ANALYSIS OF THE COVARIANCE STRUCTURE OF DIGITAL RIDGE COUNTS IN THE OFFSPRING OF MONOZYGOTIC TWINS

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1 Copyright by the Genetics Society of America ANALYSIS OF THE COVARIANCE STRUCTURE OF DIGITAL RIDGE COUNTS IN THE OFFSPRING OF MONOZYGOTIC TWINS RITA M. CANTOR, WALTER E. NANCE, LINDON J. EAVES, PHYLLIS M. WINTER AND MARSHA M. BLANCHARD Department of Human Genetics, Medical College of Virginia, Richmond, Virginia Manuscript received August 31, 1982 Revised copy accepted November 17,1982 ABSTRACT Improved methods for analysis of covariance structures now permit the rigorous testing of multivariate genetic hypotheses. Using J~RESKOG S Lisrel IV computer program we have conducted a confirmatory factor analysis of dermal ridge counts on the individual fingers of 509 offspring of 107 monozygotic twin pairs. Prior to the initiation of the model-fitting procedure, the sex-adjusted ridge counts for the offspring of male and female twins were partitioned by a multivariate nested analysis of variance yielding five 10 x 10 variance-covariance matrices containing a total of 275 distinctly observed parameters with which to estimate latent sources of genetic and environmental variation and test hypotheses about the factor structure of those latent causes. To provide an adequate explanation for the observed patterns of covariation, it was necessary to include additive genetic, random environmental, epistatic and maternal effects in the model and a structure for the additive genetic effects which included a general factor and allowed for hand assymmetry and finger symmetry. The results illustrate the value of these methods for the analysis of interrelated metric traits. HE dermatoglyphic patterns of the hands and feet are formed during early T fetal life and remain essentially unchanged thereafter. Consequently, these traits are especially useful for analysis of genetic and environmental factors that influence prenatal development. Since GALTON S time it has been recognized that many dermatoglyphic variables exhibit a high degree of genetic determination (GALTON 1892). In the early 1950s HOLT (1951, 1952, 1955) proposed that total ridge count, a quantity determined by summing the largest of the ulnar and radial counts for each finger, could be regarded as a polygenic trait with a high heritability. Her inferences were based on the pattern of correlations observed among twins, siblings and the parents and offspring in nuclear families. Although HOLT could find no significant difference between the motheroffspring and father-offspring correlations, subsequent analyses of data from maternal and paternal half-siblings have raised the possibility of a maternal effect on total ridge count (NANCE 1976). From univariate analyses this appears to be confined primarily to the ridge counts of the thumbs and fifth digits (REED et al. 1979). HOLT recognized that the ridge counts of individual fingers exhibit Current address: Division of Medical Genetics, Harbor UCLA-Medical Center, Torrance, California Genetics March

