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Doug Altman was a visionary leader and one of the most influential medical statisticians of the last 40 years. Based on a presentation in the "Invited session in memory of Doug Altman" at the 40th Annual Conference of the International Society for Clinical Biostatistics (ISCB) in Leuven, Belgium and our long‐standing collaborations with Doug, we discuss his contributions to regression modeling, reporting, prognosis research, as well as some more general issues while acknowledging that we cannot cover the whole spectrum of Doug's considerable methodological output. His statement "To maximize the benefit to society, you need to not just do research but do it well" should be a driver for all researchers. To improve current and future research, we aim to summarize Doug's messages for these three topics. [ABSTRACT FROM AUTHOR]

Non-linear exposure-outcome relationships such as between body mass index (BMI) and mortality are common. They are best explored as continuous functions using individual participant data from multiple studies. We explore two two-stage methods for meta-analysis of such relationships, where the confounder-adjusted relationship is first estimated in a non-linear regression model in each study, then combined across studies. The "metacurve" approach combines the estimated curves using multiple meta-analyses of the relative effect between a given exposure level and a reference level. The "mvmeta" approach combines the estimated model parameters in a single multivariate meta-analysis. Both methods allow the exposure-outcome relationship to differ across studies. Using theoretical arguments, we show that the methods differ most when covariate distributions differ across studies; using simulated data, we show that mvmeta gains precision but metacurve is more robust to model mis-specification. We then compare the two methods using data from the Emerging Risk Factors Collaboration on BMI, coronary heart disease events, and all-cause mortality (>80 cohorts, >18 000 events). For each outcome, we model BMI using fractional polynomials of degree 2 in each study, with adjustment for confounders. For metacurve, the powers defining the fractional polynomials may be study-specific or common across studies. For coronary heart disease, metacurve with common powers and mvmeta correctly identify a small increase in risk in the lowest levels of BMI, but metacurve with study-specific powers does not. For all-cause mortality, all methods identify a steep U-shape. The metacurve and mvmeta methods perform well in combining complex exposure-disease relationships across studies. [ABSTRACT FROM AUTHOR]

In epidemiology and clinical research, predictors often take value zero for a large amount of observations while the distribution of the remaining observations is continuous. These predictors are called variables with a spike at zero. Examples include smoking or alcohol consumption. Recently, an extension of the fractional polynomial (FP) procedure, a technique for modeling nonlinear relationships, was proposed to deal with such situations. To indicate whether or not a value is zero, a binary variable is added to the model. In a two stage procedure, called FP-spike, the necessity of the binary variable and/or the continuous FP function for the positive part are assessed for a suitable fit. In univariate analyses, the FP-spike procedure usually leads to functional relationships that are easy to interpret. This paper introduces four approaches for dealing with two variables with a spike at zero (SAZ). The methods depend on the bivariate distribution of zero and nonzero values. Bi-Sep is the simplest of the four bivariate approaches. It uses the univariate FP-spike procedure separately for the two SAZ variables. In Bi-D3, Bi-D1, and Bi-Sub, proportions of zeros in both variables are considered simultaneously in the binary indicators. Therefore, these strategies can account for correlated variables. The methods can be used for arbitrary distributions of the covariates. For illustration and comparison of results, data from a case-control study on laryngeal cancer, with smoking and alcohol intake as two SAZ variables, is considered. In addition, a possible extension to three or more SAZ variables is outlined. A combination of log-linear models for the analysis of the correlation in combination with the bivariate approaches is proposed. [ABSTRACT FROM AUTHOR]

