Prediction with single event versus aggregate data

The impact of statistical information on predictive judgment was studied in a cue probability learning (CPL) task. Two kinds of aggregate information about criterion events were used: the conditional mean (Z_M) and the conditional interquartile range (Z_R). The single event information was the exact criterion value for one randomly selected past case (Z_O). Results showed that Z_Mand Z_R increased prediction consistency and accuracy and reduced bias while Z_O led to more appropriate cue weighting but lower consistency and accuracy. When Z_M, Z_R, and Z_O were all provided, the unique benefits of single-event and aggregate data were combined. This was true even for subjects without any prior task experience. When able to select only one of Z_M, Z_R, and Z_O, judges most often chose aggregate information, particularly Z_R. However, the statistical information was underutilized. Recommendations for aiding predictive judgment are made.     

A model and algorithm for multidimensional scaling with external constraints on the distances

A method for externally constraining certain distances in multidimensional scaling configurations is introduced and illustrated. The approach defines an objective function which is a linear composite of the loss function of the point configurationX relative to the proximity dataP and the loss ofX relative to a pseudo-data matrixR. The matrixR is set up such that the side constraints to be imposed onX's distances are expressed by the relations amongR's numerical elements. One then uses a double-phase procedure with relative penalties on the loss components to generate a constrained solutionX. Various possibilities for constructing actual MDS algorithms are conceivable: the major classes are defined by the specification of metric or nonmetric loss for data and/or constraints, and by the various possibilities for partitioning the matricesP andR. Further generalizations are introduced by substitutingR by a set ofR matrices,R _i ,i=1, ...r, which opens the way for formulating overlapping constraints as, e.g., in patterns that are both row- and column-conditional at the same time.     

A new secretary problem with rank-based selection and cardinal payoffs

We present an extension of the secretary problem in which the decision maker (DM) sequentially observes up to n applicants whose values are random variables X₁,X₂,…,X_n drawn i.i.d. from a uniform distribution on [0,1]. The DM must select exactly one applicant, cannot recall released applicants, and receives a payoff of x_t, the realization of X_t, for selecting the tth applicant. For each encountered applicant, the DM only learns whether the applicant is the best so far. We prove that the optimal policy dictates skipping the first sqrt(n)-1 applicants, and then selecting the next encountered applicant whose value is a maximum.     

A discrete time model for detection of randomly presented stimuli

Considered in this paper is a discrete-time Markov decision process with unobservable states, which occurs in equipment checking, medical diagnosis, signal detection, and reaction time studies. The process is characterized by (i) a state S₀ that at some unknown time t, which is determined probabilistically, turns into an absorbing state S₁; (ii) observations that provide fallible information about the true state and which are taken sequentially to determine whether or not the change from S₀ to S₁ has occurred; and (iii) losses for a false alarm and for delays in detecting the change. A response model is proposed, prescribing a terminal decision as soon as the posterior probability that S₁ obtains reaches or exceeds a fixed probability boundary. Predictions are derived concerning the probability of false alarm and several conditional distributions of the number of observations taken.     

On the scales of measurement

Let $\text{X}$ = 〈X, ≧, R₁, R₂…〉 be a relational structure, 〈$\text{X, ≧}$〉 be a Dedekind complete, totally ordered set, and n be a nonnegative integer. $\text{X}$ is said to satisfy n-point homogeneity if and only if for each x₁,…, x_n, y₁,…, y_n such that x₁ ? x₂ ? … ? x_n and y₁ ? y₂ … ? y_n, there exists an automorphism α of $\text{X}$ such that α(x₁) = y_i. $\text{X}$ is said to satisfy n-point uniqueness if and only if for all automorphisms β and γ of $\text{X}$, if β and γ agree at n distinct points of $\text{X}$, then β and γ are identical. It is shown that if $\text{X}$ satisfies n-point homogeneity and n-point uniqueness, then n ≦ 2, and for the case n = 1, $\text{X}$ is ratio scalable, and for the case n = 2, interval scalable. This result is very general and may in part provide an explanation of why so few scale types have arisen in science. The cases of 0-point homogeneity and infinite point homogeneity are also discussed.     

Maximum likelihood estimation of the joint covariance matrix for sections of tests given to distinct samples with application to test equating

Consider an old testX consisting ofs sections and two new testsY andZ similar toX consisting ofp andq sections respectively. All subjects are given testX plus two variable sections from either testY orZ. Different pairings of variable sections are given to each subsample of subjects. We present a method of estimating the covariance matrix of the combined test (X ₁, ...,X _s ,Y ₁, ...,Y _p ,Z ₁, ...,Z _q ) and describe an application of these estimation techniques to linear, observed-score, test equating.The author is indebted to Paul W. Holland and Donald B. Rubin for their encouragement and many helpful comments and suggestions that contributed significantly to the development of this paper.This research was supported by the Program Statistics Research Project of the ETS Research Statistics Group.     

