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.

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.

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.

Personalized Model to Predict Keratoconus Progression From Demographic, Topographic, and Genetic Data.

PURPOSE: To generate a prognostic model to predict keratoconus progression to corneal crosslinking (CXL). DESIGN: Retrospective cohort study. METHODS: We recruited 5025 patients (9341 eyes) with early keratoconus between January 2011 and November 2020. Genetic data from 926 patients were available. We investigated both keratometry or CXL as end points for progression and used the Royston-Parmar method on the proportional hazards scale to generate a prognostic model. We calculated hazard ratios (HRs) for each significant covariate, with explained variation and discrimination, and performed internal-external cross validation by geographic regions. RESULTS: After exclusions, model fitting comprised 8701 eyes, of which 3232 underwent CXL. For early keratoconus, CXL provided a more robust prognostic model than keratometric progression. The final model explained 33% of the variation in time to event: age HR (95% CI) 0.9 (0.90-0.91), maximum anterior keratometry 1.08 (1.07-1.09), and minimum corneal thickness 0.95 (0.93-0.96) as significant covariates. Single-nucleotide polymorphisms (SNPs) associated with keratoconus (n=28) did not significantly contribute to the model. The predicted time-to-event curves closely followed the observed curves during internal-external validation. Differences in discrimination between geographic regions was low, suggesting the model maintained its predictive ability. CONCLUSIONS: A prognostic model to predict keratoconus progression could aid patient empowerment, triage, and service provision. Age at presentation is the most significant predictor of progression risk. Candidate SNPs associated with keratoconus do not contribute to progression risk.

External Validation of the 2003 Leibovich Prognostic Score in Patients Randomly Assigned to SORCE, an International Phase III Trial of Adjuvant Sorafenib in Renal Cell Cancer.

PURPOSE: The 2003 Leibovich score guides prognostication and selection to adjuvant clinical trials for patients with locally advanced renal cell carcinoma (RCC) after nephrectomy. We provide a robust external validation of the 2003 Leibovich score using contemporary data from SORCE, an international, randomized trial of sorafenib after excision of primary RCC. METHODS: Data used to derive the 2003 Leibovich score were compared with contemporary data from SORCE. Discrimination and calibration of the metastasis-free survival outcome were assessed in data from patients with clear-cell RCC, using Cox proportional hazards regression, Kaplan-Meier curves, and calculation of Harrell's c indexes. Secondary analyses involved three important SORCE groups: patients with any non-clear-cell subtype, papillary, and chromophobe carcinomas. RESULTS: Four hundred seven recurrences occurred in 982 patients in the Leibovich cohort and 520 recurrences were recorded in 1,445 patients in the primary SORCE cohort. Clear discrimination between intermediate-risk and high-risk SORCE cohorts was shown; hazard ratio 2.74 (95% CI, 2.29 to 3.28), c-index 0.63 (95% CI, 0.61 to 0.65). A hazard ratio of 0.61 (95% CI, 0.53 to 0.70) confirmed poor calibration of the two cohorts. Discrimination was observed in secondary populations, with c-indexes of 0.64 (95% CI, 0.59 to 0.69) for non-clear-cell RCC, 0.63 (95% CI, 0.56 to 0.69) for papillary RCC, and 0.65 (95% CI, 0.55 to 0.76) for chromophobe RCC. CONCLUSION: The 2003 Leibovich score discriminates between intermediate-risk and high-risk clear-cell and non-clear-cell RCC groups in contemporary data, supporting its use for risk stratification in adjuvant clinical trials. Over time, metastasis-free survival for patients with locally advanced RCC has improved. Contemporary data from adjuvant RCC trials should be used to improve prognostication for patients with RCC.

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.

A simulation study comparing the power of nine tests of the treatment effect in randomized controlled trials with a time-to-event outcome.

