Am J Hum Genet 34:888-903, 1982

The Cincinnati Lipid Research Clinic Family Study:Cultural and Biological Determinants of Lipids

and Lipoprotein Concentrations

D. C. RAO,' P. M. LASKARZEWSKI,2 J. A. MORRISON,2 P. KHOURY,2K. KELLY,2 R. WETTE,' J. RUSSELL,' AND C. J. GLUECK2

SUMMARY

A general linear model is described here for cultural and biological inher-itance of lipids and lipoproteins. This model involves 10 parameters to beestimated from a total of 17 correlations, leaving ample degrees of free-dom to test the goodness of fit. The model fits very well to each of thefive lipid and lipoprotein variables analyzed here from a Lipid ResearchClinic family data set. Both genetic and cultural inheritance are signifi-cant for each trait with the single exception that triglyceride levels fail tosupport genetic inheritance. Under the most parsimonious hypothesis,the genetic heritability (h2) ranges from .194 ± .092 for triglyceride to.624 ± .093 for low-density lipoprotein-cholesterol. Cultural heritabilityranges from .070 ± .030 for total cholesterol to .149 ± .034 for tri-glyceride.

INTRODUCTION

During the early seventies, the National Heart, Lung and Blood Institute estab-lished several Lipid Research Clinics (LRC) in the United States and elsewhere toassess, among other things, familial aggregation of plasma lipids and lipoproteins.

Received November 3, 1981; revised April 8, 1982.This work was supported in part by grants GM-28719 and MH-31302 from the National Institutes of

Health and National Institute of Mental Health, contract NO-1-HV-2-2914L from the National Heart,Lung and Blood Institute (Lipid Research Clinic's Program), General Clinical Research Center, andthe CLINFO center grant RR-00068-19.

Division of Biostatistics, Washington University Medical School, Box 8067, 4566 Scott Avenue, St.Louis, MO 63110.

2 The Lipid Research Clinic, The General Clinical Research Center, The CLINFO Center, Universityof Cincinnati, College of Medicine, 234 Goodman Street C2-3, Cincinnati, OH 45267.© 1982 by the American Society of Human Genetics. All rights reserved. 0002-9297/82/3406-0007$02.00

888

LIPID RESEARCH STUDY

The Cincinnati LRC is one of the clinics that collected family data followingseveral sampling schemes, including a random sample of families. Here we presentan analysis of the random sample of families from the Cincinnati LRC using pathanalysis methodology. Specifically, resolution of cultural and biological inheri-tance is attempted for each of the following five variables: total plasma cholesterol(CH); triglyceride (TG); high-density lipoprotein (HDL)-cholesterol; low-densitylipoprotein (LDL)-cholesterol; and very-low-density lipoprotein (VLDL)-choles-terol. The population under present study is culturally and ethnically very differ-ent from the population of Japanese-Americans in Hawaii reported earlier [1].The Cincinnati LRC Family Study presented here was randomly selected from alarger prevalence study [2], while the Honolulu Heart Study [1] used hyperlipi-demic probands.

Population Studied

The Cincinnati LRC Princeton School District Family Study (1976-1978) [2] was a partof the National Heart, Lung and Blood Institute's multicenter collaborative program de-signed to assess the familial aggregation of lipids and lipoprotein levels [3]. Briefly, thePrinceton School District Population Study was a two-stage epidemiological survey oflipids, lipoproteins, and other coronary heart disease risk factors in a biracial population ofschool children in grades 1-12 and their parents. Following the first two visits of theprevalence study [4], a subgroup of probands was drawn from this larger prevalence popu-lation for the Family Study. All first-degree relatives and spouses of selected probands werecontacted; socio-demographic data, fasting plasma lipids and lipoproteins, and clinicalchemistry measurements were obtained. Probands for the Family Study included bothrandomly selected subjects and hyperlipidemic subjects [2, 5, 6].

