Robert Schlaifer

On the interpretation and observation of laws

A law with factors x and concomitants z specifies a distribution given z of a potential value Yx that is defined for each x whether or not it is observed. An observed distribution of Y given x and z agrees with the law if and only if, given z, the observed x is independent of Yx or, equivalently, of the joint effect of Ux of excluded variables on Yx. To establish such independence in non-experimental data requires exhaustive exploration of the effects of concomitants, causal and non-causal; R2 and F are irrelevant. We show how the model-free theory applies to linear models, time series, and simultaneous equations, and point out its Bayesian implications.

On the Nature and Discovery of Structure

Abstract Extending principles of experimentation, we discuss conditions under which nonexperimental data allow consistent estimation of effects of the kind revealed by experimentation and relevant to decisions. We show how implications of these conditions are often overlooked and how failure to distinguish between “factors” and “concomitants” makes almost anything said about a model ambiguous if not wrong. The effects to be estimated dictate the factors to be included; consistency and efficiency determine the concomitants, whose effects are not to be estimated. Concomitants may affect but must not be affected by the factors. Effects of excluded variables on an included variable may cause inconsistency if the included variable is a factor, can only reduce inconsistency if the included variable is a concomitant. Exclusion of a variable because it is highly correlated with another may sometimes be legitimate if both variables are concomitants, never if either is a factor. A condition for consistent estimatio...

Personal probabilities of probabilities

By definition, the subjective probability distribution of a random event is revealed by the (‘rational’) subjects choice between bets — a view expressed by F. Ramsey, B. De Finetti, L. J. Savage and traceable to E. Borel and, it can be argued, to T. Bayes. Since hypotheses are not observable events, no bet can be made, and paid off, on a hypothesis. The subjective probability distribution of hypotheses (or of a parameter, as in the current ‘Bayesian’ statistical literature) is therefore a figure of speech, an ‘as if’, justifiable in the limit. Given a long sequence of previous observations, the subjective posterior probabilities of events still to be observed are derived by using a mathematical expression that would approximate the subjective probability distribution of hypotheses, if these could be bet on. This position was taken by most, but not all, respondents to a ‘Round Robin’ initiated by J. Marschak after M. H. De-Groots talk on Stopping Rules presented at the UCLA Interdisciplinary Colloquium on Mathematics in Behavioral Sciences. Other participants: K. Borch, H. Chernoif, R. Dorfman, W. Edwards, T. S. Ferguson, G. Graves, K. Miyasawa, P. Randolph, L. J. Savage, R. Schlaifer, R. L. Winkler. Attention is also drawn to K. Borchs article in this issue.

On the nature and discovery of structure

Abstract Following Marschak and a very few others who have made clear what they mean by structure, we assume that a relation may properly be called structural if and only if it in some sense predicts the effects of arbitrary, possibly experimental, manipulation of variables. We argue that the literature on structural models utterly confuses two quite distinct problems: the problem of consistently estimating statistical relations when a number of such relations hold simultaneously, and the problem of deciding whether or not the consistently estimated relations are structural or merely descriptive. We argue further that although this literature has much that is useful to say on the former problem, what it says on the latter subject is seriously misleading and in direct contradiction to principles well established in the literature on design and analysis of experiments. More specifically, we first use a well-known physical law to show that the requirement of no correlation between exogenous and unidentified excluded variables which writers on structural models take to be necessary and sufficient for learning structure from non-experimental data is actually either vacuous, impossible, or a definition rather than a requirement. We next argue that treatment coefficients derived by randomized experimentation, which are generally agreed to predict the effects of arbitrary manipulation and are therefore ‘structural’ in the sense of this paper, have a meaning totally different from that of the coefficients apparently sought by structural modellers; and we maintain that even when experimentation is not possible, it is laws of this same sort that are the proper object of inquiry in the social sciences. We further argue that the conditions which are in fact sufficient and ‘almost’ necessary for learnings such laws from data are those imposed by writers on experimentation; and we try to express these conditions in a form which makes it as easy as possible (but not easy) to decide judgmentally whether or not they are satisfied in non-experimental data. Turning to simultaneous-equations techniques, we show that, although often essential to solution of the problem that arises when data have (literally or in effect) been subjected to something akin to censoring, they are irrelevant to the problem of deciding whether the coefficients of exogenous variables represent structure or merely statistical association. Finally, we show that the structural character of coefficients of endogenous variables, e.g., price elasticities, follows directly from assumptions about the temporal order in which variables are determined, regardless of the presence or absence of correlation between exogenous and excluded variables.

A New Interpretation of the F Statistic

Abstract A new interpretation of the F statistics appearing in the ANOVA of a balanced fixed-effect model permits them to be used, not only to test null hypotheses which assert that certain effects or contrasts of effects are zero, but also to help one decide whether relations or patterns observed in estimates of three or more effects can be reasonably assumed to hold for the effects themselves. And whether or not a design is balanced, the underlying mathematical arguments hold and yield formulas for near-optimal uniform shrinkage of estimates.

Introduction to Statistical Decision Theory.

The Bayesian revolution in statistics--where statistics is integrated with decision making in areas such as management, public policy, engineering, and clinical medicine--is here to stay. Introduction to Statistical Decision Theory states the case and in a self-contained, comprehensive way shows how the approach is operational and relevant for real-world decision making under uncertainty. Starting with an extensive account of the foundations of decision theory, the authors develop the intertwining concepts of subjective probability and utility. They then systematically and comprehensively examine the Bernoulli, Poisson, and Normal (univariate and multivariate) data generating processes. For each process they consider how prior judgments about the uncertain parameters of the process are modified given the results of statistical sampling, and they investigate typical decision problems in which the main sources of uncertainty are the population parameters. They also discuss the value of sampling information and optimal sample sizes given sampling costs and the economics of the terminal decision problems. Unlike most introductory texts in statistics, Introduction to Statistical Decision Theory integrates statistical inference with decision making and discusses real-world actions involving economic payoffs and risks. After developing the rationale and demonstrating the power and relevance of the subjective, decision approach, the text also examines and critiques the limitations of the objective, classical approach.