A tutorial on frailty models Balan, Theodor A; Putter, Hein
Statistical Methods in Medical Research,
11/2020, Volume:
29, Issue:
11
Book Review, Journal Article
Peer reviewed
Open access
The hazard function plays a central role in survival analysis. In a homogeneous population, the distribution of the time to event, described by the hazard, is the same for each individual. ...Heterogeneity in the distributions can be accounted for by including covariates in a model for the hazard, for instance a proportional hazards model. In this model, individuals with the same value of the covariates will have the same distribution. It is natural to think that not all covariates that are thought to influence the distribution of the survival outcome are included in the model. This implies that there is unobserved heterogeneity; individuals with the same value of the covariates may have different distributions. One way of accounting for this unobserved heterogeneity is to include random effects in the model. In the context of hazard models for time to event outcomes, such random effects are called frailties, and the resulting models are called frailty models. In this tutorial, we study frailty models for survival outcomes. We illustrate how frailties induce selection of healthier individuals among survivors, and show how shared frailties can be used to model positively dependent survival outcomes in clustered data. The Laplace transform of the frailty distribution plays a central role in relating the hazards, conditional on the frailty, to hazards and survival functions observed in a population. Available software, mainly in R, will be discussed, and the use of frailty models is illustrated in two different applications, one on center effects and the other on recurrent events.
The topic non-parametric estimation of transition probabilities in non-Markov multi-state models has seen a remarkable surge of activity recently. Two recent papers have used the idea of subsampling ...in this context. The first paper, by de Uña Álvarez and Meira-Machado, uses a procedure based on (differences between) Kaplan–Meier estimators derived from a subset of the data consisting of all subjects observed to be in the given state at the given time. The second, by Titman, derived estimators of transition probabilities that are consistent in general non-Markov multi-state models. Here, we show that the same idea of subsampling, used in both these papers, combined with the Aalen–Johansen estimate of the state occupation probabilities derived from that subset, can also be used to obtain a relatively simple and intuitive procedure which we term landmark Aalen–Johansen. We show that the landmark Aalen–Johansen estimator yields a consistent estimator of the transition probabilities in general non-Markov multi-state models under the same conditions as needed for consistency of the Aalen–Johansen estimator of the state occupation probabilities. Simulation studies show that the landmark Aalen–Johansen estimator has good small sample properties and is slightly more efficient than the other estimators.
When analyzing correlated time to event data, shared frailty (random effect) models are particularly attractive. However, the estimation of such models has proved challenging. In semiparametric ...models, this is further complicated by the presence of the nonparametric baseline hazard. Although recent years have seen an increased availability of software for fitting frailty models, most software packages focus either on a small number of distributions of the random effect, or support only on a few data scenarios. frailtyEM is an R package that provides maximum likelihood estimation of semiparametric shared frailty models using the expectation-maximization algorithm. The implementation is consistent across several scenarios, including possibly left truncated clustered failures and recurrent events in both calendar time and gap time formulation. A large number of frailty distributions belonging to the power variance function family are supported. Several methods facilitate access to predicted survival and cumulative hazard curves, both for an individual and on a population level. An extensive number of summary measures and statistical tests are also provided.
Multi-state models are a very useful tool to answer a wide range of questions in survival analysis that cannot, or only in a more complicated way, be answered by classical models. They are suitable ...for both biomedical and other applications in which time-to-event variables are analyzed. However, they are still not frequently applied. So far, an important reason for this has been the lack of available software. To overcome this problem, we have developed the mstate package in R for the analysis of multi-state models. The package covers all steps of the analysis of multi-state models, from model building and data preparation to estimation and graphical representation of the results. It can be applied to non- and semi-parametric (Cox) models. The package is also suitable for competing risks models, as they are a special category of multi-state models.This article offers guidelines for the actual use of the software by means of an elaborate multi-state analysis of data describing post-transplant events of patients with blood cancer. The data have been provided by the EBMT (the European Group for Blood and Marrow Transplantation). Special attention will be paid to the modeling of different covariate effects (the same for all transitions or transition-specific) and different baseline hazard assumptions (different for all transitions or equal for some).
The randomized Adjuvant Chemoradiotherapy Versus Radiotherapy Alone in Women With High-Risk Endometrial Cancer (PORTEC-3) trial investigated the benefit of combined adjuvant chemotherapy and ...radiotherapy (CTRT) versus radiotherapy alone (RT) for women with high-risk endometrial cancer (EC). Because The Cancer Genome Atlas defined an EC molecular classification with strong prognostic value, we investigated prognosis and impact of chemotherapy for each molecular subgroup using tissue samples from PORTEC-3 trial participants.
Paraffin-embedded tissues of 423 consenting patients were collected. Immunohistochemistry for p53 and mismatch repair (MMR) proteins, and DNA sequencing for
exonuclease domain were done to classify tumors as p53 abnormal (p53abn),
ultramutated (
mut), MMR-deficient (MMRd), or no specific molecular profile (NSMP). The primary end point was recurrence-free survival (RFS). Kaplan-Meier method, log-rank test, and Cox model were used for analysis.
