A flexible parametric accelerated failure time model and the extension to time-dependent acceleration factors.

Accelerated failure time (AFT) models are used widely in medical research, though to a much lesser extent than proportional hazards models. In an AFT model, the effect of covariates act to accelerate or decelerate the time to event of interest, that is, shorten or extend the time to event. Commonly used parametric AFT models are limited in the underlying shapes that they can capture. In this article, we propose a general parametric AFT model, and in particular concentrate on using restricted cubic splines to model the baseline to provide substantial flexibility. We then extend the model to accommodate time-dependent acceleration factors. Delayed entry is also allowed, and hence, time-dependent covariates. We evaluate the proposed model through simulation, showing substantial improvements compared to standard parametric AFT models. We also show analytically and through simulations that the AFT models are collapsible, suggesting that this model class will be well suited to causal inference. We illustrate the methods with a data set of patients with breast cancer. Finally, we provide highly efficient, user-friendly Stata, and R software packages.

artbin: Extended sample size for randomized trials with binary outcomes.

We describe the command artbin, which offers various new facilities for the calculation of sample size for binary outcome variables that are not otherwise available in Stata. While artbin has been available since 2004, it has not been previously described in the Stata Journal. artbin has been recently updated to include new options for different statistical tests, methods and study designs, improved syntax, and better handling of noninferiority trials. In this article, we describe the updated version of artbin and detail the various formulas used within artbin in different settings.

artcat: Sample-size calculation for an ordered categorical outcome.

We describe a new command, artcat, that calculates sample size or power for a randomized controlled trial or similar experiment with an ordered categorical outcome, where analysis is by the proportional-odds model. artcat implements the method of Whitehead (1993, Statistics in Medicine 12: 2257-2271). We also propose and implement a new method that 1) allows the user to specify a treatment effect that does not obey the proportional-odds assumption, 2) offers greater accuracy for large treatment effects, and 3) allows for noninferiority trials. We illustrate the command and explore the value of an ordered categorical outcome over a binary outcome in various settings. We show by simulation that the methods perform well and that the new method is more accurate than Whitehead's method.

Investigating treatment-effect modification by a continuous covariate in IPD meta-analysis: an approach using fractional polynomials.

BACKGROUND: In clinical trials, there is considerable interest in investigating whether a treatment effect is similar in all patients, or that one or more prognostic variables indicate a differential response to treatment. To examine this, a continuous predictor is usually categorised into groups according to one or more cutpoints. Several weaknesses of categorization are well known. To avoid the disadvantages of cutpoints and to retain full information, it is preferable to keep continuous variables continuous in the analysis. To handle this issue, the Subpopulation Treatment Effect Pattern Plot (STEPP) was proposed about two decades ago, followed by the multivariable fractional polynomial interaction (MFPI) approach. Provided individual patient data (IPD) from several studies are available, it is possible to investigate for treatment heterogeneity with meta-analysis techniques. Meta-STEPP was recently proposed and in patients with primary breast cancer an interaction of estrogen receptors with chemotherapy was investigated in eight randomized controlled trials (RCTs). METHODS: We use data from eight randomized controlled trials in breast cancer to illustrate issues from two main tasks. The first task is to derive a treatment effect function (TEF), that is, a measure of the treatment effect on the continuous scale of the covariate in the individual studies. The second is to conduct a meta-analysis of the continuous TEFs from the eight studies by applying pointwise averaging to obtain a mean function. We denote the method metaTEF. To improve reporting of available data and all steps of the analysis we introduce a three-part profile called MethProf-MA. RESULTS: Although there are considerable differences between the studies (populations with large differences in prognosis, sample size, effective sample size, length of follow up, proportion of patients with very low estrogen receptor values) our results provide clear evidence of an interaction, irrespective of the choice of the FP function and random or fixed effect models. CONCLUSIONS: In contrast to cutpoint-based analyses, metaTEF retains the full information from continuous covariates and avoids several critical issues when performing IPD meta-analyses of continuous effect modifiers in randomised trials. Early experience suggests it is a promising approach. TRIAL REGISTRATION: Not applicable.

Doug Altman: Driving critical appraisal and improvements in the quality of methodological and medical research.

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.

Meta-analysis of non-linear exposure-outcome relationships using individual participant data: A comparison of two methods.

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.

Reference curves for the Australian/Canadian Hand Osteoarthritis Index in the middle-aged Dutch population.

