Statistical issues and methods in designing and analyzing survival studies

Abstract

Background

Cancer studies that are designed for early detection and screening, or used for identifying prognostic factors, or assessing treatment efficacy and health outcome are frequently assessed with survival or time‐to‐event outcomes. These studies typically require specific methods of data analysis. Appropriate statistical methods in the context of study design and objectives are required for obtaining reliable results and valid inference. Unfortunately, variable methods for the same study objectives and dubious reporting have been noticed in the survival analysis of oncology research. Applied researchers often face difficulties in selecting appropriate statistical methods due to the complex nature of cancer studies.

Recent findings

In this report, we describe briefly major statistical issues along with related challenges in planning, designing, and analyzing of survival studies. For applied researchers, we provided flow charts for selecting appropriate statistical methods. Various available statistical procedures in common statistical packages for applying survival analysis were classified according to different objectives of the study. In addition, an illustration of the statistical analysis of some common types of time‐to‐event outcomes was shown with STATA codes.

Conclusions

We anticipate that this review article assists oncology researchers in understanding important statistical concepts involved in survival analysis and appropriately select the statistical approaches for survival analysis studies. Overall, the review may help in improving designing, conducting, analyzing, and reporting of data in survival studies.

1. INTRODUCTION

Most oncology research with follow‐up studies is characterized by observable events such as death, recurrence, and response to treatment. Time‐to‐event outcomes also referred to as survival outcomes are typically observed in the follow‐up studies using either cohort or clinical trial study designs. In evidence‐based practice, cohort and clinical trial studies are considered to be the best study designs for generating quality evidence. However, some specific characteristics involved in these study designs require special statistical considerations and adjustments. Furthermore, survival studies coupled with a complex nature of oncology research pose many statistical challenges that need to be addressed in order to obtain valid and reliable results. Some of the statistical issues and challenges transpired in survival studies have been highlighted but scattered in various literature.1, 2, 3 We highlighted the most common issues and related challenges in this review and provided possible approaches to overcome or address them correctly.

Ongoing advancements in statistical methods allow applied researchers to handle the most critical issues involved in analyzing survival data in oncology research. Applied researchers most often do not apply robust statistical approaches for their research due to lack of knowledge, limited software access, or inappropriate use of these techniques.3, 4 We have therefore provided current evidence‐based statistical approaches and related statistical commands available in common statistical software for analyzing survival outcome data for different conditions and assumptions. Issues in designing and analyzing cohort or clinical trial studies in the context of oncology research are discussed in the following sections.

2. PLANNING AND DESIGNING PHASE OF THE STUDY

2.1. Selection of an appropriate cohort

The first step in any research is to identify the appropriate sample that addresses the primary research questions. The appropriate selection of a cohort depends on the study objectives. The target population should be clearly defined with the study sample being a representative of the target population. If the study's interest focuses on estimating incidence or adverse events or predictive or prognostic factors associated with the outcomes then an entire cohort can be formulated based on the risk characteristics of the subjects. However, if the interest focuses on comparing incidence rates according to exposure or intervention groups or standardized population, then there is a need to enroll a control cohort with similar characteristics of a case‐cohort except for exposure status. Clear specification of inclusion and exclusion criteria allows investigators to form appropriate cohorts. The inclusion criteria are the characteristics of subjects that are essential for recruiting participants in the study while exclusion criteria are the characteristics of subjects to exclude participants from the study. For example, if an investigator is interested in assessing the effect of neoadjuvant chemotherapy on overall survival compared with adjuvant chemotherapy in triple negative breast cancer patients then inclusion criteria may include age 25 years or older, triple negative, and stages I to III while exclusion criteria may include recurrent cancer, pregnant women, inoperable advanced, evidence of metastases, and patients already received some kind of treatments. Any characteristics that increase ethical concerns, or possibly increase the safety of patients, or do not yield reliable information, or pose a difficulty in measuring the outcome or exposure of interest, and have difficulty to interpret findings should be listed in exclusion criteria.1 Although explicit criteria for eligibility are critical in establishing the efficacy of an intervention, the study cohort should reflect reality and not be too restrictive in pragmatic studies. For example, studying the effectiveness of health services research should include pragmatic qualities to increase the generalizability of the findings.

2.2. Determining the duration of follow‐up, entry, and exit time

There are two main time points of interest in survival: the time point at which observation or follow‐up starts (entry time) such as randomization date, diagnosis date, surgery date, and treatment initiation date and the time point at which follow‐up ends (exit time) such as occurrence of the event of interest or reach end of the study. It is crucial that these two points are clearly defined at the planning and designing phases of the study and are strictly followed during the data collection period. These two time points are used to form the primary outcome of survival analysis. Therefore, it is critical that these two time points are measured consistently for all study participants. In cases failing to do so may result in inaccurate and biased estimations. Duration of the follow‐up time of the study should be based on the objective of the study, the incidence rate of the outcome, and the characteristics of the tumor such as disease severity and prognosis of the subjects. However, the length of the follow‐up time should be sufficient to capture a satisfactory number of events.2 Similar past studies or clinical knowledge about the disease can provide information about the incidence rate of the outcome of interest. A shorter length of follow‐up may suffice for an outcome with a high incidence rate compared with a low incident outcome. Generally, slow‐progressive cancers require longer follow‐up time compared with fast‐progressive cancers.