2 496 R. M. CANTOR ET AL. a complex pattern of intercorrelations not fully accounted for by her treatment of the data. Subsequent refinements have been achieved by applying the methods of exploratory factor analysis to data on individual ridge counts. These efforts have led to the recognition of a strong general genetic factor influencing all ten finger ridge counts and the existence of high correlations between homologous fingers of the two hands and adjacent fingers on the same hand (ROBERTS and COOPE 1975; ROSTRON 1977; NANCE et al. 1974; IAGOLNITZER 1978; SIERVOGEL, ROCHE and ROCHE 1979; SINCH 1979). In general, the thumbs and fifth digits have tended to show lower correlations with other fingers, suggesting a higher degree of independent determination. Some of these studies have dealt with the phenotypic correlations between unrelated individuals or population groups and, therefore, offer little potential for resolving the latent genetic and environmental causes for the observed phenotypic intercorrelations (ROBERTS and COOPE 1975; JANTZ and HAWKINSON 1980; JANTZ et al. 1982). Other studies have incorporated contrasts between related individuals such as monozygotic (MZ) and dizygotic twins but have not employed methods that permit tests of the resulting genetic and environmental factor structures (NANCE et al. 1974). However, using data on twins, MARTIN and EAVES (1977) have shown how JORESKOG S (1973) methods for the analysis of covariance structures can be applied to test specific multivariate models for the genetic and environmental determination of interrelated metric traits. In this paper we have extended their techniques to exploit the unique relationships that exist among the offspring of MZ twins. Using a widely available computer program Lisrel IV we have tested the assumptions underlying HOLT S model for the determination of ridge counts and have proposed several biologically plausible extensions to explain the latent sources of genetic and environmental variation in individual finger ridge counts, as well as the factor structure of those sources. GENETIC MODEL In Table 1, the expected values for the among half-sibship, between sibship nested within half-sibship, and within-sibship variance-covariance component matrices (E,) derived from two multivariate nested analyses of variance on data from the offspring of male and female MZ twins are expressed in terms of their latent sources of variation (NANCE and COREY 1976). In this model, VA, VAA and Vo designate the covariance matrices of additive genetic, epistatic and dominance effects, respectively, whereas VEW and VM refer to random and maternal environmental effects. Each latent source can be partitioned into one or more constituent factors of specified structure, whereras the Xs may in turn be aggregated to specify the expected values of the observed mean square and cross product matrices obtained from the multivariate nested analyses of variance as shown in Table 2. Since typical half-sib data sets are not balanced, the weighting coefficients, bl-bs, must be calculated from the distribution of sibship and half-sibship sizes (SNEDECOR 1961). We have elected to use the variance-covariance matrix derived from a pooled analysis for the within sibship matrix since the expected values for this matrix are the same for the offspring in male and female twins.

3 ANALYSIS OF DIGITAL RIDGE COUNTS 497 TABLE 1 Expected contributions of latent sources of variation to variance-covariance component matrices derived from nested multivariate analyses of variance of MZ-half-sibship data Latent sources of variation Genetic Environmental Variance component matrix V A VAA vu VM VEW EA&!4 'ne ZBd y4 %6 '/4 1 0 XW 'h % Yi 0 1 EB??4 356 '/4 0 0 EA? $2 % EA, XB, Zw = among half-sibship, between sibship nested within half-sibship, and within-sibship variance component matrices, respectively, for male (6) and female (P) twin kinships; VA, VU, VD = additive, epistatic and dominance variance-covariance matrices, respectively. VM, VE = maternal and within family environmental variance-covariance matrices, respectively. TABLE 2 Expected values of mean square and cross product matrices derived from multivariate nested analysis of variance of MZ-half-sibship data expressed in terms of constituent variance component matrices Matrix of mean square Expected value of mean square matrix d = male twin kinships; 0 = female twin kinships; A = among half-sibship component or mean square matrix; B = between sibship nested within half-sibship component or mean square matrix; W = within full-sibship mean square matrix; bl, bz,..., be = weighting coefficients determined from distribution of family sizes. In our application, using the Lisrel IV program (JORESKOG and SORBOM 1979), models were fitted jointly to the five mean sums of squares and cross products matrices. The expected values for each of the five matrices are given by where A is a y x p matrix containing the loadings of factors on the y variables, with each source of latent variation included in the model contributing at least one factor to the A matrix. In the present application of Lisrel, p is a p x p diagonal matrix whose nonzero elements correspond to the square roots of the reciprocals of coefficients for each latent source of variation. The coefficients are in turn calculated as the sum of the products of the component coefficients in Table 1 and the appropriate weighting coefficients in Table 2. The tk matrix is a p x p matrix of covariances among the postulated factors. If orthogonal factors are postulated tk is a p x p identity matrix. Finally 8, is a y x y error