Here we analyze data on Caucasian families ascertained through the randomly selectedprobands. This data set consists of 70 young probands and 55 adult probands. In all, only146 parents of the 125 probands were studied. More fathers failed to cooperate in the studythan did mothers. Most of the 55 adult probands were married, giving rise to some 3-generation families. These families were split into the component nuclear families, avoidingduplications whenever possible; that is, any given individual in a pedigree was included inonly one nuclear family, with a few exceptions. This process generated, in all, 160 nuclearfamilies containing 98 fathers, 131 mothers, and 338 children. Admittedly, there are manymissing values, but it is difficult to see how this would affect the results systematicallyexcept, perhaps, to reduce the power of hypothesis tests a little bit. In addition to the 160nuclear families, very few adopted relatives and a few half-sibs of the probands were alsostudied. Because of very small sample sizes, the adopted relatives are not analyzed here.However, half-sib correlations are estimated even though the sample sizes are small. Moredetails of the population studied can be found in Laskarzewski et al. [7] and Morrison etal. [2].

PATH ANALYSIS

Within-family associations were recently assessed among the five variables usingthe same data [2]. For parents and their pediatric and adult offspring, CH andLDL-cholesterol were, for the most part, significantly correlated. There was noconsistent pattern of higher correlations for parents and their pediatric offspringcompared with parents and their adult offspring for any of the lipid and lipopro-tein cholesterols except for HDL-cholesterol. Parents and their pediatric offspringhad higher correlations for HDL-cholesterol than did parents and their adult

889

offspring. All pediatric and adult sibling correlations for lipids and lipoproteinswere significant with the exception of HDL-cholesterol correlations in adult sib-lings. There was no consistent pattern of higher sibling lipid or lipoprotein correla-tions when pediatric siblings were compared with adult siblings, except for HDL-cholesterol correlations that were higher in the pediatric siblings of both races(Caucasians and blacks). Morrison et al. [2] concluded, "Whatever the relativecontributions of genetics and environment are to the association of lipid-lipoproteinvalues between parents and offspring and between siblings, the magnitude of theassociation (except for HDL-cholesterol) appears to outlast the period of sharedhousehold environment." Here we analyze the observed family resemblance forthe five lipid and lipoprotein variables.Age and sex have substantial effects on lipid and lipoprotein concentrations in

plasma, while behavioral, social, and physiological factors such as obesity, hemato-crit, special diet, smoking, and alcohol consumption generally have smaller ef-fects [7, 8]. The Cincinnati LRC family data was first adjusted for age and sexeffects by stepwise multiple regression, retaining only significant terms [7]. Theresiduals were then normalized for the purpose of this investigation.We assume that genotype and family environment act additively to produce a

lipid or lipoprotein phenotype. All interactions are assumed to be negligible sincenone have been demonstrated in human populations. Since family environment isnot directly measurable, an estimate of the environment, called an index, is createdfor each lipoprotein variable by regressing the variable simultaneously on mea-sured social and physiological factors, with covariance adjustment for age and sexeffects. In fact, each lipoprotein variable was separately regressed on sex, age, age2,age3, sex X age, sex X age2, sex X age3, contraceptive usage, obesity as mea-sured by Quetelet index (weight/height2), hematrocrit, and special diet usage in astepwise fashion retaining only the significant terms [7]: X = flA,S,C) + g(Z) + e,where X = lipid or lipoprotein value;f(A,S,C) = polynomial involving age, sex,and contraceptive terms; g(Z) = linear function of obesity, hematocrit, and spe-cial diet usage; and e = residual error. A cubic function of the type considered inf(A,S,C) sufficed to eliminate age and sex effects almost completely. After fittingthe above equation, we defined the age-sex-contraceptive adjusted lipid or lipopro-tein value and the index (I) as P* = X - J(A,S,C), and I = g(Z), wherej and gdenote the estimated contributions. The part of the environment so estimated bythe index (I) is called family environment or indexed environment, and the remain-der is the residual environment. We then defined the phenotype (P), for the pur-poses of the present investigation, as the normalized P* value (obtained throughinverse normal transformation of the ranked residuals).