Molecular analysis was successful in 410 high-risk EC (97%), identifying the 4 subgroups: p53abn EC (n = 93; 23%),
mut (n = 51; 12%), MMRd (n = 137; 33%), and NSMP (n = 129; 32%). Five-year RFS was 48% for patients with p53abn EC, 98% for
mut EC, 72% for MMRd EC, and 74% for NSMP EC (
< .001). The 5-year RFS with CTRT versus RT for p53abn EC was 59% versus 36% (
= .019); 100% versus 97% for patients with
mut EC (
= .637); 68% versus 76% (
= .428) for MMRd EC; and 80% versus 68% (
= .243) for NSMP EC.
Molecular classification has strong prognostic value in high-risk EC, with significantly improved RFS with adjuvant CTRT for p53abn tumors, regardless of histologic type. Patients with
mut EC had an excellent RFS in both trial arms. EC molecular classification should be incorporated in the risk stratification of these patients as well as in future trials to target specific subgroups of patients.
In this paper we study approaches for dealing with treatment when developing a clinical prediction model. Analogous to the estimand framework recently proposed by the European Medicines Agency for ...clinical trials, we propose a ‘predictimand’ framework of different questions that may be of interest when predicting risk in relation to treatment started after baseline. We provide a formal definition of the estimands matching these questions, give examples of settings in which each is useful and discuss appropriate estimators including their assumptions. We illustrate the impact of the predictimand choice in a dataset of patients with end-stage kidney disease. We argue that clearly defining the estimand is equally important in prediction research as in causal inference.
Summary Background Historical data and recent studies show that standardised extended (D2) lymphadenectomy leads to better results than standardised limited (D1) lymphadenectomy. Based on these ...findings, the Dutch D1D2 trial, a nationwide prospectively randomised clinical trial, was undertaken to compare D2 with D1 lymphadenectomy in patients with resectable primary adenocarcinoma of the stomach. The aim of the study was to assess the effect of D2 compared with D1 surgery on disease recurrence and survival in patients treated with curative intent. Methods Between August, 1989, and July, 1993, patients were entered and randomised at 80 participating hospitals by means of a telephone call to the central data centre of the trial. The sequence of randomisation was in blocks of six with stratification for the participating centre. Eligibility criteria were a histologically proven adenocarcinoma of the stomach without evidence of distance metastasis, age younger than 85 years, and adequate physical condition for D1 or D2 lymphadenectomy. Patients were excluded if they had previous or coexisting cancer or had undergone gastrectomy for benign tumours. Strict quality control measures for pathological assessment were implemented and monitored. Analyses were by intention to treat. This study is registered with the NCI trial register , as DUT-KWF-CKVO-8905, EU-90003. Findings A total of 1078 patients were entered in the study, of whom 996 were eligible. 711 patients underwent the randomly assigned treatment with curative intent (380 in the D1 group and 331 in the D2 group) and 285 had palliative treatment. Data were collected prospectively and all patients were followed up for a median time of 15·2 years (range 6·9–17·9 years). Analyses were done for the 711 patients treated with curative intent and were according to the allocated treatment group. Of the 711 patients, 174 (25%) were alive, all but one without recurrence. Overall 15-year survival was 21% (82 patients) for the D1 group and 29% (92 patients) for the D2 group (p=0·34). Gastric-cancer-related death rate was significantly higher in the D1 group (48%, 182 patients) compared with the D2 group (37%, 123 patients), whereas death due to other diseases was similar in both groups. Local recurrence was 22% (82 patients) in the D1 group versus 12% (40 patients) in D2, and regional recurrence was 19% (73 patients) in D1 versus 13% (43 patients) in D2. Patients who had the D2 procedure had a significantly higher operative mortality rate than those who had D1 (n=32 10% vs n=15 4%; 95% CI for the difference 2–9; p=0·004), higher complication rate (n=142 43% vs n=94 25%; 11–25; p<0·0001), and higher reoperation rate (n=59 18% vs n=30 8%; 5–15; p=0·00016). Interpretation After a median follow-up of 15 years, D2 lymphadenectomy is associated with lower locoregional recurrence and gastric-cancer-related death rates than D1 surgery. The D2 procedure was also associated with significantly higher postoperative mortality, morbidity, and reoperation rates. Because a safer, spleen-preserving D2 resection technique is currently available in high-volume centres, D2 lymphadenectomy is the recommended surgical approach for patients with resectable (curable) gastric cancer. Funding Dutch Health Insurance Funds Council and the Netherlands Cancer Foundation.
Abstract In recent years, multi-state models have been studied widely in survival analysis. Despite their clear advantages, their use in biomedical and other applications has been rather limited so ...far. An important reason for this is the lack of flexible and user-friendly software for multi-state models. This paper introduces a package in R , called ‘ mstate ’, for each of the steps of the analysis of multi-state models. It can be applied to non- and semi-parametric models. The package contains functions to facilitate data preparation and flexible estimation of different types of covariate effects in the context of Cox regression models, functions to estimate patient-specific transition intensities, dynamic prediction probabilities and their associated standard errors (both Greenwood and Aalen-type). Competing risks models can also be analyzed by means of mstate , as they are a special type of multi-state models. The package is available from the R homepage http://cran.r-project.org. We give a self-contained account of the underlying mathematical theory, including a new asymptotic result for the cumulative hazard function and new recursive formulas for the calculation of the estimated standard errors of the estimated transition probabilities, and we illustrate the use of the key functions of the mstate package by the analysis of a reversible multi-state model describing survival of liver cirrhosis patients.