Objective: The aim was to establish reference curves of the Australian/Canadian Hand Osteoarthritis Index (AUSCAN), a widely used questionnaire assessing hand complaints. Methods: Analyses were performed in a population-based sample, The Netherlands Epidemiology of Obesity study (n = 6671, aged 45-65 years). Factors associated with AUSCAN scores were analysed with ordered logistic regression, because AUSCAN data were zero inflated, dividing AUSCAN into three categories (0 vs 1-5 vs >5). Age- and sex-specific reference curves for the AUSCAN (range 0-60; higher is worse) were developed using quantile regression in conjunction with fractional polynomials. Observed scores in relevant subgroups were compared with the reference curves. Results: The median age was 56 [interquartile range (IQR): 50-61] years; 56% were women and 12% had hand OA according to ACR criteria. AUSCAN scores were low (median 1; IQR: 0-4). Reference curves where higher for women, and increased moderately with age: 95% percentiles for AUSCAN in men and women were, respectively, 5.0 and 12.3 points for a 45-year-old, and 15.2 and 33.6 points for a 65-year-old individual. Additional associated factors included hand OA, inflammatory rheumatic diseases, FM, socio-economic status and BMI. Median AUSCAN pain subscale scores of women with hand OA lay between the 75th and 90th centiles of the general population. Conclusion: AUSCAN scores in the middle-aged Dutch population were low overall, and higher in women than in men. AUSCAN reference curves could serve as a benchmark in research and clinical practice settings. However, the AUSCAN does not measure hand complaints specific for hand OA.

Testing many treatments within a single protocol over 10 years at MRC Clinical Trials Unit at UCL: Multi-arm, multi-stage platform, umbrella and basket protocols.

There is real need to change how we do some of our clinical trials, as currently the testing and development process is too slow, too costly and too failure-prone often we find that a new treatment is no better than the current standard. Much of the focus on the development and testing pathway has been in improving the design of phase I and II trials. In this article, we present examples of new methods for improving the design of phase III trials (and the necessary lead up to them) as they are the most time-consuming and expensive part of the pathway. Key to all these methods is the aim to test many treatments and/or pose many therapeutic questions within one protocol.

A combined test for a generalized treatment effect in clinical trials with a time-to-event outcome.

Most randomized controlled trials with a time-to-event outcome are designed and analyzed assuming proportional hazards of the treatment effect. The sample-size calculation is based on a log-rank test or the equivalent Cox test. Nonproportional hazards are seen increasingly in trials and are recognized as a potential threat to the power of the log-rank test. To address the issue, Royston and Parmar (2016, BMC Medical Research Methodology 16: 16) devised a new "combined test" of the global null hypothesis of identical survival curves in each trial arm. The test, which combines the conventional Cox test with a new formulation, is based on the maximal standardized difference in restricted mean survival time (rmst) between the arms. The test statistic is based on evaluations of rmst over several preselected time points. The combined test involves the minimum p-value across the Cox and rmst-based tests, appropriately standardized to have the correct null distribution. In this article, I outline the combined test and introduce a command, stctest, that implements the combined test. I point the way to additional tools currently under development for power and sample-size calculation for the combined test.

Model selection for univariable fractional polynomials.

Since Royston and Altman's 1994 publication (Journal of the Royal Statistical Society, Series C 43: 429-467), fractional polynomials have steadily gained popularity as a tool for flexible parametric modeling of regression relationships. In this article, I present fp_select, a postestimation tool for fp that allows the user to select a parsimonious fractional polynomial model according to a closed test procedure called the fractional polynomial selection procedure or function selection procedure. I also give a brief introduction to fractional polynomial models and provide examples of using fp and fp_select to select such models with real data.

Reconstructing time-to-event data from published Kaplan-Meier curves.

Hazard ratios can be approximated by data extracted from published Kaplan-Meier curves. Recently, this curve approach has been extended beyond hazard-ratio approximation with the capability of constructing time-to-event data at the individual level. In this article, we introduce a command, ipdfc, to implement the reconstruction method to convert Kaplan-Meier curves to time-to-event data. We give examples to illustrate how to use the command.

Augmenting the logrank test in the design of clinical trials in which non-proportional hazards of the treatment effect may be anticipated.

BACKGROUND: Most randomized controlled trials with a time-to-event outcome are designed assuming proportional hazards (PH) of the treatment effect. The sample size calculation is based on a logrank test. However, non-proportional hazards are increasingly common. At analysis, the estimated hazards ratio with a confidence interval is usually presented. The estimate is often obtained from a Cox PH model with treatment as a covariate. If non-proportional hazards are present, the logrank and equivalent Cox tests may lose power. To safeguard power, we previously suggested a 'joint test' combining the Cox test with a test of non-proportional hazards. Unfortunately, a larger sample size is needed to preserve power under PH. Here, we describe a novel test that unites the Cox test with a permutation test based on restricted mean survival time. METHODS: We propose a combined hypothesis test based on a permutation test of the difference in restricted mean survival time across time. The test involves the minimum of the Cox and permutation test P-values. We approximate its null distribution and correct it for correlation between the two P-values. Using extensive simulations, we assess the type 1 error and power of the combined test under several scenarios and compare with other tests. We investigate powering a trial using the combined test. RESULTS: The type 1 error of the combined test is close to nominal. Power under proportional hazards is slightly lower than for the Cox test. Enhanced power is available when the treatment difference shows an 'early effect', an initial separation of survival curves which diminishes over time. The power is reduced under a 'late effect', when little or no difference in survival curves is seen for an initial period and then a late separation occurs. We propose a method of powering a trial using the combined test. The 'insurance premium' offered by the combined test to safeguard power under non-PH represents about a single-digit percentage increase in sample size. CONCLUSIONS: The combined test increases trial power under an early treatment effect and protects power under other scenarios. Use of restricted mean survival time facilitates testing and displaying a generalized treatment effect.

