HeritabilityScores: Science o

The question, "To what extent canthe development of basic cognitiveskills be influenced by various kinds ofenvironmental intervention?" is cen-tral to current discussions of educa-tional and social policy. Much of thediscussion has focused on one rathernarrow, but comparatively well-defined,aspect of this question-the heritabilityof scores on standardized tests and theimplications of heritability estimates.In 1969, A. R. Jensen (1) reviewed alarge number of British and Americanstudies of the broad heritability of IQscores and concluded that, for repre-sentative British and American popula-tions, it probably lies between 0.7 and0.8, the best single estimate being closeto 0.8. Jensen argued from these figuresthat inequalities in cognitive perform-ance are largely genetic in origin andthat comparatively little can be doneto reduce them through practicable ed-ucational and social reforms, a thesishe has since developed further (2). Healso argued that the reported differencein average IQ between black and whitechildren in the United States probablyhas a substantial genetic component,another thesis elaborated later (3). Fromthe same heritability estimates, R. J.Herrnstein has argued (4, 5) that theelimination of artificial barriers tosocial and economic mobility (such asthose based on race, sex, income, andsocial class) must inevitably lead to theemergency of an hereditary meritoc-racy based on IQ.

After the publication of Jensen's1969 article, Christopher Jencks under-took a detailed analysis of the availabledata on IQ. Devoting themselves to thebroader question of how family andschooling affect social and economic, aswell as cognitive, inequality, Jencks andhis colleagues concluded (6, p. 315)that "the chances are about 2 out of 3

The author is professor of astronomy, HarvardUniversity, Camlbridge, Massachusetts 02138.

29 MARCH 1974

Analyses of IQr Numerology?

David Layzer

that [the heritability of IQ scores] isbetween 0.35 and 0.55." But Jencks'analysis also indicated that the purelyenvironmental contribution to the IQvariance probably lies between 0.25 and0.45-an estimate only mildly discor-dant with Jensen's estimate of 0.2 to0.3. Jencks and his collaborators con-cluded, moreover, that educational in-equalities accounted for only a minorpart of the environmental variance.From this brief summary, it is clear

that two methodologically distinct kindsof issues are involved in current dis-cussions of IQ heritability and its im-plications. Some of the issues are purelyscientific: What are the limitations ofconventional heritability analysis as ap-plied to IQ scores? How reliable areheritability estimates like those quotedabove? What inferences can legitimatelybe drawn from heritability estimates ofvarious kinds? Of more widespread in-terest than these technical questions arethose that involve social, political, ed-ucational, and philosophical considera-tions. Questions of the first kind can bediscussed and resolved in a value-freecontext-or at least in a context ofvalues agreed upon by members ofthe scientific community. Moreover,they must be resolved before thebroader issues can be meaningfullydebated. In this article I address myselfto that task.

Although several illuminating discus-sions of IQ heritability have alreadyappeared, it seems to me that some im-portant aspects of the problem have notyet been adequately discussed. Forexample, nearly all the published dis-cussions that I am aware of take it forgranted that some meaningful estimateof IQ heritability-high or low, roughor accurate-can be extracted from thereams of published statistics and thatrefinements of current techniques forgathering and analyzing test data maybe counted on to yield increasingly

reliable estimates. These propositionsare by no means self-evident, however,and one of my purposes here is todemonstrate that they are actually false.

This conclusion rests upon two argu-ments: One concerns the limitations ofconventional heritability analysis, theother the validity of IQ scores asphenotypic measurements. Contrary towidely held beliefs, (i) heritability analy-sis does not require the genotype-en-vironment interaction to be small, and(ii) a high phenotypic correlation be-tween separated monozygotic twinsdoes not, in general, imply that thegenotype-environment interaction issmall. If genotype-environment inter-action does contribute substantially tothe phenotypic value of a trait (as thereare strong biological reasons for sup-posing in the case of phenotypicallyplastic traits), then a necessary andsufficient condition for the applicabilityof heritability analysis is the absence ofgenotype-environment correlation. Thiscondition is rarely, if ever, met for be-havioral traits in human populations.The second argument is that IQ scorescontain uncontrollable, systematic errorsof unknown magnitude.

Limitations of the

Heritability Concept

Estimates of broad heritability (h2)answer the question: What fraction ofthe variance of a phenotypic trait ina given population is caused by (orattributable to) genetic differences? Aphenotypic value (P) that depends on agenotypic variable (or set of variables)(x) and an environmental variable (orset of variables) (y) may be expressedas the sum of a genotypic value [G(x)],an environmental value [E(y)], and aremainder [R(x,y)] that depends jointlyon x and y (see Eq. 4). Under certainconditions (which probably never ob-tain in natural human populations),there is a well-defined optimal addi-tive decomposition of this kind-that is,a decomposition that minimizes thevariance (Var) of R. Under these con-ditions, the quantities h2 =Var{G}/Var{P} and e2 = Var{E}/Var{P} (e2is the environmental fraction of pheno-typic variance) are well defined.Even when h2 and e2 are well defined,

however, their practical significance maybe obscure. Consider the hypotheticaltrait illustrated in Fig. 1. Note that thephenotypic value (P) cannot be ex-pressed as the sum of a genotypic value

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[G(x)] and an environmental value[E(y)]. Consider, for example, the pairsof points (A1, A2) and (B1, B2) in Fig.1. Since AP (the difference between thephenotypic values) = 0 for each pair,the differences AG and AE would beequal and opposite if an additive repre-sentation were possible. But AG musthave the same value for both pairs,and the values of AE must evidently beunequal [since they correspond to dif-ferent increments (Ay)]. Hence an addi-tive representation is impossible (7).

