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Research publications

The following publications describe our fitting and bootstrapping techniques, and report the results of the investigations that we have carried out using the software:

Perception & Psychophysics papers

The versions of our two papers available here are now the same as those that are published in Perception & Psychophysics. A more detailed and up-to-date account of our methods, along with various tests of their accuracy and extensions of some of the sampling issues addressed in paper II, are to be found in Hill's (2001) doctoral thesis.

Paper I
Wichmann, F. A. & Hill, N. J. (2001a): The psychometric function: I. Fitting, sampling and goodness-of-fit. Perception and Psychophysics 63(8), 1293-1313.

[pdf] wichmann_hill_2001a.pdf
283.0 kB
  • Describes a maximum-likelihood method for fitting functions to psychophysical data, constrained using Bayesian priors.
  • Investigates the biases in threshold and slope estimates that may result from stimulus-independent errors ("lapses") by the observer, and shows how to avoid such bias.
  • Demonstrates the use of Monte Carlo techniques for goodness-of-fit assessment, illustrating how asymptotic-theory methods can fail.
  • Recommends several likelihood-based goodness-of-fit tests, using the Monte Carlo method, appropriate for use with psychophysical data.

Paper II
Wichmann, F. A. & Hill, N. J. (2001b): The psychometric function: II. Bootstrap-based confidence intervals and sampling. Perception and Psychophysics 63(8), 1314-1329.

[pdf] wichmann_hill_2001b.pdf
440.2 kB
  • Describes a method for obtaining variability estimates for parameters, thresholds and slopes using bootstrap techniques.
  • Describes a method for measuring the variability estimates' sensitivity to bootstrap error.
  • Investigates the effects of one's sampling scheme (placement of samples on the stimulus axis) on confidence interval width and on sensitivity.
  • Recommmends the use of bias-corrected accelerated (BCa) confidence intervals to allow for possible bias and skewness in simulation estimates.

VSS Poster

Hill, N. J. (2001a): An Investigation of Confidence Interval Methods for the Thresholds and Slopes of Psychometric Functions. Poster presented at the first annual meeting of the Vision Sciences Society, Sarasota, FL, USA.

[pdf] vssposter_hill2001_a0.pdf
192.2 kB

Reports the results of Monte Carlo simulations which compared several different interval methods (probit methods and variations on the bootstrap) under 2AFC conditions with lapse rate included as a nuisance parameter. The results presented in the poster are a small subset of those presented in chapter 3 of the thesis.

Doctoral Thesis

Hill, N.J. (2001b): Testing Hypotheses about Psychometric Functions - an investigation of some confidence interval methods, their validity, and their use in the evaluation of optimal sampling strategies. D. Phil. thesis, University of Oxford, UK.

This study contains the most detailed account of our methods to date. It reports developments of the bootstrap confidence interval methods described in the two P & P papers, along with extensive tests of their accuracy in a number of different situations. It was accepted as a doctoral thesis by the appropriate faculty board in January 2002.

The report is available for download in the following forms:

  Adobe PDF: Plain text:
Abstract only
(1 page / 350 words)
[pdf] hill2001_abstract.pdf
46.2 kB
[txt] hill2001_abstract.txt
2.5 kB
Extended Abstract
only (10 pages)
[pdf] hill2001_longabs.pdf
110.5 kB
Complete document
(284 pages)
[pdf] hill2001.pdf
2.6 MB