mfpa: Extension of mfp using the ACD covariate transformation for enhanced parametric multivariable modeling.

In a recent article, Royston (2015, Stata Journal 15: 275-291) introduced the approximate cumulative distribution (acd) transformation of a continuous covariate x as a route toward modeling a sigmoid relationship between x and an outcome variable. In this article, we extend the approach to multivariable modeling by modifying the standard Stata program mfp. The result is a new program, mfpa, that has all the features of mfp plus the ability to fit a new model for user-selected covariates that we call fp1(p1, p2). The fp1(p1, p2) model comprises the best-fitting combination of a dimension-one fractional polynomial (fp1) function of x and an fp1 function of acd (x). We describe a new model-selection algorithm called function-selection procedure with acd transformation, which uses significance testing to attempt to simplify an fp1(p1, p2) model to a submodel, an fp1 or linear model in x or in acd (x). The function-selection procedure with acd transformation is related in concept to the fsp (fp function-selection procedure), which is an integral part of mfp and which is used to simplify a dimension-two (fp2) function. We describe the mfpa command and give univariable and multivariable examples with real data to demonstrate its use.

Reference intervals of spinal mobility measures in normal individuals: the MOBILITY study.

OBJECTIVES: To establish reference intervals (RIs) for spinal mobility measures as recommended for patients with axial spondyloarthritis, and to determine the effect of age, height and gender on spinal mobility, in normal individuals. METHODS: A cross-sectional study (MOBILITY) was conducted among normal individuals aged 20-69 years. Recruitment was stratified by gender, age (10-year categories) and height (10 cm categories). Eleven spinal mobility measures were assessed. Age specific RIs and percentiles were derived for each measure. RESULTS: 393 volunteers were included. All spinal mobility measures decreased with increasing age. Therefore, age specific RIs were developed. The 95% RIs (2.5th and 97.5th percentiles), as well as the 5th, 10th, 25th, 50th, 75th and 90th percentiles for each spinal mobility measure and different ages are presented. Mobility percentile curves were also plotted for each of the measures. For instance, the 95% RI for lateral spinal flexion was 16.2-28.0 cm for a 25-year-old subject, 13.2-25.0 cm for a 45-year-old subject and 10.1-21.9 cm for a 65-year-old subject. After adjustment for age, there was no need for gender specific RIs, while RIs of some measures are height-adjusted. CONCLUSIONS: Age specific RIs and percentiles were derived for each of the spinal mobility measures for normal individuals. These may guide clinicians when assessing the mobility of patients with axial spondyloarthritis. The RIs may serve as cut-off levels for 'normal' versus 'abnormal', whereas the mobility percentile curves may be used to assess the level of mobility of patients with axial spondyloarthritis.

Prognostic survival model for people diagnosed with invasive cutaneous melanoma.

BACKGROUND: The ability of medical practitioners to communicate risk estimates effectively to patients diagnosed with melanoma relies on accurate information about prognostic factors and their impact on survival. This study reports the development of one of the few melanoma prognostic models, called the Melanoma Severity Index (MSI), based on population-based cancer registry data. METHODS: Data from the Queensland Cancer Registry for people (20-89 years) diagnosed with a single invasive melanoma between 1995 and 2008 (n = 28,654; 1,700 melanoma deaths). Additional clinical information about metastasis, ulceration and positive lymph nodes was manually extracted from pathology forms. Flexible parametric survival models were combined with multivariable fractional polynomial for selecting variables and transformations of continuous variables. Multiple imputation was used for missing covariate values. RESULTS: The MSI contained the variables thickness (transformed, explained 40.6% of variation in survival), body site (additional 1.9% in variation), metastasis (1.8%), positive nodes (0.7%), ulceration (1.3%), age (1.1%). Royston and Sauerbrei's D statistic (measure of discrimination) was 1.50 (95% CI = 1.44, 1.56) and the corresponding RD2 (measure of explained variation) was 0.47 (0.45, 0.49), demonstrating strong explanatory performance. The Harrell-C statistic was 0.88 (0.88, 0.89). Lacking an external validation dataset, we applied internal-external cross validation to demonstrate the consistency of the prognostic information across geographically-defined subsets of the cohort. CONCLUSIONS: The MSI provides good ability to predict survival for melanoma patients. Beyond the immediate clinical use, the MSI may have important public health and research applications for evaluations of public health interventions aimed at reducing deaths from melanoma.