2.3. Events

Common events of interest studied in oncology are death, disease incidence, disease recurrence, and response to treatment. In many studies, the events of interest are derived from the objective of the study. Accordingly, events are classified into primary and secondary events of interest. The primary event of interest is typically the event associated with the greatest therapeutic importance. In addition, it should be inexpensive and measurable, easily and unbiasedly ascertainable to all patients, and sensitive to evaluate hypothesized effect. Once the primary event of interest is identified and clearly defined, the outcome of the survival analysis study can be formed. The type of time‐to‐event variable is a hybrid of time (a continuous measure) and an event (a binary measure). In many cancer studies, there are multiple instances where multiple events may occur. Multiple events can be of many types such as competing events, recurrent events, combined or composite events, events that are of equal therapeutic importance, and mixed events (longitudinal continuous and time‐to‐event). If other events obstruct the occurrence of the primary event or greatly change the chances of occurrence of the primary event, then such events are called competing events.3 In the study of assessing the effect of neoadjuvant chemotherapy on the death in triple negative breast cancer patients, the deaths due to cardiac arrest or other cancers are considered as competing events. Another situation where competing risks are present is when one is interested in the type of the event that occurs “first” among different events.4 For example, among the two events, recurrence of the breast cancer and death, if recurrence occurs first, then it precludes death from happening first. If the same type of event occurs on multiple occasions during the follow‐up, then it is referred to recurrent events. For example, multiple relapses of breast cancer over the follow‐up in triple negative breast cancer study will be considered as recurrent events. Breast cancer recurrence of different types (local, regional, and systemic) can be considered as a composite outcome (presence of any recurrence type). Longitudinal measurements of quality of life assessment with overall survival can be considered as multiple outcomes of mixed types.

2.4. Censoring

A major challenge in survival analysis is that the time at which the event occurs may be unknown due to many reasons for some individuals. Common reasons that lead to incomplete information about the time to the event are a subject is lost to follow‐up or dropped out from the study, a subject has not experienced the event before the study follow‐up period ends, or a subject has an unrelated event of interest that precludes subject for further follow‐up. This situation is called “censoring.” For censored subjects, exact survival times are not available. However, we know that the subject's survival time is at least as long as the follow‐up time available.

There are different types of censoring mechanisms. Assuming if the event of interest will occur beyond the follow‐up time, then it is referred to right censoring. Point‐right censored data are most commonly seen in the time‐to‐event analysis studies. Although it is uncommon, left censoring and interval censoring can also be observed in survival studies. Left censoring occurs when a subject has experienced the outcome of an event before the follow‐up period started, but it is not observed until the follow‐up examination has been done. In left censoring, true survival time is less than or equal to the observed survival time. Interval censoring occurs when the exact time of the event is unobserved but an interval of time during which the event occurred is known. This can be usually observed in the studies where patients are assessed in periodical examinations. A related concept of censoring is truncation. In the case of censoring, it is assumed that the event of interest may occur during the unobserved period of subjects. In the case of truncation, subjects cannot experience the event of interest after truncation from the study. Although these two concepts are different, the statistical procedures are same to deal with them.

2.5. Choice of the timescale

An important aspect of survival analysis studies is the selection of an appropriate timescale. One of the commonly used timescales in follow‐up studies is time on a study such as time since randomization/enrollment, time since registration/study entry, time from diagnosis, and time from treatment initiation. Alternatively, attained age is used as a timescale where primary time variable is the age difference at the entry time and the exit time. In addition, calendar time may be another choice in environmental exposure studies. The selection of a timescale should entirely depend on primary research questions and the nature of the data. Time on a study is a natural choice of timescale for clinical studies with well‐defined entry time point for a sample of individuals who are followed up until they experience the event of interest (death, recurrence, etc).5 However, it has been recommended that the attained age is a better choice for epidemiologic cohort studies.6 These two timescales provide equivalent results when baseline hazard is an exponential function (constant hazards) of age and when the covariate of interest and age at the entry time are independent.7

2.6. Defining covariates

Defining the choice of covariates (fixed or time‐dependent), their types (categorical or quantitative), and roles (predictor, confounder, or risk factor) are very important in survival studies like any other studies. Some covariates may change their values with respect to time due to the follow‐up of patients. The choice of covariates may influence the choice of statistical analyses. Furthermore, it is advisable to collect and analyze covariates on their original scales. The role of the covariates should be specified according to the aim of the study in the study protocol. It is recommended that the investigator collects all important covariates related to different aims of the study. Literature review and subject knowledge are helpful in constructing comprehensive up‐to‐date and valid instruments for data collection.