SummaryBackgroundThe PORTEC-3 trial investigated the benefit of combined adjuvant chemotherapy and radiotherapy versus pelvic radiotherapy alone for women with high-risk endometrial cancer. We ...updated the analysis to investigate patterns of recurrence and did a post-hoc survival analysis. MethodsIn the multicentre randomised phase 3 PORTEC-3 trial, women with high-risk endometrial cancer were eligible if they had International Federation of Gynaecology and Obstetrics (FIGO) 2009 stage I, endometrioid grade 3 cancer with deep myometrial invasion or lymphovascular space invasion, or both; stage II or III disease; or stage I–III disease with serous or clear cell histology; were aged 18 years and older; and had a WHO performance status of 0–2. Participants were randomly assigned (1:1) to receive radiotherapy alone (48·6 Gy in 1·8 Gy fractions given on 5 days per week) or chemoradiotherapy (two cycles of cisplatin 50 mg/m 2 given intravenously during radiotherapy, followed by four cycles of carboplatin AUC5 and paclitaxel 175 mg/m 2 given intravenously), by use of a biased coin minimisation procedure with stratification for participating centre, lymphadenectomy, stage, and histological type. The co-primary endpoints were overall survival and failure-free survival. Secondary endpoints of vaginal, pelvic, and distant recurrence were analysed according to the first site of recurrence. Survival endpoints were analysed by intention-to-treat, and adjusted for stratification factors. Competing risk methods were used for failure-free survival and recurrence. We did a post-hoc analysis to analyse patterns of recurrence with 1 additional year of follow-up. The study was closed on Dec 20, 2013; follow-up is ongoing. This study is registered with ISRCTN, number ISRCTN14387080, and ClinicalTrials.gov, number NCT00411138. FindingsBetween Nov 23, 2006, and Dec 20, 2013, 686 women were enrolled, of whom 660 were eligible and evaluable (330 in the chemoradiotherapy group, and 330 in the radiotherapy-alone group). At a median follow-up of 72·6 months (IQR 59·9–85·6), 5-year overall survival was 81·4% (95% CI 77·2–85·8) with chemoradiotherapy versus 76·1% (71·6–80·9) with radiotherapy alone (adjusted hazard ratio HR 0·70 95% CI 0·51–0·97, p=0·034), and 5-year failure-free survival was 76·5% (95% CI 71·5–80·7) versus 69·1% (63·8–73·8; HR 0·70 0·52–0·94, p=0·016). Distant metastases were the first site of recurrence in most patients with a relapse, occurring in 78 of 330 women (5-year probability 21·4%; 95% CI 17·3–26·3) in the chemoradiotherapy group versus 98 of 330 (5-year probability 29·1%; 24·4–34·3) in the radiotherapy-alone group (HR 0·74 95% CI 0·55–0·99; p=0·047). Isolated vaginal recurrence was the first site of recurrence in one patient (0·3%; 95% CI 0·0–2·1) in both groups (HR 0·99 95% CI 0·06–15·90; p=0·99), and isolated pelvic recurrence was the first site of recurrence in three women (0·9% 95% CI 0·3–2·8) in the chemoradiotherapy group versus four (0·9% 95% CI 0·3–2·8) in the radiotherapy-alone group (HR 0·75 95% CI 0·17–3·33; p=0·71). At 5 years, only one grade 4 adverse event (ileus or obstruction) was reported (in the chemoradiotherapy group). At 5 years, reported grade 3 adverse events did not differ significantly between the two groups, occurring in 16 (8%) of 201 women in the chemoradiotherapy group versus ten (5%) of 187 in the radiotherapy-alone group (p=0·24). The most common grade 3 adverse event was hypertension (in four 2% women in both groups). At 5 years, grade 2 or worse adverse events were reported in 76 (38%) of 201 women in the chemoradiotherapy group versus 43 (23%) of 187 in the radiotherapy-alone group (p=0·002). Sensory neuropathy persisted more often after chemoradiotherapy than after radiotherapy alone, with 5-year rates of grade 2 or worse neuropathy of 6% (13 of 201 women) versus 0% (0 of 187). No treatment-related deaths were reported. InterpretationThis updated analysis shows significantly improved overall survival and failure-free survival with chemoradiotherapy versus radiotherapy alone. This treatment schedule should be discussed and recommended, especially for women with stage III or serous cancers, or both, as part of shared decision making between doctors and patients. Follow-up is ongoing to evaluate long-term survival. FundingDutch Cancer Society, Cancer Research UK, National Health and Medical Research Council, Project Grant, Cancer Australia Grant, Italian Medicines Agency, and the Canadian Cancer Society Research Institute.
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