On pseudo-values for regression analysis in competing risks models

Visualizing hypothesis tests in survival analysis under anticipated delayed effects

What can be considered an appropriate statistical method for the primary analysis of a randomized clinical trial (RCT) with a time-to-event endpoint when we anticipate non-proportional hazards owing to a delayed effect? This question has been the subject of much recent debate. The standard approach is a log-rank test and/or a Cox proportional hazards model. Alternative methods have been explored in the statistical literature, such as weighted log-rank tests and tests based on the Restricted Mean Survival Time (RMST). While weighted log-rank tests can achieve high power compared to the standard log-rank test, some choices of weights may lead to type-I error inflation under particular conditions. In addition, they are not linked to a mathematically unambiguous summary measure. Test statistics based on the RMST, on the other hand, allow one to investigate the average difference between two survival curves up to a pre-specified time point τ -a mathematically unambiguous summary measure. However, by emphasizing differences prior to τ , such test statistics may not fully capture the benefit of a new treatment in terms of long-term survival. In this article, we introduce a graphical approach for direct comparison of weighted log-rank tests and tests based on the RMST. This new perspective allows a more informed choice of the analysis method, going beyond power and type I error comparison.

Pseudo-value regression trees

This paper presents a semi-parametric modeling technique for estimating the survival function from a set of right-censored time-to-event data. Our method, named pseudo-value regression trees (PRT), is based on the pseudo-value regression framework, modeling individual-specific survival probabilities by computing pseudo-values and relating them to a set of covariates. The standard approach to pseudo-value regression is to fit a main-effects model using generalized estimating equations (GEE). PRT extend this approach by building a multivariate regression tree with pseudo-value outcome and by successively fitting a set of regularized additive models to the data in the nodes of the tree. Due to the combination of tree learning and additive modeling, PRT are able to perform variable selection and to identify relevant interactions between the covariates, thereby addressing several limitations of the standard GEE approach. In addition, PRT include time-dependent effects in the node-wise models. Interpretability of the PRT fits is ensured by controlling the tree depth. Based on the results of two simulation studies, we investigate the properties of the PRT method and compare it to several alternative modeling techniques. Furthermore, we illustrate PRT by analyzing survival in 3,652 patients enrolled for a randomized study on primary invasive breast cancer.

Differential network connectivity analysis for microbiome data adjusted for clinical covariates using jackknife pseudo-values

BACKGROUND

A recent breakthrough in differential network (DN) analysis of microbiome data has been realized with the advent of next-generation sequencing technologies. The DN analysis disentangles the microbial co-abundance among taxa by comparing the network properties between two or more graphs under different biological conditions. However, the existing methods to the DN analysis for microbiome data do not adjust for other clinical differences between subjects.

RESULTS

We propose a Statistical Approach via Pseudo-value Information and Estimation for Differential Network Analysis (SOHPIE-DNA) that incorporates additional covariates such as continuous age and categorical BMI. SOHPIE-DNA is a regression technique adopting jackknife pseudo-values that can be implemented readily for the analysis. We demonstrate through simulations that SOHPIE-DNA consistently reaches higher recall and F1-score, while maintaining similar precision and accuracy to existing methods (NetCoMi and MDiNE). Lastly, we apply SOHPIE-DNA on two real datasets from the American Gut Project and the Diet Exchange Study to showcase the utility. The analysis of the Diet Exchange Study is to showcase that SOHPIE-DNA can also be used to incorporate the temporal change of connectivity of taxa with the inclusion of additional covariates. As a result, our method has found taxa that are related to the prevention of intestinal inflammation and severity of fatigue in advanced metastatic cancer patients.

CONCLUSION

SOHPIE-DNA is the first attempt of introducing the regression framework for the DN analysis in microbiome data. This enables the prediction of characteristics of a connectivity of a network with the presence of additional covariate information in the regression. The R package with a vignette of our methodology is available through the CRAN repository ( https://CRAN.R-project.org/package=SOHPIE ), named SOHPIE (pronounced as Sofie). The source code and user manual can be found at https://github.com/sjahnn/SOHPIE-DNA .

Multiply robust estimator for the difference in survival functions using pseudo-observations

BACKGROUND

When estimating the causal effect on survival outcomes in observational studies, it is necessary to adjust confounding factors due to unbalanced covariates between treatment and control groups. There is no study on multiple robust method for estimating the difference in survival functions. In this study, we propose a multiply robust (MR) estimator, allowing multiple propensity score models and outcome regression models, to provide multiple protection.

METHOD

Based on the previous MR estimator (Han 2014) and pseudo-observation approach, we proposed a new MR estimator for estimating the difference in survival functions. The proposed MR estimator based on the pseudo-observation approach has several advantages. First, the proposed estimator has a small bias when any PS and OR models were correctly specified. Second, the proposed estimator considers the advantage pf the pseudo-observation approach, which avoids proportional hazards assumption. A Monte Carlo simulation study was performed to evaluate the performance of the proposed estimator. And the proposed estimator was used to estimate the effect of chemotherapy on triple-negative breast cancer (TNBC) in real data.

RESULTS

The simulation studies showed that the bias of the proposed estimator was small, and the coverage rate was close to 95% when any model for propensity score or outcome regression is correctly specified regardless of whether the proportional hazard assumption holds, finite sample size and censoring rate. And the simulation results also showed that even though the propensity score models are misspecified, the bias of the proposed estimator was still small when there is a correct model in candidate outcome regression models. And we applied the proposed estimator in real data, finding that chemotherapy could improve the prognosis of TNBC.

