next up previous
Next: Results Up: The Erlang model as Previous: Structural Equations

Method

In order to make it possible to decide between the Poisson-Erlang model and the Erlang model the following procedure was followed. For each of the 185 subjects mentioned above, independent estimates of the parameters of the Poisson-Erlang model (A, $\lambda$ and $\delta$) were obtained as well as for the parameters of the Gamma model (A, $\gamma$ end $\delta$). Instead of the Erlang model with parameters A, N, and $\delta$the more general Gamma model was used with the parameters A, $\gamma$ and $\delta$. Note, that the parameter N is discrete, whereas the parameter $\gamma$is continuous. Both parameters can be written as $\lambda A$. In both models the individual working times are constant and equal to $1/\lambda$. The parameter A was estimated from the RTs 1-6, the parameter $\lambda$(and $\gamma$) from the RTs:
RT07, RT10, RT13, RT16, RT19, RT22, RT25, RT28,
and the parameter $\delta$ (in both models) from the RTs:
RT08, RT11, RT14, RT17, RT20, RT23, RT26, RT29.
For the estimation of the parameters the method of moments was used. In the case of the Poisson-Erlang model one obtains for the parameters $\lambda$ and $\delta$(by rewriting equation [*] and [*]):

 \begin{displaymath}
\hat{\lambda} = \frac{2(m-A)^2}{As^2}\end{displaymath} (15)

and

 \begin{displaymath}
\hat{\delta}(PE) = \frac{2(m-A)}{s^2}\end{displaymath} (16)

where the minimum of the RTs 1-6 is used as an estimate for A, The statistic m corresponds to the observed mean and s2 to the observed variance. For the parameters of the Gamma model one obtains (by rewriting equation [*] and [*]):

 \begin{displaymath}
\hat{\gamma} = \frac{(m-A)^2}{s^2}\end{displaymath} (17)

and

 \begin{displaymath}
\hat{\delta}(G) = \frac{(m-A)}{s^2}\end{displaymath} (18)

Using these estimators, for each model and for each subject, the predicted third central moment was computed was computed as follows: for the Poisson-Erlang:

 \begin{displaymath}
\tilde{\mu}_3(PE) = \frac{6\hat\lambda A}
{\hat\delta^3(PE)}\end{displaymath} (19)

and for the Gamma model:

 \begin{displaymath}
\tilde{\mu}_3(G) = \frac{2\hat{\gamma}}
{\hat\delta^3(G)}\end{displaymath} (20)

Note, that $\tilde{\mu}_3(PE) = \frac{3}{4}\tilde{\mu}_3(G)$and $\ln\tilde{\mu}_3(PE) = \ln\frac{3}{4} + \ln\tilde{\mu}_3(G)$Therefore, across subjects, $\ln\tilde{\mu}_3(PE)$ has the same distribution as $\ln\tilde{\mu}_3(G)$ except for a shift to the right of $\ln\frac{3}{4}$. This means that statistical tests with respect to the normality of the distribution of $\tilde{\mu}_3(PE)$ and $\tilde{\mu}_3(G)$(or $\ln\tilde{\mu}_3(PE)$ and $\ln\tilde{\mu}_3(G)$) yield identical results. Finally, for each subject, the observed third central moment was computed from the RTs: RT09, RT12, RT15, RT18, RT21, RT24, RT27, RT30. According to Kendall and Stuart (1969, Vol. I, p. 281) an unbiased estimate for the third central moment is as follows:

 \begin{displaymath}
\frac{n}{(n-1)(n-2)}\cdot \sum_{i=1}^n (x_i-m)^3
\end{displaymath} (21)


 \begin{displaymath}
\mbox{with } m=\frac{\sum_{i=1}^n x_i}{n}
\end{displaymath} (22)

For each model a separate statistical test was applied in which the mean of the observed third central moment was compared with the mean of the predicted third central moment using Student's t test for repeated measurements.


next up previous
Next: Results Up: The Erlang model as Previous: Structural Equations
AHGS VAN DER VEN
2002-01-14