Michael J. Crowther

Joint modeling of survival and longitudinal non-survival data: current methods and issues. Report of the DIA Bayesian joint modeling working group

Explicitly modeling underlying relationships between a survival endpoint and processes that generate longitudinal measured or reported outcomes potentially could improve the efficiency of clinical trials and provide greater insight into the various dimensions of the clinical effect of interventions included in the trials. Various strategies have been proposed for using longitudinal findings to elucidate intervention effects on clinical outcomes such as survival. The application of specifically Bayesian approaches for constructing models that address longitudinal and survival outcomes explicitly has been recently addressed in the literature. We review currently available methods for carrying out joint analyses, including issues of implementation and interpretation, identify software tools that can be used to carry out the necessary calculations, and review applications of the methodology.

Simulating biologically plausible complex survival data

Simulation studies are conducted to assess the performance of current and novel statistical models in pre-defined scenarios. It is often desirable that chosen simulation scenarios accurately reflect a biologically plausible underlying distribution. This is particularly important in the framework of survival analysis, where simulated distributions are chosen for both the event time and the censoring time. This paper develops methods for using complex distributions when generating survival times to assess methods in practice. We describe a general algorithm involving numerical integration and root-finding techniques to generate survival times from a variety of complex parametric distributions, incorporating any combination of time-dependent effects, time-varying covariates, delayed entry, random effects and covariates measured with error. User-friendly Stata software is provided.

The use of restricted cubic splines to approximate complex hazard functions in the analysis of time-to-event data: a simulation study

If interest lies in reporting absolute measures of risk from time-to-event data then obtaining an appropriate approximation to the shape of the underlying hazard function is vital. It has previously been shown that restricted cubic splines can be used to approximate complex hazard functions in the context of time-to-event data. The degree of complexity for the spline functions is dictated by the number of knots that are defined. We highlight through the use of a motivating example that complex hazard function shapes are often required when analysing time-to-event data. Through the use of simulation, we show that provided a sufficient number of knots are used, the approximated hazard functions given by restricted cubic splines fit closely to the true function for a range of complex hazard shapes. The simulation results also highlight the insensitivity of the estimated relative effects (hazard ratios) to the correct specification of the baseline hazard.

Individual patient data meta-analysis of survival data using Poisson regression models

BackgroundAn Individual Patient Data (IPD) meta-analysis is often considered the gold-standard for synthesising survival data from clinical trials. An IPD meta-analysis can be achieved by either a two-stage or a one-stage approach, depending on whether the trials are analysed separately or simultaneously. A range of one-stage hierarchical Cox models have been previously proposed, but these are known to be computationally intensive and are not currently available in all standard statistical software. We describe an alternative approach using Poisson based Generalised Linear Models (GLMs).MethodsWe illustrate, through application and simulation, the Poisson approach both classically and in a Bayesian framework, in two-stage and one-stage approaches. We outline the benefits of our one-stage approach through extension to modelling treatment-covariate interactions and non-proportional hazards. Ten trials of hypertension treatment, with all-cause death the outcome of interest, are used to apply and assess the approach.ResultsWe show that the Poisson approach obtains almost identical estimates to the Cox model, is additionally computationally efficient and directly estimates the baseline hazard. Some downward bias is observed in classical estimates of the heterogeneity in the treatment effect, with improved performance from the Bayesian approach.ConclusionOur approach provides a highly flexible and computationally efficient framework, available in all standard statistical software, to the investigation of not only heterogeneity, but the presence of non-proportional hazards and treatment effect modifiers.

Adjusting Survival Time Estimates to Account for Treatment Switching in Randomized Controlled Trials—an Economic Evaluation Context: Methods, Limitations, and Recommendations

Background. Treatment switching commonly occurs in clinical trials of novel interventions in the advanced or metastatic cancer setting. However, methods to adjust for switching have been used inconsistently and potentially inappropriately in health technology assessments (HTAs). Objective. We present recommendations on the use of methods to adjust survival estimates in the presence of treatment switching in the context of economic evaluations. Methods. We provide background on the treatment switching issue and summarize methods used to adjust for it in HTAs. We discuss the assumptions and limitations associated with adjustment methods and draw on results of a simulation study to make recommendations on their use. Results. We demonstrate that methods used to adjust for treatment switching have important limitations and often produce bias in realistic scenarios. We present an analysis framework that aims to increase the probability that suitable adjustment methods can be identified on a case-by-case basis. We recommend that the characteristics of clinical trials, and the treatment switching mechanism observed within them, should be considered alongside the key assumptions of the adjustment methods. Key assumptions include the “no unmeasured confounders” assumption associated with the inverse probability of censoring weights (IPCW) method and the “common treatment effect” assumption associated with the rank preserving structural failure time model (RPSFTM). Conclusions. The limitations associated with switching adjustment methods such as the RPSFTM and IPCW mean that they are appropriate in different scenarios. In some scenarios, both methods may be prone to bias; “2-stage” methods should be considered, and intention-to-treat analyses may sometimes produce the least bias. The data requirements of adjustment methods also have important implications for clinical trialists.

