Time to revision after joint arthroplasty is an important factor for assessing the quality of joint replacements, monitoring implant performance, and informing health policy planning decisions. The measure will play an increasingly important role in coming years given the growing demand for primary and revision hip and knee replacements [26, 28], particularly in younger, more physically active patients, who are likely to outlive their implants and undergo revision surgery .
Monitoring the incidence of revisions over time requires survival analysis because for some patients, time to revision is unknown because they are lost to followup, die before receiving a revision, or are alive and unrevised at the end of the observation period. Kaplan-Meier survival analysis  is often used, as seen in the orthopaedic literature and among joint replacement registries, to estimate the cumulative incidence of revision after joint arthroplasty. However, because the method was designed to estimate the time to a single event that will eventually occur for everyone (such as death), it does not consider other “competing risks” that may preclude and alter the probability of the event of interest ; for example, a patient who has died cannot subsequently undergo revision surgery, and using the Kaplan-Meier estimator in this setting violates one of its principal assumptions regarding the independence of events. Stated otherwise, when estimating time to revision, death represents an important competing risk. By treating deaths as censored observations, the Kaplan-Meier method assumes the risk of revision is independent of the risk of death. Consequently, the Kaplan-Meier method theoretically overestimates the cumulative incidence of an event in the presence of competing risks [7, 36]. This bias is particularly problematic for older arthroplasty populations with high mortality rates and in studies involving longer followup durations , in which a larger number of patients are followed until death rather than censoring.
Alternative statistical methods have been developed to estimate cumulative incidence of an event in competing-risks settings. By acknowledging that patients can no longer be revised after death, competing-risks methods provide an estimate of the number of revisions expected to occur at a specific time point. Thus, the competing-risks method may provide more accurate estimates that can be used to inform healthcare planning and policy decisions [12, 25, 38]. In contrast, the Kaplan-Meier method estimates the probability of a revision at a certain time point assuming patients cannot die and may be useful for informing individual patients of their risk of revision under the assumption they will live a certain number of years after their primary surgery [16, 25, 31, 38]. Given that these methods differ in their treatment of patients who experience a competing event before the event of interest, in situations in which no patients die before revision throughout the duration of followup, the Kaplan-Meier method and competing-risks method will produce the same estimate.
The application of competing-risks methods is now feasible using a variety of statistical software programs. However, recent studies have noted the Kaplan-Meier method continues to be used in the presence of competing risks [24, 40]. The purpose of our systematic review and meta-analysis was therefore to provide empiric evidence of the magnitude of overestimation of the Kaplan-Meier compared with competing-risks method when estimating the cumulative incidence of revision. We also sought to examine whether the extent of overestimation is influenced by duration of followup or the rate of competing events relative to events of interest.
Materials and Methods
Our search strategy was developed in consultation with a medical librarian/information scientist. We searched Ovid MEDLINE, EMBASE, BIOSIS Previews, and Web of Science from the first available date of each database (1946, 1980, 1980, and 1899, respectively) to October 26, 2013, without publication date, language, or other restrictions using Medical Subject Heading (MeSH) terms and keywords to cover the themes Kaplan-Meier and competing risks. For the Kaplan-Meier theme, we combined the MeSH term “Kaplan Meier Estimate” (Emtree term “Kaplan Meier method”) with a title and abstract keyword search for Kaplan Meier* or Kaplan-Meier* or Kaplanmeier* or Kaplan*Meyer* or censor*. For the competing-risks theme, we used a title and abstract search using the terms competing or cumulative incidence function* or cause*specific hazard*or sub*distribution*. The two themes were subsequently combined using the Boolean operator “AND”. To identify additional articles, we also used the PubMed “related articles” feature and hand-searched bibliographies of included studies and other potentially relevant citations identified during the search process.