4 498 R. M. CANTOR ET AL. matrix of residual variances and covariances that cannot be accounted for by the postulated factors. Under some circumstances, the estimates of the residual variances contained in the leading diagonal are negative. This complication can be avoided by obtaining estimates of e''' as the single loadings (Ay,p+y) on y specific factors in A, thereby constraining the error variances to be positive since e, = For a given set of postulated factors, the program estimates the maximum likelihood values for all of the elements in A, \k and 8 that have not been fixed or constrained. The log likelihood function has been derived for the program under the assumption of multivariate normality. A linear function of the log likelihood, G F = X (N,/N)[ln I Xg I + tr(sgxg-') - h I sg I - PI, (2) g= 1 is minimized. G is the number of covariance matrices, five in our case. N, is the degrees of freedom for the gth matrix and N the total degrees of freedom. S and Xg are the observed and expected covariance matrices, respectively, and p the number of variables. F is a multiple of the log likelihood ratio statistic which compares a proposed model with a perfectly fitting theoretical model. JORESKOC and SORBOM (1979) have employed a modified version of the Fletcher-Powell (1976) algorithm in their program to minimize F. Output from a Lisrel analysis includes estimates of the freed and constrained parameters as well as their standard errors and associated t-values. An overall x2 goodness-of-fit statistic is also given which is the minimum value of F with degrees of freedom df = %Gp( p + 1) - q, (3) where q is the total number of independent parameters estimated in all G of the p x p matrices of mean sums of squares and cross products. Several options are available for deciding whether a proposed factor structure is appropriate. First, the x2 statistic and its associated P-value may be used to reject models when the P-value falls below a certain criterion. Second, the value of adding parameters to a given model may be assessed by the associated reduction in the goodness-of-fit x2. For the addition of p independent parameters, the difference in xz values has p degrees of freedom. Third, the t-values associated with the specific factor loadings may be inspected to assess their relative importance. Finally, the proportions of variation explained by competing models may be estimated when compared either with a more restrictive alternative or with a simplified or baseline model. A statistic for this purpose was proposed by TUCKER and LEWIS (1973) and modified by MARTIN, EAVES and FULKER (1979). Its estimation requires the x2 value for the baseline model, such as one that contains only random environmental effects, as well as the x2 statistic for the model under question. The percentage of the variance of the observed variances and covariances unaccounted for by the simplified model but explained by the proposed model is given by the formula

5 ANALYSIS OF DIGITAL RIDGE COUNTS 499 where ~ 0is 2 the statistic under the absurd model, x12 is the statistic under the proposed model and dfo and dfl are their respective degrees of freedom. One may of course modify the procedure so that the absurd model is any one that fits more poorly and is more restrictive than the model under consideration. MATERIALS AND METHODS Sample: This analysis was based on the counts of digital ridges in 222 children of 48 male MZ twin pairs and 287 children of 59 female MZ pairs. Data from 69 of the kinships were obtained at Indiana University prior to 1976 under the supervision of one of us (W.E.N.) while he was Principal Investigator of the Indiana University Human Genetics Center. The remaining 38 kinships were studied at the Medical College of Virginia from Ascertainment of the kinships was primarily by advertisement or personal referral from previous participants. In Virginia, twins were also ascertained from birth records. Only families classified as white were included in the analysis. Rolled finger prints were taken by the inkless method using sensitized paper and developing fluid obtained from the Faurot Company. In addition, ink prints were obtained on glossy paper using Hollister disposable footprinters. The ulnar and radial ridges of each digit were counted and the largest of the two selected for analysis following the approach used by HOLT (1968) in her studies of total ridge count. Preliminary statistical analyses: The mean, standard deviation, skewness and kurtosis were calculated for the total ridge counts, as well as for individual fingers, in the sample of 509 offspring. The results are shown in Table 3. The means and standard deviations were then used to adjust for sex differences and to standardize the individual ridge counts for mean differences between the fingers. Multivariate nested analyses of variance were conducted separately for the offspring of male and female twins and for the total (pooled) sample by the Nested procedure of the SAS package. The output was assembled into five symmetric 10 X 10 matrices; each including 55 unique variances and covariances, providing a total of 275 observed statistics with which to estimate parameters and test hypotheses. The mean sums of squares and cross products matrices for the among half-sibship and between-sibship-within-half-sibship partitions were calculated separately for the offspring of male and female twins, whereas the within sibship covariance matrix was equated with the error effects (within sibship covariance matrix) derived from a multivariate analysis of variance on the entire sample. The five 10 x 10 mean square matrices that were used in the analysis are given in APPENDIX I. RESULTS To obtain a baseline against which to judge subsequent improvements, we began the analysis by testing the (absurd) hypothesis that the observed mean square and mean cross product matrices could all be attributed to random environmental variation and explained by a single parameter. To accomplish this, we set 9 as a diagonal matrix with all elements constrained to be equal. Since the input variables had all been standardized to unit variance, it was not surprising that the estimate of 9 was 0.99 or very nearly 1. Nor was it surprising that the goodness-of-fit was very poor (2 = 4044, 274 d.f.) since the proposed model assumes that both the digital ridge counts of an individual and those of genetically related individuals are uncorrelated (Table 4-a). In HOLT S classical treatment of the genetic determination of the digital ridges, the counts for individual fingers are summed to obtain a total ridge count, and a single random environmental influence is assumed, thus describing the total variation with just two parameters. In our multivariate analysis several possible approaches could have been used to specify the factor structure with two

