Regression-based variable clustering for data reduction

In many studies it is of interest to cluster states, counties or other small regions in order to obtain improved estimates of disease rates or other summary measures, and a more parsimonious representation of the country as a whole. This may be the case if there are too many to summarize concisely, and/or many regions with a small number of cases. By merging the regions into larger geographic areas, we obtain more cases within each area (and hence lower standard errors for parameter estimates), as well as fewer areas to summarize in terms of disease rates. The resulting clusters should be such that regions within the same cluster are similar in terms of their disease rates. In this paper we present a clustering algorithm which uses data at the subject-specific level in order to cluster the original regions into a reduced set of larger areas. The proposed clustering algorithm expresses the clustering goals in terms of a regression framework. This formulation of the problem allows the regions to be clustered in terms of their association with the response, and confounding variables measured at the subject-specific level may be easily incorporated during the clustering process. Additionally, this framework allows estimation and testing of the association between the areas and the response. The statistical properties and performance of the algorithm were evaluated via simulation studies, and the results are promising. Additional simulations illustrate the importance of controlling for confounding variables during the clustering process, rather than after the clusters are determined. The algorithm is illustrated with data from the Cardiovascular Health Study. Although developed with a specific application in mind, the method is applicable to a wide range of problems.     

Functional data analysis with application to periodically stimulated foetal heart rate data. II: functional logistic regression

We present a basis solution for the modelling of a binary response with a functional covariate plus any number of scalar covariates. This can be thought of as singular longitudinal data analysis as there are more measurements on the functional covariate than subjects in the study. The maximum likelihood parameter estimates are found using a basis expansion and a modified Fisher scoring algorithm. This technique has been extended to model a functional covariate with a repeated stimulus. We used periodically stimulated foetal heart rate tracings to predict the probability of a high risk birth outcome. It was found that these tracings could predict 94.1 per cent of the high risk pregnancies and without the stimulus, the heart rates were no more predictive than chance.     

Detecting dose-response using contrasts: asymptotic power and sample size determination for binomial data

Recently, Stewart and Ruberg proposed the use of contrast tests for detecting dose-response relationships. They considered in particular bivariate contrasts for healing rates and gave several possibilities of defining adequate sets of coefficients. This paper extends their work in several directions. First, asymptotic power expressions for both single and multiple contrast tests are derived. Secondly, well known trend tests are rewritten as multiple contrast tests, thus alleviating the inherent problem of choosing adequate contrast coefficients. Thirdly, recent results on the efficient calculation of multivariate normal probabilities overcome the traditional simulation-based methods for the numerical computations. Modifications of the power formulae allow the calculation of sample sizes for given type I and II errors, the spontaneous rate, and the dose-response shape. Some numerical results of a power study for small to moderate sample sizes show that the nominal power is a reasonably good approximation to the actual power. An example from a clinical trial illustrates the practical use of the results.     

A graphical sensitivity analysis for clinical trials with non-ignorable missing binary outcome

Many clinical trials are analysed using an intention-to-treat (ITT) approach. A full application of the ITT approach is only possible when complete outcome data are available for all randomized subjects. In a recent survey of clinical trial reports including an ITT analysis, complete case analysis (excluding all patients with a missing response) was common. This does not comply with the basic principles of ITT since not all randomized subjects are included in the analysis. Analyses of data with missing values are based on untestable assumptions, and so sensitivity analysis presenting a range of estimates under alternative assumptions about the missing-data mechanism is recommended. For binary outcome, extreme case analysis has been suggested as a simple form of sensitivity analysis, but this is rarely conclusive. A graphical sensitivity analysis is proposed which displays the results of all possible allocations of cases with missing binary outcome. Extension to allow binomial variation in outcome is also considered. The display is based on easily interpretable parameters and allows informal examination of the effects of varying prior beliefs.     

