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Using and Understanding Survival Statistics - or How We Learned to Stop Worrying and Love the Kaplan-Meier Estimate

2023; Elsevier BV; Volume: 115; Issue: 4 Linguagem: Inglês

10.1016/j.ijrobp.2022.11.035

1879-355X

Søren M. Bentzen, Ivan R. Vogelius,

Cancer survivorship and care

Numerous reports suggest that the Danes are the happiest people in the world. But are all Danes equally happy all the time? How about, say, lung cancer survivors? Your friend remarks that she works in a busy lung cancer clinic and that she has been wondering about the risk of her patients developing depression compared with noncancer survivors. You suggest the way to go is to estimate the cumulative incidence of depression with death as a competing event as a function of time after diagnosis in the lung cancer cohort and compare this with the incidence in population data from noncancer individuals. After a long weekend and several pots of strong coffee, you manage to complete the analysis based on electronic health record data from your hospital. Lung cancer survivors have a 5-year cumulative incidence of depression of 1.4% which is lower than the 2.7% in the general Danish population. Voila!"But wait!" Your friend hesitates when you call to tell her the result. "Wait a second… shouldn't you be using the Kaplan-Meier method?""Nah!" You say confidently. "Death is a competing risk. Dead people do not get depressed!""Hmmm… but in a way, that's exactly my point… if I could just explain it better!" She mumbles, "I think I'll run a simulation!" Lots of considerations in life involve conditional rather than absolute probabilities. For example, the lifetime risk of dying from prostate cancer for a man in the United States is 2.4%, while the risk of dying from a pancreas cancer is 1.4%. So, in a sense, you should be more concerned about dying from prostate cancer than pancreas cancer if you are male. However, that changes drastically the moment you are diagnosed with one of the cancers. A man is roughly 6 times more likely to be diagnosed with a prostate cancer than a pancreas cancer, but if you get a prostate cancer diagnosis, your risk of dying from this disease is about 21%, compared with an 84% risk of dying from a pancreas cancer. Your level of optimism after a cancer diagnosis obviously depends on the conditional risk of dying from the disease, rather than the absolute risk of dying from that cancer in the general population. From a population health perspective, the absolute estimate of 1.4%/2.4% is important, for example, in evaluating resource allocation or public health strategies, but the treating physician and patient with cancer will want to know the prognosis once you have the cancer. The distinction between conditional risk versus the absolute probability of failing in a given time interval plays a fundamental role in time-to-event statistics. Let's look at a population life table for the United States population as a concrete example.1Arias E Bastian B Xu J et al.U.S. state life tables, 2018.Natl Vital Stat Rep. 2021; 70: 1-18PubMed Google Scholar The population life table essentially follows a hypothetical population of 100,000 newborn babies. The table lists the number of individuals in that cohort who are alive at a given age, i, more specifically at his/her ith birthday. This is called the survivorship function (Fig. 1). The life table contains a column indicating how many individuals die between a specific birthday and the following birthday, that is, the probability that a newborn dies at age i. It also contains a column giving the conditional probability of dying between the ith and the i + 1 birthday given that you live until age i. The distinction between the probability that a newborn baby dies at a specific age on one hand and the conditional probability of dying at that age is important. The risk that a newborn baby dies when she is 85 years old is 3.5%, which is higher than the risk that she dies when she is 95 years old, 2.2%. The reason is not that it is less risky to be 95 rather than 85 years old, but that most people do not live to turn 95. If we consider the conditional risk of dying before your next birthday given that you lived to the age of 85, it is 8% versus 23% for the corresponding conditional risk in a 95-year-old. That the risk of dying in a given year increases with age is indeed what, in this context, we call aging. Typical health states in oncology could be all-cause mortality or cancer-specific mortality, local progression, developing depression requiring treatment, or developing a late adverse effect of therapy. These health states are often referred to as endpoints. A change in the health state of a given individual is called an event. A common feature of these health states is that they may or may not happen over time, that is, they require prolonged observation of the patient to ascertain whether they occur. For each individual patient, 2 data items are of immediate relevance: one is a variable recording if the event has occurred or not, and the other is the time when the event occurred, or if it has not, what the length of follow-up is in that individual. The health state is indicated by a binary (yes or no) variable, with a corresponding