[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