I show below that G and E can beunambiguously defined if, and only if,genotype-environment correlations areabsent. Even then, however, a certainpractical ambiguity persists. Geneticdifferences may influence the develop-ment of a trait in, qualitatively distinctways. For example, the curves labeledxl, x2, and x3 in Fig. 1 have differentthresholds, different slopes, and differ-ent final values. Heritability estimatesdo not take such qualitative distinctionsinto account. Thus, if the environmen-tal variable y is distributed in a nar--row range about the value Yi, as illus-trated in Fig. 1, h2 is close to unity.Yet in these circumstances the pheno-typic variance could reasonably beconsidered to be largely environmentalin origin since it is much greater thanthe phenotypic variance that would bemeasured in an environment (y = Y2)that permitted maximum developmentof the trait, consistent with geneticendowment. This point has been elab-orated by R. C. Lewontin (8).The conventional definitions of G

(genotypic value) and E (environmentalvalue) have specific mathematical ad-vantages in the context of a specificmathematical theory. Other ways ofassessing the effects of environment onphenotypically plastic traits may, how-ever, be more useful in other contexts.In particular, certain kinds of interven-tion studies may provide more directand more useful information about theeffects of environment on IQ than con-ventional studies of IQ heritability.

Conventional HeritabilityAnalysis and Its Limitations

Three mathematically and biologi-cally distinct effects complicateconventional heritability analysis: geno-type-environment correlation, genotype-environment interaction, and gene-geneinteraction. I now consider the theoreti-cal limitations imposed by these effects.

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Fig. 1. Phenotypic value (P) of a hypo-thetical metric trait as a function of anenvironmental variable (x) for threevalues of a genotypic variable (y). Al andA, (also B1 and B2) indicate individualswith a common phenotypic value but dis-tinct genotypes xi and x2, respectively.

Let P(x,y) denote the value of aphenotypic character that depends onn genotypic variables xl, . . . x,denoted collectively by x, and m en-vironmental variables Yi, . . . $ Yin,denoted collectively by y. The xi andy1 (where i and i are integers) arerandom variables whose joint-frequencydistribution is specified by the functionC(x,y); P is a random function of theserandom variables. e{j} is the expecta-tion value of a function f(x,y):

6{f)= f . . . ff(x,y) $(x,y)dxdy (1)where dx = dx1 . . . dx,, dy = dyI. . . dym, and the integration extendsover the ranges of all the variables.The genotypic value G and the en-

vironmental value E are defined as fol-lows (9):

G(x) = f ... fP(x,y) 4(xly)dy (2a)

E(y) = f ... fP(x,y) t(ylx)dx (2b)where the conditional frequency func-tions 4c(xly) and o1(ylx) are defined by

,k(x,y) = i(xIy) .(y) = C(yfx) +,(x)(3a)

'i'(x) = fl(x,y)dy, 4+9(y) = f4(x,y)dx(3b)

One can then write P in the form

P(x,y) = G(x) + E(y) + R(x,y) (4)

(this equation defines R), whence

Var{P} = Var{G} +Var(E} + 2 Cov{G,E} +1 (5)

where

I = 2 Cov{G + E,R} + Var{R} (6)

Here Cov denotes the covariance and Ithe contribution of genotype-environ-ment interaction to the phenotypic vari-ance.More generally, consider two sub-

populations between whose mem-bers alto-1 correspondence, defined by somegenetic or environmental relationship,has been established. For example, thetwo subpopulations might consist 'ofchildren and their respective mothersor of foster children and their respec-tive foster siblings. For each subpoptda-tion, one may write an equation withthe form of Eq. 4. Using variableswithout primes to refer to the first sub-population and variables with primesto refer to the second, multiplying thetwo equations together, and averagingthe resulting equation, one obtains

Cov{P,P'} = Cov{G,G'} + Cov(E,E'} +Cov{G,E'} + Cov{G',E) + J (7)

where

J = Cov{G + E,R'}+Cov(G' + E,R) + Cov{R,R'} (8)

where J represents the contribution ofgenotype-environment interaction to thephenotypic covariance.The phenotypic variance, along with

phenotypic covariances referring to cer-tain pairs of subpopulations, may beestimated from appropriate phenotypicmeasurements. From these data onemust try to estimate the individualcomponents of the phenotypic variancethat appear on the right side of Eq. 5.In what circumstances is this possible?

If either E or G is negligible, theproblem has a trivial solution. Other-wise, it is insoluble unless the genotype-environment covariances in Eqs. 5 and7 are negligible. For if these terms arenot negligible and their struoture is notknown a priori, the num-ber of un-knowns exceeds the number of condi-tions. One cannot even derive an upperbound for the genetic variance underthese circumstances. Suppose, to thisend, one equates all genotype-environ-ment correlations (Cor) in Eqs. 5 and7 to their theoretical upper limits,Cor{G,E} = Cor{G',E'} = 1 and Cor{G,E'} = Cor{G',E} = Cor{G,G'}. Thenumber of unknowns would still ex-ceed the number of conditions becausethere is no a priori information aboutthe magnitude, or even the sign, ofthe terms I and J.The genotype-environment covari-

ances are negligible if, and only if, thegenotypic and environmental variablesare statistically independent to a highdegree of approximation. If this is true,then the joint-frequency funct-ion ((P)assumes the form

*(X,y) = 461(X) +'(y) (9)SCIENCE, VOL. 183

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In these circumstances, not only doall genotype-environment covariancesvanish, but so do the covariancesCov{G,R}, Cov{G,R'}, Cov{E,R}, andso forth in Eqs. 6 and 8 (10). Equa-tions 5 and 7 then reduce to

-Var{P} = Var{G} + Var{E} 4. Var{R}(lOa)

Cov{P,P} = Cov{G,G'} +Cov{E,E'} + Cov{R,R'} (lOb)