dummy variable, typically coded 1 for yes and 0 for no. A patient who has not developed the health state in question at a time T is said to be at risk. A patient is said to be censored at time T if this is the time of last follow-up or the time of a competing event that prevents observation of the endpoint of interest at later times. More precisely, the observation in that individual is said to be right-censored and T is referred to as the censoring time. In this case, it is unknown if the patient will reach the endpoint at a time in the future, but it is known that if so, it would happen at a time larger than T. It is worth mentioning 2 other types of censoring that occur in biomedicine, left and interval censoring, but these will not be discussed here. Standard statistical methods such as t tests do not apply to time-to-event data. By far, the most common nonparametric method for estimating the survivorship distribution in time-to-event data is the Kaplan-Meier (K-M) estimate.2Kaplan EL Meier P Non-parametric estimation from incomplete observations.J Am Stat Soc C. 1958; 53: 457-481Crossref Scopus (48517) Google Scholar At its core, it is a simple idea. If, say, 5 patients are at risk of entering a specific health state at a given time after their treatment, and 1 of them actually does, then the probability of entering that state at the specified time is estimated as 1/5. The idea of the K-M estimate is that for each time interval we calculate the conditional probability that a patient at risk at the beginning of the interval will make it to the end of the interval without failing. The time intervals are defined as the time between events, and each time an event occurs the K-M estimator is updated by multiplying the estimator after the previous event by the conditional probability of surviving to the end of the time interval up to the current event. These conventions produce a step function that takes a constant value in the interval between 2 events. For the K-M estimate to be meaningful, 3 assumptions should be fulfilled, at least to a reasonable degree. First, censoring should be independent of the event of interest (so-called "uninformative censoring"). In other words, an individual who is censored should have the same prospect as uncensored cases for failing in the counterfactual scenario that they could be followed longer. This assumption is not easy to check and often requires a common-sense call to be made. An example could be a time-to-progression analysis using Response Evaluation Criteria in Solid Tumours criteria for progression. Censoring patients without progression at the time of the last scan may be unreasonable if patients die of clinical progression in the interval between planned scans. These patients clearly have an unfavorable prognosis and thus this censoring mechanism is informative, and it will lead to an overestimate of the time to progression. A more reasonable endpoint would be progression-free survival or even event-free-survival where clinical progression or death are included in the composite endpoint. Sometimes, especially when censoring is due to death, critics claim the K-M method assumes that a dead patient can reach the endpoint in question. This is a misunderstanding. The K-M method is based on the premise that patients who have NOT reached the endpoint at a given time carry information about the event-time distribution up to that time point only. And the likelihood of an event is updated to represent the conditional risk of the event in question using only patients who are still at risk after that time. The second assumption is that patients entering the study cohort early or late in the period covered should be comparable. This can in theory be checked in a large data set by including calendar time as a covariate in a multivariable Cox model (see the following sections). If, say, therapy or diagnostic criteria change during the study period, this assumption may break down. But then again, an estimate of the survivorship function in such a scenario would not be very useful anyway! Finally, the event time should be known. This requirement may seem obvious but if, say, the patient is seen annually and the event of interest is likely not detected in the interval between follow-ups, then using the time of the follow-up when the event is detected will bias the estimate of the event time distribution. A method handling interval censoring would be preferable. An informal sensitivity analysis could be performed by assuming that the event time is the midpoint between the 2-follow-ups. Or, one might consider using the life table method, a method not covered in this article. The product-limit estimator is named after Edward Kaplan and Paul Meier, who submitted independent manuscripts on their work to the Journal of the American Statistical Association. It was the journal editor who convinced Kaplan and Meier to combine their manuscripts into a joint article, which was published in 1958; it had taken them 4 years to resolve the differences between their approaches. It was a slow burner! From 1958 to 1968, this article was cited only 25 times, but it went on to become the most frequently cited statistics article of all—a respectable number 11 on the list of all science articles