fragmentary bones have been devised by several researchers. Prominent among these are the equations developed by Steele (1970); (also Steele and McKern, 1969) to estimate bone length for the fragmentary humerus, femur, and tibia. Criticisms of the method have focused on difficulty in identifying landmarks used in the equations, and on the possibility that the regressions may be population-specific. In this paper, we revisit Steele’s approach (1970) to long bone length estimation. We explore the issue of population specificity of bone segment proportions in greater detail. As with variation in limb bone proportions, interpopulation variation in the propor© 2003 WILEY-LISS, INC. r2 ⬎ 0.85 as reliable. Landmarks defined by muscle attachment sites were more variable in location than landmarks on articular architecture; thus we retained few equations that use these landmarks. We tested the male and combined sex equations on 36 males of unknown ethnicity exhumed from a military base in Guatemala, and found that the equations performed satisfactorily. We also evaluated the performance of equations by Steele ([1970] Personal Identification in Mass Disasters; Washington, DC: Smithsonian Institution) and Jacobs ([1992] Am J Phys Anthropol 89:333–345) on the Maya bones, and conclude that significant population variation in long bone proportions hinders their application in Central America. Am J Phys Anthropol 120:233–251, 2003. © 2003 Wiley-Liss, Inc. tions of long bone segments (or conversely, in the relative locations of bony landmarks on the bone) may be due to a diversity of factors, including aspects of genetic, climatic, and nutritional environments. However, the position of long bone landmarks may also be determined by functional stresses on the limb (Jacobs, 1992). Do landmark positions scale allometrically with bone length? Or are functional environments and activity patterns more significant? Here, we develop regression equations to estimate bone length for forensic Maya skeletons from Guatemala. Because of the extremely small stature of Maya populations, we reasoned that the proportions of long bone segments might vary substantially from published values, as a function of size alone. In addition to providing standards appropriate to this population, we use the Maya data to Grant sponsor: Office of the Vice President for Research and Graduate Studies, Texas A&M University. *Correspondence to: Lori E. Wright, Department of Anthropology, Texas A&M University, College Station, TX 77843-4352. E-mail: lwright@tamu.edu Received 26 June 2001; accepted 18 February 2002. DOI 10.1002/ajpa.10119 Published online in Wiley InterScience (www.interscience.wiley. com). 234 L.E. WRIGHT AND M.A. VÁSQUEZ examine the nature of interpopulation variation in bone segment proportions. STATURE ESTIMATION FROM FRAGMENTARY LONG BONES The estimation of bone length from incomplete long bones was pioneered by Müller (1935), who generated equations for the humerus, radius, and tibia from measurements of the skeletal collections of the Österreiches Beinhaus in Zellerndorf. Müller (1935) defined 5 segments for the humerus, 4 for the radius, and 7 for the tibia, using the margins of articular surfaces and key points of muscle attachment to define the segments. She calculated the percentage of total length represented by each section, and used this value to estimate the length of fragmentary material. Working with Mississippian archaeological material from northeast Arkansas (Steele and McKern, 1969; Steele and Bramblett, 1988) and with skeletons of American Blacks and whites from the Smithsonian’s Terry Collection (Steele, 1970), Steele generated a series of sex- and population-specific equations for the humerus, femur, and tibia. He defined landmarks to delimit 4 segments on the humerus, 4 on the femur, and 5 on the tibia, in part following the landmarks used by Müller (1935). For each segment, he generated a regression equation to estimate bone length. He also calculated multiple regression formulae to estimate bone length using a combination of segments, and provided equations that relate segment length directly to stature, to reduce the large standard error that would pertain after applying a stature regression formula to estimated bone length. Standard errors of the formulae for estimated bone length on the reference population are smaller for the multiple regressions than for individual segment equations, and span 0.20 –2.93 cm. Stature estimates using Steele’s regressions have standard errors of 3.71– 6.17 cm. Critiques of the method of Steele (1970) have primarily focused on the difficulty that some osteologists have had in locating several of the landmarks that define the bone segments (Brooks et al., 1990; Holland, 1992; Simmons et al., 1990). In a survey of members of the American Academy of Forensic Sciences, Brooks et al. (1990) found that 55% (6 of 11) of respondents who had applied the formulae to remains for which they had an independent estimate of stature found that Steele’s formulae (1970) did indeed estimate stature accurately. They also underscored the problematic nature of stature data obtained from documentary and cadaveral sources, to which skeletal estimates are routinely compared. Errors in these “known” data undoubtedly contribute to perceptions that skeletal stature estimates are inaccurate. Two studies proposed stature estimation standards that utilize transverse measurements taken on long bones. Simmons et al. (1990) used standard osteometric measurements of the femoral condyles, neck, and head to estimate stature. Using skeletons from the Terry Collection, they found reasonably strong correlations between these measures and long bone length (r ⫽ 0.22– 0.68). Similarly, Holland (1992) used the biarticular breadth of the proximal tibia to generate a stature regression equation for skeletons from the Hamann-Todd Collection at the Cleveland Museum of Natural History. Although both of these approaches obtained statistically significant correlations between the transverse measurements and maximum bone length on the reference populations from which they were developed, they may find less application than the longitudinal measurement approach of Steele (1970). The transverse measurement methods emphasize parts of bones (epiphyses and metaphyses) that are least often preserved in forensic and archaeological contexts; their utility will be greatest for recent forensic cases. Moreover, transverse measures vary substantially among populations, together with the degree of sexual dimorphism. Such transverse measurement approaches are in no real sense a “revision” of Steele’s method (1970) (contra Simmons et al., 1990), but should be seen as complementary approaches to bone length estimation. Steele (1970) had generated regression equations independently for several populations because he reasoned that bone segment proportions might differ among populations, much as total bone proportions do. In applying the equations to complete femora and tibiae of Mesolithic and Neolithic European skeletons, Jacobs (1992) demonstrated interpopulation variation in the proportions of long bone segments. He observed that some of the equations of Steele (1970) tended to overestimate bone length, while others underestimated length significantly, and he concluded that bone length could not be accurately estimated for these skeletons using these equations. However, he demonstrated that segment lengths did show significant correlations with bone length in this population, and he generated population-specific equations, thus confirming the utility of Steele’s approach. Jacobs (1992) showed that the poor performance of the femur equations of Steele (1970) on the European sample was primarily due to differences in the proportions of segments 2 and 3, which are separated by a landmark defined by the divergence of the medial and lateral supracondylar lines from the linea aspera. Similarly, he observed a different proportion of tibial segments 1 and 2, which are divided by the landmark defined by the proximal margin of the tibial tuberosity. He suggested that population differences in muscular activity would explain these differences in proportion, and recommended that researchers search out samples with diverse behavioral characteristics and adaptive strategies in order to generate regression equations that might be more appropriate analogues for both forensic and bioarchaeological applications (Jacobs, 1992). 