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Statistical regression models are used for predicting outcomes based on the values of some predictor variables or for describing the association of an outcome with predictors. With a data set at hand, a regression model can be easily fit with standard software packages. This bears the risk that data analysts may rush to perform sophisticated analyses without sufficient knowledge of basic properties, associations in and errors of their data, leading to wrong interpretation and presentation of the modeling results that lacks clarity. Ignorance about special features of the data such as redundancies or particular distributions may even invalidate the chosen analysis strategy. Initial data analysis (IDA) is prerequisite to regression analyses as it provides knowledge about the data needed to confirm the appropriateness of or to refine a chosen model building strategy, to interpret the modeling results correctly, and to guide the presentation of modeling results. In order to facilitate reproducibility, IDA needs to be preplanned, an IDA plan should be included in the general statistical analysis plan of a research project, and results should be well documented. Biased statistical inference of the final regression model can be minimized if IDA abstains from evaluating associations of outcome and predictors, a key principle of IDA. We give advice on which aspects to consider in an IDA plan for data screening in the context of regression modeling to supplement the statistical analysis plan. We illustrate this IDA plan for data screening in an example of a typical diagnostic modeling project and give recommendations for data visualizations., (© 2024. The Author(s).)

Although regression models play a central role in the analysis of medical research projects, there still exist many misconceptions on various aspects of modeling leading to faulty analyses. Indeed, the rapidly developing statistical methodology and its recent advances in regression modeling do not seem to be adequately reflected in many medical publications. This problem of knowledge transfer from statistical research to application was identified by some medical journals, which have published series of statistical tutorials and (shorter) papers mainly addressing medical researchers. The aim of this review was to assess the current level of knowledge with regard to regression modeling contained in such statistical papers. We searched for target series by a request to international statistical experts. We identified 23 series including 57 topic-relevant articles. Within each article, two independent raters analyzed the content by investigating 44 predefined aspects on regression modeling. We assessed to what extent the aspects were explained and if examples, software advices, and recommendations for or against specific methods were given. Most series (21/23) included at least one article on multivariable regression. Logistic regression was the most frequently described regression type (19/23), followed by linear regression (18/23), Cox regression and survival models (12/23) and Poisson regression (3/23). Most general aspects on regression modeling, e.g. model assumptions, reporting and interpretation of regression results, were covered. We did not find many misconceptions or misleading recommendations, but we identified relevant gaps, in particular with respect to addressing nonlinear effects of continuous predictors, model specification and variable selection. Specific recommendations on software were rarely given. Statistical guidance should be developed for nonlinear effects, model specification and variable selection to better support medical researchers who perform or interpret regression analyses., Competing Interests: The authors have declared that no competing interests exist.

In the last decades, statistical methodology has developed rapidly, in particular in the field of regression modeling. Multivariable regression models are applied in almost all medical research projects. Therefore, the potential impact of statistical misconceptions within this field can be enormous Indeed, the current theoretical statistical knowledge is not always adequately transferred to the current practice in medical statistics. Some medical journals have identified this problem and published isolated statistical articles and even whole series thereof. In this systematic review, we aim to assess the current level of education on regression modeling that is provided to medical researchers via series of statistical articles published in medical journals. The present manuscript is a protocol for a systematic review that aims to assess which aspects of regression modeling are covered by statistical series published in medical journals that intend to train and guide applied medical researchers with limited statistical knowledge. Statistical paper series cannot easily be summarized and identified by common keywords in an electronic search engine like Scopus. We therefore identified series by a systematic request to statistical experts who are part or related to the STRATOS Initiative (STRengthening Analytical Thinking for Observational Studies). Within each identified article, two raters will independently check the content of the articles with respect to a predefined list of key aspects related to regression modeling. The content analysis of the topic-relevant articles will be performed using a predefined report form to assess the content as objectively as possible. Any disputes will be resolved by a third reviewer. Summary analyses will identify potential methodological gaps and misconceptions that may have an important impact on the quality of analyses in medical research. This review will thus provide a basis for future guidance papers and tutorials in the field of regression modeling which will enable medical researchers 1) to interpret publications in a correct way, 2) to perform basic statistical analyses in a correct way and 3) to identify situations when the help of a statistical expert is required., Competing Interests: The authors have declared that no competing interests exist.

