structured exploratory data analysis (seda) of finger ridge-count inheritance: i. major gene index,...

AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 62:377-396 (1983)

Structured Exploratory Data Analysis (SEDA) of Finger Ridge-Count Inheritance: 1. Major Gene Index, Midparental Correlation, and Offspring-Between-Parents Function in 125 South Indian Families

SAMUEL KARLIN, RANAJIT CHAKRABORTY, PAUL T. WILLIAMS, AND SUSAN MATHEW Department of Mathematics, Stanford University, Stanford, California 94305 (S.K., l? TW), Center for Demographic and Population Genetics, University of Texas Health Science Center at Houston, Houston, Texas 77225 (R. C.), and Department of Physical Anthropology and Human Genetics, Andhra Uniuersity, Waltair, Andhra Pradesh, India (S. M.)

KEY WORDS analysis, MGI, OBP, MPCC, Velanadu Brahmins of South India

Finger ridge-counts, Structured exploratory data

ABSTRACT Fourteen dermatoglyphic traits measured on 125 Velanadu Brahmin families were analyzed for mode of inheritance using three Struc- tured Exploratory Data Analysis (SEDA) statistics: the major gene index, the offspring between parents function, and the traditional midparental correlation coefficient. Since the traits are integer valued with restricted ranges of variation, we simulated various transmission models with discrete expression to better understand the nature of the SEDA statistics for such variables. In addition, permutation procedures were employed to aid the interpretation of the SEDA results. These analyses suggest that corresponding homologous fingers on the left and right hands exhibit similar transmission characteristics. The relationship of the parent and child total ridge-counts of the two hands separately, as well as their combined total, virtually simulate complete Galton- ian blending inheritance. Results for the individual digital ridge-counts as well as the pattern-intensity-index variable also suggest a multifactorial mode of transmission or possibly one involving several genes.

Anthropologists and human geneticists have studied variations of finger ridge-counts within and between human racial groups for almost three-quarters of a century. On the basis of parent-offspring and sib-sib correlations it is claimed that there exists a large genetic or blending transmissible component in the determination of these dermatoglyphic variables. If indeed a major portion of the variability in dermatoglyphic features is generated by many genetic factors, then studies that contrast finger ridge-counts and related variables across populations may further clarify relationships and classifications between racial groups (Holt, 1956, 1957; Pons, 1959; Weninger, 1965; Loesch, 1971).

Identifying the mode of inheritance for these dermatoglyphic variables can be an im- portant prerequisite to such contrasts. Pen-

rose (19691, Matsunaga (19721, Matsuda (19731, and others have advocated that the genetic variation of dermal ridge-counts is due to a large number of genes each having a small effect, i.e., a multifactorial mode of inheritance. The suggestion that major genes contribute to these traits was argued by Holt (1958), Knussman (1967), and Spence et al. (1973). Holt (1968) proposed that the negative skewness of the individual digit ridge-counts could be due to a small number of genetic factors each having a significant effect. Us- ing a multivariate analysis, Spence et al. (1977) proposed that “dominance” deviations

Received January 10, 1983; revision accepted June 15,1983. Send reprint requests to Dr. Ranajit Chakraborty, Center for

Demographic and Population Genetics, University of Texas Health Science Center, P.O. Box 20334, Houston, TX 77025.

0 1983 ALAN R. LISS, INC

378 S. KARLIN, R. CHAKRABORTY, P.T. WILLIAMS, AND S. MATHEW

have a significant effect on dermal ridge- count variations. In another study, Anderson et al. (1979) surmised that the presence of an arch on any of the ten digits may be a major gene trait that is “linked” with the haptoglobin locus in a Habbanite pedigree.

The results of Penrose, Holt, Anderson and others cited above rely on classical parametric methods for assessing mode of inheritance. These necessarily involve a spectrum of innate assumptions including normality, additivity, and restricted formulations of major gene, multifactorial, and sporadic models that may not necessarily apply to dermatoglyphic variables. We employ an interactive approach to data analysis that couples an empirical and a modeling rationale that studies the data directly, but a t the same time recognizes the qualitative insights derived from a hierarchy of pertinent models. This statistical methodology, Structured Ex- ploratory Data Analysis (SEDA) (Karlin et al., 1979a, 1981b,d), assesses the nature of vertical and horizontal forms of transmission through a collection of indices, measures, and graphical representations that contrast various functions of the trait’s expressions over family members.

This paper applies three SEDA statistics: the Major Gene Index (MGI), the classical Midparental Correlation Coefficient (MPCC), and the OffspringBetween-Parents (OBP) function to the measurements of individual ridge-counts of each of the ten digits, total ridge-counts of left and right hands, total ridge-counts for both hands combined, and pattern intensity index of a Velanadu Brah- min population of South India to determine whether the familial transmission of these traits are more consistent with a pattern of (1) major gene transmission; (2) multifactorial transmission; or (3) sporadic trait expression (offsprings’ traits distributed independently of the parents’). Diagnostic prop- erties of the SEDA statistics useful for dis- criminating among the different transmission modes have been set forth for continuously distributed traits (Karlin et al., 1981b, 1983a). However, in this paper we apply SEDA methods to noncontinuous variables including ridge-counts of individual digits, which are integer-valued quantities usually assuming values between 5 and 25. The discreteness and restricted ranges of these variables affect some of the diagnostic characteristics of the SEDA methods. Therefore, we first discuss simulations of major gene,

multifactorial, and sporadic modes of transmission with discrete phenotypic expression in order to achieve a better understanding of SEDA statistics under these models with discrete phenotypic expression.

