80 Journal of The Association of Physicians of India ■ Vol. 65 ■ May 2017

STATISTICS FOR RESEARCHERS Survival Analysis NJ Gogtay, UM Thatte

Introduction and quality of life. Such studies are followed up for the remaining can last for weeks, months or even 3 years of the study duration. ften in research, we are not years. When we capture not just Patients can enter the study at any Ojust interested in knowing the the event, but also the time frame point during the first 2 years. For association of a risk factor/exposure over which the event/s occurs, this example, they can enter the study with the presence [or absence] of becomes a much more powerful right at the beginning [Month 1] or an outcome, as seen in the article tool, than simply looking at events at end of the accrual period [Month on Measures of Association,1 but alone. 24]. The first patient will undergo rather, in knowing how a risk An additional advantage with the full 5 year follow up, while the factor/exposure affects time to this type of analysis is the use patient who came into the study disease occurrence/recurrence or time of the technique of “censoring” at month 24, will only have a 3 to disease remission or time to some [described below], whereby each year follow up as the cut off that other outcome of interest. patient contributes data even if he/ we have defined for this study is 5 Survival analysis is defined she does not achieve the desired years. However, when this data is as the set of methods used for outcome of interest or drops out analyzed, regardless of the time of analysis of data where time during the course of the study for entry into the study, every patient to an event is the outcome of any reason. would be analyzed from his/her interest. Originally, this analysis point of entry into the study. This was concerned with time from Key Concepts in Survival is called “zero time” and is the time treatment until death and hence the Analysis when the patient is enrolled. name. Survival analysis however Censoring can be applied to a wide variety In order to understand survival In the above example, any one of of situations. Medical examples analysis certain concepts need to be the following could happen to any include time to metastases, time to understood before doing survival of the patients. Thus, the patient tumor recurrence, time to discharge analysis such as: Time or survival i. Would actually achieve the from the hospital, time to first time, time of entry into the study, outcome of interest [death in exacerbation after a new drug censoring, cumulative probability, this case] treatment in patients with Chronic hazards and hazard ratio and Obstructive Pulmonary Disease survival and hazard functions. We ii. Does not achieve the outcome [COPD], time to dialysis in patients discuss these briefly below. of interest although the study with renal dysfunction and so on. Time or survival time ends (i.e. 5 years are over) In the real world, survival could be The time variable in a survival iii. Is lost to follow up [so we do time to a light bulb fusing, time to analysis is called as “survival not know whether the outcome replacing the battery on the wall time”, while the event of interest has or has not occurred] clock or time to the change the gas itself is called “failure”. iv. Withdraws consent cylinder. The other terms used for Time of entry into the study- the v. Dies of some cause other survival analysis are “failure-time concept of zero time than the disease under analysis”, “reliability analysis”, Let us understand this with an investigation, in this case, lung and “event history” analysis. example of a study evaluating a cancer [for example a patient Why Survival Analysis is new medical treatment for lung enrolled in the trial dies of myocardial infarction rather Important cancer, which has a follow up duration of 5 years. The first 2 years than lung cancer. This is called Studies of how patients of this study are used for enrolling “competing risk”]. respond to treatment over time or accruing patients. These patients are fundamentally important to understanding how treatments Department of Clinical Pharmacology, Seth GS Medical College and KEM Hospital, Mumbai, Maharashtra influence both disease progression Received: 09.04.2017; Accepted: 11.04.2017 Journal of The Association of Physicians of India ■ Vol. 65 ■ May 2017 81

V5, Survival analysis, JAPI series, 12th April 2017, 930am

Figure 1 – Right Censoring True survival time the intervention/treatment under be the rate at which patients die

Observed survival time evaluation. during the course of the study [or Cumulative probability of survival the time course of their death].