In many areas of sciencewhere empirical data are analyzed, a task is often to identify important variables with influence on an outcome.Most often this is done by using a variable selection strategy in the context of a multivariable regression model. Using a study on ozone effects in children (n = 496, 24 covariates), we will discuss aspects relevant for deriving a suitable model. With an emphasis on model stability, we will explore and illustrate differences between predictive models and explanatory models, the key role of stopping criteria, and the value of bootstrap resampling (with and without replacement). Bootstrap resampling will be used to assess variable selection stability, to derive a predictor that incorporates model uncertainty, check for influential points, and visualize the variable selection process. For the latter two tasks we adapt and extend recent approaches, such as stability paths, to serve our purposes. Based on earlier experiences and on results from the example, we will argue for simpler models and that predictions are usually very similar, irrespective of the selection method used. Important differences exist for the corresponding variances, and the model uncertainty concept helps to protect against serious underestimation of the variance of a predictor-derived data dependently. Results of stability investigations illustrate severe difficulties in the task of deriving a suitable explanatory model. It seems possible to identify a small number of variables with an important and probably true influence on the outcome, but too often several variables are included whose selection may be a result of chance or may depend on a small number of observations. [ABSTRACT FROM AUTHOR]

In biomedical literature, numerous prediction models for clinical outcomes have been developed based either on clinical data or, more recently, on high-throughput molecular data (omics data). Prediction models based on both types of data, however, are less common, although some recent studies suggest that a suitable combination of clinical and molecular information may lead to models with better predictive abilities. This is probably due to the fact that it is not straightforward to combine data with different characteristics and dimensions (poorly characterized high-dimensional omics data, well-investigated low-dimensional clinical data). In this paper, we analyze two publicly available datasets related to breast cancer and neuroblastoma, respectively, in order to show some possible ways to combine clinical and omics data into a prediction model of time-to-event outcome. Different strategies and statistical methods are exploited. The results are compared and discussed according to different criteria, including the discriminative ability of the models, computed on a validation dataset. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The validity and practical utility of observational medical research depends critically on good study design, excellent data quality, appropriate statistical methods and accurate interpretation of results. Statistical methodology has seen substantial development in recent times. Unfortunately, many of these methodological developments are ignored in practice. Consequently, design and analysis of observational studies often exhibit serious weaknesses. The lack of guidance on vital practical issues discourages many applied researchers from using more sophisticated and possibly more appropriate methods when analyzing observational studies. Furthermore, many analyses are conducted by researchers with a relatively weak statistical background and limited experience in using statistical methodology and software. Consequently, even 'standard' analyses reported in the medical literature are often flawed, casting doubt on their results and conclusions. An efficient way to help researchers to keep up with recent methodological developments is to develop guidance documents that are spread to the research community at large. These observations led to the initiation of the strengthening analytical thinking for observational studies (STRATOS) initiative, a large collaboration of experts in many different areas of biostatistical research. The objective of STRATOS is to provide accessible and accurate guidance in the design and analysis of observational studies. The guidance is intended for applied statisticians and other data analysts with varying levels of statistical education, experience and interests. In this article, we introduce the STRATOS initiative and its main aims, present the need for guidance documents and outline the planned approach and progress so far. We encourage other biostatisticians to become involved. © 2014 The Authors. Statistics in Medicine published by John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In a large simulation study reported in a companion paper, we investigated the significance levels of 21 methods for investigating interactions between binary treatment and a continuous covariate in a randomised controlled trial. Several of the methods were shown to have inflated type 1 errors. In the present paper, we report the second part of the simulation study in which we investigated the power of the interaction procedures for two sample sizes and with two distributions of the covariate (well and badly behaved). We studied several methods involving categorisation and others in which the covariate was kept continuous, including fractional polynomials and splines. We believe that the results provide sufficient evidence to recommend the multivariable fractional polynomial interaction procedure as a suitable approach to investigate interactions of treatment with a continuous variable. If subject-matter knowledge gives good arguments for a non-monotone treatment effect function, we propose to use a second-degree fractional polynomial approach, but otherwise a first-degree fractional polynomial (FP1) function with added flexibility (FLEX3) is the method of choice. The FP1 class includes the linear function, and the selected functions are simple, understandable, and transferable. Furthermore, software is available. We caution that investigation of interactions in one dataset can only be interpreted in a hypothesis-generating sense and needs validation in new data. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Interactions between treatments and covariates in RCTs are a key topic. Standard methods for modelling treatment-covariate interactions with continuous covariates are categorisation or linear functions. Both approaches are easily criticised, but for different reasons. Multivariable fractional polynomial interactions, an approach based on fractional polynomials with the linear interaction model as the simplest special case, was proposed. Four variants of multivariable fractional polynomial interaction (FLEX1-FLEX4), allowing varying flexibility in functional form, were suggested. However, their properties are unknown, and comparisons with other procedures are unavailable. Additionally, we consider various methods based on categorisation and on cubic regression splines. We present the results of a simulation study to determine the significance level (probability of a type 1 error) of various tests for interaction between a binary covariate ('treatment effect') and a continuous covariate in univariate analysis. We consider a simplified setting in which the response variable is conditionally normally distributed, given the continuous covariate. We consider two main cases with the covariate distribution well behaved (approximately symmetric) or badly behaved (positively skewed). We construct nine scenarios with different functional forms for the main effect. In the well-behaved case, significance levels are in general acceptably close to nominal and are slightly better for the larger sample size ( n = 250 and 500 were investigated). In the badly behaved case, departures from nominal are more pronounced for several approaches. For a final assessment of these results and recommendations for practice, a study of power is needed. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In observational studies, many continuous or categorical covariates may be related to an outcome. Various spline-based procedures or the multivariable fractional polynomial (MFP) procedure can be used to identify important variables and functional forms for continuous covariates. This is the main aim of an explanatory model, as opposed to a model only for prediction. The type of analysis often guides the complexity of the final model. Spline-based procedures and MFP have tuning parameters for choosing the required complexity. To compare model selection approaches, we perform a simulation study in the linear regression context based on a data structure intended to reflect realistic biomedical data. We vary the sample size, variance explained and complexity parameters for model selection. We consider 15 variables. A sample size of 200 (1000) and R2 = 0.2 (0.8) is the scenario with the smallest (largest) amount of information. For assessing performance, we consider prediction error, correct and incorrect inclusion of covariates, qualitative measures for judging selected functional forms and further novel criteria. From limited information, a suitable explanatory model cannot be obtained. Prediction performance from all types of models is similar. With a medium amount of information, MFP performs better than splines on several criteria. MFP better recovers simpler functions, whereas splines better recover more complex functions. For a large amount of information and no local structure, MFP and the spline procedures often select similar explanatory models. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In epidemiology and in clinical research, risk factors often have special distributions. A common situation is that a proportion of individuals have exposure zero, and among those exposed, we have some continuous distribution. We call this a 'spike at zero'. Examples for this are smoking, duration of breastfeeding, or alcohol consumption. Furthermore, the empirical distribution of laboratory values and other measurements may have a semi-continuous distribution as a result of the lower detection limit of the measurement. To model the dose-response function, an extension of the fractional polynomial approach was recently proposed. In this paper, we suggest a modification of the previously suggested FP procedure. We first give the theoretical justification of this modified procedure by investigating relevant distribution classes. Here, we systematically derive the theoretical shapes of dose-response curves under given distributional assumptions (normal, log normal, gamma) in the framework of a logistic regression model. Further, we check the performance of the procedure in a simulation study and compare it to the previously suggested method, and finally we illustrate the procedures with data from a case-control study on breast cancer. [ABSTRACT FROM AUTHOR]

