Abstract
Most orthopedic studies involve survival analyses examining the time to an event of interest, such as a specific complication or revision surgery. Competing risks commonly arise in such studies when patients are at risk of more than one mutually exclusive event, such as death, or when the rate of an event depends on the rates of other competing events. In this paper, we briefly describe the survival analysis censoring methodology, common fatal and non-fatal competing events and define circumstances where standard survival analysis can fail in the setting of competing risks with real-world examples from orthopedics.
Keywords: survival analysis, total joint arthroplasty, competing risk
Introduction
Survival analysis, also referred to as time-to-event analysis, is a widely used statistical methodology applied in orthopedic research. Although the term survival itself implies time-to-death, in most cases, the event of interest is a clinical outcome such as time to occurrence of a specific complication or time to revision surgery. Survival analysis is used in studies of long-term prognosis following orthopedic surgery, comparisons between patients with different clinical or surgical characteristics or to determine the relationship between a risk factor and time to a clinical outcome. A related term that is unique to orthopedics is the survivorship analysis, which typically refers to time-to-event analyses at the joint-level rather than overall patient survival. One of the challenging aspects of survival analysis is competing risks. In this paper, we briefly describe survival analysis and the censoring methodology and define circumstances where standard survival analysis can fail due to competing risks with real examples from orthopedic surgery. Finally, we provide a practical guide for researchers and reviewers to avoid common pitfalls related to competing risks in survival analysis applied to orthopedics research.
Survival (time-to-event) data and types of outcomes in orthopedics
Survival analysis involves modeling and analysis of time-to-event data, i.e., data that involves an event, and the outcome of interest is the time until the event occurs. It is customary to use the terms “survival” or “survivorship” analysis regardless of the outcome of interest. Although time-to-death is a common outcome in many diseases, there are other events of interest in orthopedics such as the time between surgery to specific complications, reoperations or revision surgery. In any survival analysis, it is important to explicitly state what the event (outcome) is and when the period of observation starts and ends. For example, we may be interested in time to periprosthetic joint infections during the first year after total joint arthroplasty (TJA). Then the first diagnosis of periprosthetic joint infection is the event of interest and the time period starts on the day of surgery and ends 365 days later.
Survival analysis estimates the “survival function,” which is the probability that a patient does not experience the event of interest at a given time. Depending on the purpose of the study (e.g. descriptive or comparative), subsequent analyses may involve comparison of the survival function across patient groups and/or estimation of the effect of some patient characteristics (e.g. age, sex, socioeconomic status) or surgical factors (operative indication, surgical approach) on the survival function.
What is censoring and why it is important in survival analysis?
In orthopedics, survival analysis is often needed when analyzing follow-up data from a cohort of patients after an orthopedic procedure. Although it is relatively straightforward to identify patients who experience the outcome and calculate their survival function, it is much harder to define the survival function for patients who did not experience the event, especially over a long study period. In a typical follow-up study, only a small subset of patients will experience the outcome of interest, but the time-to-event will be unknown for the remaining patients. In survival analysis terminology, this is called “censoring”. Patients are censored when (1) they do not experience the outcome by the end of the study; (2) they are lost to follow-up or withdraw from the study before the end of study; (3) they die during follow-up. For example, when examining the risk of revisions in a cohort of arthroplasty patients, the event of interest is revision surgery. Even when none of the patients are lost to follow-up, the majority of patients will not have revision surgery after several years of follow-up. Therefore, the survival time of the patients who did not have revision surgery by the end of the study is censored. We do not know when or whether these patients will subsequently have revision surgery, but at least for the duration of the study, we do know that they were alive and did not have revision surgery. It is important to take the time we observe all patients into account in our analysis, even if they did not have an event during the study period.
Standard survival analysis methods, such as the Kaplan-Meier method, produce unbiased estimates of the survivor function only if the censoring is ‘noninformative’. Censoring is noninformative if patients who are censored at any point in time (end of study, loss to follow-up or death) are as likely to have a later event as patients who remain under observation. In other words, censoring should not occur due to reasons related to the event. If there’s a reason why patients do not return for follow-up, then censoring becomes informative. It may be difficult to assess whether censoring is informative, but when the percentage of patients lost to follow-up is small, there is less concern for informative censoring. Therefore, every effort should be made to minimize censoring due to loss to follow-up in any study.
What is competing risk?