This remarkable simplification of theterms I and J in Eqs. 5 and 7 is a con-sequence of Fisher's definitions of Gand E (9). It does not occur for otherdefinitions. An equally important con-sequence of these definitions is that theyminimize the expectation of R2 (11).That is, they provide the best possibleadditive representation of the pheno-typic value. Both properties hold onlyso long as the genotypic and environ-mental variables are independent of oneanother.The term Cov{R,R'} in Eq. lOb van-

ishes if either the genotypes or the en-vironments of the two subpopula'tionsare statistically independent. It van-ishes, in particular, for monozygotictwins reared in uncorrelated environ-ments and for genetically unrelated in-dividuals reared in identical environ-ments. For these two special cases, re-spectively, the equations are

Cov{P,P'} = Var{G}Cov{P,P'} = Var{E}

(I la)(lIlb)

If all the subpopulations under con-sideration are fair samples of the parentpopulation, then Eqs. lOa, 1 la, and1lb are three equations for three un-knowns, and Var{.G}, Var{E}, andVar{R} can all be estimated. If onlythe phenotypic variance and the vari-ance for separated monozygotic twinsare known, their difference provides anupper limit for the environmental vari-ance.Under certain assumptions, the geno-

typic covariance between related in-dividuals can be related to the geno-typic variance. Consider an equili-brated population of randomly matingdiploids, effectively infinite in size sothat inbreeding may be neglected, andassume that the trait under considera-tion is specified by an arbitrary num-ber of unlinked genes, each of whichmay exist in an arbitrary number ofstatistically independent allelic states.Under these assumptions, G may bewritten in the form

G = E G (r,s)r,s

29 MARCH 1974

(12)

where r and s, in the sum indicated byY, take on all integer values (and thevalue zero) consistent with the condi-tion r + s n, the number of pairs ofhomologous loci specifying the trait.The ordered pair (r,s) specifies a con-figuration consisting of r alleles at non-homologous loci and s pairs of allelesat homologous loci. G(r,s) denotes thesum of all contributions to G resultingfrom allelic configurations of the type(r,s):G(r,s) = mg(ii, . . ., i; jf, . . ., j,) (13)

where g is a random variable depend-ing on the indicated loci and the sumruns over all sets of indices specifyingdistinct configurations of the type (r,s).In particular, G(1,0) - GA is the so-called additive contribution to G, a sumof 2n individual allelic contributions(each represented by a random variable);G(O,1) - GD is 'the dominance contri-bution, a sum of n contributionsfrom pairs of homologous loci; G(2,O)is an epistatic contribution, made upof contributions from pairs of non-homologous loci; and so on. By virtueof the assumptions concerning statisticalindependence of allelic states, all therandom variables that figure in thedecompositions (Eqs. 12 and 13) arestatistically independent. The varianceof G is accordingly given by

Var{G} = 2, r'r,s

Var{G(r,s)}=ZVar{g(h, . .., i,; jl,.. .,j.)} ( 14)

where a2S denotes the variance of thersrandom variable G(r,s). The genotypecovariance between individuals relatedin a known way is given by

Cov{G,G'} =E P(r,s)oa,2 (15)r,s

where P(r,s) is the probability of coin-cidence between homologous allelic con-figurations of type (r,s) in the related in-dividuals. In particular, P(1,O) is theprobability that a given allele in genomeI also occurs (at one of the two cor-responding loci) in genome 2; P(O,1) isthe probalbility that a given pair ofalleles at homologous loci in genome1 also occurs in genome 2; and so on.From the assumptions about statisticalindependence, it follows at once that

P(r,s) = P1728 (16)

where

P1-P(1,0) = I( +¢")

P2-P(O,1) =oq5q' (17)

Here ¢ and q/ denote the probabilityof receiving the same gene at a givenlocus by way of the father and themother respectively. These results werefirst obtained by Kempthome (12).

If the two subpopulations are charac-terized by statistically independent en-vironmental variables, Eq. lOb reducesto

Cov{P,P} = E P(r,s)r2r,s

(18)

For the series in Eq. 14 to be usefulin practice, one must be able to truncateit after a number of terms no greaterthan the number of distinct phenotypiccovariances that can be measured. Forexample, it may be possible to repre-sent the genotypic covariance betweenparent and offspring or between half-sibs by the leading term in Eq. 15 (inboth cases P., = 0) and thereby estimateOlo2= 0,A2, the additive component ofthe genotypic variance. The ratio U2/oip2, called the narrow heritability, de-termines the rate at which phenotypicchanges can be achieved through arti-ficial selection (13).

A Conventional Applicationof Heritability Analysis

Before considering the applicabilityof the theory just sketched to IQscores, I will examine some conven-tional heritability estimates that, as wasfirst pointed out by Lerner (14), havean important bearing on the analysis ofIQ data. Figure 2 shows heritabilityestimates, derived from different kindsof correlations, for four commerciallyimportant traits in dairy cattle. Theprimary data show a high degree of in-ternal consistency, in that investiga-tors at different test stations reportsimilar measured correlations. Never-theless, heritability estimates derivedfor the same trait from different kindsof data differ suLbstantially and sys-tematically.

Heritability estimates derived fromhalf-sib correlations calculated fromfield data (indicated by open circles inFig. 2) are systematically and substan-tially lower than estimates derived fromhalf-sib correlations calculated fromtest-station data (indicated by X'sin Fig. 2). Lerner (14) has pointed outthat these differences afford a direct andstriking illustration of the importanceof genotype-environment covarianceand genotype-environment interaction.

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The fact that environments at test sta-tions are randomized (inade randomwith respect to genotype) greatly re-duces, if not entirely eliminates, botheffects. (Genotype-environment correla-tion may be expected to reduce half-sibcorrelation because it tends to amplifythe genotypic effects of the maternalcontribution to the genotype of the calf;in dairy cattle, the maternal contribu-tions to the genotypes of half-sibs areuncorrelated.)