CONCLUSIONS

The proposed estimator, allowing multiple propensity score and outcome regression models, provides multiple protection for estimating the difference in survival functions. The proposed estimator provided a new choice when researchers have a "difficult time" choosing only one model for their studies.

Predicting 10-year breast cancer mortality risk in the general female population in England: a model development and validation study

BACKGROUND

Identifying female individuals at highest risk of developing life-threatening breast cancers could inform novel stratified early detection and prevention strategies to reduce breast cancer mortality, rather than only considering cancer incidence. We aimed to develop a prognostic model that accurately predicts the 10-year risk of breast cancer mortality in female individuals without breast cancer at baseline.

METHODS

In this model development and validation study, we used an open cohort study from the QResearch primary care database, which was linked to secondary care and national cancer and mortality registers in England, UK. The data extracted were from female individuals aged 20-90 years without previous breast cancer or ductal carcinoma in situ who entered the cohort between Jan 1, 2000, and Dec 31, 2020. The primary outcome was breast cancer-related death, which was assessed in the full dataset. Cox proportional hazards, competing risks regression, XGBoost, and neural network modelling approaches were used to predict the risk of breast cancer death within 10 years using routinely collected health-care data. Death due to causes other than breast cancer was the competing risk. Internal-external validation was used to evaluate prognostic model performance (using Harrell's C, calibration slope, and calibration in the large), performance heterogeneity, and transportability. Internal-external validation involved dataset partitioning by time period and geographical region. Decision curve analysis was used to assess clinical utility.

FINDINGS

We identified data for 11 626 969 female individuals, with 70 095 574 person-years of follow-up. There were 142 712 (1·2%) diagnoses of breast cancer, 24 043 (0·2%) breast cancer-related deaths, and 696 106 (6·0%) deaths from other causes. Meta-analysis pooled estimates of Harrell's C were highest for the competing risks model (0·932, 95% CI 0·917-0·946). The competing risks model was well calibrated overall (slope 1·011, 95% CI 0·978-1·044), and across different ethnic groups. Decision curve analysis suggested favourable clinical utility across all age groups. The XGBoost and neural network models had variable performance across age and ethnic groups.

INTERPRETATION

A model that predicts the combined risk of developing and then dying from breast cancer at the population level could inform stratified screening or chemoprevention strategies. Further evaluation of the competing risks model should comprise effect and health economic assessment of model-informed strategies.

FUNDING

Cancer Research UK.

Adjusting for informative cluster size in pseudo-value-based regression approaches with clustered time to event data

Informative cluster size (ICS) arises in situations with clustered data where a latent relationship exists between the number of participants in a cluster and the outcome measures. Although this phenomenon has been sporadically reported in the statistical literature for nearly two decades now, further exploration is needed in certain statistical methodologies to avoid potentially misleading inferences. For inference about population quantities without covariates, inverse cluster size reweightings are often employed to adjust for ICS. Further, to study the effect of covariates on disease progression described by a multistate model, the pseudo-value regression technique has gained popularity in time-to-event data analysis. We seek to answer the question: "How to apply pseudo-value regression to clustered time-to-event data when cluster size is informative?" ICS adjustment by the reweighting method can be performed in two steps; estimation of marginal functions of the multistate model and fitting the estimating equations based on pseudo-value responses, leading to four possible strategies. We present theoretical arguments and thorough simulation experiments to ascertain the correct strategy for adjusting for ICS. A further extension of our methodology is implemented to include informativeness induced by the intracluster group size. We demonstrate the methods in two real-world applications: (i) to determine predictors of tooth survival in a periodontal study and (ii) to identify indicators of ambulatory recovery in spinal cord injury patients who participated in locomotor-training rehabilitation.

Development and internal-external validation of statistical and machine learning models for breast cancer prognostication: cohort study

OBJECTIVE

To develop a clinically useful model that estimates the 10 year risk of breast cancer related mortality in women (self-reported female sex) with breast cancer of any stage, comparing results from regression and machine learning approaches.

DESIGN

Population based cohort study.

SETTING

QResearch primary care database in England, with individual level linkage to the national cancer registry, Hospital Episodes Statistics, and national mortality registers.

PARTICIPANTS

141 765 women aged 20 years and older with a diagnosis of invasive breast cancer between 1 January 2000 and 31 December 2020.

MAIN OUTCOME MEASURES

Four model building strategies comprising two regression (Cox proportional hazards and competing risks regression) and two machine learning (XGBoost and an artificial neural network) approaches. Internal-external cross validation was used for model evaluation. Random effects meta-analysis that pooled estimates of discrimination and calibration metrics, calibration plots, and decision curve analysis were used to assess model performance, transportability, and clinical utility.

RESULTS

During a median 4.16 years (interquartile range 1.76-8.26) of follow-up, 21 688 breast cancer related deaths and 11 454 deaths from other causes occurred. Restricting to 10 years maximum follow-up from breast cancer diagnosis, 20 367 breast cancer related deaths occurred during a total of 688 564.81 person years. The crude breast cancer mortality rate was 295.79 per 10 000 person years (95% confidence interval 291.75 to 299.88). Predictors varied for each regression model, but both Cox and competing risks models included age at diagnosis, body mass index, smoking status, route to diagnosis, hormone receptor status, cancer stage, and grade of breast cancer. The Cox model's random effects meta-analysis pooled estimate for Harrell's C index was the highest of any model at 0.858 (95% confidence interval 0.853 to 0.864, and 95% prediction interval 0.843 to 0.873). It appeared acceptably calibrated on calibration plots. The competing risks regression model had good discrimination: pooled Harrell's C index 0.849 (0.839 to 0.859, and 0.821 to 0.876, and evidence of systematic miscalibration on summary metrics was lacking. The machine learning models had acceptable discrimination overall (Harrell's C index: XGBoost 0.821 (0.813 to 0.828, and 0.805 to 0.837); neural network 0.847 (0.835 to 0.858, and 0.816 to 0.878)), but had more complex patterns of miscalibration and more variable regional and stage specific performance. Decision curve analysis suggested that the Cox and competing risks regression models tested may have higher clinical utility than the two machine learning approaches.