ranked according to number of citations. The rapid uptick in popularity was likely, at least in part, a consequence of the introduction of statistical software that reduced the computational burden of calculating the K-M estimate as a function of time in a large sample. The fact that the K-M estimate is only updated at an actual event time generates the characteristic "staircase" appearance of the K-M plot, a sequence of piecewise constant, decreasing steps. Figure 2 shows a K-M plot generated from simulated data. Each step corresponds to 1 (or more, if events are tied on the same date) event in that group. The vertical bars are the censoring times. The figure is generated using R, the code may be found online. Note the 95% confidence bands around the curves. It should be noted that these may overlap at some time points, even when the P value for a difference between the 2 groups is statistically significant. Below the plot is a table indicating the number of patients remaining at risk in the 2 groups. This table should be included as it provides a useful impression of the sample size underlying the estimates over time. Patients leave the risk set either because they have already reached the endpoint or because of censoring. There is no hard rule for when to truncate the plot of a K-M curve, but it is clear that if there are less than, say, 10 patients remaining at risk at a given time, the estimated curve after this time point becomes unreliable. Most statistical software packages will list the survivorship estimates and the standard error of the point estimate at the time of events, that is, when the K-M estimator is updated. In the example in Fig. 2, the 4-year survival estimate in group 1 is estimated at 59.0%. We find it by going back to the event immediately before 48 months, that is, we are on the "flat part" of the K-M curve in this case. The median survival, that is, the time when the curve crosses 50% survival, is 40 months in group 2 and not reached in group 1. This is a common way of summarizing the difference in survival in the groups, but it can be very sensitive to the detailed shape of the curves. Also, mean survival time is sometimes used; again, this has a number of less attractive properties and requires a truncation of the time interval where it is estimated. A better metric for comparing groups is the survival at a given time point. This should be a time when a reasonable number of patients remain at risk and should be specified with confidence limits. A summary measure of the difference between the 2 groups that uses the full range of available observation times can be obtained by estimating the hazard ratio with 95% confidence interval in a Cox model with a dummy (0-1) variable for group membership as the only covariate (see the following sections). The log-rank test is the test of choice for testing the null hypothesis that the time-to-event curve is the same in 2 or more groups. The test is based on the same assumptions as the K-M survival curve, namely that censoring is noninformative, the survival probabilities are the same for participants recruited over the study duration, and the event time should be known. As noted by Bland and Altman,3Bland JM Altman DG The logrank test.BMJ. 2004; 328: 1073Crossref PubMed Scopus (443) Google Scholar deviations from these assumptions are most critical if they vary between the groups. Statistical software packages may give 3 different P values for 3 versions of this test, Gehan-Breslow, Mantel-Cox, or Peto's log-rank test. They differ mainly in the weight they assign to early relative to later observation times. Of these, the most common by far is the Mantel-Cox log-rank test, and it requires an extremely good argument not to use that one. And, no, you do not just pick the lowest of the 3 P values! It is possible to perform a stratified log-rank test, comparing, say, the effect of therapy across early and late-stage disease. This is often a better approach than doing 2 separate analyses with the accompanying loss of statistical power. The underlying assumption, when doing a stratified test, is that the treatment has a similar effect in the 2 or more strata but that the survival in the strata differs (including a different shape of the event time distribution between strata). Stratified testing appears to be underused in the literature. Also, if you have ordered groups of, for example, increasing radiation dose, a test-for-trend log-rank test may test for a dose-response relationship, using all the available data and without increasing the number of degrees of freedom. The workhorse in terms of multivariable analysis of time-to-event data is the semiparametric Cox proportional hazards model. As the name suggests, the Cox model, in its basic form, allows the hazard function to take any form and assumes that the effect of covariates does not depend on time, in other words, that the ratio of hazard rates for any 2 cases is constant (proportional hazards) over time. The proportional hazards assumption can, and should, be tested for a given data set. If it turns out not to be fulfilled, it may be helpful to stratify the analysis, or to introduce time-dependent covariates, or to artificially censor observations after a defined follow-up time. The underlying