Alastair Scott

The Analysis of Categorical Data from Complex Sample Surveys: Chi-Squared Tests for Goodness of Fit and Independence in Two-Way Tables

Abstract The effect of stratification and clustering on the asymptotic distributions of standard Pearson chi-squared test statistics for goodness of fit (simple hypothesis) and independence in a two-way contingency table, denoted as X 2 and XI 2, respectively, is investigated. It is shown that both X 2 and XI 2 are asymptotically distributed as weighted sums of independent χ1 2 random variables. The weights are then related to the familiar design effects (deffs) used by survey samplers. A simple correction to X 2, which requires only the knowledge of variance estimates (or deffs) for individual cells in the goodness-of-fit problem, is proposed and empirical results on the performance of corrected X 2 provided. Empirical work on XI 2 indicated that the distortion of nominal significance level is substantially smaller with XI 2 than with X 2. Some results under simple models for clustering are also given.

A simple method for the analysis of clustered binary data.

A simple method for comparing independent groups of clustered binary data with group-specific covariates is proposed. It is based on the concepts of design effect and effective sample size widely used in sample surveys, and assumes no specific models for the intracluster correlations. It can be implemented using any standard computer program for the analysis of independent binary data after a small amount of preprocessing. The method is applied to a variety of problems involving clustered binary data: testing homogeneity of proportions, estimating dose-response models and testing for trend in proportions, and performing the Mantel-Haenszel chi-squared test for independence in a series of 2 x 2 tables and estimating the common odds ratio and its variance. Illustrative applications of the method are also presented.

The Effect of Two-Stage Sampling on Ordinary Least Squares Methods

Abstract We look at the effect of intracluster correlation on standard procedures in linear regression. The ordinary least squares estimator, , of the coefficient vector performs well in most cases but the usual estimator of cov() and procedures based on this such as confidence intervals and hypothesis tests can be seriously misleading. The size of the effect, however, tends to be smaller than the corresponding effect on the variance of an estimated mean in two-stage sampling provided that the cluster sample sizes are approximately equal.

Analysis of Repeated Surveys Using Time Series Methods

Abstract Standard time series methods are applied to the analysis of repeated surveys under the assumption that the population parameters at each time period follow a stochastic model. Both overlapping and nonoverlapping surveys are considered in general terms, and specific results are obtained for the time series models assumed in the work of Patterson [10].

How much variation in clinical activity is there between general practitioners? A multi-level analysis of decision-making in primary care.

Objectives: There is considerable policy interest in medical practice variation (MPV). Although the extent of MPV has been quantified for secondary care, this has not been investigated adequately in general practice. Technical obstacles to such analyses have been presented by the reliance on ecological small area variation (SAV) data, the binary nature of many clinical outcomes in primary care and by diagnostic variability. The study seeks to quantify the extent of variation in clinical activity between general practitioners by addressing these problems. Methods: A survey of nearly 10 000 encounters drawn from a representative sample of general practitioners in the Waikato region of New Zealand was carried out in the period 1991-1992. Participating doctors recorded all details of clinical activity for a sample of encounters. Measures used in this analysis are the issuing of a prescription, the ordering of a laboratory test or radiology examination, and the recommendation of a future follow-up office visit at a specified date. An innovative statistical technique is adopted to assess the allocation of variance for binary outcomes within a multi-level analysis of decision-making. Results: As expected, there was considerable variability between doctors in levels of prescribing, ordering of investigations and requests for follow up. These differences persisted after controlling for case-mix and patient and practitioner attributes. However, analysis of the components of variance suggested that less than 10% of remaining variability occurred at the practitioner level for any of the measures of clinical activity. Further analysis of a single diagnostic group - upper respiratory tract infection - marginally increased the practitioner contribution. Conclusions: The amount of variability in clinical activity that can definitively be linked to the practitioner in primary care is similar to that recorded in studies of the secondary sector. With primary care doctors increasingly being grouped into larger professional organisations, we can expect application of multi-level techniques to the analysis of clinical activity in primary care at different levels of organisational complexity.

