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Statistical analysis Part VI Cox Proportional Hazard Model

Statistical analysis Part VI Cox Proportional Hazard Model. Presentation The Cox Proportional Hazards Model and its characteristics. focus The model form Why popular ML estimation The hazard ratio Adjusted survival curves The PH assumption. 1. Review.

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Statistical analysis Part VI Cox Proportional Hazard Model

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  1. Statistical analysisPart VICox Proportional Hazard Model

  2. PresentationThe Cox Proportional Hazards Model and its characteristics focus The model form Why popular ML estimation The hazard ratio Adjusted survival curves The PH assumption

  3. 1. Review Survival analysis: outcome variable - time until an event occurs follow-up start TIME event event: death, disease, relapse time survival time event failure

  4. A computer example using the Cox PH model

  5. A computer example using the Cox PH model cont’d + denotes censored observation

  6. T = weeks until going out of remission X1 = group status = E X2 = log WBC (confounding?) X3 = X1 * X2 (interaction?) = group status x log WBC

  7. Model 1 N:42 %Cen:28.571 -2logL:172.759

  8. Model 2 N:42 %Cen:28.571 -2logL:144.559

  9. Model 3 N:42 %Cen:28.571 -2logL:144.131

  10. Same data set for each model n=42 subjects T=time (weeks) until out of remission Model 1: Rx only Model 2: Rx and logWBC Model 3: Rx, logWBC and Rx x logWBC

  11. ML estimation used for all modelsModel 3: N:42 %Cen:28.571 -2logL:144.131

  12. P=.510: -0.342 = -0.66 = Z - Wald statistic 0.520 LR statistic: uses -2logL Log likelihood

  13. Model 2 N:42 %Cen:28.571 -2logL:144.559

  14. LR (interaction) = -2logLmodel 2 - (-2logL model 3) = 144.559 - 144.131 = 0.428 (LR is 2 with 1 df under H0: no interaction) .40 < P < .50 not significant

  15. LR  Wald When in doubt use the LR test

  16. Model 2 N:42 %Cen:28.571 -2logL:144.559 3 statistical objectives: 1) test for significance of effect 2) point estimate of effect 3) confidence interval for effect

  17. Test for treatment effect: Wald statistic: P=.002 (strongly significant) LR statistic: compare -2logL from model 2 with -2logL from model without treatment variable (printout not provided here) Conclusion: treatment effect is significant, after adjusting for logWBC

  18. Point estimate “RRisk” = HR = 3.648 = e1.294 coefficient of treatment variable

  19. Model 2: Confidence interval: N:42 %Cen:28.571 -2logL:144.559

  20. 95% confidence interval for the HR: (1.505, 8.343) 1 1.505 3.648 8.343 95% CI: e11.96*s1 ^ ^

  21. Model 1 N:42 %Cen:28.571 -2logL:172.759 Model 2 N:42 %Cen:28.571 -2logL:144.559 HR for model 1 (4.523) is higher than HR for model 2 (3.648)

  22. Confounding: crude versus adjusted HR’s are meaningfully different Confounding due to logWBC must control for logWBC, i.e. prefer model 2 to model 1

  23. If no confounding, then consider precision: e.g., if 95% CI is narrower for model 2 than model 1, we prefer model2. Model 1: Column name . . . 0.95 CI Rx 2.027 - 10.094 Model 2: Column name . . . 0.95 CI LogWBC 2.609 - 9.486 Rx 1.505- 8.343

  24. Summary Model 2 is best model HR = 3.648 statistically significant 95% CI: 1.5, 8.3

  25. Model 2: Column name . . . P(PH) LogWBC 0.469 Rx 0.497 P(PH): gives P-value for evaluating PH assumption for each variable in model; derived from N(0,1) statistic P(PH) large PH satisfied (e.g., P>.10) P(PH) small PH not satisfied (e.g., P<.05)

  26. Model 2: P(PH) non-significant for both variables, i.e., PH is satisfied Three approaches for evaluating PH Procedures when PH not satisfied

  27. The formula for the Cox PH model p h(t,X) = h0(t) e iXi X = (X1, X2, …, Xp) explanatory/predictor variables i=1

  28. p i=1 h0(t) x e iXi baseline hazard exponential involves t but involves X’s not X’s but not t (X’s are time-independent) X’s involving t: time dependent requires extended Cox model (no PH)

  29. Time-independent variable: Values for a given individual do not change over time e.g., SEX and SMK assumed not to change once measured AGE and WGT values do not change much, or effect on survival depends on one measurement

