Survival Analysis. Key variable = time until some event time from treatment to death time for a fracture to heal time from surgery to relapse.

Survival Analysis

Key variable = time until some event

• time from treatment to death

• time for a fracture to heal

• time from surgery to relapse

Censored observations

• subjects removed from data set at some stage without suffering an event[lost to follow-up or died from unrelated event]

• study period ends with some subjects not suffering an event

Example

Patient Time at entry

(months)

Time at death/

censoring

Dead or censored

Survival time

1 0.0 11.8 D 11.8

2 0.0 12.5 C 12.5 *

3 0.4 18.0 C 17.6*

4 1.2 6.6 D 5.4

5 3.0 18.0 C 15.0*

Survival analysis uses information about subjects who suffer an event and subjects who do not suffer an

event

Life Table

• Shows pattern of survival for a group of subjects

• Assesses number of subjects at risk at each time point and estimates the probability of survival at each point

Motion sickness data

N=21 subjects placed in a cabin and subjected to vertical motion

Endpoint = time to vomit

Motion sickness data

• 14 survived 2 hours without vomiting

• 5 subjects vomited at 30, 50, 51, 82 and 92 minutes respectively

• 2 subjects requested an early stop to the experiment at 50 and 66 minutes respectively

Life tableSubject Survival time

(min)Survival proportion

1 30 0.952

2 50 0.905

3 50 *

4 51 0.855

5 66*

6 82 0.801

7 92 0.748

8 – 21 120*

Calculation of survival probabilities

pk = pk-1 x (rk – fk)/ rk

where p = probability of surviving to time k

r = number of subjects still at risk

f = number of events (eg. death) at

time k

Calculation of survival probabilities

Time 30 mins : (21 – 1)/21 = 0.952

Time 50 mins : 0.952 x (20 – 1)/20 = 0.905

Time 51 mins : 0.905 x (18 – 1)/18 = 0.854

Kaplan-Meier survival curve

• Graph of the proportion of subjects surviving against time

• Drawn as a step function (the proportion surviving remains unchanged between events)

Survival Curve

TIME (mins)

1209060300

Su

rviv

al p

rob

ab

ility

1.0

.8

.6

.4

.2

0.0

Kaplan-Meier survival curve

• times of censored observations indicated by ticks

• numbers at risk shown at regular time intervals

Summary statistics

1. Median survival time

2. Proportion surviving at a specific time point

Survival Curve

TIME (mins)

1209060300

Su

rviv

al p

rob

ab

ility

1.0

.8

.6

.4

.2

0.0

Comparison of survival in two groups

Log rank test

Nonparametric – similar to chi-square test

SPSS Commands

• Analyse – Survival – Kaplan-Meier

• Time = length of time up to event or last follow-up

• Status = variable indicating whether event has occurred

• Options – plots - survival

SPSS Commands(more than one group)

• Factor = categorical variable showing grouping

• Compare factor – choose log rank test

Example

RCT of 23 cancer patients 11 received chemotherapy

Main outcome = time to relapse

Chemotherapy example

Time (weeks)

180160140120100806040200

Pro

po

rtio

n r

ela

pse

-fre

e1.0

.8

.6

.4

.2

0.0

Chemotherapy

Yes

Yes-censored

No

No-censored

Chemotherapy example

No chemotherapy

Median relapse-free time = 23 weeks

Proportion surviving to 28 weeks = 0.39

Chemotherapy

Median relapse-free time = 31 weeks

Proportion surviving to 28 weeks = 0.61

The Cox modelProportional hazards regression analysis

Generalisation of simple survival analysis to allow for multiple independent variables which can be binary, categorical and continuous

The Cox Model

Dependent variable = hazard

Hazard = probability of dying at a point in time, conditional on surviving up to that point in time

= “instantaneous failure rate”

The Cox Model

Log [hi(t)] =

log[h0(t)] + ß1x1 + ß2x2 + …….. ßkxk

where [h0(t)] = baseline hazard

and x1 ,x2 , …xk are covariates associated with subject i

The Cox Model

hi(t) =

h0(t) exp [ß1x1 + ß2x2 + …….. ßkxk]

where [h0(t)] = baseline hazard

and x1 ,x2 , …xk are covariates associated with subject i

The Cox Model

Interpretation of binary predictor variable defining groups A and B:

Exponential of regression coefficient, b, = hazard ratio (or relative risk)= ratio of event rate in group A and event rate in

group B= relative risk of the event (death) in group A

compared to group B

The Cox Model

Interpretation of continuous predictor variable:

Exponential of regression coefficient, b,

refers to the increase in hazard (or relative risk) for a unit increase in the variable

The Cox Model

Model fitting:

• Similar to that for linear or logistic regression analysis

• Can use stepwise procedures such as ‘Forward Wald’ to obtain the ‘best’ subset of predictors

The Cox modelProportional hazards regression analysis

Assumption:

Effects of the different variables on event occurrence are constant over time

[ie. the hazard ratio remains constant over time]

SPSS Commands

• Analyse – Survival – Cox regression

• Time = length of time up to event or last follow-up• Status = variable indicating whether event has

occurred• Covariates = predictors (continuous and categorical)• Options – plots and 95% CI for exp(b)

The Cox model

Check of assumption of proportional hazards (for categorical covariate):

• Survival curves• Hazard functions• Complementary log-log curves

For each, the curves for each group should not cross and should be approximately parallel