Trichotomous preferences for gambles

Let G_X be the set of all gambles on a set X of decision outcomes. Trichotomous preferences divide G_X into three disjoint subsets, viz., the set P⁺ of preferable gambles, the set P⁰ of indifferent gambles, and the set P^- of unpreferable gambles. Those preference comparisons are made against the status quo, i.e., the do-nothing alternative. This paper presents and discusses dyadic representations of such trichotomous preferences and identifies necessary and sufficient axioms for the existence of those representations.     

Reasoning defeasibly about probabilities

In concrete applications of probability, statistical investigation gives us knowledge of some probabilities, but we generally want to know many others that are not directly revealed by our data. For instance, we may know prob(P/Q) (the probability of P given Q) and prob(P/R), but what we really want is prob(P/Q&;R), and we may not have the data required to assess that directly. The probability calculus is of no help here. Given prob(P/Q) and prob(P/R), it is consistent with the probability calculus for prob(P/Q&;R) to have any value between 0 and 1. Is there any way to make a reasonable estimate of the value of prob(P/Q&;R)? A related problem occurs when probability practitioners adopt undefended assumptions of statistical independence simply on the basis of not seeing any connection between two propositions. This is common practice, but its justification has eluded probability theorists, and researchers are typically apologetic about making such assumptions. Is there any way to defend the practice? This paper shows that on a certain conception of probability—nomic probability—there are principles of “probable probabilities” that license inferences of the above sort. These are principles telling us that although certain inferences from probabilities to probabilities are not deductively valid, nevertheless the second-order probability of their yielding correct results is 1. This makes it defeasibly reasonable to make the inferences. Thus I argue that it is defeasibly reasonable to assume statistical independence when we have no information to the contrary. And I show that there is a function Y(r, s, a) such that if prob(P/Q) = r, prob(P/R) = s, and prob(P/U) = a (where U is our background knowledge) then it is defeasibly reasonable to expect that prob(P/Q&;R) = Y(r, s, a). Numerous other defeasible inferences are licensed by similar principles of probable probabilities. This has the potential to greatly enhance the usefulness of probabilities in practical application.     

Indirect representation and the self-representational theory of consciousness

According to Uriah Kriegel’s self-representational theory of consciousness, mental state M is conscious just in case it is a complex with suitably integrated proper parts, M ₁ and M ₂, such that M ₁ is a higher-order representation of lower-order representation M ₂. Kriegel claims that M thereby “indirectly” represents itself, and he attempts to motivate this claim by appealing to what he regards as intuitive cases of indirect perceptual and pictorial representation. For example, Kriegel claims that it’s natural to say that in directly perceiving the front surface of an apple one thereby perceives the apple itself. Cases such as this are supposed to provide intuitive support for the principle that if X represents Y, and Y is highly integrated into complex object Z, then X indirectly represents Z. In this paper I provide counterexamples to Kriegel’s principle of indirect representation, before going on to argue that we can explain what is going on in those cases in which the subject seems to represent a complex whole by representing one its parts without positing indirect representations anyway. I then argue that my alternative approach is superior to Kriegel’s in a number of ways, thereby rendering his theory of consciousness implausible.     

Real Interval Representations

This paper presents several necessary and sufficient conditions for real interval representability of biorders, interval orders, and semiorders. Let A and X be nonempty sets. We consider two types of interval representations for P⊂A×X. The first concerns the existence of two mappings, F: A→J_↓ and F: X→J_↑, such that, for all (a, x)∈A×X, (a, x)∈P⇔F(a)∩F (x)= ∅, where J_↓ and J_↑ respectively denote the set of all real intervals that are unbounded below and the set of all real intervals that are unbounded above. The second yields two mappings, F: A→J_↓ and G: X→J_↓, such that, for all (a, x)∈A×X, (a, x)∈P⇔F(a)⊂G(x). Specializations of those representations include the cases of A=X for interval orders and semiorders.     

The adequacy of the theories of deductive systems with respect to sentential calculi

Summary The sentential calculiR, under discussion, are axiomatizable and implication is among their primitive terms. The modus ponens and the rule of substitution are their primitive rules. ByS _r is denoted the set of sentences obtained from the formulae of the calculusR by substituting sentences of a given language for all variables. The variablesx, y, z ... represent the elements of the setS _r , the variablesX, Y, Z ... represent the subsets ofS _R . The formulacxy designates an implication withx as its antecedent andy as its consequent,cxy is always an element ofS _R δ(X) means, thatX is closed with respect to the modus ponens rule.A _R designates the class of allS _R -substitutions of the axioms of the sentential calculusR.     

Decision policies minimizing risk in a multistage betting game

Considered in this paper is a decision task which has been employed to study multistage betting behavior. When the task commences, a decision maker (DM) is provided with some capital x (x > 0) which he is required to allocate over m (m > 1) mutually exclusive and collectively exhaustive alternatives, each of which occurs with probability p_i (p_i > 0, i = 1,…, m; Σ_i=1^mp_i = 1). If the amount y_i is allocated to alternative i (y_i ≥ 0, Σ_i=1^my_i = x) and alternative i obtains, DM's capital for the next stage of the game becomes y_ir_i, where r_i (r_i > 0) is the return per unit allocated to alternative i. The task consists of N stages.Defining risk in terms of the mean and variance of DM's bets, and assuming that the minimization of risk is DM's objective, decision policies satisfying this objective are derived in closed form and their testable properties are briefly discussed.     