When several studies are available, a meta-analytic assessment of the effect of a risk or prognostic factor on an outcome is often required. We propose a new strategy, requiring individual participant data, to provide a summary estimate of the functional relationship between a continuous covariate and the outcome in a regression model, adjusting for confounding factors. Our procedure comprises three steps. First, we determine a confounder model. Ideally, the latter should include the same variables across studies, but this may be impossible. Next, we estimate the functional form for the continuous variable of interest in each study, adjusted for the confounder model. Finally, we combine the individual functions by weighted averaging to obtain a summary estimate of the function. Fractional polynomial methodology and pointwise weighted averaging of functions are the key components. In contrast to a pooled analysis, our approach can reflect more variability between functions from different studies and more flexibility with respect to confounders. We illustrate the procedure by using data from breast cancer patients in the Surveillance, Epidemiology, and End Results Program database, where we consider data from nine individual registries as separate studies. We estimate the functional forms for the number of positive lymph nodes and age. The former is an example where a strong prognostic effect has long been recognized, whereas the prognostic effect of the latter is weak or even controversial. We further discuss some general issues that are found in meta-analyses of observational studies. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Extreme values in predictors often strongly affect the results of statistical analyses in high-dimensional settings. Although they frequently occur with most high-throughput techniques, the problem is often ignored in the literature. We suggest to use a very simple transformation, proposed before in a different context by Royston and Sauerbrei, as an intermediary step between array preprocessing and high-level statistical analysis. This straightforward univariate transformation identifies extreme values in continuous features and can thus be used as a diagnostic tool for outliers. The use of the transformation and its effects is demonstrated for diverse univariate and multivariate statistical analyses using nine publicly available microarray data sets. [ABSTRACT FROM AUTHOR]

