1 / 31

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)

Download Presentation

Assessing Survival: Cox Proportional Hazards Model

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


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

  2. 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

  3. Modelling: Detecting signal from background noise

  4. 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

  5. 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….

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

  7. 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’

  8. Linear model of hazard Hazard Age in years

  9. 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)

  10. Exponential model of hazard Hazard Age in years

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

  12. 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

  13. 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

  14. Fitting Gender in Cox Model in SPSS

  15. 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

  16. 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

  17. 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

  18. 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

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

  20. 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

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

  22. Interpretation of the Hazard Ratio For a continuous variable such as age, HR represents the incremental increase in hazard per unit increase in agei.eHR=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

  23. 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

  24. 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

  25. 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

  26. 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

  27. 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

  28. 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)

  29. Proportional hazards holds for Duke’s Staging

  30. 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

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

More Related