Paul H. Garthwaite

Investigation of the single case in neuropsychology: confidence limits on the abnormality of test scores and test score differences.

Neuropsychologists often need to estimate the abnormality of an individual patients test score, or test score discrepancies, when the normative or control sample against which the patient is compared is modest in size. Crawford and Howell [The Clinical Neuropsychologist 12 (1998) 482] and Crawford et al. [Journal of Clinical and Experimental Neuropsychology 20 (1998) 898] presented methods for obtaining point estimates of the abnormality of test scores and test score discrepancies in this situation. In the present study, we extend this work by developing methods of setting confidence limits on the estimates of abnormality. Although these limits can be used with data from normative or control samples of any size, they will be most useful when the sample sizes are modest. We also develop a method for obtaining point estimates and confidence limits on the abnormality of a discrepancy between a patients mean score on k-tests and a test entering into that mean. Computer programs that implement the formulae for the confidence limits (and point estimates) are described and made available.

Statistical Methods for Eliciting Probability Distributions

Elicitation is a key task for subjectivist Bayesians. Although skeptics hold that elicitation cannot (or perhaps should not) be done, in practice it brings statisticians closer to their clients and subject-matter expert colleagues. This article reviews the state of the art, reflecting the experience of statisticians informed by the fruits of a long line of psychological research into how people represent uncertain information cognitively and how they respond to questions about that information. In a discussion of the elicitation process, the first issue to address is what it means for an elicitation to be successful; that is, what criteria should be used. Our answer is that a successful elicitation faithfully represents the opinion of the person being elicited. It is not necessarily “true” in some objectivistic sense, and cannot be judged in that way. We see that elicitation as simply part of the process of statistical modeling. Indeed, in a hierarchical model at which point the likelihood ends and the prior begins is ambiguous. Thus the same kinds of judgment that inform statistical modeling in general also inform elicitation of prior distributions. The psychological literature suggests that people are prone to certain heuristics and biases in how they respond to situations involving uncertainty. As a result, some of the ways of asking questions about uncertain quantities are preferable to others, and appear to be more reliable. However, data are lacking on exactly how well the various methods work, because it is unclear, other than by asking using an elicitation method, just what the person believes. Consequently, one is reduced to indirect means of assessing elicitation methods. The tool chest of methods is growing. Historically, the first methods involved choosing hyperparameters using conjugate prior families, at a time when these were the only families for which posterior distributions could be computed. Modern computational methods, such as Markov chain Monte Carlo, have freed elicitation from this constraint. As a result, now both parametric and nonparametric methods are available for low-dimensional problems. High-dimensional problems are probably best thought of as lacking another hierarchical level, which has the effect of reducing the as-yet-unelicited parameter space. Special considerations apply to the elicitation of group opinions. Informal methods, such as Delphi, encourage the participants to discuss the issue in the hope of reaching consensus. Formal methods, such as weighted averages or logarithmic opinion pools, each have mathematical characteristics that are uncomfortable. Finally, there is the question of what a group opinion even means, because it is not necessarily the opinion of any participant.

Uncertain Judgements: Eliciting Experts' Probabilities: O'Hagan/Uncertain Judgements: Eliciting Experts' Probabilities

Elicitation is the process of extracting expert knowledge about some unknown quantity or quantities, and formulating that information as a probability distribution. Elicitation is important in situations, such as modelling the safety of nuclear installations or assessing the risk of terrorist attacks, where expert knowledge is essentially the only source of good information. It also plays a major role in other contexts by augmenting scarce observational data, through the use of Bayesian statistical methods. However, elicitation is not a simple task, and practitioners need to be aware of a wide range of research findings in order to elicit expert judgements accurately and reliably. Uncertain Judgements introduces the area, before guiding the reader through the study of appropriate elicitation methods, illustrated by a variety of multi-disciplinary examples.