2.7. Sample size

As for any other type of study, survival studies should have also an adequate sample size to get reliable results and power to detect significant contributions of covariates on survival outcome. In contrast to other types of studies, the validity and power of survival analysis depend on enough number of events rather than the total sample size. A rule of thumb is to have at least 10 events for each covariate.8, 9 Small sample size studies limit the generalizability of the findings, produce less reliable estimates, less imprecise and overoptimistic findings, and less power to detect significant findings. Appropriate sample size estimate specific to study aims under different assumptions and requirements of the statistical analyses is required prior to conducting any retrospective or prospective survival studies.

3. ANALYSIS PHASE OF THE STUDY

An assumption of noninformative censoring is common to all statistical methods developed for time‐to‐event data analysis meaning the distribution of survival times is not related to the distribution of censored times.10 Subjects' censoring should not occur due to reasons related to their outcome (unobserved event) or the study. In addition, the distribution of survival times should not be related to the entry time of the participants in the study. Informative censoring occurs usually when subjects drop out of the study due to disease worsening or drug toxicity or any specific reasons related to the study. Although this assumption is difficult to assess in practice, some methods have been suggested to evaluate and deal with the informative censoring data.11, 12, 13 In survival analysis, censoring should be noninformative and possible efforts should be made to minimize censoring due to loss to follow‐up. Typically, any analysis starts with the descriptive analysis followed by univariate (bivariate) and multivariable analysis. In addition, some sensitivity or ancillary analyses depending on the study characteristics or statistical methods may be conducted to validate the findings obtained in the study. The choice of statistical tests, statistical models, and procedure for developing statistical models are highly dependent on the type of outcome (single or multivariable events, right or left censoring) and aim of the study. In the following subsections, we describe the choice of common statistical methods depending on the type of outcome of the study. A summary of the statistical methods described in the following sections is presented in Table 1 with the corresponding statistical software procedures. Figure 1 displays a flow chart for choosing appropriate statistical methods for single event outcome while Figure 2 shows a flow chart for selecting appropriate statistical methods for multiple event outcomes.

Table 1.

Summary of statistical methods and corresponding software procedures for time‐to‐event data