For each of the four traits, the heri-tability estimate derived from separatedmonozygotic twins is close to unity.This finding can be reconciled with thecomparatively low estimates derivedfrom half-sib correlations in two ways:

1) If environmental randomizationhas, in fact, effectively eliminated cor-relations between genotypic and en-vironmental variables, so that Eqs.lOa and 1 la are applicable, the correla-tions between separated monozygotictwins are estimates of broad heritability,and the difference 1 - h2 provides anupper limit for e2, the environmentalfraction of the phenotypic variance.The half-sib correlations, on the otherhand, yield estimates of narrow herita-bility, and the differences between theseestimates and those derived fromsplit monozygotic pairs represent thenonadditive genotypic contribution toh2.

2) If, however, the environmentalvalue (E) contains a sizable componentcorrelated with the animals' genotype,Eqs. lOa and 1 la do not apply. Such acomponent would be present if the en-vironmental variables were imperfectlycontrolled, allowing the animals to ex-ercise a modicum of environmentalselection-with respect to diet, for ex-ample. To the ex-tent that such selec-tion was genetically determined, itwould give rise to a "hidden" geno-type-environment correlation. And sinceseparated monozygotic twins would tendto select the same environments (to theextent permitted by the controls), theirenvironments would be correlated. Inthese circumstances, a high phenotypiccorrelation between monozygotic twinswould not entail a correspondingly lowvalue for the environmental componentof the phenotypic variance. Hiddengenotype-environment interaction wouldsimilarly inflate estimates of otherphenotypic covariances.

In the absence of additional in-formation, it does not seem possible tochoose unequivocally between these ex-planations, or even to be sure that gene-

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gene interactions and hidden genotype-environment correlations are jointly re-sponsible for the systematic differencesunder discussion. Explanation 1 is per-haps the less plausible of the two be-cause it requires the nonadditive geno-typic variance to be greatest when theadditive genotypic variance is least andbecause it fails to explain why theirsum is nearly the same for traits whosenarrow heritabilities span a wide rangeof values. Until explanation 2 can beruled out, it must be assumed thatestimates of broad heritability based onphenotypic correlations between sep-arated monozygotic twins may begrossly unreliable.

IQ Scores as Measurementsof a Phenotypic Character

Before examining the applicabilityof heritability analysis to IQ scores, onemust ask whether such scores can legit-imately be assimilated to measure-ments of a phenotypic character. Testsof IQ differ from measurements of con-ventional phenotypic characters in twoimportant respects.

In the physical and biological sci-ences, every measurable quantity is de-fined in the context of a definite theoret-ical structure that, in general, serves togenerate a variety of distinct opera-tional definitions. For example, thephysical geometry of rigid bodies pro-vides the basis for several operationallydistinct ways of measuring length anddistance. Electromagnetic theory pro-vides a second set of operational def-initions of distance and length; thetheory of sound a third set; and soon. The flexibility conferred by theexistence of operationally distinct waysof measuring the same quantity is im-portant because it makes possible thedetection and eventual elimination ofsystematic errors; and the existence ofdistinct operational definitions impliesan underlying theoretical structure.The definition of IQ has no theoreti-

cal context or substratum. Tests of IQmeasure what they measure. They areprecisely analogous to physical read-ings made with a black box-a devicewhose internal working is unknown.Because we do not know what an IQtest or a black box is supposed tomeasure or how it works, we cannotknow to what extent measurementscarried out on different subjects arecomparable or to what extent they areinfluenced by extraneous factors. Thus

IQ scores contain uncontrollable, sys-tematic errors of unknown magnitude.This helps to explain why different

investigators frequently report suchwidely differing estimates of the sameIQ correlation. For example, reportedestimates of the parent-child correla-tion range from .2 to .8, while estimates of the correlation between same-sex dizygotic twins range from .4 to .9(IS). According to Jensen (16), thereare no objective criteria (other thansample size) for weighting discrepantestimates of the same correlation.

Because the definition of IQ is purelyinstrumental, it fails to confer the mostessential attribute of a scientific mea-surement-objectivity. To measure asubject's Stanford-Binet IQ, one mustadminister a specific test in a specificway under specific conditions. By con-trast, a well-equipped physics labora-tory does not need to have replicas ofthe standard meter and the standardkilogram to measure length and mass,and the physicist or biologist is free todevise his own techniques for measur-ing such quantities. Systematic dis-crepancies between measurements ofthe same quantity are never ignoredin the physical and biological sciences,because they signal the presence of un-suspected systematic errors or of de-fects in the theory underlying the mea-surements.

IQ scores also differ from conven-tional measurements in that they haveno strict quanitative meaning. The IQis an index of rank order on a standard-ized test, expressed according to a con-venient but essentially arbitrary conven-tion (I7). In effect, the intervals of theIQ scale are chosen in such a way as tomake the frequency distribution of testscores in a reference population ap-proximately normal, but other methodsof defining the scale could claim equalprior justification.

These considerations show that IQscores are not phenotypic measure-ments in the usual sense. This is notto say that they have no scientific valueor practical utility, or that certainaspects of intelligent behavior cannot,in principle, be adequately defined,quantified, and measured. It is evenconceivable that valid measurementsof some well-defined behavioral traitwould rank subjects in roughly thesame way as IQ tests and that theywould turn out to be normally dis-tributed. In these circumstances, IQscores would indeed approximate validmeasurements of a phenotypic trait,

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although they would continue to be af-flicted by uncontrollable systematicerrors of unknown magnitude. For themoment, however, I ignore these short-comings of the primary data, and turnto the methods available for analyzingtlem.

