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# Survival Analysis

Cox regression

by

Tweet## Jan Vojtíšek

on 17 September 2011#### Transcript of Survival Analysis

Sir David Roxbee Cox

(b.1924 Birmingham, England)

is a prominent British statistician.

He has made pioneering and important contributions to numerous areas of statistics and applied probability, of which the best known is perhaps the proportional hazards model, which is widely used in the analysis of survival data. From Wikipedia, the free encyclopedia Survival Analysis Cox regression Jan Radim Vojtíšek SA examines and models the time it takes for events to occur. Key variable is TIME Any questions References TARLING, Roger. Statistical modelling for social researchers : principles and practice. Routledge, 2008. 120 s.

NORUŠIS, Marija, J. Advanced Statistical Procedures Companion. NJ : Prentice Hall Press, 2008.

HENDL, Jan. Přehled statistických metod zpracování dat : analýza a metaanalýza dat. Jan Hendl. Vyd. 1. Praha : Portál, 2004.

FOX, John. An R and S-PLUS Companion to Applied Regression. Sage Publications, 2008. Cox Proportional-Hazards Regression for Survival Data. WWW: <http://cran.r-project.org/doc/contrib/Fox-Companion/appendix-cox-regression.pdf>. - Life tables: estimate survival at fixed time points

- Kaplan-Mayer: descriptive technique, comparing survival curves - graphically, significance test

- Cox Regression: predicts a dependent variable as a function of a set of indipendent variables. Proportional Hazards model Semiparametric model Censoring What is the problem? Censoring Time-dependent covariates Terminology Hazard function Baseline hazard Parameters effect Survival function Hazard function Cox regression models effect of predictors on the survival function curves.

predictors (independent/explanatory variables) = covariates

dependent variable = hazard function Assuptions Procedures Results Diagnostics Which cases are censored? to describe, show and test the event process to find the predictors and build models FOX 2008 Survival analysis - medical science (death, relapse...)

Event history analysis - social sciece (divorce, unemployment...)

Reliability analysis - engineering science (failure in mechanical systems)

(Duration analysis, Time-failure analysis...) S (t) = P (T > t) h (t) = P (x < t < y I t > x) Survival function is the probability, that the time of death (T) is later than some specified time (t). From Wikipedia, the free encyclopedia Event rate in time interval (x;y) conditional on survival until time x. h (t) = [log S(t)]’ Censoring is the inability to observe events before of after a defined cut-off point. TARLING 2008 Left censoring

Right censoring depend only on time

the same for all cases (not dependent on the covariates)

similar to the constant in ordinary regression depend only on the values of the covariates and the regression coeficients.

how the hazard changes in response to explanatory varibles

not dependent on time For any two cases, the ratio of their predicted hazards is constant fol all time points. Because although it doesn`t make assumptions about the underlying cumulative hazard function, it assumes that covariates are additive and linearly related to the log of the hazard function. NORUŠIS 2008 No assumptions about the distribution of hazard (the baseline hazard function).

Assumption of proportional hazards Time-constant covariates

Time dependent covariate However, you should use the Cox Regression procedure if your model contains only time-constant covariates, since this procedure allows you tu save diagnostic variables and obtain plots that are not available in Time-Dependent Cox Regression procedure. NORUŠIS 2008 violation of assumption of proportional hazards - e.g. LML plot (log minus log) - graphically (strata in cox reg.)

nonlinearity in the relationship between the log hazard and the covariates

influential data SPSS creates coding of categorical variables (e.g. dummy variables coding). Test of model.

Is this model better than model without predictors? Exp(B) = hazard rate

Full transcript(b.1924 Birmingham, England)

is a prominent British statistician.

He has made pioneering and important contributions to numerous areas of statistics and applied probability, of which the best known is perhaps the proportional hazards model, which is widely used in the analysis of survival data. From Wikipedia, the free encyclopedia Survival Analysis Cox regression Jan Radim Vojtíšek SA examines and models the time it takes for events to occur. Key variable is TIME Any questions References TARLING, Roger. Statistical modelling for social researchers : principles and practice. Routledge, 2008. 120 s.

NORUŠIS, Marija, J. Advanced Statistical Procedures Companion. NJ : Prentice Hall Press, 2008.

HENDL, Jan. Přehled statistických metod zpracování dat : analýza a metaanalýza dat. Jan Hendl. Vyd. 1. Praha : Portál, 2004.

FOX, John. An R and S-PLUS Companion to Applied Regression. Sage Publications, 2008. Cox Proportional-Hazards Regression for Survival Data. WWW: <http://cran.r-project.org/doc/contrib/Fox-Companion/appendix-cox-regression.pdf>. - Life tables: estimate survival at fixed time points

- Kaplan-Mayer: descriptive technique, comparing survival curves - graphically, significance test

- Cox Regression: predicts a dependent variable as a function of a set of indipendent variables. Proportional Hazards model Semiparametric model Censoring What is the problem? Censoring Time-dependent covariates Terminology Hazard function Baseline hazard Parameters effect Survival function Hazard function Cox regression models effect of predictors on the survival function curves.

predictors (independent/explanatory variables) = covariates

dependent variable = hazard function Assuptions Procedures Results Diagnostics Which cases are censored? to describe, show and test the event process to find the predictors and build models FOX 2008 Survival analysis - medical science (death, relapse...)

Event history analysis - social sciece (divorce, unemployment...)

Reliability analysis - engineering science (failure in mechanical systems)

(Duration analysis, Time-failure analysis...) S (t) = P (T > t) h (t) = P (x < t < y I t > x) Survival function is the probability, that the time of death (T) is later than some specified time (t). From Wikipedia, the free encyclopedia Event rate in time interval (x;y) conditional on survival until time x. h (t) = [log S(t)]’ Censoring is the inability to observe events before of after a defined cut-off point. TARLING 2008 Left censoring

Right censoring depend only on time

the same for all cases (not dependent on the covariates)

similar to the constant in ordinary regression depend only on the values of the covariates and the regression coeficients.

how the hazard changes in response to explanatory varibles

not dependent on time For any two cases, the ratio of their predicted hazards is constant fol all time points. Because although it doesn`t make assumptions about the underlying cumulative hazard function, it assumes that covariates are additive and linearly related to the log of the hazard function. NORUŠIS 2008 No assumptions about the distribution of hazard (the baseline hazard function).

Assumption of proportional hazards Time-constant covariates

Time dependent covariate However, you should use the Cox Regression procedure if your model contains only time-constant covariates, since this procedure allows you tu save diagnostic variables and obtain plots that are not available in Time-Dependent Cox Regression procedure. NORUŠIS 2008 violation of assumption of proportional hazards - e.g. LML plot (log minus log) - graphically (strata in cox reg.)

nonlinearity in the relationship between the log hazard and the covariates

influential data SPSS creates coding of categorical variables (e.g. dummy variables coding). Test of model.

Is this model better than model without predictors? Exp(B) = hazard rate