The focus of many medical applications is to model the impact of several factors on time to an event. A standard approach for such analyses is the Cox proportional hazards model. It assumes that the factors act linearly on the log hazard function (linearity assumption) and that their effects are constant over time (proportional hazards (PH) assumption). Variable selection is often required to specify a more parsimonious model aiming to include only variables with an influence on the outcome. As follow-up increases the effect of a variable often gets weaker, which means that it varies in time. However, spurious time-varying effects may also be introduced by mismodelling other parts of the multivariable model, such as omission of an important covariate or an incorrect functional form of a continuous covariate. These issues interact. To check whether the effect of a variable varies in time several tests for non-PH have been proposed. However, they are not sufficient to derive a model, as appropriate modelling of the shape of time-varying effects is required. In three examples we will compare five recently published strategies to assess whether and how the effects of covariates from a multivariable model vary in time. For practical use we will give some recommendations. [ABSTRACT FROM AUTHOR]

A common task in epidemiology is to estimate the dose-response function for a continuous exposure. Often a proportion of subjects is unexposed. Typical examples are cigarette consumption, alcohol intake, or occupational exposures. The question arises as to how to model such variables statistically. The fractional polynomial method of modelling continuous exposure variables is extended to allow for a proportion unexposed. A binary variable for the unexposed fraction is added to the model. In a two-stage procedure, we assess whether the binary variable and/or the continuous function for the exposed individuals is required for a good fit to the data. Extension to the multivariable situation is described. Three data sets with different characteristics are used as illustrations. The analyses of the three studies using the proposed procedure give differing results. In one example, only the binary variable seems to be required. In the other two examples, the binary variable and fractional polynomial functions of the exposure variable are needed. One function is monotonic and the other has a minimum. In the third example, adjusting for confounders has almost no effect on the function selected. In conclusion, the new procedure offers a worthwhile extension of dose-response modelling with an unexposed fraction. It is simple to carry out with standard software. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In developing regression models, data analysts are often faced with many predictor variables that may influence an outcome variable. After more than half a century of research, the 'best' way of selecting a multivariable model is still unresolved. It is generally agreed that subject matter knowledge, when available, should guide model building. However, such knowledge is often limited, and data-dependent model building is required. We limit the scope of the modelling exercise to selecting important predictors and choosing interpretable and transportable functions for continuous predictors. Assuming linear functions, stepwise selection and all-subset strategies are discussed; the key tuning parameters are the nominal P-value for testing a variable for inclusion and the penalty for model complexity, respectively. We argue that stepwise procedures perform better than a literature-based assessment would suggest. Concerning selection of functional form for continuous predictors, the principal competitors are fractional polynomial functions and various types of spline techniques. We note that a rigorous selection strategy known as multivariable fractional polynomials (MFP) has been developed. No spline-based procedure for simultaneously selecting variables and functional forms has found wide acceptance. Results of FP and spline modelling are compared in two data sets. It is shown that spline modelling, while extremely flexible, can generate fitted curves with uninterpretable 'wiggles', particularly when automatic methods for choosing the smoothness are employed. We give general recommendations to practitioners for carrying out variable and function selection. While acknowledging that further research is needed, we argue why MFP is our preferred approach for multivariable model building with continuous covariates. [ABSTRACT FROM AUTHOR]