Heritability of IQ

In applications of heritability analy-sis to metric (continuously variable)characters in plants and animals, at leastsome of the relevant genetic and en-vironmental factors are under experi-mental control. In particular, plant andanimal geneticists can minimize geno-type-environment correlation by ran-domizing environments. As I haveshown, this step is indispensable for theapplication of heritability analysis. Un-less Eqs. 5 and 7 can be replaced byEqs. 1 Oa and lOb, there is no hope ofdisentangling the genotypic and en-vironmental contributions to phenotypicvariances. Recent discussions of the ap-plicability of heritability analysis to IQscores have failed to stress this point.The applicability of heritability analysisdoes not, as is commonly assumed,hinge on the smallness of -the interactionterm (R) relative to the terms G and Ein Fisher's decomposition of the pheno-typic value. In fact, one may reasonablyassume on biological grounds thatgenotype-environment interaction makesa substantial contribution to the pheno-typic value of every phenotypicallyplastic trait, except in populations wherethe ranges of genetic and environmentalvariation are severely restricted. Evenso, heritability analysis can be appliedto phenotypically plastic traits, providedthat the relevant genetic and environ-mental variables are statistically uncor-related. When this condition is notsatisfied, the contributions of interac-tion to phenotypic variances and covari-ances cannot, in general, be separatedfrom the contributions of genotype andcnvironment, and heritability analysiscannot, therefore, be applied meaning-fully.

In adult subpopulations, IQ and en-vironment are well known to be moreor less strongly correlated. Since dif-ferences in IQ are undeniably relatedto genetic differences (although not,perhaps, in a very simple way), one maysafely assume that genotype-environ-ment correlation is significant in adultsubpopulations and in subpopulationscomposed of children reared by their29 MARCH 1974

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gical parents or by close relatives. 53 pairs included in the study wase, no valid estimate of IQ heri- reared by his or her natural parent (19).ty can be based on data that refer Since reliable inferences about thech subpopulations. heritability of behavioral traits cannott data of precisely this kind make be drawn from data referring to sub-le bulk of the available material, populations of adults or of childrenmany published heritability esti- reared by their biological parents or bys have been based on them. Burt close relatives, one must turn to studiesJensen (1), and Herrnstein (4), for in which the subjects are adopted chil-ple, all cite kinship correlation dren. For example, phenotypic correla-as evidence for a high value of h2. tions between half-sibs reared in fosterese authors, among others, rely homes would yield estimates of narrowially on IQ correlations between heritability analogous to those indicatedated monozygotic twins. Such cor- by X's in Fig. 2-provided that thereons, however, are highly sensitive was no selective placement and thatistortion by genotype-environment the range and distribution of relevantlation. As I have shown, significant environmental factors was the same forrtion may occur even under ex- the subpopulation of foster homes asnental conditions in which the con- for the reference population. Analogousble aspects of the environment data on siblings would be somewhat lessbeen randomized, because selec- useful, because gene-gene interactions-of "microenvironments" is geneti- in particular, the dominance covarianceinfluenced. The development of -could contribute substantially to the

in cognitive skills is presumably phenotypic covariance. However, nomore sensitive to such selection suitable data for either siblings or half-the development of physiological sibs seem to exist (6).icters of dairy cattle. In published Finally, the IQ correlation betweenes of separated monozygotic twins, unrelated, randomly selected childrenerious attempts have been made reared in the same foster home could,inimize the effects of genotype-en- in principle, provide an estimate of e2ment interaction-something that (Eq. 1 lb). The phenotypic covarianced be very difficult to do under the might substantially underestimate e2,of circumstances. For example, in however, because the environments ofargest and the most homogeneous children reared together are nevere four major twin studies, that of identical. Age differences may give rise(18), one member of each of the to significant "macroenvironmental" dif-

ferences, while differences in genotypemay give rise to significant "microen-

1 - vironmental" differences, as discussedabove.

+ t. t Jencks et al. (6) cite four studies of| x pairs of unrelated foster children rearedO together. The reported IQ (Stanford-X IBinet) correlations, corrected for un-

0.5 - + reliability and restriction of range, areas follows: .17 (10 cases), .29 (21 cases),.46 (93 cases), and .72 (41 cases). UsingFisher's method (20) to combine theseestimates, one arrives at the value r =.5 ± .05. The differences between theindividual estimates of r, although large,

o are not statistically significant, accord-A BC D ing to the test described by Fisher (20).

2. Heritability estimates for four For the reasons explained above, thec traits in dairy cattle: (A) calf environments of unrelated foster chil-it and measure, (B) milk yield, (C)-rcentages, and (D) body weight and dren reared together may be signi-ure at 2 years. The estimates are de- icantly different. On the other hand,from comparisons among separated selective placement may introduce a'zygotic twins (closed circle), mono- correlation between the genotypes of.ic and dizygotic twins (cross), half the foster children. The two effectss at testing stations in which en- went is randomized (x), and half ork In opposltedlrections, and neither

s at field stations (open circle). can be reliably estimated. Making thece: Donald (25), reproduced by not-very-safe assumption that they can-r (/4 p. 4l)1 cel, one arrives at the highly tentative

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estimate, e2 -.5 + .05. This estimaterefers to the purely environmental frac-tion of the phenotypic variance. Ofgreater relevance to the social and edu-cational issues mentioned at the begin-ning of this article is the nongeneticfraction of the phenotypic variance,given by 1 - h2 = e2 +i2, where i2 =Var{R}/ Var{P} is the fraction of thephenotypic variance relating to interac-tion. (As noted earlier, this decompo-sition of the phenotypic variance pre-supposes that genotype-environmentcorrelations are absent.) Since thereare no available data that would per-mit an independent estimate of the in-teraction contribution, the above esti-mate of e2 provides only a lower limitfor I -h2.Thus the available data for unrelated

foster children raised together yieldthe following, highly tentative estimateof the broad heritability:

O-h' .5 (19)It is important to notice that this esti-mate refers specifically to the subpopu-lation of unrelated foster children rearedtogether. [For this subpopulation, it isconsistent with the estimates of Jenckset al. (6).] It does not apply to popula-tions composed of children reared bytheir biological parents or by near rela-tives, because, as shown above, boththe conceptual and operational defini-tions of heritability break down in thepresence of significant genotype-en-vironment correlation. This explainswhy Jencks et al. (6) were unable toreconcile their heritability estimate'based on unrelated foster childrenreared together with estimates derivedfrom other kinds of IQ data. The pres-ent considerations show that the esti-mates based on other kinds of dataare, in a strict sense, meaningless.

