Survival Analysis Project

German Breast Cancer Survival

AnalysisDavid Schuler, Ankur Verma, Udyot Kumar,

Uma Lalitha-Chockalingham

Objectives

●Understand the length of time between breast cancer diagnoses and specific events

●Understand what factors play a role in determining these lengths

About Survival Analysis

●Predicting the time until an event of interest occurs

●Applications in Medicine, Manufacturing, Sociology, Sports, and many more

●Right Censored Data – an observation where the event has not yet occurred

● Survival Function - probability that at a given time, t, an event of interest has not yet occurred

Kaplan-Meier Estimator

●Non-parametric estimator of the survival function

●Time on the X-axis

●Percentage surviving on Y axis

●Tick marks represent right-censored observations

Cox Proportional Hazard Regression●Used to look at the relationship between the survival of a patient

and various explanatory variables

●Each explanatory variable is given a coefficient

○ HR = 1 : No effect

○ HR < 1 : Reduction in hazard ( Death)

○ HR > 1 : Increase in Hazard ( Death)

German Breast Cancer Data●Retrieved from UMass Amherst’s Statistics website

●Data collected from clinical trials performed by the German Breast Cancer Study Group

●Total of 686 observations conducted between July 1984 and December 1989● 16 variables, including censoring and time-length fields for death and cancer

recurrence

Exploratory Data Analysis

Correlation

● menopause and age

● Estrogen and Progesterone

Preliminary Survival Analysis - KM Curves

Preliminary Survival Analysis - KM Curves

Probability density function f(t)

Survival function S(t) = P(T>=t)

Hazard function h(t) = f(t) / S(t)

A way to compare two hazard functions:Hazard ratio : HR(t) = h0 (t) / h1(t)

Proportional hazard assumption : The hazard ratio does not vary with time, i.e. HR(t) = HR

This is an important assumption for Cox PH model

Preliminary Survival Analysis - Cox PH

○ HR = 1 : No effect

○ HR < 1 : Reduction in hazard ( Death)

○ HR > 1 : Increase in Hazard ( Death)

● Age1 = [21,30]● Age2 = (30,50]● Age3 =(50,80]

COX Regression Modelling with phreg in SAS

Survival Curve Estimate ( Test Data - 3 rows)

Comparing R, Python & SAS Code

Comparing R, Python, & SAS: Plots

SAS Plot

Comparing Regression Models: Tumor Grade

Comparing Regression Models: Hormone

Comparing Regression Models: Menopause

References

●http://www.medicine.ox.ac.uk/bandolier/painres/download/whatis/cox_model.pdf

●https://media.readthedocs.org/pdf/lifelines/latest/lifelines.pdf

●https://www.cscu.cornell.edu/news/statnews/stnews78.pdf