235 ESTIMATING LENGTH OF INCOMPLETE LONG BONES TABLE 1. Mean stature for males and females in the Guatemalan forensic remains, and in comparative series used for estimation of long bone length and stature Male stature Female stature Skeletal series or reference sample Source Mean (cm) SD (cm) N Mean (cm) SD (cm) N Forensic, Mayan1 Forensic, El Chal1 Mexican cadavers Terry Collection, White Terry Collection, Black Mississippian1 This study This study Genovés, 1967 Steele, 1970 Steele, 1970 Steele and McKern, 1969 158.25 160.78 163.99 168.44 172.02 165.35 4.48 4.05 5.11 8.11 7.84 — 66 33 22 61 42 72 147.29 5.45 32 152.30 157.62 159.88 154.75 6.71 7.96 6.22 — 15 52 57 29 1 2 Stature is calculated from the measured maximum length of the femur, using the femur equation of Genovés (1967). Unreported. Although he confirmed the success of Steele’s methodological approach, Jacobs (1992) demonstrated that the standards of (Steele, 1970) may not accurately estimate bone length for all skeletal populations. This is likely to be especially true for populations that diverge significantly in completed adult stature or body proportions from reference populations of Steele’s. In both our forensic and bioarchaeological research in Guatemala, we are seldom able to estimate stature from intact bones due to the rapid deterioration of bone in this tropical environment, and due to other perimortem and postdepositional factors. We began our investigation with the suspicion that the existing standards would not be appropriate for Maya skeletal remains because of the extremely diminutive stature of Maya populations. Twentieth century Maya populations in Guatemala and Mexico experience marked stunting of stature due to nutritional insufficiency during the period of childhood growth (Bogin and MacVean, 1984; Danforth, 1994; Furbee et al., 1988; Martorell, 1995; Martorell et al., 1994; McCullough, 1982). Table 1 illustrates that the forensic Maya skeletons from Guatemala used in this study fall about 10 cm short of the average stature of the skeletal series from which Steele (1970) generated his regression equations. Maya skeletal remains are also very slight; established standards for sex estimation from bone dimensions typically misclassify many males as females. Therefore, most bioarchaeologists working on Maya remains have generated series-specific discriminant functions for metric sex estimation (Whittington, 1989; Wright, 1994). Both the stature and robusticity of Maya skeletons raise the question of whether Steele’s formulae would work on Maya remains. We report on new regression equations derived from measurements of forensic skeletons from Guatemala. The skeletons were exhumed as part of the ongoing process of reconciliation following Guatemala’s long civil war, which spanned the 1954 CIAsponsored coup d’etat to a final peace agreement signed in 1996. Many of the remains that have been the subject of forensic investigation derive from a period of brutal “scorched earth campaigns” carried out by the military between 1982–1984 (Arzobispado de Guatemala, 1998; Carmack, 1988; Equipo de Antropologı́a Forense de Guatemala, 1997). Al- though interments have also been recovered from military bases and other contexts, most of the civilian victims of the conflict were rural subsistence farmers of Maya descent and had been hastily interred in small “clandestine” cemeteries in remote areas of the Guatemalan highlands. To investigate the applicability of Steele’s approach to estimating bone length from long bone segments, we generate regression equations from measurements of these forensic Maya remains. We test the regressions on an independent forensic sample, and also evaluate the accuracy of the regression equations of Steele (1970; also Steele and Bramblett, 1988) and Jacobs (1992) on forensic Maya remains. MATERIALS AND METHODS The skeletal remains used for this study are treated as two separate samples. The largest sample, hereafter referred to as the “forensic Maya sample,” consists of some 100 human skeletons of rural highland Maya villagers. We use this skeletal sample to generate regression equations to estimate long bone length. The forensic Maya sample comprises 76 skeletons recovered by exhumations carried out by the Area of Exhumations of the Office of Human Rights of the Archbishop of Guatemala (ODHAG), together with 24 skeletons recovered by the Foundation for Forensic Anthropology of Guatemala (FAFG). Although the specific identity of many individual skeletons is not known, all of the remains derive from clandestine cemeteries near remote rural communities. The circumstances of the massacres and subsequent interment are now well-documented, and permit identification of the remains as to village and/or ethnicity. Positive identities were achieved for a sizable proportion of the remains (Arzobispado de Guatemala, 1998; Equipo de Antropologı́a Forense de Guatemala, 1997; unpublished data). We considered only adult skeletons, which showed complete fusion of all long bone epiphyses. Males outnumbered females almost 2:1 in the sample because females were less often the targets of military action than males. We measured all complete adult remains that were available for study during the period of data collection. The remains include skeletons from five major highland Maya ethnic groups (Table 2); there is no ethnic difference in skeletal stature (as estimated from femur 236 L.E. WRIGHT AND M.A. VÁSQUEZ TABLE 2. Maya forensic sample by ethnic group and sex Ethnic group Achi Ixil Kakchiquel Keqchi Quiche Total Male Female Total 1 3 17 14 33 68 5 5 3 16 3 32 6 8 20 30 36 100 lengths) among males of Kakchiquel, Keqchi, and Quiche descent in the sample (ANOVA, P ⫽ 0.64). Unfortunately, cell sizes are too small to test statural differences among ethnic groups for the females. The second forensic sample consists of 36 male skeletons exhumed by the FAFG from the grounds of the military base at El Chal, Petén. No females were recovered from this site. These remains are of “disappeared” individuals, so their ethnic identity is less well-constrained than in the Maya sample. The mean stature of the El Chal remains is approximately 2.5 cm taller than the mean stature of males in the Maya sample (Table 1); thus it is likely that the sample includes a greater proportion of Ladino individuals (of mixed Spanish and indigenous Maya ancestry). We use this sample to test the applicability of regression equations developed from the forensic Maya sample to forensic remains of unknown ethnicity in Guatemala and Central America at large. A standard set of measurements was taken on four major long bones for each skeleton. All of the bones were measured by M.V., using the same osteometric board, between June–December 1998. He measured the left bone for each skeleton, but substituted the right side for skeletons in which the right bone was better preserved than the left. All measurements were recorded in millimeters. The landmarks we measured are described in Table 3 and illustrated in Figure 1. For the most part, the landmarks measured follow those defined by Steele (1970). However, we eliminated one landmark on the tibia (no. 3) that we could not locate confidently, and we added several new landmarks. In addition to the humerus, femur, and tibia, we also measured the locations of several landmarks on the fibula. Rather than numbering the bone segments (Steele and McKern, 1969), we refer to the segment lengths by a code that incorporates a letter representing the bone and the landmark numbers. For instance, “F2–5” represents the segment between landmarks F2 and F5 on the femur. Likewise, “H0 –2” represents the humerus segment from landmarks H0 to H2, and “T3–5” represents the tibia segment from landmarks T3 to T5. To avoid confusion between the femur and fibula, we designate fibular landmarks with “P,” from the Spanish, peroné. The equations we derived relate the distances between these landmarks to the maximum length of each long bone, measured following the standard definitions (Bass, 1987). To measure the locations of landmarks, we place each bone on the osteometric board, with the long axis of the diaphysis parallel to the length of the osteometric board. We use clear plastic triangles to extrapolate the location of landmarks relative to the metric scale on the board. For the humerus, femur, and fibula, we align the bone on the osteometric board with the most proximal portion of the bone that is present at the zero stop; thus, the landmarks are numbered from proximal to distal. For the tibia, the points are numbered from distal to proximal because the maximum length (landmark T7) must be measured with the block of the board on the lateral malleolus in order to exclude the cruciate eminence. For the femur, all measurements are taken with