Lecture 4 - Survival outcomes

Question 1

Describe what survival analysis is.

Answer 1

These kinds of analyses deal with ‘time to events’.

Question 2

Some common terms used in survival analyses are:
* event
* time
* censoring
* survival function
* cumulative incidence function
Describe the definition/use of these terms.

Answer 2

• event: death, disease occurrence, disease recurrence, recovery, or other experience of interest.
• time: the time from the beginning of an observation period to an event, or end of the study, or loss of contact or withdrawal from the study.
• censoring: occurs when information about an individual’s survival time is available, but the exact survival time is unknown. The subject is censored in the sense that nothing is observed or known about that subject after the time of censoring.
• survival function: the probability that a subject survives longer than time t (i.e. the probability that the event does not occur before time t)
• cumulative incidence function: the probability that the event occurs before time t.

Question 3

Survival function can be defined as S(t). Describe the associated formula for the cumulative incidence function (i.e. the probability that the event occurs before time t).

Answer 3

F(t) = 1 - S(t)

Question 4

Which ‘formula’ can you use to calculate survival function S(t)?

Answer 4

Amount of e.g. patients that have not reached their ‘event’ divided by total amount of patients that will eventually reach their ‘event’. So if 9 patients will eventually go into remission after cancer treatment and at time t=2, 2 out of 9 patients will have reached remission, S(t) = 7/9 = 0.78.

Question 5

A survival analysis is performed to analyse the survival function of the amount of weeks in remission among leukemia patients treated with 6-MP. In total 9 patients were followed with the following weeks in remission (i.e. event): 6, 6, 6, 7, 10, 13, 16, 22, and 23. Calculate the survival function (S(t)) for these 9 patients.

Answer 5

• On t=0, all patients (9) were still in remission so S(t)= 9/9 = 1
• On t=1, all 9 patients were still in remission so S(t) = 9/9 = 1
• On t=6, 6 patients were still in remission, while 3 patients already recovered, so S(t) = 6/9 = 0.67
• On t=7, 5 patients were still in remission, while 4 patients already recovered, so S(t) = 5/9 = 0.67.
• On t=10, 4 patients were still in remission, while 5 patients already recovered, so S(t) = 4/9 = 0.45.
• On t=13, 3 patients were still in remission, while 6 patients already recovered, so S(t) = 3/9 = 0.33
• On t=16, 2 patients were still in remission, while 7 patients already recovered, so S(t) = 2/9 = 0.22
• On t=22, 1 patient was still in remission, while 8 patients already recovered, so S(t) = 1/9 = 0.11
• On t=23, 0 patients were still in remission and all 9 patients already recovered, so S(t) = 0

Question 6

What is the use of the Kaplan-Meier estimator in survival analysis?

Answer 6

A non-parametric statistic used to estimate the survival function from lifetime data. It is often used to measure the fraction of patients living for a certain amount of time after the treatment.

Question 7

Which variables are needed for survival analysis with the use of the Kaplan-Meier estimator?

Answer 7

• ti, time when at least one event happened.
• di, the number of events that happened at time ti.
• ni, the individuals known to have survived (or have not yet had an event or been censored) up to time ti.

Question 8

A survival analysis is performed to analyse the survival function of the amount of weeks in remission among leukemia patients treated with 6-MP. In total 9 patients were followed with the following weeks in remission (i.e. event): 6, 6, 6, 7, 10, 13, 16, 22, and 23.

Calculate the survival function (S(t)), di, the number of events that happened at time ti, and ni, the individuals known to have survived (or have not yet had an event or been censored) up to time ti.
(Only calculate these variables for t=0, t=6, and t=7)

Answer 8

t=0:
* S(t) = 9/9 =1
* ni = 9
* di = 0

t=6:
* S(t) = 6/9 = 0.67
* ni = 6
* di = 3

t=7:
* S(t) = 5/9 = 0.56
* ni = 5
* di = 4

Question 9

How can you calculate the survival fraction at ti?

Answer 9

(ni - di) / ni

Question 10

Given the following information, calculate the survival fraction for t=0, t=6, and t=7:

A survival analysis is performed to analyse the survival function of the amount of weeks in remission among leukemia patients treated with 6-MP. In total 9 patients were followed with the following weeks in remission (i.e. event): 6, 6, 6, 7, 10, 13, 16, 22, and 23.

t=0:
* S(t) = 9/9 =1
* ni = 9
* di = 0

t=6:
* S(t) = 6/9 = 0.67
* ni = 6
* di = 3

t=7:
* S(t) = 5/9 = 0.56
* ni = 5
* di = 4

Answer 10

t=0:
* (9-0)/9 = 1

t=6:
* (6-3)/6 = 0.5

t=7:
* (5-4)/5 = 0.2

Question 11

Given the following information, calculate the probability of being at least 6 and 7 weeks in remission.