Condition	Methods	Reference	SAS	R	Stata
Describing time‐to‐event outcome	Kaplan‐Meier estimator, Nelson‐Aalen estimator, Breslow estimator	8	Proc LIFETEST: METHOD option	survfit function	sts
Comparing groups under proportional hazards assumption	Log‐rank test	14	Proc LIFETEST: TEST statement	survfit function	sts test
Comparing groups without satisfying proportional hazards assumption	Tarone‐Ware test, Peto‐Peto Prentice test, Fleming‐Harrington test	10, 15	Proc LIFETEST: TEST statement	logrank_test function	sts test, tware peto fh()
Unadjusted and adjusted analyses under proportional hazards assumption	Cox proportion hazards model or parametric proportional hazards model with exponential or Weibull or Gompertz distributions or accelerated failure time model with Weibull or exponential distributions	16	Proc PHREG: MODEL statement; Proc LIFEREG: MODEL statement with DISTRIBUTION option	coxph function or survreg function in package survival; aftgee package	stcox or streg, distribution()
Testing proportional hazards assumption	Log‐negative‐log survival plot, scaled Schoenfeld residual plot with test	17	Proc PHREG: ASSESS statement	cox.zph function	stphplot, by(); estat phtest, plot(); estat phtest, detail
Unadjusted and adjusted analyses under proportional hazards assumption but different baseline risks due to different stratums	Stratified Cox proportion hazards model	10	Proc PHREG: MODEL statement and STRATA statement	coxph function with strata term	stcox with strata() option
Unadjusted analysis and adjusted analyses with nonproportionality data	Time‐dependent coefficients Cox model	2	Proc PHREG: MODEL statement	timecox function in timereg package	For continuous time‐varying effects: stcox with tvc() texp()
Unadjusted analysis and adjusted analyses with external time‐dependent covariates	Time‐dependent covariates Cox model	2	Proc PHREG: MODEL statement	coxph, tmerge, survSplit functions	For continuous time‐varying covariates: stcox with tvc() texp()
Unadjusted analysis and adjusted analyses with internal time‐dependent covariates	G‐estimation methods	17, 18, 19	SAS macros available in Witteman19; (https://www.hsph.harvard.edu/causal/software/ )	NA	stgest
Unadjusted analysis and adjusted analyses with a time‐dependent response with or without internal time‐dependent covariates	Joint modeling	20, 21, 22	JMFit macro	JM, joineR packages	stjm
Unadjusted and adjusted parametric analyses with nonproportional hazards data	Accelerated failure time with log‐normal, log‐logistic, and generalized gamma; Aalen's additive hazards model	23, 24, 25	Proc LIFEREG: MODEL statement with DISTRIBUTION option	survreg function in package survival; aftgee package	streg
Analyses in the presence of competing events under proportional hazards assumption	Descriptive: Cumulative incidence function (CIF), Unadjusted/adjusted analysis: Cause‐specific Cox model or Fine and Gray subdistribution hazards model	4, 26	CIF: %CIF macro; Cause‐specific hazard model: Proc PHREG; Subdistribution hazard model: Proc PHREG “eventcode” option	CIF: function cuminc in package cmprsk; Cause‐specific hazard model: survival package coxph function; Subdistribution hazard model: crr function in the cmprsk package	CIF: stcurve; stcox, failure (event type); Fine and Gray subdistribution hazards model: stcrreg
Unadjusted and adjusted survival analyses on entire distribution of time‐to‐event data	Censored quantile regression	27, 28	Proc QUANTLIFE	crq using quantreg package	stqkm
Unadjusted and adjusted analyses for clustered event data under proportional hazards assumption	Marginal Cox model (with robust variance estimator) or frailty model	2, 29	Marginal Cox model: Proc SURVEYPHREG; Frailty model: Proc PHREG‐RANDOM statement	coxph function with robust argument; frailtyPenal using frailtypack package	stcox with vce(); (data need to be stset accordingly); streg, frailty() or shared(); mestreg, distribution()
Unadjusted and adjusted survival analyses for multiple events (recurrent same event)	Frailty or random effects survival analysis, Anderson‐Gill model, marginal risk set model, the conditional risk set model (time from entry), and the conditional risk set model (time from the previous event)	2, 17, 19, 30, 31, 32	Frailty model: Proc PHREG‐RANDOM statement; Other models: Proc PHREG with appropriate options after data has been arranged according to a specific method	coxph function in the survival package after arranging data according to a specific method; cph function of the rms package; frailtyPenal using frailtypack package	stcox with vce(); (data need to be arranged for different methods differently and stset accordingly); streg, frailty () or shared(); mestreg, distribution()
Unadjusted and adjusted survival analyses for multiple events (distinct events)	Multivariate frailty models or marginal risk set model	17	Frailty model: Proc PHREG‐RANDOM statement; Proc PHREG with appropriate options after arranging data according to a specific method	coxph function in the survival package after arranging data according to a specific method; cph function of the rms package or frailtyPenal using frailtypack package	stcox with vce() strata(); (data need to be arranged for different methods differently and stset accordingly); streg, frailty() or shared(); mestreg, distribution()
Unadjusted and adjusted survival analyses with lead‐time or length‐survival bias	Corrected Cox model for lead‐time or length bias analysis	33	NA	NA	Stata routine for the lead‐time correction is available in Duffy et al35
Unadjusted and adjusted survival analyses with nonrandom assignment of treatments	Propensity score matched robust Cox model or endogenous selection survival model or two stage residual inclusion models	34, 35, 36, 37	Propensity score matched robust Cox model: %BLINPLUS	Propensity score matched robust Cox model: twang package to calculate propensity score weights (ps) and use in coxph by setting weights = ps	Propensity score matched robust Cox model: stcox with psmatch2; Two stage residual inclusion model: glm
Summarizing effect size in survival analysis in randomized trials	Restricted mean survival time analysis	38	Proc RMSTREG or Proc LIFETEST with RMST option	rmst2 using survRM2 package	strmst2 and strmst2pw
Assessing the model adequacy	Cox‐Snell residuals, Martingale residuals, and scaled Schoenfeld residuals	39	Proc PHREG with ASSESS Statement; OUTPUT Statement with options LOGLOGS, LOGSURV, RESMART, RESSCH, WTRESSCH	residuals.coxph	stcox postestimation commands
Discriminatory power	c‐statistic (Harrell's and Uno's)	18, 40	Proc PHREG: CONCORDANCE=HARRELL option, CONCORDANCE=UNO option	summary (coxph())$concordance, survConcordance in survival package, UnoC in survAUC package	stcox postestimation command: estat concordance, somersd package

Abbreviation: NA: Not available.

Figure 1.

Flow chart for selecting statistical models for analyzing time‐to‐event data with single outcome

Figure 2.

Flow chart for selecting statistical models for analyzing multiple time‐to‐event outcomes

3.1. Analysis of time to a single event outcome

3.1.1. Descriptive measures of time‐to‐event data

One of the goals of survival analysis is to estimate and interpret survival (survivor) and/or hazard functions from the survival data. These two are related functions and commonly used for describing and modeling survival data. The survivor function (or survival probability) provides the probability that a person survives longer than a specified time point. The hazard function or instantaneous failure rate is the instantaneous rate of occurrence of an event for a subject at a particular time point, given that the subject has survived up to that specified time.10 The cumulative hazard function determines the cumulative instantaneous risk up to a specified time.

