Donald Bamber

The area above the ordinal dominance graph and the area below the receiver operating characteristic graph

Abstract Receiver operating characteristic graphs are shown to be a variant form of ordinal dominance graphs. The area above the latter graph and the area below the former graph are useful measures of both the size or importance of a difference between two populations and/or the accuracy of discrimination performance. The usual estimator for this area is closely related to the Mann-Whitney U statistic. Statistical literature on this area estimator is reviewed. For large sample sizes, the area estimator is approximately normally distributed. Formulas for the variance and the maximum variance of the area estimator are given. Several different methods of constructing confidence intervals for the area measure are presented and the strengths and weaknesses of each of these methods are discussed. Finally, the Appendix presents the derivation of a new mathematical result, the maximum variance of the area estimator over convex ordinal dominance graphs.

State-trace analysis: A method of testing simple theories of causation☆

Abstract An independent variable measures some aspect of a treatment applied to a person; a dependent variable measures some aspect of the treatments effect upon the person. Two dependent variables will often covary with each other because they are affected by a common independent variable. A state trace is a graph which plots the value of one dependent variable as a function of another. (Thus, a state trace is a generalization of the yes-no receiver-operating-characteristic curve.) By appropriately analyzing state traces, it is possible to test theories concerning (a) the mechanisms by which independent variables affect dependent variables and (b) how these mechanisms differ from one population of persons to another. As an illustration, a study of long-term memory in normal persons, schizophrenics, and persons with organic brain syndrome is presented. The effects of number of word presentations and length of retention interval upon probability of word recall and recognition were investigated. The results were analyzed by state-trace analysis.

How many parameters can a model have and still be testable

Abstract A standard rule of thumb states that a model has too many parameters to be testable if and only if it has at least as many parameters as empirically observable quantities. We argue that when one asks whether a model has too many parameters to be testable, one implicitly refers to a particular type of testability, which we call quantitative testability . A model is defined to be quantitatively testable if the models predictions have zero probability of being correct by chance. Next, we propose a new rule of thumb, based on the rank of the Jacobian matrix of a model (i.e., the matrix of partial derivatives of the function that maps the models parameter values onto predicted experimental outcomes). According to this rule, a model is quantitatively testable if and only if the rank of the Jacobian matrix is less than the number of observables. (The rank of his matrix can be found with standard computer algorithms.) Using Sards theorem, we prove that the proposed new rule of thumb is correct provided that certain “smoothness” conditions are satisfied. We also discuss the relation between quantitative testability and reparameterization, identifiability, and goodness-of-fit testing.

Cognitive psychometrics: assessing storage and retrieval deficits in special populations with multinomial processing tree models.

This article demonstrates how multinomial processing tree models can be used as assessment tools to measure cognitive deficits in clinical populations. This is illustrated with a model developed by W. H. Batchelder and D. M. Riefer (1980) that separately measures storage and retrieval processes in memory. The validity of the model is tested in 2 experiments, which show that presentation rate affects the storage of items (Experiment 1) and part-list cuing hurts item retrieval (Experiment 2). Experiments 3 and 4 examine 2 clinical populations: schizophrenics and alcoholics with organic brain damage. The model reveals that each group exhibits deficits in storage and retrieval, with the retrieval deficits being stronger and occurring more consistently over trials. Also, the alcoholics with organic brain damage show no improvement in retrieval over trials, although their storage improves at the same rate as a control group.

Reaction times in a task analogous to “same”-“different” judgment

Subjects responded “yes” if two equal-length strings of letters contained a common letter in a common position; otherwise they responded “no.” Thus, the task was to judge whether all or not all of the letters in one string differed from the letter occupying the corresponding position in the other string. Conversely, in “same”-“different” judgment, the task is to judge whether all or not all of the letters in one string match the corresponding letter in the other string. Thus, common-letter judgment and “same”-“different” judgment are symmetrically related with “no” analogous to “same” and “yes” analogous to “different.” The response “same” is often faster than the response “different.” However, in the common-letter task, “no” was slower than “yes.” More specifically, both the “yes” and “no” reaction times were consistent with a serial self-terminating search. This is precisely what would be expected from Bamber’s (1969) two-process model.