CONCLUSION

In women with breast cancer of any stage, using the predictors available in this dataset, regression based methods had better and more consistent performance compared with machine learning approaches and may be worthy of further evaluation for potential clinical use, such as for stratified follow-up.

Bivariate pseudo-observations for recurrent event analysis with terminal events

The analysis of recurrent events in the presence of terminal events requires special attention. Several approaches have been suggested for such analyses either using intensity models or marginal models. When analysing treatment effects on recurrent events in controlled trials, special attention should be paid to competing deaths and their impact on interpretation. This paper proposes a method that formulates a marginal model for recurrent events and terminal events simultaneously. Estimation is based on pseudo-observations for both the expected number of events and survival probabilities. Various relevant hypothesis tests in the framework are explored. Theoretical derivations and simulation studies are conducted to investigate the behaviour of the method. The method is applied to two real data examples. The bivariate marginal pseudo-observation model carries the strength of a two-dimensional modelling procedure and performs well in comparison with available models. Finally, an extension to a three-dimensional model, which decomposes the terminal event per death cause, is proposed and exemplified.

Fast approximations of pseudo-observations in the context of right censoring and interval censoring

In the context of right-censored and interval-censored data, we develop asymptotic formulas to compute pseudo-observations for the survival function and the restricted mean survival time (RMST). These formulas are based on the original estimators and do not involve computation of the jackknife estimators. For right-censored data, Von Mises expansions of the Kaplan-Meier estimator are used to derive the pseudo-observations. For interval-censored data, a general class of parametric models for the survival function is studied. An asymptotic representation of the pseudo-observations is derived involving the Hessian matrix and the score vector. Theoretical results that justify the use of pseudo-observations in regression are also derived. The formula is illustrated on the piecewise-constant-hazard model for the RMST. The proposed approximations are extremely accurate, even for small sample sizes, as illustrated by Monte Carlo simulations and real data. We also study the gain in terms of computation time, as compared to the original jackknife method, which can be substantial for a large dataset.

A pseudo-value regression approach for differential network analysis of co-expression data

BACKGROUND

The differential network (DN) analysis identifies changes in measures of association among genes under two or more experimental conditions. In this article, we introduce a pseudo-value regression approach for network analysis (PRANA). This is a novel method of differential network analysis that also adjusts for additional clinical covariates. We start from mutual information criteria, followed by pseudo-value calculations, which are then entered into a robust regression model.

RESULTS

This article assesses the model performances of PRANA in a multivariable setting, followed by a comparison to dnapath and DINGO in both univariable and multivariable settings through variety of simulations. Performance in terms of precision, recall, and F1 score of differentially connected (DC) genes is assessed. By and large, PRANA outperformed dnapath and DINGO, neither of which is equipped to adjust for available covariates such as patient-age. Lastly, we employ PRANA in a real data application from the Gene Expression Omnibus database to identify DC genes that are associated with chronic obstructive pulmonary disease to demonstrate its utility.

CONCLUSION

To the best of our knowledge, this is the first attempt of utilizing a regression modeling for DN analysis by collective gene expression levels between two or more groups with the inclusion of additional clinical covariates. By and large, adjusting for available covariates improves accuracy of a DN analysis.

Propensity score matching and stratification using multiparty data without pooling

Matching and stratification based on confounding factors or propensity scores (PS) are powerful approaches for reducing confounding bias in indirect treatment comparisons. However, implementing these approaches requires pooled individual patient data (IPD). The research presented here was motivated by an indirect comparison between a single-armed trial in acute myeloid leukemia (AML), and two external AML registries with current treatments for a control. For confidentiality reasons, IPD cannot be pooled. Common approaches to adjusting confounding bias, such as PS matching or stratification, cannot be applied as 1) a model for PS, for example, a logistic model, cannot be fitted without pooling covariate data; 2) pooling response data may be necessary for some statistical inference (e.g., estimating the SE of mean difference of matched pairs) after PS matching. We propose a set of approaches that do not require pooling IPD, using a combination of methods including a linear discriminant for matching and stratification, and secure multiparty computation for estimation of within-pair sample variance and for calculations involving multiple control sources. The approaches only need to share aggregated data offline, rather than real-time secure data transfer, as required by typical secure multiparty computation for model fitting. For survival analysis, we propose an approach using restricted mean survival time. A simulation study was conducted to evaluate this approach in several scenarios, in particular, with a mixture of continuous and binary covariates. The results confirmed the robustness and efficiency of the proposed approach. A real data example is also provided for illustration.

Development and external validation of a risk prediction model for falls in patients with an indication for antihypertensive treatment: retrospective cohort study

OBJECTIVE

To develop and externally validate the STRAtifying Treatments In the multi-morbid Frail elderlY (STRATIFY)-Falls clinical prediction model to identify the risk of hospital admission or death from a fall in patients with an indication for antihypertensive treatment.

DESIGN

Retrospective cohort study.

SETTING

Primary care data from electronic health records contained within the UK Clinical Practice Research Datalink (CPRD).