Construction of the indices (I) needs some clarification. The first question thatoften arises is what variables the index should be based on. Ideally, we prefer to

include in the above regression [g(Z)] only relevant environmental variables. Thehigher the correlation of the index variables with the phenotype, the more inform-ative the index is. However, often we do not have many relevant variables toderive very informative indices, or we are not sure of the genetic basis of some ofthese variables, or both. In our study, perhaps the only relevant environmental

RAO ET AL.890

LIPID RESEARCH STUDY

variable is special diet usage, which was measured as a binary variable: peoplewere simply asked whether or not they were on a "special diet." "Yes" and "no"were recorded as 2 and 1, respectively. Since this alone was highly uninformative,we included obesity (as defined by Quetelet index) and hematocrit on the groundsof minimal genetic involvement, if any. In a recent review, Garn et al. [9] con-cluded that fat people have fat relatives whether they are genetically related or not,suggesting that familial fatness is the result of living together and not of genes heldin common. If this is true, then obesity is an ideal index variable. In any case, thegenetic component appears very small [10]. Therefore, an assumption in thisanalysis is that the index (I), based on special diet usage, hematocrit, and obesity,is an estimate of the family environment alone. This neglects possible geneticcorrelation between the lipid/lipoprotein variable and the index. If such geneticcorrelation exists, we expect our analysis to reduce the genetic heritability (h2) andenhance the cultural heritability (c2). However, so long as the genotype for thelipid/lipoprotein variable and the genotype for the index are uncorrelated, theseheritabilities will not be altered (even if the index variables, such as obesity, arepartly genetically determined).The model we used here is presented in figure 1, which is a special case of the

one presented by Rao et al. [11]. The 10 parameters of this model are defined intable 1. Marital resemblance is treated in terms of correlated environments ofspouses (u). This parameter u reflects the combined effects of secondary assorta-tive mating (called social homogamy) and cohabitation, even though we oftenrefer to it as the effect of assortative mating. Intergenerational differences areretained: whereas h2 and c2 are the genetic and cultural heritabilities in children,

z Cy v z

MS ~~~~VtCIV

FIG. 1.-Path diagram showings cultural and biological inheritance of lipids and lipoproteins innuclear families. The subscripts F. M, Cl, and C2 denote father, mother, and two children, respective-ly. P is phenotype, G is genotype. C is transmissible environment with index 1, and B is nontransmittedcommon sibship environment.

891

TABLE I

DEFINITION OF THE PARAMETERS OF THE MODEL (FIG. 1)

Parameter Definition

h.Effect of genotype on child's phenotype (square root of genetic heritability)hz........... Effect of genotype on adult's phenotype

c.Effect of environment on child's phenotype (square root of cultural heritability)cy........... Effect of environment on adult's phenotypeu............ Correlation between parental environments

b.Effect of nontransmitted common sibship environment on child's environment

fF........... Effect of father's environment on that of a child he rears

fM. Effect of mother's environment on that of a child she rearsi............ Effect of environment on child's indexiv. Effect of environment on adult's index

they are h2z2 and c2y2 in adults. Maternal effects are also retained by distinguishingthe effects of paternal (fF) and maternal (fM) environments on that of their child.Indices of father and mother are treated similarly: the path coefficient from in-dexed environment to index is i for children, and iv for adults (regardless of sex).Whereas the previous model [1] postulated a single indexed environment commonfor the entire sibship, the present one incorporates separate indexed environmentsand indices for each child, in addition to a nontransmitted common sibship envi-ronment (B).

There are 15 correlations between pairs of the following six variables: PF =phenotype of father, IF = index of father, PM = phenotype of mother, IM =index of mother, PC = phenotype of a child, and IC = index of a child. However,since the expected correlations are identical for (PF' IF) and (PM, IM)' and also for(PF IM) and (PM' IF)' the number of correlations with distinct expectations re-duces to 13. Sibships generate three more correlations: between phenotypes ofsibs, between indices of sibs, and phenotype of a child with sib's index. This makesthe total number of correlations 17, including a half-sib correlation (betweenphenotypes). All the 17 expected correlations are presented in table 2. The corre-sponding observed correlations were estimated by the method of maximum likeli-hood as described in the APPENDIX. The estimated correlations (r) and theirsample sizes (n) for each of the five lipid and lipoprotein variables are given intable 3.The statistical method of analysis utilizes the distributional properties of Fisher's

z-transformation. An observed correlation ri is first converted into its z transformzi = ½2 ln[(1 +ri)/(l - ri)], which asymptotically follows a normal distributionwith mean zi = l2 In[(1 + pi)/(l - pi)], and approximate variance 1/ni, wherepi is the corresponding expected correlation given in table 2 (i = 1, 2, . . ., 17).Therefore, assuming all 17 observed correlations to be independent, we take theoverall log likelihood (approximately) as ln L = -X2/2 + constant, where