BACKGROUND: The logrank test is routinely applied to design and analyse randomized controlled trials (RCTs) with time-to-event outcomes. Sample size and power calculations assume the treatment effect follows proportional hazards (PH). If the PH assumption is false, power is reduced and interpretation of the hazard ratio (HR) as the estimated treatment effect is compromised. Using statistical simulation, we investigated the type 1 error and power of the logrank (LR)test and eight alternatives. We aimed to identify test(s) that improve power with three types of non-proportional hazards (non-PH): early, late or near-PH treatment effects. METHODS: We investigated weighted logrank tests (early, LRE; late, LRL), the supremum logrank test (SupLR) and composite tests (joint, J; combined, C; weighted combined, WC; versatile and modified versatile weighted logrank, VWLR, VWLR2) with two or more components. Weighted logrank tests are intended to be sensitive to particular non-PH patterns. Composite tests attempt to improve power across a wider range of non-PH patterns. Using extensive simulations based on real trials, we studied test size and power under PH and under simple departures from PH comprising pointwise constant HRs with a single change point at various follow-up times. We systematically investigated the influence of high or low control-arm event rates on power. RESULTS: With no preconceived type of treatment effect, the preferred test is VWLR2. Expecting an early effect, tests with acceptable power are SupLR, C, VWLR2, J, LRE and WC. Expecting a late effect, acceptable tests are LRL, VWLR, VWLR2, WC and J. Under near-PH, acceptable tests are LR, LRE, VWLR, C, VWLR2 and SupLR. Type 1 error was well controlled for all tests, showing only minor deviations from the nominal 5%. The location of the HR change point relative to the cumulative proportion of control-arm events considerably affected power. CONCLUSIONS: Assuming ignorance of the likely treatment effect, the best choice is VWLR2. Several non-standard tests performed well when the correct type of treatment effect was assumed. A low control-arm event rate reduced the power of weighted logrank tests targeting early effects. Test size was generally well controlled. Further investigation of test characteristics with different types of non-proportional hazards of the treatment effect is warranted.

State of the art in selection of variables and functional forms in multivariable analysis-outstanding issues.

BACKGROUND: How to select variables and identify functional forms for continuous variables is a key concern when creating a multivariable model. Ad hoc 'traditional' approaches to variable selection have been in use for at least 50 years. Similarly, methods for determining functional forms for continuous variables were first suggested many years ago. More recently, many alternative approaches to address these two challenges have been proposed, but knowledge of their properties and meaningful comparisons between them are scarce. To define a state of the art and to provide evidence-supported guidance to researchers who have only a basic level of statistical knowledge, many outstanding issues in multivariable modelling remain. Our main aims are to identify and illustrate such gaps in the literature and present them at a moderate technical level to the wide community of practitioners, researchers and students of statistics. METHODS: We briefly discuss general issues in building descriptive regression models, strategies for variable selection, different ways of choosing functional forms for continuous variables and methods for combining the selection of variables and functions. We discuss two examples, taken from the medical literature, to illustrate problems in the practice of modelling. RESULTS: Our overview revealed that there is not yet enough evidence on which to base recommendations for the selection of variables and functional forms in multivariable analysis. Such evidence may come from comparisons between alternative methods. In particular, we highlight seven important topics that require further investigation and make suggestions for the direction of further research. CONCLUSIONS: Selection of variables and of functional forms are important topics in multivariable analysis. To define a state of the art and to provide evidence-supported guidance to researchers who have only a basic level of statistical knowledge, further comparative research is required.

Combined test versus logrank/Cox test in 50 randomised trials.

BACKGROUND: The logrank test and the Cox proportional hazards model are routinely applied in the design and analysis of randomised controlled trials (RCTs) with time-to-event outcomes. Usually, sample size and power calculations assume proportional hazards (PH) of the treatment effect, i.e. the hazard ratio is constant over the entire follow-up period. If the PH assumption fails, the power of the logrank/Cox test may be reduced, sometimes severely. It is, therefore, important to understand how serious this can become in real trials, and for a proven, alternative test to be available to increase the robustness of the primary test. METHODS: We performed a systematic search to identify relevant articles in four leading medical journals that publish results of phase 3 clinical trials. Altogether, 50 articles satisfied our inclusion criteria. We digitised published Kaplan-Meier curves and created approximations to the original times to event or censoring at the individual patient level. Using the reconstructed data, we tested for non-PH in all 50 trials. We compared the results from the logrank/Cox test with those from the combined test recently proposed by Royston and Parmar. RESULTS: The PH assumption was checked and reported only in 28% of the studies. Evidence of non-PH at the 0.10 level was detected in 31% of comparisons. The Cox test of the treatment effect was significant at the 0.05 level in 49% of comparisons, and the combined test in 55%. In four of five trials with discordant results, the interpretation would have changed had the combined test been used. The degree of non-PH and the dominance of the p value for the combined test were strongly associated. Graphical investigation suggested that non-PH was mostly due to a treatment effect manifesting in an early follow-up and disappearing later. CONCLUSIONS: The evidence for non-PH is checked (and, hence, identified) in only a small minority of RCTs, but non-PH may be present in a substantial fraction of such trials. In our reanalysis of the reconstructed data from 50 trials, the combined test outperformed the Cox test overall. The combined test is a promising approach to making trial design and analysis more robust.

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.

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.

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.

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.