PARTICIPANTS

Patients aged 40 years or older with at least one blood pressure measurement between 130 mm Hg and 179 mm Hg.

MAIN OUTCOME MEASURE

First serious fall, defined as hospital admission or death with a primary diagnosis of a fall within 10 years of the index date (12 months after cohort entry). Model development was conducted using a Fine-Gray approach in data from CPRD GOLD, accounting for the competing risk of death from other causes, with subsequent recalibration at one, five, and 10 years using pseudo values. External validation was conducted using data from CPRD Aurum, with performance assessed through calibration curves and the observed to expected ratio, C statistic, and D statistic, pooled across general practices, and clinical utility using decision curve analysis at thresholds around 10%.

RESULTS

Analysis included 1 772 600 patients (experiencing 62 691 serious falls) from CPRD GOLD used in model development, and 3 805 366 (experiencing 206 956 serious falls) from CPRD Aurum in the external validation. The final model consisted of 24 predictors, including age, sex, ethnicity, alcohol consumption, living in an area of high social deprivation, a history of falls, multiple sclerosis, and prescriptions of antihypertensives, antidepressants, hypnotics, and anxiolytics. Upon external validation, the recalibrated model showed good discrimination, with pooled C statistics of 0.833 (95% confidence interval 0.831 to 0.835) and 0.843 (0.841 to 0.844) at five and 10 years, respectively. Original model calibration was poor on visual inspection and although this was improved with recalibration, under-prediction of risk remained (observed to expected ratio at 10 years 1.839, 95% confidence interval 1.811 to 1.865). Nevertheless, decision curve analysis suggests potential clinical utility, with net benefit larger than other strategies.

CONCLUSIONS

This prediction model uses commonly recorded clinical characteristics and distinguishes well between patients at high and low risk of falls in the next 1-10 years. Although miscalibration was evident on external validation, the model still had potential clinical utility around risk thresholds of 10% and so could be useful in routine clinical practice to help identify those at high risk of falls who might benefit from closer monitoring or early intervention to prevent future falls. Further studies are needed to explore the appropriate thresholds that maximise the model's clinical utility and cost effectiveness.

Analysis of survival data with cure fraction and variable selection: A pseudo-observations approach

In biomedical studies, survival data with a cure fraction (the proportion of subjects cured of disease) are commonly encountered. The mixture cure and bounded cumulative hazard models are two main types of cure fraction models when analyzing survival data with long-term survivors. In this article, in the framework of the Cox proportional hazards mixture cure model and bounded cumulative hazard model, we propose several estimators utilizing pseudo-observations to assess the effects of covariates on the cure rate and the risk of having the event of interest for survival data with a cure fraction. A variable selection procedure is also presented based on the pseudo-observations using penalized generalized estimating equations for proportional hazards mixture cure and bounded cumulative hazard models. Extensive simulation studies are conducted to examine the proposed methods. The proposed technique is demonstrated through applications to a melanoma study and a dental data set with high-dimensional covariates.

Regression modeling of restricted mean survival time for left-truncated right-censored data

The restricted mean survival time (RMST) is a clinically meaningful summary measure in studies with survival outcomes. Statistical methods have been developed for regression analysis of RMST to investigate impacts of covariates on RMST, which is a useful alternative to the Cox regression analysis. However, existing methods for regression modeling of RMST are not applicable to left-truncated right-censored data that arise frequently in prevalent cohort studies, for which the sampling bias due to left truncation and informative censoring induced by the prevalent sampling scheme must be properly addressed. The pseudo-observation (PO) approach has been used in regression modeling of RMST for right-censored data and competing-risks data. For left-truncated right-censored data, we propose to directly model RMST as a function of baseline covariates based on POs under general censoring mechanisms. We adjust for the potential covariate-dependent censoring or dependent censoring by the inverse probability of censoring weighting method. We establish large sample properties of the proposed estimators and assess their finite sample performances by simulation studies under various scenarios. We apply the proposed methods to a prevalent cohort of women diagnosed with stage IV breast cancer identified from surveillance, epidemiology, and end results-medicare linked database.

Doubly-robust estimator of the difference in restricted mean times lost with competing risks data

In the context of competing risks data, the subdistribution hazard ratio has limited clinical interpretability to measure treatment effects. An alternative is the difference in restricted mean times lost (RMTL), which gives the mean time lost to a specific cause of failure between treatment groups. In non-randomized studies, the average causal effect is conventionally used for decision-making about treatment and public health policies. We show how the difference in RMTL can be estimated by contrasting the integrated cumulative incidence functions from a Fine-Gray model. We also show how the difference in RMTL can be estimated by using inverse probability of treatment weighting and contrasts between weighted non-parametric estimators of the area below the cumulative incidence. We use pseudo-observation approaches to estimate both component models and we integrate them into a doubly-robust estimator. We demonstrate that this estimator is consistent when either component is correctly specified. We conduct simulation studies to assess its finite-sample performance and demonstrate its inherited consistency property from its component models. We also examine the performance of this estimator under varying degrees of covariate overlap and under a model misspecification of nonlinearity. We apply the proposed method to assess biomarker-treatment interaction in subpopulations of the POPLAR and OAK randomized controlled trials of second-line therapy for advanced non-small-cell lung cancer.

