Falconer’s Classical Twin Designs & Structural Equation Modelling

Genetics have “a substantial role in the origins of individual differences” (Rutter, 2002: 2; In Bouchard, 2004: 148). In fact, the literature is very confident in the belief that all human behavioural traits have a genetic basis (Johnson, et.al, 2009: 217). This is not to say that genes determine behaviour, but rather that there are “intrinsic factors” to consider in explaining behavioural traits (ibid: 220). As a research endeavour, we need to untangle what proportions of individual differences are causally traceable to a person’s genotype as opposed to environmental conditions. The total variance in this measure is constituted by genetic (G) and environmental factors (E), the interaction between genotype and the environment (GxE), as well as measurement error (Falconer, 1960; In Jang, 2005: 19). This presents us with a problem: Given the environmental and genetic variance between individuals, how can we make judgements about how a genotype influences behaviour, ceteris paribus?

Classical twin studies are one tool that can be used to address this issue. To begin, we must distinguish between dizygotic (DZ) twins, i.e. those who share essentially the same detrimental aspects of early developmental environment, birthday and exact age, and a mean genetic identity of 50%; as well as monozygotic (MZ) twins, who differ from DZ twins only in their genetic identity and originate from the same ovum (MacGregor, et.al, 2000: 131; Jang, 2005: 21). Any divergence in individual differences between MZ twins cannot be due to genetic factors, which means that they allow us to trace environmental influences. The reason that we want to examine DZ twins as well is that the difference between MZ and DZ twin similarity gives an indication of the size of the environmental effect. In the classical twin design, we measure the correlation of the presence of a trait between MZ as well as DZ twins. As an example we could correlate a conscientiousness and openness measure between twins of both types across a sample. To illustrate, consider the following matrix:

observed correlations

Let us assume for simplicity that the results of measuring this phenotype led to normally distributed data allowing the use of Pearson’s coefficient. In the example, greater genetic similarity (being MZ rather than DZ) increases the coefficients, entailing greater phenotypical likeness (Boomsma, et.al, 2002: 873). In analysing such data, Falconer has provided arithmetic tools for separating the sources of variance between the traits, splitting them into three: The variability proportion in behaviour due to the shared “family environment between households”, e.g. family income, denoted by c2 (Jang, 2005: 22); variance due to unique experiences of the individual, the e2 term, which characterises non-shared environment (ibid: 23); as well as broad heritability (h2b or a2). Their sum is the total variance Vp.

The notion of heritability requires explanation. We are interested in the proportion of variance in the behavioural phenotype of a population that is due to that population’s genotype (MacGregor, et.al, 2000: 131). 1 minus this term conversely gives the common and non-shared environmental contribution to the trait plus error variance (Bouchard, 2004: 148). One problem in trying to assess this number is that many genes interact. For instance, your hair may be darker by carrying a certain gene. For now we assume that if this gene is present twice (as is the case in MZ twins), then your hair should be twice as dark. In this narrow sense, heritability is only additive. However, this is a gross oversimplification since it ignores both dominance of genes as well as the interaction with other genes that control pigmentation. Considering all of these at once would be heritability in a broad sense. The narrow sense is still a useful measure as the non-additive effects are not strictly speaking heritable, since a dominant or recessive relationship cannot be passed down from a single parent (Jang, 2005: 20). However, Falconer’s formula cannot distinguish additive from other genetic factors, hence at the moment we must consider only broad heritability. Let us solve for the three variables.

For trait 1:for trait 1

 

For trait 2:for trait 2

To begin, let us note that since correlations overall were higher for trait 1, we can assume a greater role of the shared family environment in bringing it about in comparison to trait 2. It is hence possible that trait 1 corresponds to some lifestyle measure (Boomsma, et.al, 2002: 873). Note also that trait 1 has a broad heritability of 40% and trait 2 of 100%. The shared environment for trait 2 accounts for negative 40% of behavioural variance. Here we find easily a limitation of Falconer’s formula, namely that it can produce uninterpretable values, such as negative variance and total explanation by genome alone. Falconer’s formula also assumes equality of the shared environments between MZ and DZ twins. Violation hereof is not necessarily problematic (see Borkenau et.al, 2002; in Jang, 2005: 24), who found that reports of being treated more similarly by MZ twins in comparison to DZ twins did not affect personality measures). Still, its inability to distinguish additive genetic effects should compel us to consider an alternative approach.

In trying to explain the sources of variance between the two traits in the data, we must consider a less statistically simplistic strategy. Preferably, we desire a modelling technique that allows for the inclusion of covariates, such as sex (a great many genes of interest are only expressed in one sex, the classical design cannot include this as predictor variable). Also it should be able to model the difference between the additive only and broad characterisations of heritability within our data.

One way to achieve this is through structural equation modelling (SEM) (Boomsma, et.al, 2002: 874). Here we define unobserved latent variables, to represent the sources of variance. Consider figure 1 below. Underlying all four score categories (for each twin in MZ pairs and again in DZ twins), there are four factors explaining its variance that we model (note that we can model arbitrarily more): Additive genetic (A), dominant genetic to account for gene interaction (D), common environment (C), as well as non-shared environment (E) (Jang, 2005: 27; Snieder, et.al, 1999: 429). The observed scores are now permitted to load onto these latents. Their factor loadings, or relationship strengths (a+d), c, and e, when squared become estimates of the previously introduced h2b, c2 and e2. (It would not be appropriate to plug the output of Falconer’s formula in as factor loading though.) The loadings are also allowed to correlate. Here we find a difference between MZ and DZ twins: for MZ pairs, A, D and C are assumed to be equal, hence have correlations of 1.0. For DZ twins, A will be on average .5. D, since it requires two chromosome pairs lies at .25, while the environment is again assumed to be the same. Extending this model would also include more observed variables, such as sex to be allowed to load on the latents.