Despite its importance in the theoretical literature, the bootstrap appears to play a negligible role in pharmaceutical research, as will be demonstrated by a brief literature review. As will be shown by examples, the bootstrap is a useful tool in the planning and analysis of clinical trials. The first example shows that some important information required in the design of a study can best be gained by using the bootstrap. It is argued from two further examples that more information can be extracted from large clinical trials by data-dependent modelling. This is shown by identifying a prognostic factor that may play a role as an inclusion criterion of a new study and by an interaction of a continuous predictor with treatment. To protect against erroneous conclusions from data-dependent modelling in a multivariable context, detailed checks of the results and stability analyses should be performed by the bootstrap. To conclude, a discussion of the future of modelling and the future of the bootstrap is given. [ABSTRACT FROM AUTHOR]

In medical research, continuous variables are often converted into categorical variables by grouping values into two or more categories. We consider in detail issues pertaining to creating just two groups, a common approach in clinical research. We argue that the simplicity achieved is gained at a cost; dichotomization may create rather than avoid problems, notably a considerable loss of power and residual confounding. In addition, the use of a data-derived 'optimal' cutpoint leads to serious bias. We illustrate the impact of dichotomization of continuous predictor variables using as a detailed case study a randomized trial in primary biliary cirrhosis. Dichotomization of continuous data is unnecessary for statistical analysis and in particular should not be applied to explanatory variables in regression models. [ABSTRACT FROM AUTHOR]

We consider modelling interaction between a categoric covariate T and a continuous covariate Z in a regression model. Here T represents the two treatment arms in a parallel-group clinical trial and Z is a prognostic factor which may influence response to treatment (known as a predictive factor). Generalization to more than two treatments is straightforward. The usual approach to analysis is to categorize Z into groups according to cutpoint(s) and to analyse the interaction in a model with main effects and multiplicative terms. The cutpoint approach raises several well-known and difficult issues for the analyst. We propose an alternative approach based on fractional polynomial (FP) modelling of Z in all patients and at each level of T. Other prognostic variables can also be incorporated by first constructing a multivariable adjustment model which may contain binary covariates and FP transformations of continuous covariates other than Z. The main step involves FP modelling of Z and testing equality of regression coefficients between treatment groups in an interaction model adjusted for other covariates. Extensive experience suggests that a two-term fractional polynomial (FP2) function may describe the effect of a prognostic factor on a survival outcome quite well. In a controlled trial, this FP2 function describes the prognostic effect averaged over the treatment groups. We refit this function in each treatment group to see if there are substantial differences between groups. Allowing different parameter values for the chosen FP2 function is flexible enough to detect such differences. Within the same algorithm we can also deal with the conceptually different cases of a predefined hypothesis of interaction or searching for interactions. We demonstrate the ability of the approach to detect and display treatment/covariate interactions in two examples from controlled trials in cancer. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

When investigating the effects of potential prognostic or risk factors that have been measured on a quantitative scale, values of these factors are often categorized into two groups. Sometimes an 'optimal' cutpoint is chosen that gives the best separation in terms of a two-sample test statistic. It is well known that this approach leads to a serious inflation of the type I error and to an overestimation of the effect of the prognostic or risk factor in absolute terms. In this paper, we illustrate that the resulting confidence intervals are similarly affected. We show that the application of a shrinkage procedure to correct for bias, together with bootstrap resampling for estimating the variance, yields confidence intervals for the effect of a potential prognostic or risk factor with the desired coverage. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Multivariable prognostic models are widely used in cancer and other disease areas, and have a range of applications in clinical medicine, clinical trials and allocation of health services resources. A well-founded and reliable measure of the prognostic ability of a model would be valuable to help define the separation between patients or prognostic groups that the model could provide, and to act as a benchmark of model performance in a validation setting. We propose such a measure for models of survival data. Its motivation derives originally from the idea of separation between Kaplan-Meier curves. We define the criteria for a successful measure and discuss them with respect to our approach. Adjustments for 'optimism', the tendency for a model to predict better on the data on which it was derived than on new data, are suggested. We study the properties of the measure by simulation and by example in three substantial data sets. We believe that our new measure will prove useful as a tool to evaluate the separation available-with a prognostic model. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

A letter to the editor is presented in response to the article "Performance of Using Multiple Stepwise Algorithms for Variable Selection," by R. E. Wiegard in the 2010 issue.