Assessing Survival: Cox Proportional Hazards Model Peter T. Donnan Professor of Epidemiology and Biostatistics Statistics for Health Research

Objectives of Workshop Understand the general form of Cox PH model Understand the general form of Cox PH model Understand the need for adjusted Hazard Ratios (HR) Understand the need for adjusted Hazard Ratios (HR) Implement the Cox model in SPSS Implement the Cox model in SPSS Understand and interpret the output from SPSS Understand and interpret the output from SPSS

Modelling: Detecting signal from background noise

Survival Regression Models Expressed in terms of the hazard function formally defined as: The instantaneous risk of event (mortality) in next time interval t, conditional on having survived to start of the interval t

Survival Regression Models The Cox model expresses the relationship between the hazard and a set of variables or covariates These could be arm of trial, age, gender, social deprivation, Dukes stage, co-morbidity, etc….

How is the relationship formulated? Simplest equation is: h is the hazard K is a constant e.g. 0.3 per Person-year Age in years HazardHazardHazardHazard

How is the relationship formulated? Next Simplest is linear equation: h is the outcome; a is the intercept; β is the slope related to x the explanatory variable and; e is the error term or ‘noise’

Linear model of hazard Hazard Age in years

Cox Proportional Hazards Model (1972) h 0 is the baseline hazard; r ( β, x) function reflects how the hazard function changes (β) according to differences in subjects’ characteristics (x)

Exponential model of hazard Hazard Age in years

Cox Proportional Hazards Model: Hazard Ratio Consider hazard ratio for men vs. women, then -

Cox Proportional Hazards Model: Hazard Ratio If coding for gender is x=1 (men) and x=0 (women) then: where β is the regression coefficient for gender

Hazard ratios in SPSS SPSS gives hazard ratios for a binary factor coded (0,1) automatically from exponentiation of regression coefficients (95% CI are also given as an option) Note that the HR is labelled as EXP(B) in the output

Fitting Gender in Cox Model in SPSS

Output from Cox Model in SPSS Variable in model Regression Coefficient Standard error Test Statistic ( β/se(β) ) 2 Degrees of freedom p- value HR for men vs. women

Logrank Test: Null Hypothesis The Null hypothesis for the logrank test: Hazard Rate group A = Hazard Rate group A = Hazard Rate for group B Hazard Rate for group B = HR = O A / E A = 1 = HR = O A / E A = 1 O B / E B O B / E B

Wald Test: Null Hypothesis The Null hypothesis for the Wald test: Hazard Ratio = 1 Hazard Ratio = 1 Equivalent to regression coefficient β=0 Note that if the 95% CI for the HR includes 1 then the null hypothesis cannot be rejected

Hazard ratios for categorical factors in SPSS Enter factor as before Enter factor as before Click on ‘categorical’ and choose the reference category (usually first or last) Click on ‘categorical’ and choose the reference category (usually first or last) E.g. Duke’s staging may choose Stage A as the reference category E.g. Duke’s staging may choose Stage A as the reference category HRs are now given in output for survival in each category relative to Stage A HRs are now given in output for survival in each category relative to Stage A Hence there will be n-1 HRs for n categories Hence there will be n-1 HRs for n categories

Fitting a categorical variable: Duke’s Staging Reference category B vs. A C vs. A D vs. A UK vs. A

One Solution to Confounding Use multiple Cox regression with both predictor and confounder as explanatory variables i.e fit: x 1 is Duke’s Stage and x 2 is Age

Fitting a multiple regression: Duke’s Staging and Age Age adjusted for Duke’s Stage

Interpretation of the Hazard Ratio For a continuous variable such as age, HR represents the incremental increase in hazard per unit increase in age i.e HR=1.024, increase 2.4% for a one year increase in age For a categorical variable the HR represents the incremental increase in hazard in one category relative to the reference category i.e. HR = 6.66 for Stage D compared with A represents a 6.7 fold increase in hazard

First steps in modelling What hypotheses are you testing?What hypotheses are you testing? If main ‘exposure’ variable, enter first and assess confounders one at a timeIf main ‘exposure’ variable, enter first and assess confounders one at a time Assess each variable on statistical significance and clinical importance.Assess each variable on statistical significance and clinical importance. It is acceptable to have an ‘important’ variable without statistical significanceIt is acceptable to have an ‘important’ variable without statistical significance

Summary The Cox Proportional Hazards model is the most used analytical tool in survival research The Cox Proportional Hazards model is the most used analytical tool in survival research It is easily fitted in SPSS It is easily fitted in SPSS Model assessment requires some thought Model assessment requires some thought Next step is to consider how to select multiple factors for the ‘best’ model Next step is to consider how to select multiple factors for the ‘best’ model

Check assumption of proportional hazards (PH) Proportional hazards assumes that the ratio of hazard in one group to another remains the same throughout the follow-up period Proportional hazards assumes that the ratio of hazard in one group to another remains the same throughout the follow-up period For example, that the HR for men vs. women is constant over time For example, that the HR for men vs. women is constant over time Simplest method is to check for parallel lines in the Log (-Log) plot of survival Simplest method is to check for parallel lines in the Log (-Log) plot of survival

Check assumption of proportional hazards for each factor. Log minus log plot of survival should give parallel lines if PH holds Hint: Within Cox model select factor as CATEGORICAL and in PLOTS select log minus log function for separate lines of factor

Check assumption of proportional hazards for each factor. Log minus log plot of survival should give parallel lines if PH holds Hint: Within Cox model select factor as CATEGORICAL and in PLOTS select log minus log function for separate lines of factor

Proportional hazards holds for Duke’s Staging Categorical Variable Codings(b) Frequency(1)(2)(3)(4) dukes(a)0=A =B =C =D =UK a Indicator Parameter Coding b Category variable: dukes (Dukes Staging)

Proportional hazards holds for Duke’s Staging

Summary Selection of factors for Multiple Cox regression models requires some judgement Selection of factors for Multiple Cox regression models requires some judgement Automatic procedures are available but treat results with caution Automatic procedures are available but treat results with caution They are easily fitted in SPSS They are easily fitted in SPSS Check proportional hazards assumption Check proportional hazards assumption Parsimonious models are better Parsimonious models are better

Practical Read in Colorectal.sav and try to fit a multiple proportional hazards model Read in Colorectal.sav and try to fit a multiple proportional hazards model Check proportional hazards assumption Check proportional hazards assumption