The type of analysis of time‐to‐event data depends on the type of censoring. Methods that can be used to deal the left censored data include substitution, Kaplan‐Meier, and multiple imputation methods.15, 41, 42 The simplest way to treat interval censored data is to assume that the event took place at the beginning, midpoint, or at the end of each interval to get a point of the event time. Thereafter, the standard survival analysis methods for point censored data can be applied to analyze interval censored data. However, this approach is not recommended as it tends to produce invalid results. A reader can get a more in‐depth understanding of how to handle interval censored data from the published studies.43, 44, 45

The most commonly used method of describing and estimating survival probability for right censored data is the nonparametric Kaplan‐Meier or product limit estimator.8 Other methods include Nelson‐Aalen estimate of the cumulative hazard function and Breslow estimator of survivor function.9, 20 The survival times are summarized using median or mean survival time and expected future lifetime. The median survival time is the time by which 50% of the participants will have the event of interest while mean survival time is the expected event time. The expected future lifetime estimates the future lifetime provided survival up to a specified time. Kaplan‐Meier curve is often used to describe survival probability against follow‐up time. This curve can be used to estimate survival probability at a given time or median survival time. Survival probability along with a 95% confidence interval at a clinically important time should be used to describe survival data.

3.1.2. Comparing survival differences between groups (univariable analysis)

In many randomized cancer trials, investigators are interested in comparing time‐to‐event of interest differences between two or more groups that can easily be sufficed by simple univariable analysis due to the absence of any confounders. The log‐rank test is the most commonly used method of survival comparison.14 This method is more suitable when proportional hazards assumption, described later, for the group variable is satisfied. Variations of the log‐rank test, the Tarone‐Ware, the Peto‐Peto Prentice, and the Fleming‐Harrington tests are preferred when the proportional hazards assumption is not satisfied.23

3.1.3. Adjusted (multivariable) analysis

Multivariable model building process includes several steps: (a) selection of a most appropriate multivariable model, (b) selection of appropriate variables in the model, (c) selection of appropriate forms of variables, (d) model fit assessment, and (e) summarizing and reporting results from the developed model. The selection of covariates in a statistical model and model development are highly dependent on the objective of the study or aim of the model. The aim of the model can have three types: (a) to assess the effect of an exposure on survival time (inferential model), (b) to assess the relationship of explanatory variables with survival time (descriptive model), or (c) to develop a model using covariates to predict survival time (predictive).

Given the structure and distribution of the data and process of data collection, the most appropriate multivariable model is usually selected. Once the appropriate model is selected, the selection of an appropriate form of each variable is required based on the linearity assumption involved in most statistical models. Categorization of a quantitative covariate should be avoided in the analysis unless it is clinically and objectively required. An appropriate transformation may be made to improve linearity between a quantitative variable and a time‐to‐event outcome. The selection of covariates in the model highly depends on the aim of the model. For example, the variables other than exposure should be treated as probable confounding variables in the inferential model analysis, and all significant confounders must be adjusted in the model while assessing the effect of exposure on the outcome. If the aim of the study is descriptive model development, then various automated variable selection approaches (typically backward selection) may be used to finalize the list of prognostic factors included in the multivariable model. If the aim of the study is the predictive model development, then variables that predict survival should only be retained in the final model and predictors should not be selected based on mere P value from the univariable analysis. Once the model is finalized, then the adequacy of the model fit should be assessed. There are a number of approaches that can be used to assess the overall goodness of the model fit. A brief description of model assessment and summarizing of results from the model is provided in the later section of this review. There exist various statistical methodologies to address a wide range of aspects in statistical modeling of survival data. The choices of statistical models differ according to the satisfaction of the proportionality assumption.

Unadjusted or multivariable analysis under proportionality assumption

A semiparametric Cox proportional hazards model is the most widely used for analyzing time‐to‐event data under the proportionality assumption.16 The most attractive feature of this model is that it does not require to have a known function of the underlying probability distribution of the survival times. It provides the risks associated with exposures or prognostic factors relative to corresponding baseline levels of the variables. The results of the Cox proportional model are typically summarized with hazards ratio and 95% confidence interval. As mentioned, this model produces valid results only when the proportionality assumption is satisfied. In other words, the hazard ratio (HR) for two individuals with different covariate levels does not change over time. It is critical that this assumption is assessed before making any inferences from the model; otherwise, it will produce misleading interpretations. There exist formal and informal methods to assess the proportional hazards assumption. A simple method is to assess whether the log‐negative‐log survival curves among the levels of the covariate are parallel. However, quantitative variables need to be categorized into groups for this analysis. More formal methods to assess the proportional hazards assumption include smoothed scaled Schoenfeld residual plot and Schoenfeld test46 and simulated score residual plot and score test,47 which are available in common statistical packages like STATA, R, and SAS. It is recommended that the functional form of the covariates is appropriately chosen prior to assessing the proportional hazards assumption. The parametric survival model with Weibull, exponential, and Gompertz distributions can also be used under the proportionality assumption.29, 48 The exponential and Weibull models can be parameterized as either a proportional hazards model or an accelerated failure time model.