6 500 R. M. CANTOR ET AL. TABLE 3 Descriptive statistics for digital ridge count data from 270 male and 253 female offspring Digit Sex Mean S.D. Skewness Kurtosis Thumb d 0 2nd d P 3rd 6 P 4th d P 5th d Q Thumb 2nd 3rd 4th 5th * P < d P d 0 d 0 d 0 d Q Left hand Right hand Total ridge count -0.53* -0.40* * -0.68* * 1.03* -0.54* d * parameters. For example, separate but uniform genetic and environmental factors could be assumed to influence each of the ten fingers by constraining the factor loadings within each latent source of variation to be equal. Alternatively, one or both sources of variation could be assumed to operate through a singie common factor with equal factor loadings on every finger. HOLT showed that the unweighted sum of the digital ridge counts was much more heritable than was any individual ridge count, thus indicating the existence of a substantial number of positive correlations between the genetic determinants of digital ridge counts and fewer environmental correlations. We, therefore, chose to represent the HOLT model in our multivariate analysis by a single additive genetic factor with equal weights on every digit and ten separate (specific) environmental factors, with their loadings also constrained to be equal. As shown in Table 4-b, this model resulted in a significant reduction in the 2 goodness-of-fit statistic and accounted for no less than 78% of the variation that was not explained by the simple environmental model. Nevertheless, the model

7 ANALYSIS OF DIGITAL RIDGE COUNTS 501 m u E 0 Q M a M 'ii - 2,- U 2 +- rn a, :.- a s 3 2 m 4 2:.-. 0) -0 - E Y) W.-..a a 0 c? E 5 v)