We also employ familial permutation procedures for contrasting the observed MGI, MPCC, and OBP statistics with their reali- zations under certain reconstructions of the data (Karlin et al., 1983a). These procedures permute one or more family members across families in order to expose the effects of closeness of parents, parent-offspring asymmetry (for assessing the strength of maternal effect, in particular), generational differences, sex differences, population heterogeneity, and other factors that may affect the SEDA statistics. All permutations preserve the family size distribution. The dermatoglyphic data were analyzed using spouse-pair permutations that keep spouse-pairs and sibships intact but reassign the spouse-pairs to the sibships, and total permutations that randomly permute all individuals without regard to sex or relationship.

The issues to be addressed in this paper include: (1) What modes of transmission are suggested for the individual finger ridge- counts, composite ridge-count variables (total finger ridge counts) and pattern intensity index? (2) Do homologous fingers of the left and right hand have similar modes of inheritance? (3) Is the mode of inheritance different among fingers of the same hand? (4) We also examine how the permutations aid in the interpretation of the SEDA statistics.

MATERIALS AND METHODS Population and variables

Our data come from 125 families from an endogamous group of Velanadu Brahmins from Andhra Pradesh, India. The Velanadu Brahmins belong to the Vaidiki sect of Brah- mins, the highest caste group of South India. These families may be regarded as a random sample from this caste community of South India with respect to their dermatoglyphic features. The rigid marital practice in this patrilineal society makes Velanadu Brah- mins a strict endogamous group, with few if any genes infiltrating from other populations. Detailed descriptions of these people and their dermatoglyphic traits are presented by Mathew (1980). Table 1 presents the family size distribution for these families.

Finger prints were obtained for 125 parental pairs and 375 children (186 sons and 189

SEDA STATISTICS OF FINGER-COUNTS 379

TABLE 1. Family size distribution in I25 Velanadu Brahmin families of Andhra Pradesh, India

No. of children' 0 1 2 3 4 5 6 7 Total

Males 22 54 27 15 3 3 1 0 125 Females 28 36 36 19 6 0 0 0 125 All 13 34 40 24 8 4 2 125 'Average family consists of 3.00 chlldren per family with equal representation of male and female offspring.

-

daughters). Children less than 1 year old were excluded. The standard ink and roller method was used to obtain bilateral finger and palm prints. The ridge-counts were made following the procedure described by Holt (1949, 1958) and the pattern types (whorl, arch, or loop) were identified and scored using the protocol of Cummins and Midlo (1976). These 625 individuals did not have any of the rare aberrant pattern types and therefore no special adjustments were re- quired for scoring the variables.

The variables of focus are ridge-counts (RC: the larger of the two counts, ulnar and ra- dial) of each of the ten digits of the two hands, which are denoted by RCL (i), RCR (i); i = 1, 2, ..., 5; RCL (i) being the ridge-count of the i- th digit of the left hand and RCR (i) the ridge- count of the i-th digit of the right hand. We follow the convention that the thumb is digit 1 and the little finger is digit 5. The total ridge-count of the left and right hands (TRCL and TRCR, respectively) are the sum of the ridge-counts of the individual digits of the respective hands (e.g., TRCL = RCL(1) + ... + RCL (5)). In addition, we shall also con- sider the traits total finger ridge-count (TRC = TRCL + TRCR) and pattern intensity index (PII), which is the total number of tri- radii for all ten digits. Pattern intensity index necessarily takes on values between 0 and 20 since a digit usually has zero (in the case of an arch), one (in the case of a loop), or two (in the case of a whorl) tri-radii.

Table 2 provides the descriptive statistics of the 14 finger dermatoglyphic variables by sex and generation.

Males and females showed differences for most of the trait values. Therefore, the data were adjusted for sex differences using me- dian and interquartile range adjustment (Karlin et al., 1981d).

Simulation of genetic models with discrete phenotype expression

We examined simulated models of major gene, multifactorial, and sporadic transmis-

sion in which the phenotypic expression of the traits are made discrete to aid in the interpretation of the SEDA statistics when applied to the dermatoglyphic variables.

The simplest major gene model postulates that the genetic components of a trait are specified at a single locus with two alleles. The alleles one and zero occur with probabil- ities p and q (q = 1 - p), respectively. We assume random mating, no sex differences after suitable age-sex adjustments, and Hardy-Weinberg equilibrium. Under these conditions the parents of genotypes (1,l) occur with probability p2, genotypes (0, 1) and (1, 0) each occur with probability pq and parental genotypes (0,O) occur with probability q2. The offspring are formed by Mendelian segregation. Specifically, if the male and female genotypes correspond to (Xl, X2) and (Y1, Y2), respectively, then from the male parent we select either XI or Xz with probability 112, and similarly from the female parent we select either Y1 or Y2 with probability 112 to produce the offspring genotype.

A genetic main effect a is assigned to the (1, 1) genotype, the effect ad to the (0, 1) and (1, 0) genotypes, and the effect -a to the (0,O) genotype. We refer to a as the displacement parameter at the major locus and to d as the dominance parameter. In particular, d = 0 refers to an additive dose model and d = 1 and - 1 correspond to dominance models. The trait of each individual is then determined by adding an independent random residual term ti (i = 1, 2, or 3, corresponding to the genotype) to the genetic main effect. The ti's are independently generated from distributions (which need not necessaril be normal) with mean zero and variance u .