Unadjusted or multivariable analysis under nonproportional hazards

If the proportional hazards violation is detected, the standard Cox proportional hazards model needs to be adjusted to account for nonproportional hazards. A simple and easy approach is to use the stratified Cox proportional hazards model by allowing different baseline hazards for each level of a covariate that violates the proportional hazards assumption.10 This method has several drawbacks such as loss in estimation efficiency, more suitable for a limited number of categorical variables, requires categorization of a quantitative covariate with nonproportional hazards, and does not allow estimating the effect of stratified variables. Another approach for adjusting nonproportional hazards is to include an interaction term between the covariate with nonproportional hazards and a function of time in the multivariable Cox model. This model is called the time‐dependent coefficients Cox model.24 The appropriate function of time can be chosen by the theoretical knowledge or by the pattern of smoothed Schoenfeld residual plot. A detailed illustration of assessing proportional hazards and extending the Cox model for nonproportional hazards using breast cancer data can be found in a published study.49 Alternatively, one can use parametric survival models such as accelerated failure time models with log‐normal, log‐logistic, and generalized gamma models instead of the Cox proportional hazards model under nonproportional hazards.50, 51 Aside from identifying an appropriate distribution form of survival function, parametric accelerated failure time survival models allow more flexibility, ease of interpretation of estimated effect, and prediction of survival at any time.

3.2. Analysis of multiple outcomes

In some survival analysis studies, multiple events, or multivariate survival data can occur due to clustering or repeated measurements, which occur more frequently due to longitudinal follow‐up of the subjects. Clustered event times typically occur when subjects are grouped as twins, family members, study centers, etc whereas repeated event times occur when multiple episodes of an event or multiple types of events are recorded within an individual.30 Due to the correlation of observations within a group or individual, special methods are required to handle these data. In these studies, events could be unordered events of different types (competing risks, composite outcomes, and individual outcomes of equivalent importance), ordered events (recurrent events), unordered events of the same type (different types of recurrence and composite outcomes), and mixed type of outcome (longitudinal and time‐to‐event outcomes).29, 48

3.2.1. Competing events

Failure to account for the competing risks in the analysis would result in overestimation of the risk. For example, as elderly people might die due to reasons of old age before they experience a cancer‐specific death. Geriatric cancer research is more prone to have competing events.52, 53 To obtain valid cancer‐specific risks, noncancer‐related deaths or events have to be adjusted in the analysis instead of considering such events as censoring events.54, 55, 56 In the presence of competing risks, conventional Kaplan‐Meier method overestimates the risk of a primary event of interest.57 An alternative method is to use a cumulative incidence function (CIF). CIF estimates the marginal probability of an event in the presence of competing risks, and it does not require the independence of competing events.10 There are two ways of modeling competing events; one is the use of cause‐specific hazard function, and the other one is the use of subdistribution hazard function.4 Fine and Gray developed a model that is based on the subdistribution hazard to account for competing risks.26 Austin et al have recommended the use of cause‐specific hazard model to answer etiologic research questions and the use of Fine and Gray subdistribution hazard model to estimate incidence or predicting prognosis.4

3.2.2. Composite outcomes (unordered events of same or different types)

Creating a composite outcome is an alternative approach to handle multiple events in a study. It is more applicable when a primary event of interest is rare so that multiple surrogate or secondary events of interest can be combined to make a composite outcome. If there are several important outcomes in the study, then investigator can combine and create a composite outcome instead of making an arbitrary choice between outcomes. In addition, forming a composite outcome is also useful in a study in which multiple aspects are required to assess patient outcome status. Investigators should be cautious about the relative importance of each component in the composite outcome. If some components are more important than others, then weights can be assigned to reflect the relative importance. More details about weighting can be found in the literature.58, 59 There are important advantages of using composite outcomes. A composite outcome usually yields a higher event rate due to a combination of multiple outcomes, which improves statistical efficiency by reducing sample size, time, and costs.

The standard methods used for analyzing single outcome can be applied to analyze a composite time‐to‐event data as well. Sometimes, interpretation of results from composite outcome analysis can be difficult and misleading especially when the treatment or exposure effects considerably differ across different components. Results must be carefully interpreted and discussed in the presence of differential treatment effects on different components of a composite outcome. The need for a composite outcome, method for forming a composite outcome, and clinically meaningful effect for each component should be specified in the study in view of the objective of the study. It is recommended that the investigator should consistently report each component of the composite outcome and clearly report the frequency of events and missing data for all individual components to avoid misinterpretations.

3.2.3. Multiple time‐to‐event outcomes

Broadly, two main approaches have been used for modeling multiple time‐to‐event outcomes (unordered or ordered): (a) variance‐corrected models and (b) frailty models. Variance‐corrected models adjust variance estimates by accounting for dependencies between event times while frailty models adjust the dependencies between event times by introducing a random‐effect covariate into the model. Often a gamma distribution is used for the frailties (random effect). The variance‐corrected models such as marginal Cox model with jackknife variance estimator or robust sandwich estimator can be used for unordered or ordered events.