An Interpretation of Partial Least Squares

Abstract Univariate partial least squares (PLS) is a method of modeling relationships between a Y variable and other explanatory variables. It may be used with any number of explanatory variables, even far more than the number of observations. A simple interpretation is given that shows the method to be a straightforward and reasonable way of forming prediction equations. Its relationship to multivariate PLS, in which there are two or more Y variables, is examined, and an example is given in which it is compared by simulation with other methods of forming prediction equations. With univariate PLS, linear combinations of the explanatory variables are formed sequentially and related to Y by ordinary least squares regression. It is shown that these linear combinations, here called components, may be viewed as weighted averages of predictors, where each predictor holds the residual information in an explanatory variable that is not contained in earlier components, and the quantity to be predicted is the vecto...

Testing for suspected impairments and dissociations in single-case studies in neuropsychology: Evaluation of alternatives using Monte Carlo simulations and revised tests for dissociations

In neuropsychological single-case studies, a patient is compared with a small control sample. Methods of testing for a deficit on Task X, or a significant difference between Tasks X and Y, either treat the control sample statistics as parameters (using z and zD) or use modified t tests. Monte Carlo simulations demonstrated that if z is used to test for a deficit, the Type I error rate is high for small control samples, whereas control of the error rate is essentially perfect for a modified t test. Simulations on tests for differences revealed that error rates were very high for zD. A new method of testing for a difference (the revised standardized difference test) achieved good control of the error rate, even with very small sample sizes. A computer program that implements this new test (and applies criteria to test for classical and strong dissociations) is made available.

Obesity, Weight Loss and Prognosis in Type 2 Diabetes

Medical records were reviewed of all 263 Type 2 diabetic patients from the Aberdeen diabetic clinic who were known to have died in 1985 or 1986. Mean age was 65 years (interquartile range 57–75 years) at diagnosis and 72 (66–80) years for men, 75 (72–83) years for women, at death. Life expectancy at age 65 was 35% less than published figures for the general population. Analysis of survival in 233 patients who lived more than 1 year (189 overweight) using stepwise multiple regression indicated as significant (p < 0.05) adverse independent variables: age at diagnosis, presence of clinical ischaemic heart disease at diagnosis, plasma glucose at diagnosis; and as significant favourable variables: oral hypoglycaemic drug therapy, weight loss in the first year, and an interaction between weight loss and BMI for patients with BMI > 25 kg m−2. Changes in fashions over the years are likely to have biased these results towards including oral hypoglycaemic therapy and excluding the expected adverse effect of smoking. Mean weight loss at 1 year was 2.6 kg for those with BMI 25–30 kg m−2, 6.8 kg with BMI > 30 kg m−2, following standard dietetic advice. For the average patient each 1 kg weight loss was associated with 3–4 months prolonged survival.

Point and interval estimates of effect sizes for the case-controls design in neuropsychology: Rationale, methods, implementations, and proposed reporting standards

It is increasingly common for group studies in neuropsychology to report effect sizes. In contrast this is rarely done in single-case studies (at least in those studies that employ a case-controls design). The present paper sets out the advantages of reporting effect sizes, derives suitable effect size indexes for use in single-case studies, and develops methods of supplementing point estimates of effect sizes with interval estimates. Computer programs that implement existing classical and Bayesian inferential methods for the single case (as developed by Crawford, Garthwaite, Howell, and colleagues) are upgraded to provide these point and interval estimates. The upgraded programs can be downloaded from www.abdn.ac.uk/~psy086/dept/Single_Case_Effect_Sizes.htm

Comparison of a single case to a control or normative sample in neuropsychology: development of a Bayesian approach.