The model is specified by the parameters p, d, and varying u2 while a is standardized at a = 1. Results are presented for p = 0.1 with d = 0, 1 and -1 and for p = 0.5 with d = 0 and 1 (by symmetry the results for p = 0.5 with d = 1 are identical to those for p = 0.5 with d = -1). The distributions of the random error terms ti were taken independent

s

380 S. KARLIN, R. CHAKRABORTY, P.T. WILLIAMS. A N D S. MATHEW

normally distributed with standard devia- tion CT = $4, which includes moderate overlap among the trait expressions for the three genotypes.

Multifactorial inheritance engenders a hierarchy of models that blend paternal and maternal phenotypes symmetrically or asymmetrically and in either linear or non- linear combinations and may include sex-de- pendent offspring expression. We restrict our present analysis to the specific case of midparental transmission. The Galtonian (Gaus- sian) blending model is constructed by sampling parents from a normal distribution with mean zero and variance 1 and then con- structing offspring by adding an independent normally distributed residual term with mean zero and variance Yi to the average of the parental trait values. Other blending models can be similarly constructed for other background (residual) distributions (Karlin et al., 1981b). For further discussion of multifactorial models see Karlin (197913).

Sporadic trait expression is a special limit- ing case of both major gene and multifactorial transmission in which the individual variation term overwhelms the transmitted component.

The discrete phenotypes for all three models were formed by replacing the continuous trait values (as generated by the above procedures) with discrete values formed by (1) rounding off the continuous values to the nearest integer; (2) rounding off to the nearest half-integer; (3) rounding off the nearest quarter-integer; and (4) rounding off to the nearest eighth-integer value. In each case, we generated 200 families with a single child per family.

Structured exploratory data analysis (SEDA) In the vein of the SEDA approach (Karlin

et al., 1981b-e, 1983a) we employ three classes of statistics: Major Gene Index (MGI), Offspring-Between-Parents Function (OBP), and Midparental Correlation Coeficient (MPCC) to assess the relative degree of con- sistency of the data with major gene, multifactorial, or sporadic transmission. (Analysis of these data involving other SEDA statistics such as the offspring near mother (ONM) and offspring near father (ONF) functions will be presented elsewhere.) For easy reference the motivations and definitions of these statistics are briefly reviewed along with their ex- pectations for continuous traits.

Major gene index (MGI): This class of indices contrasts midparental offspring devia-

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tions with a function of the separate parent- offspring deviations. If a major gene is seg- regating in families, then on the average, the children's values should be closer to the val-

values than with an average from both parents. The opposite ordering is expected for multifactorial traits. These indices are a one parameter family in a, where the parameter a! is used to emphasize small (a! = %), moderate (a! = l), and large (a! = 2) deviations, i.e.,

occur in intervals of varying length /3 that are symmetric about the midparental value. Specifically, we let

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where Zij is the trait value of the j-th offspring in family i, Xi is the trait value of the father in family i; Yi is the trait value of the mother in family i; and Ki is the number of children in family i, 1 < i < n.

Under purely deterministic models, the value of the MGI for a major gene tends to exceed 1, multifactorial models generally produce values less than 1, and sporadic traits produce values close to 1. Studies on continuous models (Karlin et al., 1981b) show that the statistic may be influenced by many factors confounding their interpretation (particularly in the case of a major gene). For example, superimposing residual terms onto genetic main effects in a major gene model may decrease the index, particularly under allelic effects and an allele frequency of %. Dominance, dissimilar allele frequencies, and residual (noise) perturbations from skewed probability density functions elevate the MGI for traits which are controlled by a major gene, whereas the equilibrium distribution generated by multifactorial transmission of a skewed distribution decreases the MGI even further below 1. Skewed sporadic traits and asymmetric transmission of multifactorial traits may yield values which are similar to MGI values under major gene inheritance, and therefore by itself the MGI may not ac- curately distinguish major gene from other transmission forms (Karlin et al., 1981b, 1983a).

Offspring between parents (OBP) function: Under multifactorial inheritance a greater number of offspring are expected to have trait values closer to the midparental value than for inheritance through a major gene. We therefore count the number of children which

where the summation is taken over all offspring, and Zij, Xi, Yi, and Ki are as defined for equation 1. We examine the plot of OBP (0) as a function of /3 over a reasonable range of /3 (0 < /3 < 3). Earlier analyses (Karlin et al., 1981d,e) emphasized primarily three fac- ets of the OBP (p) functions-the level, the regions of steep increase, and the nature of the undulations to guide their interpretations. In particular, OBP functions for multifactorial inheritance of a continuous trait were expected to lie above the OBP functions of both major gene and sporadic models, to rise more sharply for small values of /3 (0 < /3 < %I, and to be usually concave for all /3. The corresponding curves for continuous sporadic traits were expected to start initially with a less-steep increase relative to multifactorial traits and remain much below these curves for 0 < /3 < 3. Sporadic OBP functions tend to have a low elevation and may exhibit moderate undulations (i.e., a portion convex, then concave). Guidelines for dis- criminating continuous major gene models emphasized OBP functions occurring at an intermediate level relative to the continuous multifactorial and sporadic models and which displayed a steeper increase in their slope in the neighborhood of /3 = 1.

Midparental correlation coefficient (MP- CC): For definiteness we report the family weighted midparental correlation coefficient (Karlin et al., 1981a), which is expected to be relatively high for complete multifactorial transmission, mostly intermediate for a major gene, and zero for sporadic traits.

