The PXL distribution diskettes contain several demo and test programs that
are not full applications but only demonstrate the use of certain features.
The demo programs have
their own command line options. Options that are identical to those of PXL
applications are those for setting `screenwidth` and
`screenheight` by `-w n` and `-h n` respectively, `-v n` for
setting `videocode`, and `-t n` for setting `switchtype`. In
addition to that the demos have `-g filename` for explicitly setting the
file name of the gamma table to be used.
Some of the programs take additional options. In order to get a complete
list of options and their meaning, start the programs with option `-?`.

Some programs have parameters which require keyboard codes as values.
Examples are the parameters `yeskey`, `nokey`, and `stopkey`. The
program `event` may be used to find the appropriate key code values.

The program `ttcheck` tests the proper working of PXL's internal time
measurement functions. It also gives an idea of how precise the time
measurement routines are. `ttcheck` can measure the duration of the
following events (the numbers give the arguments for `ttcheck` necessary
to measure the respective event):

- 0:
*Video Frame Duration.*This is the duration of a single video frame in the current video mode. - 1:
*Vertical Retrace Pulse.*This is an extremely short time interval of approximately 0.064 ms duration. It corresponds to the duration of the pulse which initiates the vertical retrace. - 2:
*BIOS Timer Tick Interval.*The BIOS timer tick is derived from the system timer tick which is run at a frequency of 1193180 Hz. The BIOS timer tick is incremented every 65536 system ticks. Thus the BIOS timer tick interval should be 54.925 ms.

The psychometric function type used is a logistic function with the
parameters *a*, corresponding to the just noticable difference and the
parameter *c* corresponding to the constant error:
P(yes|x_j) = (1-exp(-(x_j - c)/a))^(-1)
The minimization function is the *Chi^2*:
Chi^2 = Sum(j=1,M) ((n_j - N_j P(yes|x_j))^2)/(N_j P(yes|x_j))
where *x_j* is the *j*th value of the independent variable, *N_j* is the
number of times that *x_j* was presented and *n_j* is the number of
"yes"-responses found with that value *x_j*. *M* is the number of
levels of *x_j*. Note that minimizing *Chi^2* is equivalent to a
maximum likelihood estimation of the parameters *a* and *c* but is more
save to compute than the original likelihood equation. `logpmf` is
able to find minima even if there are many different values of *x_j* and
only few observations per value. This may be useful for adaptive
procedure data where the whole data set me be used to get a final
estimate.

The output of `logpmf` is written to stdout. By default the values of
the point of subjective equality *c*, the just noticable difference
*a*, and the final value of *Chi^2* are given as output.
If the command line option `-g` is given then the estimated
psychometric function is printed out. This means that the output is a
series of lines, each one containing a pair of values for the level of
the independent variable and the response probability. If the option
`-d` is given then the data are echoed to the output.

In some cases the psychometric function as given in Equation
\eqrefLogisticPMF is inappropriate since there is a nonzero guessing
probability involved. The probability of guessing may be given by the
command line option `-a g`, where *g* is the guessing parameter. In
this case the actual psychometric function used is
P(yes|x_j) = g + (1-g) * (1-exp(-(x_j-c)/a))^(-1).
This results in a psychometric function which has *g* as its
left asymptotic value. In case the option `-a` is used it does not
make sense to consider *c = x_0.5* with *P(*yes*|x_0.5) =
0.5* to be the point of subjective equality.
Thus one may use the option `-p e` to define the probability value
*e* which should be considered to be that probability which corresponds
to the point of subjective equality. Its actual definition is
P(yes|x_e) = e,
where by default *e=0.5* is assumed, but may be changed with the option
`-p`. This does not affect the estimation
procedure at all. It only affects the value which is printed as point of
subjective equality to the output. The option `-h` gives a short help
text for `logpmf`.

The minimization algorithm used by `logpmf` is based on the principal
axis algorithm PRAXIS of Brent(1973) as implemented by Gegenfurtner
(1993).

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Author: Hans Irtel

irtel@psychologie.uni-mannheim.de