172 -)X = ni(zi - Zi)2

i=l

892 RAO ET AL.

LIPID RESEARCH STUDY

TABLE 2

EXPECTED CORRELATIONS FOR NUCLEAR FAMILIES AND HALF-SIBS DERIVED USING FIGURE 1

Relation Variables Correlation (p)

Marital ......F' PM Uc2y 2

IF, IM UiIV2(PF' IM) or (PM, IF) ucyiv

Parental ......................... (PF' IF) or (PM, IM) cyivFull-sibs ......................... PCI' PC2 h2/2 + c2q,

IC1, IC2 1i2

PCI Ic Ci(PCI,1IC2) or (PC2 ICId) 41i

Parent-offspring .*.----.---- PF' PC h2z/2 + c2y(fF + UfM)IF' PC CiV(fF + UfM)PF' IC cyi(fF + UfM)IF' IC i2V(fF + UfM)PM' PC h2z/2 + C2y(fM + UfF)IM' PC CiV(fM + UfF)PM' IC cyi(fM + UfF)IM' ' C i2V(fM + UfF)

Half-sibs ........................ PHI' PH2 h2/4 + c24i

NOTE: P = phenotype, I = index. Subscripts F, M, and C refer to father, mother, and child, respective-ly. Cl, C2 denote full sibs, and Hl, H2 denote half sibs. if = b2 + fF2 + fM2 + 2ufFfM; half-sibs arereared together by parents of one.

The residual x2 after estimating k parameters follows a chi-square distributionwith 17 - k df. By specifying pi's as functions of parameters (see table 2), wemake x2 a function of the parameters. Therefore, by minimizing x2 we can esti-mate all or some parameters and use the residual x2 for tests of hypotheses. Thegeneral model, in 10 parameters, is tested by the residual x2 with 17 - 10 = 7 df.If X27 is the value of x2 after estimating all the 10 parameters, and X27+w is anothervalue after estimating only 10 - w of the 10 parameters, the other w parametersbeing fixed under a null hypothesis, X2W = X2 7+w - X27 provides the likelihood-ratio test for the null hypothesis on the w other parameters.

All these methods have been implemented in PATHMIX, a computer programwritten in FORTRAN for HARRIS computers [12]. Analyses presented here wereperformed using PATHMIX.A fundamental assumption of this analysis is that the different sample correla-

tion coefficients are independent. Clearly, estimates of correlations obtained fromthe same data are not all independent of one another. Our previous methodsincorporated correlations between correlations to account for this nonindepen-dence [1, 13]. However, several analyses performed under independence and nonin-dependence assumptions gave essentially identical results [1, 14-17]. We thereforefeel justified in adapting the simpler approach here.

893

RESULTS

All five lipoprotein variables were analyzed using the estimated correlations oftable 3. Goodness-of-fit x2 values and estimates of parameters are presented intables 4-8 under various subhypotheses, separately for each variable. In whatfollows, y = z = 1, u = 0, andfF = fM correspond to the three hypotheses of nointergenerational differences in heritabilities, no effect of assortative mating andcohabitation, and no maternal effects, respectively.

Total cholesterol (CH) (table 4)

The general model fits impressively well (X27 = 4.74, P > .65), unlike for theHonolulu Heart Study (HHS), which required a special transient maternal effectto give a marginal fit [1]. There is no significant evidence for intergenerationaldifferences in heritabilities, marital resemblance, or maternal effects. Other sourcesof family resemblance have significant effects.

Triglyceride (TG) (table 5)

The general model fits very well (X27 = 5.92, P > .50). There is no evidence forintergenerational differences or maternal effects, but the evidence is marginallysignificant for marital resemblance (x2, = 9.97 - 5.92 = 4.05, P < .05). Where-as we obtained overwhelming significance for genetic determination of TG levelsin the HHS, genetic inheritance does not approach significance in this Caucasianpopulation (X22 = 10.45 - 5.92 = 4.53, P > .10).