Microalbuminuria in a Middle-Aged Workforce: Effect of hyperglycemia and ethnicity

OBJECTIVE To determine the prevalence of microalbuminuria in a mixed, thnic population and to find the extent that ethnic variation in microalbuminuria can be explained by abnormal glucose metabolism, obesity, hypertension, hypertriglyceridemia, and life-style factors. RESEARCH DESIGN AND METHODS Urinary albumin concentrations were measured in 5467 middle-aged Maori, Pacific Islander, and European workers who participated in a health-screening survey of 46 New Zealand companies. Participants provided a first-voided, morning urine sample; had a 75-g oral glucose tolerance test; had weight, height, and blood pressure measured; and completed a self-administered questionnaire about past medical history and sociodemographic status. RESULTS A significantly higher prevalence of microalbuminuria was found in individuals with new cases of diabetes mellitus (24.1%), in cases of diabetes mellitus previously diagnosed (20.6%), and in those with impaired glucose tolerance (16.1%) compared with nondiabetic individuals (4.0%). Moreover, in the general population, a piecewise linear relationship was detected between albuminuria and plasma glucose with significant changes of slope corresponding with 2 h plasma glucose concentrations (95% confidence interval) of 6.7 (6.4–7.0) and 9.2 (8.6–9.8) mM, respectively. After adjusting for sex, obesity, hypertension, hypertriglyceridemia, cigarette smoking, and heavy alcohol consumption in a multivariate model, glycemia was the most significant determinant of urinary albumin concentrations in all three ethnic groups. However, blood glucose concentrations did not completely explain the higher relative risk (95% confidence interval) of microalbuminuria in Maori (5.97; 4.48–7.78) and Pacific Islander (5.33; 4.13–6.87) workers compared with European workers. CONCLUSIONS Of the variables investigated, hyperglycemia was the most important factor explaining the high prevalence of microalbuminuria in Maori and Pacific Islander workers compared with the European workers. However, only 14.9% of the variation in urinary albumin concentrations was found in our multivariate model, and we have speculated on the contribution of other factors such as diet and coexisting renal diseases.

Transformations and R 2

Abstract The coefficient of determination, R 2, is widely used to compare the fit of competing regression models in spite of repeated warnings about the potential dangers. The use of R 2 is particularly inappropriate if the models are obtained by different transformations of the response scale. In this article we give a real example where two models are identical for all practical purposes and yet have very different values of R 2 calculated on the transformed scales.

Fitting Logistic Regression Models in Stratified Case-Control Studies

SUMMARY Methods are developed for fitting logistic models to data in which cases and/or controls are sampled from the available cases and controls within population strata. Particular attention is paid to models in which stratum differences are modelled as well as the effects of the different risk factors experienced by individuals within strata. Maximum likelihood estimation is developed for discrete explanatory variables and is compared with the method of Fears and Brown (1986, Biometrics 42, 955-960) and Breslow and Cain (1988, Biometrika 75, 11-20), in which the prospective logistic model is fitted with a fixed offset. An example is explored in which maximum likelihood estimation proves to be substantially more efficient than the Fears-Brown method and the modelling of the stratum effects leads to much more efficient estimates of regression coefficients for the remaining variables.

Quick Simultaneous Confidence Intervals for Multinomial Proportions

Abstract Opinion polls often give an indication of sampling error based on the standard conservative confidence intervals for a single binomial proportion. We give a lower bound for the simultaneous coverage probability when such intervals are applied to all of the proportions in a poll. This bound is equal to 1 – 2α. for small values of α. The same result is shown to apply to the standard confidence intervals for changes in proportions between surveys.

On the Asymptotic Distribution of Ratio and Regression Estimators

We give a formal derivation of the asymptotic normality of ratio and regression estimators of a finite population total for simple random sampling and for sampling with probability proportional to aggregate size. The results are all well known and widely used, of course, but a rigorous development, spelling out relatively simple sufficient conditions under which the standard results are valid, does not seem to be available at present.