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A Primer of Signal Detection Theory A PRIMER OF SIGNAL DETECTION THEORY D. McNicol Lecturer in Applied Psychology, University of New South Wales LONDON. GEORGE ALLEN & UNWIN L.TD SYDNEY. AUSTRALASIAN PUBLISHING COMPANY First published in 1972 Preface This book is copyright under the Berne Convention. All rights are reserved. Apart from any fair dealing for the purposes of private study, research, criticism or review, as permitted under the Copyright Act, 1956, no part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, electrical, chemical, mechanical, optical, photocopying, There is hardly a field in psychology in which the effects of signal recording or otherwise, without the prior permission of the copyright owner. Enquiries should be addressed to the detection theory have not been felt. The authoritative work on the publishers. subject, Green's & Swets' Signal Detection Theory and Psycho­ © George Allen & Unwin Ltd 1972 physics (New York: Wiley) appearedjn 1966, and is having a profound influence on method and theory in psychology. All this British ISBN 0 04 152006 8 cased makes things exciting but rather difficult for undergraduate students o 04 152007 6 paper and their teachers, because a complete course in psychology now requires an understanding ofthe concepts ofsignal detection theory, Australasian Publishing Company, Sydney SBN 900882 34 4 and many undergraduates have done no mathematics at university ' level. Their total mathematical skills consist of dim recollections of secondary school algebra coupled with an introductory course in statistics taken in conjunction with their studies in psychology. This book is intended to present the methods of signal detection theory to a person with such a mathematical background. It assumes a know­ ledge only ofelementary algebra and elementary statistics. Symbols and terminology are kept as close as possible to those of Green & Swets (1966) so that the eventual and hoped for transfer to a more advanced text will be accomplished as easily as possible. The book is best considered as being divided into two main sections, the first comprising Chapters 1 to 5, and the second, Chapters 6 to 8. The first section introduces the basic ideas of detection theory, and its fundamental measures. The aim is to enable the reader to be able to understand and compute these measures. The section ends with a detailed working through of a typical experiment and a discussion of some of the problems which can arise for the potential user of detection theory. The second section considers three more advanced topics. The first ofthese, which is treated thoroughly elsewhere in the literature, is threshold theory. However, because this contender against signal Prin ted in Great Britain detection theory has been so ubiquitous in the literature of experi­ in 10 on 12pt 'Monophoto' Times Mathematics Series 569 mental psychology, and so powerful in its influence both in the by Page Bros (Norwich) Ltd., Norwich PREFACE construction oftheories and the design ofexperiments, it is discussed Contents again. The second topic concerns the extension of detection theory, which customarily requires experiments involving recognition tests, to experiments using more open-ended procedures, such as recall; and the third topic is an examination of Thurstonian scaling procedures which extend signal detection theory in a number of useful ways. An author needs the assistance of many people to produce his Preface page book, and I have been no exception. I am particularly beholden to WHAT ARE STATISTICAL DECISIONS? 1 David Ingleby, who, when he was working at the Medical Research An example 1 Council Applied Psychology Unit, Cambridge, gave me much useful Some definitions 3 advice, and who was subsequently most generous in allowing me to read a number of his reports. The reader will notice frequent Decision rules and the criterion 6 reference to his unpublished Ph.D. thesis from which I gained Signal detection theory and psychology 10 considerable help when writing Chapters 7 and 8 ofthis book. Many 2 NON-PARAMETRIC MEASURES OF SENSITIVITY 18 of my colleagues at Adelaide have helped me too, and I am grateful The yes-no task 18 to Ted Nettelbeck, Ron Penny and Maxine Shephard, who read and The rating scale task 25 commented on drafts of the manuscript, to Su Williams and Bob Area estimation with only a single pair of hit and false Willson, who assisted with computer programming, and to my alarm rates 31 Head of Department, Professor A. T. Welford for his encourage­ The forced-choice task 40 ~ ment. I am equally indebted to those responsible