  30. If X1=X2=…=Xk = 0, h(t,X) = h0(t) e iXi = h0(t) e0 = h0(t) baseline No X’s in model: h(t,X) = h0(t) p i=1

  31. If H0(t) is unspecified then: Cox model: nonparametric Example: Parametric model Weibull: h(t,X) = t-1e iXi where h0(t) = t-1 p i=1

  32. Non-parametric property popularity of the Cox model

  33. Why the Cox PH model is popular Cox PH model is “robust”: will closely approximate correct parametric model If correct model is: Weibull Cox model will approximate Weibull Exponential Cox model will approximate Exponential

  34. Prefer parametric model if sure of correct model, e.g., use goodness of fit test (Lee, 1982) However, when in doubt, the Cox model is a “safe” choice

  35. Even though h0(t) is unspecified, can estimate the ’s Measure of effect: hazard ratio (HR) involves only ’s, without estimating h0(t)

  36. Can estimated h(t,X) and S(t,X) for Cox model using a minimum of assumptions Cox model preferred to logistic model uses survival uses (0,1) times outcome, ignores survival times

  37. ML estimation of the Cox PH model p p h(t,X) = h0(t) e  iXi ML estimate: I Column Name Coeff StErr Pvalue RRisk Rx 1. 294 0.422 0.002 3.648 Log wbc 1.604 0.329 0.000 4.975 n:42 %Cen:28.571 -2LogL:144.559 i=1 ^

  38. Model 2: h(t,X) = h0(t) e 1 Rx + 2 logwbc Estimated model: h (t,X) = h0(t)e1.294 Rx + 1.604 logwbc ^ ^

  39. Computing the hazard ratio ^ ^ HR = h(t,X*) h (t,X) where X* = (X1*, X2*, …, Xp*) and X = (X1, X2, …, Xp) denote the set of X’s for two individuals ^

  40. HR = h(t,X*) = h0(t) e iX*i h(t,X) h0(t) e iXi ^ ^ ^ p ^ i=1 ^ ^ p ^ i=1

  41. p ^ ^ p ^ ^ HR = h0(t) e iX*i = e i (X*i-Xi) h0(t) e iXi HR = exp [i (X*i-Xi)] i=1 i=1 ^ p ^ i=1 ^ p ^ i=1

  42. Example X=(X1) where X1 denotes (0,1) exposure status (p=1) X1* = 1, X1=0 HR = exp [1(X1*-X1)] = exp [1 (1-0)] = e1 Model 1: Column name Coeff StErr Pvalue Rrisk Rx 1.509 0.410 0 4.523 ^ ^ ^ ^

  43. Example 2 Model 2: Column name Coeff StErr Pvalue RRisk Rx 1.294 0.422 0.002 3.648 Log wbc 1.604 0.329 0.000 4.975 Want HR for effect of Rx adjusted for logWBC X* = (1, logWBC) X = (0,logWBC) HR = exp [1(X*1-X1)+2(X2*-X2)] = exp [1.294(1-0)+1.604(logWBC-logWBC)] = exp [1.294(1)+1.604(0)] = e1.294 ^ ^ ^

  44. General rule: If X1 is a (0,1) exposure variable then HR = e1(effect of exposure adjusted for other X’s) provided no other X’s are product terms involving exposure ^ ^

  45. Example 3 Model 3: Column name Coeff StErr Pvalue Rrisk Rx 2.355 1.681 0.161 10.537 Log wbc 1.803 0.447 0.000 6.067 Rx x lgwbc -0.342 0.520 0.510 0.710 Want HR for effect of Rx adjusted for logWBC Treated subject: X* = (X1*=1, X2*=logWBC, X3*=1 x logWBC) Placebo subject: X = (X1=0, X2=logWBC, X3=0 x logWBC)

  46. ^ 3 ^ HR = exp [ i(Xi*-Xi)] HR = exp [2.355(1-0) + 1.803(logWBC-logWBC) (-.342) (1 x logWBC- 0 x logWBC) = exp [2.355 -.342 x log WBC] i=1 ^

  47. logWBC= 2: HR = exp [2.355 - .342 (2)] = e1.671 = 5.32 logWBC=4: HR = exp [2.355 - .342(4)] = e0.987 = 2.68 ^ ^

  48. General rule for (0,1) exposure variables when there are product terms: HR = exp [ + jWj] where  = coefficient of exposure (E) j=coeffeicient of product term E x Wj (HR does not contain coefficients of non-product terms) ^ ^ ^ ^ ^

  49. Example Model 3: E W1  = coefficient of Rx 1 = coefficient of Rx x log WBC HR(model 3) = exp [ + 1logWBC)] = exp [2.355 - .342logWBC)] ^ ^ ^ ^ ^

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