Several extensions of the Cox model that includes Anderson‐Gill model, marginal risk set model, the conditional risk set model (time from entry) and the conditional risk set model (time from the previous event), or the frailty models can be used for analyzing recurrent events (ordered events).48 In addition, marginal models for means/rates or multistate models can also be used for modeling recurrent data.29 Clustered event data can be analyzed using multivariate frailty models.2, 27, 33, 60 Unordered‐distinct event times can be analyzed using either multivariate frailty model or marginal risk set model.29 Multistate models can be used for an unordered dataset; however, it is rarely used due to lack of programs in the standard statistical software. The choice of statistical models depends on many aspects of survival data. In the case of a few events per subjects and varying risks between events, conditional risk set model is typically preferred. In the case of frequent events per subjects and constant hazard between events, frailty models are preferred. For unordered events, marginal risk set model may be preferred when interest in on population averageeffect. Random effects or frailty model preferably with accelerate failure times may be preferred when theinterest is on conditional effect of covariates on unordered event times.28

3.2.4. Longitudinal and time‐to‐event data

Some studies collect repeated measurements on a continuous response and time‐to‐event data depending on the research questions to be addressed. One of the objectives of such studies includes identification of how changes in a biomarker affect or associated with the occurrence of a certain event. For example, a study of prostate cancer may record repeated measurements on prostate‐specific antigen (PSA) along with time to recurrence of the disease. These longitudinal measurements of the PSA and time‐to‐recurrence data can be modeled jointly to observe how PSA process and cancer recurrence is associated.38 Another objective of studies that collect longitudinal and time‐to‐event outcomes might be to evaluate the changes in the biomarker also by taking the informative dropout into account. This kind of multiple outcomes should be modeled using joint models approach where longitudinal mixed effects model and survival model is modeled simultaneously.20, 34 The joint model approach assumes that the distribution of time‐to‐event and longitudinal measurements depend on a common set of latent random effects.

3.3. Additional issues involved in the survival analysis

3.3.1. Lead‐time bias and length‐time bias

Lead‐time bias and length‐time bias are two types of biases that can be seen typically in cancer screening studies. Lead time is the increased survival from detecting disease at an early stage. When survival time is measured from the date of diagnosis, lead‐time overestimates the survival time of screen‐detected individuals compared with the individuals detected by a symptomatic presentation. Selection of an appropriate timescale such as attained age in cancer screening studies usually helps to avoid the lead‐time bias. Progression speed of tumors can vary by the individual to individual. Screening tools applied at pre‐specified time points have a greater chance of detecting slow‐progression tumors than the rapidly growing tumors. Hence, it overestimates the survival time among slow progressing individuals. This is called length‐time bias.35

3.3.2. Nonrandom assignment of treatments or interventions

In practice, there can be situations where patients are not randomly assigned to treatments/interventions. This nonrandom assignment of patients to treatments will confound the treatment effect on the outcome. This will cause a correlation between treatment and the error in the outcome. Conventional methods used to estimate treatment effects are not appropriate for this type of situation. Special techniques designed to account for the differences in confounding factors should be used to obtain accurate estimates of treatment effects such as endogenous selection survival model,36 propensity score matched robust Cox model,61 or two‐stage residual inclusion models.62

3.3.3. Censored quantile regression

Censored quantile regression is an alternative method to model survival data. Unlike in the Cox model that models the hazard function, the quantile regression model can be used to model survival times itself. It models quantiles of the time‐to‐event data distribution.40, 63 This is typically used for understanding the effects of covariates on the entire survival distribution and help the investigator to explore heterogeneity in effects.

3.3.4. Time‐dependent covariates

Most of the time, the covariates being studied are fixed at the entry of the study. But, sometimes, there can be covariates that change over time such as measurements at a series of time points. These time‐dependent variables have to be modeled and interpreted differently than conventional survival models. There are two classifications of time‐dependent variables: internal and external. Internal time‐dependent covariate generates due to a behavior of the subject (eg, tumor shrinkage, smoking status, and blood counts). External time‐dependent variable arises due to an external process such as the dose of a drug, air temperature, etc. More details about handling time‐dependent covariates can be found in Hosmer et al7, 39 and Collet.6, 64 Internal time‐dependent covariates are usually analyzed by G‐estimation methods.17, 18, 65 External time‐dependent covariates can be handled by an extended Cox model with time‐dependent covariates.2

3.3.5. Summarizing effect size in survival analysis in randomized trials

Restricted mean survival time (RMST) is a novel method of summarizing survival time data. It has been found that the RMST is a better approach to summarize treatment effect than the HR. There exists a statistical test to assess the difference between restricted means of treatment arms. A major advantage of this method is that it works well in situations where the proportional hazards assumption is violated. RMST can be successfully applied to obtain valid estimates of treatment effects of cancer trials that have nonproportional hazards. Recently, regulatory agencies have also approved the use of RMST in oncology trials instead of HR.50

3.3.6. Missing values

Subjects with missing covariate data is a common issue in cancer research. The best way to handle missing data is taking an effort to minimize missing data in the design and execution stages of the research projects.66 Recommendations to minimize the amount of missing data can be found in published studies.67, 68, 69 If missing values for important variables occur in the dataset, then the data have to be examined carefully to identify the mechanism of the missingness prior to employing any strategies to impute missing values. Missing can occur due to missing completely at random (MCAR), missing at random (MAR), or missing not at random (MNAR).70 MCAR implies that the missing does not depend on the covariates or outcomes whereas MAR assumes that the missing depends on the observed outcome and the covariates. MNAR assumes that missing is dependent on the values of the outcome and the missing covariates. The usual practice is to omit the individuals with missing data and perform the analysis on complete data provided missing are due to MCAR. In case of low proportion of missing data, piecewise deletion method can be used when data are MCAR or MAR. Mean substitution, regression imputation, and multiple imputations can also be used to replace missing values with an estimated value for MAR data. Kang has described details about how to handle missing data at the analysis stage.70

