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## PowerPoint Slideshow about ' Survival Analysis' - norman-mayer

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### Key variable = time until some event

### Censored observations

### Kaplan-Meier survival curve

time from treatment to death

time for a fracture to heal

time from surgery to relapse

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

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

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)

times of censored observations indicated by ticks

numbers at risk shown at regular time intervals

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

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

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