Caveats pertaining to the interpretation of SEDA statistics: The accumulated experi- ence from theoretical and computer simula-


TABLE 3. SEDA statistics for a multifactorial and a sporadic trait with discrete phenotypic expression as compared to continuous phenotypic expression

Integer l/a-integer 1/4-integer 118-integer Continuous

Multifactorial MGI(Yz) MGI(1)

2.35 1.87

MGI(2) 1.29 MPCC 0.54 OBP jump a t 0 = 0 0.05 OBP jump at p = 1 0.44

MGIW 1.57 MGI(1) 1 3 7

Sporadic

1.18 -0.17

OBP jump a t = 0 0.03 OBP iumD a t L3 = 1 0.35

1.54 1.25 0.96 0.63 0.09 0.45

1.26 1.17

1.08 0.98 0.94 0.87 0.78 0.76 0.71 0.69 0.08 0.06 0.29 0.20

1.12 1.07 1.07 1.04 1.07 1.06 1.10

-0.07 -0.05 -0.08 0.06 0.06 0.04 0.29 0.25 0.19

0.89 0.84 0.78 0.69 -

0.95 0.96 1.01 - -

tion studies (Karlin et al., 1981b,c, 1983a) and from applying the SEDA statistics to a variety of biochemical and physiological variables (Karlin et al., 1981d,e) has shown us that caution must be exercised in their interpretation. For example, parental closeness, generational differences, population heterogeneity, and asymmetric parent-offspring transmission are factors that are frequently encountered with real data, and are known to effect the behavior of SEDA statistics. Per- mutation sampling routines can significantly aid the interpretation of the SEDA statistics under these conditions, which we now describe.

Permutation procedures: Permutation procedures are reconstructions of the data by randomly permuting trait values across families while preserving the family size distribution. The term coupled with “permutation” defines the class(es) of individuals being permuted. For example, spouse-pair permutations randomly assign spouse-pairs to families but leave the offspring set intact, father permutations permute the fathers across families but keep the mother and her children together, and total permutations reassign all individuals without regard to sex or their position in the family. The SEDA statistics are computed with respect to parents and children on several permuted samples of the data set. Comparisons are made between the SEDA statistics computed on the original and the permuted data and also between the different types of permutation samples. For example, we expect that spouse- pair permutations retain the effects of closeness of parents, sibship closeness, and generation and sex differences but eliminate the

effect of vertical transmission on SEDA statistics. Comparing the SEDA results for the original data versus spouse-pair permutation samples may provide insights into the form of the vertical transmission while controlling for these other effects. Also, mother permutations may be contrasted with father permutations to reveal possible asymmetries in parent-offspring transmission.

The present study examines spouse-pair and total permutations in the analysis of dermatoglyphic features. We focus on the results for total permutations, which primarily ad- justs for distribution effects (including discreteness and restricted range of variation) to aid in the interpretation of the SEDA statistics. Five permutations of each type were performed for selected variables. Examples of other permutation methods in assessing the nature and form of familial lipid and lipoprotein similarities are described by Kar- lin et al. (1983a).

RESULTS

Although the primary focus of this paper is the analysis of dermatoglyphic variables, we first describe the results of the simulations which aid the interpretations of the SEDA statistics for discrete data.

Simulation studies Table 3 presents the SEDA statistics pro-

duced under discrete multifactorial and sporadic models. Table 4 displays the SEDA results for discrete major gene models.

Figures 1-3 detail the shape of the OBP curves for multifactorial transmission (Fig. 11, for sporadic traits (Fig. l), and for major gene transmission (Fig. 2 for p = 0.5, d = 0,

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Fig. 1. OBP curves for simulated normal Galtonian multifactorial and sporadic transmission models with discrete phenotypic expression. a. Multifactorial transmission with phenotypic scores rounded off to one-quarter integers. b. Multifactorial transmission with phenotypic scores rounded off to one-eighth integers. c. Sporadic

transmission with phenotypic scores rounded off to one- quarter integers. d. Sporadic transmission with phenotypic scores rounded off to one-eighth integers. The lower and upper continuous concave curves are the theoretical curves for continuous normal sporadic and Galtonian multifactorial inheritance.

and Fig. 3 for p = 0.1, d = 0) with discrete phenotypic expression.

The key points from these simulations are now summarized.

Major gene index: Tables 3 and 4 and more extensive simulations reported elsewhere (Karlin et al., 1981b) show that the MGI(cr) statistics are generally decreasing functions in a for continuous multifactorial models but increasing functions in a for continuous spo-

radic and major gene models. These tables show that the expected values for MGI(cr) are elevated by discreteness compared to the corresponding continuous models. The MGI(cr) statistics for discretely expressed data are generally decreasing functions in cr for sporadic and major gene models as well as for multifactorial models.

When the traits’ expression is moderately discrete (rounded to the nearest quarter or

SEDA STATISTICS OF FINGER-COUNTS

', 385

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Fig. 2. OBP curves for the major gene model with discrete phenotypic expression: (a) allele frequency, p = 0.5 and dominance parameter, d = 0 with phenotypic scores rounded off to one-quarter integers; (b) p = 0.5, d = 0 with phenotypic scores rounded off to one-eighth

integers; (c) p = 0.5, d = 1 with phenotypic scores rounded off to one-quarter integers; and (d) p = 0.5, d = 1 with phenotypic scores rounded off to one-eighth integers. The continuous concave curves are as in Figure 1.

eighth integer) the observed values for MGI(2) are very similar to MGI(2) values for continuous trait expression and this appears to be the most useful choice of a for discrimi- nating among transmission models. MGI(2) for multifactorial traits are substantially lower than MGI(2) for sporadic and major gene expression. There is less distinction between the MGI statistics for sporadic and major gene inheritance models. As in the

case of a continuous trait, dominance and unequal allele frequencies tend to elevate the MGI statistics relative to additive major gene models that have similar allele frequencies.