Two independent reviewers (SL, TW) screened all identified titles and abstracts. Abstracts deemed potentially relevant by either reviewer were subsequently read in full. Full-text articles were included if: (1) both Kaplan-Meier and competing-risks methods, as defined subsequently, were applied to estimate the cumulative incidence of revision after joint arthroplasty; (2) cumulative incidence estimates were provided for both methods (either as point estimates or graphically); and (3) studies involved humans. Conference abstracts, unpublished studies, and studies using simulated data sets were excluded. In situations where multiple studies analyzed the same data or data subsets, we included the study that reported the most detailed information with respect to requirements for our meta-analysis (eg, count of events, number at risk) or the study with the earliest publication date. Agreement between reviewers was quantified using the κ statistic . Disagreements were resolved by consensus.
The Kaplan-Meier method was defined as the Kaplan-Meier failure function (complement of the Kaplan-Meier survival function), which estimates the probability of an event of interest occurring at a specific time point among those who had not already experienced that event. Patients who die are excluded from the at-risk population at the time of their deaths and, similar to those lost to followup, are assumed to have the same probability of revision as those remaining in the risk set . The competing-risks method was defined as the cumulative incidence function using the approach of Kalbfleisch and Prentice , which estimates the probability of the event of interest occurring at a specific time point given that neither the event of interest nor the competing event has yet occurred. Thus, the competing-risks method depends on the risk of the event of interest and the competing event, whereas the Kaplan-Meier estimate considers only the event of interest.
Among 1162 unique citations identified by our search strategy, 101 full-text articles were assessed for eligibility. Seven cohort studies compared the Kaplan-Meier and competing-risks methods when estimating the time to revision after joint arthroplasty and were included in our systematic review (κ statistic = 1) [6, 7, 16, 17, 24, 38, 41], of which six included enough data to be included in our meta-analysis [6, 7, 16, 17, 24, 41] (Fig. 1). Publication years ranged from 2001 to 2012 (Table 1). Five studies assessed time to revision after partial or total hip arthroplasty (THA) [6, 16, 17, 38, 41]; one assessed time to revision after acetabular revision  and one assessed time to revision after total knee arthroplasty (TKA) with a megaprosthesis after bone tumor resection . Death was identified as a competing risk in all studies. One study also considered amputation as a competing risk .
The same two reviewers (SL, TW) independently extracted data using a predesigned and pilot-tested data extraction tool. We extracted data regarding author, year of publication, study design, sample size, age of the population, followup time, type and number of events of interest, competing events, and the statistical software package used.
The primary data elements extracted from each study were the cumulative incidence estimates obtained for the Kaplan-Meier and competing-risks methods. These outcomes are often reported at multiple time points throughout a followup period; therefore, we extracted estimates and 95% confidence intervals (CIs), when reported, across all reported time points for each study. For studies reporting multiple stratified analyses, we extracted data for each stratum. To ensure only mutually exclusive strata from each study were included, we conducted two separate analyses for strata containing: (1) the largest number of events of interest (ie, revisions); and (2) the highest rate of competing risks relative to events of interest (ie, number of competing risks observed/number of events of interest observed). For example, Gillam et al.  analyzed three subsets of data. The subset with the largest number of events of interest compared the cumulative incidence of revision for patients receiving THA for osteoarthritis who were younger than 70 years old with patients who were aged 70 years and older. The subset with the highest proportion of competing risks compared the cumulative incidence of revision after THA for two types of monoblock prostheses.