Finite and infinite state confusion models

A confusion model is defined as a model that decomposes response probabilities in stimulus identification experiments into perceptual parameters and response parameters. Historically, confusion models fall into two groups. Models in Group I, which includes Townsends (Perception and Psychophysics, 1971, 9, 40–50) overlap model, were developed on the basis of the notion that stimulus identification is mediated by a finite number of internal states. We call the general class of models that have this processing interpretation finite state confusion models. Models in Group II, which includes Luces (R. O. Luce et al., Eds., Handbook of Mathematical Psychology (Vol. I), New York: Wiley, 1963) biased choice model, were not developed on the basis of an explicit processing interpretation. It is shown here that models in Group II are not finite state confusion models. We prove in addition that except for Falmagnes (Journal of Mathematical Psychology, 1972, 9, 206–224) simply biased model models in Group II belong to a certain class of infinite state confusion models, namely, models asserting that stimulus identification is mediated by a continuous space of vectors representing detector activation levels.

Evaluation of a method for studying forgetting: Is data from split-half recognition tests contaminated by test interference?

Split-half recognition testing is a method of investigating forgetting. In this method, a subject studies a list of items and then is given a recognition test covering half the items on the list. At a later time, a final recognition test covering the other half of the items is given. The present study investigates whether data from split-half recognition tests is contaminated by test interference. Specifically, this study investigates whether taking the first recognition test affects the subject’s recognition accuracy on the second test. It is concluded that the first test does not affect performance on the second test provided that the two tests are separated by 24 h or more.

A method for finding the maximum of a function of several variables suitable for use on a programmable desk calculator

Abstract It is often useful to have a procedure for locating the point where a function of several variables attains its maximum value. Numerous computer programs have been developed to perform this task. However, most of these programs are not suitable for use on a desk calculator as they require more memory space for program storage and numerical data storage than is often available inthe limited memory of a desk calculator. A program which locates the maximum of a multivariate function and which is short enough to fit in the memory of a desk calculator is presented here.

Estimators for functions of the probability of success in a sequence of bernoulli trials

Suppose that a sample of N experimental subjects are ran-daaly selected from a population of potential subjects. Sacs,sequence of T independent Bernoulli trials. The summary data from this experiment consists of the number of successes out of T trials attained by each of the N subjects. Each subject say be characterized by his probability p of success on a Bernoulli trial. This probability remains constant over successive trials,. but varies from subject to subject. Let g be a continuous function defined over the interval zero to one., Then, g(p) is a random variable over the population of potential subjects. It is desired to utilise the summary data from the experiment to estimate to- both the sample mean and the population mean of g(p).Two estimation procedures are proposed: a general method and an alternative method for use only with large N. The first method Minimizes the maximum possible mean square error. The second method consists of approximating the function g with a polynomial and, then, constru...

Comment on Bernbach's prediction of the invariance of Type 2 d′ in confidence-rated recall

Abstract Bernbach Bernbach, 1967 , Bernbach, 1971 has claimed that empirical findings of Type 2 d′ invariance in confidence-rated recall experiments confirm a prediction of his two-state theory of recognition and disprove strength theories of memory. In order to derive this prediction from his recognition theory, Bernbach (1967) found it necessary to add to the theory an assumption concerning recall processes. However, this recall assumption is untenable, as it leads to an empirically false prediction concerning accuracy on recognition and recall tests. Moreover, an alternative recall assumption proposed by Bernbach and Kupchak (1972) does not lead to a prediction of Type 2 d′ invariance. Thus, Bernbachs theory predicts the invariance of Type 2 d′ only with the aid of an untenable recall assumption. Consequently, empirical findings of Type 2 d′ invariance cannot be regarded as supportive of Bernbachs theory.