In the case-crossover design, only cases are sampled and the hazard ratio is estimated from within-subject comparisons of exposures at the event time and in M control periods prior to the event. We consider the effect of within-subject dependence of exposures in successive time periods. We show that estimates obtained from the conditional logistic model are biased. This bias disappears if the distribution of exposures in the M+1 successive time intervals is exchangeable. In contrast, the Mantel-Haenszel estimator for the odds ratio is approximately unbiased provided that exposures are stationary. Suitable methods of analysis of case-crossover designs using maximum likelihood may be derived from cohort rather than case-control models.

We introduce an algorithm for producing simple approximate principal components directly from a variance±covariance matrix. At the heart of the algorithm is a series of`simplicity preserving' linear transformations. Each transformation seeks a direction within a two-dimensional subspace that has maximum variance. However, the choice of directions is limited so that the direction can be represented by a vector of integers whenever the subspace can also be represented by vectors of integers. The resulting approximate components can therefore always be represented by integers. Furthermore the elements of these integer vectors are often small, particularly for the ®rst few components. We demonstrate the performance of this algorithm on two data sets and show that good approximations to the principal components that are also clearly simple and interpretable can result.

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The short insulin tolerance test is a simple method of estimating insulin resistance by measuring the rate of fall of blood glucose following the intravenous administration of insulin. To determine its reproducibility, 18 healthy subjects underwent duplicate insulin tolerance tests separated by at least 1 week. Intravenous insulin (0.05 units kg-1) was administered into an antecubital vein and arterialized venous samples were obtained from a retrogradely cannulated vein on the dorsum of the hand on the same side. The test was terminated with an intravenous glucose injection 15 min after the administration of insulin. The mean whole blood glucose concentration fell from 4.6 mmol l-1 to 2.8 mmol l-1 while plasma insulin concentrations rose to supraphysiological levels and declined exponentially. The glucose disappearance rate (Kitt) calculated from the slope of the fall in log transformed blood glucose between 3 and 15 min after insulin injection ranged from 2.1 to 6.5 (mean 4.4) % min-1 during the first visit and 1.7 to 7.4 (mean 4.4) % min-1 during the second. The ratio of the within-subject to between-subject variance of the test was 0.24, the within-subject coefficient of variation was 13% and the between-subject coefficient of variation 26%. The short insulin tolerance test is reproducible and could be used to measure insulin resistance in large-scale epidemiological studies.

Bayesian random effects models may be fitted using Gibbs sampling, but the Gibbs sampler can be slow mixing due to what might be regarded as lack of model identifiability. This slow mixing substantially increases the number of iterations required during Gibbs sampling. We present an analysis of data on immunity after Rubella vaccinations which results in a slow-mixing Gibbs sampler. We show that this problem of slow mixing can be resolved by transforming the random effects and then, if desired, expressing their joint prior distribution as a sequence of univariate conditional distributions. The resulting analysis shows that the decline in antibodies after Rubella vaccination is relatively shallow compared to the decline in antibodies which has been shown after Hepatitis B vaccination.

Principal component analysis (PCA) is widely used in atmospheric science, and the resulting empirical orthogonal functions (EOFs) are often rotated to aid interpretation. In this paper 3 methods are described which provide alternatives to the standard 2-stage procedure of PCA followed by rotation. The techniques are illustrated on a small example involving sea-surface temperatures in the Mediterranean. Each method is shown to give different simplified interpretations for the major sources of variation in the data set. All 3 techniques have advantages over standard rotation.

In this article, we introduce and study local constant and our preferred local linear nonparametric regression estimators when it is appropriate to assess performance in terms of mean squared relative error of prediction. We give asymptotic results for both boundary and non-boundary cases. These are special cases of more general asymptotic results that we provide concerning the estimation of the ratio of conditional expectations of two functions of the response variable. We also provide a good bandwidth selection method for our estimator. Examples of application and discussion of related problems and approaches are also given.

When the difference between samples is measured using a Euclidean-embeddable dissimilarity function, observations and the associated variables can be displayed on a nonlinear biplot. Furthermore, a nonlinear biplot is predictive if information on variables is added in such a way that it allows the values of the variables to be estimated for points in the biplot. In this paper an r dimensional biplot which maps the predicted value of a variable for every point in the plot, is introduced. Using such maps it is shown that even with continuous data, predicted values do not always vary continuously across the biplot plane. Prediction trajectories that appropriate for summarising such non-continuous prediction maps are also introduced. These prediction trajectories allow information about two or more variables to be estimated even when the underlying predicted values do not vary continuously.