Right censoring Mathematically, it is expressed Probability is the chance of a as the hazard function [described single event occurring whereas FigureFig. 2 – Data 1: of nRight =100 patients Censoring depicting Survival after surgery for prostate cancer below]. Since studies on survival if you want to calculate the analysis involve the comparison chance of two, three, or more When a patient achieves the of two or more groups, the hazard events happening, we measure outcome of interest (i), it is useful in one group is compared with the the “Cumulative probability”. to the researcher as it contributes other group and expressed as a Two caveats need to be fulfilled valuable data. But what if the ratio called the hazard ratio. This is for calculating Cumulative patient experiences any of the other defined as the ratio of the hazard probability: 1) each event needs to situations (ii to v)? in the experimental to the hazard be independent of the other and 2) Survival analysis is unique in in the control arm. The distinction outcome of the first event should that it “allows” the researcher to between hazard ratio and odds not influence the probability of use data from such patients up ratio [or relative risk] lies in the fact occurrence of the second. until the point of their last follow that the latter are simply ratios of up by using a method called as This concept can be best proportions, while hazard, which censoring. There are three main understood with an unbiased coin incorporates time, is a ratio of types of censoring: right, left, and toss experiment. Let us say we want incidence rates. to answer the question “When a interval. The most commonly used Survival and hazard functions one is the “right censoring” where coin is flipped twice, what is the Both of these are crucial to censoring occurs after the patient probability9 of getting heads on both the analysis of survival data and has entered the study (Figure 1) occasions?” Each toss will have are related to each other. They because the participant has left two outcomes (H or T) and two describe the distribution of event the study for any of the reasons consecutive tosses will have four times. The survival function s(t) is mentioned above. possible outcomes: the probability that an individual • HH Let us take another example survives from a specified time of a breast-feeding survey done • TT point (e.g., the diagnosis of cancer) monthly. Two types of mothers • HT to a specified future time t. It can enter the study: those who • TH directly summarizes time to event are breast feeding at the time of experience of a group of patients And the answer to our question entering the study and those who and is crucial to analyzing time is therefore one in four or 25%. have stopped breast feeding at to event data. Hazard function is When the coin is tossed the first the point of entry into the study. denoted as h(t) and represents the time, the probability of getting For the former, right censoring probability that an individual who heads is ½ or 50%. When it is tossed can be done, while for the latter is under observation at a time t, has the second time, the probability of as we do not know exactly when an event at that time t. It can range heads is again ½ as this outcome they stopped breast feeding before from o to infinity. In other words, is independent of the first and its entering the study, we need to it represents the instantaneous probability is not influenced by the do “left censoring” for the point event rate [at time t], given that probability of the first. Thus, the where they stopped breast-feeding. the individual has already survived cumulative probability of getting Interval censoring occurs if the upto that time t. The distinction two consecutive heads is calculated breast-feeding ended between between the two functions lies in as the product of the probabilities two successive surveys since one the fact that the survival function of each event – i.e. ½ x ½ or ¼ or can only say that breast feeding relates to not having the event, 25%. We will see how this is applied ended somewhere between the two while the hazard function relates to 2 to calculate the cumulative survival surveys. the probability of the event occurring and survival and draw the survival For censoring to have validity, per unit time. 3 Mathematical curve [see below]. when a patient is censored, the relationships between the two Hazard and hazard ratio risk for achieving the outcome for functions have been defined and the remainder of the patients who A “hazard” is simply the rate computer software can return the continue on the study, should be at which a particular event occurs. value of one function, given the unchanged. In addition, censoring If we were to take the example of value of the other. should be randomly distributed lung cancer presented earlier, and over time as its main assumption is we are interested in death as the that it is independent of time and outcome, then the hazard would 82 Journal of The Association of Physicians of India ■ Vol. 65 ■ May 2017

