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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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Adaptive Methods

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Adaptive methods l.jpg

Adaptive Methods

Research Methods

Fall 2010

Tamás Bőhm


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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


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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


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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


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Notations

Xnstimulus level at trial n

Znresponse at trial n

Zn= 1detected / correct

Zn = 0not 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%

xφthreshold


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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)


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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


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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


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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 fore.g. 2AFC

Staircase method


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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


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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%


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Nonparametric up-down method


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Weighted up-down method

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


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‘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


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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


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Stochastic approximation


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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


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Accelerated stochastic approximation


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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


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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


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Parametric adaptive methods

  • A template for the psychometric function is chosen:

    • Cumulative normal

    • Logistic

    • Weibull

    • Gumbel


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Parametric adaptive methods

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

    • Threshold

    • Slope


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Fitting the psychometric function

  • Linearization (inverse transformation)of data points

    • Inverse cumulative normal (probit)

    • Inverse logistic(logit)


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Fitting the psychometric function

  • Linear regression

  • Transformation of regression line parameters

X-intercept & linear slope

Threshold & logistic slope


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slope = -0.6

slope = 0.3

D = 2

D = 65

Contour integration experiment


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Contour integration experiment

5-day perceptual learning


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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


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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


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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


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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)

---+

--++

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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


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