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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

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Key variable time until some event

Key variable = time until some event

time from treatment to death

time for a fracture to heal

time from surgery to relapse


Censored observations

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



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


Life table
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
Motion sickness data

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

Endpoint = time to vomit


Motion sickness data1
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
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 probabilities1
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
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)



Kaplan meier survival curve1

Kaplan-Meier survival curve

times of censored observations indicated by ticks

numbers at risk shown at regular time intervals


Summary statistics

Summary statistics

Median survival time

Proportion surviving at a specific time point



Comparison of survival in two groups
Comparison of survival in two groups

Log rank test

Nonparametric – similar to chi-square test


Spss commands
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
SPSS Commands(more than one group)

  • Factor = categorical variable showing grouping

  • Compare factor – choose log rank test


Example1

Example

RCT of 23 cancer patients

11 received chemotherapy

Main outcome = time to relapse



Chemotherapy example1
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 model proportional hazards regression analysis
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
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 model1
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 model2
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 model3
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 model4
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 model5
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 model proportional hazards regression analysis1
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 commands1
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 model6
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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