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Survival Analysis NJ Gogtay, UM Thatte 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.
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