7 Identification of Models and Parameters

One key issue with structural equation modeling is whether a model, or a parameter within a model is identified. We say that the free parameters of a model are either (i) overidentified; (ii) just identified; or (iii) underidentified. If all of the parameters fall into the first two classes, we say that the model as a whole is identified, but if one or more parameters are in class (iii), we say that the model is not identified. In this section, we briefly address the identification of parameters in structural equation models, and illustrate how data from additional types of relative may or may not identify the parameters of a model. When we applied the rules of standardized path analysis to the simple path coefficient model for twins (Figure 5.3a), we obtained expressions for MZ and DZ covariances and the phenotypic variance:

$$\mbox{Cov}(MZ) = c^2+a^2+d^2$$ (33)

$$\mbox{Cov}(DZ) = c^2+.5a^2+.25d^2$$ (34)

$$V_P = c^2+a^2+d^2+e^2$$ (35)

These three equations have four unknown parameters $c,a,d$ and $e$, and illustrate the first point about identification. A model is underidentified if the number of free parameters is greater than the number of distinct statistics that it predicts. Here there are four unknown parameters but only three distinct statistics, so the model is underidentified. One way of checking the identification of simple models is to represent the expected variances and covariances as a system of equations in matrix algebra:

$${\bf Ax} = \bf b$$

where $\bf x$ is the vector of parameters, $\bf b$ is the vector of observed statistics, and $\bf A$ is the matrix of weights such that element $A_{ij}$ gives the coefficient of parameter $j$ in equation $i$. Then, if the inverse of $\bf A$ exists, the model is identified. Thus in our example we have:

$$\left(\begin{array}{rrrr} 1&1&1&0\ 1&.5&.25&0\ 1&1&1&1 \e... ... \left(\begin{array}{r} b_1\ b_2\ b_3 \end{array}\right).$$ (36)

where $b_1$ is Cov(MZ), $b_2$ is Cov(DZ), and $b_3$ is $V_P$. Now, what we would really like to find here is the left inverse, $\bf L$, of $\bf A$ such that ${\bf LA}=\bf I$. However, it is easy to show that left inverses may exist only when $\bf A$ has at least as many rows as it does columns (for proof see, e.g., Searle, 1982, p. 147). Therefore, if we are limited to data from a classical twin study, i.e. MZ and DZ twins reared together, it is necessary to assume that one of the parameters $a,c$ or $d$ is zero to identify the model. Let us suppose that we have reason to believe that $c$ can be ignored, so that the equations may be rewritten as:

$$\left(\begin{array}{rrr} 1&1&0\ .5&.25&0\ 1&1&1 \end{arra... ...)= \left(\begin{array}{r} b_1\ b_2\ b_3 \end{array}\right)$$

and in this case, the inverse of $\bf A$ exists. Another, generally superior, approach to resolving the parameters of the model is to collect new data. For example, if we collected data from separated MZ or DZ twins, then we could add a fourth row to $\bf A$ in equation 5.11 to get (for MZ twins apart)

$$\left(\begin{array}{rrrr} 1&1&1&0\ 1&.5&.25&0\ 1&1&1&1\ ... ...t(\begin{array}{r} b_1\ b_2\ b_3\ b_4 \end{array}\right)$$ (37)

where $b_4$ is Cov(MZA), and again the inverse of $\bf A$ exists. Now it is not necessarily the case that adding another type of relative (or type of rearing environment) will turn an underidentified model into one that is identified! Far from it, in fact, as we show with reference to siblings reared together, and half-siblings and cousins reared apart. Under our simple genetic model, the expected covariances of the siblings and half-siblings are

$$\mbox{Cov}(Sibs) = c^2+.5a^2+.25d^2$$ (38)

$$\mbox{Cov}(Half-sibs) = .25a^2$$ (39)

$$\mbox{Cov}(Cousins) = .125a^2$$ (40)

$$V_P = c^2+a^2+d^2+e^2$$ (41)

as could be shown by extending the methods outlined in Chapter 3. In matrix form the equations are:

$$\left(\begin{array}{rrrr} 1&.5&.25&0\ 0&.25&0&0\ 0&.125&0... ...(\begin{array}{r} b_1\ b_2\ b_3\ b_4 \end{array}\right).$$ (42)

where $b_1$ is Cov(Sibs), $b_2$ is Cov(Half-sibs), $b_3$ is Cov(Cousins), and $b_4$ is $V_P$. Now in this case, although we have as many types of relationship with different expected covariance as there are unknown parameters in the model, we still cannot identify all the parameters, because the matrix $\bf A$ is singular. The presence of data collected from cousins does not add any information to the system, because their expected covariance is exactly half that of the half-siblings. In general, if any row (column) of a matrix can be expressed as a linear combination of the other rows (columns) of a matrix, then the matrix is singular and cannot be inverted. Note, however, that just because we cannot identify the model as a whole, it does not mean that none of the parameters can be estimated. In this example, we can obtain a valid estimate of additive genetic variance $a^2$ simply from, say, eight times the difference of the half-sib and cousin covariances. With this knowledge and the observed full sibling covariance, we could estimate the combined effect of dominance and the shared environment, but it is impossible to separate these two sources. Throughout the above examples, we have taken advantage of their inherent simplicity. The first useful feature is that the parameters of the model only occur in linear combinations, so that, e.g., terms of the form $c^2a$ are not present. While true of a number of simple genetic models that we shall use in this book, it is not the case for them all (see Table  for example). Nevertheless, some insight may be gained by examining the model in this way, since if we are able to identify both $c$ and $c^2a$ then both parameters may be estimated. Yet for complex systems this can prove a difficult task, so we suggest an alternative, numerical approach. The idea is to simulate expected covariances for certain values of the parameters, and then see whether a program such as Mx can recover these values from a number of different starting points. If we find another set of parameter values that generates the same expected variances and covariances, the model is not identified. We shall not go into this procedure in detail here, but simply note that it is very similar to that described for power calculations in Chapter 7.