## Model fitting

Although we can simply estimate h2, c2, and e2 from the equations above for more complex data, solving multiple linear equations becomes difficult. We may also wish to test alternative models, for example one where there is no genetic contribution or one where shared environmental influences are dropped. Model fitting allows us to first statistically test how well a given model explains the observed data and to then compare different models.

Computer packages such as Mx1 are all based on the same principles. The raw data are read into the program and the researcher supplies the initial starting values for the unknown parameters (h, c, and e for a full genetic model). The program then iterates with different parameter estimates until values are found which give an optimum fit (usually this involves maximizing a likelihood function or minimizing a c2). The goodness of fit of the model is then assessed by examining the c2 goodness of fit where a smaller value indicates a better fit.

The fit of a reduced model (R) can then be compared against the full model (F) by subtracting the c2 values (R - F). Alternatively the fit of models can be compared by using the likelihood ratio test where twice the difference between the log likelihoods for each model (this approximates a c 2 distribution) is calculated.

An example of model fitting is shown in Table,,,!,.

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Table 1 Data on depressive symptoms for school-aged twins: example of model fitting

A full ACE model (additive genes, common/shared environment, and non-shared environment) gives a good fit (c2 = 0.6, df = 3, P = 0.90) with estimates of 0.59 for h2, 0.15 for c2, and 0.26 for e2. When we compare the fit of the reduced AE model where we have dropped shared environmental effects, there is no significant difference in the fit of the model (c2 = 1.36 - 0.6 = 0.76, df = 1, P < 0.001). However, there is a significant worsening of fit for the CE model (c2 = 15.18) and a very poor fit for a model of no familial transmission E (c2 = 102.3). Thus, on the grounds of goodness of fit and parsimony (accepting the simplest model), an AE model provides the most satisfactory explanation of the data.

### Multivariate model fitting

So far we have considered the influence of genes and environment on variation in a single phenotype. This type of analysis is known as univariate genetic analysis.

Increasingly we have become more interested in the contribution of genes and environment to the covariation or correlation of two or more phenotypes (multivariate genetic analysis). This strategy allows us to examine to what extent the covariation or correlation between two phenotypes, for example depression and anxiety, is due to genetic factors that influence both phenotypes and how much can be explained by an environmental aetiology common to both disorders. Thus, multivariate genetic analysis can be used to examine the co-occurrence or comorbidity of different psychiatric disorders. This method, for example, has shown that the same set of genes, but different non-shared environmental factors influence anxiety and depressive disorders (and symptoms). Similarly this method is increasingly being utilized to examine the relationship between environmental risk factors and psychopathology and the role of genetic and environmental mediating factors.

### Multiple regression analysis

Another commonly used method of analysing twin data is multiple regression analyses.(3) Here the score of the co-twin C is predicted by the score of the proband twin P, the coefficient of the relationship or zygosity R and an interaction term PR. The partial regression coefficients provide direct estimates of heritability and shared environment. The advantage of this method is that it can then also be used to test whether the genetic aetiology of extreme scores for a continuous trait differs from scores within the normal range. 