8 502 R. M. CANTOR ET AL. may still be regarded as an inadequate explanation for the data because of the poor fit (xz = 1089, 273 d.f.). At least two general strategies could have been employed to develop a more detailed model for the determination of finger ridge counts. We could have proceeded directly to include additional sources of latent variation or we could have attempted to refine the factor structure of the two sources of variation postulated by HOLT before invoking other latent sources. We chose the latter course. Guided by biological intuition and a knowledge of previously reported intercorrelations, we first attempted to specify a plausible factor structure for the additive genetic effects. As a first approximation, it seemed reasonable to postulate at least one general additive genetic factor influencing all of the digits, with other additive factors whose effects were restricted to individual hands and still others restricted to homologous fingers. As shown in Table 4-c, a model that included a general factor, two hand factors and five specific finger factors markedly improved the goodness-of-fit. However, since the ridge counts of homologous fingers are known to exhibit a high degree of bilateral symmetry, we constrained the weights on homologous fingers in the general and specific finger factors to be equal, thus achieving a more parsimonious solution, which incorporates finger symmetry but still makes allowance for hand asymmetry. We tested several other constraints on this model including the use of equal weights for all variables loading on the general and hand factors, but these modifications significantly worsened the goodness-of-fit. We next proceeded to model the effect of the uterine environment by first relaxing the constraint that all of the environmental factor loadings on individual fingers be equal as shown in Table 4-d. This modification of the model plus the inclusion of a single general factor to account for correlated effects on individual fingers of the random environmental variation among individuals resulted (Table 4-e) in significant improvement in the goodness-of-fit. The full additive genetic, random environmental model described in Table 4-f accounts for 98% of the variation not explained by the baseline environmental model, as well as 90% of the variation not explained by the HOLT formulation. To estimate how much additional improvement might be expected from further refinement of the factor structure of the additive genetic and environmental effects, we obtained maximum likelihood estimates of the full 10 x 10 additive genetic and within-sibship environment covariance matrices which presuppose no explicit factor structure. These matrices were estimated with the Lisrel program by recognizing that any Gramian matrix may be decomposed into the product of a triangular matrix and its transpose. Thus, we proceeded to estimate simultaneously two 10 X 10 triangular matrices of 55 unconstrained factor loadings for the additive genetic and environmental effects. The resulting additive genetic and environmental correlation matrices are given in APPENDIX 11. As shown in Table 4-g, when 110 degrees of freedom were used to fit these two completely unconstrained additive genetic and environmental matrices, there was a significant improvement in fit over our full or rational model in which a specific factor structure for the genetic and environmental effects was specified (xz = 100.1, d.f. 70, P < 0.01). However, when both solutions were

9 ANALYSIS OF DIGITAL RIDGE COUNTS 503 compared we found that with the use of just 40 parameters the rational model accounted for 99% of the variation that was explained by the complete 110 parameter triangular model. These observations suggest that, although our final additive genetic and environmental model (4-f) is still inadequate as judged by the x test, further elaboration of its constituent factor structure is not likely to explain a substantially greater proportion of the overall variation and covariation in digital ridge counts. Consequently, we next explored the effect of including other general factors to account for two additional potentially significant latent sources of variation: additive epistasis and maternal effects. The results of these analyses are summarized in Table 5. Supplementation of the full additive genetic random environmental model by ten additional parameters to estimate a general epistatic factor caused a highly significant improvement in the goodness-of-fit (x2 = 29.9, 10 d.f., P < ), whereas the addition of a general maternal factor alone resulted in only a marginal improvement (xz = 19.0, 10 d.f., P < 0.041). However, in the presence of the general factor for epistasis, the addition of maternal effects did make a significant further improvement in the model (x = 26.0, 10 d.f., P = ) and yielded a solution that could not be rejected at the 0.05 level, which accounted for 99% of the variation not explained by the baseline model. The estimated factor loadings for the final model are given in Table 6, along with their associated t-statistics. The substitution of dominance as a latent source of variation in the place of epistasis produced no obvious difference in fit, confirming that these two potential sources of genetic nonadditivity cannot be resolved with data from the offspring of MZ twins alone. The parameter estimates given in Table 6 can be used to compute ratios that reflect the contributions of different sources of variation to the ten finger ridge counts separately or in combination. To achieve this goal one first computes the phenotypic covariance matrix Q=M, where A is the y x (y + p) augmented matrix of factor loadings of the latent sources of variation and resulting residuals. The diagonal elements of Q are the sums of all sources of variation in the model for each variable. One may define the coefficients of any linear combination of the observed ridge counts by a y element vector a I, Thus, total ridge count would be represented by a vector of ten units, and the total for the left hand would be specified by the vector ( ). Individual fingers would be specified by a vector in which all elements were zero, apart from the element corresponding to the specified finger, which would be unity. The phenotypic variance of the combination is then u,2 = a Qa. Thus, we see that ap2 for total ridge count includes the covariances between fingers as well as their variances. Now suppose we wish to obtain the contribution to a, of certain selected