Unfortunately the data available does not lend itself to performing SEM, but we anticipate more sophisticated and reasonable heritability estimates than we could calculate above. We would calculate an initial model fit, which would be moulded into a best fit given our data using maximum likelihood estimates (if the data is normal). The SEM approach makes a number of important assumptions that must be considered though. Just like Falconer’s approach, we presume equality of shared environments. More importantly however, we still cannot account for the GxE interaction. Since we believe that some environments influence gene expression (epigenetics) or that gene expression changes one’s sensitivity to certain environments (Boomsma, et.al, 2002: 873), this is limiting. Still, this modelling technique for classical twin studies has found useful application in what Wendy Johnson calls a “sexy example” (et.al, 2009: 219): Running a multiple regression between age of first sex and delinquency reveals a negative relationship: The earlier an individual engages in sexual activity, the more likely the person is to become delinquent (Armour & Haynie, 2007). However, using twins within this data set, it was shown that after controlling for environmental and genetic influences using SEM, twins that had sex earlier were in fact less likely to become delinquent later (Harden, et.al, 2008).

 

SEM path diagram

Figure 1: SEM path diagram for heritability estimation

In essence, the classical twin design consists of finding correlations between MZ and DZ twins in respect to a trait. These can be used to calculate broad heritability as well as shared and non-shared environmental influences using Falconer’s formula as we have done for the sample data. Furthermore, using SEM we can control for covariates and untangle the components of heritability into a more comprehensive list of sources of variance, while avoiding some of the pitfalls of Falconer’s method. These potential sources of variance in traits 1 and 2 are 1) the common shared environment of individuals, such as culture, diet or income; 2) the non-shared unique experience of each individual which shapes our behaviour, 3) dominance and gene combination effects that bring about traits contingent on the presence of multiple genes or two sets of the same gene, 4) the interaction of environment and genotype as well as the 5) additive effect of single genes.

References

Armour, S., & Haynie, D. L. 2007. Adolescent Sexual Debut and Later Delinquency. Journal of Youth and Adolescence 36: 141-152.

Boomsma, D., Busjahn, A., & Peltonen, L. 2002. Classical Twin Studies and Beyond. Nature Reviews Genetics 3: 872-882.

Borkenau, P., Riemann, R., Angleitner, A., & Spinath, F. M. 2002. Similarity of Childhood Experiences and Personality Resemblance in Monozygotic and Dizygotic Twins: A Test of the Equal Environments Assumption. Personality and Individual Differences 33: 261-269.

Bouchard, T. J. 2004. Genetic Influence on Human Psychological Traits. Current Directions in Psychological Science 13(4): 148-151.

Duffy, D. L., Martin, N. G., Battistutta, D., Hopper, J. L., & Mathews, J. D. 1990. Genetics of Asthma and Hay Fever in Australian Twins. American Review of Respiratory Disease 142: 1351-1358.

Falconer, D. S. 1960. Introduction to Quantitative Genetics. New York: Ronald Press.

Harden, K. P., Mendle, J., Hill, J. E., Turkheimer, E., & Emery, R. E. 2008. Rethinking Timing of First Sex and Delinquency. Journal of Youth and Adolescence 37: 373-385.

Hardoon, D. C., Ettinger, U., Mourão-Miranda, J., Antonova, E., Collier, D., Kumari, V., Williams, S. C. R. & Brammer, M. 2009. Correlation-based Multivariate Analysis of Genetic Influence on Brain Volume. Neuroscience Letters 450: 281-286.

Jang, K. L. 2005. The Behavioral Genetics of Psychopathology: A Clinical Guide. London: Routledge.

Johnson, W., Turkheimer, E., Gottesmann, I. I. & Bouchard, T. J. 2009. Beyond Heritability: Twin Studies in Behavioural Research. Current Directions in Psychological Science 18(4): 217-220.

MacGregor, A. J., Snieder, H., Schork, N. J. & Spector, T. D. 2000. Twins: Novel Uses to Study complex Traits and Genetic Diseases. Trends in Genetics 16(3): 131-134.

Martin, N., Boomsma, D. & Machin, G. 1997. A Twin-pronged Attack on Complex Traits. Nature Genetics 17: 387-392.

Neale, M. C. & Kendler, K. S. 1995. Models of comorbidity for multifactorial disorders. American Journal of Human Genetics. 57: 935-953.

Posthuma, D. Multivariate Genetic Analysis. In Kim, Y. K. 2009. Handbook of behavior genetics. 47-59. New York: Springer.

Rutter, M. 2002. Nature, Nurture, and Development: From Evangelism through Science toward Policy and Practice. Child Development 73: 1-21.

Schmitz, S., Cherny, S. S. & Fulker, D. W. 1998. Increase in Power through Multivariate Analysis. Behavior Genetics 28(5): 357-363.

Snieder, H., Boomsma, D. I., van Doornen, L. J. P., & Neal, M. C. 1999. Bivariate Genetic Analysis of Fasting Insulin and Glucose Levels. Genetic Epidemiology 16: 426-446.

Trubnikov, V. I., Alfimova, M. V., Uvarova, L. G., Orlova, V. A. 1995. A Multivariate Genetic Analysis of the Data from a Complex Study of the Predisposition of Schizophrenia. [article published in Russian]. Zh Nevrol Psikhiatr Im S S Korsakova 95(2): 50-56.

Wassermann, E. M. Individual Differences in Response to Transcranial Magnetic Stimulation of the Motor Cortex. In Hallett, M. & Chokroverty, S. (Eds.). 2005. Magnetic stimulation in clinical neurophysiology. Philadelphia: Elsevier Health Sciences.

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