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 PatientTime at entry (months) Time at death/ censoring Dead or censored Survival time D C12.5 * C17.6* D C15.0*

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 table SubjectSurvival time (min) Survival proportion * * – 21120*

Calculation of survival probabilities p k = p k-1 x (r k – f k )/ r k 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 = Time 50 mins : x (20 – 1)/20 = Time 51 mins : 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

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

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

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 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 [h i (t)] = log[h 0 (t)] + ß 1 x 1 + ß 2 x 2 + …….. ß k x k where [h 0 (t)] = baseline hazard and x 1,x 2, …x k are covariates associated with subject i

The Cox Model h i (t) = h 0 (t) exp [ß 1 x 1 + ß 2 x 2 + …….. ß k x k] where [h 0 (t)] = baseline hazard and x 1,x 2, …x k 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 model Proportional 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