3.3.7. Assessing the adequacy of the multivariable model

As for any other statistical models, assessing the fit of the survival model is important before making any inferences from the model. The different types of residuals for the Cox proportional hazard model include Cox‐Snell residuals, Martingale residuals, and scaled Schoenfeld residuals.39 Cox‐Snell residual plot is used to assess the overall goodness‐of‐fit of the model, and Martingale residual plot can be used to identify the appropriateness of the functional form of the covariates included in the model. Scaled Schoenfeld residual plot is used to assess the nonproportional hazard assumption of the covariates. The log‐negative‐log‐survival plots and Akaike's information criteria (AIC) are used for assessing fit for parametric survival modeling. Discriminatory power is also an important aspect of a fitted model. It quantifies the model's ability to correctly classify subjects for their actual outcomes. Harrell's c‐statistic and Uno's c‐statistic are two such discrimination measures. Higher c‐statistic values imply better discriminatory power.31, 32 The external validation of model performance is typically required for predictive survival models. The results of the multivariable model have summarized with the regression coefficient (or hazards ratio or survival ratio), 95% confidence interval, and P value.

4. DATA ANALYSIS

We utilized The Cancer Genome Atlas (TCGA) platform to extract breast cancer (BRCA) data generated by the TCGA Research Network (https://www.cancer.gov/tcga) to illustrate the statistical analysis of some common types of time‐to‐event outcomes. The TCGA‐BRCA cohort includes 30 datasets. The clinical data such as menopause status (premenopause, perimenopause, postmenopause, and indeterminate status), age (years), weight (pounds), stage (1/2/3/4) along with two time‐to‐event outcomes (recurrence type: locoregional disease, locoregional recurrence, distant metastasis, new primary tumor, and no recurrence; death: death and no death) were extracted from the UCSC Xena website (https://xenabrowser.net/). Time is recorded in days. We used IlliminalHiSeq TCGA hub to download data. After removing duplicates and missing, there were 1097 breast cancer patients. Variable descriptions of the example excel dataset are included in Appendix SA1, and the data file is included in Appendix SA2. We illustrated unadjusted and adjusted associations of menopause status, age, weight, and stage with time‐to‐death (death versus no death) using proportional and nonproportional hazards models. We also demonstrated adjusted associations of considered cofactors with event‐free survival defined as a composite outcome (relapse or death versus no death by considering no death as a censored event), where relapse could be of any type (locoregional disease, locoregional recurrence, distant metastasis, and new primary tumor). To appropriately estimate the risk of relapse, time‐to‐relapse data were also analyzed for any relapse (locoregional disease, locoregional recurrence, distant metastasis, new primary tumor, and no relapse by considering death as a competing event). All these analyses were conducted using STATA 15.1, and the corresponding codes are given in Appendix SB.

5. SUMMARY

In implementing statistical concepts and methodologies in cancer research, there are issues and challenges that oncologists may encounter. The purpose of this review article is to highlight systematically and comprehensively the current statistical issues and challenges involved in designing and analyzing follow‐up studies, in order to help applied researchers overcome or correctly address them. As the primary outcome in many of the oncology research is a time‐to‐an‐event of interest, we focused on the main issues/challenges in designing time‐to‐event (survival) analysis studies. Although our focus of the review is for cancer research, our review is applicable to any cohort studies. We have also provided a flow chart and a summary of statistical models depending on the variety of conditions for applied researchers. In addition, an illustration of the statistical analysis of some common types of time‐to‐event outcomes is presented along with STATA codes using breast cancer (BRCA) data generated by the TCGA Research Network. Our synthesis of current evidence‐based concepts and statistical methods may help applied researchers to understand important concepts involved in survival analysis and appropriately select the statistical approaches for their analysis. Overall, the review may help in improving designing, conducting, analyzing, and reporting of data in survival studies.

CONFLICT OF INTEREST

The authors declare no conflict of interest.

AUTHORS' CONTRIBUTIONS

All authors had full access to the data in the study and take responsibility for the integrity of the data and the accuracy of the data analysis. Conceptualization, A.K.D.; Investigation, M.P.; Resources, M.P.; Writing ‐ Original Draft, M.P.; Writing ‐ Review & Editing, A.K.D.; Supervision, A.K.D.

Supporting information

Data S1. Supporting information

Data S2. Supporting information

Data S3. Supporting information

Perera M, Dwivedi AK. Statistical issues and methods in designing and analyzing survival studies. Cancer Reports. 2020;3:e1176. 10.1002/cnr2.1176