for the production An overall view of non-parametric sensitivity measures 45 of the final manuscript which was organised by Margaret Blaber ably assisted by Judy Hallett. My thanks also to Sue Thom who 3 GAUSSIAN DISTRIBUTIONS OF SIGNAL AND prepared the diagrams, and to my wife Kathie, who did the proof NOISE WITH EQUAL VARIANCES 50 reading. The ROC curve for the yes-no task 50 The impetus for this work came from a project on the applications Double-probability scales 53 of signal detection theory to the processing of verbal information, The formula for d' 57 supported by Grant No A67/16714 from the Australian Research The criterion 58 Grants Committee. I am also grateful to St John's College, Camb­ Forced-choice tasks 64 bridge, for making it possible to return to England during 1969 to 4 GAUSSIAN DISTRIBUTIONS OF SIGNAL AND work on the book, and to Adelaide University, which allowed me to NOISE WITH UNEQUAL VARIANCES 79 take up the St John's offer. ROC curves for unequal variance cases 80 A final word of thanks is due to some people who know more Sensitivity measures in the unequal variance case 86 about the development of this book than anyone else. These are Measuring the signal distribution variance 91 the Psychology III students at Adelaide University who have served as a tolerant but critical proving ground for the material which The measurement of response bias 92 follows. 5 CONDUCTING A RATING SCALE EXPERIMENT 99 Adelaide University D. MCNICOL Experimental design 100 September 1970 CONTENTS Analysis of data 105 Chapter 1 Measures of sensitivity 113 Measures of bias 119 6 CHOICE THEORY APPROXIMA TIONS TO WHAT ARE STATISTICAL SIGNAL DETECTION THEOR Y 131 DECISIONS? The logistic distribution 134 Determining detection measures from logistic distributions 136 The matrix of relative response strengths 139 Open-ended tasks 141 A summary of the procedure for an open-ended task 147 AN EXAMPLE 7 THRESHOLD THEORY 157 Often we must make decisions on the basis of evidence which is Classical psychophysics 157 less than perfect. F or instance, a group of people has heights ranging High threshold theory and the yes-no task 162 from 5 ft 3 in. to 5 ft 9 in. These heights are measured with the group Forced-choice tasks 172 members standing in bare feet. When each person wears shoes his height is increased by 1 inch, so that the range of heights for the Other evidence favouring detection theory 180 group becomes 5 ft 4in. to 5 ft 10 in. The distributions of heights for 8 THE LAWS OF CATEGORICAL AND members of the group with shoes on and with shoes off are illustrated COMPARATIVE JUDGEMENT 185 in the histograms of Figure 1.1. Antecedents of signal detection theory 185 Solid line: Distribution 5 -shoes on Category rating tasks 186 Dotted line: Distribution n-shoes off Forced-choice tasks and the Law of Comparative 4 - - --I i6- r Judgement 206 1 I I I I I Bibliography 215 I 1 3 ~ r---- - -- - ~ f6 c I I ~ I 1 o::::J I o I Appendix 1 Answers to problems 219 o I I ~ 2 --- ~ - -- -- Appendix 2 Logarithms 223 o i6 r - ~ I I :.::: Appendix 3 Integration of the expression for the logistic curve 225 z I I o I .0 I Appendix 4 Computer programmes for signal detection analysis 226 o I I - - - et J.. r - - - - - - Appendix 5 Tables 229 II 16 I ~ I I I I Index 239 I I I I I I I I o 63 64 65 66 67 68 69 70 x = Height in inches FIGURE 1.1 1 A PRIMER OF SIGNAL DETECTION THEORY WHA TARE ST A TISTICAL DECISIONS? You can see that the two histograms are identical, with the excep­ of the group was obtained with shoes on. This is done in Table 1.1. tion that s, the 'Shoes on' histogram, is 1 in. further up the X-axis The probabilities in columns 2 and 3 have been obtained from Figure than n, the 'Shoes off histogram. 1.1. Given these two distributions you are told that a particular F or the sake of brevity we will refer to the two states of affairs person is 5 ft 7 in. tall and from this evidence you must deduce 'Shoes on' and 'Shoes off as states sand n respectively. whether the measurement was taken with shoes on or with shoes off. It can be seen that the odds favouring hypothesis s are calculated A look at these histograms in Figure 1.1 shows that you will not be in the following way: able to make a decision which is certain to be correct. The histograms For a particular height, which we will call x, we take the proba­ reveal that 3/16ths of the group is 5 ft 7 in. tall with shoes off and bility that it will occur with shoes on and divide it by the probability that 4/16ths of the group is 5 ft 7 in. tall with shoes on. The best bet that it will occur with shoes off.
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