Offspring-between-parents plots: The levels of the OBP curves are generally lowered by transforming the trait values from a continuous to a discrete distribution due to a higher probability of the parents being iden-


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tical. The OBP(0) curves for discrete traits will also often show jumps for all transmission models because the discreteness of phenotypic scores entails a higher probability of the child assuming the value of one of the two parents, producing a jump at 0 = 1. Jumps at other values may occur as well; e.g., at 0 = 0 (if 22 = X + Y), at 0 = 1/3 (if 32 = 2X + Y or 2Y + X), a t 6 = 2(if 22 = 3X - Y or 3Y - X) , or at 0 = 3 (if Z = 2X - Y). The jump at 0 = 1 lowers the OBP curve for 0 slightly less than 1 and raises the OBP curve for slightly greater than 1 relative to the OBP curve for the continuous expression of the trait.

The jumps in the OBP function for a major gene trait are not clearly different from those observed for curves computed on the total permutation of these data. However, the level of the OBP function for the major gene model is higher than the level of the curves for the permuted observations. The level of the OBP function for multifactorial traits is higher than the level observed for sporadic or major gene expression, but reduced parent-offspring transmission would be expected to lower the level of the OBP function.

Midparental correlation coefficient: The MPCC value is slightly decreased by transforming the phenotypic trait from a continuous to a discrete distribution as seen in Tables 3 and 4. The decrease is substantial when the trait assumes a very limited number of possible values.

Analysis of Dermatoglyphic Variables The MGI and midparental correlation coef-

ficients (MPCC) are presented in Table 5 along with descriptions of the OBP curves. The comparison of MGI, MPCC, and OBP statistics with those obtained for the permutation samples are shown in Table 6 for

Fig. 3. OBP curves for the major gene model with skewed gene frequency and discrete phenotypic expression: (a) allele frequency, p = 0.1 and dominance parameter, d = 0 with phenotypic scores rounded off to one- quarter integers; @) p = 0.1, d = 0 with phenotypic scores rounded off to one-eighth integers; (c) p = 0.1, d = 1 with phenotypic scores rounded off to one-quarter integers; (d) p = 0.1, d = 1 with phenotypic scores rounded off to one-eighth integers; (e) p = 0.1, d = -1 with phenotypic scores rounded off to one-quarter integers, and (0 p = 0.1, d = -1 with phenotypic scores rounded off to one-eighth integers. The continuous concave curves are as in Figure 1.

RCL(11, RCR(l), RCL(51, RCR(5), TRC, and PII. Figure 4 illustrates the OBP curves for some of these variables relative to the curves for five total permutations of these data. We summarize the findings from these analyses in tabular form.

Individual digit ridge-counts: The ridge- counts RCL(2), RCL(3), RCR(B), RCR(3), and RCR(5) have MGI scores near unity whereas the other individual digits yield MGI values less than 1 and that lie clearly outside the range of MGI values calculated from the re- constructed data sets generated from spouse- pair and total permutations. The MPCC values for individual digits are large (0.42-0.57). Offspring between parents curves of the ridge-counts of individual digits show jumps at 0 = 1, 2, and 3, and occasionally an im- mediate increase at 0 = 0. The especially large jump at 0 = 1 is at least partly due to discreteness as observed in the simulations. The jumps observed at 0 = 1 for RCL(1) and RCR(5) are somewhat larger than those observed in totally permuted families indicat- ing some transmission influences (perhaps suggestive of some major gene component).

Total ridge-counts of the left, right, and the hands combined MGI scores for the total ridge-counts of the right hand (TRCR) and the left hand (TRCL) are much less than 1 and are nearly identical to each other. The MGI for the total ridge-count of the two hands combined (TRC) is virtually identical to that of the theoretical model describing complete Galtonian blending inheritance involving normal residual distributions.

All midparental correlation coefficients for these variables are large. In considering the relative orderings of the MPCC value, we observe, however, that the correlations for the individual digits tend to be somewhat smaller than midparental correlation of the total ridge-count of the left and right hands (TRCL and TRCR), which in turn are somewhat smaller than the MPCC for the total ridge-count for both hands combined (TRC). The midparental correlation for TRC has a value that is almost identical to the expected value for complete Galtonian inheritance (close to 0.71). The MPCC values for TRCL and TRCR are slightly smaller, 0.6 compared - .

with 0.71. The left (TRCL), right (TRCR), and com-

bined total ridge-counts (TRC) are more continuous in their distributions and for all practical purposes do not exhibit jumps in their OBP curves. Further, these three vari-


TABLE 5. Summary of the SEDA statistics for 14 finger ridge-count variables as observed in the 125 Velanadu Brahmin families

Traits u = ? 4 f f = l u = 2 MPCC OBP curve

RCL(1) 0.95 0.90 0.90

RCL(2) 1.01 0.97 0.98

RCL(3) 1.00 0.99 1.00

RCL(4) 0.95 0.89 0.85

RCL(5) 0.98 0.93 0.93

TRCL RCR(1)

RCR(2)

RCR(3)

