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PubMed1997

Linearly divergent treatment effects in clinical trials with repeated measures: efficient analysis using summary statistics.

In many randomized clinical trials with repeated measures of a response variable one anticipates a linear divergence over time in the difference between treatments. This paper explores how to make an efficient choice of analysis based on individual patient summary statistics. With the objective of estimating the mean rate of treatment divergence the simplest choice of summary statistic is the regression coefficient of response on time for each subject (SLOPE). The gains in statistical efficiency imposed by adjusting for the observed pre-treatment levels, or even better the estimated intercepts, are clarified. In the process, we develop the optimal linear summary statistic for any repeated measures design with assumed known covariance structure and shape of true mean treatment difference over time. Statistical power considerations are explored and an example from an asthma trial is used to illustrate the main points.

Analysis of Variance↗
PubMed1997

Variation in baseline risk as an explanation of heterogeneity in meta-analysis.

A relationship of baseline risk to treatment effect size has been suggested as a possible explanation of between-study heterogeneity in meta-analyses. To address this question, we develop regression models to examine the relationship between the logits (or other response measure) in the intervention and control groups. A weighted least squares (WLS) approach is described that allows for the heterogeneous sampling variation in the two groups, together with a correction of the coefficients for sampling error. Two approximate maximum likelihood (ML) solutions are also obtained, with or without an assumption of equal variances between groups within studies. A closed form ML solution exists with the assumption of equal variances. Both methods appear preferable to a previously suggested regression model of the log odds ratio on the control event rate; the methods proposed here use the same scale of measurement for both study groups, and eliminate an artifactual correlation in the regression error structure. The ML approach may be preferable because of its symmetric treatment of study groups, but WLS is more easily implemented with standard software. The methods are illustrated with data from meta-analyses on pre-term delivery and on therapies to lower serum cholesterol.

Anticholesteremic Agents↗
PubMed1997

A prospectively planned cumulative meta-analysis applied to a series of concurrent clinical trials.

Sequential designs are now a familiar part of clinical trial methodology. In particular, the triangular test has been used in several individual studies. Methods of combining studies are also well-known from the literature on meta-analysis. However, the combination of the two approaches is new. Consider the situation where a series of studies is to be conducted, following broadly similar protocols comparing a new treatment with a control treatment. In order to obtain an answer as quickly as possible to an efficacy or safety question it may be desirable to perform a cumulative meta-analysis on one particular variable. This could, for example, be the primary efficacy variable, an expensive assessment conducted in only a subgroup of patients, or a serious side-effect. To allow for the size of the treatment difference varying from study to study we might wish to provide a global estimate. Hence a random effects combined analysis, within a sequential framework, would appear to be appropriate. A methodology which utilizes efficient score statistics and Fisher's information is presented. Simulations show that the proposed methodology will achieve the specified error probabilities with reasonable accuracy provided that any random effect is relatively small. Ignoring random effects when they are present can lead to inaccuracies. A simulated example illustrates a number of practical issues.

Clinical Trials as Topic↗
PubMedDate not supplied

Some controversies in planning and analysing multi-centre trials.

It is shown that a rational approach to planning multi-centre trials will lead to an unequal distribution of patients across centres. Consequently different approaches to estimation will yield different estimates. However, some such approaches are not reasonable and it is concluded that multi-centre trials are less problematic than is commonly supposed.

Bias↗
PubMedDate not supplied

A comparison of various estimators of a treatment difference for a multi-centre clinical trial.