A competing risk is an event that prevents the occurrence of the event of interest. Competing events violate the noninformative censoring assumption because they modify the chance that the study outcome occurs. Table 1 provides some examples of potential competing events in orthopedics. The most obvious competing event is death because when patients die before experiencing the outcome of interest, their chance of experiencing the study outcome is zero. For example, when examining the long-term risk of revision as the outcome following TJA, death before revision competes with revision. Alternatively, when examining cancer mortality, deaths from other causes compete with cancer-related deaths. Fortunately, mortality is not high in most elective TJA studies and competing risk of death is rarely a challenge in standard survival analyses with shorter follow-up. In contrast, hip fracture patients have high mortality; it is estimated that a quarter of patients are deceased within one year of their hip fracture[1]. In that case, the competing risk of death needs to be considered in a study involving hip fracture, such as comparison of the risk of revision in hemiarthroplasty versus total hip arthroplasty patients.
Table 1.
Examples of competing events in orthopedics
| Condition | Outcome(s) of interest and comparison groups | Competing event |
|---|---|---|
| TJA | Complication, revision | Death |
| TJA | Cancer mortality | Death from other causes |
| TJA | Revision in patients with osteoarthritis versus tumors | Death |
| Hip fracture | Revision risk in hemiarthroplasty versus THA | Death |
| Conversion hemiarthroplasty | Complication, revision | Death |
| TJA | Hospital-acquired infection | Discharge or death without an infection |
| TJA | Cause-specific revisions | Revisions for instability and aseptic loosening |
| Hip fracture | Inpatient mortality | Discharge alive is a non-fatal competing event |
| Lower limb amputation | Cardiac mortality | Death from other causes |
| Lower limb amputation | Reamputation | Revascularization is a non-fatal competing event |
Non-fatal competing events can also change the probability of experiencing the outcome of interest, even in studies with relatively short follow-up. For example, if the outcome of interest is inpatient mortality of hospitalized hip fracture patients and patients are followed up until discharge or death, discharge (alive) is a competing event for death because discharged hip fracture patients are usually in better health and have better survival than hospitalized patients. If hospital-acquired infection is the event of interest, discharge or death without an infection are competing events because discharge usually precludes the observation of an infection. This was observed in a large clinical trial examining the effectiveness of chlorhexidine bathing on hospital-acquired infections[2]. The reported cumulative risk of infection was 25%, even though only 4% of patients actually acquired infections in the hospital. The cumulative risk of infection was overestimated because data from patients who were discharged without having an infection were censored. Another situation where competing risks are present is when one is interested in the type of the event that occurs “first” among different events. For example, in a study of cause-specific revisions following TJA, revisions for instability and aseptic loosening are competing events; if a patient first undergoes revision for instability, then it precludes revision for aseptic loosening first.
How competing risk analysis is different from the Kaplan-Meier method
The Kaplan-Meier method estimates the survival function as the fraction of patients living without the event (outcome) of interest for a certain amount of time following surgery. The survival time of patients who experience the event ends at the time they experience the event whereas the survival time of patients who do not experience the event is censored and all follow-up time until censoring is included in the analysis. Censoring of patients who experience competing events creates two problems: (a) they are censored in an informative way; and (b) the probability of experiencing the outcome of interest is estimated in an unrealistic setting in which the competing events cannot occur.
The competing risk equivalent of the Kaplan-Meier method is the Aalen-Johansen cumulative incidence method, but the name of this method is rarely seen in the medical literature. The most common analysis adjusting for competing risks uses the cumulative incidence function. The cumulative incidence function estimates the probability of experiencing the event of interest before the occurrence of a competing event. When there are no competing risks, the cumulative incidence function can be correctly estimated as one minus the Kaplan-Meier estimate of the survival function. When completing risks are present, the Kaplan-Meier estimate for the cumulative incidence is too large. Under the Aalen-Johansen method, patients who experience competing events are considered no longer to be at risk of developing the event of interest. As a result of this, the cumulative incidence function is lowered by the occurrence of a competing events. In other words, the Aalen-Johansen method estimates a smaller cumulative incidence of the event than the Kaplan-Meier method. However, the extent of overestimation depends upon the proportion of patients experiencing the event of interest and the type of competing event. Age is the most important determinant of death and comparisons across age groups are illustrative of the effects of competing risk of death. When examining the risk of revision in old versus young total knee arthroplasty patients[3], the incidence of mortality in elderly patients is much higher than the incidence of revision (Figure 1). As a result, the event rate is overestimated using the Kaplan-Meier method. For example, in patients aged 65–74 years, the incidence of mortality (green line) was higher than the incidence of revision, and the Kaplan-Meier method overestimated the risk of revision by 11% at 10 years and by 45% at 20 years. In contrast, in younger patients aged 45–54 years, the incidence of mortality is lower than the incidence of revision throughout follow-up, and the Kaplan-Meier and cumulative incidence estimates (accounting for competing risk of death) did not differ as much, at least with shorter follow-up. In this younger age group, the Kaplan-Meier method overestimated the risk of revision by only 3% at 10 years and 16% at 20 years. As a rule of thumb, competing risk analysis should be considered when the proportion of patients experiencing the competing event, such as death, is equal to or greater than the proportion of patients experiencing the outcome of interest or when the absolute percentage of competing events is greater than 10%.
