Behavior Genetics, Vol. 16, No. 1, 1986

Estimating Genetic Correlations Among Discontinuous Phenotypes: An Analysis of Criminal Convictions and Psychiatric-Hospital Diagnoses in Danish Adoptees L a u r a A . B a k e r ~

A biometrical model-fitting approach was applied to data from a full adoption design to study phenotypic variation and covariation among multiple discontinuous traits. Using statistical methods available for fac- tor analyses o f dichotomous-item data, general&ed least-squares esti- mates were obtained for parameters o f additive polygenic and environ- mental influences on criminal convictions and psychiatric-hospital di- agnoses in 2532 Danish male adoptees and their family members. The procedure also provided estimates o f genetic correlations, environmental correlations, and genotype-environment correlations among the meas- ures while taking into account the effects of selective placement, assor- tative mating, and cultural transmission from parental phenotypes to off- spring environments. Significant heritable influences were found for property crime and for a composite psychiatric variable pertaining to hospital admissions for alcohol abuse, drug abuse, and personalty dis- orders. Both correlated environmental factors and correlated genetic fac- tors important to the different variables accounted for the observed phen- otypic relationship between property crime and these antisocial disorders. No cultural transmission was present for the adoptees' anti- social disorders, although significant paternal cultural transmission was found for property crimes. Patterns of assortative mating differed for the

This work was supported by NIMH Grant MH39904-01 and PHS Grant 31353 from the Center for Studies of Crime and Delinquency.

Department of Psychology, SGM 501, Uniyersity of Southern California, Los Angeles, California 90089-1061.

127

0001-8244/86/0100-0127505.00/0 �9 1986 Plenum Publishing Corporation

128 Baker

biological versus adoptive parents, and selective placement was found to be negligible.

KEY WORDS: genetic correlations; environmental correlations; threshold characters; prop- erty crime; psychiatric illness.

INTRODUCTION

Multivariate behavioral genetics is concerned with the study of pheno- typic variation and covariation among two or more behavioral charac- teristics and the contributions of genetic and environmental factors to observed relationships among such variables. In addition to determining the relative proportions of phenotypic variance in each characteristic at- tributable to genetic differences among individuals [i.e., heritability (Lush, 1940)], multivariate biometrical models also provide a framework for determining coefficients of relationship among genetic factors impor- tant to different phenotypes [i.e., genetic correlations (Hazel, I943)]. Al- though considerable advances have been made in the development of such models since Vandenberg's (1965) early work in the area, the focus has been almost entirely on the analysis of continuously distributed or quan- titative phenotypes [for a review see DeFries and Fulker (1986, this issue)]. Much less attention has been paid to the analysis of two or more qualitative phenotypes which may have multifactorial patterns of inherit- ance (Bonett et al., 1986). When multiple measures of a discontinuous nature are available for individuals in a behavioral genetic design, the contributions of polygenic and environmental influences on each measure, as well as the correlations among such influences, may also be obtained; this is the focus of the present paper.

These qualitative phenotypes have been referred to as "threshold characters" (Falconer, 1965), which may be dichotomous observed var- iables characterized by underlying continuous distributions, or liabilities. Only when an individual's liability exceeds some threshold value is the trait expressed. In some instances there may be several thresholds which correspond to different levels of severity in liability, so that the observed phenotype may be an ordered polychotomy with more than two cate- gories. In any case the concern is with the underlying liabilities which are presumed to be influenced by multifactorial (both polygenic and en- vironmental) influences. Correlations among liabilities in different family members may be estimated on the basis of risk rates in relatives of affected individuals in a proband selection design (Rice et al., 1979; Rao et al., 1981) and parameters of genetic and environmental influence may be es- timated from such correlations. In a cohort design where individuals are randomly sampled with respect to the threshold characters under study,

Crime and Psychiatric Illness !29

tetrachoric correlations may be obtained directly from contingency tables among the measured phenotypes in the different family members in the design using maximum-likelihood estimation procedures described by Olsen (1979). A model-fitting procedure may then be used to estimate the multifactorial transmission parameters underlying the expected correla- tions among these threshold characters in the different relatives. A general fitting function (F) may be minimized to obtain the parameter estimates which best reproduce the observed tetrachoric correlations:

F = [r - E(r)]'W-~[r -E( r ) ] , (1)

where r is a p ( p - 1)/2 • 1 vector containing the sample correlations, and E(r) is a corresponding vector of expected correlations expressed in terms of the genetic and environmental parameters to be estimated. W is a weight matrix which varies depending upon the method of estimation. W is the identity matrix in ordinary least-squares estimation, or W may be a diagonal matrix containing, for example, the respective variance of each tetrachoric correlation (see Bishop et al . , 1975, p. 381) in a weighted least-squares estimation. Moreover, when W is a full rank error matrix of the sample thresholds and correlations and is iteratively updated during the estimation procedure, a generalized least-squares estimation of the parameters and their standard errors is achieved. The latter approach has the advantage of also providing a statistical test of model fit.

Procedures for estimating the error matrix for tetrachoric correlations have been described by Christoffersson (1975) and Muthen (1978) for the factor analysis of dichotomous-item data. Muthen's method is compu- tationally simpler, involving only univariate standard normal integration and calculation of second-, third-, and fourth-order probabilities from the sample data. That method is used in the present paper in a model-fitting approach to the biometrical analysis of tetrachoric correlations among qualitative indicators of criminality and psychiatric illness obtained in a study of Danish adoptees and their natural and adoptive parents. The relative importance of polygenic and environmental factors for each mea- sure is estimated in addition to the genetic, environmental, and genotype- environment correlations between the measures. In addition, both selec- tive placement and assortative mating are taken into account.

METHODS

Subjects and Measures

Measures of criminality and psychiatric illness were constructed using record data from the study of Danish adoptees obtained through the register established by Kety et al. (1968) and more recently described

130 Baker

Table I. Offenses Included as Property Crimes Under Danish Criminal Law

Burglary of an apartment Burglary of a business, office, or bank Burglary of an inhabited summerhouse Other burglary Unspecified burglary Thievery of a motor vehicle for use Thievery of a bicycle Stealing from a motor vehicle Stealing from a place of work Stealing from family or friends during a visit Shoplifting, stealing from a vending machine Other thievery Unspecified thievery Fraud by intent Other embezzlement Fraud Forgery Other swindling Unspecified swindling Extortion, blackmail Fencing stolen goods Malicious property destruction Other property crimes

by Hutchings and Mednick (1975) and Mednick et al. (1984). Of the 6700 nonfamilial adoptions of males which took place in Denmark between 1924 and 1947, families were included in the present analyses if there was adequate information concerning birthplace and birthdate to identify each of the five family members (male adoptee, biological mother, biological father, adoptive mother, and adoptive father) in the Danish government's conviction records at the Ministry of Justice. Furthermore, only those families in which the adoptee was transferred into the adoptive home within 12 months after birth were included, which resulted in a total of 2532 families with complete data concerning criminal convictions. The adoptees were between 29 and 52 years old at the time the data were collected for the present study and thus had achieved the age of risk for criminal behavior and many serious mental illnesses.

This report is concerned with relationships among criminality and mental illness in these individuals. Court convictions, which may be con- sidered a conservative measure since there are many points (reporting, detection, apprehension, arrest, charging, etc.) at which individuals per- haps guilty of crimes may be excluded, were utilized as an index of crim- inal involvement. Only nonviolent property crimes (see Table I) are ex-

Crime and Psychiatric Illness 131

amined here since patterns of transmission for violent and nonviolent crimes appear to differ (Mednick et al., 1984; Cloninger and Gottesman, 1986). An individual is considered to be "criminal" if at least one con- viction for any property crime listed in Table I was received prior to the data collection in 1974. For those individuals who were convicted of vi- olent and property crimes, only their property offenses are considered in these analyses.

Information regarding mental illness for each individual in the study was obtained by a search of the Danish national register of all admissions to and discharges from psychiatric inpatient facilities, which is maintained by the Institut for Psykiatrisk Demografi in Risskov, Denmark. The search was completed between 1976 and 1978, and the information for each sub- ject contains the dates of all admissions to any psychiatric hospital in Denmark. Diagnoses received were assigned diagnostic labels from the eighth edition of the World Health Organization (1968) International Clas- sification o f Diseases (ICD-8). Of the 109 different diagnoses found in the data set, 13 major category diagnoses were then formed, including organic psychoses, schizophrenia, affective psychoses, paranoid psychoses, re- active psychoses, personality disorders, psychosomatic disorders, mental retardation, neuroses, alcohol and drug use, suicide attempts, and sexual disorders.