hazard function is a nonparametric K-M estimate from the data set. Building a multivariable regression model for time-to-event data relies on a number of considerations, including how to code categorical and continuous covariates. It is not unlikely that 2 researchers analyzing the same data set will end up with different models. It is beyond the scope of the present note to discuss multivariable model building in detail. Kaplan-Meier estimates are widely used and remain a powerful tool for summarizing time-to-event data. However, there are situations in which uncritical interpretation of the K-M estimates would be misleading (see Table 1). Of the 3 assumptions underlying this method, the assumption of noninformative censoring is often the most critical. If disease tends to progress in both distant and local sites, treating distant progression as a censoring event when estimating the time-to-local progression is clearly not reasonable. A practical and simple solution is often to use the endpoint time-to-(any)-progression or progression-free survival. These composite endpoints clearly lack specificity, but they may still provide the most meaningful signal for detecting, say, a patient-level benefit from, for example, a change in therapy. Kaplan-Meier estimates quantify the time to the first occurrence of an event. If the endpoint is death, then this is clear. However, this may or may not be a useful descriptor, for example, of intermittent episodes of a condition or a symptom. In such cases, the time spent with symptoms could be a relevant measure. But there are many examples where the K-M really is the go-to method. In some cases, it may be helpful to "reset the clock," for example in estimating the time to resolution of a symptom in those who develop it. The 0 time will then be the date when the symptom occurred. These kinds of data are often right-censored, as there will likely be a subgroup of cases where the symptom had not resolved when they were last seen. The K-M estimate would again be the method of choice in this situation. We generally encourage authors carefully to define/select meaningful and informative endpoints and effect measures and to apply established statistical methods for analysis of these endpoints rather than using unconventional or informal analytical methods. Late effects occur late. And 1000 patients, followed for a few months, will not provide useful information on late adverse effects seen years after therapy. Any report on the outcome of a cancer therapy should include data on both efficacy and toxicity.4Trotti A Bentzen SM The need for adverse effects reporting standards in oncology clinical trials.J Clin Oncol. 2004; 22: 19-22Crossref PubMed Scopus (136) Google Scholar Indeed, the efforts to develop more rigorous and comprehensive systems for recording and grading the severity of toxicity have been pioneered by radiation oncologists. Still, incomplete and potentially misleading methods for summarizing toxicity after various therapies remain a problem. A recent review5Vittrup AS Kirchheiner K Fokdal LU et al.Reporting of late morbidity after radiation therapy in large prospective studies: A descriptive review of the current status.Int J Radiat Oncol Biol Phys. 2019; 105: 957-967Abstract Full Text Full Text PDF PubMed Scopus (12) Google Scholar of 60 publications on radiation therapy, including >200 patients and published between December 2015 and November 2017 in 10 high-impact journals, showed that late effects were summarized as a crude incidence (events/total sample size) in 43 publications (72%); in 26 publications (43% of all publications) this was the only metric reported. Crude incidence is clearly an inadequate metric for summarizing late toxicity as it does not take the duration of follow-up into consideration. Even the most meticulous recording and grading of toxicity is futile if the data analysis and presentation are inadequate. Observations of late effects should be censored at the time of last follow-up, and the K-M estimate should be used to summarize the survivorship function for those living to a given time. Death is, in one sense, a competing risk, as it prevents the observation of a subsequent adverse event. However, from a pathophysiology perspective, exceeding the normal-tissue tolerance in an organ at risk is not in general mechanistically related to death of the index malignancy or of intercurrent disease. This is the reason for treating death as a censoring event, essentially truncating the possible follow-up time for normal tissue injury in an individual. As discussed by Bentzen et al,6Bentzen SM Vaeth M Pedersen DE et al.Why actuarial estimates should be used in reporting late normal-tissue effects of cancer treatment ... now!.Int J Radiat Oncol Biol Phys. 1995; 32: 1531-1534Abstract Full Text PDF PubMed Scopus (78) Google Scholar this is, in particular, a problem in populations with a poor prognosis and may essentially mask the fact that a given therapy is too toxic in the sense that long-term survivors develop an unacceptable rate of late effects. Use of the K-M method, adjusting for time at risk for an event, facilitates a comparison of late effects across various cancer types or stages of disease. Cumulative incidence estimates from competing