Because no validated tool is available to assist in examining the quality of reporting specifically for survival analysis studies, we developed criteria based on recommendations and guidelines for reporting these types of analyses [1, 3, 9, 12, 34, 35, 38]. Nine criteria were assessed independently by the same two reviewers (SL, TW), who asked: (1) Was the number of patients at risk presented at each followup time? (2) Was the observed number of events of interest and competing events provided? (3) Were losses to followup clearly described? (4) Was the handling of losses to followup described (eg, treated as censored at the time of loss to followup)? (5) Was a description of censoring provided? (6) Were graphic representations of cumulative incidence of the event of interest and competing event(s) provided for the Kaplan-Meier method? (7) Were graphic representations of cumulative incidence of the event of interest and competing event(s) provided for the competing-risks method? (8) Were estimates of precision of the cumulative incidence provided (ie, SEs or CIs)? (9) Was the name of the statistical software provided? Questions answered “yes” received one point and those answered “no” received zero points. We calculated the percentage of studies that received points for each criterion to assess the overall quality of reporting for the body of our study literature and identified inconsistencies in reporting. Of the seven studies included, three (43%) provided the number at risk at each followup time or the name of the statistical software used (Table 2). The number of events was provided by six studies (86%). Five studies (71%) reported the number of losses to followup, four of which described how losses to followup were accounted for in their analysis. All seven studies described the censoring mechanisms used, although only three studies (43%) reported the number of censored observations. Cumulative incidence curves were provided in all seven studies. Only three studies (43%) provided CIs for both Kaplan-Meier and competing-risks methods. We calculated risk ratios (RRs) to compare the cumulative incidence estimated using the Kaplan-Meier method with the competing-risks method for each study, where:
Because we did not have individual patient data required to calculate the variance around the RRs, we used an approximation that has been proposed to estimate the variance (var) of a hazard ratio (HR) using summary data , where:
Because we could not find an approximation for the variance of the ratio of cumulative incidences, we used this approximation for the log HR given that both the RR and HR compare the measure of occurrence of events over time, while accounting for censoring, in the form of a ratio. We also performed a sensitivity analysis using an alternative approximation , where:
It is important to note that these variances were primarily used for the purposes of weighting each individual study for our meta-analysis. Therefore, the CIs estimated using this variance approximation must be interpreted carefully.
A DerSimonian and Laird  random-effects model was used to pool RR estimates across studies. RR estimates were log-transformed before being entered into the model. As we anticipated, the time points at which estimates were reported varied across studies, so we included estimates reported at the longest followup time point for each study. To assess interstudy heterogeneity, we inspected forest plots stratified by followup time (< 10 years, ≥ 10 years) and the rate of competing risks relative to events of interest (< 1, 1-10, > 10). We did not observe differences in the magnitude of overestimation of the Kaplan-Meier method when assessing these forest plots (data not shown). We used univariate metaregression to examine the effect of the covariates on the estimated pooled RR with p values < 0.10 considered significant given the low power of these tests . All analyses were performed using Stata/SE Version 12.0 (StataCorp, College Station, TX, USA).
To What Extent Does the Kaplan-Meier Method Overestimate the Cumulative Incidence of Revision?
Kaplan-Meier survivorship resulted in a larger estimate of the risk of revision than did the competing-risks estimator when we considered the seven strata within the population of included studies that contained a high proportion of patients who had died during the followup period. The pooled RR was 1.55 (95% CI, 1.43-1.68; p < 0.001), indicating that the cumulative incidence of revision estimated using the Kaplan-Meier approach was 55% greater than that obtained using the competing-risks estimator (Fig. 2A). The RRs for these six studies, including seven mutually exclusive strata, ranged from 1.15 (95% CI, 0.82-1.62; p = 0.429), demonstrating no difference in RR between Kaplan-Meier and competing-risks estimators, to 1.79 (95% CI, 1.43-2.24; p < 0.001), demonstrating a significant difference in RR (Fig. 2A).
When we considered the seven strata that recorded the largest number of revisions, the Kaplan-Meier estimate of revision risk was once again greater than the competing-risks method. The pooled RR was 1.07 (95% CI, 1.00-1.14; p = 0.049), demonstrating that the cumulative incidence estimated using the Kaplan-Meier method was 1.07 times greater than the competing-risks method, corresponding to a relative increase in estimation of 7% (Fig. 2B). RRs for these studies ranged from 1.02 (95% CI, 0.96-1.08; p = 0.540) to 1.62 (95% CI, 1.00-2.63; p = 0.051), both of which demonstrate no difference in RR between Kaplan-Meier and competing-risks estimators.
Is the Extent of Overestimation Influenced by Followup Time or Frequency of Competing Risks?
Increasing duration of followup was not associated with an increase in the amount of overestimation of revision risk by the Kaplan-Meier method. This may be due to the small number of studies that met the inclusion criteria and conservative variance approximation. Using metaregression, we found the RR comparing the Kaplan-Meier estimator with the competing-risks estimator for studies with followup times less than 10 years was not different than the RR obtained for studies with followup times greater than or equal to 10 years in either our analysis of strata containing the largest number of revisions (p = 0.125) or our analysis of strata containing the highest proportion of competing risks (p = 0.203) (Table 3).
