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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,
and )
were obtained as
well as for
the parameters of the Gamma model (A,
end ).
Instead of the Erlang model with parameters A, N, and the more general Gamma model was used with the
parameters A,
and .
Note, that the parameter N is discrete, whereas the parameter is continuous. Both parameters can be written as .
In both
models the individual working times are constant and equal to
.
The parameter A was estimated from the RTs 1-6,
the parameter (and )
from the RTs:
RT07, RT10, RT13, RT16, RT19, RT22, RT25, RT28,
and the parameter
(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
and (by rewriting equation and ):
|
(15) |
and
|
(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 ):
|
(17) |
and
|
(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:
|
(19) |
and for the Gamma model:
|
(20) |
Note, that
and
Therefore,
across subjects,
has the same distribution
as
except for a shift to the right
of
.
This means that statistical tests
with respect to the normality of the distribution of
and
(or
and
)
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:
|
(21) |
|
(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: Results
Up: The Erlang model as
Previous: Structural Equations
AHGS VAN DER VEN
2002-01-14