When a clinical trial is conducted at more than one centre it is likely that the true treatment effect will not be identical at each centre. In other words there will be some degree of treatment-by-centre interaction. A number of alternative approaches for dealing with this have been suggested in the literature. These include frequentist approaches with a fixed or random effects model for the observed data and Bayesian approaches. In the fixed effects model, there are two common competing estimators of the treatment difference, based on weighted or unweighted estimates from individual centres. Which one of these should be used is the subject of some controversy and we do not intend to take a particular methodological position in this paper. Our intention is to provide some insight into the relative merits of the indicated range of possible estimators of the treatment effect. For the fixed effects model, we also look at the merits of using a preliminary test for interaction assuming a 10 per cent significance level for the test. In order to make comparisons we have simulated a 'typical' trial which compares an active drug with a placebo in the treatment of hypertension, using systolic blood pressure as the primary variable. As well as allowing the treatment effect to vary between centres, we have concentrated on the particular case where one centre is out of line with the others in terms of its true treatment difference. The various estimators that result from the different approaches are compared in terms of mean squared error and power to reject the null hypothesis of no treatment difference. Overall, the approach that uses the fixed effects weighted estimator of overall treatment difference is recommended as one that has much to offer.

Analysis of Variance↗
PubMedDate not supplied

Multi-centre trial analysis revisited.

Analyses of multi-centre trials must consider the effects of the individual centres and the possibility of non-constancy of treatment effect differences among centres. This usually means an ANOVA with terms for centres, treatments, and centre x treatment interactions in practice, at least in the U.S.A. Empirical and conventional Bayes methods provide attractive alternatives to conventional ANOVAs for analysing and reporting the findings from multi-centre trials and do not require more restrictive assumptions than the ANOVA approach. These approaches require regarding the centre effects as random instead of fixed, a view which often will reasonably describe outcomes of clinical trials in spite of the fact that the individual centres certainly do not comprise a random sample of all possible centres. The components of these approaches are well understood and have been employed in related applications such as meta-analysis. Combining them in a way that makes their application to routine multi-centre trial analysis relatively straightforward does not appear to have been described previously, and is what forms the topic of this paper. The empirical Bayes approach leads to useful graphical displays, including one with the data superimposed on probability contours of the joint distribution of the individual centre means and standard deviations, which provides a handy way to identify possible outliers. Covariates can be incorporated without difficulty. The Bayes approach, implemented with Gibbs sampling, provides a convenient way to construct posterior and predictive distributions for a variety of useful statistics. We compare the result of empirical and conventional Bayes analyses with the result of fixed and mixed model ANOVAs applied to data from a multi-centre trial.

5-alpha Reductase Inhibitors↗
PubMedDate not supplied

Providing evidence of efficacy for a new drug.

There are many issues to consider when designing an efficacy package for drug registration. Generally in Europe and the United States, two or more confirmatory trials demonstrating efficacy (p < 0.025, one-tailed) of the test treatment versus a suitable control group must be conducted with a priori definition of a primary efficacy endpoint. Exceptions are possible, and there is always extensive discussion whenever less is proposed or more is required. Every aspect of the basic requirement can be questioned: number of trials; choice of control groups; selection of primary efficacy variables(s); levels of significance; one-tailed versus two-tailed test. These issues will be discussed, and justification is given when proposals are made for deviations from standard practice. Differences between Europe and the U.S. are discussed for certain disease entities. Because the assessment of the weight of evidence in favour of a drug effect is difficult to quantitate, if not impossible, no definitive guidance can be given that is suitable for all circumstances and countries.

Bias↗
PubMedDate not supplied

Issues for covariance analysis of dichotomous and ordered categorical data from randomized clinical trials and non-parametric strategies for addressing them.