Table 1: Hypothetical survival data for n = 100 patients with prostate cancer following surgery Time interval No. No. censored (Data No. who achieved the No. survived (years) at risk not available beyond outcome of interest a point) (died) 1 100 3 5 95 2 92 3 10 82 3 79 3 15 64 4 61 3 20 41 5 38 3 25 13 Fig. 2: Data of n =100 patients depicting Survival after Table 2: Kaplan Meier cumulative survival estimates in hypothetical cases of surgery for prostate cancer prostate cancer (n=100 at start) Time interval No. No. censored No. who No. Kaplan-Meier Types of survival analysis (years) at risk (Data not achieved the survived cumulative available beyond outcome of survival A number of models are available a point) interest (died) estimate to analyse the relationship of a set 1 100 3 5 95 (95/100)=0.95 of predictor variables with the 2 92 3 10 82 0.95 survival time. The methods for x(82/92)=0.8467 doing a survival analysis fall into 3 79 3 15 64 0.8467 x (64/79)=0.70 three broad categories 4 61 3 20 41 0.7 x • Non-parametric (41/61)=0.4611 • Semi-parametric, and 5 38 3 25 13 0.4611 x (13/38)=0.1577 • Parametric. 6 10 2 8 0 0 The difference between these analyze survival data and the Cox Before we can do any survival methods lies in the assumptions proportional hazards regression analysis, we need to make sure that that we make regarding the (see below) is representative of our data is structured in a manner distribution of survival data. In these methods. When we assume that makes analysis easy. This nonparametric analysis, there is that survival times conform to some needs to be done for each and every absolutely no assumption made specific statistical distribution patient in the study. Thus, the first and these methods can be used (e.g. exponential, Weibull, and thing to do is to organize the data. for all types of data. The Kaplan lognormal distributions) we use Each patient as discussed earlier Meier method is a widely used non- parametric survival analysis, less would enter the study at time zero. parametric method in medicine for commonly seen in medicine. Then, we look at each patient to see survival analysis. It can be used whether he/she has achieved the for study of a variable in a single Analysis of Survival Data outcome of interest or needs to be group over time (e.g. group of censored. Let us understand this Several techniques exist for smokers developing lung cancer with an example. over time), but it also serves the analyzing survival data. In the A hypothetical research question purpose when you want to compare Life table technique [also called we are trying to answer could two or more groups over time actuarial analysis] the data is be “How many years do patients (e.g. progression free survival divided into fixed time intervals with prostate cancer survive after with Trastuzumab innovator and for each time interval, we they undergo surgery”? We lay vs. Trastuzumab biosimilar in assess cumulative probability with down the following unequivocal metastatic breast cancer over patients who have achieved the boundaries for the study right at time). This method plots a curve outcome of interest and those who the beginning: 1) all patients who (Kaplan Meier curve; see Figure 2) have been censored. The Kaplan- undergo surgery in the first 1 year of cumulative probability against Meier technique described above, after the start of the study would be time and can be used to obtain on the other hand, uses the time included i.e., 1 year would be the univariate descriptive statistics for intervals that are data driven and total accrual period 2) The study survival data, e.g. median survival not necessarily fixed as in Life would be done for a total of 5 years time. Inferential statistics can be Table Methods (although they 4 regardless of when they enter the applied using several tests and the can be fixed as well). Another study. The data for 100 patients log-rank test is the most popular technique is the Cox Proportional The data for 100 patients is given (see below). Hazard Method, used when there are several predictor variables in Table 1. Semi-parametric tests are equally impacting the event of interest. The Kaplan-Meier procedure commonly used in medicine to Journal of The Association of Physicians of India ■ Vol. 65 ■ May 2017 83 then calculates the cumulative remaining and thus survival occurring at any time point is the probability (calculated as the number estimates at the end of the same for each of the two groups of subjects surviving divided by the curve are less accurate. being studied). Survival curves number of patients at risk) for each • When patients are censored, are estimated separately for each of the t time periods, except the first we do not know whether they group, using the Kaplan-Meier (Table 2). have actually experienced the method and compared statistically using this test to give a chi-square Kaplan Meier / Product- outcome of interest. Thus, more the number of patients that are value. Limit Curve censored, the less reliable is the Assumptions of the log rank test Kaplan Meier curve. The Kaplan-Meier survival curve • Survival times are ordinal or is the curve that results at the end Cox Regression or continuous. of the survival analysis and after Proportional