Introducing Survival Analysis

Survival analysis is a branch of statistics that deals with modeling of time-to-event. In the context of “survival,” the most common event studied is death; however, any other important biological event could be analyzed in a similar fashion (e.g., spreading of the primary tumor or occurrence/relapse of disease). Survival analysis tries to answer questions such as: What is the proportion of a population who will survive past a certain time (i.e., what is the 5-year survival rate)? What is the rate at which the event occurs? Do particular characteristics have an impact on survival rates (e.g., are certain genes associated with survival)? Is the 5-year survival rate improved in patients treated by a new drug? Cox regression and Kaplan-Meier analysis are two techniques which are commonly used to assess survival analysis. 

In survival analysis, the event should be well-defined with two levels and occur at a specific time. Because the primary outcome of the event is typically unfavorable (e.g., death, metastasis, relapse, etc.), the event is called a “hazard.” The hazard ratio is used to assess the likelihood of the event occurring while controlling for other co-predictors (co-variables/co-factors) if added to the model. In other words the hazard ratio is how rapidly an event is experienced and is a comparison of the hazard between groups. A hazard ratio greater than 1 indicates a shorter time-to-event, a hazard ratio less than 1 is associated with a greater time-to-event, and a hazard ratio of 1 indicates no effect on time-to-event. For example, if the hazard ratio is 2 then there is twice a chance of occurrence compared to the other group. 

An important aspect of survival analysis is “censored” data. Censored data refers to subjects that have not experienced the event being studied. For example, medical studies often focus on survival of patients after treatment so the survival times are recorded during the study period. At the end of the study period, some patients are dead, some patients are alive, and the status of some patients is unknown because they dropped out of the study. Censored data refers to the latter two groups. The patients who survived until the end of the study or those who dropped out of the study have not experienced the study event "death" and are listed as "censored". 

Cox Regression

Cox regression (Cox proportional-hazards model) tests the effects of factors (predictors) on survival time. Predictors that lower the probability of survival at a given time are called risk factors; predictors that increase the probability of survival at a given time are called protective factors. The Cox proportional-hazards model are similar to a multiple logistic regression that considers time-to-event rather than simply whether an event occurred or not. Cox regression should not be used for a small sample size (n < 40) because the events could accidently concentrate into one of the cohorts leading to an infinite hazard ratio which will not produce meaningful results (Xu R, Shaw PA, Mehrotra DV. Hazard ratio estimation in small samples. Stat Biopharm Res. https://shawstat.org/wp-content/uploads/2021/03/Hazard-ratio-estimation-in-small-samples.pdf)

Configuring the Cox Regression Dialogue 

• Open the Cox Regression task in the task menu under Statistics. 

• Next, select the Time, Event, and Event status using the drop-down window. Partek Flow will automatically guess factors that are appropriate for these options.  

• The predictors and co-predictors (factors) in the model must be defined. Co-predictors are numeric or categorical factors that will be included in the regression model. Time-to-event will be performed on features (e.g. genes) by default. Select a factor that is not features to model a different variable which will disable each feature as a factor. Next, choose the factor of interest from the drop-down menu. Add factors will add factors to the model that act to explain the relationship for time-to-event (co-factors) in addition to features. Choose Add interaction to add co-predictors with known dependencies. If factors are added here, they cannot be added as stratification factors.
• Next, the user can define comparisons. Configure contrasts by moving factors into the numerator (e.g. experimental factor) or denominator (e.g. control factor / reference), choose Combine or Pairwise, and add the comparison which will be displayed below. Combine all numerator levels and combine all denominator levels in a single comparison or Pairwise to split all numerator levels and split all denominator levels into a factorial set of comparisons meaning every numerator will be paired with every denominator. Multiple comparisons from different factors can be added. 
• Select categorical factors to perform stratification. Stratification is used when proportional hazard assumptions are violated or not constant over time with co-predictors. Stratified Cox regression accounts for non-proportional hazards over time by optimizing hazard strata then fitting the stratified Cox regression model. In other words, the data is split into subgroups based on the categorical variable and the model is re-estimated. This accounts for the effect of a co-predictor that varies over time. 
• The results of Cox regression analysis provide key information to interpret, including:
  • Hazard ratio (HR): if the HR = 0.5 then half as many patients are experiencing the event compared to the control group, if the HR = 1 the event rates are the same in both groups, and if the HR = 2 then twice as many are experiencing an event compared to the control group. 
  • HR limit: 
  • P-value: the lower the p-value, the greater the significance of the observation. 

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