Analysis of covariance is an effective method for addressing two considerations for randomized clinical trials. One is reduction of variance for estimates of treatment effects and thereby the production of narrower confidence intervals and more powerful statistical tests. The other is the clarification of the magnitude of treatment effects through adjustment of corresponding estimates for any random imbalances between the treatment groups with respect to the covariables. The statistical basis of covariance analysis can be either non-parametric, with reliance only on the randomization in the study design, or parametric through a statistical model for a postulated sampling process. For non-parametric methods, there are no formal assumptions for how a response variable is related to the covariables, but strong correlation between response and covariables is necessary for variance reduction. Computations for these methods are straightforward through the application of weighted least squares to fit linear models to the differences between treatment groups for the means of the response variable and the covariables jointly with a specification that has null values for the differences that correspond to the covariables. Moreover, such analysis is similarly applicable to dichotomous indicators, ranks or integers for ordered categories, and continuous measurements. Since non-parametric covariance analysis can have many forms, the ones which are planned for a clinical trial need careful specification in its protocol. A limitation of non-parametric analysis is that it does not directly address the magnitude of treatment effects within subgroups based on the covariables or the homogeneity of such effects. For this purpose, a statistical model is needed. When the response criterion is dichotomous or has ordered categories, such a model may have a non-linear nature which determines how covariance adjustment modifies results for treatment effects. Insight concerning such modifications can be gained through their evaluation relative to non-parametric counterparts. Such evaluation usually indicates that alternative ways to compare treatments for a response criterion with adjustment for a set of covariables mutually support the same conclusion about the strength of treatment effects. This robustness is noteworthy since the alternative methods for covariance analysis have substantially different rationales and assumptions. Since findings can differ in important ways across alternative choices for covariables (as opposed to methods for covariance adjustment), the critical consideration for studies with covariance analyses planned as the primary method for comparing treatments is the specification of the covariables in the protocol (or in an amendment or formal plan prior to any unmasking of the study.

Analysis of Variance↗
PubMed1999

Estimation of the average correlation coefficient for stratified bivariate data.

If the relationship between two ordered categorical variables X and Y is influenced by a third categorical variable with K levels, the Cochran-Mantel-Haenszel (CMH) correlation statistic QC is a useful stratum-adjusted summary statistic for testing the null hypothesis of no association between X and Y. Although motivated by and developed for the case of K I x J contingency tables, the correlation statistic QC is also applicable when X and Y are continuous variables. In this paper we derive a corresponding estimator of the average correlation coefficient for K I x J tables. We also study two estimates of the variance of the average correlation coefficient. The first is a restricted variance based on the variances of the observed cell frequencies under the null hypothesis of no association. The second is an unrestricted variance based on an asymptotic variance derived by Brown and Benedetti. The estimator of the average correlation coefficient works well in tables with balanced and unbalanced margins, for equal and unequal stratum-specific sample sizes, when correlation coefficients are constant over strata, and when correlation coefficients vary across strata. When the correlation coefficients are zero, close to zero, or the cell frequencies are small, the confidence intervals based on the restricted variance are preferred. For larger correlations and larger cell frequencies, the unrestricted confidence intervals give superior performance. We also apply the CMH statistic and proposed estimators to continuous non-normal data sampled from bivariate gamma distributions. We compare our methods to statistics for data sampled from normal distributions. The size and power of the CMH and normal theory statistics are comparable. When the stratum-specific sample sizes are small and the distributions are skewed, the proposed estimator is superior to the normal theory estimator. When the correlation coefficient is zero or close to zero, the restricted confidence intervals provide the best performance. None of the confidence intervals studied provides acceptable performances across all correlation coefficients, sample sizes and non-normal distributions.

Aged↗
PubMed1999

Combining classification trees using MLE.

We propose a probability distribution for an equivalence class of classification trees (that is, those that ignore the value of the cutpoints but retain tree structure). This distribution is parameterized by a central tree structure representing the true model, and a precision or concentration coefficient representing the variability around the central tree. We use this distribution to model an observed set of classification trees exhibiting variability in tree structure. We propose the maximum likelihood estimate of the central tree as the best tree to represent the set. This MLE retains the interpretability of a single tree model and has excellent generalizability. We implement an ascent search for the MLE tree structure using a data set of 13 classification trees that predict the presence or absence of cancer based on immune system parameters.

Classification↗
PubMed1999

Exact test size and power of a Gaussian error linear model for an internal pilot study.

