A Generalization of Gurland's Theorem,with Applications to Economic Behavior under Uncertainty

We illustrate the use of the Mathematica software system (programming language) for performing Bayesian calculations of the sort encountered in introductory presentations of Bayesian statistics. In particular, we show the ease with which one can do numerical computation, graphics, and symbolic computation to analyze one- and two-dimensional probability density functions. Our illustrations involve a time series problem using oxygen uptake data taken from a burn patient.     

A Tale of Two Matrix Factorizations

In statistical practice, rectangular tables of numeric data are commonplace, and are often analyzed using dimension-reduction methods like the singular value decomposition and its close cousin, principal component analysis (PCA). This analysis produces score and loading matrices representing the rows and the columns of the original table and these matrices may be used for both prediction purposes and to gain structural understanding of the data. In some tables, the data entries are necessarily nonnegative (apart, perhaps, from some small random noise), and so the matrix factors meant to represent them should arguably also contain only nonnegative elements. This thinking, and the desire for parsimony, underlies such techniques as rotating factors in a search for “simple structure.” These attempts to transform score or loading matrices of mixed sign into nonnegative, parsimonious forms are, however, indirect and at best imperfect. The recent development of nonnegative matrix factorization, or NMF, is an attractive alternative. Rather than attempt to transform a loading or score matrix of mixed signs into one with only nonnegative elements, it directly seeks matrix factors containing only nonnegative elements. The resulting factorization often leads to substantial improvements in interpretability of the factors. We illustrate this potential by synthetic examples and a real dataset. The question of exactly when NMF is effective is not fully resolved, but some indicators of its domain of success are given. It is pointed out that the NMF factors can be used in much the same way as those coming from PCA for such tasks as ordination, clustering, and prediction. Supplementary materials for this article are available online.     

Parameter estimation in a highly non-linear model using simultaneous perturbation stochastic approximation

Often it is necessary to estimate the parameters of a model or unknown system. Various techniques exist to accomplish this task, including Kalman and Wiener filtering, least-mean-square (LMS) algorithms, and the Levenberg-Marquardt(L-M) algorithm. These techniques require an analytic form of the gradient of the function of the parameters to be estimated. A key feature of the simultaneous perturbation stochastic approximation (SPSA) method is that it is a gradient-free optimization technique (Spall; 1992,1998a,b, 1999). In the current problem, the function of parameters to be identified is highly non-linear and of sufficient difficulty that obtaining an analytic form of the gradient is impractical. Therefore, in this paper the

performance of the SPSA algorithm will be examined in terms of parameter selection, data requirements, and convergence performance on this non-linear problem. Results will be reported on both a first-order "standard" implementation of SPSA and on a second-order version of SPSA that tends to enhance convergence.     

Improved bootstrap confidence intervals in certain toxicological experiments

The bootstrap, the jackknife, and classical methods are compared through their confidence intervals for the proportion of affected fetuses in a common type of animal experiment. Specifically, suppose that for the ith of M pregnant animals, there are x _i affected fetuses out of n _i total in the litter. The conditional distribution of x _i given n _i is sometimes modeled as binomial (n _i p _i ), where p _i is a realization from some unknown continuous density. The p _i are not observable and it is of interest in some toxicological experiments to find confidence intervals for E(p). Theory suggests that the proposed parametric bootstrap should produce higher order agreement between the nominal and actual coverage than that exhibited by the usual nonparametric bootstrap. Some simulation results provide additional evidence of this superiority of the modified parametric bootstrap over the jack-knife and classical approaches. The proposed resampling is flexible enough to handle a more general model allowing correlation between p _i and n _i .     

On Testing Sample Selection Bias Under the Multicollinearity Problem

ABSTRACT

This paper reviews and extends the literature on the finite sample behavior of tests for sample selection bias. Monte Carlo results show that, when the “multicollinearity problem” identified by Nawata (1993 Nawata , K. ( 1993 ). A note on the estimation of models with sample-selection biases . Economics Letters 42 : 15 – 24 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) is severe, (i) the t-test based on the Heckman–Greene variance estimator can be unreliable, (ii) the Likelihood Ratio test remains powerful, and (iii) nonnormality can be interpreted as severe sample selection bias by Maximum Likelihood methods, leading to negative Wald statistics. We also confirm previous findings (Leung and Yu, 1996 Leung , S. F. , Yu , S. ( 1996 ). On the choice between sample selection and two-part models . Journal of Econometrics 72 : 197 – 229 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) that the standard regression-based t-test (Heckman, 1979 Heckman , J. J. ( 1979 ). Sample selection bias as a specification error . Econometrica 47 : 153 – 161 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) and the asymptotically efficient Lagrange Multiplier test (Melino, 1982 Melino , A. ( 1982 ). Testing for sample selection bias . Review of Economic Studies 49 : 151 – 153 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), are robust to nonnormality but have very little power.     

Mark-resighting analysis of a California gull population

California gulls ( Larus californicus ) of known age and sex were censused on their breeding colony in 1979, 1980 and 1984 through 1993. Ages of 235 males and 196 females ranged from 4 to 27 years. Age classes used in the analysis were limited to 17, 4 through 19, and 20 or more as a final age category because data on gulls over 20 were sparse. Survival declined with age in a way that was parsimoniously modelled with a quadratic function. Other factors, sex and time, did not explain any variation in survival. Resighting depended on age, sex and time. Younger adults skipped breeding more frequently than did older adults, and females skipped breeding more frequently than did males. There was also good evidence for time dependence in resighting probability, but its inclusion in the model occurred at the expense of interpretability and precision. In a data set such as this, resighting probability may assume more importance than a mere 'nuisance parameter'. In this study, resighting history measured attendance at the breeding ground. In turn, attendance rates may be a manifestation of reproductive strategy, which can also have consequences for survival. In this situation, there may be heterogeneity in both survival and resighting probability that is unexplained by the model. While such complexity may well be a nuisance to deal with, it can also point to important biological questions.     

The confidential patient: the social construction of therapeutic relationships in general medical practice

The interpersonal relationship between doctor and patient is fundamental to general medical practice. In this paper we explore the ways in which general practitioners make sense of the changing political economy of this relationship, as it is restructured by ideas about the patient as consumer, and as it increasingly constitutes the consultation as a point of interaction that may be intrinsically therapeutic. In particular, we explore the ways in which the consultation is the site of negotiated power relations between doctor and patient, and is the site of the doctor's negotiation of powerful discourses of professional and institutional identity.