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# Adaptive Methods - PowerPoint PPT Presentation

Adaptive Methods. Research Methods Fall 2010 Tamás Bőhm. Adaptive methods. Classical (Fechnerian) methods: stimulus is often far from the threshold inefficient A daptive methods: accelerated testing Modifications of the method of constant stimuli and method of limits. Adaptive methods.

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Presentation Transcript

Research Methods

Fall 2010

Tamás Bőhm

• Classical (Fechnerian) methods: stimulus is often far from the thresholdinefficient

• Modifications of the method of constant stimuli and method of limits

• Classical methods: stimulus values to be presented are fixed before the experiment

• Adaptive methods: stimulus values to be presented depend critically on preceding responses

• 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

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

• 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)

• 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

• 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

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

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

• 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%

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

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

• 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

• 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

• 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

• 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

• A template for the psychometric function is chosen:

• Cumulative normal

• Logistic

• Weibull

• Gumbel

• Only the parameters of the template need to be measured:

• Threshold

• Slope

• Linearization (inverse transformation)of data points

• Inverse cumulative normal (probit)

• Inverse logistic(logit)

• Linear regression

• Transformation of regression line parameters

X-intercept & linear slope

Threshold & logistic slope

slope = 0.3

D = 2

D = 65

Contour integration experiment

5-day perceptual learning

• 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

• 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

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

Maximum likelihood estimation experimenter

• 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)

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Bayes’ estimation experimenter

• 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