HC8+9: Survival analysis

When to use survival analysis

Survival is used to determine when a certain event happens, such as:

• When a tumor develops after exposure to a carcinogen
• When a patient dies after a cancer diagnosis
• When a patient becomes pregnant after the start of fertility treatment
• When a graft is rejected after transplantation

Possible questions that can be answered with survival analysis are:

• What is the distribution of the survival times?
• Is there a difference in expected life-time between different treatments?
• What is the fastest way to become pregnant?
• Which factors predict 5-year survival probabilities?

Example:

For patients with end stage renal disease, it needs to be determined which treatment gives better survival:

• Hemodialysis (HD)
• Peritoneal dialysis (PD)

Survival probabilities and survival times after start of dialysis can possibly be compared with help of T-tests of Chi square tests. However, there are several problems:

• Not all patients died
• Not all patients have the same follow-up time
  • Some started dialysis a while ago, some started dialysis recently
• Patients get lost (e.g. migration)

Censoring

Because the time of death of all patients isn’t known, some patients have censored survival times. In an ideal world, patients that are still alive in each group are compared after a certain time period. However, some patients may be lost to follow-up. This can be solved by censoring survival times → the time of event (e.g. death) is not observed, it is only known when the patient was last seen alive. Reasons for censoring are:

• Administrative censoring: individuals do not have the event (death) before the end of the study
• Lost to follow-up: the patient moves or does not show up for appointments
• The patient dies of another disease

Example:

In the peritoneal dialysis group, 207 patients died and 446 are still alive and were last seen between 0,8 and 5 years after the start of dialysis → the follow-up varies. Not only the data of the 207 patients should be used because it is very informative that someone is still alive after 5 years → all patients must be used.

This can be shown in a graph with on the y-axis the a line for each patient, with the length of the line corresponding to the time after start of dialysis on the x-axis. Red dots indicate that the patient has died, green dots indicate that the patient is still alive or, if the dots are before the time of measurement, that the patient was lost to follow-up.

The survival function

The aim of survival analysis is to estimate the survival function S(t):

• S(t) = probability that an event occurs after time (t) → the probability to be “alive” at time (t)
• Always starts at 1 → everyone is alive at the start of study

Kaplan-Meier method:

Survival probabilities can be estimated with the Kaplan-Meier method. Information of each patient is used until death/censoring, for instance the information of 10 persons:

1. 3
2. 4
3. 7+
4. 9
5. 10
6. 11+
7. 12
8. 20
9. 20
10. 25+

The “+” indicates censored → the patient was still alive at the indicated time point, but it is unknown what happened afterwards. This information can be converted into a curve:

• Y-axis: survival probability
  • How many % of the patients are still alive
• X-axis: time
  • Stops after 25 → all the patients have died/are censored

Because there are only 10 observations, there is a lot of uncertainty in the estimate → there is a high rate of standard errors.

95% confidence intervals (CI) for the survival probability S(t) at each time point are computed separately for each time point. The CI can be calculated with the estimate standard error (se):

• (S(t) – 1,96 x se, S(t) + 1,96 x se)

In this case, the 95% CI at t=3 is (0,71 - 1,09), which is impossible → it is rounded down to (0,71 – 1,00). This can occur when the sample is small. Each 95% CI can be shown on the Kaplan-Meier curve. CI’s become wider as uncertainty increases. This typically happens as time increases, because the number of patients in the follow-up decreases.

Median survival time:

The median is the value that half of the people are above and half of the people are above →  the median survival time is the time at which 50% is still alive. It is the value which corresponds with a survival probability of 0,5.

Assumptions

Assumptions that are made when using the Kaplan-Meier method are:

• Survival probabilities are the same for individuals recruited early and late in the study
• At any time censored patients have the same survival prognosis as those in study
• The time of events is known exactly

Informative censoring:

Censoring should be independent, meaning that censored patients are at the same risk of the event (e.g. death) as the other patients. In case of informative censoring, this isn’t the case → the Kaplan-Meier will be biased.