Hazards • The risk of an event in one the calculation of the survival group relative to the other does probabilities. The generation of the Regression not change with time curve makes certain assumptions It is intuitive that time to an The Mantel Cox test is also • At any time, patients who event/outcome is influenced by another commonly used test. are censored have the same not one by multiple predictor Conclusions survival prospects as those variables. For example, coronary The fundamental reason why who continue to follow up on artery disease [CAD] we know is survival analysis is done is that time the study (non-informative). associated with a high Body mass to an event is a far more powerful • Survival probabilities are the index [BMI], gender, hypertension, tool than simply looking at events same for patients regardless and smoking. Cox regression alone. Several examples of these of the time point at which they also called proportional hazards can be found in medical literature. enter the study regression [semi-parametric] helps One recent example is that of • The event of interest happens investigate the effect and impact association of atypical femoral at the time or time interval of several predictor variables on shaft fractures with long term specified. the time to event or the variable bisphosphonate use. We would of interest. It has three critical thus be answering the question A Kaplan Meier curve generated assumptions – 1) The effects of the “What is the time to developing on the hypothetical data is described predictor variables upon survival atypical femoral fractures after in Figure 2. Time is given on the x remain constant over time, the the use of bisphosphonates for axis and cumulative probability on concept of proportional hazards (the several years” and how would the y axis. Censored patients are parametric component). In other it change the way we precribe shown as vertical lines or “ticks” on words, in this example, the impact bisphosphonates? This would then the curve while the deaths/outcome of BMI, gender and hypertension drive evidence based practice. of interest are the dips in the curve all remain constant or do not Thus, the recommendation of a Pitfalls of the Kaplan Meier curve change over the duration of the task force of the American Society • Curves that have many small study. 2) It makes no assumptions for Bone and Mineral Research is steps usually have a higher regarding the baseline hazard (the to give a “bisphosphonate holiday” number of participating non-parametric component) 3) It of 2-3 years in postmenopausal subjects, whereas curves with assumes that the censoring is non- women at low risk of fractures large steps usually have a informative. who have received 3-5 years of limited number of subjects and treatment with bisphosphonates.6 tend to be less accurate. The Log Rank Test – Censoring is an integral • Most studies have a minimum Comparing Two Survival component of survival analysis. duration of follow-up based on Curves However, this must not mean knowledge of disease biology that the researcher should relax and overall survival. At this This is a popular test that is with respect to follow up as minimum duration of follow closely related to the chi-square completeness of follow-up is up, the status of each patient test and tests the null hypothesis of crucial so as not to miss events is known. The survival rate at no difference in survival between and lose patients as this can lead to this point becomes the most two or more independent groups. biased results. Unequal follow-up accurate reflection of the The null hypothesis would be that between different treatment groups survival rate of the group. At there is no difference between may also produce biased results. the end of the survival curve, the population survival curves Finally, it must be ensured that there are far fewer patients (i.e. the probability of an event when Cox regression is used, the 84 Journal of The Association of Physicians of India ■ Vol. 65 ■ May 2017 proportional hazards assumption 2016; 64:70-3. Applied Science 2013; 1:17-21. is met with. 2. https://www.cscu.cornell.edu/news/ 5. Edwards BJ, Bunta AD, Lane J, et al. th Acknowledgments statnews/stnews67.pdf, accessed on 7 Bisphosphonates and Nonhealing Femoral April 2017. Fractures: Analysis of the FDA Adverse The authors are grateful to 3. George B, Seals S, Aban I. Survival analysis Event Reporting System (FAERS) and Dr. L. Jeyaseelan, Dr. Girish and regression models. Journal of Nuclear International Safety Efforts: A Systematic Chinnaswamy and Dr. Manju Cardiology: official publication of the Review from the Research on Adverse Drug Sengar for helpful inputs in refining American Society of Nuclear Cardiology Events And Reports (RADAR) Project. The 2014; 21:686-694. Journal of Bone and Joint Surgery American the manuscript. volume 2013; 95:297-307. 4. Sambath AK, Ramanujam R, Chinnaiyan References P. Survival analysis: Kaplan Meier and life 6. https://www.asbmr.org/About/detail. table estimates for time to event clinical aspx?cid=9caf8b31-8157-407e-a61d- th 1. Gogtay NJ, Deshpande S, Thatte UM. trial tuberculosis data. Concepts in Pure and e9b34065e3b4, accessed on 7 April 2017. Measures of Association. J Assoc Phy Ind