[Martin Taylor 2009.23.10.21]
[From Bill Powers (2009.02.23.0010 MST)]
Martin Taylor 2009.02.21.13.49 –
I 'd like to know a little more about the timing plot. d^2 runs from 0
to 8, so d runs from 0 to about 2.8. That doesn’t seem to be related
directly to “probability of a correct press.” Could you translate the
numbers on the y axis into fraction of incorrect presses? That is form
of the raw data, isn’t it? From that I can use my Chemical Rubber table
to estimate the magnitude of the hypothetical perceptual signal as a
number of standard deviations above the noise for each point on the
plot, assuming Gaussian noise distribution.
d’ is just the number of standard deviations, so no need to look up
tables. It’s not signal power compared to noise power, but total signal
energy compared to noise power per unit bandwidth (at least in
acoustics, where those quantities are measurable or computable). Maybe
one could use the same concepts when considering the signals in the
neural pathways, but I think it’s not a real issue. If you think only
of the magnitude of the hypothetical perceptual signal when there is a
real signal as compared to its magnitude when there is no real signal,
that’s close to what signal detection psychophysicists consider.
I think you would like the logic of how to get between this and the
probability of a correct response in a forced-choice experiment, and
it’s pretty simple, so here it is in excruciating detail 
.
We don’t have to assume Gaussian noise, since the real conditions are
not so stringent, but Gaussian noise works, so let’s assume it. First
let’s consider the case of detecting whether a signal exists or not
(whether a light turned on, for example). The hypotheses predict a
Gaussian distribution of likelihoods (that a signal was present) for
the “signal here” case, and another with the same standard deviation
for the “signal not here” case. We usually call these “signal plus
noise” (S+N) and “noise” (N) likelihood distributions. Any observation
of data can be placed somewhere along the likelihood axis.
I should emphasize that I’m not saying people actually perceive or
compute likelihoods. A mathematically ideal observer would, but that
says nothing about people. In practice, what we are talking about is
modelling – people’s responses do match what a model of this kind
does, but with a sensitivity (d’) less than a mathematically ideal
observer would achieve. The same argument can be made for any way of
creating the perceptual magnitude, provided that it is continuous (at
least on the scale of the observations) and subject to noise
fluctuations. I use likelihoods because that is what the mathematically
ideal observer would use as a perceptual signal.
Looking at the diagram (forgive my freehand Gaussians), an observation
that gave rise to a likelihood as marked would be extremely improbable
if there had been no signal, but quite probable if there had been a
signal. The ratio of likelihoods for these two possibilities gives the
posterior probability of the signal having been present. If that is
high enough, the observer will report that a signal was presented. d’
is the separation of the means of the two distributions, measured in
standard deviations.
The foregoing is just for a single presentation, in which either there
was or there was not a signal. In the Schouten experiment, we are
talking about a “forced-choice” situation. Rather than the question
being whether a signal exists or not, the question is in which of
exactly one of two possible slots there is a signal, knowing that there
is a signal in one of them (in psychoacoustics, the slots are usually
one of two time intervals, but in the Schouten experiment it’s whether
the light is to the left or
right). The diagram above still applies in each of the two slots, but
now the choice is in which of the two intervals the likelihood was
higher (that a signal was presented). This question leads to a new
Gaussian distribution, the distribution of (SN-N). That’s a
distribution with twice the variance, or sqrt(2) times the SD of the
originals. Its mean is d’, and the probability of a correct response is
the probability that SN-N > 0.
So to get the probability of a correct response, you just see how much
of a normal distribution that has SD sqrt(2) and mean d’ lies above
zero. Conversely, to get from forced-choice probability of a correct
response to d’, find the location of the mean of a normal distribution
of SD sqrt(2) that has that much of it above zero. Examples, from the
Wikipedia page on Gaussian distribution, at d’^2 = 2, p correct = 0.84,
and at d’^2 = 8, p correct = 0.978 (approximately).
I hope this is more useful than confusing. And I hope my calculations
don’t include some stupid error like inverting a fraction!
Martin
.