Minimum sample size calculations for external validation of a clinical prediction model with a time-to-event outcome

Previous articles in Statistics in Medicine describe how to calculate the sample size required for external validation of prediction models with continuous and binary outcomes. The minimum sample size criteria aim to ensure precise estimation of key measures of a model's predictive performance, including measures of calibration, discrimination, and net benefit. Here, we extend the sample size guidance to prediction models with a time-to-event (survival) outcome, to cover external validation in datasets containing censoring. A simulation-based framework is proposed, which calculates the sample size required to target a particular confidence interval width for the calibration slope measuring the agreement between predicted risks (from the model) and observed risks (derived using pseudo-observations to account for censoring) on the log cumulative hazard scale. Precise estimation of calibration curves, discrimination, and net-benefit can also be checked in this framework. The process requires assumptions about the validation population in terms of the (i) distribution of the model's linear predictor and (ii) event and censoring distributions. Existing information can inform this; in particular, the linear predictor distribution can be approximated using the C-index or Royston's D statistic from the model development article, together with the overall event risk. We demonstrate how the approach can be used to calculate the sample size required to validate a prediction model for recurrent venous thromboembolism. Ideally the sample size should ensure precise calibration across the entire range of predicted risks, but must at least ensure adequate precision in regions important for clinical decision-making. Stata and R code are provided.

A comparison of statistical methods to predict the residual lifetime risk

Lifetime risk measures the cumulative risk for developing a disease over one's lifespan. Modeling the lifetime risk must account for left truncation, the competing risk of death, and inference at a fixed age. In addition, statistical methods to predict the lifetime risk should account for covariate-outcome associations that change with age. In this paper, we review and compare statistical methods to predict the lifetime risk. We first consider a generalized linear model for the lifetime risk using pseudo-observations of the Aalen-Johansen estimator at a fixed age, allowing for left truncation. We also consider modeling the subdistribution hazard with Fine-Gray and Royston-Parmar flexible parametric models in left truncated data with time-covariate interactions, and using these models to predict lifetime risk. In simulation studies, we found the pseudo-observation approach had the least bias, particularly in settings with crossing or converging cumulative incidence curves. We illustrate our method by modeling the lifetime risk of atrial fibrillation in the Framingham Heart Study. We provide technical guidance to replicate all analyses in R.

Direct modeling of the crude probability of cancer death and the number of life years lost due to cancer without the need of cause of death: a pseudo-observation approach in the relative survival setting

In population-based cancer studies, net survival is a crucial measure for population comparison purposes. However, alternative measures, namely the crude probability of death (CPr) and the number of life years lost (LYL) due to death according to different causes, are useful as complementary measures for reflecting different dimensions in terms of prognosis, treatment choice, or development of a control strategy. When the cause of death (COD) information is available, both measures can be estimated in competing risks setting using either cause-specific or subdistribution hazard regression models or with the pseudo-observation approach through direct modeling. We extended the pseudo-observation approach in order to model the CPr and the LYL due to different causes when information on COD is unavailable or unreliable (i.e., in relative survival setting). In a simulation study, we assessed the performance of the proposed approach in estimating regression parameters and examined models with different link functions that can provide an easier interpretation of the parameters. We showed that the pseudo-observation approach performs well for both measures and we illustrated their use on cervical cancer data from the England population-based cancer registry. A tutorial showing how to implement the method in R software is also provided.

Weighted pseudo-values for partly unobserved group membership in paediatric stem cell transplantation studies

Generalised pseudo-values have been suggested to evaluate the impact of allogeneic stem cell transplantation on childhood leukaemia. The approach compares long-term survival of two cohorts defined by the availability or non-availability of suitable donors for stem cell transplantation. A patient's cohort membership becomes known only after completed donor search with or without an identified donor. If a patient suffers an event during donor search, stem cell transplantation will no longer be indicated. In such a case, donor search will be ceased and cohort membership will remain unknown. The generalised pseudo-values approach considers donor identification as binary time-dependent covariate and uses inverse-probability-of-censoring weighting to adjust for non-identified donors. The approach leads to time-consuming computations due to multiple redefinitions of the risk set for pseudo-value calculation and an explicit adjustment for waiting-time bias. Here, the problem is looked at from a different angle. By considering the probability that a donor would have been identified after ceasing of donor search, weights for common pseudo-values are defined. This leads to a faster alternative approach as only a single risk set is necessary. Extensive computer simulations show that both, the generalised and the new weighted pseudo-values approach, provide approximately unbiased estimates. Confidence interval coverage is satisfactory for typical clinical scenarios. In situations, where donor identification takes considerably longer than usual, the weighted pseudo-values approach is preferable. Both approaches complement each other as they have different potential in addressing further aspects of the underlying medical question.

A semiparametric modeling approach for analyzing clinical biomarkers restricted to limits of detection

Before biomarkers can be used in clinical trials or patients' management, the laboratory assays that measure their levels have to go through development and analytical validation. One of the most critical performance metrics for validation of any assay is related to the minimum amount of values that can be detected and any value below this limit is referred to as below the limit of detection (LOD). Most of the existing approaches that model such biomarkers, restricted by LOD, are parametric in nature. These parametric models, however, heavily depend on the distributional assumptions, and can result in loss of precision under the model or the distributional misspecifications. Using an example from a prostate cancer clinical trial, we show how a critical relationship between serum androgen biomarker and a prognostic factor of overall survival is completely missed by the widely used parametric Tobit model. Motivated by this example, we implement a semiparametric approach, through a pseudo-value technique, that effectively captures the important relationship between the LOD restricted serum androgen and the prognostic factor. Our simulations show that the pseudo-value based semiparametric model outperforms a commonly used parametric model for modeling below LOD biomarkers by having lower mean square errors of estimation.