Moffitt (1984) recently examined the relationships between a subset of the aforementioned category diagnoses and criminal convictions. A composite diagnostic variable based on personality disorders, alcohol use, and drug use was significantly related to nonviolent property crimes in the individuals in that study. Those persons who received at least one hospital admission for any of these three category diagnoses were thus considered to have an "antisocial disorder" and were found in these data to have increased rates of nonviolent property offenses rel- ative to those without such diagnoses. Analyses presented here were in- tended to investigate further the etiology of this relationship and to de- termine the extent to which it may be mediated by correlated genetic and/ or correlated environmental factors.

Analyses

Maximum-likelihood estimates of the sample thresholds for I0 ob- served variables (property criminality and antisocial disorder for each of the five family members) and the tetrachoric correlations among them were obtained using Olsen's (1979) iterative estimation procedure imple- mented in the LISREL VI computer program (J6reskog and S6rbom, 1983). The 10 thresholds and 45 correlations are contained in Table II in

1 3 2 B a k e r

T a b l e II. Maximum-Likelihood Estimates of Thresholds (on Diagonal) and Correlations Between Property Crime (C) and Antisocial Disorders (A) in Male Adoptees and Their

Family Members a

ACA BMA BFA AMA AFA ACc BMc BFc AMc AFc

- 1 . 7 3

0.01 - 1 . 9 4

0.04 -0 .06 -1 .75 0.11 -0.01 0.12 -2 .14 0.08 0.03 0.09 0.11 0.37 0.17 0.04 -0 .06 0.16 0.34 0.12 -0 .17

-0 .05 0.09 0.26 -0 .05 -0 .09 -0.01 0.05 0.22 -0 .08 -0 .04 -0 .07 -0 .02

-2 .25 0.06 -1 .13

-0 .12 0.20 - 1.54 0.14 0.17 0.11 -0 .74 0.12 -0 .05 0.11 0.10 0.19 0.14 0.07 0.10

-2 .16 0.29 - 1.61

a Adopted children (AC); biological mothers (BM); biological fathers (BF); adoptive mothers (AM); adoptive fathers (AF).

matrix form. These were then represented in a 55 x 1 vector (r) to be used in the model-fitting analyses.

Expected correlations among these variables were derived on the basis of a bivariate path model of polygenic and environmental influences in a full adoption design described by Baker (1983). The model takes into account the relative importance of genetic and environmental influences for each character and includes parameters of phenotypic transmission from parental phenotypes to child's environment, as well as assortative mating and selective placement (both isomorphic and heteromorphic) for property crime and antisocial disorders. The expectations are presented here in a more convenient and general matrix form using the notation described subsequently by Vogler and DeFries (1984).

The structural equations for an individual's two (standardized) phenotypes may be expressed in a 2 x 1 column vector (P) as

P = hG + eE, (2)

where G and E are 2 x 1 vectors of latent additive genetic and environ- mental deviations corresponding to each phenotype, and h and e are 2 x 2 diagonal matrices of path coefficients pertaining to square roots of her- itability and environmentality, respectively, for each character. Equation (2) is used to represent the set of structural equations for vectors of pheno- types in each of the five individuals in the full adoption design: adopted children (PAc), biological mothers (PBM), biological fathers (PBF), adop- tive mothers (PAM), and adoptive fathers (PAY), in which each individual has his or her own corresponding vectors of genetic and environmental

Crime and Psychiatric Illness t33

Table III. Matrices of Expected Correlations Among Family Members in a Full Adoption Design

Phenotypes a Expected correlation matrix

PBMP~F MS PAMP~F MA PAFP~F Tll PAvPI3M T12 PAMP~F T21 PAMP~M T22 PAcP~M �89 + se') + FM~] + e(m'T22 + f'T12) PAcP~F �89 + se') + FMB] + e(m'T2~ + f 'Tll) PAcP~,M e(m'Rv + f'M,~) + �89 F (T~2 + T:~) PAcP.~.F e(f'Rp + m'MA) + �89 F (Ti2 + Til) PAcP.~c = PBMPhM =