Verification of algebra step problems: A chronometric study of human problem solving

A class of simple problem solving tasks requiring fast accurate solutions is introduced. In an experiment subjects memorized a mapping rule represented by lists of words labeled by cue words and made true/false decisions about conjunctions of propositions of the form “Y is in the list labeled by X”, written “X → Y”. Response times are analyzed using a “stage modeling” technique where problem solving algorithms are composed using a small set of psychological operations that have real time characteristics specified parametrically. The theoretical analysis shows that response time performance is adequately described in terms of the sequential application of elementary psychological operations. Unexpectedly, it was found that the proposition “X → YandX → Z” was verified as quickly as the apparently simpler “X → Y”. A case is presented for the modeling technique as applied to memory and problem solving tasks in terms of theoretical parsimony, statistical simplicity, and flexibility in investigative empirical research. Suggestions are made as to possible theoretical relations among fast problem solving, more complex and slower problem solving, and research in fundamental memory processes.     

The center and range of the probability interval as factors affecting ambiguity preferences

Ambiguous decision situations are characterized as having probabilities that are uncertain. The uncertainty is due to the common, real-world deficiency of information about the process by which the outcomes are determined. Thirty lotteries having uncertain probabilities were constructed by varying the centers and the ranges of the intervals within which the imprecise probabilities of winning could lie. Pairs of the lotteries were presented as choice alternatives to subjects, with each pair having lotteries with the same interval center but differing interval ranges. Ambiguity avoidance, the selection of the less ambiguous option, was found to increase with the interval center C, with ambiguity indifference occurring for values of C ? 0.40. No evidence of ambiguity seeking as the prevalent behavior was obtained. Ambiguity avoidance did not significantly increase with the interval range R, but an interaction effect between C and the ranges R₁ and R₂ of the choice pair was obtained. This effect of the ranges could not be described simply by knowledge of the difference R₁ ? R₂; knowledge of both individual values was necessary. The theoretical implications of these results are discussed.     

Further test of a subjective probability model of social inference

Subjects read paragraphs suggesting that a fictitious person (O) either did or did not possess an attribute X. Then, they reported their beliefs that they would like O (P_L), that O possessed X (P_X), that O was intelligent (P_I) and that O was sarcastic (P_s). Other beliefs necessary to test the predictive accuracy of Wyer's subjective probability model of cognitive organization were also reported. The model accurately predicted P_L on the basis of beliefs about X (the attribute to which information about O directly pertained); this accuracy was greater than that obtained in an earlier study in which real persons were evaluated, and did not depend upon the favorableness of X. However, predictions of P_L based upon beliefs about O's intelligence overestimated obtained values, while predictions based upon beliefs about O's sarcasm underestimated these values; moreover, P_I and P_S were underestimated and overestimated, respectively, by beliefs about X. It was speculated that these discrepancies were due to temporary inconsistencies among beliefs resulting from the information presented which did not have time to be resolved before beliefs were reported. Changes in P_L, P_I, and P_S resulting from additional information about X were accurately predicted in all cases. Implications of the model for other formulations of social evaluation processes are noted.     

Feature cues in probability learning: Data base information and judgment

Judgments of probabilistic events are often based partly on some information about past similar events. This study investigates the impact of summarized historical data termed a feature cue on performance in a cue probability learning task. Judges (n = 64) made 150 predictions of a criterion variable (Y_e) from a single cue variable (X). The feature cue variable (Z) provided judges with the “average past criterion” for the cue value on trial i, i.e., the conditional mean . Availability of the feature cue was varied with an AB-BA transfer design. Results demonstrate that the presence of the feature cue greatly imporved prediction achievement and accuracy. Under certain conditions, consistency and cue weighting were also improved by the feature cue aid. Although the feature cue value itself was not used as a prediction, it served as an anchor, around which judgments were dispersed. Implications for decision making with data base information are discussed.     

An interaction model for odor quality and intensity

The perceived intensity and quality of binary odor mixtures are studied in relation to bow their components are perceived when presented separately. Subjects judged the perceived intensity and quality of 6 concentrations of pyridine, 6 concentrations ofn-butanol, and their 36 possible combinations. The results show that the perceived intensity of the mixture can be predicted from the perceived intensity of its components presented separately (R _A andR _B) by the Euclidian arithmetic model. The maximum probability of identifying a mixture as a “mixture” is reached whenR _A andR _B are equally strong. An interaction model for mixtures that relates perceived intensity and quality is presented and tested. The test reveals that both the perceived intensity of a mixture and the probability that it will be identified as the one or the other component can, by a simple interaction model, be successfully predicted fromR _A andR _B.