HDL-Cholesterol (table 6)The general model fits very well (X27 = 6.33, P > .49). Once again, intergener-

ational differences and maternal effects are not significant, but marital resemblanceis marginally significant (X21 = 4.06, P < .05). Other sources of family resem-blance are all highly significant.

LDL-Cholesterol (table 7)

The general model fits the data very well (X27 = 5.88, P > .54). Again, inter-generational differences and maternal effects are not significant, but marital re-semblance is (X2, = 4.78, P < .03). Other sources are highly significant.

VLDL-Cholesterol (table 8)

The general model fits very well (X27 = 5.55, P > .58). Once again, whereasintergenerational differences and maternal effects are not significant, marital re-semblance is (X21 = 4.47, P < .04). Other sources of family resemblance are veryhighly significant.On the whole, every trait gives highly significant evidence for both genetic and

cultural inheritance with the single exception that TG fails to support geneticeffects. The nontransmitted sibship environment (B) has significant effects on alltraits. Combined effect of assortative mating and cohabitation (u) is marginallysignificant for all traits except CH. Neither intergenerational differences in herit-

894 RAO ET AL.

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arrived at the most parsimonious hypothesis for each trait. This was achieved bycombining several hypotheses that were not rejected. Since neither y = z = 1 nor

fF = fM was rejected for any trait, we took y = z = 1 and fF = fM as a par-simonious hypothesis, which provided excellent fits to all five variables. Further,since the parameter u was only marginally significant, and for reasons given in thenext section, we set u = 0 in addition toy = z = 1 andfF = fM. Not only did sucha parsimonious hypothesis give very good fits to all data, but the parameter esti-mates were also remarkably similar: the only exceptions were that when u was set tozero, estimates off(= fF = fM) increased a little bit, and estimates of b dropped alittle bit. In particular, estimates of the genetic (h2) and cultural (c2) heritabilitieshardly changed, and that too only in the third decimal place. Therefore, we tooky = z = 1, fF = fMq and u = 0 (no intergenerational differences in heritabilities,no maternal effects, and no marital resemblance) as the most parsimonious hy-pothesis for each trait. Goodness-of-fit x2 values under the most parsimonioushypothesis, the corresponding estimates of relative variance components (h2 andc2), and the two other parameters of interest are presented in table 9. Geneticheritability (h2) is least for TG (.194) and highest for LDL-cholesterol (.624).Cultural heritability (c2) is least for CH (.070) and highest for TG (.149). In thispopulation also, the family resemblance is largely genetic for most lipoproteins.For ready reference, we also included in table 9 the heritability estimates from theHHS [1].

DISCUSSION

In 1979 path analysis methodology was applied to the HHS family data onlipids and lipoproteins [1, 8]. Here we applied the same methodology to anotherfamily study generated as a part of the LRC program [2]. These two investigationsproduced comparable results to a large degree, yielding a few minor discrepanciesthat may partly be due to differences between the two studies.The HHS first selected a cohort of adult males and determined their lipid and

lipoprotein levels. A group of probands were then selected from this cohort (withelevated lipid levels), and the lipid and lipoprotein levels of their wives and chil-dren were determined a few years later, giving a total of 496 nuclear families. Onthe other hand, the Cincinnati LRC study represents a younger population fromwhich a random sample of families was selected. In almost all cases, the lipid andlipoprotein analyses of different members of a family were carried out at the sametime. In a few exceptional cases, such values were taken from a previous visitblood sample. There were a few 3-generation families that were split into thecomponent nuclear families, giving a total of 160 nuclear families.Such differences between the two studies account for most of the discrepancies

between the two analyses. Whereas the HHS concluded in favor of significant

898 RAO ET AL.

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genetic heritability for TG, we failed to obtain this evidence in the LRC sample.This may well have been because of reduced power (smaller sample size). How-ever, since the heritability is known to increase with age [18], the evidence may bediluted in the LRC study that represents a pediatric population. Unlike in HHS,we detect significant assortative mating for most traits in the LRC sample. Thisevidence seems to come primarily from indices, and not from phenotypes; esti-mates of phenotypic marital correlations are much smaller than the marital corre-lations for indices. In the HHS, marital correlations for indices were mostly nega-tive, and not as large as here. The indices of the HHS utilized information onsmoking and alcohol consumption. Perhaps more couples are discordant for theseattributes, contributing a negative value to the marital correlation. Even thoughobesity alone contributes a positive marital correlation, the net resultant is anegative marital correlation for indices in the HHS. In any case, the most parsi-monious hypotheses exclude assortative mating.