10 ~ 504 R. M. CANTOR ET AL. TABLE 5 Comparison of fitting identical genetic models to sex-adjusted and transformed sex-odjusted data Sources of variation included in model General factor Sex-adjusted data Transformed ad,usted data Maternal Additive Random cnvi- Additive environeenetic ronmental epistasis ment d.f. Xd P XL P Triangular Triangular No No <0.001 Full model Full model No No < <0.008 Full model Full model Yes No < <0.073 Full model Full model No Yes < <0.012 Full model Full model Yes Yes < <0.135 columns of A. We define a diagonal selection matrix, A, whose elements are zero apart from the diagonal elements corresponding to the selected factors from A. These elements are set to unity. The contribution of the specified orthogonal factors in the selection matrix to the phenotypic variation of the combination is now or2 = a 'AAA's. The proportion of the total variation accounted for by the specified factors, U,"/ up2, may then be computed. In Table 7, the contribution of common and specific genetic and environmental factors to the variance of individual fingers to total ridge count and to the separate totals for left and right hands are given. DISCUSSION Data from the families of twins have previously been used for univariate analyses of a variety of traits, including birth weight (NANCE 1979), blood pressure (EWELL, COREY and WINTER 1978; ROSE et al. 1979c), stature (NANCE, COREY and EAVES 1980), serum cholesterol (CHRISTIAN and KANG 1977; NANCE, COREY and BOUGHMAN 1978), uric acid (RICH, COREY and NANCE 1978), immunoglobulin levels (ESCOBAR, COREY and BIXLER,1979) and several reproductive (GOLDEN 1980), dermatoglyphic (NANCE 1976; REED et al. 1979) and psychological variables (NANCE 1977; ROSE et al. 1979a,b; ROSE, BOUGHMAN and COREY 1980). The present study demonstrates how this research design can be extended to permit multivariate analyses of interrelated metric traits. HOLT'S simple two-parameter model for the genetic control of digital ridge counts was found to account for 78% of the variation in the observed statistics that was not explained by a random environmental model but did not provide an adequate explanation for the phenotypic and familial correlations between fingers. We have tested specific factor patterns for several combinations of latent sources of genetic and environmental variation and have been abie to explain the covariance matrices of finger ridge counts by the assumption of a

11 ~~~~ ANALYSIS OF DIGITAL RIDGE COUNTS 505 TABLE 6 Factor loadings and random environmental variances for the final model with their associoted t-statistics Random Additive genetic Epistatic Maternal environmental General Hand Finger General General General Specific (n = 1) (n = 2) (n = 5) (n = 1) (n = I) (n = 1) (n = 10) Thumb 2nd 3rd 4th 5th Thumb AIJ t ZJ t A3J 2nd 71 t 3rd t 4th A91 t 5th t t t t hl0, Right hand Left hand n = number of constituent orthogonal factors general additive genetic factor loading on all fingers, two factors loading onto the fingers of each hand separately and five factors loading onto the homologous fingers. A more parsimonious and biologically plausible structure includes the constraint of bilateral symmetry in the general and finger-specific factors. Since previous analyses have shown that the ridge counts on adjacent fingers are generally more correlated than are those of nonadjacent fingers (HOLT 1968), we tested the fit of models containing factor structures that allowed for covariation between adjacent fingers. We estimated parameters representing additional weights on adjacent fingers in the specific additive finger factors, as well as off-diagonal correlations between the specific finger factors in the 9 matrix, but neither of these approaches improved the fit. Presumably, the tendency for the ridge counts of adjacent fingers to be correlated must be explained by the general genetic factor, where, in our analysis, higher loadings have been estimated for the three middle digits. Although our final model accounts for a large percentage of the total variation and covariation in digital ridge counts, we clearly have not exhausted the possibilities for describing the covariance struc-