Data integration, i.e. the use of different sources of information for data analysis, is becoming one of the most important topics in modern statistics. Especially in, but not limited to, biomedical applications, a relevant issue is the combination of low-dimensional (e.g. clinical data) and high-dimensional (e.g. molecular data such as gene expressions) data sources in a prediction model. Not only the different characteristics of the data, but also the complex correlation structure within and between the two data sources, pose challenging issues. In this paper, we investigate these issues via simulations, providing some useful insight into strategies to combine low- and high-dimensional data in a regression prediction model. In particular, we focus on the effect of the correlation structure on the results, while accounting for the influence of our specific choices in the design of the simulation study., (© The Author(s) 2019. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oup.com.)

Background: Omics data can be very informative in survival analysis and may improve the prognostic ability of classical models based on clinical risk factors for various diseases, for example breast cancer. Recent research has focused on integrating omics and clinical data, yet has often ignored the need for appropriate model building for clinical variables. Medical literature on classical prognostic scores, as well as biostatistical literature on appropriate model selection strategies for low dimensional (clinical) data, are often ignored in the context of omics research. The goal of this paper is to fill this methodological gap by investigating the added predictive value of gene expression data for models using varying amounts of clinical information., Methods: We analyze two data sets from the field of survival prognosis of breast cancer patients. First, we construct several proportional hazards prediction models using varying amounts of clinical information based on established medical knowledge. These models are then used as a starting point (i.e. included as a clinical offset) for identifying informative gene expression variables using resampling procedures and penalized regression approaches (model based boosting and the LASSO). In order to assess the added predictive value of the gene signatures, measures of prediction accuracy and separation are examined on a validation data set for the clinical models and the models that combine the two sources of information., Results: For one data set, we do not find any substantial added predictive value of the omics data when compared to clinical models. On the second data set, we identify a noticeable added predictive value, however only for scenarios where little or no clinical information is included in the modeling process. We find that including more clinical information can lead to a smaller number of selected omics predictors., Conclusions: New research using omics data should include all available established medical knowledge in order to allow an adequate evaluation of the added predictive value of omics data. Including all relevant clinical information in the analysis might also lead to more parsimonious models. The developed procedure to assess the predictive value of the omics data can be readily applied to other scenarios.

In most epidemiologic studies and in clinical research generally, there are variables with a spike at zero, namely variables for which a proportion of individuals have zero exposure (e.g., never smokers) and among those exposed the variable has a continuous distribution. Different options exist for modeling such variables, such as categorization where the nonexposed form the reference group, or ignoring the spike by including the variable in the regression model with or without some transformation or modeling procedures. It has been shown that such situations can be analyzed by adding a binary indicator (exposed/nonexposed) to the regression model, and a method based on fractional polynomials with which to estimate a suitable functional form for the positive portion of the spike-at-zero variable distribution has been developed. In this paper, we compare different approaches using data from 3 case-control studies carried out in Germany: the Mammary Carcinoma Risk Factor Investigation (MARIE), a breast cancer study conducted in 2002-2005 (Flesch-Janys et al., Int J Cancer. 2008;123(4):933-941); the Rhein-Neckar Larynx Study, a study of laryngeal cancer conducted in 1998-2000 (Dietz et al., Int J Cancer. 2004;108(6):907-911); and a lung cancer study conducted in 1988-1993 (Jöckel et al., Int J Epidemiol. 1998;27(4):549-560). Strengths and limitations of different procedures are demonstrated, and some recommendations for practical use are given., (© The Author 2017. Published by Oxford University Press on behalf of the Johns Hopkins Bloomberg School of Public Health. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com.)

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., (© 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.)

In recent years, increasing attention has been devoted to the problem of the stability of multivariable regression models, understood as the resistance of the model to small changes in the data on which it has been fitted. Resampling techniques, mainly based on the bootstrap, have been developed to address this issue. In particular, the approaches based on the idea of "inclusion frequency" consider the repeated implementation of a variable selection procedure, for example backward elimination, on several bootstrap samples. The analysis of the variables selected in each iteration provides useful information on the model stability and on the variables' importance. Recent findings, nevertheless, show possible pitfalls in the use of the bootstrap, and alternatives such as subsampling have begun to be taken into consideration in the literature. Using model selection frequencies and variable inclusion frequencies, we empirically compare these two different resampling techniques, investigating the effect of their use in selected classical model selection procedures for multivariable regression. We conduct our investigations by analyzing two real data examples and by performing a simulation study. Our results reveal some advantages in using a subsampling technique rather than the bootstrap in this context., (© 2015, The International Biometric Society.)

In many areas of science where 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., (© 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.)

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.)

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.)

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 R(2)  = 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.)

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., (© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.)

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.)

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.)

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., (Copyright (c) 2007 John Wiley & Sons, Ltd.)

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.

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.

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.)