the bone in a prone position on the osteometric board (dorsal surface up). The fibula is placed on its dorsolateral surface, such that the distal articulation is facing to one side and slightly upwards. Because some of the landmarks on the humerus and tibia are located on the ventral aspect of the bone, while others are dorsal, the bone must be turned over to measure several landmarks. After turning the bone, we verify the total length of the bone (or portion present) in order to ensure accurate alignment. Rather than the multiple regression approach used by Steele (1970; also Steele and McKern, 1969; Jacobs, 1992), we follow a univariate approach to the regression analyses, calculating regression equations against bone length for the distance between pairs of landmarks on each bone. Therefore, it is possible to estimate length for bones that suffer from erosion of intermediate landmarks, which would not be possible using the multiple regression approach. We calculate the regression equation for a given pair of points for each sex separately, and for both sexes combined. In addition, we regress fibula segment length against the maximum length of the tibia, since fibula and tibia length are highly correlated (r ⫽ 0.982, n ⫽ 76 for Maya males and females, combined), and standards for stature estimation from the fibula are scarce. Instead of regressing bone length (y) on segment length (x), as is commonly done in forensic stature estimation, we regress segment length (y) on bone length (x) and then solve for bone length. This method, known as classical calibration, has been applied to age estimation (Konigsberg et al., 1997), and is the approach favored in the allometry literature. The former “inverse calibration” method approaches the problem from the perspective that bone length is a product of segment length, an essentially Bayesian approach. Classical calibration takes a maximum likelihood approach, and sees segment length as constrained by bone length. For reconstructing the length of fragmentary bones the latter approach makes logical sense, especially for landmarks defined by muscle attachment sites, since these features are determined by muscle size and action in the context of a complete functioning limb. Konigsberg et al. (1998); (also Hens et al., 2000) have shown that inverse calibration is appropriate ESTIMATING LENGTH OF INCOMPLETE LONG BONES 237 TABLE 3. Definitions of bone landmarks used in estimating bone length, with correspondence to landmarks of Steele (1970) Bone Humerus Femur Tibia Landmark (this study) Landmark (Steele, 1970) H0 H1 H2 1 Prone Supine Supine H3 H41 2 Prone Supine H5 H6 H7 3 4 5 Prone Prone Prone F0 F1 F2 F32 1 Prone Prone Prone Prone F41 3 Prone F5 4 Prone 2 F6 5 Prone T7 1 Supine T6 T51 2 Supine Supine T41 T31 Fibula 1 2 Position of bone Prone Prone T21 4 Supine T1 5 Supine T0 6 Supine P0 P1 Supine Supine P22 P3 P4 Supine Supine Supine Definition of landmark Most proximal point on the head Most proximal point of the greater tuberosity Most projecting proximal point on the lesser tuberosity, along its lateral border Most distal point of circumference of the head Most distal point of the deltoid tuberosity, where the two deltoid lines join Proximal margin of the olecranon fossa Distal margin of the olecranon fossa Most distal point of the trochlea Most proximal point on the head Most proximal point on the greater trochanter Midpoint of the lesser trochanter Distal limit of smooth bone between the union of the pectineal line and linea aspera, at which point the intersection of the lines is filled with rough bone Most proximal extension of the popliteal surface at point where the medial and lateral supracondylar lines become parallel below the linea aspera Most proximal point on margin of the intercondylar fossa Most distal point of the medial condyle Most prominent point on the lateral half of the lateral condyle Most proximal point of the tibial tuberosity Point at which the anterior crest crosses the central axis of tibia, as drawn through the tibial tuberosity Nutrient foramen Point on the popliteal line where it crosses over the medial angle of the diaphysis Point where the anterior crest crosses over to the medial border of the shaft above the medial malleolus Proximal margin of the distal articular surface, at a point opposite the tip of the medial malleolus Most distal point of the medial malleolus Most proximal point of the head Most laterally projecting point on the head, opposite the proximal articulation Nutrient foramen Proximal border of the distal articular facet Most distal point on the lateral malleolus Landmark is not used in any of the equations listed in Table 5. However, it is used in equations listed in Appendix. Landmark is not used in any of the equations generated and deemed acceptable in this study. to contexts where the specimen is known to belong to the same statistical distribution as that from which the estimator was developed. But if the specimen is not known to have come from a given distribution, classical calibration is the preferred method. Equations generated by classical calibration will have smaller standard errors than would those generated by inverse calibration when applied to skeletons with bone lengths that are substantially different from the reference sample mean (Konigsberg et al., 1998). Although inverse calibration might be more appropriate for the narrow context of forensic Maya skeletons in Guatemala, we hope that the equations may find application in other closely related populations (e.g., forensic Ladinos in Central America, or archaeological Maya remains) which are likely to be taller than the forensic Maya sample. Thus, classical calibration is the preferred method (Konigsberg et al., 1998). We reject regression equations that show r2 ⬍ 0.85, and retain only those equations with high correlations and highly significant F values (P ⬍ 0.0001). RESULTS Table 4 contains Pearson’s correlation coefficients, together with sample sizes, for the correlation of each possible segment with maximum bone length in the forensic Maya sample, for males, for females, and for the two sexes pooled. The correlations range from essentially 0 – 0.99. Approximately half of the 238 L.E. WRIGHT AND M.A. VÁSQUEZ Fig. 1. Landmarks used to estimate bone length from the humerus (a), femur (b), tibia (c), and fibula (d). Left bones are illustrated, with ventral aspect at left, and dorsal aspect at right. 239 ESTIMATING LENGTH OF INCOMPLETE LONG BONES TABLE 4. Pearson’s r for correlation of individual segment lengths with maximum bone length Bone and Segment Humerus H0–1 H0–2 H0–3 H0–4 H0–5 H0–6 H1–2 H1–3 H1–4 H1–5 H1–6 H1–7 H2–3 H2–4 H2–5 H2–6 H2–7 H3–4 H3–5 H3–6 H3–7 H4–5 H4–6 H4–7 H5–6 H5–7 H6–7 Femur F0–1 F0–2 F0–3 F0–4 F0–5 F1–2 F1–3 F1–4 F1–5 F1–6 F2–3 F2–4 F2–5 F2–6 F3–4 F3–5 F3–6 F4–5 F4–6 F5–6 Tibia T0–1 T0–2 T0–3 T0–4 T0–5 T0–6 T1–2 T1–3 T1–4 T1–5 T1–6 T1–7 T2–3 T2–4 T2–5 T2–6 T2–7 T3–4 T3–5 T3–6 T3–7 T4–5 T4–6 Maya males Maya females Combined sex r N r N r N 0.256* 0.185* 0.355 0.827 0.981 0.991 0.048* 0.150* 0.841 0.975 0.985 0.993 0.159* 0.785 0.941 0.966 0.977 0.773 0.948 0.974 0.981 0.621 0.679 0.731 0.219* 0.390 0.284* 54 57 56 57 58 58 53 52 53 54 54 54 55 56 57 57 57 55 56 56 56 57 57 57 58 58 58 0.015* ⫺0.068* 0.140* 0.836 0.976 0.984 ⫺0.080* 0.112* 0.845 0.970 0.976 0.992 0.159* 0.873 0.959 0.965 0.976 0.806 0.952 0.953 0.974 0.012* 0.177* 0.265* 0.613 0.569 0.132* 23 23 24 23 24 24 23 23 23 23 23 23 23 23 23 23 23 23 24 24 24 23 23 23 24 24 24 0.179* 0.368 0.578 0.868 0.989 0.994 0.308 0.448 0.875 0.986 0.991 0.996 0.258* 0.830 0.969 0.980 0.986 0.789 0.967 0.979 0.986 0.694 0.748 0.789 0.371 0.513 0.294 77 80 80 80 82 82 76 75 76 80 80 77 78 79 80 80 80 78 80 77 80 76 82 80 80 82 77 0.198* 0.378 0.255* 0.798 0.984 0.358 0.213* 0.792 0.963 0.981 0.113 0.743 0.926 0.953 0.529 0.703 0.727 0.241* 0.305* 0.216* 64 64 59 64 65 62 57 62 63 64 58 62 63 64 58 58 59 63 64 65 0.491 0.679 0.544 0.839 0.986 0.559 0.370* 0.779 0.970 0.984 0.244 0.761 0.956 0.975 0.556 0.803 0.842 0.472 0.530 0.270* 32 31 32 32 32 31 32 32 32 32 31 31 31 31 32 32 32 32 32 32 0.268 0.628 0.513 0.862 0.988 0.637 0.459 0.852 0.976 0.988 0.292 0.789 0.944 0.970 0.524 0.717 0.772 0.314 0.445 0.450 96 95 91 96 97 93 89 94 95 96 89 93 94 95 90 90 91 95 96 97 0.167* 0.505 0.815 0.761 0.853 0.981 0.494 0.792 0.749 0.846 0.973 0.992 0.499 0.390 0.596 0.734 0.774 ⫺0.179* 0.048* 0.302* 0.424 0.223* 0.419 65 63 64 65 63 60 63 64 65 63 60 65 63 63 63 60 60 64 63 60 64 63 60 0.115* 0.112* 0.731 0.832 0.808 0.981 0.095 0.722 0.801 0.816 0.975 0.996 0.580 0.652 0.841 0.849 0.879 ⫺0.226* 0.009* 0.105* 