Maternal effects raise several points. First, the prior testing of fathers in theHHS must have reduced the father-child correlations (relative to mother-childcorrelations), exaggerating the evidence for maternal effects. Secondly, families inHHS were selected through fathers with elevated lipid levels. Ignoring this consid-eration also results in the underestimation of father-child correlation (relative tomother-child correlation), once again exaggerating the evidence for maternal ef-fects. In any case, maternal effects do exist in the LRC sample, although they areless pronounced than in the HHS sample; lack of significance must be partly dueto reduced power.

APPENDIX

MAXIMUM-LIKELIHOOD ESTIMATION OF FAMILIAL CORRELATIONSFROM QUANTITATIVE DATA ON NUCLEAR FAMILIES

Since the statistical method of analysis considers the estimates of various familial correla-tions to be independent, even when estimated from the same data, here we rely on estimat-ing the correlations individually as opposed to the simultaneous estimation in Rao et al. [1];this also resolves the numerical problems associated with simultaneous estimation. For thisreason, we shall first describe a general algorithm to estimate correlations, and then discusshow the different correlations are estimated using this algorithm.

General Algorithm

Consider N individual or joint observations on two vectored random variables X and Y.For example, X may denote observations on parents and Y may denote the same onchildren. Since the number of parents and sibship size vary among families, X and y can beof varying dimensions. The ith unit of observation (a family in the above example) hasinformation on

(XI, X2, .... Xm) and (YI,Y2*, Yni),

For a family of one parent and three children in the example, mi = 1 and ni = 3. Careshould be taken to use X and Y consistently. For example, parents in each family should be

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represented by the same vector, X say. The conditions on the dimensionality of the vectorsare: mi > 0 ni > 0 but mi + ni > 1. Moments of X and Y are specified as follows:E(Xj) = ml, V(Xj) = 12, p(XjXk) = pl;j # k = 1,2, m,; E(Yj) = 12,V(Y) = a22,p(Yp, Yk) = P2; i $ k = 1, 2., ni; and

Xp12ifj = k j 1,2,... ,mp(X},Yk)- P12 ifij k k =1, 2,. ** ni

where E( ) stands for expectation, V( ) stands for variance, and p(-, ) stands for correlationcoefficient. PI, and P12 are equal in all but one case (described later). We assume thatthe variables

(X I. Xmi, Y,*, Yni)

jointly follow an (mi + ni)-variate normal distribution, and write the log-likelihood of theith observation as In Li = -½2(lnIYll + D'U 'D), where 111 = determinant of X,

D' =(X I I, **,Xm, Al, Y. - I2,..., Yni-I2) = transpose of D

each diaga e nE * = covariance matrix of the X's and Y's;

each diagonal element of y;j =aj2, j = 1, 2, each off-diagonal element of Xj = pj Uj2,] = 1, 2, each (j, k)th element of ,12 = Pu1u2, where p = pi, if j = k and P = P12otherwise. If a sample consists of N observations, the overall log-likelihood is

N

In L = LIn Lii= 1

The total number of unknowns in in L is 8: 1A], ,u2, a21 U22, P,, P2, PI,, and P12. As discussedlater, Pl = P12 except in one case, reducing the number of parameters to seven. Some or allof these eight parameters can be estimated by maximizing ln L. A FORTRAN program calledMLECOR is developed for this purpose which uses GEMINI, a generalized minimizationroutine [19].

