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

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

What is hazard? Hazard rate is an instantaneous rate of events as a function of time Survival is one measure of outcome but cancer studies also use hazards. Hazards intuitively sounds like the inverse of survival which is true but it is not quite as simple as that as they are related through the exponential function The hazard is an instantaneous rate of events. The hazard ratio (HR) (see later) is the ratio of hazards in two groups, often trial arms and is a measure of the efficacy in a cancer trial. A HR = 1 indicates no difference between the two arms, a HR > 1 indicates a greater hazard for the numerator arm and a HR < 1 indicates a lower hazard. Often in cancer trials of a new drug the trialists would be seeking a HR < 1 indicating a survival benefit by reducing deaths relative to the comparator.

Plot of hazard Note that the hazard changes over time denoted by h(t) The hazard rate is a function of time and tends to increase over time. Think of the risk of death from birth to old age which is high just after birth, declines through childhood and increases from middle-age. time Birth Old age

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: Age in years Hazard h is the hazard K is a constant e.g. 0.3 per Person-year

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 Age in years Hazard

Cox Proportional Hazards Model (1972) h0 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 Age in years

What is Hazard Ratio? Hazard Ratio (HR) is ratio of hazards in two groups e.g. men vs women, new drug vs. BSC N.B. It is the improvement in one group over the other in terms of rate at which events will occur from a particular time point to another time point Note that although the hazard is changing over time the ratio of the hazard in two groups is often assumed to be constant for most of the models we use in cancer research. This is the property of proportional hazards.

What is Hazard Ratio? Hazard Ratio (HR) is ratio of hazards in two groups and remains constant over time (n.b. survival curve widens) Survival Constant ratio of hazards means the survival curves gradually widen. Hazard and survival are related through the log/ exponential functions. The HR is a summary of the difference in effect across the whole time course but at any time point the HR is the same if we have proportional hazards. Time

Interpretation of HR comparing two groups HR = 1 ; Do NOT reject null hypothesis (i.e. no difference) HR < 1; Reduction in Hazard relative to comparator (e.g. HR = 0.6 is 40% reduction) HR > 1; Increase in Hazard relative to comparator (e.g. HR = 1.7 is 70% increase) A HR = 1 indicates no difference between the two groups. HR < 1 is generally what we aim for in cancer trials as mainly seeking to reduce events such as death. A HR > 1 indicates a factor that increases the hazard in relation to the comparator.

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

Logrank Test: Null Hypothesis The Null hypothesis for the logrank test: Hazard Rate group A = Hazard Rate for group B = HR = OA / EA = 1 OB / EB

Wald Test: Null Hypothesis The Null hypothesis for the Wald test: 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 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 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

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: x1 is Duke’s Stage and x2 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? If main ‘exposure’ variable, enter first and assess confounders one at a time Assess each variable on statistical significance and clinical importance. It 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 It is easily fitted in SPSS Model assessment requires some thought 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 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

Check assumption of proportional hazards for each 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

Check assumption of proportional hazards for each 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 18 1 0 0 0 1=B 107 0 1 0 0 2=C 188 0 0 1 0 3=D 123 0 0 0 1 9=UK 40 0 0 0 0 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 Automatic procedures are available but treat results with caution They are easily fitted in SPSS Check proportional hazards assumption Parsimonious models are better

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