Multicollinearity in prognostic factor analyses using the EORTC QLQ-C30: identification and impact on model selection

Clinical and quality of life (QL) variables from an EORTC clinical trial of first line chemotherapy in advanced breast cancer were used in a prognostic factor analysis of survival and response to chemotherapy. For response, different final multivariate models were obtained from forward and backward selection methods, suggesting a disconcerting instability. Quality of life was measured using the EORTC QLQ-C30 questionnaire completed by patients. Subscales on the questionnaire are known to be highly correlated, and therefore it was hypothesized that multicollinearity contributed to model instability. A correlation matrix indicated that global QL was highly correlated with 7 out of 11 variables. In a first attempt to explore multicollinearity, we used global QL as dependent variable in a regression model with other QL subscales as predictors. Afterwards, standard diagnostic tests for multicollinearity were performed. An exploratory principal components analysis and factor analysis of the QL subscales identified at most three important components and indicated that inclusion of global QL made minimal difference to the loadings on each component, suggesting that it is redundant in the model. In a second approach, we advocate a bootstrap technique to assess the stability of the models. Based on these analyses and since global QL exacerbates problems of multicollinearity, we therefore recommend that global QL be excluded from prognostic factor analyses using the QLQ-C30. The prognostic factor analysis was rerun without global QL in the model, and selected the same significant prognostic factors as before.     

Estimating risk and rate levels,ratios and differences in case-control studies

Classic (or 'cumulative') case-control sampling designs do not admit inferences about quantities of interest other than risk ratios, and then only by making the rare events assumption. Probabilities, risk differences and other quantities cannot be computed without knowledge of the population incidence fraction. Similarly, density (or 'risk set') case-control sampling designs do not allow inferences about quantities other than the rate ratio. Rates, rate differences, cumulative rates, risks, and other quantities cannot be estimated unless auxiliary information about the underlying cohort such as the number of controls in each full risk set is available. Most scholars who have considered the issue recommend reporting more than just risk and rate ratios, but auxiliary population information needed to do this is not usually available. We address this problem by developing methods that allow valid inferences about all relevant quantities of interest from either type of case-control study when completely ignorant of or only partially knowledgeable about relevant auxiliary population information.     

Aggregate data meta-analysis with time-to-event outcomes

In a meta-analysis of randomized controlled trials with time-to-event outcomes, an aggregate data approach may be required for some or all included studies. Variation in the reporting of survival analyses in journals suggests that no single method for extracting the log(hazard ratio) estimate will suffice. Methods are described which improve upon a previously proposed method for estimating the log(HR) from survival curves. These methods extend to life-tables. In the situation where the treatment effect varies over time and the trials in the meta-analysis have different lengths of follow-up, heterogeneity may be evident. In order to assess whether the hazard ratio changes with time, several tests are proposed and compared. A cohort study comparing life expectancy of males and females with cerebral palsy and a systematic review of five trials comparing two anti-epileptic drugs, carbamazepine and sodium valproate, are used for illustration.     

Fixed-effect versus random-effect models for evaluating therapeutic preferences

A preference trial is a special form of cross-over trial where clinical conditions determine when patients change treatment, in a prescribed order. This leads to binary responses with variable lengths. In cross-over trials with normal responses, patient effect may be treated as either fixed or random. However, with binary responses, random- and fixed-effect assumptions may lead to very different conclusions, so that one is no longer an alternative to the other.     

Jointly modelling the relationship between survival and pulmonary function in cystic fibrosis patients

Modelling the relationship between pulmonary function and survival in cystic fibrosis (CF) is complicated by the fact that measures of pulmonary function commonly used such as the forced expiratory volume in one second (FEV(1)) are measured with error, and patients with the poorest lung function are increasingly censored by death, that is, data are available only for the patients who have survived to the current age. We assume a linear random effects model for FEV1 per cent predicted, where the random intercept and slope of FEV(1) per cent predicted, along with a specified transformation of the age at death follow a trivariate normal distribution. We illustrate how this model can be used to describe the relationship between age at death and parameters of the individual patient's regression of FEV(1) per cent predicted versus age, such as the slope and the intercept or true value of FEV(1) per cent predicted at a given age. We also illustrate how the model provides empirical Bayes estimates of these individual parameters. In particular, we explore how the predicted value of the age at death might be used as a prognostic or severity index. The model and methods are illustrated on a cohort of 188 cystic fibrosis patients with a common genotype (homozygous for the DeltaF508 mutation), born on or after 1965 and followed at the CF Center at the Rainbow Babies and Children's Hospital, Cleveland, OH, U.S.A.     

Analysis of censored and incomplete survival data using flowgraph models

Multi-state stochastic models are widely used to model stages of disease progression in survival analysis. This paper develops flowgraph models for data analysis in survival analysis. We illustrate these methods using data from a study of diabetic retinopathy consisting of 277 subjects with insulin-dependent (type I) diabetes mellitus (IDDM). These data were collected at the Eye-Kidney Clinic of the Barbara Davis Center for Childhood Diabetes at the University of Colorado Health Sciences Center.