Frequentist methods are available for comparison of a patients test score (or score difference) to a control or normative sample; these methods also provide a point estimate of the percentage of the population that would obtain a more extreme score (or score difference) and, for some problems, an accompanying interval estimate (i.e., confidence limits) on this percentage. In the present paper we develop a Bayesian approach to these problems. Despite the very different approaches, the Bayesian and frequentist methods yield equivalent point and interval estimates when (a) a cases score is compared to that of a control sample, and (b) when the raw (i.e., unstandardized) difference between a cases scores on two tasks are compared to the differences in controls. In contrast, the two approaches differ with regard to point estimates of the abnormality of the difference between a cases standardized scores. The Bayesian method for standardized differences has the advantages that (a) it can directly evaluate the probability that a control will obtain a more extreme difference score, (b) it appropriately incorporates error in estimating the standard deviations of the tasks from which the patients difference score is derived, and (c) it provides a credible interval for the abnormality of the difference between an individuals standardized scores; this latter problem has failed to succumb to frequentist methods. Computer programs that implement the Bayesian methods are described and made available.

Quantifying Precision of Mark-Recapture Estimates Using the Bootstrap and Related Methods

SUMMARY Bootstrap techniques yield variance estimates under any model for which parameter estimates can be calculated, and are useful in cases where analytic variances are not available in closed form, or are available only if more restrictive assumptions are made. Here the application of bootstrap techniques to mark-recapture models is discussed. The approach also allows generation of robust confidence intervals, which extend beyond the permissible parameter range only if the mark-recapture model itself allows out-of-range parameter estimates. If an animal population is assumed to be closed (i.e., no death, birth, or migration), two further methods of obtaining confidence limits for population size are suggested. The first is based on a Robbins-Monro search for each limit, and the second applies the concept of a randomisation or permutation test. In the absence of nuisance parameters, both methods are exact apart from Monte Carlo variation and the limitations imposed by a discrete distribution. For the second, if all possible permutations are enumerated, Monte Carlo variation is eliminated.

The effect of dietary molybdenum and iron on copper status, puberty, fertility and oestrous cycles in cattle

Two experiments were conducted to examine the effects of supplementation of a control diet of barley grain and barley straw containing 4 mg copper (Cu)/kg dry matter (D.M.) either with 5 mg molybdenum (Mo)/kg D.M. or with 500 or 800 mg iron (Fe)/kg D.M. on puberty, fertility and oestrous cycles of cattle. Puberty occurred normally in control, Fesupplemented and control animals on a restricted intake whereas it was delayed by 12 and 8 weeks respectively by Mo supplementation. This effect of Mo was not due to the low Cu status since this was equally low in the Fe-supplemented animals, nor was it due to the reduced growth rate since puberty occurred normally in control animals that had a similar live-weight gain. A significant reduction in the pulsatile release of luteinizing hormone was observed within 11 weeks of the Mo supplementation and before any of the other clinical signs were evident, suggesting that Mo may be affecting puberty by altering the release of luteinizing hormone either directly or indirectly. Mo supplementation significantly reduced the percentage conception rate to 12–33% compared with 57–80% in control and Fe-supplemented animals. This effect was not dependent on the rate of live-weight gain which was standardized across the different treatments at approximately 0·6 kg/day. Within 12 weeks of the replacement of dietary Fe by Mo a lower conception rate occurred; replacing dietary Mo by Fe led to a normal conception rate within 12 weeks without any accompanying changes in Cu status or in the rate of live-weight gain. The plasma Mo concentrations, however, changed significantly during these alterations in dietary supplementation. The pre-ovulatory peak height of luteinizing hormone was significantly lower in animals on the Mo-supplemented diet compared with control and Fe-supplemented animals, but the administration of LHRH did not alter the conception rate. More Mo-supplemented animals failed to ovulate following prostaglandin induced synchronization in comparison with the other treatments, and by the 84th week a significantly greater number of Mo-supplemented animals (12/18) had become anoestrous compared with the other groups (2/30). Cu repletion of these anoestrous Mo animals for a period of 20 weeks did not result in resumption of normal oestrous cycles, but ovulation and oestrus were induced by progesterone and LHRH treatment. Results in the latter part of the study indicated that Mo caused superovulation. These data show that Mo supplementation delayed the onset of puberty, decreased the conception rate and caused anovulation and anoestrus in cattle without accompanying changes in Cu status or in live-weight gain. It is suggested that these effects of Mo are associated with a decreased release of luteinizing hormone that might be due to an altered ovarian steroid secretion.