Increasing the ratio of competing risks to events of interest was also not associated with an increase in the amount of overestimation of revision risk by the Kaplan-Meier method. Again, this may be due to the small number of studies that met the inclusion criteria and conservative variance approximation. When we considered the seven strata with the largest number of revisions, there were no differences between the RR comparing the Kaplan-Meier and competing-risks estimators for studies with a ratio of competing risks to events of interest less than one compared with the RR for studies with ratios between one and 10 (p = 0.342) or greater than 10 (p = 0.581) (Table 3). Similarly, when we considered the seven strata that contained a high proportion of patients who had died during the followup period, there were no differences between the RRs obtained for studies with a ratio of competing risks to events of interest between one and 10 compared with the RR for studies with ratios greater than 10 (p = 0.161). There were no strata with ratios less than one for our analysis of strata containing the highest proportion of patients who died.
Applying an alternative variance approximation (defined in the Materials and Methods) produced similar results for all analyses (data not shown).
The rapidly increasing demand for joint replacements has placed growing importance on our ability to accurately monitor the cumulative incidence of revisions to assess implant quality, predict future demand for revisions, and inform clinical and health policy decisions [10, 26, 28]. Because the Kaplan-Meier method theoretically overestimates the cumulative incidence of events in the presence of competing risks, alternative competing-risks methods provide more accurate estimates of the cumulative incidence of revisions [18, 22]. However, competing-risks methods have yet to be widely reported within the orthopaedic literature and in joint replacement registries . Our systematic review and meta-analysis aimed to determine the degree of overestimation of the Kaplan-Meier method compared with the competing-risks method when estimating the cumulative incidence of revision and to examine whether followup time and the rate of competing risks influenced this bias.
The articles included in our study conducted analyses of cohort and joint replacement registry data. Although randomized controlled trials are considered the highest level of evidence, registries have recently gained recognition as credible data sources [13, 19, 32, 37]. However, our assessment of the quality of reporting of these studies identified deficiencies similar to those previously identified in a review of survival analyses . For example, only 43% of studies included in our review reported the number of patients that were at risk of revision at each followup time, estimates of precision (such as SEs or CIs), or the statistical software used. Only three of the nine quality of reporting criteria assessed were fulfilled by all studies included in our review, reflecting the need for adherence to and strict enforcement of guidelines, perhaps through the development of a checklist, to improve the standards of reporting of survival analyses. However, it is important to note that, given that the goal of the studies included in our review was to summarize differences between the Kaplan-Meier and competing-risks methods, several studies did not conduct a full survival analysis using original data. Therefore, our assessment may underestimate the quality of reporting. Furthermore, given that our findings are based on a small number of studies (n = 7), caution is needed in interpreting these results. Nevertheless, clear guidance on the reporting of survival analyses is needed, specifically to address complications that arise in the analysis of competing-risks data. For example, reporting the number of patients at risk of revision becomes ambiguous in competing-risks situations as a result of differences in the censoring procedures between the Kaplan-Meier and competing-risks methods. Although the Kaplan-Meier method censors and removes patients from the risk set at their time of death, the competing-risks method includes patients who die in the risk set for the remainder of the observation period.
Individual patient data are considered the “gold standard” for meta-analyzing survival data [8, 39, 44]. Thus, the use of summary data is a limitation of our study. As a result of a lack of individual patient data, we were unable to examine factors that may have impacted the magnitude of overestimation such as patient age and comorbidities. Furthermore, we were unable to derive a variance estimate for our effect measure. We therefore used a variance approximation primarily for the purpose of assigning weights to the individual studies in our meta-analysis. Because this approximation does not take into account the covariance between the Kaplan-Meier and competing-risks estimators (which are correlated given that both estimators are calculated using the same data), it likely overestimated the variance and width of the associated CIs around the RR estimates. This overestimation reduces the chance of observing statistically significant differences between the Kaplan-Meier and competing-risks estimators, resulting in a conservative estimate of our findings. For instance, the 95% CI around the RR of 1.30 obtained for Fennema and Lubsen  ranges from 0.63 to 2.72. The lower bound of this CI suggests that the Kaplan-Meier estimator may be less than the competing-risks estimator. This is mathematically incorrect given that the KM estimator must always be greater than or equal to the competing-risks estimator. Thus, the CIs around individual study and our pooled RR estimates must be interpreted with caution.