PBFP~F = PAMP,~M = PAFP~,F Rp

a Adopted children (AC); biological mothers (BM); biological fathers (BF); adoptive mothers (AM); adoptive fathers (AF).

deviations (e.g., GBM, GBF, etc.). The adopted child's genetic and en- vironmental deviations may be further expressed in terms of the parental variables as

GAC = �89 + GBF) (3)

and

Eric = m'PAM + f'PAF (4)

where m and f are square nonsymmetric matrices containing maternal and paternal (respectively) cultural transmission parameters from the adoptive parent phenotypes (rows) to the child's latent environment (columns). Thus, substituting Eqs. (3) and (4) into Eq. (2) for the adopted child pro- vides a set of five structural equations in matrix form which may be used algebraically to derive expected correlations among the observed varia- bles (see Vogler, 1985).

The matrices of expected correlations are presented in Table III for all family members in the full adoption design. Rp, Re, and R~ are sym- metric matrices of correlations among the phenotypic, additive genotypic, and environmental variables, respectively, and each is constrained tO be equal across all types of individuals. Rp is expressed in terms of h, e, Re, RE, and a nonsymmetric, square matrix of genotype-environment cor- relations (s):

Rp = hR~h' + eREe' + hse' + es'h' (5)

134 Baker

F is a diagonal matrix containing the univariate path coefficients (h #- se) for each character. Nonsymmetric, square matrices of assortative mating correlations among maternal (rows) and paternal (columns) phenotypes are represented separately by MA for adoptive parents and MB for bio- logical parents. The model is similar to that described in this issue by Rice et al. (1986), with the exception that the expectations in Table III allow for different patterns of assortative mating among bioli~gical and adoptive parents and, additionally, take into account effects of selective placement parameters contained in TH, Tl2, T21, and T22. The subscripts for each matrix distinguish between selective placement parameters for parents of different sexes, where the first subscript denotes the sex (male, 1; female, 2) of the adoptive parent (rows) and the second subscript refers to the sex of the biological parent (columns). For example, T12 is a 2 X 2 matrix containing correlations among adoptive fathers and biological mothers.

The expectations in Table III were subsequently transformed along with 10 threshold parameters into vector form E(r) corresponding ele- mentwise to r. Intergenerational equilibrium constraints regarding s pro- vided the derivation of one of the matrices of cultural transmission (i.e., m or f) in terms of other parameters in the model, so that a total of 46 parameters was free to be estimated.

Estimation

The MINUIT (CERN, 1977) package of optimization and error anal- ysis routines was used to minimize the generalized least-squares (GLS) fitting function in Eq. (1), using an estimated variance-covariance matrix of the residuals [r - E(r)] as the weight matrix (W). As described by Muthen (1978), W was defined as

W = GSG',

where S is a consistent estimate of the variancc-covariance matrix of the deviations of expected and observed first- and second-order marginal pro- portions (see Christoffersson, 1975, Appendix 2). G contains consistent estimates of first-order derivatives of sample thresholds and correlations with respect to first- and second-order marginal proportions in the sample.

GLS estimates of the 46 free parameters and their standard errors were obtained for both the full and the reduced models. Although twice the minimum value of F is distributed as chi-square with [p(p + 1)/2 - m] degrees of freedom (where p is the number of observed variables and m is the number of estimated parameters), the large sample size guar- anteed high fit statistics for even the full model with as few restrictions

Crime and Psychiatric Illness 135

Table IV. Rates (%) of Psychiatric Admission and Prop- erty Crime in Members of 2532 Families of Male Adoptees

Antisocial Property disorder crime r a

Biological mothers 2,89 6,15 0.34 Adoptive mothers 1.76 1.32 0.22 Biological fathers 4.39 23.92 0.26 Adoptive fathers 1.21 4.80 0.19 Male adoptees 4.19 12.88 0.37

a Tetrachoric correlation between antisocial disorders and property crime.

as possible. Thus, normed fit indices which are not dependent on such factors as sample size and assumptions of multivariate normality (Bentler and Bonett, 1980) were used to evaluate the fit of each model and to compare the importance of different restrictions in the models. For the full and reduced models, the corresponding normed fit index (Ao) was computed for each model as

Ao = (Fo - Fr)/Fo, (6)

where Fr is the chi-square variate for the restricted model and Fo is the chi-square computed for the null model of complete independence among the observations. Although no statistical evaluation of A o can be made, its value necessarily falls between 0 and 1 and may be used as an indication of the adequacy of the model in accounting tbr the observed covariance structure. High values of Ao (about 0.9) generally indicate an adequate fit and suggest that a less constrained model will not provide much improvement.