12 506 R. M. CANTOR ET AL. TABLE 7 Percentage contribution of latent sources and their factors to variation in ridge counts on individual fingers, hands and total ridge count Epis- Mater- Additive genetic tatic nal Random environmental General Hand Finger Total General General General Specific Total (n = 1) (n = 2) (n = 5) (n = 8) (n = 1) (n = 1) (n = 1) (n = 10) (n = 11) Right hand Thumb nd rd th th 43.7 Hand Left hand Thumb nd rd th th Hand Total ridge count n = number of constituent orthogonal factors. ture, and it is possible that a nonlinear model could provide a better explanation for the observed relationship between the magnitude of ridge count correlations and the physical propinquity of digits on the hand. A unique feature of our research design is the ability of observations on the offspring of MZ twins to estimate maternal and nonadditive genetic effects from observations on individuals who are members of the same generation. Thus, although we were able to account for 90% of the residual variation beyond the baseline model by additive genetic effects alone, we also tested for the presence of dominance, epistatic, maternal and random environmental effects. The goodness-of-fit and parameter estimates for models including dominance and epistasis were virtually identical, and it is possible that the inclusion of data on twins or parent-offspring relationships might allow a clear choice between these two alternatives (MATHER 1974). The rationale for the inclusion of maternal environmental effects in our model is certainly credible, since the dermal ridges are formed during early intrauterine life. In an overall analysis, we found significant (P < 0.01) evidence for a maternal effect that accounted for from 5.1 to 23.2% of the variation in the dermal ridge counts of the individual fingers. Although the maternal factor loadings were larger for the thumbs and fifth fingers, significant values (t > 1.96) were estimated for all of the fingers. The intrauterine maternal influence

13 ANALYSIS OF DIGITAL RIDGE COUNTS 507 detected in this analysis reflects effects that are common to all pregnancies of an individual mother as well as to those of her identical twin. Whether the residual within-sibship environmental variation includes additional intrauterine influences that are specific to a single pregnancy remains to be determined. In addition to including systematic variation in the intrauterine environment attributable to the genotype of the mother, the final model postulates two types of random environmental variation, some of these effects being specific to individual fingers, whereas others are more generalized in their influence. The percentage of variance explained by the general additive factor is substantially larger for the summed hand counts than it is for any individual finger, whereas the proportion is larger still for total ridge count. In contrast, the proportions of variation in summed hand and total ridge count that are explained by the general factors for epistasis, maternal effects and random environmental effects fall within the range of values exhibited by individual fingers. In addition, the specific random environmental effects on the hand and total ridge counts are generally much lower than those observed for individual fingers. These findings imply that the environmental effects on individual finger ridge counts are largely uncorrelated, whereas the additive genetic effects are highly correlated. The application of maximum likelihood confirmatory factor analysis to investigate the genetic determination of multivariate traits permits the specification and testing not only of the number of factors but their explicit structure and thus has compelling advantages over exploratory factor analysis where the only hypotheses that can be tested concern the number of factors present. Statistical tests of proposed models, however, are based on large sample theory and depend upon the assumption of multivariate normality. Although the number of half-sibships included in our sample is relatively small, the analysis was based on the counts from more than 5000 fingers, and among the marginal distributions of ridge counts for individual fingers, only those of the thumbs showed statistically significant departures from normality (Table 3). TO investigate the possible influence that departures from multivariate normality may have had on the conclusions drawn from our model fitting, we repeated the analyses after transforming the marginal distributions to normality using the power transformation proposed by Box and Cox (1964). Although this procedure does not guarantee multivariate normality, it may provide some insight into the magnitude and direction of biases arising from failure of the normality assumption. The results of the analyses of the transformed data are given in the last two columns of Table 5. With these data, comparable models showed a modest improvement in the goodness-of-fit except for the full triangular decompositions of additive genetic and environmental effects where the variation and covariation of the transformed scores were less well explained. Since this model completely exhausts the ability of additive genetic and random environmental factors to account for the pattern of covariation, its failure tends to support the view that latent sources of variation other than additive genetic and random environmental effects do in fact contribute to providing a more adequate explanation of the covariance structure of digital ridge counts. The inclusion of

