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## Über dieses Buch

Many of the commonly used methods for modeling and fitting psychophysical data are special cases of statistical procedures of great power and generality, notably the Generalized Linear Model (GLM). This book illustrates how to fit data from a variety of psychophysical paradigms using modern statistical methods and the statistical language R. The paradigms include signal detection theory, psychometric function fitting, classification images and more. In two chapters, recently developed methods for scaling appearance, maximum likelihood difference scaling and maximum likelihood conjoint measurement are examined. The authors also consider the application of mixed-effects models to psychophysical data.

R is an open-source programming language that is widely used by statisticians and is seeing enormous growth in its application to data in all fields. It is interactive, containing many powerful facilities for optimization, model evaluation, model selection, and graphical display of data. The reader who fits data in R can readily make use of these methods. The researcher who uses R to fit and model his data has access to most recently developed statistical methods.

This book does not assume that the reader is familiar with R, and a little experience with any programming language is all that is needed to appreciate this book. There are large numbers of examples of R in the text and the source code for all examples is available in an R package MPDiR available through R.
Kenneth Knoblauch is a researcher in the Department of Integrative Neurosciences in Inserm Unit 846, The Stem Cell and Brain Research Institute and associated with the University Claude Bernard, Lyon 1, in France.

Laurence T. Maloney is Professor of Psychology and Neural Science at New York University. His research focusses on applications of mathematical models to perception, motor control and decision making.

## Inhaltsverzeichnis

### Chapter 1. A First Tour Through R by Example

In this chapter we illustrate how Rcan be used to explore and analyze psychophysical data. We examine a data set from the classic article by Hecht et al. [79] in order to introduce basic data structures and functions that permit us to examine and model data in R.
Kenneth Knoblauch, Laurence T. Maloney

### Chapter 2. Modeling in R

One of the many strengths of R is in the diversity and convenience of its modeling functions. In this chapter, we describe several standard statistical models and show how to fit them to data using R.
Kenneth Knoblauch, Laurence T. Maloney

### Chapter 3. Signal Detection Theory

Signal detection theory (SDT), developed in the 1950s, is a framework of statistical methods used to model how observers classify sensory events [72, 169]. In this chapter we describe commonly used signal detection models and methods for fitting them. A recurring theme in this book is the use of the generalized linear model (GLM) to fit psychophysical data, and here we describe the use of the GLM to fit signal detection data.
Kenneth Knoblauch, Laurence T. Maloney

### Chapter 4. The Psychometric Function: Introduction

The psychometric function is a summary of the relation between performance in a classification task (such as the ability to detect or discriminate between stimuli) and stimulus level [59, 176]. Stimulus level is typically a measure of magnitude of a physical stimulus along a single physical dimension such as size, distance, light or sound intensity, concentration, or frequency. We will use the terms “stimulus level and “stimulus intensity interchangeably. We gave an example of a psychometric function in Chaps. 1 and Chaps. 2, we discussed the close relationship between the psychometric function and the generalized linear model. While there is no need to use the GLM in fitting psychometric functions we will see that doing so makes it very convenient to apply advanced statistical methods to psychophysical data.
Kenneth Knoblauch, Laurence T. Maloney

### Chapter 5. The Psychometric Function: Continuation

In the previous chapter we showed how to use direct optimization methods and the generalized linear model (GLM) to fit psychometric functions to Yes–No data. In an extended example using GLM, we illustrated the procedure for selecting a model to fit multiple psychometric functions, across a series of experimental conditions and how to evaluate the goodness of fit of the model. In this chapter, we continue the exploration of fitting psychometric functions and demonstrate methods required when the observer’s task is to select one among many alternatives, a type of experiment referred to as m-alternative forced choice (mAFC). In addition, we illustrate methods for assigning standard errors and confidence limits to estimated parameters. Finally, we end with a short discussion of non- (or semi-)parametric methods for fitting psychometric functions.
Kenneth Knoblauch, Laurence T. Maloney

### Chapter 6. Classification Images

In 1996, Ahumada [2] reported the following experiment. On each trial, the observer was presented with two small horizontal bars separated by a small, lateral gap (of 1 pixel width, corresponding to 1.26 min on his display) on a background. The bars could be either aligned or offset vertically (1 pixel) as shown in the top two images of Fig. 6.1, respectively. The observer’s task on each trial was to judge whether an offset was present or not. This is a localization judgment measuring what is called Vernier acuity. Under appropriate conditions, humans are remarkably good at this, detecting offsets that can be an order of magnitude or more finer than the minimum separation that they can resolve between two adjacent bars, hence the term hyperacuity [186]. One innovation for this type of experiment in Ahumada’s design was that stimuli were presented in “noise,” i.e., the luminance of each pixel in the image (128 ×128 pixels) was increased or decreased by a random amount, as illustrated in the bottom two images of Fig. 6.1.
Kenneth Knoblauch, Laurence T. Maloney

### Chapter 7. Maximum Likelihood Difference Scaling

In previous chapters we focused on psychophysical methods such as discrimination and models such as signal detection theory which allow us to measure the ability to discriminate stimuli that are not very different. In this chapter and the following (Chap. 8), we focus on assessing perceptual differences that are readily discriminable, well above the “threshold.” The methods we present are, historically, part of the scaling literature. This literature has been closely connected to psychophysics since its inception [59, 165] and the interested reader can read about scaling and its controversies in any of the several excellent references (e.g., [68, 91, 115]).
Kenneth Knoblauch, Laurence T. Maloney

### Chapter 8. Maximum Likelihood Conjoint Measurement

Conjoint measurement allows the experimenter to estimate psychophysical scales that capture how two or more physical dimensions contribute to a perceptual judgment. It is most easily explained by an example. Figure 8.1 shows a range of stimuli taken from a study by Ho et al. [80]. Each stimulus was the image of an irregular surface (the original stimuli were rendered for binocular viewing at a higher resolution than those shown in the figure). The experimenters were interested in how the perceived roughness (which they refer to as “bumpiness) and glossiness of a surface were affected by variations in two physical parameters that plausibly affect perceived roughness and glossiness. They reported two experiments that differed only in the judgments observers made. In the first, observers judged perceived roughness, in the second, perceived glossiness.
Kenneth Knoblauch, Laurence T. Maloney

### Chapter 9. Mixed-Effects Models

With one exception the models that we have treated before this chapter contain a single source of variability. In the linear models with Gaussian error, the variance of the population was estimated from the residuals. In the GLM models used to describe the judgments of observers, the variability of observer’s choices is given by that of the binomial distribution and depends on the estimated probability of a particular choice, $$\hat{p}$$, and the number of trials, n.
Kenneth Knoblauch, Laurence T. Maloney

### Backmatter

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