Some Explanations about Survival Analysis or Time to Event Analysis

The concept of Survival analysis came from the old days for cancer study where the measurement is the length of the survival or time to death. The concept of survival analysis has since been generalized to ‘time to event’ analysis. ‘Time to death’ is just one type of time to event variables. There are a lot of events that can be analyzed in the same way. For time to event measures, the event can be positive and negative. For example, in a clinical trial for recurrent herpes labialis (or cold sores), the event (healing of the primary lesion complex) is positive. The time to event is measured as time to healing of the primary lesion complex. Perhaps, more clinical trials have the negative event (event is bad). In oncology studies, we measure survival (or time to death). In COPD studies, we may measure time to exacerbation. In a PAH study, we may measure time to clinical worsening.

In short, in Time to Event analysis, the analysis relates not just to whether an event occurs but also when.

The planning (for example sample size estimation) and analysis of the time to event study, several important concepts and their relationship are important. These concepts and their relationships are explained below:

Time to Event is just the measure from the start of an intervention to the time when an event occurs. The start of an intervention could be the randomization, start of the treatment, date of surgery,…

Event Rate is the proportion of subjects or patients in a group in whom an event is observed.  Event rate is usually measured for a period of the time from t to t + Dt. For example, if the Dt = 12 months, the event rate will be for one year. Event rate is also given as the event rate for the entire study period.

Hazard Rate is the probability of an event occurring given that it hasn’t occurred up to the current point in time. Hazard rate is the instantaneous risk of a patient experiencing a particular event at each specified time. The instantaneous rate with which an event occurs at a single point in time. It is the probability that the event occurs between time t and time t+delta given that it has not yet occurred by time t, divided by delta, as delta becomes vanishingly small. Note that rates, unlike probabilities, can exceed 1.0 because they are quotients.

Hazard Ratio is a measure of effect produced by a survival analysis. This represents the increased risk with which one group is likely to experience the outcome of interest.  For example, if the hazard ratio for death for a treatment is 0.5, then we can say that treated patients are likely to die at half the rate of untreated patients.

Hazard ratio is calculated as the ratio of hazard rates at a single time t, for two groups of subjects (treatment versus control group). Hazard ratios are in the interval [0, infinity), and they are frequently good ways to summarize the relative effects of two treatments at a specific time t. Like odds ratios, hazard ratios can apply to any level of outcome probability for the reference group. Note that a hazard ratio is distinct from a risk ratio, the latter being the ratio of two simple probabilities and not the ratio of two rates.

The Median Event Time is calculated as the smallest even time for which the event function is less or equal to 0.5.

When the event is death, the median event time is called the median survival time. The median survival time is calculated as the smallest survival time for which the survivor function is less than or equal to 0.5. In oncology study, median survival time the time from either diagnosis or treatment at which half of the patients with a given disease are found to be, or expected to be, still alive. In a clinical trial, median survival time is one way to measure the effectiveness of a treatment to see how well a new treatment works. Median survival time may be called median overall survival or simply median survival.  

Censoring is a form of missing data problem which is common in survival analysis and time to event analysis. In clinical trials, we usually have to deal with the right censoring. In the situation of the right censoring, the event did not occur when subjects are lost to follow-up or when the study ends. A patient might be known not to have had the event only up to a particular point in time, so ‘time to event’ or ‘survival time’ is censored at this point.

Lost to Follow-up refers to patients who at one point in time were actively participating in a clinical trial, but have become lost (either by error in a computer tracking system or by being unreachable) at the point of follow-up in the trial. These patients can become lost for many reasons. Without properly informing the investigator associated with the clinical trial, they may have opted to withdraw from the clinical trial, moved away from the particular study site during the clinical trial, or become ill and unable to communicate or are deceased.

Attrition: The loss of participants during the course of a study. Participants that are lost during the study are often call dropouts.

Accrual time or accrual period is recruitment period during which subjects are being enrolled (recruited) into a study.

Follow-up time or follow-up period is the period after the last subject entered the study until the end of the study. The follow-up defines the phase of a study during which subjects are under observation and no new subjects enter the study.

If T is the total duration of a study, and R is the accrual period of the study, then follow-up period f is equal to T – R.

Event Rate =  1 -  Non Event Rate

Mortality Rate =  1 - Survival Rate

Given the MET (median event time), we can calculate the hazard rate and the event rate, and hazard ratio.

If METc is the Median event time for control group and METt is the Median event time for treatment group, HAZARDc and HAZARDt are Hazard rates for control group and treatment groups, we will have:

        HAZARDc = log(2)/METc       
        HAZARDt = log(2)/METt
        METc  = log(2)/HAZARDc       
        METt = log(2)/HAZARDt

    Event rate at month 12 for control group is

      Ec = 1 - exp(-12*HAZARDc) 

    Event rate at month 12 for treatment group is  

      Et = 1 - exp(-12*HAZARDt);   

    HAZARD rate can be calculated from Event rate (for example event rate at month 12)

      HAZARDc = -ln(1-Ec) / t   (for example t=12)
      HAZARDt  = -ln(1-Et) / t   (for example t=12)  
   
The hazard ratio is:
      HAZARDt/HAZARDc

If the given parameter is event rate over the entire course of the study (for example, 5 years), the event rate for one year can be calculated using the following formula:

        1 - (1 - event rate)^(1/t)

        where t=5

 If the given parameter is event rate over one year period, the event rate for five year study period can be calculated using the formula below:


       1 - exp(t*ln(1 - annual event rate))

        where t=5

 
The formula above can also be used to convert the loss follow up rate from the entire treatment period to one year or vice versa.

References:

•   Clinical trial glossary 1
•   Clinical trial glossary 2
•   Janet Wittes (2002) Sample Size Calculations for Randomized Controlled Trials
•    Lakatos (1988) Sample Sizes Based on the Log-Rank Statistic in Complex Clinical Trial
•    Lachin and Foulkes (1986) Evaluation of Sample Size and Power for analyses of survival with allowance for nonuniform patient entry, losses to follow-up, noncompliance, and stratification
•    Lakatos and Lan (1992) A comparison of sample size methods for the log-rank statistic
•    An introduction to survival analysis by Maarten Buis
•    Survival models