Wittes and Brittain recommended using an 'internal pilot study' to adjust sample size. The approach involves five steps in testing a general linear hypothesis for a general linear univariate model, with Gaussian errors. First, specify the design, hypothesis, desired test size, power, a smallest 'clinically meaningful' effect, and a speculated error variance. Second, conduct a power analysis to choose provisionally a planned sample size. Third, collect a specified proportion of the planned sample as the internal pilot sample, and estimate the variance (but do not test the hypothesis). Fourth, update the power analysis with the variance estimate to adjust the total sample size. Fifth, finish the study and test the hypothesis with all data. We describe methods for computing exact test size and power under this scenario. Our analytic results agree with simulations of Wittes and Brittain. Furthermore, our exact results apply to any general linear univariate model with fixed predictors, which is much more general than the two-sample t-test considered by Wittes and Brittain. In addition, our results allow for examination of the impact on test size of internal pilot studies for more complicated designs in the framework of the general linear model. We examine the impact of (i) small samples, (ii) allowing the planned sample size to decrease, (iii) the choice of internal pilot sample size, and (iv) the maximum allowable size of the second sample. All affect test size, power and expected total sample size. We present a number of examples including one that uses an internal pilot study in a three-group analysis of variance.

Analysis of Variance↗
PubMed1999

A threshold causal model for clinical trials with departures from intended treatment.

Randomized clinical trials often are planned to study a specific intervention. However, the collection of data on treatment actually received often reveals variable levels of treatment exposure (or 'dose') across subjects, due to non-compliance or other reasons. This paper presents a new method, using such 'dose' data as well as control group responses, to assess a causal dose-response relationship. The specific model utilizes a threshold function and incorporates a random effect term to allow for heterogeneous treatment responses among subjects. Further modelling of the random effects allows for reduction of error variance and control for potential confounders. The threshold dose is estimated using a residual variance criterion based on a transformed model. Estimates of standard errors and confidence intervals are obtained using a bootstrap procedure. The method is applied to data from an AIDS clinical trial. A simulation study demonstrates the adequacy of the threshold estimates for particular sample sizes and error variances. The limitations of this essentially exploratory method, as well as some possible extensions, are discussed. Published in 1999 by John Wiley & Sons, Ltd. This article is a US Government Work and is in the public domain in the United States.

CD4 Lymphocyte Count↗
PubMed1998

Meta-analysis methodology for combining non-parametric sibpair linkage results: genetic homogeneity and identical markers.

Meta-analysis methodology is developed for combining sibpair linkage results across multiple studies employing different study designs, some employing quantitative traits (e.g., blood pressure) and some employing qualitative traits (e.g., clinical hypertension), under the assumption that the underlying (disease) trait loci are the same. Pooling results based on three commonly used sibpair methods is considered: the affected sibpair method for dichotomous traits and, for quantitative traits, the Haseman-Elston regression method and the Risch-Zhang extremely discordant sibpair method. The proportion of genes shared identical by descent (IBD) by a sibpair of certain trait outcomes is chosen as a common effect to be pooled across studies. Variation in the observed IBD proportions among individual studies is modeled using a random effects model. A heterogeneity test is provided to assess the variability among individual studies. When results from all three types of studies are available, we derive pooled estimates of IBD proportions both for sibpairs with extremely concordant trait values and for sibpairs with extremely discordant trait values, and construct a combined test of linkage based on the difference of the two estimates. Simulation studies demonstrate the need for and the advantage of meta-analysis of linkage results. We also present some guidelines for reporting linkage studies bearing potential future meta-analysis in mind.

Chromosome Mapping↗
PubMed2000

Age-related changes in initiation and maintenance of sleep: a meta-analysis.

The purpose of this meta-analysis was to determine the magnitude of change over the adult life span in four key sleep characteristics and to explore research design features that may account for variability in reported age-related sleep change. Forty-one published studies (combined N = 3293) provided 99 correlational effect sizes. Waking frequency and duration increased with age as previously concluded by narrative reviewers. Although narrative reviewers were less certain whether nighttime sleep amount or the ability to initiate sleep decreased with age, the meta-analysis suggested that both decreased. When sleep variables were measured by polysomnography rather than self-report, larger age-related changes were found. Few researchers who studied normal sleep controlled for important health moderators or studied women.

Adult↗
PubMedDate not supplied

Cluster effects and simultaneity in multilevel models.

For small group sizes, the GLS estimator in multilevel models is biased and inconsistent when the random cluster effects are correlated with the regressors. A fixed effects approach, conditioning on the cluster effects, provides consistent estimates for the slope parameters. The two estimators are equivalent when group sizes are large. The same results obtain for two-stage estimation procedures that allow for some of the regressors to be simultaneously determined with the dependent variable. The GLS and fixed effects estimators are applied to data on acute care hospital utilization in the UK, allowing for health authority district effects.

