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Adaptive Methods

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Adaptive Methods

Research Methods

Fall 2010

Tamás Bőhm

- Classical (Fechnerian) methods: stimulus is often far from the thresholdinefficient
- Adaptive methods: accelerated testing
- 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

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

- 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 fore.g. 2AFC

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%)

reversal points

- 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%

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

‘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

- 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 = -0.6

slope = 0.3

D = 2

D = 65

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

- 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

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

- 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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- 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