Figure 1: Cumulative incidence of death (Death CI) and revision probabilities using the Kaplan-Meier method (Revision KM) and the cumulative incidence function (Revision CI) in four age groups.
(published with permission from JBJS[3])
In summary, the cumulative incidence function estimates the probability of an outcome in a clinically relevant setting in which patients may also die or experience another competing event. When there are no competing events or when the proportion of patients experiencing competing events is relatively low, the cumulative incidence function gives the same results as the Kaplan-Meier method. The choice between the Kaplan-Meier and competing risk methods depends on the study questions, as has been discussed extensively in the literature[3–8]. Nevertheless, overestimation of the cumulative incidence of an outcome may have practical and/or public health implications. For example, surgical decisions are often guided by formal or informal risk predictions, and ignoring competing risks, among other things, can lead to possible overtreatment or undertreatment of patients. Furthermore, when communicating long-term risks to patients, estimates from cumulative incidence are more appropriate. In comparison of different implant types, it is generally assumed that censoring of deaths satisfies the uninformative censoring assumption because mortality is independent of whether implants are revised or not[4]. Fortunately, the magnitude of competing risk of death has been consistently shown to be relatively small in several survival analyses in arthroplasty. However, relatively little is known about other competing events in orthopedics.
Guidelines for the Researcher
1. How to collect data and prepare the dataset for survival analysis
In collection of survival analysis data, the first task is identifying the study population and patients who experience the outcome of interest during follow-up. If you are examining an outcome that occurs several years after surgery, then you must have sufficiently long follow-up to capture outcomes to perform analyses with enough power. For example, Journal of Arthroplasty typically requires at least 2 years of follow-up. Furthermore, you need to ensure completeness of follow-up, especially if you are planning to compare two or more groups of patients. You need to resolve any differences in the percentage of patients lost to follow-up across groups. In the case of pooling data from multiple centers, you also need to ensure there is consistency in treatments and follow-up intervals and mechanisms across centers.
2. How to identify patients to be censored and their censoring times
Once you identify the study cohort and patients who experienced the outcome of interest, the next step is to review the last follow-up dates and censoring times for the remaining patients in the dataset. The censoring date for patients who did not yet experience the outcome is their last in-clinic or other follow-up date. The censoring date for patients who are deceased without experiencing the outcome is their death date.
3. How to check assumptions for noninformative censoring
Unfortunately, there is no systematic way to check for informative censoring. Assessing the possibility of informative censoring requires a careful review of the data collection procedures to make sure data were collected uniformly on those with and without events. Are you concerned that patients are lost to follow-up because of a higher risk of the outcome? You can graphically compare the event times and the censoring times to see if the distributions differ. If the follow-up is good, the event times may be shorter than the censoring times. However, if the censoring times are shorter than the event times, this may indicate informative censoring.
4. How to know if you need to account for competing risks
The next step is to evaluate whether there are any competing risks. What is the proportion of patients who are deceased and is it higher than the proportion of patients who experienced the outcome of interest? Competing risk analysis may be needed when the proportion of deceased patients is equal to or greater than the proportion of patients who experienced the outcome or when the percentage of deceased patients is 10% or greater.
5. How to check for other assumptions related to survival analysis
If your study spans several years, patients who are recruited in later years may not have the same survival probability as patients who are recruited in earlier years due to changes in treatments or other reasons. Therefore, it is important to examine cohort effects by comparing survival of patients who are recruited early versus late. Additionally, if you are planning to use a Cox regression model, it is advised to check the proportional hazards assumption graphically or by using statistical tests.
Guidelines for the Reviewers and Readers
Determine the purpose of the analysis and whether there is only one group or multiple groups of patients.
Determine whether the analysis is at the patient or joint level.
Determine the outcome of interest and how it is ascertained. Is it one outcome or multiple outcomes?
How are patients followed-up over time? Is the duration of follow-up adequate for the outcome of interest?
How complete is follow-up? Are there differences in the follow-up mechanisms between groups?
Are the censoring methods sufficiently described in the methods section?
Determine whether competing risks is a potential concern. If so, how are they accounted for in the analysis?
Supplementary Material
Funding:
This work was funded by a grant from the National Institute of Arthritis and Musculoskeletal and Skin Diseases (NIAMS) grant P30AR76312 and the American Joint Replacement Research-Collaborative (AJRR-C). The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
Footnotes
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