Examples of informative censoring are:

• Patients are taken off a study because of inadequate response to treatment
  • Are likely to have a higher risk than the patients who remain in the study → the Kaplan-Meier estimate may underestimate the risk of death
• Patients have been discharged from the hospital
  • Are likely to have a (much) lower risk than the patients who remain in the study → the Kaplan-Meier estimate may overestimate the risk of death

The log-rank test

By comparing 2 survival (Kaplan-Meier) curves, it can be concluded which treatment works better. In case of dialysis, it is peritoneal dialysis. Comparing survival curves can be done with the log-rank test, where observed curves are compared with what would be expected if H0is true:

• Null hypothesis H0: 2 curves are equal
• Alternative H1: curves differ

Log-rank test statistic X². X² has a chi-square distribution with 1 degree of freedom under H0→ used to calculate the p-value.

Example:

If the log-rank test is applied to the dialysis example, p <0,001. This is a very small p-value → the observed differences are unlikely to occur under H0 (only in less than 1 in 1000 of cases). Because the p-value <0,05, survival on peritoneal dialysis is statistically significantly different from survival on hemodialysis.

Immortal time bias

Researchers need to be very careful when comparing subgroups based on characteristics that only manifest later, for instance in case of responder versus non-responder → responders need to survive long enough to be a responder. Therefore, special statistical methods are required. For example, it seems as if patients who have received a heart transplantation is higher than of those who didn’t. This is caused by patients dying while they are on the waiting list, making it seem like survival for heart transplantation patients is better while this is not the case.

The hazard ratio

Hazard ratios are often used to quantify the difference between survival functions. They can be used as effect size measures. Log-rank tests give P-values, which can be problematic because they depend on the sample size:

• In small studies, large effects can still be not statistically significant
• In large studies, small, clinically irrelevant effects may be statistically significant

Therefore, it is necessary to look at effect sizes and precision. For survival data, the effect size measure is the hazard ratio.

Hazard function:

For discrete time, the hazard function is the probability that a person, still alive just before time (t), dies at time (t), for example:

• S(12) = 0,40 and S(13) = 0,20
  • 40% of the people are still alive after 12 months, and 20% are still alive after 13 months
• H(13) = (S(12) – S(13))/S(12) = (0,40-0,20)/0,40 = 0,50
  • 50% of the people have died in the 13^th month → the hazard is 50%

In continuous time, the hazard is also called the instantaneous rate of failure.

Hazard functions can be converted into survival curves, and the other way around, via the following relationship:

• h(t) =   = -

Calculation:

High hazard is associated with a decrease in the survival curve. The hazard ratio shows the difference in survival, for example of hemodialysis and peritoneal dialysis patients:

• HR =
  • h1(t)= hemodialysis patients
  • h0(t) = peritoneal dialysis patients

Interpretation:

The HR is often assumed to be constant over time and can be interpreted as follows:

• HR >1 → the survival of group 1 is worse than of group 0
• HR <1 → the survival of group 1 is better than of group 0
  • A HR will never be lower than 0
• HR = 1 → the survival in both groups is similar

In SPSS, the HR can also be made visible:

• B: the logarithm of the HR → ln(HR)
  • Easier to calculate with because ln(HR) is symmetric around 0
• Exp(B): the HR

However, hazard ratios may change over time → are not proportional:

• Tumor size is very prognostic for the first years of cancer survival, but less so later on
• Lymph node dissection as treatment of gastric cancer gives better survival than extensive lymph node dissection in the first 2 years, but later on the latter is better
  • The Kaplan-Meier survival curves cross each other → indicates violation of the proportional hazards assumption

A possible solution for non-proportional HRs is to calculate the HR in a limited period of time, such as per year.

Is survival analysis appropriate?

Sometimes, survival analysis can be used as a research method and sometimes it cannot. Examples of such research questions are:

• Do patients who have had a heart attack score lower on a depression survey than patients who have not had a heart attack?
  • Survival analysis cannot be used → the mean scores between the groups need to be compared, which require standard methods such as T-tests
• How long until patients who have had a heart attack have a second heart attack?
  • Survival analysis can be used → the event is a second heart attack

Standard deviation

The standard deviation (σ) can be calculated by multiplying the standard error (e) of a mean with the square root of the sample size (N):

• σ = ex √N

When the group is large, σ can also be calculated when the 95% CI is known:

• σ = √N x (upper limit – lower limit)/(2 x 1,96)