The impact of atrial fibrillation and long-term oral anticoagulant use on all-cause and cardiovascular mortality: A 12-year evaluation of the prospective Brazilian Study of Stroke Mortality and Morbidity

BACKGROUND

Atrial fibrillation is a predictor of poor prognosis after stroke.

AIMS

To evaluate atrial fibrillation and all-cause and cardiovascular mortality in a stroke cohort with low socioeconomic status, taking into consideration oral anticoagulant use during 12-year follow-up.

METHODS

All-cause mortality was analyzed by Kaplan-Meier survival curve and Cox regression models to estimate hazard ratios and 95% confidence intervals (95% CI). For specific mortality causes, cumulative incidence functions were computed. A logit link function was used to calculate odds ratios (OR) with 95% CIs. Full models were adjusted by age, sex, oral anticoagulant use (as a time-dependent variable) and cardiovascular risk factors.

RESULTS

Of 1121 ischemic stroke participants, 17.8% had atrial fibrillation. Overall, 654 deaths (58.3%) were observed. Survival rate was lower (median days, interquartile range-IQR) among those with atrial fibrillation (531, IQR: 46-2039) vs. non-atrial fibrillation (1808, IQR: 334-3301), p-log rank < 0.0001). Over 12-year follow-up, previous atrial fibrillation was associated with increased mortality: all-cause (multivariable hazard ratios, 1.82; 95% CI: 1.43-2.31) and cardiovascular mortality (multivariable OR, 2.07; 95% CI: 1.36-3.14), but not stroke mortality. In the same multivariable models, oral anticoagulant use was inversely associated with all-cause mortality (oral anticoagulant time-dependent effect: multivariable hazard ratios, 0.47; 95% CI: 0.30-0.50, p = 0.002) and stroke mortality (oral anticoagulant time-dependent effect ≥ 6 months: multivariable OR, 0.09; 95% CI: 0.01-0.65, p-value = 0.02), but not cardiovascular mortality.

CONCLUSIONS

Among individuals with low socioeconomic status, atrial fibrillation was an independent predictor of poor survival, increasing all-cause and cardiovascular mortality risk. Long-term oral anticoagulant use was associated with a markedly reduced risk of all-cause and stroke mortality.

Regression models using parametric pseudo-observations

Pseudo-observations based on the nonparametric Kaplan-Meier estimator of the survival function have been proposed as an alternative to the widely used Cox model for analyzing censored time-to-event data. Using a spline-based estimator of the survival has some potential benefits over the nonparametric approach in terms of less variability. We propose to define pseudo-observations based on a flexible parametric estimator and use these for analysis in regression models to estimate parameters related to the cumulative risk. We report the results of a simulation study that compares the empirical standard errors of estimates based on parametric and nonparametric pseudo-observations in various settings. Our simulations show that in some situations there is a substantial gain in terms of reduced variability using the proposed parametric pseudo-observations compared with the nonparametric pseudo-observations. The gain can be measured as a reduction of the empirical standard error by up to about one third; corresponding to an additional 125% larger sample size. We illustrate the use of the proposed method in a brief data example.

Causal inference for recurrent event data using pseudo-observations

Recurrent event data are commonly encountered in observational studies where each subject may experience a particular event repeatedly over time. In this article, we aim to compare cumulative rate functions (CRFs) of two groups when treatment assignment may depend on the unbalanced distribution of confounders. Several estimators based on pseudo-observations are proposed to adjust for the confounding effects, namely inverse probability of treatment weighting estimator, regression model-based estimators, and doubly robust estimators. The proposed marginal regression estimator and doubly robust estimators based on pseudo-observations are shown to be consistent and asymptotically normal. A bootstrap approach is proposed for the variance estimation of the proposed estimators. Model diagnostic plots of residuals are presented to assess the goodness-of-fit for the proposed regression models. A family of adjusted two-sample pseudo-score tests is proposed to compare two CRFs. Simulation studies are conducted to assess finite sample performance of the proposed method. The proposed technique is demonstrated through an application to a hospital readmission data set.

Using pseudo-observations for estimation in relative survival

A common goal in the analysis of the long-term survival related to a specific disease is to estimate a measure that is comparable between populations with different general population mortality. When cause of death is unavailable or unreliable, as for example in cancer registry studies, relative survival methodology is used-in addition to the mortality data of the patients, we use the data on the mortality of the general population. In this article, we focus on the marginal relative survival measure that summarizes the information about the disease-specific hazard. Under additional assumptions about latent times to death of each cause, this measure equals net survival. We propose a new approach to estimation based on pseudo-observations and derive two estimators of its variance. The properties of the new approach are assessed both theoretically and with simulations, showing practically no bias and a close to nominal coverage of the confidence intervals with the precise formula for the variance. The approximate formula for the variance has sufficiently good performance in large samples where the precise formula calculation becomes computationally intensive. Using bladder cancer data and simulations, we show that the behavior of the new approach is very close to that of the Pohar Perme estimator but has the important advantage of a simpler formula that does not require numerical integration and therefore lends itself more naturally to further extensions.

Regression analysis of recurrent-event-free time from multiple follow-up windows

This research develops multivariable restricted time models appropriate for analysis of recurrent events data, where data is repurposed into censored longitudinal time-to-first-event outcomes in τ-length follow-up windows. We develop two approaches for addressing the censored nature of the outcomes: a pseudo-observation (PO) approach and a multiple-imputation (MI) approach. Each of these approaches allows for complete data methods, such as generalized estimating equations, to be used for the analysis of the newly constructed correlated outcomes. Through simulation, this manuscript assesses the performance of the proposed PO and MI methods. Both PO and MI approaches show attractive results with either correlated or independent gap times in an individual. We also demonstrate how to apply the proposed methods in the data from azithromycin in Chronic Obstructive Pulmonary Disease Trial.