Estimation of Correlation Coefficients from Nuclear Family Data

Another FORTRAN program called PATHMIX [12] was developed for analysis of nuclearfamily data. The general algorithm described above constitutes a core part of PATHMIX. Byrepeated use of this algorithm, we estimate various familial correlations as described below.Consider the phenotypic value (P) on each member of a family (father, mother, and chil-dren). It is assumed that these data are covariance-adjusted to remove the effect of suchvariables as age and sex, and suitably transformed (such as inverse normal transformationof the residuals). Create an index (I) for each member as described in the text. For a nuclearfamily with s children, define the following 2s + 4 variables: PF = phenotype of father,IF = index of father, PM = phenotype of mother, IM = index of mother, Pi = phenotypeof the ith child (i = 1, 2, . . , s), and Ii = index of the ith child (i = 1, 2, . .s). Sample

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means are computed for each variable, noting that all sibling phenotypes have the samemean, and likewise all sibling indices have the same mean. Also, sample variances ofPF F' IF'PM' and IM are computed. These six means and four variances are taken as known values,

estimating the remaining relevant parameters by repeated use of the general algorithm. For

example, we define IF and PM as X and Y, respectively (mi . 1, ni . 1). fix the twomeans and two variances at the sample values, and estimate P12 by settingppa =P12P Notethat pi and P2 do not arise since mi < 1 and ni < 1. This way we estimate the maritalphenotypic correlation in one use of the general algorithm. Sibling correlations are estimat-ed in three steps: (1) We define P1, P2. P, as X (mi = s, and ni = 0), and estimate the

sibling phenotypic variance (01') and sibling phenotypic correlation (p,) by fixing the meanat the sample value. Note that P2' 022, P2,P',, and P12 do not arise since ni = 0; (2)WedefineI , 2.. . I, as X(mi = s, ni = 0), and estimate the variance and correlation as

above, this time for indices; (3) Finally, we defineP ,.. . P, as X andI , ,I, asY, fixmeans at sample values and fix variances and correlations at the values estimatedabove,and estimatep,, and P12 (this is the only case wherepl,# P12)-

Parent-offspring correlations are estimated one at a time. For example, we define P F as X

(mi . 1) and P1, P2,. ., P, as Y(ni = s), fix means and the variance ofPF at the samplevalues, fix the variance and correlation of the Pi's at the values estimated above, and finallyestimate the father-child correlationpl,(= P12). Note that p, does not arise because mi < 1.Approximate Sample Sizes

If a correlation is based on n pairs of observations, the sample size associated with thecorrelation is simply the number of pairs used. For correlations such as the sibling correla-tion, however, data on all sibship sizes are used. Since not all sibpairs are independent, totalnumber of sibpairs would be an overestimate of the "sample size." Similarly, the totalnumber of sibships used (each with at least two sibs) provides an underestimate. In suchcases, we propose to estimate the sample sizes as follows. The relevant correlation estimatedis eitherp, or P2 orp,l or P12Ifit is Pi, we are dealing with N X vectors of varying size thatwould involve only three parameters: .LI, u12, andpi. At the end of estimatinga,2 andp,(fixing MI), we calculate the empirical information matrix between all the three estimates,say K (3X 3). The diagonal element corresponding top, in the inverse of the K-matrix, K-'(3 X 3), denoted by V(p1), gives the empirical variance ofp,. Since the largevariance of A, based on n pairs is (1 - p12)2/n [20], we define the approximate number of

pair equivalents used in estimatingp, as [1] i = (1 - p2)2/V(p,). The above argumentholds for P2 also, with the subscript 1 replaced by 2. When pij(pll or P12) is estimated, all

eight parameters are involved. At the end of estimating the relevant parameters, we calcu-late one empirical information matrix between all eight estimates, say K (8 X 8). If the

diagonal element corresponding topi in K-} (8 X 8) is denoted by V(pij), the above

equation yields the desired sample size withpi in place ofp. In our judgment, this

provides an objective way of computing sample sizes, not necessarily always correct. In any

case, reciprocal of n provides empirical variance of the z-transform, and therefore it may becalled a "weight" rather than a sample size.