Systematic effects of genotype-en-vironment correlation are by no meansthe only obstacles to meaningful analy-ses of IQ correlations. Additional seri-omis difficulties arise from:

1) Ignorance of the specific environ-mnental factors affecting cognitive de-velopment. Because social scientistshave not yet identified the specific en-vironmental factors most relevant tocognitive development, they are unableto assess environmental similarities ordifferences objectively, even at a quali-tative level. For example, some authors(1, p. 52; 4, p. 55) have assumed thatthe within-pair environmental differ-ences between separated monozygotictwins in Burt's study (18) are representa-

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tive of those between randomly selectedsubjects of the same age. This as-sumption is based on a reported lackof correlation between the occupa-tional statuses of the biological andadoptive fathers. As far as I know, noevidence has been adduced to supportthe implied assumption that the occupa-tional status of the father plays a cru-cial role in cognitive development. Cur-rent studies suggest that specific kindsof mother-child interaction during in-fancy and early childhood do play asignificant role in cognitive develop-ment (21), but the study of such inter-actions is still in a primitive stage.

2) Nonrandoin mating. If one wishesto analyze phenotypic correlations be-tween related persons other thanmonozygotic twins, one must allow forassortative mating. This introduces twomore unknowns into the analysis: theenvironmental and the phenotypic co-variances between mates. In the absenceof reliable assessments of relevant en-vironmental factors, the environmentalcovariance is not measurable, so theonly additional datum would be thephenotypic covariance. Thus assorta-tive mating introduces a further un-known into an already top-heavy analy-sis. Moreover, Eq. 15 (for the geno-typic covariance) applies only to popu-lations with random mating, and anappropriate generalization of it that al-lows for assortative mating has not yet,to my knowledge, been made.

3) Gene-gene interactions. The pos-sibility of evaluating the narrow herita-bility of a trait hinges on how rapidlythe series in Eq. 15 converges, thenumber of unknown components of thegenotypic variance one can hope toestimate being limited by the numberof measured kinship correlations. Now,for a trait specified by n pairs of genes,ErS is made up of 2"n!/r!s! (n-r-s)! sep-arate contributions. This suggests that,as n increases, the relative importanceof gene-gene interactions of a given kindalso increases: the greater the numberof genes contributing to the specifica-tion of a trait, the more likely it isthat nonadditive genetic effects will playan important role. These considerationsraise the possibility that human intelli-gence, however it may be defined, coulddepend on the total genotype in a man-ner too complex to permit the applica-tion of conventional heritability analy-sis. In any case, one may reasonablyexpect to find substan;tial differencesbetween the broad and narrow herita-bilities of cognitive traits (22).

In view of the difficulties discussedabove, studies of unrelated foster chil-dren reared together and of half-sibsreared in foster homes seem to offer'the only realistic prospects for esti-mating, respectively, the environmentalfraction of the IQ variance and thenarrow heritability.

Alternative Quantitative Approaches

Heritability analysis was devised tohelp answer one of the central practicalquestions of plant and animal genetics:Under given environmental conditions,how rapidly can systematic changes ina metric character be produced by arti-ficial selection? Fisher's "fundamentaltheorem of natural selection" (23) im-plies that the rate of evolution inquestion is proportional to the additivegenotypic variance and hence to thenarrow heritability of the character.For obvious reasons, this question-and, therefore, the value of the narrowheritability-is not of comparable im-portance for human behavioral traits.On the other hand, the phenotypic plas-ticity of human behavioral traits is ofconsiderable interest both to geneticistsand to students of human behavior.Estimates of the broad heritability-or, better still, of e2-do tell us some-thing about the sensitivity of a traitto environmental variation, but theythrow little light on what are perhapsthe most important questions: To whatsorts of environmental changes are be-havioral traits most sensitive? To whatextent can cognitive performance beimproved by appropriate forms of envi-ronmental intervention? How do genet-ic differences affect levels of cognitiveperformance attainable under optimalenvironmental conditions? These andsimilar questions are not of less sci-entific interest than those to whichheritability estimates provide answers;and they are considerably more relevantto the educational, social, and politicalissues mentionied at thc beginning ofthis article.

It may turn out to be less difficultto find semiquantitative or even quan-titative answers to such questions thanto obtain relialble heritability estimates.Notable progress along these lines hasalready been made. The remarkableachievements of the Milwaukee Project(24), to cite a single example, afforda direct and dramatic demonstration ofthe efficacy of appropriate environ-mental modifications in accelerating

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cognitive development. In this study,now in its sixth year, a comprehensivefamily intervention program produceda- sustained, 30-point difference in IQbetween an experimental group and acontrol group, each composed of 20randomly selected children of motherswith tested IQ's of under 75. Over a5-year period, the average IQ of theexperimental group remained close to125. Children in the experimentalgroup were evaluated through indepen-dent tests administered by psychologistsnot connected with the study.The methodological difficulties of in-

tervention studies should not be mini-mized. Nevertheless, the nature of thequestions such studies are trying toanswer makes these difficulties inher-ently less formidable than those be-setting the application of conventionalheritability analysis to IQ scores. Forexample, the shortcomings of IQ as aphenotypic measurement, although theycast serious doubt on the meaningful-ness of heritability estimates, do notimpair the usefulness of IQ tests forassessing differences in cognitive per-formance between an experimentalgroup and a control group in studieslike that of Heber and his colleagues.