RESULTS

Rates of property crime and antisocial disorders for each of the five family members in the full adoption design are shown in Table IV. As previously reported by Mednick et al. (1984), crime rates differ consid- erably between sexes, generations, and parental types. Crime rates in the birth parents significantly exceed those in the adoptive parents, and males have significantly higher property-crime rates than females. Admission rates to mental hospitals for antisocial disorders range from 1.21% in the adoptive fathers to 4.39% in the biological fathers. Also shown in Table IV are the tetrachoric correlations between property crime and antisocial disorders, which are significant for each family member (P < 0.05).

136 Baker

Table V. Normed Fit Indices ~ (Ao) for Full and Reduced Models

Model Ao

1. Full 0.99 2. s = 0 b 0.99 3. m = f = 0 0.89 4. TI1 = 0 b 0.97 5. T21 = 0 b 0.94 6. T12 = 0 b 0.93 7. T22 = 0 b 0.92 8. MB = 0 0.83 9. MA = 0 0.74

10. MA = MB 0.75

Indicate a re la t ive improvement over the null model.

b Const ra in t re ta ined in all subsequent models .

The GLS estimation procedure was used to fit the full model of ge- netic and environmental influences to the observed correlations, which yielded a • of 3683.27 (df = 9, P < 0.001). As mentioned earlier, this highly significant summary statistic is considerably affected by the large sample size. The • obtained for the null model of complete independence among the observations was 498,036.90 (df = 45, P < 0.001), and the normed fit index (A o) comparing the full and null models was thus cal- culated as (498,036.90 - 3683.27)/498,036.90 = 0.99, which indicates that the full model is remarkably accurate in reproducing the observed cor- relations. In fact, the expected correlations based on the full model deviate from the observed correlations by less than 0.01, further suggesting that the full model quite adequately accounts for the observed pattern of cor- relations among property crime and antisocial disorders in these adoptees and their families.

Several constraints were subsequently placed on the model, and these are summarized in Table V along with their corresponding normed fit indices. First, the matrix of genotype-environment correlations was fixed as a zero matrix, which resulted in a change of less than 0.01 in the normed fit index. Thus, this constraint was deemed an inconsequential one in terms of the relative goodness of fit and was retained in all subsequent models in Table V. However, the next constraint of no cultural trans- mission (i.e., m = f = 0) resulted in a more substantial reduction in the fit (Ao = 0.89), indicating that at least some parameters of maternal and/ or paternal cultural transmission may be important in these data, and they were thus left free to be estimated in subsequent models.

Crime and Psychiatric Illness 137

The next four models include fixing matrices of selective placement as zero matrices, and the fit was relatively unchanged in the absence of each of the matrices Tn , TI2, Tzl, and T22. Assortative mating for prop- erty crime and antisocial disorders was apparently present in these data for both biological and adoptive parents, as suggested by the considerable reduction in the normed fit index when setting MB to zero in Model 8 (Ao = 0.83) and when setting MA to zero in Model 9 (Ao = 0.74). Moreover, constraining assortative mating parameters to be equal in biological and adoptive parents in Model 10 (i.e., MA = MB) also led to a poorer fit (Ao = 0.75), indicating that at least some of these parameters vary between the two types of parents.

