Pitfalls and perils of survival analysis under incorrect assumptions: the case of Covid-19 data

Non-parametric survival analysis has become a very popular statistical method in current medical research. Employing, however, survival methodology when its fundamental assumptions are not fulfilled can severely bias the results. Currently, hundreds of clinical studies are using survival methods to investigate factors potentially associated with the prognosis of Corona Virus Disease 2019 (Covid-19), and test new preventive and therapeutic strategies. In the pandemic era, it is more critical than ever that decision-making is evidence-based and relies on solid statistical methods. However, this is not always the case. Serious methodologic errors have been identified in recent seminal studies about Covid-19: one reporting outcomes of patients treated with remdesivir, and another one on the epidemiology, clinical course and outcomes of critically-ill patients. High-quality evidence is essential to inform clinicians about optimal Covid-19 therapies, and policymakers about the true effect of preventive measures aiming to tackle the pandemic. Though timely evidence is needed, we should encourage the appropriate application of survival analysis methods and careful peer-review to avoid publishing flawed results, which could affect decision-making. In this paper, we recapitulate the basic assumptions underlying non-parametric survival analysis and frequent errors in its application, and discuss how to handle data of Covid-19.

Epidemiology of moderately severe and severe non-proliferative diabetic retinopathy in South West England

Aims To estimate the incidence of early treatment diabetic retinopathy study (ETDRS) level 47 and 53 and progression to treatment with panretinal photocoagulation (PRP) for proliferative DR (PDR). Methods Log-linear regression was used to estimate the incidence of level 47–53 or worse for 33,009 people with diabetes (PWD) in Gloucestershire during 2013–2016 by calendar year and diabetes type, based on the first recording. Progression was analysed in Gloucestershire and Bristol with a parametric survival analysis examining the association of baseline and time-varying demographic and clinical factors on time to PRP after the first recording of level 47–53. Results Incidence decreased from 0.57 (95% confidence intervals (CI) 0.48–0.67) per 100 PWD in 2013 to 0.35 (95% CI 0.29–0.43) in 2016 (p < 0.001). For progression, 338 eligible PWD from Gloucestershire and 418 from Bristol were followed for a median of 1.4 years; 78 and 83% had Type 2 diabetes and a median (interquartile range) of 15 (10–22) and 17 (11–25) years duration of diagnosed diabetes respectively. Three years from the incident ETDRS 47–53, 18.9% and 17.2% had received PRP respectively. For Gloucestershire, severe IRMA and updated mean HbA1c were associated with an increase in the risk of initiating PRP (hazard ratio 3.14 (95% CI: 1.60–6.15) and 1.21 (95% CI: 1.06–1.38 per 10 mmol/mol) respectively). Conclusion This study provides additional understanding of this population and shows that a high proportion of patients with ETDRS levels 47–53 need to be monitored as they are at high risk of progressing to PDR.

Can’t see the wood for the trees: Using common likelihoods to share information regarding selection of functional form in survival analysis

Background The estimation of mean survival is a key consideration when estimating the cost-effectiveness of interventions. Estimation typically requires extrapolation using parametric survival analysis and may be sensitive to choice of functional form. We demonstrate a simple method that allows goodness-of-fit information to be assessed jointly across studies, where such information is believed likely to be exchangeable, to help inform the selection of functional form for parametric survival analysis. Methods Individual patient data for survival was estimated from digitised Kaplan-Meier curves for four trials in advanced soft tissue sarcoma. A range of parametric survival models were fitted. Two approaches were explored for identifying the preferred parametric model: (i) selecting functional forms independently for each study; and (ii) selecting a common functional form across studies. Akaike’s Information Criterion (AIC) or Bayesian Information Criterion (BIC) were used to select optimal functional form. For approach (ii) joint BIC or AIC statistics were calculated for each model by combining the components of the AIC or BIC, respectively, across studies. A bootstrap analysis was conducted to estimate the uncertainty in selection of functional form. Results Estimates of mean survival varied markedly according to choice of functional form. Independent selection led to different functional forms being selected for each study with considerable uncertainty regarding the optimum functional form according to AIC or BIC. In this case study, selecting an optimum functional form based on joint AIC or BIC across studies reduced uncertainty in model selection and variance in mean survival compared to selection of functional form independently. Conclusion Extrapolation of estimates of survival may be sensitive to choice of functional form. If it is believed, a priori, reasonable to share information regarding the choice of functional form for survival analysis across studies, estimating joint goodness-of-fit information across studies could reduce uncertainty and lead to more reliable estimates of mean survival and cost-effectiveness.

High dimensional nuisance parameters: an example from parametric survival analysis

Parametric statistical problems involving both large amounts of data and models with many parameters raise issues that are explicitly or implicitly differential geometric. When the number of nuisance parameters is comparable to the sample size, alternative approaches to inference on interest parameters treat the nuisance parameters either as random variables or as arbitrary constants. The two approaches are compared in the context of parametric survival analysis, with emphasis on the effects of misspecification of the random effects distribution. Notably, we derive a detailed expression for the precision of the maximum likelihood estimator of an interest parameter when the assumed random effects model is erroneous, recovering simply derived results based on the Fisher information in the correctly specified situation but otherwise illustrating complex dependence on other aspects. Methods of assessing model adequacy are given. The results are both directly applicable and illustrate general principles of inference when there is a high-dimensional nuisance parameter. Open problems with an information geometrical bearing are outlined.

Multilevel mixed-effects parametric survival analysis: Estimation, simulation, and application

In this article, I present the community-contributed stmixed command for fitting multilevel survival models. It serves as both an alternative to Stata’s official mestreg command and a complimentary command with substantial extensions. stmixed can fit multilevel survival models with any number of levels and random effects at each level, including flexible spline-based approaches (such as Royston–Parmar and the log-hazard equivalent) and user-defined hazard models. Simple or complex time-dependent effects can be included, as can expected mortality for a relative survival model. Left-truncation (delayed entry) is supported, and t-distributed random effects are provided as an alternative to Gaussian random effects. I illustrate the methods with a commonly used dataset of patients with kidney disease suffering recurrent infections and a simulated example illustrating a simple approach to simulating clustered survival data using survsim (Crowther and Lambert 2012, Stata Journal 12: 674–687; 2013, Statistics in Medicine 32: 4118–4134). stmixed is part of the merlin family (Crowther 2017, arXiv Working Paper No. arXiv:1710.02223; 2018, arXiv Working Paper No. arXiv:1806.01615).