Flexible parametric joint modelling of longitudinal and survival data.

The joint modelling of longitudinal and survival data is a highly active area of biostatistical research. The submodel for the longitudinal biomarker usually takes the form of a linear mixed effects model. We describe a flexible parametric approach for the survival submodel that models the log baseline cumulative hazard using restricted cubic splines. This approach overcomes limitations of standard parametric choices for the survival submodel, which can lack the flexibility to effectively capture the shape of the underlying hazard function. Numerical integration techniques, such as Gauss-Hermite quadrature, are usually required to evaluate both the cumulative hazard and the overall joint likelihood; however, by using a flexible parametric model, the cumulative hazard has an analytically tractable form, providing considerable computational benefits. We conduct an extensive simulation study to assess the proposed model, comparing it with a B-spline formulation, illustrating insensitivity of parameter estimates to the baseline cumulative hazard function specification. Furthermore, we compare non-adaptive and fully adaptive quadrature, showing the superiority of adaptive quadrature in evaluating the joint likelihood. We also describe a useful technique to simulate survival times from complex baseline hazard functions and illustrate the methods using an example data set investigating the association between longitudinal prothrombin index and survival of patients with liver cirrhosis, showing greater flexibility and improved stability with fewer parameters under the proposed model compared with the B-spline approach. We provide user-friendly Stata software.

When is a further clinical trial justified

High quality randomised trials provide unbiased estimates of the effects of health interventions, but the findings of a single trial are rarely conclusive. Usually data from several similar trials must be pooled together to draw firm conclusions about the effectiveness of an intervention. This article considers how to determine whether data from existing trials are conclusive and, if not, whether a further trial is justified.

A general framework for parametric survival analysis

Parametric survival models are being increasingly used as an alternative to the Cox model in biomedical research. Through direct modelling of the baseline hazard function, we can gain greater understanding of the risk profile of patients over time, obtaining absolute measures of risk. Commonly used parametric survival models, such as the Weibull, make restrictive assumptions of the baseline hazard function, such as monotonicity, which is often violated in clinical datasets. In this article, we extend the general framework of parametric survival models proposed by Crowther and Lambert (Journal of Statistical Software 53:12, 2013), to incorporate relative survival, and robust and cluster robust standard errors. We describe the general framework through three applications to clinical datasets, in particular, illustrating the use of restricted cubic splines, modelled on the log hazard scale, to provide a highly flexible survival modelling framework. Through the use of restricted cubic splines, we can derive the cumulative hazard function analytically beyond the boundary knots, resulting in a combined analytic/numerical approach, which substantially improves the estimation process compared with only using numerical integration. User-friendly Stata software is provided, which significantly extends parametric survival models available in standard software.

Total hip replacement and surface replacement for the treatment of pain and disability resulting from end-stage arthritis of the hip (review of technology appraisal guidance 2 and 44): systematic review and economic evaluation

