Psychological Bulletin 1975, Vol. 82, No. S, 711-719
A Model for Interpreting Genetic Correlations with Estimates of Parameters Hoben Thomas Pennsylvania State University A general additive model for kinship correlational structures which includes genetic, environmental, covariance, and measurement error components is parameterized. The properties and assumptions of the model and other models which are viewed as special cases are discussed. A procedure is described for estimating the model parameters from sample correlation coefficients. The procedure results in numerical estimates of the ratio of genetic to environmental variance, estimates of the correlation between genetic and environmental factors, and estimates of the correlation between environments. A simple procedure for partitioning a correlation coefficient into its variance components is suggested.
Figure 1 from Erlenmeyer-Kimling and Jarvik's (1963) classic paper clearly illustrates how correlation coefficients computed from IQ test scores obtained from pairs of individuals tend to increase as the within-pair genetic similarity of the individuals increases. Erlenmeyer-Kimling and Jarvik suggested only that the correlations are consistent with a "polygenie hypothesis" and thus that genetic factors play a role in determining intelligence. Probably few would disagree with this interpretation since the orderliness of the data is striking. However, recent interest has focused on the interpretation of certain kinship correlations as estimates of heritability, that is, the ratio of genetic variance to total phenotypic variance. Constructing theoretical models for interpreting genetic correlations presents a dilemma. To be useful a model must, of course, be applicable to data. However, sometimes such applicability is achieved by making simplifying model assumptions about the values of certain unknown parameters such as, for example, that intelligence-influencing factors in different environments are correlated zero. If such assumed values are psychologically unreasonable, as has been argued, what more realistic values should the parameters take on? Adequate answers to such questions have been difficult to provide
because it has not been clear how to proceed. Yet such answers are critical if progress is to be made and more realistic models are to be constructed. In 1971 Jensen sketched two simple models for interpreting certain correlation coefficients as estimates of heritability. Subsequently, Miller and Levine (1973) and Linn (1974) have argued that Jensen's (test theory) model makes unrealistic assumptions about certain parameter values, thus leading to erroneous heritability estimates. But neither Miller and Levine nor Linn have suggested what they might view as more appropriate parameter values, and thus it is not clear what they would propose as a more reasonable interpretation of the correlational data. In the spirit of rapprochement, an alternative to these positions is offered here. A hopefully more realistic model of genetic correlations is parameterized. Then a procedure is described for obtaining estimates of the model parameters. Estimates of between-environment correlations, genetic-environment correlations, and the relative size of genetic and environmental variances are provided. The estimates appear to be psychologically reasonable and were obtained from kinship correlational data given by Erlenmeyer-Kimling and Jarvik (1963) and Jensen (1969). Finally, a procedure for estiThe author thanks A. R. Jensen and a referee for mating the variance components from any constructive comments on an earlier draft. Requests for reprints should be sent to Hoben kinship correlation is suggested. Thomas, Department of Psychology, 513 B. V. Moore First, however, it is necessary to consider Building, Pennsylvania State University, University Park, Pennsylvania 16802. the general model structure, examine some of 711
712
HOBEN THOMAS
its properties and discuss the assumptions required. The Model Let GI and G2 denote genetic factors which are interpreted as random variables, each with unknown means p,oi and HGZ, respectively, and with common variance ere2. Let E\ and -E2 be the environmental factors, also considered as random variables with unknown means jtt^i and HEZ, respectively, and common variance o-.E2. Finally, let ei and e2 be (test measurement) error random variables with common means and variances jue = 0 and p* and pa > 0, then pz< p* if
A Model for Interpreting Genetic Correlations with Estimates of Parameters Hoben Thomas Pennsylvania State University A general additive model for kinship correlational structures which includes genetic, environmental, covariance, and measurement error components is parameterized. The properties and assumptions of the model and other models which are viewed as special cases are discussed. A procedure is described for estimating the model parameters from sample correlation coefficients. The procedure results in numerical estimates of the ratio of genetic to environmental variance, estimates of the correlation between genetic and environmental factors, and estimates of the correlation between environments. A simple procedure for partitioning a correlation coefficient into its variance components is suggested.
Figure 1 from Erlenmeyer-Kimling and Jarvik's (1963) classic paper clearly illustrates how correlation coefficients computed from IQ test scores obtained from pairs of individuals tend to increase as the within-pair genetic similarity of the individuals increases. Erlenmeyer-Kimling and Jarvik suggested only that the correlations are consistent with a "polygenie hypothesis" and thus that genetic factors play a role in determining intelligence. Probably few would disagree with this interpretation since the orderliness of the data is striking. However, recent interest has focused on the interpretation of certain kinship correlations as estimates of heritability, that is, the ratio of genetic variance to total phenotypic variance. Constructing theoretical models for interpreting genetic correlations presents a dilemma. To be useful a model must, of course, be applicable to data. However, sometimes such applicability is achieved by making simplifying model assumptions about the values of certain unknown parameters such as, for example, that intelligence-influencing factors in different environments are correlated zero. If such assumed values are psychologically unreasonable, as has been argued, what more realistic values should the parameters take on? Adequate answers to such questions have been difficult to provide
because it has not been clear how to proceed. Yet such answers are critical if progress is to be made and more realistic models are to be constructed. In 1971 Jensen sketched two simple models for interpreting certain correlation coefficients as estimates of heritability. Subsequently, Miller and Levine (1973) and Linn (1974) have argued that Jensen's (test theory) model makes unrealistic assumptions about certain parameter values, thus leading to erroneous heritability estimates. But neither Miller and Levine nor Linn have suggested what they might view as more appropriate parameter values, and thus it is not clear what they would propose as a more reasonable interpretation of the correlational data. In the spirit of rapprochement, an alternative to these positions is offered here. A hopefully more realistic model of genetic correlations is parameterized. Then a procedure is described for obtaining estimates of the model parameters. Estimates of between-environment correlations, genetic-environment correlations, and the relative size of genetic and environmental variances are provided. The estimates appear to be psychologically reasonable and were obtained from kinship correlational data given by Erlenmeyer-Kimling and Jarvik (1963) and Jensen (1969). Finally, a procedure for estiThe author thanks A. R. Jensen and a referee for mating the variance components from any constructive comments on an earlier draft. Requests for reprints should be sent to Hoben kinship correlation is suggested. Thomas, Department of Psychology, 513 B. V. Moore First, however, it is necessary to consider Building, Pennsylvania State University, University Park, Pennsylvania 16802. the general model structure, examine some of 711
712
HOBEN THOMAS
its properties and discuss the assumptions required. The Model Let GI and G2 denote genetic factors which are interpreted as random variables, each with unknown means p,oi and HGZ, respectively, and with common variance ere2. Let E\ and -E2 be the environmental factors, also considered as random variables with unknown means jtt^i and HEZ, respectively, and common variance o-.E2. Finally, let ei and e2 be (test measurement) error random variables with common means and variances jue = 0 and p* and pa > 0, then pz< p* if