Meta-analysis of time-to-event outcomes from randomized trials using restricted mean survival time: application to individual participant data.

Meta-analysis of time-to-event outcomes using the hazard ratio as a treatment effect measure has an underlying assumption that hazards are proportional. The between-arm difference in the restricted mean survival time is a measure that avoids this assumption and allows the treatment effect to vary with time. We describe and evaluate meta-analysis based on the restricted mean survival time for dealing with non-proportional hazards and present a diagnostic method for the overall proportional hazards assumption. The methods are illustrated with the application to two individual participant meta-analyses in cancer. The examples were chosen because they differ in disease severity and the patterns of follow-up, in order to understand the potential impacts on the hazards and the overall effect estimates. We further investigate the estimation methods for restricted mean survival time by a simulation study.

Combining fractional polynomial model building with multiple imputation.

Multivariable fractional polynomial (MFP) models are commonly used in medical research. The datasets in which MFP models are applied often contain covariates with missing values. To handle the missing values, we describe methods for combining multiple imputation with MFP modelling, considering in turn three issues: first, how to impute so that the imputation model does not favour certain fractional polynomial (FP) models over others; second, how to estimate the FP exponents in multiply imputed data; and third, how to choose between models of differing complexity. Two imputation methods are outlined for different settings. For model selection, methods based on Wald-type statistics and weighted likelihood-ratio tests are proposed and evaluated in simulation studies. The Wald-based method is very slightly better at estimating FP exponents. Type I error rates are very similar for both methods, although slightly less well controlled than analysis of complete records; however, there is potential for substantial gains in power over the analysis of complete records. We illustrate the two methods in a dataset from five trauma registries for which a prognostic model has previously been published, contrasting the selected models with that obtained by analysing the complete records only.

Discrimination-based sample size calculations for multivariable prognostic models for time-to-event data.

BACKGROUND: Prognostic studies of time-to-event data, where researchers aim to develop or validate multivariable prognostic models in order to predict survival, are commonly seen in the medical literature; however, most are performed retrospectively and few consider sample size prior to analysis. Events per variable rules are sometimes cited, but these are based on bias and coverage of confidence intervals for model terms, which are not of primary interest when developing a model to predict outcome. In this paper we aim to develop sample size recommendations for multivariable models of time-to-event data, based on their prognostic ability. METHODS: We derive formulae for determining the sample size required for multivariable prognostic models in time-to-event data, based on a measure of discrimination, D, developed by Royston and Sauerbrei. These formulae fall into two categories: either based on the significance of the value of D in a new study compared to a previous estimate, or based on the precision of the estimate of D in a new study in terms of confidence interval width. Using simulation we show that they give the desired power and type I error and are not affected by random censoring. Additionally, we conduct a literature review to collate published values of D in different disease areas. RESULTS: We illustrate our methods using parameters from a published prognostic study in liver cancer. The resulting sample sizes can be large, and we suggest controlling study size by expressing the desired accuracy in the new study as a relative value as well as an absolute value. To improve usability we use the values of D obtained from the literature review to develop an equation to approximately convert the commonly reported Harrell's c-index to D. A flow chart is provided to aid decision making when using these methods. CONCLUSION: We have developed a suite of sample size calculations based on the prognostic ability of a survival model, rather than the magnitude or significance of model coefficients. We have taken care to develop the practical utility of the calculations and give recommendations for their use in contemporary clinical research.

The extension of total gain (TG) statistic in survival models: properties and applications.

BACKGROUND: The results of multivariable regression models are usually summarized in the form of parameter estimates for the covariates, goodness-of-fit statistics, and the relevant p-values. These statistics do not inform us about whether covariate information will lead to any substantial improvement in prediction. Predictive ability measures can be used for this purpose since they provide important information about the practical significance of prognostic factors. R (2)-type indices are the most familiar forms of such measures in survival models, but they all have limitations and none is widely used. METHODS: In this paper, we extend the total gain (TG) measure, proposed for a logistic regression model, to survival models and explore its properties using simulations and real data. TG is based on the binary regression quantile plot, otherwise known as the predictiveness curve. Standardised TG ranges from 0 (no explanatory power) to 1 ('perfect' explanatory power). RESULTS: The results of our simulations show that unlike many of the other R (2)-type predictive ability measures, TG is independent of random censoring. It increases as the effect of a covariate increases and can be applied to different types of survival models, including models with time-dependent covariate effects. We also apply TG to quantify the predictive ability of multivariable prognostic models developed in several disease areas. CONCLUSIONS: Overall, TG performs well in our simulation studies and can be recommended as a measure to quantify the predictive ability in survival models.