Adaptive Methods

1. Adaptive Methods Research Methods Fall 2010 Tamás Bőhm

2. Adaptive methods • Classical (Fechnerian) methods: stimulus is often far from the thresholdinefficient • Adaptive methods: accelerated testing • Modifications of the method of constant stimuli and method of limits

3. Adaptive methods • Classical methods: stimulus values to be presented are fixed before the experiment • Adaptive methods: stimulus values to be presented depend critically on preceding responses

4. Adaptive methods • Constituents • Stepping rule: which stimulus level to use next? • Stopping criterion: when to finish the session? • What is the final threshold estimate? • Performance • Bias: systematic error • Precision: related to random error • Efficiency: # of trials needed for a specific precision; measured by the sweat factor

5. Notations Xn stimulus level at trial n Zn response at trial n Zn= 1 detected / correct Zn = 0 not detected / incorrect φtarget probability absolute threshold: φ = 50% difference threshold: φ = 75% 2AFC: φ = 50% + 50% / 2 = 75% 4AFC: φ = 25% + 75% / 2 = 62.5% xφ threshold

6. Adaptive methods • Classical methods: stimulus values to be presented are fixed before the experiment • Adaptive methods: stimulus values to be presented depend critically on preceding responses Xn+1 = f(φ, n, Zn, Xn, Zn-1, Xn-1,…, Z1, X1)

7. Adaptive methods • Nonparametric methods: • No assumptions about the shape of the psychometric function • Can measure threshold only • Parametric methods: • General form of the psychometric function is known, only its parameters (threshold and slope) need to be measured • If slope is also known: measure only threshold

8. Nonparametric adaptive methods • Staircase method (aka. truncated method of limits, simple up-down) • Transformed up-down method • Nonparametric up-down method • Weighted up-down method • Modified binary search • Stochastic approximation • Accelerated stochastic approximation • PEST and More Virulent PEST

9. Stepping rule:Xn+1 = Xn - δ(2Zn - 1) fixed step size δ if response changes:direction of steps is reversed Stopping criterion:after a predetermined number of reversals Threshold estimate: average of reversal points(mid-run estimate) Converges to φ = 50% cannot be used for e.g. 2AFC Staircase method

10. Improvement of the simple up-down (staircase) method Xn+1 depends on 2 or more preceding responses E.g.1-up/2-down or 2-step rule: Increase stimulus level after each incorrect response Decrease only after 2 correct responses φ = 70.7% Threshold:mid-run estimate 8 rules for 8 different φ values(15.9%, 29.3%, 50%, 70.7%, 79.4%, 84.1%) Transformed up-down method reversal points

11. Nonparametric up-down method • Stepping rule: Xn+1= Xn- δ(2ZnSφ - 1) • Sφ: random number p(Sφ=1) = 1 / 2φ p(Sφ=0) = 1 – (1 / 2φ) • After a correct answer: stimulus decreased with p = 1 / 2φ stimulus increased with p = 1 - (1 / 2φ) • After an incorrect answer: stimulus increased • Can converge to any φ≥ 50%

12. Nonparametric up-down method

13. Weighted up-down method • Different step sizes for upward (δup) and downward steps (δdown)

14. ‘Divide and conquer’ Stimulus interval containing the threshold is halved in every step(one endpoint is replaced by the midpoint) Stopping criterion: a lower limit on the step size Threshold estimate:last tested level Heuristic, no theoreticalfoundation Modified binary search Figure from Sedgewick & Wayne

15. Stochastic approximation • A theoretically sound variant of the modified binary search • Stepping rule: • c: initial step size • Stimulus value increases for correct responses,decreases for incorrect ones • If φ = 50%: upward and downward steps are equal; otherwise asymmetric • Step size (both upward and downward) decreases from trial to trial • Can converge to any φ

16. Stochastic approximation

17. Accelerated stochastic approximation • Stepping rule: • First 2 trials: stochastic approximation • n > 2:step size is changed only when response changes (mreversals: number of reversals) • Otherwise the same as stochastic approximation • Less trials than stochastic approximation

18. Accelerated stochastic approximation

19. Parameter Estimation by Sequential Testing (PEST) • Sequential testing: • Run multiple trials at the same stimulus level x • If x is near the threshold, the expected number of correct responses mc after nx presentations will be around φnx the stimulus level is changed if mcis not in φnx ± w • w: deviation limit; w=1 for φ=75% • If the stimulus level needs to be changed:step size determined by a set of heuristic rules • Variants: MOUSE, RAT, More Virulent PEST

20. Adaptive methods • Nonparametric methods: • No assumptions about the shape of the psychometric function • Can measure threshold only • Parametric methods: • General form of the psychometric function is known, only its parameters (threshold and slope) need to be measured • If slope is also known: measure only threshold

21. Parametric adaptive methods • A template for the psychometric function is chosen: • Cumulative normal • Logistic • Weibull • Gumbel

22. Parametric adaptive methods • Only the parameters of the template need to be measured: • Threshold • Slope

23. Fitting the psychometric function • Linearization (inverse transformation)of data points • Inverse cumulative normal (probit) • Inverse logistic(logit)

24. Fitting the psychometric function • Linear regression • Transformation of regression line parameters X-intercept & linear slope Threshold & logistic slope

25. slope = -0.6 slope = 0.3 D = 2 D = 65 Contour integration experiment

26. Contour integration experiment 5-day perceptual learning

27. Adaptive probit estimation • Short blocks of method of constant stimuli • Between blocks: threshold and slope is estimated (psychometric function is fitted to the data) and stimulus levels adjusted accordingly • Assumes a cumulative normal function probit analysis • Stopping criterion: after a fixed number of blocks • Final estimate of threshold and slope: re-analysis of all the responses

28. Adaptive probit estimation • Start with an educated guess of the threshold and slope • In each block: 4 stimulus levels presented 10 times each • After each block: threshold ( ) and slope ( ) is estimatedby probit analysis of the responses in block • Stimulus levels for the next block are adjusted accordingly • Estimated threshold and slopeapplied only through correctionfactors  inertia

29. Function shape (form & slope) is predetermined by the experimenter Only the position along the x-axis (threshold) needs to be measured Iteratively estimating the threshold and adapting the stimulus levels Two ways to estimate the threshold: Maximum likelihood (ML) Bayes’ estimation QUEST, BEST PEST, ML-TEST, Quadrature Method, IDEAL, YAAP, ZEST Measuring the threshold only

30. Maximum likelihood estimation • Construct the psychometric function with each possible threshold value • Calculate the probability of the responses with each threshold value (likelihood) • Choose the threshold value for which the likelihood is maximal (i.e. the psychometric function that is the most likely to produce such responses) - - - + - - + + - + + +

31. Bayes’ estimation • Prior information is also used • Distribution of the threshold in the population(e.g. from a survey of the literature) • The experimenter’s beliefs about the threshold values of the psychometric functions at the tested stimulus levels a priori distribution of the threshold