Effect Modifier, Epidemiologic↗
PubMed1998

Drinking patterns within households: the estimation and interpretation of individual and group variables.

Levels of alcohol consumption tend to be similar for individuals living in the same household. This may be because: (a) individuals with similar characteristics collect in households (correlated effects); (b) individuals in the same household are influenced by common factors (exogenous effects); and/or (c) the consumption levels of an individual directly influences the consumption levels of other individuals in the same household (endogenous effects). Whichever of these three possibilities is the principal reason underlying household clustering of consumption levels has important policy implications. In this paper we propose a testing strategy to distinguish between the three types of effect in a cross-sectional data-set. Allowing for exogenous or endogenous effects shows that the significant socio-economic gradient in a model containing only individual variables arises because of misspecification. However, because we find significant evidence of correlated effects, we cannot identify whether it is endogenous or exogenous effects which give rise to statistically significant group level variables. The results indicate the possible pitfalls of omitting group level influences.

Adult↗
PubMed2000

Heterogeneity and statistical significance in meta-analysis: an empirical study of 125 meta-analyses.

For meta-analysis, substantial uncertainty remains about the most appropriate statistical methods for combining the results of separate trials. An important issue for meta-analysis is how to incorporate heterogeneity, defined as variation among the results of individual trials beyond that expected from chance, into summary estimates of treatment effect. Another consideration is which 'metric' to use to measure treatment effect; for trials with binary outcomes, there are several possible metrics, including the odds ratio (a relative measure) and risk difference (an absolute measure). To examine empirically how assessment of treatment effect and heterogeneity may differ when different methods are utilized, we studied 125 meta-analyses representative of those performed by clinical investigators. There was no meta-analysis in which the summary risk difference and odds ratio were discrepant to the extent that one indicated significant benefit while the other indicated significant harm. Further, for most meta-analyses, summary odds ratios and risk differences agreed in statistical significance, leading to similar conclusions about whether treatments affected outcome. Heterogeneity was common regardless of whether treatment effects were measured by odds ratios or risk differences. However, risk differences usually displayed more heterogeneity than odds ratios. Random effects estimates, which incorporate heterogeneity, tended to be less precisely estimated than fixed effects estimates. We present two exceptions to these observations, which derive from the weights assigned to individual trial estimates. We discuss the implications of these findings for selection of a metric for meta-analysis and incorporation of heterogeneity into summary estimates. Published in 2000 by John Wiley & Sons, Ltd.

Bias↗
PubMed2000

Modelling covariance structure in the analysis of repeated measures data.

The term 'repeated measures' refers to data with multiple observations on the same sampling unit. In most cases, the multiple observations are taken over time, but they could be over space. It is usually plausible to assume that observations on the same unit are correlated. Hence, statistical analysis of repeated measures data must address the issue of covariation between measures on the same unit. Until recently, analysis techniques available in computer software only offered the user limited and inadequate choices. One choice was to ignore covariance structure and make invalid assumptions. Another was to avoid the covariance structure issue by analysing transformed data or making adjustments to otherwise inadequate analyses. Ignoring covariance structure may result in erroneous inference, and avoiding it may result in inefficient inference. Recently available mixed model methodology permits the covariance structure to be incorporated into the statistical model. The MIXED procedure of the SAS((R)) System provides a rich selection of covariance structures through the RANDOM and REPEATED statements. Modelling the covariance structure is a major hurdle in the use of PROC MIXED. However, once the covariance structure is modelled, inference about fixed effects proceeds essentially as when using PROC GLM. An example from the pharmaceutical industry is used to illustrate how to choose a covariance structure. The example also illustrates the effects of choice of covariance structure on tests and estimates of fixed effects. In many situations, estimates of linear combinations are invariant with respect to covariance structure, yet standard errors of the estimates may still depend on the covariance structure.

Analysis of Variance↗