risks analysis have been proposed as an alternative descriptor of late adverse events after radiation therapy alone or combined with other modalities.7Caplan RJ Pajak TF Cox JD Analysis of the probability and risk of cause-specific failure.Int J Radiat Oncol Biol Phys. 1994; 29: 1183-1186Abstract Full Text PDF PubMed Scopus (99) Google Scholar However, the lack of specificity of the cumulative incidence estimate makes it a less useful summary description of time-to-late-effects data.6Bentzen SM Vaeth M Pedersen DE et al.Why actuarial estimates should be used in reporting late normal-tissue effects of cancer treatment ... now!.Int J Radiat Oncol Biol Phys. 1995; 32: 1531-1534Abstract Full Text PDF PubMed Scopus (78) Google Scholar,8Chappell R Re: Caplan et al. IJROBP 29:1183-1186; 1994, and Bentzen et al. IJROBP 32:1531-1534; 1995.Int J Radiat Oncol Biol Phys. 1996; 36: 988-989Abstract Full Text PDF PubMed Scopus (10) Google Scholar This point is further illustrated by our clinical vignette (see next section). Now, let's revisit our clinical vignette. Are Danish lung cancer survivors happier than the happiest people in the world? Figure 3 shows a simulation of time-to-depression data from a hypothetical lung cancer survivor population and a general population of Danish men more than 60 years old. We simulate time-to-depression data sets for 2 cohorts of 2000 patients with identical literature estimates of rates of depression in the 2 groups and the competing risk of mortality in noncancer individuals and individuals with lung cancer. The 5-year K-M estimate of depression is 2.75% in the noncancer population versus 2.72% in the non-small cell lung cancer survivors. In other words, there is no difference, which is consistent with the assumptions made when simulating the data. As noted in the vignette, this is in contrast to the cumulative incidence estimates from competing risk analysis, showing that the 5-year cumulative incidence of depression in the lung cancer survivors is roughly half of that in the general population. The idea of competing risk analysis may sound attractive to many investigators, but it does come at a price, namely a lack of specificity of the estimate of a given event type. As shown previously, the cumulative incidence of depression is lowered by the presence of the competing risk of all-cause mortality. Although this is true in a sense, it does lead to the false impression that a given pathology, in the example, depression, is less likely to occur in individuals with a high rate of competing events. Developing depression as a lung cancer survivor obviously requires that you are alive. Kaplan-Meier estimates and competing risks analyses are alternative approaches to the analysis of time-to-event data. In most practical situations, you cannot say that one approach is "right" and the other is "wrong"—they answer different questions. A patient being treated for a life-threatening disease with intensive therapy may want to know the likelihood that he/she develops a chronic late effect. It may be of little comfort to the patient that the cumulative incidence of this late effect is low if this is the result of a high risk of dying of disease before the typical latent time of the late effect. Likewise, in most considerations of biology, for example dose-fractionation biology of a normal tissue effect or when studying the effect of a new technology or a modified prescription, it is the K-M estimate that is most relevant. In contrast, cumulative incidence estimates from competing risk analysis may be helpful in health care policy considerations as it reflects the incidence of competing endpoints in a population8Chappell R Re: Caplan et al. IJROBP 29:1183-1186; 1994, and Bentzen et al. IJROBP 32:1531-1534; 1995.Int J Radiat Oncol Biol Phys. 1996; 36: 988-989Abstract Full Text PDF PubMed Scopus (10) Google Scholar and could well be the relevant measure for allocating resources. Some journals, including the Red Journal, have editorial guidelines recommending specific methods for defined endpoints. We would maintain, however, that while respecting these guidelines, authors should appreciate the difference between the 2 methodologies and how to interpret the resulting estimates. The estimates obtained from the K-M or the cumulative incidence method are complementary rather than contrariant. We suggest that the critical reader of any report on time-to-event outcomes from clinical studies in oncology should look for K-M estimates with 95% confidence intervals or standard error of the estimate and with clear definition of endpoints, censoring events, and the starting point of the time axis. And we encourage authors, in addition to whatever metric you may find adds additional information, do report the K-M estimate!Table 1The dos and don'ts of analyzing endpoints requiring prolonged follow-upIssueExampleWhy problematic?Suggested alternativeCommentsInformative censoringTime to local progression estimated with censoring of observations at the time of distant progressionPatients with distant progression are likely at higher risk of local progression; censoring these cases will overestimate the likelihood of local controlSwitch endpoint to TTP or to PFSTTP or PFS are less-specific endpoints but are more robust