Adaptive Methods

Research Methods

Fall 2010

Tamás Bőhm

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

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

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

Notations

Xn stimulus level at trial n

Zn response at trial nZn = 1 detected / correct

Zn = 0 not detected / incorrect

φ target probabilityabsolute threshold: φ = 50%

difference threshold: φ = 75%2AFC: φ = 50% + 50% / 2 = 75%4AFC: φ = 25% + 75% / 2 = 62.5%

xφ threshold

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)

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

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

Staircase method

• 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

Transformed up-down method

• 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%)

5 10 15 200

10

20

30

40

50

trial

stim

ulu

s le

vel

reversal points

Nonparametric up-down method

• Stepping rule: Xn+1 = Xn - δ(2ZnSφ - 1)

– Sφ: random numberp(Sφ=1) = 1 / 2φp(Sφ=0) = 1 – (1 / 2φ)

– After a correct answer:stimulus decreased with p = 1 / 2φstimulus increased with p = 1 - (1 / 2φ)

– After an incorrect answer: stimulus increased

• Can converge to any φ ≥ 50%

Nonparametric up-down method

5 10 15 200

10

20

30

40

50

trial

stim

ulu

s le

vel

Weighted up-down method

• Different step sizes for upward (δup) and downward steps (δdown)

1

downup

Modified binary search

• ‘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

Figure from Sedgewick & Wayne

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

)(1 nnn Zn

cXX

Stochastic approximation

5 10 15 200

10

20

30

40

50

trial

stim

ulu

s le

vel

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

)(21

nreversals

nn Zm

cXX

Accelerated stochastic approximation

5 10 15 200

10

20

30

40

50

trial

stim

ulu

s le

vel

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 mc is 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

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

Parametric adaptive methods

• A template for the psychometric function is chosen:– Cumulative normal– Logistic– Weibull– Gumbel

x

xp

exp1

1)(

Parametric adaptive methods

• Only the parameters of the template need to be measured:– Threshold– Slope

x

xp

exp1

1)(

Fitting the psychometric function

1. Linearization (inverse transformation)of data points

• Inverse cumulative normal (probit)• Inverse logistic (logit) p

pp

1log)logit(

2 4 6 80

0.2

0.4

0.6

0.8

1

2 4 6 8

-8

-6

-4

-2

0

2

4

Fitting the psychometric function

2. Linear regression

3. Transformation of regression line parameters

0 2 4 6 8 10

-8

-6

-4

-2

0

2

4

6

2 4 6 80

0.2

0.4

0.6

0.8

1

X-intercept & linear slope Threshold & logistic slope

Contour integration experiment

0 7.5 11.5 15.5 19.5 23.5

0

0.25

0.5

0.75

1

orientation jitter ( )

ratio

of

corr

ect

resp

onse

s

3=15.14

D = 2

slope = -0.6

0 7.5 11.5 15.5 19.5 23.5

0

0.25

0.5

0.75

1

orientation jitter ( )

ratio

of

corr

ect

resp

onse

s

3=20.29

D = 65

slope = 0.3

Contour integration experiment

0 7.5 11.5 15.5 19.5 23.5

0

0.25

0.5

0.75

1

orientation jitter ( )

ratio

of

corr

ect

resp

onse

s

1=17.43

2=18.65

3=19.37

4=19.44

5=20.23

0 7.5 11.5 15.5 19.5 23.5

0

0.25

0.5

0.75

1

orientation jitter ( )

ratio

of

corr

ect

resp

onse

s

1=13.84

2=14.84

3=15.14

4=15.44

5=14.43

5-day perceptual learning

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

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 estimated by probit analysis of the responses in block

• Stimulus levels for the next block are adjusted accordingly– Estimated threshold and slope

applied only through correctionfactors inertia

rrrrrrrr

cc ,3

,3

,

)ˆ(1 rrr corr

)ˆ(1 rrr corr

r̂ r̂/1

rr/1

Measuring the threshold only

• 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

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)

)|,()( thresholdZXpthresholdL

- - - +

- - + +

- + + +

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

)()|,()( thresholdpthresholdZXpthresholdL

a priori distribution of the threshold

values of the psychometric

functions at the tested stimulus

levels