0.298* 0.255* 0.614 30 30 29 30 30 29 30 29 30 30 29 30 29 30 30 29 27 29 29 28 29 30 29 0.294 0.525 0.814 0.845 0.883 0.988 0.497 0.792 0.828 0.877 0.982 0.996 0.510 0.539 0.711 0.822 0.859 ⫺0.080* 0.124* 0.457 0.559 0.224* 0.536 95 93 93 95 93 89 93 93 95 93 89 95 92 93 93 89 87 93 92 88 93 93 89 (continued) 240 L.E. WRIGHT AND M.A. VÁSQUEZ TABLE 4. (Continued) Bone and Segment T4–7 T5–6 T5–7 T6–7 Fibula P0–1 P0–2 P0–3 P1–2 P1–3 P1–4 P2–3 P2–4 P3–4 Tibia from fibula P0–1 P0–2 P0–3 P1–2 P1–3 P1–4 P2–3 P2–4 P3–4 Maya males Maya females Combined sex r N r N r N 0.580 0.257* 0.407 0.434 65 60 63 60 0.728 0.302* 0.364* 0.356* 30 29 30 29 0.692 0.361 0.517 0.561 95 89 93 89 0.079* 0.335* 0.989 0.345* 0.964 0.981 0.444 0.460 0.322* 45 46 51 40 45 45 46 46 51 0.046* 0.110* 0.993 0.165* 0.984 0.990 0.449* 0.453* 0.162* 23 24 25 23 23 23 24 24 25 0.073* 0.353 0.992 0.398 0.979 0.990 0.459 0.501 0.538 68 70 76 63 68 68 70 70 76 0.081* 0.376 0.972 0.401 0.947 0.947 0.392 0.396 0.228* 45 46 51 40 45 45 46 46 51 0.061* 0.090* 0.984 0.150* 0.972 0.966 0.462* 0.459* 0.052* 23 24 25 23 23 23 24 24 25 0.076* 0.364 0.982 0.416 0.969 0.970 0.441 0.475 0.474 68 70 76 63 68 68 70 70 76 * P ⱖ 0.01. segments show correlations greater than 0.9. In general, longer segments are better correlated with bone length than are shorter segments, as might be expected. Many shorter segments show nonsignificant correlations with bone length. Segments defined by landmarks that lie on articular architecture or secondary centers of ossification also show higher correlations than segments defined by muscle markings or nutrient foramina. For instance, segments defined in part by H4 (the most distal point on the deltoid tuberosity) show a lower correlation than several other humeral segments. Likewise, correlations are low for segments defined by femoral landmarks F3 and F4, as well as tibial segments T5, T3, and T2. Indeed, segment T3– 4 is negatively (but not significantly) correlated with tibia length. This is largely due to the variable location of landmark T4, the nutrient foramen; other tibial segments delimited by the nutrient foramen also show poor correlations with bone length. The nutrient foramen on the fibula, P2, is also fairly variable in location, as indicated by the low correlations of segments that it defines. Correlations between fibular segment lengths and tibial length are comparable to those for fibular length. We generated regression equations for segment length vs. total bone length for each of the segments that showed a statistically significant correlation at P ⱕ 0.01 (Table 4). In total, we generated some 215 regression equations: 73 for males, 57 for females, and 85 for sexes combined. Of these, 75 equations have r2 ⬎ 0.85, and highly significant F values (P ⬍ 0.0001); they are listed in Table 5. Because the articular architecture of the humerus lends itself to the definition of several landmarks, the humerus presents the largest number of successful equations, providing 11 equations each for males, females, and sexes combined. For the femur, we accepted five equations for each sex. The tibia and fibula each resulted in three equations per sex, as was also the case for the estimation of tibial length from fibular measurements. Table 5 also includes the standard errors of the estimated bone lengths. For most equations, the error is quite small, ranging between 2.86 –18.43 mm. Standard errors are smaller for the combined sex equations, because of the larger sample size on which the regressions were based. The Appendix contains 19 additional equations that have r2 between 0.6 – 0.85. Although these have significant F values (P ⬍ 0.0001), they predict bone length with less success than the equations in Table 5, as indicated by the lower r2 and larger standard errors. Because these equations are defined by muscle attachment sites, they should be applied only with extreme caution. We rejected 121 equations with r2 ⬍ 0.6, and those with standard errors ⬎20 mm. To evaluate the applicability of the regression equations in Table 5 on Guatemalan remains of unknown ethnicity, we apply the equations to complete bones from 36 male skeletons from the military base at El Chal, Petén. Figure 2 illustrates the relationship between measured bone length and bone length estimated using a selection of the Maya male regression equations. Estimates derived with all three equations are represented for the tibia and fibula. The slopes of regression lines drawn through these data range from 0.9 –1.1 for all of the equations, including those not illustrated in Figure 2. Following Feldesman and Fountain (1996), we use 241 ESTIMATING LENGTH OF INCOMPLETE LONG BONES TABLE 5. Regression lines for estimation of total bone length for forensic Maya skeletons1 Regression line A. Humerus Sexes combined (H0–5) ⫽ ⫺7.561 ⫹ 0.918 * Humerus length (H0–6) ⫽ ⫺4.805 ⫹ 0.968 * Humerus length (H1–5) ⫽ ⫺8.145 ⫹ 0.903 * Humerus length (H1–6) ⫽ ⫺4.803 ⫹ 0.951 * Humerus length (H1–7) ⫽ ⫺0.361 ⫹ 0.984 * Humerus length (H2–5) ⫽ ⫺4.255 ⫹ 0.848 * Humerus length (H2–6) ⫽ ⫺3.072 ⫹ 0.904 * Humerus length (H2–7) ⫽ 1.328 ⫹ 0.937 * Humerus length (H3–5) ⫽ ⫺7.229 ⫹ 0.808 * Humerus length (H3–6) ⫽ ⫺4.192 ⫹ 0.857 * Humerus length (H3–7) ⫽ ⫺0.191 ⫹ 0.892 * Humerus length Males (H0–5) ⫽ ⫺8.976 ⫹ 0.923 * Humerus length (H0–6) ⫽ ⫺3.008 ⫹ 0.962 * Humerus length (H1–5) ⫽ ⫺7.817 ⫹ 0.902 * Humerus length (H1–6) ⫽ 1.191 ⫹ 0.931 * Humerus length (H1–7) ⫽ 4.032 ⫹ 0.97 * Humerus length (H2–5) ⫽ ⫺10.958 ⫹ 0.87 * Humerus length (H2–6) ⫽ ⫺7.981 ⫹ 0.92 * Humerus length (H2–7) ⫽ ⫺5.936 ⫹ 0.961 * Humerus length (H3–5) ⫽ ⫺20.641 ⫹ 0.851 * Humerus length (H3–6) ⫽ ⫺14.351 ⫹ 0.89 * Humerus length (H3–7) ⫽ ⫺12.205 ⫹ 0.931 * Humerus length Females (H0–5) ⫽ 6.396 ⫹ 0.866 * Humerus length (H0–6) ⫽ ⫺7.124 ⫹ 0.977 * Humerus length (H1–5) ⫽ 5.387 ⫹ 0.852 * Humerus length (H1–6) ⫽ ⫺4.765 ⫹ 0.95 * Humerus length (H1–7) ⫽ ⫺4.223 ⫹ 0.998 * Humerus length (H2–5) ⫽ ⫺9.534 ⫹ 0.869 * Humerus length (H2–6) ⫽ ⫺19.686 ⫹ 0.967 * Humerus length (H2–7) ⫽ ⫺19.144 ⫹ 1.016 * Humerus length (H3–5) ⫽ ⫺13.503 ⫹ 0.834 * Humerus length (H3–6) ⫽ ⫺27.023 ⫹ 0.945 * Humerus length (H3–7) ⫽ ⫺19.899 ⫹ 0.968 * Humerus length B. Femur Sexes combined (F0–5) ⫽ ⫺4.716 ⫹ 0.932 * Femur max. length (F1–5) ⫽ ⫺2.152 ⫹ 0.890 * Femur max. length (F1–6) ⫽ 0.138 ⫹ 0.960 * Femur max. length (F2–5) ⫽ ⫺4.139 ⫹ 0.757 * Femur max. length (F2–6) ⫽ ⫺3.780 ⫹ 0.832 * Femur max. length Males (F0–5) ⫽ ⫺18.842 ⫹ 0.964 * Femur max. length (F1–5) ⫽ ⫺17.956 ⫹ 0.928 * Femur max. length (F1–6) ⫽ ⫺1.231 ⫹ 0.965 * Femur max. length (F2–5) ⫽ ⫺42.396 ⫹ 0.846 * Femur max. length (F2–6) ⫽ ⫺27.202 ⫹ 0.887 * Femur max. length Females (F0–5) ⫽ ⫺13.195 ⫹ 0.957 * Femur max. length (F1–5) ⫽ 9.809 ⫹ 0.861 * Femur max. length (F1–6) ⫽ 22.378 ⫹ 0.901 * Femur max. length (F2–5) ⫽ ⫺6.179 ⫹ 0.769 * Femur max. length (F2–6) ⫽ 2.260 ⫹ 0.820 * Femur max. length C. Tibia Sexes combined (T0–6) ⫽ 10.807 ⫹ 0.904 * Tibia length (T1–6) ⫽ 8.847 ⫹ 0.873 * Tibia length (T1–7) ⫽ ⫺2.770 ⫹ 0.972 * Tibia length Males (T0–6) ⫽ 7.407 ⫹ 0.914 * Tibia length (T1–6) ⫽ 2.803 ⫹ 0.890 * Tibia length (T1–7) ⫽ ⫺5.478 ⫹ 0.979 * Tibia length Females (T0–6) ⫽ 3.281 ⫹ 0.930 * Tibia length (T1–6) ⫽ ⫺5.925 ⫹ 0.923 * Tibia length (T1–7) ⫽ ⫺8.034 ⫹ 0.990 * Tibia length D. Fibula Sexes combined (P0–3) ⫽ 2.877 ⫹ 0.924 * Fibula length (P1–3) ⫽ ⫺5.000 ⫹ 0.909 * Fibula length (P1–4) ⫽ ⫺9.305 ⫹ 0.990 * Fibula length r2 N F SE 0.978 0.989 0.973 0.982 0.993 0.939 0.961 0.972 0.935 0.958 0.971 82 82 77 77 77 80 80 80 80 80 80 3,567.4 7,036.4 2,709.7 4,180.6 9,972.8 1,191.7 1,916.2 2,714.7 1,126.1 1,760.9 2,653.0 4.46 3.35 5.03 4.27 2.86 7.12 5.98 5.21 6.98 5.92 5.02 0.963 0.983 0.951 0.970 0.986 0.886 0.933 0.955 0.899 0.949 0.963 58 58 54 54 54 57 57 57 56 56 56 1,441.2 3,200.8 1,000.6 1,668.2 3,783.4 426.2 765.8 1,162.1 481.2 999.3 1,401.0 7.24 5.06 8.49 6.79 4.70 12.51 9.87 8.37 11.55 8.37 7.40 0.952 0.969 0.941 0.953 0.983 0.919 0.932 0.952 0.905 0.907 0.949 24 24 23 23 23 23 23 23 24 24 24 437.1 682.7 333.7 426.4 1,222.4 238.2 286.0 414.3 210.7 215.4 407.4 11.24 10.14 12.61 12.44 7.72 15.24 15.47 13.50 15.59 17.48 13.02 0.977 0.952 0.976 0.890 0.941 97 95 96 94 95 3,999.0 1,852.7 3,893.0 751.8 1,484.7 6.01 8.44 6.28 11.25 8.81 0.968 0.928 0.963 0.858 0.909 65 63 64 63 64 1,899.8 789.7 1,624.7 367.6 617.9 9.24 13.80 10.01 18.43 14.91 0.973 0.941 0.968 0.915 0.948 32 32 32 31 31 1,076.5 478.7 911.9 311.5 557.2 11.28 15.22 11.55 16.78 13.39 0.976 0.965 0.991 89 89 95 3,551.3 2,402.1 10,473.2 5.04 5.92 3.16 0.963 0.948 0.985 60 60 65 1,504.2 1,047.4 4,042.0 8.04 9.39 5.26 0.962 0.951 0.992 29 29 30 684.6 521.1 3,488.0 11.09 12.62 5.24 0.984 0.958 0.980 