IQ and Race

Jensen (2) and others have arguedthat reported differences in average IQbetween black and white children areprobably attributable in part to sys-tematic genetic differences. Jensen ex-plicitly states the "rule of inference"used to draw this conclusion (16, p.438): "The probability that a pheno-typic mean difference between twogroups is in the same direction as agenotypic mean difference is greaterthan the probability that the phenotypicand genotypic mean differences are inopposite directions." This rule fails,however, when systematic effects whosemagnitude cannot be estimated areknown to contribute to the phenotypicmean differences. (Unfortunately, theseare the only circumstances in whichthe rule might be useful.) In order toestimate the probabilities mentioned inthe rule, one would need to estimatethe probability that the observed pheno-typic mean difference exceeds the con-tribution of systematic effects-which,by assumption, is impossibAle.Among the relevant systematic dif-

ferences between blacks and whites arecultural differences and differences in29 MARCH 1974

psychological environment. Both influ-ence the development of cognitive skillsin complex ways, and no one has suc-ceeded in either estimating or elimina-ting their effects. "iCulture-free" testsdeal with this problem only on themost superficial level, for culture-freeand "culture-bound" aspects of cogni-tive development are inseparable. Thedifficulties cannot be overcome by re-fined statistical analyses. As long assystematic differences remain and theireffects cannot be reliably estimated, novalid inference can be drawn concern-ing genetic differences among races.

Precisely the same arguments andconclusions apply to the interpretationof IQ differences between socioeco-nomic groups.

Summary and Conclusions

Estimates of IQ heritability are sub-ject to a variety of systematic errors.The IQ scores themselves contain

uncontrollable, systematic errors of un-known magnitude. These arise becauseIQ scores, unlike conventional physicaland biological measurements, have apurely instrumental definition. The ef-fects of these errors are apparent inthe very large discrepancies among IQcorrelations measured by different in-vestigators.

Genotype-environment correlations,whose effects can sometimes be mini-mized, if not wholly eliminated, in ex-periments with plants and animals, arenearly always important in humanpopulations. The absence of significanteffects arising from genotype-environ-ment correlations is a necessary condi-tion for the applicability of conven-tional heritability analysis to pheno-typically plastic traits. When this con-dition fails, no quantitative inferencesabout heritability can be drawn frommeasured phenotypic variances and co-variances, except under special condi-tions that are unlikely to be satisfiedby phenotypically plastic traits in humanpopulations.

Inadequate understanding of theprecise environmental factors relevantto the development of specific behav-ioral traits is an important source ofsystematic errors, as is the inability toallow adequately for the effects of as-sortative mating and gene-gene interac-tion.

Systematic cultural differences anddifferences in psychological environ-ment among races and among sociocco-

nomic groups vitiate any attempt todraw from IQ data meaningful infer-ences about genetic differences.

Estimates based on phenotypic cor-relations between separated monozygot-ic twins-usually considered to be themost reliable kind of estimates-arevitiated by systematic errors inherentin IQ tests, by the presence of genotype-environment correlation, and by thelack of detailed understanding of en-vironmental factors relevant to the de-velopment of behavioral traits. Otherkinds of estimates are beset, in addi-tion, by systematic errors arising fromincomplete allowance for the effects ofassortative mating and from gene-geneinteractions. The only potentially usefuldata are phenotypic correlations be-tween unrelated foster children rearedtogether, which could, in principle,yield lower limits for e2. Available dataindicate that, for unrelated foster chil-dren reared together, the broad herita-bility (h2) may lie between 0.0 and 0.5.This estimate does not apply to popula-tions composed of children reared bytheir biological parents or by nearrelatives. For such populations the heri-tability of IQ remains undefined.The only data that might yield

meaningful esti'mates ot narrow herita-bility are phenotypic correlations be-tween half-sibs reared in statisticallyindependent environments. No usefuldata of this kind are available.

Intervention studies like Heber'sMilwaukee Project afford an alterna-tive and comparatively direct way ofstudying the plasticity of cognitive andother behavioral traits in human popu-lations. Results obtained so far stronglysuggest that the development of cogni-tive skills is highly sensitive to varia-tions in environmental factors.

These conclusions have three obviousimplications for the broader issues men-tioned at the beginning of this article.

1) Published analyses of IQ dataprovide no support whatever for Jen-sen's thesis that inequalities in cogni-tive performance are due largely togenetic differences. As Lewontin (8)has clearly shown, the value of thebroad heritability of IQ is in any caseonly marginally relevant to this ques-tion. I have argued that conventionalestimates of the broad heritability ofIQ are invalid and that the only dataon which potentially valid estimatesmight be based are consistent with abroad heritability of less than 0.5. Onthe other hand, intervention studies, iftheir findings prove to be replicable,

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would directly establish that, undersuitable conditions, the offspring ofparents whose cognitive skills are sopoorly developed as to exclude themfrom all but the most menial occupa-tions can achieve what are regarded asdistinctly high levels of cognitive per-formance. Thus, despite the fact thatchildren differ suibstantially in cogni-tive aptitudes and appetites, and despitethe very high probability that thesedifferences have a substantial geneticcomponent, available scientific evidencestrongly suggests that environmentalfactors are responsible for the failureof children not suffering from specificneurological disorders to achieve ade-quate levels of cognitive performance.

2) Under prevailing social condi-tions, no valid inferences can be drawnfrom IQ data concerning systematicgenetic differences among races orsocioeconomic groups. Research alongpresent lines directed toward this end-whatever its ethical status-is scientif-ically worthless.

3) Since there are no suitable datafor estimating the narrow heritabilityof IQ, it seems pointless to speculateabout the prospects for a hereditarymeritocracy based on IQ.