TRCR

TRC

PI1

0.92 0.95

1.04

1.01

0.90

1.02

0.91

0.90

1.02

0.87 0.90

1.01

0.98

0.84

1.00

0.87

0.85

0.96

0.80 0.88

1.04

1.01

0.81

1.03

0.81

0.78

0.49

0.45

0.52

0.59

0.45

0.58 0.51

0.51

0.45

0.42

0.47

0.60

0.65

0.96 0.62

Sharp initial increase with moderate jump at (3 = 1: level of curve very much above the curves of the permuted families

moderate jumps at p = 3

1, moderate jump at

slight jump at = 1

minor jumps at 6 = 0,2, and 3; lies above curve for permuted families over most of its range

Essentially linear for 0 < 0 < 1, = 1 and

Only slightly concave for 0 < < Concave form, high elevation and

Moderate jumps at p = 1 with

= 1

Smooth concave form Rises sharply for small and

remains above the curve for permuted families: jump at 0 = 1 similar to the jump observed in the OBP curve for the permuted families

= 1 and smaller increases at p = 0,2, and 3

Similar to OBP curve of the second digit (R2) in level and location of jumps

Curve shows an initial steep rise, and a minor jump at

Curve for sampled families lies slightly above curve for the permuted families and has a larger jump a t permutation curves: both the original and permuted curves have jumps at p = 0 and = 1

Level of curve high with smooth concave form

High elevation with smooth concave form (shape and elevation similar to form expected for Galtonian polygenic inheritance): clearly more elevated than curves for permuted families

Slightly concave for 0 < /3 < 1: large jump at than observed for any of the permutations), relatively flat for 1 Q B < 3

Pronounced jump at

= 1

= 1 than

= 1 (larger jump

TAB

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9 -1

-A4 LI

M U .L

0 0 0 5 1 0 1 5 2 0 2.5 3.0

P

0

-1

m i

- . 0.0 0 . 5 c.0 1.5 2 . 0 2 . 5 i . 0

P

Fig. 4. OBP curves for some finger dermatoglyphic variables (A points) in 125 Velanadu Brahmin families compared to the OBP curves (step functions) when all individuals are permuted across families (total permuta-

ables have OBP curves of a smooth concave form and high level which are expected under the supposition of a complete Galtonian transmission.

Pattern intensity index: Although the MGI scores for PI1 are near 1, MGI(1) and MGI(2) fall clearly outside the range of MGI scores corresponding to the total and spouse-pair permutations. The MPCC for PI1 is nearly as large as the MPCC obtained for the total ridge-count of the two hands combined. The

0

;1 ?. 0 -

cl -

0.0 0 .5 1.0 1'5 21.0 21.5 i . 0

d

i t

P 0.0 0.5 1.0 1 . 5 2.0 2.5 3.0

tion): (a) RCR (11, (b) RCR (51, (c) TRC (both hands combined), and (d) PII. Five replications of total permutations are represented on each diagram.

OBP curve shows a large jump at /3 = 1 and lies substantially above OBP curves associ- ated with the total and spouse-pair permutations of the PI1 scores. The jump in the OBP curve at /3 = 1 is greater for the original PI1 values than for their total or spouse-pair permutations.

Comparison of homologous digits: To determine whether corresponding digits show the same pattern of genetic mode of inheritance we display in Figure 5 the OBP curves for


the ridge-counts of homologous digits and total ridge-counts of two hands (TRCL and TRCR) juxtaposed to each other. We also plot in this figure the expected OBP curve for a normally distributed sporadic trait (dotted line) and the same for a Galtonian blending (multifactorial) trait. The OBP curves show nearly symmetric bilateral ridge-count transmissions from the parental to the offspring generation. The bilateral symmetry of total ridge-counts for the right and left hand is almostperfect as far as the pattern of transmission is concerned as depicted by the shape and rise of the OBP function. Furthermore, the total ridge-counts for each hand (TRCR and TRCL) virtually coincide with the OBP curves for complete Galtonian blending transmission (Fig. 1).

Comparison among digits: The MGI values for the first and fourth digits tend to be smaller than the MGI's observed for other digits. MPCC values are generally similar for the different digits, but the level of the OBP curve is highest for the thumb (digit 1) and becomes progressively lower for digits 2 through 5. Consistent with the lower MGI values for digits 1 and 4, the OBP curves show smaller jumps for these digits compared with the other digits.

DISCUSSION

We first address the more general issue of whether corresponding fingers of the left and right hands are determined by similar mechanisms before attempting to identify their specific modes of transmission. Traditionally, bilateral symmetry of dermatoglyphic traits is examined via paired t-tests for comparing the mean ridge-counts of homologous digits of the two hands (i.e., RCR(i) vs RCL(i), i= 1- 5) and via correlations. The largest correlations for ridge-counts between digits are reported for homologous fingers (Mavalwala, 1962). There are also reports suggesting that fingers of the right hand tend to have higher mean ridge-counts than those of the left (Jantz, 1974). A higher mean ridge-count for the right vis-a-vis the left hand is evident in our data as well (Table 2). However, correlations across digits and comparison of mean ridge-count values of homologous digits do not directly assess bilateral symmetry of homologous digits with respect to their transmission mode. The more conclusive result of nearly identical OBP curves and similar MGI and MPCC values for corresponding fingers of the left and right hands (Fig. 5 , Table 5) suggests that homologous digits are determined by similar transmission mechanisms.