14 508 R. M. CANTOR ET AL. a general epistatic factor, alone or in combination with a general maternal factor, resulted in statistically significant improvements in the goodness-of-fit with both the raw and transformed data. However, with the transformed data, the addition of the maternal factor resulted in a statistically significant improvement in fit (P < 0.05) only in the presence of the general epistatic factor. The general agreement between the conclusions drawn from the two approaches suggests that the analysis is sufficiently robust to tolerate the modest distributional departures observed in the present data. In view of the small size of the observed maternal and epistatic effects it would clearly be desirable to replicate these findings with other data sets to be sure that they do not reflect sampling error or a subtle residual bias arising from failure of the data to conform to the assumption of multivariate normality. Previous workers have reported that, as the number of variables included in a confirmatory factor analysis rises, it becomes increasingly difficult to specify causal factor models that cannot be rejected at conventional levels of statistical significance (MARTIN, EAVES and FULKER 1979). Our data set permitted the calculation of 275 independent variances and covariances, and our relative success in explaining the interrelationships implied by this complex pattern of covariation with a genetic model that included only 55 unknown parameters may well be attributable to the nature of the variables and the structure of the data set. Because of the natural grouping of the variables into hands and homologous pairs of fingers and the strong tendency for bilateral symmetry in the digital ridge counts, biological intuition appears to have been a reasonable guide for postulating factors that make a major contribution to the pattern of covariation. The ready availability of software for fitting structural models to multivariate data encourages the search for simple paradigms in which biological considerations can serve as a reliable guide to the formulation of plausible causal models. Data from twins and their offspring are especially valuable for such studies because they permit the detection of latent sources of variation that cannot be resolved by studies of nuclear sibships. As the present work suggests, small, but potentially important, biological effects can be identified with this research design. The methods we have employed may also prove to be of value in experimental systems for the genetic analysis of interrelated biochemical, morphogenic or physiological traits in which the knowledge of metabolic pathways, developmental sequences or regulatory control mechanisms may permit the formulation of explicit hypotheses about the interrelationships between multiple variables. Dermal ridge counts constitute a multivariate system that is largely free from many of the scalar problems associated with other variables and that carries real potential for helping to chart the course of early development. For this reason we believe they remain valuable objects of genetic investigation that have still to be exploited to the fullest. This is paper #138 from the Department of Human Genetics of the Medical College of Virginia. The work was supported in part by National Institutes of Health grants HD 15838, HD and GM Reprint requests should be addressed to the Medical College of Virginia.

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17 APPENDIX I Mean sums of squares and cross products of sex-adjusted digital ridge counts from multivariate nested analyses of variance of data from the offspring of MZ twins Among half-sibships: Male twin kinships bz b BA BAA Bo = P-M = PEW = 1.OOO d.f. = Between sibships-within half-sibship: Male twin kinships bi = PA = BAA = Po = BM = PEW = d.f. = 48 Within sibships: Male and female twin kinships, pooled PA BAA = Po = psw = d.f. = Between sibships-within half-sibship: Female twin kinships bs = PA PAA PD psw = d.f. = Among half-sibships: Female twin kinships bs = bs Pa = PAA = PD = DM = PEW = d.f. =

18 512 R. M. CANTOR ET AL. APPENDIX I1 Genetic and environmental correlation matrices for digital ridge counts Additive genetic correlation matrix Right hand Left hand Thumb 2nd digit 3rd digit 4th digit 5th digit Thumb 2nd digit 3rd digit 4th digit 5th digit RT R2 R3 R4 R5 LT L2 L3 L4 L Symmetric ~- Random environmental correlation matrix RT R2 R3 R4 R5 LT LZ L3 L4 L5 Right hand Thumb nd digit 3rd digit 4th digit th digit Symmetric Left hand Thumb nd digit 3rd digit th digit th digit RT, R2,.., R5, LT, L2..., L5 refer to thumbs and digits of right and left hands, respectively.