76 68 68 4,463.6 1,504.4 3,173.4 4.55 7.73 5.79 (continued) 242 L.E. WRIGHT AND M.A. VÁSQUEZ TABLE 5. (Continued) Regression line Males (P0–3) ⫽ ⫺6.583 ⫹ 0.951 * Fibula length (P1–3) ⫽ ⫺13.376 ⫹ 0.933 * Fibula length (P1–4) ⫽ ⫺7.512 ⫹ 0.984 * Fibula length Females (P0–3) ⫽ ⫺13.445 ⫹ 0.981 * Fibula length (P1–3) ⫽ ⫺23.848 ⫹ 0.974 * Fibula length (P1–4) ⫽ ⫺10.529 ⫹ 0.993 * Fibula length E. Tibia from fibula Sexes combined (P0–3) ⫽ ⫺6.216 ⫹ 0.941 * Tibia length (P1–3) ⫽ ⫺11.450 ⫹ 0.918 * Tibia length (P1–4) ⫽ ⫺13.239 ⫹ 0.990 * Tibia length Males (P0–3) ⫽ ⫺20.435 ⫹ 0.981 * Tibia length (P1–3) ⫽ ⫺24.067 ⫹ 0.954 * Tibia length (P1–4) ⫽ ⫺13.214 ⫹ 0.990 * Tibia length Females (P0–3) ⫽ ⫺9.256 ⫹ 0.953 * Tibia length (P1–3) ⫽ ⫺15.667 ⫹ 0.934 * Tibia length (P1–4) ⫽ 1.646 ⫹ 0.941 * Tibia length 1 r2 N F SE 0.977 0.930 0.962 51 45 45 2,097.4 573.4 1,089.5 7.01 13.21 10.11 0.986 0.968 0.979 25 23 23 1,569.9 626.6 998.0 7.66 12.09 9.77 0.965 0.938 0.942 76 68 68 2,033.5 1,003.4 1,067.6 6.94 9.67 10.11 0.944 0.896 0.897 51 45 45 826.5 370.3 373.7 11.65 16.99 17.56 0.967 0.945 0.932 25 23 23 684.1 362.2 290.0 11.45 15.47 17.42 All measurements, estimated bone lengths, and standard errors are in mm. max., maximum. the mean absolute deviation (MAD)1 and the mean squared error (MSE)2 for the difference between measured and estimated bone length to evaluate the accuracy of the Maya male and combined sex equations on the El Chal remains. Smaller MAD and MSE values indicate better accuracy of the equations. Table 6 contains the r2, sample size, MAD, and MSE, as well as the mean differences and probability for a paired t-test between estimated and measured lengths for each equation. Positive mean differences indicate a tendency to underestimate bone length using a given equation, while a negative mean difference indicates that the regression equation overestimates bone length. Given the large number of equations for which we calculated t-tests, we consider these differences to be significant only when P ⬍ 0.001, to minimize the likelihood of type I errors. For all bones, estimated length is highly correlated with measured length in the El Chal remains, with values of r2 above 0.84 for all of the equations in Table 5. For the humerus, mean absolute deviations are less than 3 mm and are well within the standard error of the estimating equations, and estimated humeral lengths do not differ significantly from measured length for any equation. However, all mean differences are positive, indicating slight underestimation of humerus length. Although a few of 1 Mean absolute deviation is calculated as follows (after Feldesman and Fountain, 1996): MAD ⫽ 冘 兩共estimated ⫺ measured兲兩 n 2 Mean squared error is calculated as follows (after Feldesman and Fountain, 1996): MSE ⫽ 冘 共estimated ⫺ measured兲2 n the femoral equations show statistically significant mean differences, and larger values of MAD and MSE than the humeral equations, the MAD are substantially smaller than the standard error of the estimating equations. The Maya equations perform less well on El Chal tibiae, and tend to overestimate tibia length. Tibial length estimates using the T1– 6 equations differ significantly from measured tibial lengths. For the fibula, none of the equations produced estimates that differed significantly from measured fibular length; MAD and MSE are very small. Estimates of tibial length from fibular measurements at El Chal appear to work better than estimates based on tibial measurements. Overall, the absolute differences between measured and estimated lengths of the El Chal long bones are quite small, the largest MAD being only 6.4 mm. Despite the significance of some mean differences, we believe that the forensic Maya equations estimate bone length with sufficient accuracy for the El Chal sample, and can be applied in diverse forensic contexts in Guatemala with confidence. We used the equations of Steele (1970), Steele and McKern (1969), also Steele and Bramblett (1988), and Jacobs (1992) to examine the applicability of other standards for long bone length estimation on Maya remains. We limit our comparison to Steele’s equations based on segment combinations that were shown to be highly correlated (r ⱖ 0.9) with bone length in the original studies, in order to assess equations of similar accuracy to those generated here. Because we did not measure Steele’s tibia landmark 3, direct comparison with many of the tibia equations of Steele (1970) and of Jacobs (1992) is not possible. Single segment regressions for the tibia have P ⬍ 0.9, so we compare only equations for humerus length and femur length. Although it may seem obvious that these equations should be less ESTIMATING LENGTH OF INCOMPLETE LONG BONES 243 Fig. 2. Comparison of measured and estimated bone length for El Chal male skeletons, with lengths estimated using male forensic Maya regression equations. accurate on Maya remains than the Maya equations are, this exercise is a useful demonstration of the magnitude of population differences. For the forensic Maya males and females, Figure 3 illustrates the relationship of measured humerus length to humerus length estimated using the “best” (segment 1 ⫹ 2 ⫹ 3) and “worst” (segment 2) of the humerus equations of Steele (1970) as judged by MAD and MSE. Table 7 lists the r2, sample sizes, MAD, MSE, mean differences, and paired t-test probabilities for all the regression lines that we compared. Figure 4 and Table 8 provide comparable data for the femur. On the Maya skeletons, the equations of Steele’s give high MAD and MSE, and almost all of the lengths estimated using published equations are significantly different from measured lengths. Yet several equations give estimated lengths that are strongly correlated with measured length, for which the regression lines pass through zero and have slopes close to one. For the humerus, the white male segment 1 ⫹ 2 ⫹ 3 equation appears to give a fairly good estimate of Mayan male humerus length (Fig. 3a), but the Black male 1 ⫹ 2 ⫹ 3 equation consistently overestimates humerus length, while the Mississippian male 1 ⫹ 2 ⫹ 3 equation grossly underestimates humerus length. White and Black male equations for segments 1 ⫹ 2, 2 ⫹ 3, and 2 ⫹ 3 ⫹ 4 overestimate humerus length consistently, while the Mississippian equations substantially underestimate humerus length (Table 7). These population differences are less evident for Maya females, for whom many of 244 L.E. WRIGHT AND M.A. VÁSQUEZ TABLE 6. Comparison of measured bone length to estimated bone length for male skeletons from El Chal Equation Humerus Maya male H0–5 Maya male H0–6 Maya male H1–5 Maya male H1–6 Maya male H1–7 Maya male H2–5 Maya male H2–6 Maya male H2–7 Maya male H3–5 Maya male H3–6 Maya male H3–7 Maya combined sex H0–5 Maya combined sex H0–6 Maya combined sex H1–5 Maya combined sex H1–6 Maya combined sex H1–7 Maya combined sex H2–5 Maya combined sex H2–6 Maya combined sex H2–7 Maya combined sex H3–5 Maya combined sex H3–6 Maya combined sex H3–7 Femur Maya male F0–5 Maya male F1–5 Maya male F1–6 Maya male F2–5 Maya male F2–6 Maya combined sex F0–5 Maya combined sex F1–5 Maya combined sex F1–6 Maya combined sex F2–5 Maya combined sex F2–6 Tibia Maya male T0–6 Maya male T1–6 Maya male T1–7 Maya combined sex T0–6 Maya combined sex T1–6 Maya combined sex T1–7 Fibula Maya male P0–3 Maya male P1–3 Maya male P1–4 Maya combined sex P0–3 Maya combined sex P1–3 Maya combined sex P1–4 Tibia from fibula Maya male P0–3 tibia Maya male P1–3 tibia Maya male P1–4 tibia Maya combined sex P0–3 tibia Maya combined sex P1–3 tibia Maya combined sex P1–4 tibia r2 N 0.97 0.98 0.94 0.95 0.98 0.93 0.95 0.96 0.93 0.96 0.96 0.97 0.98 0.94 0.95 0.98 0.93 0.95 0.96 0.93 0.96 0.96 32 32 30 30 30 30 30 30 31 31 31 32 32 30 30 30 30 30 30 31 31 31 0.9 0.1 1.5 0.6 0.6 1.8 1.3 1.0 0.9 0.4 0.3 0.8 0.1 1.4 0.6 0.5 1.9 1.4 1.1 1.5 0.7 0.6 0.99 0.95 0.97 0.84 0.88 0.99 0.95 0.97 0.84 0.88 34 32 32 34 34 34 32 32 34 34 0.95 0.93 0.98 0.95 0.93 0.98 MAD MSE 0.0270 0.6571 0.0172 0.2769 0.0724 0.0076 0.0232 0.0321 0.1367 0.3401 0.4527 0.0486 0.6146 0.0194 0.2475 0.1897 0.0062 0.0155 0.0296 0.0300 0.1623 0.2011 1.8 1.1 2.7 2.1 1.4 2.5 2.1 2.0 1.9 2.9 2.6 1.7 1.1 2.7 2.1 1.4 3.0 2.5 2.2 2.8 2.1 2.2 5.5 2.1 11.4 7.3 3.6 9.2 6.8 6.4 5.6 14.5 12.0 5.3 2.2 11.3 7.4 3.6 15.6 9.5 7.0 15.4 7.3 6.8 1.3 0.1 ⫺1.3 ⫺1.4 ⫺2.5 1.8 ⫺0.4 ⫺2.1 ⫺1.3 ⫺2.8 0.0006 0.9253 0.0332 0.2774 0.0238 ⬍0.0001 0.5610 0.0011 0.3275 0.0148 1.9 3.2 5.4 5.9 5.2 2.1 3.2 3.1 6.4 5.5 5.6 16.0 38.0 52.7 42.0 7.0 16.6 14.6 60.8 46.4 29 29 29 29 29 29 ⫺2.4 ⫺4.0 ⫺1.2 ⫺2.4 ⫺4.0 ⫺1.1 0.0017 ⬍0.0001 0.0067 0.0021 0.0001 0.0128 3.7 5.0 2.0 3.7 5.1 1.9 18.8 36.4 6.1 