to detect, say, the effect of modified treatment. Possibly use cumulative incidence estimates of competing failure modes in addition to K-M.Change in case mix over timeXerostomia as a function of time after RT for head and neck cancerSurvival time bias: Locally advanced disease may have larger irradiated volumes; long-term survivors are likely to have less-advanced diseaseIn some cases, selecting patients who survived a minimum of, say, 5 y would allow studying xerostomia over time in a fixed populationSimilar issues may arise for endpoints requiring very long observation timesEndpoint is transientRectal bleedingK-M provides an estimate of the distribution of time to first onset, but this does not capture the possible clinical resolution of the symptomIn addition to time to onset, provide K-M estimate of time to resolution in patients with bleeding by "resetting the clock" to the time of onset of bleeding; patients without resolution at last follow-up are censoredAvoid "home-made" solutions and solve together with collaborators with statistical and domain knowledgeEndpoint is episodic or varies in gradeFatigueK-M estimates may be less informativePoint or period prevalence of high-grade fatigue or proportion of days with fatigue within a time windowCensoring will still need to be consideredIndependent variable is time dependentOS compared in responders versus nonrespondersImmortal time bias: Verification of complete response requires prolonged observation of the patientUse time-dependent PHMAll variables should be known at time 0 in standard K-M/PHM analysisBackdatingASTRO definition: PSA failure after RT for prostate cancer defined as occurring after 3 consecutive PSA rises after a nadir with date of failure defined as the point halfway between the nadir date and the first riseWith, say, 6-monthly PSA measurements, patients censored early will not have met the criterion for failing. Backdating biases the K-M estimate of relapse-free survival; see Fig. 1 in Vicini et al.11Vicini FA Kestin LL Martinez AA The importance of adequate follow-up in defining treatment success after external beam irradiation for prostate cancer.Int J Radiat Oncol Biol Phys. 1999; 45: 553-561Abstract Full Text Full Text PDF PubMed Scopus (125) Google ScholarPhoenix criteria: A rise by 2 ng/mL or more above the nadir. No backdating.Survivorship biasConvenience sampling of patients who are seen in survivorship clinicPatients who die shortly after treatment are not in the risk set. Patients with medical issues may be more likely to show up in the follow-up clinic.Although descriptive studies of such a population may be of interest from the perspective of the follow-up clinic, risk estimates will likely be biasedAbbreviations: ASTRO = American Society for Radiation Oncology; K-M = Kaplan-Meier; OS = overall survival; PFS = progression-free survival; PHM = proportional hazards model; PSA = prostate-specific antigen; RT = radiation therapy; TTP = time to any progression. Open table in a new tab Abbreviations: ASTRO = American Society for Radiation Oncology; K-M = Kaplan-Meier; OS = overall survival; PFS = progression-free survival; PHM = proportional hazards model; PSA = prostate-specific antigen; RT = radiation therapy; TTP = time to any progression. Box 1. The mathIt may be useful for some readers to get a more precise look at the equations underpinning this note. If you would not include yourself in this subset, you can skip reading this box without missing any of the points in the main text.Let T be a random variable denoting the event time; for simplicity, consider all-cause mortality as the event. We define 4 functions:The survival function:S(t)=Pr(T>t)The lifetime distribution:F(t)=1−S(t)The density function of the lifetime distribution:f(t)=F′(t)=dF(t)dtThus, the density function is the derivative of the lifetime distribution, F(t).The hazard function:λ(t)=limΔt→0Pr(t≤T t) The lifetime distribution:F(t)=1−S(t) The density function of the lifetime distribution:f(t)=F′(t)=dF(t)dt Thus, the density function is the derivative of the lifetime distribution, F(t). The hazard function:λ(t)=limΔt→0Pr(t≤T<t+Δt|T≥t)Δt=f(t)S(t) In other words, λ(t)·∆t is the conditional probability that death occurs in the interval from t to t +Δt, given that the patient is alive at time, t. The product-limit or K-M estimator is a nonparametric estimate of the survival function:S^(t)=∏i:ti≤t(1−dini)where ni are the number of individuals at risk at time ti and di are the number of events at that time. S(t) is a random variable itself. The most common variance estimator for S(t) is Greenwood's formula:Var^(S^(t))=S^(t)2∑i:ti≤tdini(ni−di) The Cox proportional hazards model is a semiparametric model specified by the following parameterization of the hazard function in a patient with covariates z1, z2, …λ(t,z¯)=λ0(t)·exp(β1·z1+β2·z2+…)where λ0(t) is estimated from the data using the nonparametric K-M method, and the βs are regression coefficients estimated from the data. The time dependence is modeled by the common underlying hazard λ0(t). It follows that the ratio of hazards in 2 patients in the basic formulation of the model is constant over time, hence the name of the model:λ(t,z¯1)λ(t,z¯2)=exp(β1·z11+β2·z12+…)exp(β1·z21+β2·z22+…) Abbreviation: K-M = Kaplan-Meier.