1、Survival Analysis,Key variable = time until some event,time from treatment to deathtime for a fracture to healtime 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 eventstudy period ends wit
2、h some subjects not suffering an event,Example,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 probabili
3、ty of survival at each point,Motion sickness data,N=21 subjects placed in a cabin and subjected to vertical motionEndpoint = 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
4、the experiment at 50 and 66 minutes respectively,Life table,Calculation of survival probabilities,pk = pk-1 x (rk fk)/ rkwhere p = probability of surviving to time kr = number of subjects still at riskf = number of events (eg. death) at time k,Calculation of survival probabilities,Time 30 mins : (21
5、 1)/21 = 0.952Time 50 mins : 0.952 x (20 1)/20 = 0.905Time 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,Kaplan-Meier survi
6、val curve,times of censored observations indicated by ticksnumbers at risk shown at regular time intervals,Summary statistics,Median survival timeProportion surviving at a specific time point,Survival Curve,Comparison of survival in two groups,Log rank testNonparametric similar to chi-square test,SP
7、SS Commands,Analyse Survival Kaplan-MeierTime = length of time up to event or last follow-up Status = variable indicating whether event has occurredOptions plots - survival,SPSS Commands (more than one group),Factor = categorical variable showing groupingCompare factor choose log rank test,Example,R
8、CT of 23 cancer patients11 received chemotherapyMain outcome = time to relapse,Chemotherapy example,Chemotherapy example,No chemotherapy Median relapse-free time = 23 weeks Proportion surviving to 28 weeks = 0.39Chemotherapy Median relapse-free time = 31 weeks Proportion surviving to 28 weeks = 0.61
9、,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 = hazardHazard = probability of dying at a point in time, conditional on sur
10、viving up to that point in time= “instantaneous failure rate”,The Cox Model,Log hi(t) =logh0(t) + 1x1 + 2x2 + kxkwhere h0(t) = baseline hazardand x1 ,x2 , xk are covariates associated with subject i,The Cox Model,hi(t) = h0(t) exp 1x1 + 2x2 + kxkwhere h0(t) = baseline hazardand x1 ,x2 , xk are covar
11、iates 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 comp
12、ared 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 us
13、e 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 timeie. the hazard ratio remains constant over time,SPSS Commands,Analyse Surv
14、ival Cox regressionTime = 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 curvesFor each, the curves for each group should not cross and should be approximately parallel,
