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Survival Analysis

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

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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

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Example PatientTime at entry (months) Time at death/ censoring Dead or censored Survival time 10.011.8D 20.012.5C12.5 * 30.418.0C17.6* 41.26.6D5.4 53.018.0C15.0*

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

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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

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Motion sickness data N=21 subjects placed in a cabin and subjected to vertical motion Endpoint = time to vomit

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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

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Life table SubjectSurvival time (min) Survival proportion 1300.952 2500.905 350 * 4510.855 566* 6820.801 7920.748 8 – 21120*

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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

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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

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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)

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Survival Curve

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Kaplan-Meier survival curve times of censored observations indicated by ticks numbers at risk shown at regular time intervals

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Summary statistics 1.Median survival time 2.Proportion surviving at a specific time point

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Survival Curve

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Comparison of survival in two groups Log rank test Nonparametric – similar to chi-square test

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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

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SPSS Commands (more than one group) Factor = categorical variable showing grouping Compare factor – choose log rank test

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Example RCT of 23 cancer patients 11 received chemotherapy Main outcome = time to relapse

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Chemotherapy example

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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

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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

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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

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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

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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

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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

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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

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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

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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]

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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)

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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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