A lack of standardized analysis and reporting of revision rates within the arthroplasty literature and among joint replacement registries currently limits our ability to accurately monitor and compare outcomes across patient populations [29, 33]. Although registries have begun reporting the cumulative incidence of revision rather than person-time incidence rates , given the former provides more information regarding how the risk of revision changes over time, the International Society of Arthroplasty Registries has recently called for improvements in the standardization of survival analysis methods used to estimate these measures . Because the choice of method depends on the study objective and audience (such as health policy planner versus patient perspective), establishing detailed guidelines for the approach to survival analysis may help address these issues. However, because questions have been raised regarding whether the difference between the Kaplan-Meier and competing-risks methods is clinically significant [17, 38], there is first a need for consensus among experts regarding the appropriateness of these methods.
Our study provides evidence that the Kaplan-Meier method overestimates the cumulative incidence of revision after hip or knee arthroplasty compared with the competing-risks method. The overestimation observed in individual studies ranged from 2% to 79% and in aggregate was approximately 55%. The magnitude of this bias is consistent with what has been observed across several other clinical areas when estimating the time to an event in the presence of competing risks [2, 4, 15, 40, 42, 46]. An alternative approach to exploring the magnitude of bias of the Kaplan-Meier method in the setting of competing risks that might be considered would be to compare the original Kaplan-Meier estimate with an estimate obtained when all patients who experienced a competing event (such as death) before the event of interest (like revision) are presumed to have infinite followup. These patients would therefore be included in the risk set for the remainder of the followup period after their death, which is similar to how patient deaths are handled using the competing-risks method.
In general, we did not find the rate of competing risks or the duration of followup to influence the degree of overestimation. This may be the result of the small number of studies included in our meta-analysis or the conservative variance approximation that likely biased our results toward showing nonsignificant differences. It should be noted that the duration of followup directly influences the rate of competing events given that studies that follow patients over a longer period of time are more likely to follow patients until death rather than administrative censoring (that is, being unrevised at the end of the study period). However, based on the RR point estimates obtained for our stratified analyses, we speculate that the incidence of competing risks has a greater influence on the overestimation of the Kaplan-Meier method compared with the length of followup. For instance, in our analysis of strata containing the highest ratios of competing risks to revisions, the RR point estimate indicated the overestimation of the Kaplan-Meier method was greater for followup times less than 10 years compared with 10 or more years. This may be the result of the relatively high incidence of competing risks for the Gillam et al.  stratum compared with the other strata, despite a relatively shorter followup.
Overestimation of the Kaplan-Meier method may have important implications when monitoring the incidence of revision after hip or knee arthroplasty and is particularly concerning when using these estimates to inform healthcare planning and policy decisions. Although our study provides strong support for increased use of competing-risks methods to more accurately estimate the absolute risk of revision, further investigation into factors influencing the overestimation of the Kaplan-Meier method such as the rate of competing events is required to better understand in which circumstances the bias of the Kaplan-Meier method becomes significant. We agree with the recommendations that Clinical Orthopaedics and Related Research® made in their editorial earlier this year on this topic , which suggest the use of competing-risks estimators when competing risks (such as death) might preclude the occurrence of important events of interest (such as revision surgery). Going forward, we urge journals to develop and encourage improved survival analysis guidelines to ensure appropriate methods are applied to produce unbiased estimates of the risk of revision that can be used to monitor the safety of joint replacements and deliver relevant information to patients, clinicians, and health policymakers.
We thank Diane Lorenzetti, research librarian in the Department of Community Health Sciences, Cumming School of Medicine, University of Calgary, Calgary, Alberta, Canada, and the Institute of Health Economics, Edmonton, Alberta, Canada, for her assistance in the development of our literature search strategy.
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