Examination of the remaining free-parameter matrices (h, Re, RE, MA, MB, and either f or m) and their standard errors revealed several path coefficients which were negligible (absolute value less than 0.1) or small relative to their standard errors. These included one of the cross- assortative mating parameters (between the adoptive mother's antisocial disorders and the adoptive father's property crime), indicating possible asymmetry in mate selection since the correlation between adoptive fa- thers' antisocial disorders and adoptive mothers' property crime was sig- nificant in these data (r = 0.12 + 0.0004). Of the four assortative mating parameters contained in MB, the only important correlation was the iso- morphic one between property crime in the biological mothers and prop- erty crime in the biological fathers. Thus, unlike other findings of equiv- alent assortative mating for cognitive variables in biological and adoptive couples (Rice et al., 1986), these data indicate a somewhat reduced sim- ilarity among biological couples with respect to property crime convic- tions and antisocial disorders.

Cultural transmission for these variables, at least for male adoptees, appeared to be primarily of paternal origin, since each of the four para- meters contained in m was negligible. Furthermore, only the isomorphic parameter for the criminality variable was important in f (fc = 0.14 _+ 0.02), suggesting an influence of the adoptive father's phenotype for prop- erty crime on the environmental factors important to his adopted son's criminal outcome. This result is consistent with Hutchings and Mednick's (1975) earlier finding of adoptive-parent contributions to criminality in their adopted offspring. It should also be noted that other analyses of the entire cohort of male adoptees (which includes those transferred into the adoptive home later than I year after birth, in addition to those considered in the present analyses) by Mednick et al. (1984) did not reveal any sig- nificant adoptive-parent effect in criminal-law convictions. It is therefore suggested that any factors associated with the adoptive father's crimi- nality exert their influence fairly early in life; and, of course, it remains

138 Baker

Table VI. Generalized Least-Squares (GLS) Parameter Estimates for the Final Reduced ModeP

GLS estimate • SE

Parameter matrix C A

h I0.45 • 0 ] 0 0.27 • 0.01

[ , ] RE 0.22 • 0.003 1

[ ,.0 1 RG 0.90 -+ 0.002 1.0

f [0.14• 0 0 ] I0.29 • 0.002 0 ]

MA [_0.12 • 0.0004 0.11 • 0.0004

MB [0.11 • 00 1

Ao = 0.90. C, property crime; A, antisocial disorders.

possible that such effects are mediated by social-class effects also found in these data (Van Dusen et al., 1983).

Constraining these negligible and nonsignificant parameters to zero yielded a normed fit index of Ao = 0.90, which reflected an adequate fit of the final reduced model. Generalized least-squares parameter estimates obtained in the reduced model are presented in Table VI along with their corresponding standard errors. These indicate significant additive genetic influences on property crime (hc = 0.45 _+ 0.01), which is consistent with earlier reports based on other samples from this Danish adoption cohort (Hutchings and Mednick, 1977; Mednick et al., 1984). Significant genetic influence was also found for antisocial disorders (hA = 0.27 -- 0.01); furthermore, the genetic factors important to this psychiatric composite variable were found to be highly correlated with the genetic influence on property crime (rG = 0.90 ----- 0.003). Environmental factors important to these variables were also significantly but moderately correlated (rE = 0.22 ----- 0.003). It is noteworthy, however, that although the environmental correlation is much smaller in absolute value than the genetic correlation, over half of the observed phenotypic relationship between property crime and antisocial disorders may be attributed to correlated environmental factors. That is, expressing the observed correlation (rp = 0.27) in terms of these genetic and environmental parameters - -hchAr~ + e c e A r E [see.

Crime and Psychiatric Illness 139

Eq. (5)I--revealed that the genetic parameters (hchara = 0.45 x 0.27 x 0.90 = 0.11) account for 0.11/0.27 = 40.5% of this observed relation- ship, whereas the environmental parameters (ecearE = 0.80 X 0.93 • 0.22 = 0.16) account for 0.16/0.27 = 59.5%. Thus, although the genetic correlation appeared high in this case, due to the relatively low propor- tions of genetic influences on each variable, these factors appeared no more important than environmental factors in explaining the observed covariation among property crime and antisocial disorders.