The results from simulating discrete major gene, multifactorial, and sporadic transmission models significantly aid the interpretation of the SEDA statistics in their application to finger ridge-counts. The MGI scores are increased for discrete traits relative to their expected values for continuous distributions so that even multifactorial parent-offspring transmission may produce MGI scores near 1. The three summed ridge-counts TRCL, TRCR, and TRC, each having a broad range of possible values, have low MGI scores, but MGI statistics for individual ridge- counts tend to be higher. Digits 1 and 4 generally have higher ridge-counts (digit 1 having generally the largest mean) and greater variability between individuals than the other digits (Holt, 1958; Malhotra et al., 1980). Interestingly, the ridge-counts RCL(l), RCL(4), RCR(l), and RCR(4) also have MGI values significantly smaller than 1, suggesting that some of the individual digit levels are under multifactorial control or that at least several genes are involved. The other three digits are probably also inherited through multigene mechanisms and their high MGI values (larger than 1) probably reflect their smaller mean and more restricted ranges of possible ridge-count scores. Moreover, the decreasing trend in a for MGI value for the PI1 variable also appear to indicate multifactorial transmission for this trait as well, although its MGI statistics are close to 1. The high value (0.62) of MPCC for PI1 may also be regarded as close to the ex- pectation under a multifactorial model since 0.62 probably represents a slightly deflated value due to the discrete phenotypic expression of PI1 scores.

The SEDA statistics for the total ridge- counts for each hand (TRCL and TRCR) separately as well as when combined together (TRC) fit extremely well to the classical Gal- tonian multifactorial model of inheritance. In fact, a Galtonian continuous trait with complete transmission is expected to have an MGI(1/2) of 0.89, MGI(1) of 0.85, and MGI(2) of 0.78, and an OBP curve

n L

T OBP(P) = - arctan 0,

and these values are identical (up to two decimal places) to the corresponding values obtained for the combined total ridge-count of the two hands for the data analyzed. We conclude then that these results are consistent with the hypothesis that total ridge-

392 0

S. KARLIN, R. CHAKRABORTY, P.T. WILLIAMS, AND S. MATHEW

(a1 0 (b)

m l

P

( e l

I

0.0 0.5 1.0 1.5 2.0 2.5 '3.0

?- 0

'9- 0 - Ql

m ?-

- 0 0 0

2-

?- 0.0 d.5 i.0 i . 5 2.0 2.5 3.0

0

7 P

( f )

P P

Fig. 5 . Comparison of OBP curves for ridge-counts of homologous digits and total ridge-counts of left and right hands: (a) RCR (I), (b) RCL (11, (c) RCR (2), (d) RCL (2), (e) RCR (3), (0 RCL (3), (g) RCR (4), 01) RCL (4), (i) RCR (51,

(i) RCL (51, (k) TRCR, and 0) TRCL. A represents points for right hand and + represents points for left hand. The continuous concave curve are as in Figure 1.

3 -1

SEDA STATISTICS OF FINGER-COUNTS

0

- 0'

=- 0 -

Q

m 7-

- 0 00

9 01 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0

P

( i )

9 -1

m I 393

P

Fig. 5. continued

P


count is the sum of many small independent random variables presumably reflecting separate genetic determination. Penrose (1969) came up with a similar conclusion of multifactorial transmission of TRC based on the parent-child and sib-sib correlations using data collected by Holt (1968) on British populations. The multifactorial mode of inheritance of the ridge-counts of the ten individual digits has also been suggested by Holt, although the agreement of the observation and the theory in these cases are not clear as in the case of TRC (Holt, 1968).

The larger MPCC value for the total ridge- count of the two hands combined relative to the MPCC values for the total ridge-counts of the separate hands does not necessarily suggest that a greater number of genes are involved in determining TRC than in determining TRCL or TRCR. The high correlation and the similarity in the SEDA statistics for ridge-counts of homologous digits suggest common genes may determine the ridge- counts for corresponding digits of the left and right hands. The larger MPCC value for TRC than for TRCL or TRCR can be due to an averaging effect that reduces the random variation component in TRC relative to TRCL and TRCR.

Weninger (1964, 1965, 1976) has criticized the additivity model for summed ridge-count measures because she believes (1) the TRC is the sum of a mixed set of values from individual digits; and (2) the distribution of TRC is negatively skewed and flattened. It is true that there is considerable heterogeneity of ridge-count values among individual digits; the use of SEDA statistics whose statistical behavior is largely distribution-free (does not resuppose the shape and parametric form of the distribution) largely absolves criticism 2. The first criticism, however, is inherent in the definition of ridge-count and has been debated by Holt (1977) and Smith (1977). The suggestion of multifactorial mode of transmission is also provided in our analysis of individual ridge-counts where summations of heterogeneous entities are not involved.

Although postnatal environmental effects are virtually irrelevant for ridge-count variations, the early fetal development aberra- tions (during the first 4 months of embryonic and fetal life) may produce some profound environmental effects at the individual digital level (Mulvihill and Smith, 1969). Some indications of low ridge-counts (arising from a high frequency of arches) have been re-

ported in spontaneously aborted fetuses during the 11th through the 25th week of gestation (Babler, 1978).

The inheritance of arch patterns on the digital tips may also be relevant. Anderson et al. (1979), in a study of a large Israeli Habbanite pedigree, proposed that the presence of an “arch on any of the ten digits” behaves like a dominant gene with almost complete penetrance (95%) linked with the Haptoglobin locus. However, as shown in a separate communication (Chakraborty et al., 1982), the present set of data does not corro- borate their findings at least with regard to the genetics of the trait, suggesting that zero ridge-count (resulting from the presence of an arch), though in theory it can produce departure from multifactorial mode of inheritance, does not necessarily behave like a major gene.