19.1 37.3 6.0 0.99 0.95 0.96 0.99 0.95 0.96 15 12 12 15 12 12 ⫺1.8 ⫺1.3 0.5 ⫺1.7 ⫺1.2 0.8 0.0025 0.2112 0.5565 0.0066 0.2480 0.3756 1.8 3.2 2.6 1.9 2.6 2.7 6.3 12.5 8.4 6.8 10.2 8.8 0.94 0.91 0.93 0.94 0.91 0.93 13 10 10 13 10 10 ⫺3.6 ⫺2.9 ⫺1.3 ⫺3.4 ⫺2.8 ⫺1.4 0.0067 0.0830 0.3220 0.0136 0.1021 0.3134 4.8 5.0 3.4 4.8 4.8 3.4 27.9 29.3 16.5 27.0 27.7 16.5 the equations give comparably good results, but tend toward slightly overestimating humerus length (Fig. 3b, Table 7). All of the equations for segment 2 overestimate humerus length substantially for both Maya males and females, especially for individuals with shorter humeri (Fig. 3c,d). The segment 2 equations stand out as the least successful predictors of humerus length on the Mayan remains. Figure 4 illustrates the comparison of measured maximum femur length with maximum femur length, estimated using the 1 ⫹ 2 and 1 ⫹ 2 ⫹ 3 equations of Steele (1970), Steele and Bramblett (1988), and Jacobs (1992) on the forensic Maya sample. Table 8 lists the r2, MAD, MSE, mean differ- Mean difference P ences, and paired t-test probabilities for all of the femoral regression lines that we compared. The equations for segments 1 ⫹ 2 perform very poorly on Maya femora. They show very low correlations, low slopes, large MAD, and large MSE. The white and Black 1 ⫹ 2 equations dramatically underestimate maximum length for most femora, while the Mississippian and European equations overestimate maximum length. The nonsignificant mean difference and small MAD and MSE shown for males using the Mississippian 1 ⫹ 2 equation is misleading; it is an artifact of the low slope and sample mean. The equations based on femoral segments 1 ⫹ 2 ⫹ 3 perform much better. Although they too show statistically ESTIMATING LENGTH OF INCOMPLETE LONG BONES 245 Fig. 3. Comparison of measured and estimated humerus length for forensic Maya skeletons, with lengths estimated using regression equations of Steele (1970) and Steele and Bramblett (1988, p. 166). significant mean differences from measured femur length, these are smaller than for most other equations. For both Maya males and females, the white, Black, and European equations are coincident, and slightly overestimate femoral length, while the Mississippian 1 ⫹ 2 ⫹ 3 equations underestimate femoral length consistently. Femoral length estimates based on equations using segments 2 ⫹ 3 and 2 ⫹ 3 ⫹ 4 also show high correlations with measured femur length and small MAD. Yet the equations based on segments 2 ⫹ 3 tend to overestimate femur length slightly, while the equations based on segments 2 ⫹ 3 ⫹ 4 tend to underestimate femur length. The Mississippian male 2 ⫹ 3 and Mississippian female 1 ⫹ 2 ⫹ 3 equations perform well, but in general, the Mississippian equations show larger deviations than the white, Black, or European equations. The inaccuracies in estimating bone length using the published equations must be due to different proportions of the bone in the reference populations from which the published equations were derived. Table 9 contains the mean proportions of each of Steele’s segments in the Maya forensic sample, together with the mean proportions of the segments in both the reference samples of Steele (1970), Steele and McKern (1969), and in the prehistoric European sample of Jacobs (1992). For the humerus, segment proportions are quite similar among the Terry whites, Blacks, and the Maya. The Mississippians have a slightly longer humerus segment 3. However, the generally similar proportions imply that the in- 246 L.E. WRIGHT AND M.A. VÁSQUEZ TABLE 7. Comparison of measured humerus length to estimated humerus length of Maya skeletons, using equations of Steele (1970) and Steele and Bramblett (1988, p. 166) Equation Males White male 2 White male 1 ⫹ 2 White male 2 ⫹ 3 White male 1 ⫹ 2 ⫹ 3 White male 2 ⫹ 3 ⫹ 4 Black male 2 Black male 1 ⫹ 2 Black male 2 ⫹ 3 Black male 1 ⫹ 2 ⫹ 3 Black male 2 ⫹ 3 ⫹ 4 Mississippian male 2 Mississippian male 1 ⫹ 2 Mississippian male 2 ⫹ 3 Mississippian male 1 ⫹ 2 ⫹ 3 Mississippian male 2 ⫹ 3 ⫹ 4 Females White female 2 White female 1 ⫹ 2 White female 2 ⫹ 3 White female 1 ⫹ 2 ⫹ 3 White female 2 ⫹ 3 ⫹ 4 Black female 2 Black female 1 ⫹ 2 Black female 2 ⫹ 3 Black female 1 ⫹ 2 ⫹ 3 Black female 2 ⫹ 3 ⫹ 4 Mississippian female 2 Mississippian female 1 ⫹ 2 Mississippian female 2 ⫹ 3 Mississippian female 1 ⫹ 2 ⫹ 3 Mississippian female 2 ⫹ 3 ⫹ 4 r2 N Mean difference P MAD MSE 0.899 0.962 0.943 0.984 0.956 0.899 0.962 0.941 0.984 0.955 0.899 0.964 0.949 0.984 0.968 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 ⫺10.6 ⫺4.8 ⫺5.5 ⫺0.9 ⫺2.9 ⫺12.4 ⫺6.0 ⫺7.0 ⫺4.4 ⫺5.8 ⫺10.8 10.0 16.4 7.2 7.4 ⬍0.0001 ⬍0.0001 ⬍0.0001 0.0003 ⬍0.0001 ⬍0.0001 ⬍0.0001 ⬍0.0001 ⬍0.0001 ⬍0.0001 ⬍0.0001 ⬍0.0001 ⬍0.0001 ⬍0.0001 ⬍0.0001 15.5 10.0 10.7 6.7 8.2 17.2 11.1 12.0 9.5 10.8 15.6 14.8 21.1 12.2 12.6 1,720.2 1,622.5 1,633.4 1,597.0 1,609.4 1,760.1 1,635.7 1,650.8 1,615.5 1,633.5 1,723.5 1,695.4 1,862.7 1,647.0 1,652.3 0.906 0.924 0.906 0.980 0.968 0.906 0.939 0.906 0.968 0.945 0.906 0.931 0.904 0.968 0.943 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 ⫺10.8 0.7 ⫺3.2 2.1 ⫺3.6 ⫺6.5 ⫺1.1 ⫺5.3 ⫺2.4 ⫺2.5 ⫺11.8 ⫺3.7 0.1 1.0 1.7 ⬍0.0001 0.3160 0.0004 0.0003 0.0015 ⬍0.0001 0.0820 ⬍0.0001 ⬍0.0001 0.0004 ⬍0.0001 ⬍0.0001 0.9066 0.0324 0.0125 10.3 2.4 4.1 2.2 2.6 6.4 2.3 5.8 2.7 2.9 11.3 4.1 2.5 1.8 2.3 125.2 10.5 23.1 9.5 13.3 53.9 9.1 41.5 9.8 13.9 149.4 22.9 13.2 5.0 10.9 accuracies of the humerus equations on Maya remains are due more to the absolute difference in bone size than in the relative length of the segments. For humerus segment 2 in both sexes, this is suggested by the lower slope of the regression (Fig. 3c,d), which illustrates that the equation overestimates the length of shorter humeri more than it does for longer humeri. This may simply be due to the fact that the mean length of Maya humeri is substantially less than the mean length of humeri in any of the reference series. Table 9 illustrates that Maya femora have a proportionately shorter segment 2 and longer segment 3 than the white, Black, and Mississippian series, but the prehistoric European series have a still shorter segment 2 and longer segment 3 than the Maya. These differences undoubtedly account for the poor performance of the segment 1 ⫹ 2 equations on Maya remains. Equations that incorporate both segments 2 and 3 perform better because these differences of proportion are minimized by including both in the calculation. Although it was not possible to estimate Maya tibia length using the equations of Steele (1970) or Jacobs (1992) because we did not measure landmark 3, the proportions of the segments can be compared by adding segments 2 and 3 together. For both males and females, Maya tibia have shorter segments 1 and 2 ⫹ 3, but a longer segment 4 than in other reference series. This suggests that the tibia equa- tions would also provide unreliable estimates of Maya tibia length. DISCUSSION The results of this study indicate that long bone length can be estimated from incomplete bones with considerable accuracy using longitudinal measurements, and provides strong support for Steele and McKern’s (1969) methodological approach. Standard errors for the forensic Maya regression equations defined here are slightly larger than those of the better-performing equations of Steele (1970), in part due to our use of classical calibration rather than inverse regression. The broader standard errors for the Maya female equations are due to the smaller sample size of Maya females that we were able to measure. Like Steele (1970), also Steele and McKern (1969), we found that landmarks defined by joint architecture and secondary centers of ossification were easier to identify than those identified by muscle markings. Moreover, segments defined by articular landmarks are more highly correlated with bone length than are segments defined by muscle attachment sites. Intrapopulation variation in muscle mass and activity presumably accounts for this lower correlation. Therefore, we believe that many of the regression equations that use these landmarks cannot be used with confidence. Where possible, we recommend that bone length estimates be ESTIMATING LENGTH OF INCOMPLETE LONG BONES 247 Fig. 4. Comparison of measured and estimated femur length for forensic Maya skeletons, with lengths estimated using regression equations of Steele (1970), Steele and Bramblett (1988, p. 230), and Jacobs (1992). limited to those equations presented in Table 5. The regression equations given in the Appendix should be used only with extreme caution; investigators who wish to apply them should first evaluate the accuracy of the equations