# Behavioural Genetic Interactive Modules

## Multivariate Analysis

### Overview

This module aims to demonstrate some of the principles underlying multivariate genetic analysis using twins. The ACE model, introduced in previous modules, is extended to the bivariate case: analysing two variables to decompose the covariance between measures into genetic and environmental components.

### Tutorial

The Multivariate Analysis module consists of three main panels. The panel at the top of the display contains sliders which control nine parameters of the bivariate ACE model. Adjusting these parameters will cause the Expected correlations and Bivariate statistics (displayed in the two panels below) to vary. The nine parameters of the bivariate ACE model are divided into three colour-coded groups: three univariate parameters for the first trait (in red) which represent the additive genetic, shared environmental and nonshared environmental components of variance. Note that these parameters are expressed as proportions of variance - that is, they will always sum to 1 (which is why increasing one slider will cause the two other values to drop slightly).

In a similar manner, there are three univariate parameters that can be specified for the second trait (coded in green). In this case, the heritability of Trait 2 is set to just under 42%, the shared environment to just under 5% and the nonshared environmental to just under 54%. Note: you may find it difficult when trying to specify exact values for the three parameters because of the inter-dependency amongst them that makes them all change when only one is changed (to ensure they always sum to 1, as mentioned).

The remaining three parameters represent the correlations between univariate parameters (coded white). The concept of a genetic, shared environmental or nonshared environmental correlation is explained in the Appendix. As these parameters are correlations, not proportions of variance, they range from -1 to +1 instead of 0 to 1. A high, positive genetic correlation would imply that the genetic influences on one trait tend to overlap with the genetic influences on the second trait, independent of the actual magnitude of the genetic influence for either trait.

The aim of multivariate analysis would be to determine these nine parameters based on comparison between the three types of multivariate twin correlation:

• cross-trait, within-individual
• within-trait, cross-twin (separately for MZ and DZ twin pairs)
• cross-trait, cross-twin (separately for MZ and DZ twin pairs)

This process is reversed in the current module: we can specify the nine parameters to determine what type of correlation structure we would expect to observe for different underlying aetiologies.

The three types of correlation above are expressed, in a slightly different manner, in the Expected correlations panel of the module. This panel represents three correlation matrices. The first matrix is labelled Phenotypic correlations and represents the within-individual correlations. Of course, within-individual, within-trait correlations will always be 1 - these are represented by the diagonal elements of this matrix. The off-diagonal element is of primary interest in this matrix: it represents the cross-trait, within-individual correlation. That is, this is the correlation we would expect to find between Trait 1 and Trait 2 if we measured both traits in a large number of unrelated individuals to see how the one individual's score on Trait 1 is associated with their score of Trait 2.

The elements of this correlation matrix are colour-coded (as are the other two correlation matrices). The top diagonal element is coded red as it represents Trait 1. The second diagonal element is coded green as it represents Trait 2. The off-diagonal element is coded white as it represents a correlation between the two traits. An important point to note, however, is that the "white" elements of all three expected correlation matrices are determined not just by the three Correlation parameters in the panel above: they will also depend on the values of the six univariate parameters. Exploring the modules, you will be able to examine these relationships further.

The final panel represents the bivariate statistics. These three statistics would often be calculated after multivariate genetic analysis has been performed as they provide useful information regarding the extent to which the phenotypic correlation between two traits is mediated by genetic and/or environmental factors. For example, the bivariate heritability is a measure of the extent to which shared genetic influence generates a correlation between two traits. If two traits had a phenotypic correlation of 0.5 and bivariate heritability of 0.4, we could conclude that 80% of the correlation is mediated by shared genetic influence. Bivariate heritability is therefore distinct from the genetic correlation. For example, even if two traits have a very high genetic correlation, if neither trait is strongly heritable then shared genetic influences are unlikely to explain much of the observed correlation between the two traits. The calculation of the bivariate heritability is outlined in the Appendix : it is essentially a function of the two univariate heritabilities and the genetic correlation. The bivariate statistics for the shared and nonshared environment are equivalent to the bivariate heritability in calculation and interpretation.

### Example

Use the module to verify the example of multivariate analysis in the Appendix, where three correlation matrices were given (phenotypic, MZ and DZ) from which the univariate statistics, the genetic and environmental correlations and the bivariate statistics were calculated. Although the example was for three traits the module can only deal with two at a time, the bivariate case. (Multivariate analysis can, in principle, deal with any number of variables.) However, it is perfectly valid to extract the appropriate pairwise values for only two of the three traits and we should expect to find a similar pattern.

Instead of calculating the parameters given the observed correlations, we can use the module to calculated the expected correlations, given certain parameter values: this should give similar results.

Parameters

Univariate statistics

 Trait X Trait Y Trait Z A 74% 60% 23% C 4% 31% 47% E 22% 9% 30%

Genetic correlations

Shared environmental correlations

Nonshared environmental correlations

 1 0.44 1.00 0.11 0.75 1.00

 1 0.98 1.00 0.17 0.26 1.00

 1 0.1 1.00 0.89 0.46 1.00

Expected

Phenotypic correlations

 Trait X Trait Y Trait Z Trait X 1.00 Trait Y 0.42 1.00 Trait Z 0.30 0.45 1.00

MZ correlations  DZ correlations

 Trait X Trait Y Trait Z Trait X 0.78 Trait Y 0.40 0.91 Trait Z 0.08 0.39 0.70

 Trait X Trait Y Trait Z Trait X 0.40 Trait Y 0.26 0.61 Trait Z 0.04 0.23 0.58

For example, to confirm the relationship between X and Y, enter the relevant univariate parameters as Trait 1 and Trait 2, enter the genetic, shared environmental and nonshared correlations between X and Y and confirm that the expected correlations are similar to those observed in the example. Note the bivariate statistics. What conclusions can you draw from these about the relationships between these three traits?

### Questions

Question 1.

Using the module, do you notice any special relationship between the three bivariate statistics and the phenotypic correlation between Trait 1 and Trait 2? If so, why does this relationship exist?

Question 2.

Is there any scenario in which we might observe little or no correlation between the two traits despite a high genetic correlation? Verify your answer using the module.

Question 3.

Is there any scenario where we might expect DZ twins to be more similar than MZ twins? Verify your answer using the module.

The bivariate statistics always sum to the phenotypic correlation. This is because they represent the proportion of between-trait shared variance which is shared due to either additive genetic, shared environmental or nonshared environmental influences. The phenotypic correlation, by definition, represents the sum of these three sources of covariation between the two traits.

If two traits have a high positive genetic correlation but a negative (non)shared environmental correlation (or vice versa) then we might expect to find near zero phenotypic correlations (depending on the relative balance of genetic and environmental influences for the two traits).

This point is important because it shows that two traits that appear phenotypically unrelated might actually share genetic or environmental causes.