As Meier (1980) argued, simple Mendelian inheritance (major gene transmission) for ridge-count variation seems implausible for a t least two reasons. First, it is unlikely that gene effects can be finger specific as well as pattern-type specific. Second, it is improba- ble that a one-to-one correspondence exists in a direct pattern-type-to-gene relationship, but rather indirect genetic effects are exerted through preceding as well as concurrent developmental events which themselves are de- pendent upon an attendant complexity of biochemical and physiological processes. Furthermore, the reference to the number of “genes” attributable to the mode of transmission of quantitative traits (e.g., skin color) is also based on a number of oversimplifying assumptions that are not beyond caveat (see, e.g., Byard and Lees, 1981, for their criticism of estimating the number of loci determining skin color).

SEDA methods were used by Young et al. (1981) to examine 48 digital dermatoglyphic traits in 192 nuclear families selected through a twin registry. Their conclusions are essentially in agreement with our own findings; they concluded that the ridge-count variables are consistent with a multifactorial model but there are some suggestions that the pattern-type variables may be under the control of major genes. Their interpretation of the MGI and OBP statistics as based on guidelines derived for continuous data. Their discussion did not account for the higher- than-1 values for the MGI or the discrete jumps that occurred for the OBP with trait values of very restricted ranges over very few


values. The pattern-type variable for individual digits analyzed by Young et a1 (1981) has a discrete distribution with only four possible values. Our simulations show the SEDA statistics may be very much influenced by such extreme discreteness. Their conclusion of a possible major gene for individual digit patterns may not be supportable by the SEDA methods and requires further investigation.

In conclusion, we state that even if the early developmental noises as well as other complicating forces (maternal effects, pleio- tropic effects, etc.) are responsible for making the individual digits distinctly different from each other in terms of their dermatoglyphic features, the finger ridge-count variations when summed over digits are in accordance with a multifactorial mode of transmission. The number of genetic factors influencing the ridge-count may, however, vary in different digits. For some fingers there may be fewer genes involved (the second and third fingers) whereas other fingers exhibit SEDA statistics more consistent with sufficient blending transmission (first and fourth fingers). Overall, the total ridge-count, as our SEDA statistics indicate, stands out as a classical example of a complete Galtonian trait.

LITERATURE CITED

Anderson, MW, Bonne-Tamir, B, Carmelli, D, and Thompson, EA (1979) Linkage analysis and the inheritance of arches in a Habbanite isolate. Am. J. Hum. Genet. 31:620-629.

Babler, WJ (1978) Prenatal selection and dermatoglyphic patterns. Am. J. Phys. Anthropol. 48:21-27.

Byard, PJ, and Lees, FC (1981) Estimating the number of loci determining skin color in a hybrid population. Ann. Hum. Biol. 8:49-58.

Carmelli, D, Karlin, S, and Williams, RR (1979) A class of indices to assess major-gene versus polygenic inheritance of distributed variables. In CF Sing and M Skol- nick (eds): Genetic Analysis of Common Diseases: Application to Predictive Factors in Coronary Disease. New York: Alan R. Liss, Inc. pp. 259-270.

Chakraborty, R, Mathew, S, Satyanarayana, M, and Ma- jumder, PP (1982) Inheritance of digital arches in man: Is the major gene fully penetrant? Am. J. Phys. An- thropol. 58:413-418.

Cummins, H, and Midlo, C (1976) Finger Prints, Palms and Soles. South Berlin, MA: Research Publishing Co., Inc.

Holt, SB (1949) A quantitative survey of the finger-prints of a small sample of British population. Ann. Eugen. (Lond) 14:329-338.

Holt, SB (1956) Genetics of dermal ridges: Parent-child correlations for total finger ridge-count. Ann. Eugen. (Lond) 20:270-281.

Holt, SB (1957) Genetics of dermal ridges: Sib pair correlations for total finger ridge-count. Ann. Eugen. (Lond) 21r352-362.

Holt, SB (1958) Genetics of dermal ridges: The relation between total ridgecount and the variability of counts from finger to finger. Ann. Hum. Genet. 22r323-341.

Holt, SB (1968) The Genetics of Dermal Ridges. Spring- field, IL: Charles C. Thomas.

Holt, SB (1977) Reply to Margarete Weninger’s “Consid- erations to total finger ridge-count.“ Bull. Int. Derma- toglyphics Assoc. VE5-7.

Jantz, RL (1974) Finger ridgecounts and inter-finger variability in Negroes and Whites. Hum. Biol. 46:662- 675.

Karlin, S (1979a) Approaches in modelling mode of inheritance with distributed traits. In CF Sing and M Skolnick (eds): Genetic Analysis of Common Diseases: Applications to Predictive Factors in Coronary Dis- ease, Alan R. Liss, Inc. pp. 229-257.

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Karlin, S, Carmelli, D, and Williams, R (1979) Index measures for the mode of inheritance of continuously distributed traits. I. Theory and justifications. Theor. Pop. Biol. 16:81-106.

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Karlin, S, Williams, PT, and Carmelli, D (1981b) Struc- tured exploratory data analysis (SEDA) for determining mode of inheritance of quantitative traits. I. Simulation studies on the effect of background distributions. Am. J. Hum. Genet. 33:262-281.

Karlin, S, and Williams, IT (1981~) Structured exploratory data analysis (SEDA) for determining mode of inheritance of quantitative traits. 11. Simulation studies on the effect of ascertaining families through high- valued probands. Am. J. Hum. Genet. 33r282-292.

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