DISCUSSION

The biometrical model-fitting approaches that have been extensively applied in the study of continuous phenotypes are also useful in deter- mining polygenic and environmental influences on discontinuous meas- ures, or threshold characters. Statistical procedures used to fit factor models to dichotomous-item data provide a framework for estimating ge- netic and environmental parameters on the basis of multiple familial re- lationships so that such effects as assortative mating and selective place- ment may be simultaneously taken into account. In the present analyses it seems reassuring that, even after removing any biases in the biometrical parameters resulting from such effects, the results are consistent with other reports of heritable influences on property criminality in both twin and adoption studies (see Cloninger and Gottesman, 1985). Also, Hutch- ings and Mednick's (1975) earlier report of shared family environmental influences on criminality may not necessarily be the result of correlations among adoptive and biological parents, since these data also indicate pa- ternal influences which are independent of selective placement effects. Although significant genetic influences were also found for antisocial dis- orders considered in these analyses, this effect was rather small, ac- counting for only 7.3% of the phenotypic variance in this psychiatric com- posite. This might possibly be attributed to heterogeneity of transmission patterns in different kinds of substance abuse (alcohol versus other drugs) and personality disorders, suggesting the need for considering more spe- cific psychiatric diagnoses and their possible relationships to criminality. It must also be kept in mind that hospital admissions were used here as criteria for determining the mental status of these individuals, which may have a reduced reliability and sensitivity for genetic epidemiological in- vestigations. One may consider the need for obtaining more detailed psy- chiatric evaluations in order to establish more clearly relationships among criminality and mental illnesses in future studies.

When vectors of observations for several phenotypes are available for each family member in a genetic design, matrices of genetic corre-

140 Baker

lations and environmental correlations may also be estimated as dem- onstrated in this report. Such parameters provide important information concerning the etiology of observed relationships among variables of in- terest. Specifically, the extent to which correlations among phenotypes may be attributed to similar genetic and/or similar environmental factors that influence the different characteristics may be determined. Regarding the moderate relationship between property-crime convictions and anti- social disorders found in these data, both correlated genetic and correlated environmental factors accounted for the observed relationship. Since cul- tural transmission is primarily isomorphic for property crime, the cor- related environmental influences which contributed to this relationship must be primarily due to the correlation among nonshared environmental influences important to property criminality and antisocial disorders.

The application presented here for the full adoption design is only one of several which are possible for the analysis of threshold characters. Of course, these procedures may be used in any genetic design in which expected correlations among several relatives may be derived, such as family, twin, and extended twin designs. Multivariate extensions for these designs and the resulting matrices of expectations may be used in the same approach for studying genetic and environmental covariance struc- tures among several phenotypes. Although single-gene models are often of major importance in the analysis of truly discontinuous phenotypes, biometrical models such as that presented here may be useful alternatives if there is uncertainty about the nature of the underlying liability.

Most recent advances in statistical procedures for fitting structural models to combinations of dichotomous, polychotomous, and continuous traits (Muthen, 1984) allow for considerably more complex analyses of multiple phenotypes with differing levels of measurement. Of particular significance may be Muthen's implementation of these statistical advances in a general program for the analysis of linear structural equation models in continuous and discontinuous indicators (LACCI). The framework pro- vided by such a program allows the analysis of covariance components in behavioral genetic designs to be achieved through straightforward spec- ification of multivariate models completely analogous to that described by Fulker et al. (1983) for continuous phenotypes using the LISREL pro- gram (J6reskog and S6rbom, 1983). Thus, the opportunity is available for multivariate biometrical analyses of interesting combinations of contin- uously distributed behavioral and physiological variables (e.g., heart-rate response) and discontinuous phenotypes typically studied in epidemiol- ogy. Such an approach may be especially useful for studying genetic and environment factors underlying relationships between the autonomic ner-

Crime and Psychiatric Illness 141

vous sy s t em and cr iminal i ty (Mednick and Chr is t iansen, 1977), or be- tween H L A B-27 and sch izophren ia (Gat taz et al., 1985).

A C K N O W L E D G M E N T S

The au tho r wishes to thank T h o m a s Eppe l and Toshiaki Sasao for ass i s tance in da ta ana lyses , Dr. Douglas Bone t t fo r p rov id ing useful dis- cuss ions regard ing the statistical es t imat ion p rocedure , and T. N g u y e n for typing the manuscr ip t . R e b e c c a G. Miles and Drs. J. C. DeFr ies , D. W. Fulker , I. I. G o t t e s m a n , and S. A. Me dn ick con t r ibu ted useful and m u c h apprec ia ted c o m m e n t s regarding ear ly vers ions o f the manuscr ip t .

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