### Survival analysis

The study of recurring events is relevant in systems reliability , and in many areas of social sciences and medical research. The survival function must be non-increasing: May I commend the clarity in which you have presented Hazard Ratio.

Adios to Winter Bash Cross Validated works best with JavaScript enabled. These three tests are asymptotically equivalent. For large enough N, they will give similar results. For small N, they may differ somewhat. The Likelihood ratio test has better behavior for small sample sizes, so it is generally preferred.

The Cox model extends the log-rank test by allowing the inclusion of additional covariates. Regression models, including the Cox model, generally give more reliable results with normally-distributed variables. For this example use a log transform. The log of the thickness of the tumor looks to be more normally distributed, so the Cox models will use log thickness.

The Cox PH analysis gives the results in the box. The p-value for all three overall tests likelihood, Wald, and score are significant, indicating that the model is significant. The p-value for log thick is 6. Because the confidence interval for HR includes 1, these results indicate that sex makes a smaller contribution to the difference in the HR after controlling for the thickness of the tumor, and only trend toward significance. Examination of graphs of log thickness by sex and a t-test of log thickness by sex both indicate that there is a significant difference between men and women in the thickness of the tumor when they first see the clinician.

The Cox model assumes that the hazards are proportional. The proportional hazard assumption may be tested using the R function cox. A p-value is less than 0.

Additional tests and graphs for examining a Cox model are described in the textbooks cited. The Cox PH regression model is a linear model. It is similar to linear regression and logistic regression. Specifically, these methods assume that a single line, curve, plane, or surface is sufficient to separate groups alive, dead or to estimate a quantitative response survival time.

In some cases alternative partitions give more accurate classification or quantitative estimates. One set of alternative methods are tree-structured survival models, including survival random forests.

Tree-structured survival models may give more accurate predictions than Cox models. Examining both types of models for a given data set is a reasonable strategy.

This example of a survival tree analysis uses the R package "rpart". The example is based on stage C prostate cancer patients in the data set stagec in rpart. Each branch in the tree indicates a split on the value of a variable. The terminal nodes indicate the number of subjects in the node, the number of subjects who have events, and the relative event rate compared to the root. An alternative to building a single survival tree is to build many survival trees, where each tree is constructed using a sample of the data, and average the trees to predict survival.

This is the method underlying the survival random forest models. The randomForestSRC package includes an example survival random forest analysis using the data set pbc. In the example, the random forest survival model gives more accurate predictions of survival than the Cox PH model. The prediction errors are estimated by bootstrap re-sampling.

The object of primary interest is the survival function , conventionally denoted S , which is defined as. That is, the survival function is the probability that the time of death is later than some specified time t.

The survival function is also called the survivor function or survivorship function in problems of biological survival, and the reliability function in mechanical survival problems. In the latter case, the reliability function is denoted R t. The survival function must be non-increasing: This reflects the notion that survival to a later age is only possible if all younger ages are attained. Given this property, the lifetime distribution function and event density F and f below are well-defined.

The survival function is usually assumed to approach zero as age increases without bound, i. For instance, we could apply survival analysis to a mixture of stable and unstable carbon isotopes ; unstable isotopes would decay sooner or later, but the stable isotopes would last indefinitely. The lifetime distribution function , conventionally denoted F , is defined as the complement of the survival function,. If F is differentiable then the derivative, which is the density function of the lifetime distribution, is conventionally denoted f ,.

The function f is sometimes called the event density ; it is the rate of death or failure events per unit time. The survival function can be expressed in terms of probability distribution and probability density functions. In other fields, such as statistical physics, the survival event density function is known as the first passage time density.

Suppose that an item has survived for a time t and we desire the probability that it will not survive for an additional time dt:. The term hazard rate is another synonym. The force of mortality is also called the force of failure. It is the probability density function of the distribution of mortality.

In actuarial science, the hazard rate is the rate of death for lives aged x. The hazard rate is also called the failure rate.

Hazard rate and failure rate are names used in reliability theory. Any function h is a hazard function if and only if it satisfies the following properties:. In fact, the hazard rate is usually more informative about the underlying mechanism of failure than the other representatives of a lifetime distribution. An example is the bathtub curve hazard function, which is large for small values of t , decreasing to some minimum, and thereafter increasing again; this can model the property of some mechanical systems to either fail soon after operation, or much later, as the system ages.

The expected future lifetime is the expected value of future lifetime. Hazard ratios become meaningless when this assumption of proportionality is not met. If the proportional hazard assumption holds, a hazard ratio of one means equivalence in the hazard rate of the two groups, whereas a hazard ratio other than one indicates difference in hazard rates between groups.

The researcher indicates the probability of this sample difference being due to chance by reporting the probability associated with some test statistic. Conventionally, probabilities lower than 0. The proportional hazards assumption for hazard ratio estimation is strong and often unreasonable. For instance, a surgical procedure may have high early risk, but excellent long term outcomes. If the hazard ratio between groups remain constant, this is not a problem for interpretation. However, interpretation of hazard ratios become impossible when selection bias exists between group.

For instance, a particularly risky surgery might result in the survival of a systematically more robust group who would have fared better under any of the competing treatment conditions, making it look as if the risky procedure was better.

Follow-up time is also important. A cancer treatment associated with better remission rates, might on follow-up be associated with higher relapse rates. The researchers' decision about when to follow up is arbitrary and may lead to very different reported hazard ratios.

Hazard ratios are often treated as a ratio of death probabilities. In the Cox-model, this can be shown to translate to the following relationship between group survival functions: The corresponding death probabilities are 0.

While hazard ratios allow for hypothesis testing , they should be considered alongside other measures for interpretation of the treatment effect, e. If the analogy of a race is applied, the hazard ratio is equivalent to the odds that an individual in the group with the higher hazard reaches the end of the race first. The probability of being first can be derived from the odds, which is the probability of being first divided by the probability of not being first:.

The hazard ratio does not convey information about how soon the death will occur. Treatment effect depends on the underlying disease related to survival function, not just the hazard ratio. Since the hazard ratio does not give us direct time-to-event information, researchers have to report median endpoint times and calculate the median endpoint time ratio by dividing the control group median value by the treatment group median value.

While the median endpoint ratio is a relative speed measure, the hazard ratio is not. A statistically important, but practically insignificant effect can produce a large hazard ratio, e. It is unlikely that such a treatment would have had much impact on the median endpoint time ratio, which likely would have been close to unity, i. If it takes ten weeks for all cases in the treatment group and half of cases in the control group to resolve, the ten-week hazard ratio remains at two, but the median endpoint time ratio is ten, a clinically significant difference.