## Path Analysis

If instead of latent variables for measuring stress and health in the path diagram of Fig. 57.1 we had a single indicator for each one, the resulting model would be a path analysis model shown in Fig. 57.2. Path analysis is a special case of SEM where all variables are observed (except for the errors) and assumed to be perfectly reliable. To the extent the reliability assumption is true, the path coefficients from the path model versus the structural model should be the same. But when reliability is not perfect, and the indicators contain measurement error, the path coefficients from the path model will be biased, typically underestimated.

Fig. 57.2 Path diagram of mediators of stress and health with single indicators

rather on estimation of model parameters, their test of significance, and the explanatory power of the model. When df is greater than 0, as is the case in our example, the model is overidentified and model fit can be tested. The 2 df (10 - 8) imply that there are two different ways in which our model could be incorrect. In our example, they come from the fact that we specified complete mediation between stress and health, i.e., there is no direct effect linking those two variables (1 df). Also, the two mediators are not specified to be correlated in this model beyond what results from sharing the common predictor of stress (1 df). There is neither a single-headed nor a double-headed arrow linking those two. If either one of those conditions is true our model will be rejected.

Path analysis models have the direction of effects going one way only. These are called recursive models. Recursive models are all identified, meaning solutions may be obtained for all of the parameters in the model. When effects go both ways, say X to Y and also Y to X, the models are no longer path analysis models. These models are called nonrecursive. Under certain conditions nonrecursive models can be identified and analyzed in SEM. While the issue of identification will not be covered in this chapter, it is worth pointing out that a necessary condition for identification of any model is that the number of parameters to be estimated be less than or equal to the number of variances and covariances in the data. This will always be true for path analysis models. If our model has, for example, p = 4 variables, there will be p x (p + 1)/2 or 4 x (4 + 1)/2 = 10 unique variances and covariances. This is the information used for model estimation. In our model in Fig. 57.2 there are q = 8 parameters to be estimated. The difference between p and q is the degrees of freedom (df). When df is 0, the model is just identified and cannot be tested for model fit because it fits the data perfectly, meaning the model-implied variance-covariance matrix perfectly matches the one obtained from the data. These models are also called saturated models. In this case the focus is not on model fit but

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