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Survival analysis is used when the researcher is interested in whether or not an event happens, and also when it happens. Many social-science phenomena are about causal relationships, that is, that they have a progress over time. The common denominator for the above mentioned examples is that they possess the same logical structure. Survival analysis is a statistical methodology to study the occurrence of an event over time.
A graphical representation of typical survival data is depicted in Figure 1, which shows study recruitment over time. Survival analysis regression aims at investigating and quantifying the impact subject and study factors have on the time until the event occurs. Our New BMJ website does not support IE6 please upgrade your browser to the latest version or use alternative browsers suggested below. Survival analysis is concerned with studying the time between entry to a study and a subsequent event. When the outcome of a study is the time between one event and another, a number of problems can occur.
We cannot afford to wait until events have happened to all the subjects, for example until all are dead. Mclllmurray and Turkie (2) describe a clinical trial of 69 patients for the treatment of Dukes' C colorectal cancer. The calculation of the Kaplan-Meier survival curve for the 25 patients randomly assigned to receive 7 linoleic acid is described in Table 12.2 . Figure 12.1 Survival curve of 25 patients with Dukes' C colorectal cancer treated with linoleic acid.
To compare two survival curves produced from two groups A and B we use the rather curiously named log rank test,1 so called because it can be shown to be related to a test that uses the logarithms of the ranks of the data. At each event (death) at time we consider the total number alive and the total number still alive in group A up to that point. The effect of the censored observations is to reduce the numbers at risk, but they do not contribute to the expected numbers. In the same way that multiple regression is an extension of linear regression, an extension of the log rank test includes, for example, allowance for prognostic factors. Not necessarily, you could use a rank test such as the Mann-Whitney U test, but the survival method would yield an estimate of risk, which is often required, and lends itself to a useful way of displaying the data.
12.1 Twenty patients, ten of normal weight and ten severely overweight underwent an exercise stress test, in which they had to lift a progressively increasing load for up to 12 minutes, but they were allowed to stop earlier if they could do no more.
12.2 What is the risk of stopping in the normal weight group compared with the overweight group, and a 95% confidence interval? Survival analysis is a collection of statistical procedures for data analysis for which the outcome variable of interest is time until an event occurs.


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Standard practice for classification based machine learning involves feeding a model loan characteristics with its final outcome (fully paid or charged off). A common tool for the medicine, the social scientist community is starting to see the opportunities provided by this tool. The best non-experimental technique to study these processes of causality is by the use of survival analysis. A sample of subjects is considered a group at risk where events, like getting married, giving birth, change job, or die from an illness, can happen in a period of risk. It is referred to as survival analysis because it was originally derived in contexts where the event was death, but the event under study need not be death. These factors are often measured at study entry (t0) for each individual participant, and their effect on time to event is quantified via the hazard function of the survival time distribution. Originally the analysis was concerned with time from treatment until death, hence the name, but survival analysis is applicable to many areas as well as mortality. The survival curve is unchanged at the time of a censored observation, but at the next event after the censored observation the number of people "at risk" is reduced by the number censored between the two events. Thus if linoleic acid reduces the risk of death in patients with colorectal cancer, then this risk reduction does not change with time (the so called proportional hazards assumption ). As for the Kaplan-Meier survival curve, we now consider each event in turn, starting at time t = 0. If we had a total of events at time then, under the null hypothesis, we consider what proportion of these would have been expected in group A. The log rank test is quite "robust" against departures from proportional hazards, but care should be taken. Design and analysis of randomized clinical trials requiring prolonged observation of each patient: II. The strength of survival analysis is that the observations over time enable us to estimate the chain of causality with a large degree of confidence.
In sociology one can mention studies of unemployment, careers, marriage, divorce, and child birth.
For each subject one records whether or not the event took place within the time scope of the study, and also how long it took from when that subject entered into the risk phase. Examples from the social sciences where survival analysis can be used are studies that investigate time from marriage until separation or divorce and intervals between births. The hazard function models the rates at which events occur as a function of subject and study factors.
Recent examples include time to discontinuation of a contraceptive, maximum dose of bronchoconstrictor required to reduce a patient's lung function to 80% of baseline, time taken to exercise to maximum tolerance, time that a transdermal patch can be left in place, time for a leg fracture to heal. Thus the only information we have about some patients is that they were still alive at the last follow up.


Clearly the more people at risk in one group the more deaths (under the null hypothesis) we would expect. If the Kaplan-Meier survival curves cross then this is clear departure from proportional hazards, and the log rank test should not be used. If no event is observed during the study period, the last known event-free time point is marked with a circle. This can happen, for example, in a two drug trial for cancer, if one drug is very toxic initially but produces more long term cures. We can now calculate the survival times , for each value of i from 1 to n by means of the following recurrence formula. At time 6 months two subjects have been censored and so the number alive just before 6 months is 23.
As one can see the effect of the censored observations is to reduce the number at risk without affecting the survival curve S(t). In this case there is no simple answer to the question "is one drug better than the other?", because the answer depends on the time scale. The most frequently used model for analyzing survival data is the Cox proportional hazards model (a semiparametric model). A censored observation can arise from the fact that a participant is lost to follow-up during the observation period or from a limited observation period, that is, the event might occur some time after the observation period has ended.
It assumes that hazard rates are proportional over time but does not make distributional assumptions regarding survival times. Examples of parametric methods are the Weibull and accelerated failure time models, which assume specific statistical distributions for survival times in addition to assuming proportional hazards. A right censored observation indicates that occurrence of the event, if it happens, will take place after the time that contact is lost with the participant or after the end of the observation period.
Standard models assume independence between observations, but extensions of the models are available to accommodate dependencies (frailty models) between observations. Extensions also exist to accommodate multiple events, competing events, and factors that might change over time. On one hand the extension to multiple events and competing events is conceptually straightforward. Analysis involving factors that might change over time, on the other hand, are both technically and conceptually more involved.
Survival analysis regression has been used extensively and successfully in various fields to quantify the impact of different factors on time to event.



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