References and Notes

1. A. R. Jensen, Harv. Educ. Rev. 39, 1 (1969).2. , Genetics and Edutcation (Harper &

Row, New York, 1972).3. , Educability and Group Differences

(Methuen, London, 1972).4. R. J. Herrnstein, Ati. Month. 228, 43 (1971).5. , IQ and the Meritocracy (Little, Brown,

Boston, 1973).6. C. Jencks et al., Inequality: A Reassessment

of the Effects of Family and Schooling inAmerica (Basic Books, New York, 1972),appendix A, pp. 266-319.

7. A similar argument was given as long agoas 1933 by Hogben [cited by C. H. Wadding-ton, The Strategy of the Genes (Allen &Unwin, London, 1957), pp. 92-94].

8. R. C. Lewontin, Btull. At. Sci. 26, 2 (March1970); in The Fallacy of IQ, C. Senna, Ed.

(Third press, New York, 1973), pp. 1-17.9. R. A. Fisher, Trans. R. Soc. Edinb. 52, 399

(1918). In this fundamental paper, Fisherexpresses the genotypic value (G) of a traitas the sum of additive allelic contributions(G1,G ... G,) and dominance contributions(G,d,s . . .) in such a way as to minimizethe contribution of dominance to the geno-typic variance (pp. 404, 415). The decomposi-tion (Eq. 4) of the phenotypic value (P)into additive genotypic and environmentalvalues (G and E) and a residual interactionterm (R) is formally analogous: E (in theformula E = E1 + E, + El.) corresponds to P(in the formula P = G + E+ R), E1 to G,E, to E, and E12 to R. The formal analog ofthe requirement of statistical independence be-tween genotype and environment (Eq. 9) isFisher's assumption that genes assort inde-pendently. Thus the effects of linkage areformally analogous to those of genotype-en-vironment correlation.

10. Consider first the case of a single genotypicvariable x x,, and a single environmentalvariable y - y1. That the covariances inquestion do indeed vanish follows immedi-ately from the power-series representations ofG, E, and R:

G =E 1 ! (xL-6 (xk)) Pk.o(0,0)k

E =E j - 6 (Y)) Pe J(O,O)

R=E E 1 [rys~r > 1 s>1 (r+s)![

6(xr) 64(y8)]Pr,8(O,0)

where

Pr, (0,0)= L.acrav' P(X,Y)

These formulas follow from the Taylor ex-pansion of P (assumed to be valid for allx,y) and the defining conditions (2). In thegeneral case, the proof is notationally morecomplex but essentially the same.

11. For notational simplicity, let x = x1, y = yl.The most general form of R (assuming thatit has derivatives of all orders at x = y = 0)is

R = R*(x,y) +f(x) + g(y)

where R* is the expression for R given earlier(10), g is a random variable depending onthe indicated loci, and 6 If F = e6 g F =0.By expanding f and g, one finds that 6 R*f=6 R*g =0. Hence

6 1 R2 E=e R*2 + e f2 +e6 g2 Fe6 R*2S

12. 0. Kempthorne, Proc. R. Soc. Lond. Ser. BBiol. Sci. 143, 103 (1954).

13. Narrow heritability is also the kind of herita-bility that is relevant to Herrnstein's (4)meritocracy argument.

14. I. M. Lerner, Evol. Biol. 6, 399 (1972).15. L. Erlenmeyer-Kimling and L. F. Jarvik,

Science 142, 1477 (1963).16. A. R. Jensen, Cognition 1, 427 (1973); see

also D. Layzer, ibid., p. 453.17. This point has been made by S. S. Stevens,

Science 103, 677 (1946).18. C. Burt, Br. J. Psychol. 57, 137 (1966).19. Jencks et al. (6) recognized that estimates of

h3 derived from twin studies must make allo*v-ance for genotype-environment correlation,and they have, in fact, attempted to do this.However, once the effects of genotype-environ-ment correlation are admitted to be signifi-cant, the problem of estimating h2 becomesmathematically indeterminate: the number ofunknowns exceeds the number of availableconditions. The analysis of Jencks et al., al-though considerably more elaborate and real-istic than previous analyses of the same data,depends on plausible but ad hoc assumptionsto close the gap between the number of un-knowns and the number of conditions.

20. R. A. Fisher, Statistical Methods for ResearchWorkers (Oliver & Boyd, Edinburgh, ed. 13,1958), p. 204.

21. U. Bronfenbrenner, Is Early Intervention Ef-fective? (Cornell Univ. Press, Ithaca, N.Y.,in press).

22. Thus Jensen's estimate of broad heritability,whatever its validity, is not directly relevantto the argument by which Herrnstein (4) pre-dicts the emergence of an hereditary meritoc-racy. This is not to say that the argument isfree from other serious defects.

23. R. A. Fisher, The Genetical Theory of NaturalSelection (Dover, New York, ed. 2, 1958),p. 22.

24. R. Heber, H. Garber, S. Harrington, C. Hoff-man, "Rehabilitation of families at risk formental retardation" (unpublished researchreport). Some writers [for example, Jensen(16, p. 427)] have expressed skepticism con-cerning the reported gain of IQ because ofits size. Recent findings by R. B. McCall,M. I. Appelbaum, and P. S. Hogarty [Mono-graphs of the Society for Research in ChildDevelopment 38, 1 (1973)] may help to putthose of Heber and his colleagues in perspec-tive: "Normal home-reared middle-class chil-dren change in IQ performance during child-hood, some a substantial amount . . . the aver-age individual's range of IQ between 2½ and17 years of age was 28.5 IQ points, one ofevery three children displayed a progressivechange of more than 30 points, and one inseven shifted more than 40 points. Rare in-dividuals may alter their performance as muchas 74 points" (p. 70).

25. H. P. Donald, in Proceedings of the TenthInternational Congress of Genetics (Univ. ofToronto Press, Toronto, 1959), vol. 1, pp.225-235; reproduced by Lerner (14, p. 412).

26. I am indebted to L. Cavalli-Sforza, E. R.Dempster, and I. M. Lerner for helpfulcomments, criticism, and suggestions.

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