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Model-free estimation of a psychometric function

Example 7. Auditory detection of a gap in noise

Baker, R. J., Jayewardene, D., Sayle, C., & Saeed, S. “Failure to find asymmetry in auditory gap detection”, Laterality: Asymmetries of Body, Brain and Cognition, 13, 1-21, 2008.

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A 300-ms noise burst containing a gap of 2–8 ms duration or no gap was presented to one ear of a subject. The symbols in the figure below show the proportion of responses "gap" as a function of gap duration. There were 12 trials with each gap duration and 84 trials with no gap.

Parametric and local linear fitting

Three different parametric models and the local linear fitting are used and fits are plotted against the measured psychometric data. Three different parametric models are fitted to these data: Gaussian (probit), Weibull, and reverse Weibull. Local linear fitting is also performed with the bandwidth bwd chosen by the minimising cross-validated deviance.

Load the data and plot the measured psychometric data (black dots):

data("07_Baker")
x = example07$x
r = example07$r
m = example07$m
plot( x, r / m, xlim = c( 0.16, 7.83 ), ylim = c( -0.01, 1.01 ), type = "p", pch="*" ) # Limits set to match the MatLab ones

1. For the Gaussian cumulative distribution function (black curve):

val <- binomfit_lims( r, m, x, link = "probit" )
numxfit <- 199; # Number of new points to be generated minus 1
xfit <- (max(x)-min(x)) * (0:numxfit) / numxfit + min(x)
# Plot the fitted curve
pfit<-predict( val$fit, data.frame( x = xfit ), type = "response" )
lines(xfit, pfit )

2. For the Weibull function (red curve):

val <- binom_weib( r, m, x )
# Plot the fitted curve
pfit<-predict( val$fit, data.frame( x = xfit ), type = "response" )
lines(xfit, pfit, col = "red" )

3. For the reverse Weibull function (green curve):

val <- binom_revweib( r, m, x )
# Plot the fitted curve
pfit<-predict( val$fit, data.frame( x = xfit ), type = "response" )
lines(xfit, pfit, col = "green" )

4. For the local linear fit (blue curve):

bwd_min <- min( diff( x ) )
bwd_max <- max( x ) - min( x )
bwd <- bandwidth_cross_validation( r, m, x, c( bwd_min, bwd_max ) )
# Plot the fitted curve
bwd <- bwd$deviance # choose the estimate based on cross-validated deviance
pfit <- locglmfit( xfit, r, m, x, bwd )$pfit
lines(xfit, pfit, col = "blue" )
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