on more complete skeletons from the same skeletal series. Critiques of Steele’s method have focused on the difficulty that some investigators have had in identifying landmarks, and on inaccuracy in bone length estimates (Bass, 1987; Brooks et al., 1990; Simmons et al., 1990). We discarded only one of Steele’s points that we could not identify with confidence (tibia landmark 3). We suspect that the difficulties other investigators may have encountered with the method center on locating those points that are defined by muscle attachment sites, some of which we also found difficult to pinpoint. Because these points also show greater intrapopulation variation than articular points, uncertainty in identifying these points probably contributes to perceptions of inaccuracy of the equations overall. Together these factors might contribute to an exaggerated bias against the method. For instance, Simmons et al. (1990; p 629) cite comments by Steele (1970; p 87) regarding a landmark that he discarded,3 to support their claim 3 This landmark, defined as the point of narrowest anterior-posterior diameter of the humerus proximal to the deltoid tuberosity, was proposed by Steele and McKern (1969), but discarded by Steele (1970). Modified versions of the 1969 regression equations for Mississippians that do not use this landmark were published by Steele and Bramblett (1988, p. 166). 248 L.E. WRIGHT AND M.A. VÁSQUEZ TABLE 8. Comparison of measured femur length to estimated femur length on Maya skeletons, using equations of Steele (1970), Steele and Bramblett (1988, p. 230), and Jacobs (1992) Equation Males White male 1 ⫹ 2 White male 2 ⫹ 3 White male 1 ⫹ 2 ⫹ 3 White male 2 ⫹ 3 ⫹ 4 Black male 1 ⫹ 2 Black male 2 ⫹ 3 Black male 1 ⫹ 2 ⫹ 3 Black male 2 ⫹ 3 ⫹ 4 Mississippian male 1 ⫹ 2 Mississippian male 2 ⫹ 3 Mississippian male 1 ⫹ 2 ⫹ 3 Mississippian male 2 ⫹ 3 ⫹ 4 European male 1 ⫹ 2 European male 2 ⫹ 3 European male 1 ⫹ 2 ⫹ 3 European male 2 ⫹ 3 ⫹ 4 Females White female 1 ⫹ 2 White female 2 ⫹ 3 White female 1 ⫹ 2 ⫹ 3 White female 2 ⫹ 3 ⫹ 4 Black female 1 ⫹ 2 Black female 2 ⫹ 3 Black female 1 ⫹ 2 ⫹ 3 Black female 2 ⫹ 3 ⫹ 4 Mississippian female 1 ⫹ 2 Mississippian female 2 ⫹ 3 Mississippian female 1 ⫹ 2 ⫹ 3 Mississippian female 2 ⫹ 3 ⫹ 4 European female 1 ⫹ 2 European female 2 ⫹ 3 European female 1 ⫹ 2 ⫹ 3 European female 2 ⫹ 3 ⫹ 4 r2 N Mean difference P MAD MSE 0.640 0.852 0.966 0.899 0.640 0.856 0.968 0.899 0.623 0.861 0.968 0.908 0.554 0.861 0.968 0.909 62 61 61 61 62 61 61 61 62 61 61 61 62 61 61 61 25.2 ⫺9.2 ⫺5.5 0.5 39.7 ⫺10.8 ⫺4.1 3.7 ⫺1.4 ⫺4.0 13.1 14.7 ⫺31.5 ⫺4.5 ⫺6.1 5.9 ⬍0.0001 ⬍0.0001 ⬍0.0001 0.5420 ⬍0.0001 ⬍0.0001 ⬍0.0001 ⬍0.0001 0.3767 0.0003 ⬍0.0001 ⬍0.0001 ⬍0.0001 0.0001 ⬍0.0001 ⬍0.0001 25.1 16.3 11.9 11.7 39.2 17.3 10.7 12.5 9.8 13.4 19.1 20.7 30.8 13.7 12.4 13.5 780.3 2,899.9 2,791.7 2,791.5 1,753.2 2,924.2 2,777.9 2,802.9 146.3 2,828.8 2,926.6 2,992.5 1,130.9 2,837.8 2,798.2 2,824.6 0.736 0.908 0.968 0.939 0.769 0.914 0.966 0.924 0.783 0.914 0.968 0.943 0.787 0.912 0.969 0.939 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 30.1 ⫺0.7 ⫺4.2 3.5 24.6 ⫺10.9 ⫺7.1 7.9 ⫺13.8 ⫺10.0 3.4 1.0 ⫺31.8 ⫺2.6 ⫺3.1 4.6 ⬍0.0001 0.5505 ⬍0.0001 0.0006 ⬍0.0001 ⬍0.0001 ⬍0.0001 ⬍0.0001 ⬍0.0001 ⬍0.0001 ⬍0.0001 0.3702 ⬍0.0001 0.0237 0.0001 ⬍0.0001 30.1 4.6 4.7 5.0 24.6 10.9 7.2 8.1 14.7 10.2 3.9 4.8 31.8 4.9 4.2 5.5 1,008.8 36.2 29.3 36.6 693.5 153.0 64.1 91.2 281.3 140.9 24.0 33.9 1,100.5 43.6 24.7 45.2 that the landmarks he retained were impossible to locate. Although Simmons et al. (1990) suggested that the method is “plagued” by difficulty and therefore is seldom used, recall that Brooks et al. (1990) surveyed a small number of forensic investigators who do use the method and who found it to have considerable accuracy. In contrast, we found the points defined by articular architecture and secondary centers of ossification to be extremely easy to identify. Greater reliance on these articular landmarks, coupled with the linear regression approach used here (vs. Steele’s multiple regression approach) may improve the ease of use and accuracy of longitudinal measurements for bone length estimation. By using the distance between pairs of landmarks in linear regressions on bone length, we eliminate error that would be introduced by imprecision in locating intermediate landmarks, as well as error due to inter- and intrapopulation variation in the location of intermediate landmarks. That we found estimates of Maya bone length using the equations of Steele (1970) to be more accurate for equations based on longer segments than those based on shorter segments provides some support for this argument. By discarding landmarks defined by muscle attachment sites, however, the length of bone required to obtain an estimate becomes larger. Our equations demand that the entire diaphysis as well as some marginal articular bone be present in order to apply the equations. Thus, they will not permit estimation of extremely fragmentary remains. Nonetheless, the equations can be used to estimate the length of bones with broken or eroded articular surfaces, such as femora that lack the head and neck or condyles. Standard errors of the equations described in this study are smaller than those obtained by Simmons et al. (1990) for transverse measurements of small fragments of the femur. Holland (1992) generated equations that relate proximal tibial measures to stature. Standard errors for his equations are comparable to the regressions of Simmons et al. (1990) against stature for femora, which indicates that regression of proximal tibia measures on tibia length would also be larger than the standard errors obtained here for longitudinal measurements. Thus, if sufficient length of the diaphysis is available, estimates of bone length from longitudinal measurements should be superior to those from transverse measurements. Of course, the longitudinal and transverse measurement regressions should be seen as complementary methods. We observed that estimates of bone length using the equations of Steele (1970), also Steele and Bramblett (1988), and Jacobs (1992) for the Maya remains often differed significantly from measured bone lengths. This finding confirms the observation by Jacobs (1992) that significant interpopulation differ- 249 ESTIMATING LENGTH OF INCOMPLETE LONG BONES TABLE 9. Proportions of long bone segments in forensic Maya sample, and in comparative series1 Bone Males Humerus Femur Tibia Females Humerus Femur Tibia Forensic Maya Proportion of total bone length Segment (Steele, 1970) Segment (this study) Mean SD N Maya Terry White Terry Black Mississippian European Total length 1 2 3 4 Max length 1 2 3 4 Total length 1 2⫹3 4 5 Total length H0–3 H3–5 H5–6 H6–7 Max length F0–2 F2–4 F4–5 F5–6 Max length T6–7 T2–6 T1–2 T0–1 297.4 32.8 232.4 17.7 14.2 417.6 72.7 207.8 102.2 32.3 340.9 22.1 202.2 104.7 12.5 14.0 2.7 12.0 2.4 1.8 20.0 6.2 19.0 13.0 3.6 18.0 3.5 14.0 11.0 2.2 58 56 56 58 58 66 64 62 63 65 65 60 60 63 65 1.00 0.11 0.78 0.06 0.05 1.00 0.17 0.50 0.24 0.08 1.00 0.06 0.59 0.31 0.04 1.00 0.11 0.77 0.06 0.05 1.00 0.17 0.59 0.15 0.08 1.00 0.09 0.62 0.27 0.04 1.00 0.11 0.78 0.06 0.05 1.00 0.17 0.59 0.16 0.08 1.00 0.08 0.62 0.28 0.04 1.00 0.10 0.77 0.08 0.04 1.00 0.16 0.56 0.19 0.08 1.00 0.07 0.65 0.25 0.04 1.00 0.16 0.47 0.28 0.09 1.00 0.07 0.62 0.26 0.04 Total length 1 2 3 4 Max length 1 2 3 4 Max length 1 2⫹3 4 5 Total length H0–3 H3–5 H5–6 H6–7 Max length F0–2 F2–4 F4–5 F5–6 Max length T6–7 T2–6 T1–2 T0–1 271.2 28.5 212.6 16.5 13.5 386.3 65.3 191.3 99.4 28.2 312.3 18.6 185.1 97.2 11.2 12.0 2.7 10.0 2.2 2.1 21.0 5.2 15.0 11.0 3.2 16.0 3.2 17.0 8.6 1.5 24 24 24 24 24 32 31 31 32 32 30 29 29 30 30 1.00 0.11 0.78 0.06 0.05 1.00 0.17 0.50 0.26 0.07 1.00 0.06 0.59 0.31 0.04 1.00 0.11 0.78 0.06 0.05 1.00 0.17 0.60 0.15 0.08 1.00 0.08 0.64 0.25 0.04 1.00 0.11 0.78 0.06 0.05 1.00 0.16 0.60 0.16 0.08 1.00 0.08 0.65 0.26 0.04 1.00 0.10 0.77 0.08 0.04 1.00 0.16 0.57 0.20 0.08 1.00 0.07 0.66 0.25 0.04 1.00 0.16 0.46 0.30 0.08 1.00 0.07 0.62 0.26 0.04 1 Data for comparative series are from Steele (1970), Steele and McKern (1969), and Jacobs (1992). All measurements are in mm. Max, maximum. ences exist in the proportions of segments. Much as he found that several of Steele’s (1970) White equations performed poorly on Mesolithic Europeans, we observed little correlation between the success of Steele’s equations and any a priori expectations about affinity between the reference populations and our Maya sample. The Mississippian equations often estimate Maya bone length with greater error than the Black and white equations, despite common Native American ancestry, shared maize agricultural