Survival analysis

First example of the day Small cell lungcanser Meadian survival time: 8-10 months 2-year survival is 10% New treatment showed median survival of 13.2months

Progressively censored observations Current life table Completed dataset Cohort life table Analysis “on the fly”

Problem Do patients survive longer after treatment 1 than after treatment 2? Possible solutions: ANOVA on mean survival time? ANOVA on median survival time? 100 person years of observation: How long has the average person been in the study. 10 persons being observed for 10 years 100 persons being observed for 100 years

Life table analysis A sub-set of 13 patients undergoing the same treatment

Life table analysis Time interval chosen to be 3 months n i number of patients starting a given period

Life table analysis d i number of terminal events, in this example; progression/response w i number of patients that have not yet been in the study long enough to finish this period

Life table analysis Number exposed to risk: n i – w i /2 Assuming that patients withdraw in the middle of the period on average.

Life table analysis q i = d i /(n i – w i /2) Proportion of patients terminating in the period

Life table analysis p i = 1 - q i Proportion of patients surviving

Life table analysis S i = p i p i-1...p i-N Cumulative proportion of surviving Conditional probability

Survival curves How long will a lung canser patient keep having canser on this particular treatment?

Kaplan-Meier Simple example with only 2 ”terminal-events”.

Confidence interval of the Kaplan-Meier method Fx after 32 months

Confidence interval of the Kaplan-Meier method Survival plot for all data on treatment 1 Are there differences between the treatments?

Comparing Two Survival Curves One could use the confidence intervals… But what if the confidence intervals are not overlapping only at some points? Logrank-stats Hazard ratio Mantel-Haenszel methods

Comparing Two Survival Curves The logrank statistics Aka Mantel-logrank statistics Aka Cox-Mantel-logrank statistics

Comparing Two Survival Curves Five steps to the logrank statistics table 1.Divide the data into intervals (eg. 10 months) 2.Count the number of patients at risk in the groups and in total 3.Count the number of terminal events in the groups and in total 4.Calculate the expected numbers of terminal events e.g. (31-40) 44 in grp1 and 46 in grp2, 4 terminal events. expected terminal events 4x(44/90) and 4x(46/90) 5.Calculate the total

Comparing Two Survival Curves Smells like Chi-Square statistics

Comparing Two Survival Curves Hazard ratio

Comparing Two Survival Curves Mantel Haenszel test Is the OR significant different from 1? Look at cell (1,1) Estimated value, E(a i ) Variance, V(a i ) row total * column total grand total

Comparing Two Survival Curves Mantel Haenszel test df = 1; p>0.05

Hazard function d is the number of terminal events  f is the sum of failure times  c is the sum of censured times

Logistic regression Who survived Titanic?

25 The sinking of Titanic Titanic sank April 14th 1912 with 2228 souls 705 survived. A dataset of 1309 passengers survived. Who survived?

26 The data Sibsp is the number of siblings and/or spouses accompanying Parsc is the number of parents and/or children accompanying Some values are missing Can we predict who will survive titanic II? pclasssurvivednamesexagesibspparch 11Allen, Miss. Elisabeth Waltonfemale Allison, Master. Hudson Trevormale Allison, Miss. Helen Lorainefemale212 10Allison, Mr. Hudson Joshua Creightonmale Allison, Mrs. Hudson J C (Bessie Waldo Daniels)female Anderson, Mr. Harrymale Andrews, Miss. Kornelia Theodosiafemale Andrews, Mr. Thomas Jrmale Appleton, Mrs. Edward Dale (Charlotte Lamson)female5320

27 Analyzing the data in a (too) simple manner Associations between factors without considering interactions

28 Analyzing the data in a (too) simple manner Associations between factors without considering interactions

29 Analyzing the data in a (too) simple manner Associations between factors without considering interactions

30 Could we use multiple linear regression to predict survival? multiple linear regressionLogistic regression Response variable is defined between –inf and +inf Response variable is defined between 0 and 1 Normal distributedBernoulli distributed

31 Logit transformation is modeled linearly The logistic function

32 The sigmodal curve

33 The sigmodal curve The intercept basically just ‘scale’ the input variable

34 The sigmodal curve The intercept basically just ‘scale’ the input variable Large regression coefficient → risk factor strongly influences the probability

35 The sigmodal curve The intercept basically just ‘scale’ the input variable Large regression coefficient → risk factor strongly influences the probability Positive regression coefficient → risk factor increases the probability

36 Logistic regression of the Titanic data

37 Logistic regression of the Titanic data – passenger class 1.Summary of data 2.Coding of the dependent variable 3.Coding of the categorical explanatory variable: First class: 1 Second class: 2 Third class: reference

38 Logistic regression of the Titanic data – passenger class A fit of the null-model, basically just the intercept. Usually not interesting The total probability of survival is 500/1309 = Cutoff is 0.5 so all are classified as non- survivers. Basically tests if the null-model is sufficient. It almost certainly is not. Shows that survival is related to pclass (which is not in the null- model)

39 Logistic regression of the Titanic data – passenger class 1.Omnibus test: Uses LR to describe if the adding the pclass variable to the model makes it better. It did! But better than the null-model, so no surprise. 2.Model Summary. Other measures of the goodness of fit. 3.Classification table: By including pclass 67.7 passengers were correctly categorized. 4.Variables in the equation: first line repeats that pclass has a significant effect on survival. B is the logistic fittet parameter. Exp(B) is the odds rations, so the odds of survival is 4.7 ( ) times higher than passengers on third class (reference class)

40 Logistic regression of the Titanic data – Adding age to the model Ups… Some data points are missing And the null model is poorer

41 Logistic regression of the Titanic data – Adding age to the model Cox and Senll’s R-square increased from to 0.141, indicating a better model By this model we can classify 69.1% passenger class only classified 67.7%

42 Logistic regression of the Titanic data – Adding age to the model Age has a significant influence on survival. The odds ratio of age is So the odds of a 31 year old is times the odds of a 30 year old. Or the odds for a 30 year old to survive is 1/0.963 = times larger than that of a 31 year old

43 Logistic regression of the Titanic data – Age alone The model is extremely poor Consequently age appear to be insignificant in estimating survival.

44 Logistic regression of the Titanic data – Adding family and sex The model is becoming better

45 Logistic regression of the Titanic data – Using the model as to predict What is the probability that a 25 year old woman accompanied only by her husband holding a second class ticket would survive Titanic? z = * =

46 Using the model to predict survival What is the probability that a 25 year old woman accompanied only by her husband holding a second class ticket would survive Titanic? z = *(-5)/ =

47 Is it realistic that Leonardo survives and the chick dies?