BACKGROUND Total hip replacement (THR) involves the replacement of a damaged hip joint with an artificial hip prosthesis. Resurfacing arthroplasty (RS) involves replacement of the joint surface of the femoral head with a metal surface covering. OBJECTIVES To undertake clinical effectiveness and cost-effectiveness analysis of different types of THR and RS for the treatment of pain and disability in people with end-stage arthritis of the hip, in particular to compare the clinical effectiveness and cost-effectiveness of (1) different types of primary THR and RS for people in whom both procedures are suitable and (2) different types of primary THR for people who are not suitable for hip RS. DATA SOURCES Electronic databases including MEDLINE, EMBASE, The Cochrane Library, Current Controlled Trials and UK Clinical Research Network (UKCRN) Portfolio Database were searched in December 2012, with searches limited to publications from 2008 and sample sizes of ≥ 100 participants. Reference lists and websites of manufacturers and professional organisations were also screened. REVIEW METHODS Systematic reviews of the literature were undertaken to appraise the clinical effectiveness and cost-effectiveness of different types of THR and RS for people with end-stage arthritis of the hip. Included randomised controlled trials (RCTs) and systematic reviews were data extracted and risk of bias and methodological quality were independently assessed by two reviewers using the Cochrane Collaboration risk of bias tool and the Assessment of Multiple Systematic Reviews (AMSTAR) tool. A Markov multistate model was developed for the economic evaluation of the technologies. Sensitivity analyses stratified by sex and controlled for age were carried out to assess the robustness of the results. RESULTS A total of 2469 records were screened of which 37 were included, representing 16 RCTs and eight systematic reviews. The mean post-THR Harris Hip Score measured at different follow-up times (from 6 months to 10 years) did not differ between THR groups, including between cross-linked polyethylene and traditional polyethylene cup liners (pooled mean difference 2.29, 95% confidence interval -0.88 to 5.45). Five systematic reviews reported evidence on different types of THR (cemented vs. cementless cup fixation and implant articulation materials) but these reviews were inconclusive. Eleven cost-effectiveness studies were included; four provided relevant cost and utility data for the model. Thirty registry studies were included, with no studies reporting better implant survival for RS than for all types of THR. For all analyses, mean costs for RS were higher than those for THR and mean quality-adjusted life-years (QALYs) were lower. The incremental cost-effectiveness ratio for RS was dominated by THR, that is, THR was cheaper and more effective than RS (for a lifetime horizon in the base-case analysis, the incremental cost of RS was £11,284 and the incremental QALYs were -0.0879). For all age and sex groups RS remained clearly dominated by THR. Cost-effectiveness acceptability curves showed that, for all patients, THR was almost 100% cost-effective at any willingness-to-pay level. There were age and sex differences in the populations with different types of THR and variations in revision rates (from 1.6% to 3.5% at 9 years). For the base-case analysis, for all age and sex groups and a lifetime horizon, mean costs for category E (cemented components with a polyethylene-on-ceramic articulation) were slightly lower and mean QALYs for category E were slightly higher than those for all other THR categories in both deterministic and probabilistic analyses. Hence, category E dominated the other four categories. Sensitivity analysis using an age- and sex-adjusted log-normal model demonstrated that, over a lifetime horizon and at a willingness-to-pay threshold of £20,000 per QALY, categories A and E were equally likely (50%) to be cost-effective. LIMITATIONS A large proportion of the included studies were inconclusive because of poor reporting, missing data, inconsistent results and/or great uncertainty in the treatment effect estimates. This warrants cautious interpretation of the findings. The evidence on complications was scarce, which may be because of the absence or rarity of these events or because of under-reporting. The poor reporting meant that it was not possible to explore contextual factors that might have influenced study results and also reduced the applicability of the findings to routine clinical practice in the UK. The scope of the review was limited to evidence published in English in 2008 or later, which could be interpreted as a weakness; however, systematic reviews would provide summary evidence for studies published before 2008. CONCLUSIONS Compared with THR, revision rates for RS were higher, mean costs for RS were higher and mean QALYs gained were lower; RS was dominated by THR. Similar results were obtained in the deterministic and probabilistic analyses and for all age and sex groups THR was almost 100% cost-effective at any willingness-to-pay level. Revision rates for all types of THR were low. Category A THR (cemented components with a polyethylene-on-metal articulation) was more cost-effective for older age groups. However, across all age-sex groups combined, the mean cost for category E THR (cemented components with a polyethylene-on-ceramic articulation) was slightly lower and the mean QALYs gained were slightly higher. Category E therefore dominated the other four categories. Certain types of THR appeared to confer some benefit, including larger femoral head sizes, use of a cemented cup, use of a cross-linked polyethylene cup liner and a ceramic-on-ceramic as opposed to a metal-on-polyethylene articulation. Further RCTs with long-term follow-up are needed. STUDY REGISTRATION This study is registered as PROSPERO CRD42013003924. FUNDING The National Institute for Health Research Health Technology Assessment programme.

Setting benchmark revision rates for total hip replacement: analysis of registry evidence

Objective To compare 10 year revision rates for frequently used types of primary total hip replacement to inform setting of a new benchmark rate in England and Wales that will be of international relevance. Design Retrospective cohort study. Setting National Joint Registry. Participants 239 000 patient records. Main outcome measures Revision rates for five frequently used types of total hip replacement that differed according to bearing surface and fixation mode, encompassing 62% of all primary total hip replacements in the National Joint Registry for England and Wales. Revision rates were compared using Kaplan-Meier and competing risks analyses, and five and 10 year rates were estimated using well fitting parametric models. Results Estimated revision rates at 10 years were 4% or below for four of the five types of total hip replacement investigated. Rates differed little according to Kaplan-Meier or competing risks analysis, but differences between prosthesis types were more substantial. Cemented prostheses with ceramic-on-polyethylene bearing surfaces had the lowest revision rates (1.88-2.11% at 10 years depending on the method used), and cementless prostheses with ceramic-on-ceramic bearing surfaces had the highest revision rates (3.93-4.33%). Men were more likely to receive revision of total hip replacement than were women, and this difference was statistically significant for four of the five prosthesis types. Conclusions Ten year revision rate estimates were all less than 5%, and in some instances considerably less. The results suggest that the current revision rate benchmark should be at least halved from 10% to less than 5% at 10 years. This has implications for benchmarks internationally.