Structural Equation Modeling Using the PE Model

According to the Poisson-Erlang model, expectation and variance are as follows:

$$E(T) = A + \frac{\lambda}{\delta}A$$ (6)

and

$$Var(T) = 2\frac{\lambda}{\delta^2}A$$ (7)

Taking the natural logarithm of both sides one obtains:

$$\ln E(T) = \ln(A+\frac{\lambda}{\delta}A) = \ln A + \ln(1+\frac{\lambda}{\delta})$$ (8)

and

$$\ln Var(T) = \ln(2\frac{\lambda}{\delta^2}A) = \ln 2 + \ln \lambda + \ln A - 2 \ln \delta$$ (9)

The logarithm of the variance is a linear function of the logarithm of the parameters A, $\lambda$ and $\delta$. However this is not the case with respect to the logarithm of the expectation. However, the difference between the expectation E(T) and the real total working time A: E(T)-A, is a linear function of the log of the PE-parameters. So, according to the PE model, the correlation across subjects between $\ln(E(T)-A)$ and $\ln Var(T)$ is linearly dependent on the log parameters $\ln A, \ln \lambda$ and $\ln \delta$. This correlation can now easily be described in terms of the variances and covariances of these variables across subjects. However, this is a correlation between true scores. In actual practice only observed scores (estimates) for $\ln(E(T)-A)$ and $\ln Var(T)$are available. Moreover, $\ln \lambda$ and $\ln \delta$ are latent variables. Therefore, the decision was made to use structural equation modeling to test the above mentioned linear equations using the computer program EQS (see Bentler, 1989).