A survival analysis is performed to analyse the survival function of the amount of weeks in remission among leukemia patients treated with 6-MP. In total 9 patients were followed with the following weeks in remission (i.e. event): 6, 6, 6, 7, 10, 13, 16, 22, and 23.

t=0:
* S(t) = 9/9 =1
* ni = 9
* di = 0
* survival fraction: 1

t=6:
* S(t) = 6/9 = 0.67
* ni = 6
* di = 3
* survival fraction: 0.5

t=7:
* S(t) = 5/9 = 0.56
* ni = 5
* di = 4
* survival fraction: 0.2

Answer 11

t=6:
* 0.5 (survival fraction at t=6) x 1 (survival probability at t=0) = 0.5

t=7:
* 0.5 (survival probability at t=6) x 0.2 (survival fraction at t=7) = 0.1

Question 12

Given the following information, calculate the probability of relapse within 6 and 7 weeks.

A survival analysis is performed to analyse the survival function of the amount of weeks in remission among leukemia patients treated with 6-MP. In total 9 patients were followed with the following weeks in remission (i.e. event): 6, 6, 6, 7, 10, 13, 16, 22, and 23.

t=0:
* S(t) = 9/9 =1
* ni = 9
* di = 0
* survival fraction: 1

t=6:
* S(t) = 6/9 = 0.67
* ni = 6
* di = 3
* survival fraction: 0.5
* survival probability: 0.5

t=7:
* S(t) = 5/9 = 0.56
* ni = 5
* di = 4
* survival fraction: 0.2
* survival probability: 0.1

Answer 12

t = 6:
* 100% - 50% (survival probability at t = 6) = 50%

t = 7:
* 100% - 10% (survival probability at t = 7) = 90%

Question 13

After having analysed the survival function in the amount of weeks in remission among leukemia patients treated with 6-MP, a placebo group is added to compare the two groups to each other. When analyzing the survival curve in SPSS, the Overall Comparisons Table can be used. Which variable in this table can be used to determine whether the null hypothesis is true (i.e. no difference in survival between both groups)?

Answer 13

The LogRank p-value. If this p-value is <0.05, the null hypothesis should be rejected and vice versa.

Question 14

After having analysed the survival function in the amount of weeks in remission among leukemia patients treated with 6-MP, a placebo group is added to compare the two groups to each other. When analyzing the survival curve in SPSS, the Overall Comparisons Table can be used.

After you have established that there is a significant LogRank p-value, what can you do to investigate how survival times between both groups differ?

Answer 14

Use the median survival time

Question 15

Survival function: the probability that the event does not occur before time t.
Cumulative incidence function: the probability that the evenet occurs before time t.
How can you define hazard function h(t)?

Answer 15

The instantaneous ‘event-rate’ at time t for subjects who have not yet experienced the event.

Question 16

How can you calculate the hazard ratio between two groups?

Answer 16

HR = h1(t) / h0(t)

Question 17

What is the use of a Cox regression model?

Answer 17

Used for researching the association between the survival time of patients and one or more predictor variables.

Question 18

Which assumptions need to be met in order to perform a Cox regression analysis?

Answer 18

• Hazard Ratio is constant over time
• Hazards are proportional, i.e. the relative hazard remains constant over time with different predictors or covariate levels.

Question 19

The regression formulas are different when hazard are proportional over time (proportionality) and when this is not the case (non-proportionality). Name the associated formulas.

Answer 19

• Proportionality: HR = exp(b1) -> the Kaplan-Meier Curve graph of survival function vs. survival time should have parallel curves.
• Non-proportionality: HR(t)= exp(b1(t))

Question 20

Just read.

Understanding the proportional hazard assumption:
For this section, an example is used where the survival curve of a new treatment is compared to the survival curve of the standard treatment. By running the Kaplan-Meier curve and Cox-regression analysis, you can estimate whether the hazards are proportional compared to the time. Here, the hazard ratio (exp(B)) of the Cox-regression analysis gives an estimation of what effect the new treatment has on survival. The p-value of the new treatment can be used to investigate whether the new treatment is also significantly associated with the hazard ratio of the new treatment. In this example, our exp(B)=0.746 and our p-value=0.467, indicating that the new treatment is beneficial on the survival outcome, but at the same time this beneficial effect is not significant.
After this, we can have a look if we also see this in the Kaplan-Meier curve. In this case, the Kaplan-Meier curve of the new treatment shows that the medication has a disadvantageous effect on the survival outcome in the beginning (steep decline of the curve). While if we have a look at the Kaplan-Meier curve of the new treatment later in time, the curve is more flat and therefore it shows a beneficial effect on the survival. This finding is not in line with what is shown in the Cox-regression analysis and therefore, the survival data does not fit the proportional hazard assumption.
OR
If the Kaplan-Meier curves for the different groups/categories cross each other.

Question 21

What assumption exists for the Cox regression model with a continuous X?

Answer 21

The determinant needs to meet the assumption of linearity.

Question 22

How can linearity for a continuous X in a Cox regression model best be tested?

Answer 22

The best way this can be done is by grouping the continuous determinant (into quartiles) and then making an analysis with a categorical X. By doing this, you can depict the means of the four quartiles and with this you can conclude whether the, now categorical, determinant is linear.