Modeling marginal features in studies of recurrent events in the presence of a terminal event

We study models for recurrent events with special emphasis on the situation where a terminal event acts as a competing risk for the recurrent events process and where there may be gaps between periods during which subjects are at risk for the recurrent event. We focus on marginal analysis of the expected number of events and show that an Aalen-Johansen type estimator proposed by Cook and Lawless is applicable in this situation. A motivating example deals with psychiatric hospital admissions where we supplement with analyses of the marginal distribution of time to the competing event and the marginal distribution of the time spent in hospital. Pseudo-observations are used for the latter purpose.

Regression analysis in an illness-death model with interval-censored data: A pseudo-value approach

Pseudo-values provide a method to perform regression analysis for complex quantities with right-censored data. A further complication, interval-censored data, appears when events such as dementia are studied in an epidemiological cohort. We propose an extension of the pseudo-value approach for interval-censored data based on a semi-parametric estimator computed using penalised likelihood and splines. This estimator takes interval-censoring and competing risks into account in an illness-death model. We apply the pseudo-value approach to three mean value parameters of interest in studies of dementia: the probability of staying alive and non-demented, the restricted mean survival time without dementia and the absolute risk of dementia. Simulation studies are conducted to examine properties of pseudo-values based on this semi-parametric estimator. The method is applied to the French cohort PAQUID, which included more than 3,000 non-demented subjects, followed for dementia for more than 25 years.

Goodness of fit tests for estimating equations based on pseudo-observations

We study regression models for mean value parameters in survival analysis based on pseudo-observations. Such parameters include the survival probability and the cumulative incidence in a single point as well as the restricted mean life time and the cause-specific number of years lost. Goodness of fit techniques for such models based on cumulative sums of pseudo-residuals are derived including asymptotic results and Monte Carlo simulations. Practical examples from liver cirrhosis and bone marrow transplantation are also provided.

Testing for center effects on survival and competing risks outcomes using pseudo-value regression

In multi-center studies, the presence of a cluster effect leads to correlation among outcomes within a center and requires different techniques to handle such correlation. Testing for a cluster effect can serve as a pre-screening step to help guide the researcher towards the appropriate analysis. With time to event data, score tests have been proposed which test for the presence of a center effect on the hazard function. However, sometimes researchers are interested in directly modeling other quantities such as survival probabilities or cumulative incidence at a fixed time. We propose a test for the presence of a center effect acting directly on the quantity of interest using pseudo-value regression, and derive the asymptotic properties of our proposed test statistic. We examine the performance of our proposed test through simulation studies in both survival and competing risks settings. The proposed test may be more powerful than tests based on the hazard function in settings where the center effect is time-varying. We illustrate the test using a multicenter registry study of survival and competing risks outcomes after hematopoietic cell transplantation.

Dynamic prediction of repeated events data based on landmarking model: application to colorectal liver metastases data

BACKGROUND

In some clinical situations, patients experience repeated events of the same type. Among these, cancer recurrences can result in terminal events such as death. Therefore, here we dynamically predicted the risks of repeated and terminal events given longitudinal histories observed before prediction time using dynamic pseudo-observations (DPOs) in a landmarking model.

METHODS

The proposed DPOs were calculated using Aalen-Johansen estimator for the event processes described in the multi-state model. Furthermore, in the absence of a terminal event, a more convenient approach without matrix operation was described using the ordering of repeated events. Finally, generalized estimating equations were used to calculate probabilities of repeated and terminal events, which were treated as multinomial outcomes.

RESULTS

Simulation studies were conducted to assess bias and investigate the efficiency of the proposed DPOs in a finite sample. Little bias was detected in DPOs even under relatively heavy censoring, and the method was applied to data from patients with colorectal liver metastases.

CONCLUSIONS

The proposed method enabled intuitive interpretations of terminal event settings.

Effectiveness of the "Elderly Activity Performance Intervention" on elderly patients' discharge from a short-stay unit at the emergency department: a quasi-experimental trial

Purpose

To examine the effectiveness of the Elderly Activity Performance Intervention on reducing the risk of readmission in elderly patients discharged from a short-stay unit at the emergency department.

Patients and methods

The study was conducted as a nonrandomized, quasi-experimental trial. Three hundred and seventy-five elderly patients were included and allocated to the Elderly Activity Performance Intervention (n=144) or usual practice (n=231). The intervention consisted of 1) assessment of the patients’ performance of daily activities, 2) referral to further rehabilitation, and 3) follow-up visit the day after discharge. Primary outcome was readmission (yes/no) within 26 weeks. The study was registered in ClinicalTrial.gov (NCT02078466).

Results

No between-group differences were found in readmission. Overall, 44% of the patients in the intervention group and 42% in the usual practice group were readmitted within 26 weeks (risk difference=0.02, 95% CI: [−0.08; 0.12] and risk ratio=1.05, 95% CI: [0.83; 1.33]). No between-group differences were found in any of the secondary outcomes.

Conclusion

The Elderly Activity Performance Intervention showed no effectiveness in reducing the risk of readmission in elderly patients discharged from a short-stay unit at the emergency department. The study revealed that 60% of the elderly patients had a need for further rehabilitation after discharge.