REFERENCES1. RAO DC, MORTON NE, GULBRANDSEN CL, RHOADS GG, KAGAN A, YEE S: Cultural and

biological determinants of lipoprotein concentrations. Ann Hum Genet 42:467-477,1979

2. MORRISON JA, KELLY KK, HORVITz R, ET AL.: Parent-offspring and sibling lipid and

lipoprotein associations during and after sharing of household environments. ThePrinceton School District Family Study. Metabolism 131:158-167, 1982

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3. HEiss G, TAMIR I, DAVIS CE, ET AL.: Lipoprotein-cholesterol distributions in- selectedNorth American populations: The Lipid Research Clinics Program Prevalence Study.Circulation 61:302-315, 1980

4. MORRISON JA, KELLY KK, MELLIES MJ, DEGROOT I, GLUECK CJ: Parent-child associa-tions at upper and lower ranges of plasma cholesterol and triglyceride. Pediatrics62:468-478, 1978

5. KELLY KK, AUSTIN M, MACIOLOWSKI M, ETAL.: The Collaborative Lipid Research Clin-ics Family Study: design, ascertainment, lipids and lipoproteins, 1982. In preparation

6. TYROLER HA, ANDERSON T, CHASE G, ELLIS L, MOWERY R, VALULICK D: The LipidResearch Clinics Population Based Family Study, in Genetic Analysis of CommonDiseases: Application to Predictive Factors in Coronary Disease, edited by SING C,SKOLNICK M, New York, Alan R. Liss, 1979, pp 647-652

7. LASKARZEWSKI PM, RAO DC, MORRISON JA, KHOURY P, GLUECK CJ: The CincinnatiLipid Research Clinic Family Study: social and physiological determinants of lipidsand lipoprotein concentrations. Hum Hered. In press, 1983

8. GULBRANDSEN CL, MORTON NE, RHOADS GG, KAGAN A, LEW R: Behavioral, social andphysiological determinants of lipoprotein concentrations. Soc Biol 24:289-293, 1977

9. GARN SM, BAILEY SM, HIGGENS ITT: Effects of socio economic status, family line andliving together on fatness and obesity, in Childhood Prevention of Atherosclerosis andHypertension, edited by LAUER RM, SHEKELLE RB, New York, Raven Press, 1980

10. GLUECK CJ, LASKARZEWSKI PM, RAO DC, MORRISON JA: Familial aggregation of coro-nary risk factors, in Complications in Coronary Heart Disease, edited by CONNOR W,BRISTOW D, New York, LIPPINCOTT. In press, 1982

11. RAO DC, MORTON NE, CLONINGER CR: Path analysis under generalized assortativemating. I. Theory. Genet Res 33:175-188, 1979

12. MORTON NE, RAO DC, LALOUEL JM: Methods in Genetic Epidemiology. Basel, Switzer-land, and New York, Karger AG. In press, 1983

13. ELSTON RC: Correlations between correlations. Biometrika 62:133-148, 197514. RAO DC, CHUNG CS, MORTON NE: Genetic and environmental determinants of perio-

dontal disease. Am J Med Genet 4:39-45, 197915. MORTON NE, GULBRANDSEN CL, RAO DC, RHOADS GG, KAGAN A: Determinants of

blood pressure in Japanese-American families. Hum Genet 53:261-266, 198016. GULBRANDSEN CL, MORTON NE, RAO DC, RHOADS GG, KAGAN A: Determinants of

plasma uric acid. Hum Genet 50:307-312, 197917. KRIEGER H, MORTON NE, RAO DC, AZEVEDO E: Familial determinants of blood pressure

in Northeastern Brazil. Hum Genet 53:261-266, 198018. GOLDSTEIN JL, SCHROTT HG, HAZZARD WR, BIERMAN EL, MOTULSKY AG: Hyperlipidemia

in coronary heart disease. II. Genetic analysis of lipid levels in 176 families anddelineation of a new inherited disorder, combined hyperlipidemia. J Clin Invest 52:1544-1568, 1973

19. LALOUEL JM: GEMINI-A computer program for optimization of general nonlinearfunctions. Technical Report No. 14, Department of Medical Biophysics and Comput-ing, Salt Lake City, Univ. of Utah, 1979

20. FISHER RA: On the probable error of a coefficient of correlation deduced from a smallsample. Metron 1: 1-32, 1921

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