The psychometric function

Question 1

What is a psychometric function

Answer 1

is a summary of the relation between performance in a classification task (such as the ability to detect or
discriminate between stimuli) and stimulus level/intensity

Question 2

3 steps of modelling data

Answer 2

-Collect psychophysical data
-Estimate model parameters from data
-Model evaluation and criticism (goodness-of-fit, systematic prediction
errors, confidence regions, bias, …)

Question 3

Measures of psychophysical performance

Answer 3

• Response thresholds
• Rate of improvement as stimulus intensity increases

These can be derived from the psychometric function

Question 4

Important!

Answer 4

Psychometric function fitting is much more than fitting a sigmoidal curve through a number of data points

Question 5

The psychometric function as a psychophysical model

Answer 5

• Psychophysical model of the psychological mechanism and stochastic
  process underlying psychophysical data
• Relatively simple, providing a performance description but no explanation of the signal-to-noise ratio of the psychological mechanism
• Of course, comparing performances in different experimental conditions
  can lead to an explanation

Question 6

Formalizing the psychometric function

Answer 6

Suppose we have K signal levels (independent variables) yielding a vector
For each signal level we have K proportions of correct responses

If we have n trials for each signal xi level , the number of correct responses is equal to yi*ni

Question 7

Interpreting: ψ(x | α, ß, γ, λ)= γ+(1 - γ - λ)F(x;α, ß)

Answer 7

F is a sigmoidal function of stimulus intensity ranging from 0 to 1 (e.g.,
cumulative Gaussian, logistic or Weibull) with mean α and ß standard deviation, describing the performance of the underlying psychological mechanism

γ specifies the guess rate

λ specifies the rate of lapsing, i.e. stimulus-independent errors

Question 8

Parameter estimation

Answer 8

For which values of α, β, γ, λ does the psychometric function Ψ provide a good
description of the data?

Question 9

The psychometric function as a stochastic model

Answer 9

Psychophysical responses are the result of a stochastic process, meaning that these responses are to some extent random instead of deterministic

The amount of randomness depends on the amount of stimulus information and amount of trials per block

The psychometric function model contains specific assumptions to capture the stochastic process

Question 10

Assumption

Answer 10

Psychological trials are Bernoulli processes
This means
f(y) provides the probability of obtaining the proportion y correct responses given n trials and a succes probibilty p
But we want likelihood
L(p | y,n)
Likelihood provides the likelihood of the observer having a succes probability given n trials and an observed proportion y of correct responses
in other words a loaded coinflip is assimed to underlie psychophysical responses

Question 11

how to choose the values of α, ß, γ, λ

Answer 11

for wich the overall likelihood is maximal = maximum-likelihood estiomation

Question 12

Goodness-of- fit assessment and model checking

Answer 12

Likelihood maximization yields the best-fitting parameter combination for the model under consideration

Question 13

how can the model fail?

Answer 13

The failure of goodness of fit may result from failure of one or more of the assumptions of one’s model:
- Inappropriate functional form
- Assumption that observer responses are binomial may be false
-The observer’s psychometric function may be nonstationary
Results in overdispersion or extra-binomial variation:
Data points are (significantly) further away from the fitted curve than expected

Question 14

Assessing over dispersion: Deviance

Answer 14

D= 2log(L(θmax;y)/L(θest;y))

Question 15

Monte Carlo generation of the deviance distribution

Answer 15

What distribution of deviance values would we expect, for a given n, if the model is true ?

1. Obtain best fitting psychometric function
2. generate a large number of new datasets using function assume binomal var
3. Calcualte deviance for each simulated dataset
4. compare the empirical deviance to the distrubution of simulated deviance

Question 16

Assessing systematic prediction errors

Answer 16

Correlation between deviance residuals and model prediction reveals
systematic deviations

Monte-Carlo simulation indicates that such a high correlation is unlikely
to occur when the model is true (but, beware for the opposite!)

Question 17

Obtaining confidence regions

Answer 17

The same Monte-Carlo methods can be used to obtain confidence intervals on the estimated parameters = parametric bootstrap

1. Obtain best fitting psychometric function
2. Generate a large number of new datasets using psi
3. for each data set refit psi

Question 18

ψ versus explanatory models

Answer 18

• 2-AFC method of constant stimuli allows us to model the stochastic process underlying psychophysical responses quantitatively
• The statistical techniques described in this presentation can be easily used for more complex explanatory psychophysical model
• We simply use these models instead of ψ to specify the success probability p, assuming the same stochastic process

Question 19

2-AFC versus method of adjustment

Answer 19

Each adjustment (e.g., left or right) can be seen as one two-alternative forced-choice response based on a sample from a Bernoulli process

Amount of adjustments is typically uncontrolled and samples are not independent. Therefore, the binomial distribution cannot be assumed and you lose considerable statistical power

Question 20

Method of constant stimuli versus adaptive

procedures

Answer 20

Similar objections can be made against adaptive procedures such as
staircases
Response on any given trial depends per definition on the response on the
previous trial
Sequence of Bernoulli random variables is again not i.i.d. distributed, the
binomial distribution cannot be assumed