Assessing the effect of a partly unobserved, exogenous, binary time-dependent covariate on survival probabilities using generalised pseudo-values

Background

Investigating the impact of a time-dependent intervention on the probability of long-term survival is statistically challenging. A typical example is stem-cell transplantation performed after successful donor identification from registered donors. Here, a suggested simple analysis based on the exogenous donor availability status according to registered donors would allow the estimation and comparison of survival probabilities. As donor search is usually ceased after a patient’s event, donor availability status is incompletely observed, so that this simple comparison is not possible and the waiting time to donor identification needs to be addressed in the analysis to avoid bias. It is methodologically unclear, how to directly address cumulative long-term treatment effects without relying on proportional hazards while avoiding waiting time bias.

Methods

The pseudo-value regression technique is able to handle the first two issues; a novel generalisation of this technique also avoids waiting time bias. Inverse-probability-of-censoring weighting is used to account for the partly unobserved exogenous covariate donor availability.

Results

Simulation studies demonstrate unbiasedness and satisfying coverage probabilities of the new method. A real data example demonstrates that study results based on generalised pseudo-values have a clear medical interpretation which supports the clinical decision making process.

Conclusions

The proposed generalisation of the pseudo-value regression technique enables to compare survival probabilities between two independent groups where group membership becomes known over time and remains partly unknown. Hence, cumulative long-term treatment effects are directly addressed without relying on proportional hazards while avoiding waiting time bias.

Electronic supplementary material

The online version of this article (10.1186/s12874-017-0430-5) contains supplementary material, which is available to authorized users.

The Cox model is better than the Fine and Gray model when estimating relative revision risks from arthroplasty register data

Background and purpose - Analysis of the revision-free survival of knee and hip prostheses has traditionally been performed using Kaplan-Meier analysis and Cox regression. The competing risk problem that is related to patients who die during follow-up has recently been increasingly discussed, not least with regard to the problem of choosing a suitable statistical method for the analysis. We compared the results from analyses of Cox models and Fine and Gray models. Methods - We used data simulation based on parameter estimates from the Swedish Knee Arthroplasty Register and assessed hypothetical effects of the studied risk factors. Results - The Cox model provided more adequate results. Interpretation - The parameter estimates from the Fine and Gray model can be misleading if interpreted in terms of relative risk.

Prediction errors for state occupation and transition probabilities in multi-state models

In this paper, we consider the estimation of prediction errors for state occupation probabilities and transition probabilities for multistate time-to-event data. We study prediction errors based on the Brier score and on the Kullback-Leibler score and prove their properness. In the presence of right-censored data, two classes of estimators, based on inverse probability weighting and pseudo-values, respectively, are proposed, and consistency properties of the proposed estimators are investigated. The second part of the paper is devoted to the estimation of dynamic prediction errors for state occupation probabilities for multistate models, conditional on being alive, and for transition probabilities. Cross-validated versions are proposed. Our methods are illustrated on the CSL1 randomized clinical trial comparing prednisone versus placebo for liver cirrhosis patients.

A new approach to regression analysis of censored competing-risks data

An approximate likelihood approach is developed for regression analysis of censored competing-risks data. This approach models directly the cumulative incidence function, instead of the cause-specific hazard function, in terms of explanatory covariates under a proportional subdistribution hazards assumption. It uses a self-consistent iterative procedure to maximize an approximate semiparametric likelihood function, leading to an asymptotically normal and efficient estimator of the vector of regression parameters. Simulation studies demonstrate its advantages over previous methods.

Causal inference in survival analysis using pseudo-observations

Causal inference for non-censored response variables, such as binary or quantitative outcomes, is often based on either (1) direct standardization ('G-formula') or (2) inverse probability of treatment assignment weights ('propensity score'). To do causal inference in survival analysis, one needs to address right-censoring, and often, special techniques are required for that purpose. We will show how censoring can be dealt with 'once and for all' by means of so-called pseudo-observations when doing causal inference in survival analysis. The pseudo-observations can be used as a replacement of the outcomes without censoring when applying 'standard' causal inference methods, such as (1) or (2) earlier. We study this idea for estimating the average causal effect of a binary treatment on the survival probability, the restricted mean lifetime, and the cumulative incidence in a competing risks situation. The methods will be illustrated in a small simulation study and via a study of patients with acute myeloid leukemia who received either myeloablative or non-myeloablative conditioning before allogeneic hematopoetic cell transplantation. We will estimate the average causal effect of the conditioning regime on outcomes such as the 3-year overall survival probability and the 3-year risk of chronic graft-versus-host disease. Copyright © 2017 John Wiley & Sons, Ltd.

Temporal Prediction of Future State Occupation in a Multistate Model from High-Dimensional Baseline Covariates via Pseudo-Value Regression

In many complex diseases such as cancer, a patient undergoes various disease stages before reaching a terminal state (say disease free or death). This fits a multistate model framework where a prognosis may be equivalent to predicting the state occupation at a future time t. With the advent of high throughput genomic and proteomic assays, a clinician may intent to use such high dimensional covariates in making better prediction of state occupation. In this article, we offer a practical solution to this problem by combining a useful technique, called pseudo value regression, with a latent factor or a penalized regression method such as the partial least squares (PLS) or the least absolute shrinkage and selection operator (LASSO), or their variants. We explore the predictive performances of these combinations in various high dimensional settings via extensive simulation studies. Overall, this strategy works fairly well provided the models are tuned properly. Overall, the PLS turns out to be slightly better than LASSO in most settings investigated by us, for the purpose of temporal prediction of future state occupation. We illustrate the utility of these pseudo-value based high dimensional regression methods using a lung cancer data set where we use the patients' baseline gene expression values.