subsistence practices, and a similarly warm climate. Jacobs (1992) proposed that differences in muscle activity may explain the variation, and suggested that data be collected on populations with diverse behavioral adaptations. Indeed, we note that the most evident differences in segment proportion illustrated in Table 9 are those determined by the position of landmarks defined by muscle attachment points. However, if we limit bone length estimation to those equations defined in Table 5, which exclude most points defined by muscle action, these population differences in proportion may be less problematic. While activity differences likely contribute to these interpopulation differences, it may be significant to note that most of the equations overestimate bone length when applied to Maya remains. The diminutive stature of Maya skeletons is perhaps too far removed from the mean stature of the series on which these equations were derived to permit an unbiased estimate of bone length. We suggest that bone proportions are also significantly constrained by body size. Regardless of whether the segment proportion differences are due to activity differences and/or allometric considerations, it is clear that estimates of bone length of Maya remains cannot be made with confidence using the majority of the previously published equations. The equations we have provided here are best suited to the specific forensic context of Guatemala, and presumably adjacent Central American nations. The success of the El Chal tests demonstrates that the equations perform adequately on the modern Guatemalan population, even when Maya ethnicity is not known. We hope that they will aid stature estimation and the identification of forensic remains in the ongoing human rights investigations in this region. We also expect that the equations will perform well on archaeological human remains of the Maya and other Mesoamerican cultures, many of whom were taller than are the modern Maya. Additional research might be directed at collecting measurements from other populations, in order to determine whether interpopulation differences in bone proportions hinder estimation of bone length using the forensic Maya equations, and whether it would 250 L.E. WRIGHT AND M.A. VÁSQUEZ be necessary to generate regression equations for each population to be studied. Before applying our equations to distant populations, and especially to significantly taller populations, we recommend that osteologists verify the results with any complete long bones that might be available for the population in question. Complementary research on skeletal series with differing behavioral adaptations and mean statures is needed to provide researchers with a variety of options for estimating the length of fragmentary long bones. ACKNOWLEDGMENTS We thank the members of the Área de Exumaciones and the Fundación de Antropologı́a Forense de Guatemala for their kind collaboration in our study. This research would have been impossible without the various institutions that have supported their forensic work, both financially and politically. We thank Miguel Angel Morales for assistance with data recording. Last, but not least, we thank Gentry Steele for inspiration, for his helpful suggestions during the research, and for his comments on the manuscript. LITERATURE CITED Arzobispado de Guatemala, Oficina de Derechos Humanos. 1998. Guatemala: nunca más. Informe Proyecto Interdiocesano de Recuperación de la Memoria Historica. Guatemala: ODHAG. Bass WM. 1987. Human osteology: a laboratory and field manual of the human skeleton. Columbia, MO: Missouri Archaeological Society. Bogin B, MacVean RB. 1984. Growth status of non-agrarian, semi-urban living indians in Guatemala. Hum Biol 56:527–538. Brooks S, Steele DG, Brooks RH. 1990. Formulae for stature estimation on incomplete long bones: a survey of their reliability. J Forensic Med 6:167–170. Carmack RM. 1988. Harvest of violence. The Maya Indians and the Guatemalan crisis. Norman, OK: University of Oklahoma Press. Danforth ME. 1994. Stature change in prehistoric Maya of the Southern Lowlands. Latin Am Antiq 5:206 –211. Equipo de Antropologı́a Forense de Guatemala. 1997. Las masacres en Rabinal: estudio histórico antropológico de las masacres de Plan de Sánchez, Chichupac y Rı́o Negro. Guatemala City: Editorial Serviprensa. Feldesman MR, Fountain RL. 1996. “Race” specificity and the femur/stature ratio. Am J Phys Anthropol 100:207–225. Furbee L, Thomas JS, Lynch HK, Benfer RA. 1988. Tojolabal Maya population response to stress. Geosci Man 26:17–27. Genovés S. 1967. Porportionality of the long bones and their relation to stature among Mesoamericans. Am J Phys Anthropol 26:67–78. Hens SM, Konigsberg LW, Jungers WL. 2000. Estimating stature in fossil hominids: which regression model and reference sample to use? J Hum Evol 38:767–784. Holland TD. 1992. Estimation of adult stature from fragmentary tibias. J Forensic Sci 37:1223–1229. Jacobs K. 1992. Estimating femur and tibia length from fragmentary bones: an evaluation of Steele’s (1970) method using a prehistoric European sample. Am J Phys Anthropol 89:333– 345. Konigsberg LW, Frankenberg SR, Walker RB. 1997. Regress what on what? Paleodemographic age estimation as a calibration problem. In: Paine RR, editor. Integrating archaeological demography: multidisciplinary approaches to prehistoric population. Carbondale, IL: Southern Illinois University. p 64 – 88. Konigsberg LW, Hens SM, Jantz LM, Jungers WL. 1998. Stature estimation and calibration: Bayesian and maximum likelihood perspectives in physical anthropology. Yrbk Phys Anthropol 41:65–92. Martorell R. 1995. Results and implications of the INCAP follow-up study. J Nutr 125:1127–1138. Martorell R, Kettel Khan L, Schroeder DG. 1994. Reversibility of stunting: epidemiological findings in children from developing countries. Eur J Clin Nutr 48:45–57. McCullough JM. 1982. Secular trend for stature in adult male Yucatec Maya to 1968. Am J Phys Anthropol 58:221–225. Müller G. 1935. Zur bestimmung der Länge beschädigter Extremitatenknochen. Anthropol Anzeig 12:70 –72. Simmons T, Jantz RL, Bass WM. 1990. Stature estimation from fragmentary femora: a revision of the Steele method. J Forensic Sci 35:628 – 636. Steele DG. 1970. Estimation of stature from fragments of long limb bones. In: Stewart TD, editor. Personal identification in mass disasters. Washington, DC: Smithsonian Institution. p 85–97. Steele DG, Bramblett CA. 1988. The anatomy and biology of the human skeleton. College Station, TX: Texas A&M University Press. Steele DG, McKern TW. 1969. A method for assessment of maximum long bone length and living stature from fragmentary long bones. Am J Phys Anthropol 31:215–228. Whittington SL. 1989. Characteristics of demography and disease in low status Maya from Classic Period Copan, Honduras. Ph.D. dissertation, Pennsylvania State University. Wright LE. 1994. The sacrifice of the earth? Diet, health, and inequality in the Pasión Maya lowlands. Ph.D. dissertation, University of Chicago. 251 ESTIMATING LENGTH OF INCOMPLETE LONG BONES APPENDIX. Additional regression lines for estimation of total bone length, derived from forensic Maya skeletons1 Regression line A. Humerus (sexes combined) (H0–4) ⫽ ⫺16.558 ⫹ 0.577 * Humerus length (H1–4) ⫽ ⫺17.334 ⫹ 0.562 * Humerus length (H2–4) ⫽ ⫺18.046 ⫹ 0.524 * Humerus length (H3–4) ⫽ ⫺17.743 ⫹ 0.472 * Humerus length (H4–7) ⫽ 16.558 ⫹ 0.423 * Humerus length B. Humerus (males) (H0–4) ⫽ ⫺17.746 ⫹ 0.579 * Humerus length (H1–4) ⫽ ⫺17.525 ⫹ 0.561 * Humerus length (H2–4) ⫽ ⫺28.425 ⫹ 0.556 * Humerus length (H3–4) ⫽ ⫺30.899 ⫹ 0.513 * Humerus length C. Femur (sexes combined) (F0–4) ⫽ ⫺42.021 ⫹ 0.774 * Femur max. length (F1–4) ⫽ ⫺39.495 ⫹ 0.734 * Femur max. length (F2–4) ⫽ ⫺43.683 ⫹ 0.606 * Femur max. length D. Tibia (sexes combined) (T0–3) ⫽ ⫺21.602 ⫹ 0.675 * Tibia length (T0–4) ⫽ 17.393 ⫹ 0.623 * Tibia length (T0–5) ⫽ 1.196 ⫹ 0.757 * Tibia length (T1–4) ⫽ 14.623 ⫹ 0.594 * Tibia length (T1–5) ⫽ ⫺1.561 ⫹ 0.729 * Tibia length (T2–6) ⫽ ⫺16.913 ⫹ 0.644 * Tibia length (T2–7) ⫽ ⫺25.335 ⫹ 0.732 * Tibia length 1 r2 N F SE 0.754 0.766 0.688 0.623 0.622 80 76 79 78 80 239.1 242.3 170.1 125.4 128.4 10.83 10.48 11.64 12.23 10.83 0.685 0.707 0.616 0.598 57 57 56 55 119.5 122.8 86.6 78.0 15.78 15.09 17.77 17.20 0.743 0.726 0.623 96 94 93 271.2 243.6 150.2 19.16 19.16 20.11 0.663 0.714 0.780 0.686 0.769 0.675 0.738 93 95 93 95 93 89 87 178.8 232.2 322.7 202.9 303.8 181.1 239.5 16.80 13.58 14.01 13.87 13.91 15.90 15.76 All measurements, estimated bone lengths, and standard errors are in mm. Max., maximum.

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