For data constrained to lie between limits 0 to 1 as in our case or any
forced choice experiment, the data are not Gaussian distributed, and the
basic assumption for the use of ANOVA (Analysis of Variance) is violated. In
this case the logistic regression transformation (Aldrich and Nelson, 1984)
can be used. Here a transformation of the type log(p/(1-p)) is used so that
the variable goes from - to
as p goes from 0 to 1, and one
can then perform an ANOVA on the transformed data.
In the case of our data, a further problem arises in that any time a subject had eight identical responses ( p = 1 or p = 0 ) the transformed variable diverges. This was particularly troublesome for the data in Experiment 3, where the subjects were both young and highly experienced listeners. Some of these subjects had 8 identical responses even in the case of 3 cents difference in the reference and target pitches. We compensated for this by use of the ``uncertain" response (See previous section) in the data, and weighted this as 75 %of a certain response. We also subtracted ``lower" responses from ``higher" responses in order to weight the uncertainties of the lower responses equally. This number was then divided by 16 and the quotient was augmented by .5 so that the result would have limits 0 to 1 as before.
Even so it was necessary to choose a value for log(p/(1-p)) at the limits or to discard data from some of the subjects. This was done by determining the limit of the above function as responses changed from 8 ``maybe higher'' to 7 ``definitely higher'' and 1 ``maybe higher'' with one response changing at a time. See Figure 3. One can then extrapolate the curve graphically to obtain 5 or take the value of the last point plus the difference plus half the second order difference to obtain 4.993. The latter value was used in the statistical calculations.