Advanced Statistics for Interventional Cardiologists

1. Advanced Statistics for Interventional Cardiologists

2. What you will learn • Introduction • Basics of multivariable statistical modeling • Advanced linear regression methods • Logistic regression and generalized linear model • Multifactor analysis of variance • Cox proportional hazards analysis • Propensity analysis • Bayesian methods • Resampling methods • Meta-analysis • Most popular statistical packages • Conclusions and take home messages

3. What you will learn • Cox proportional hazards analysis • Checking assumptions • Variable selection methods

4. Survival analysis • A collection of statistical procedures for data analysis for which: • - the outcome variable is: time until event occurs • the study design has: follow up • event: dichotomous (e.g. death, TLR, MACE...)For combined endpoints (e.g. MACE), 1 event counts: hierarchical (most severe first, e.g. in MACE: 1° death, 2° MI, 3° TVR) or temporal (first to happen) order • time (survival or failuretime): days, weeks, years…

5. Survival analysis T ? Study start - Study end - Lost to f.u. - Withdrawal Event • KEY PROBLEM • Censored data • We don’t know their survival time exactly • Who are the censored? • The study ends and no event occurs • The patient is lost to follow up • The patient withdraws from the study

6. Survival analysis • Assumptions about censoring: • non-informative (no info about patient outcome) • Patients censored and non censored should have the same chance of failure • Chance of censoring independent of failure • Censored patients should be representative of those at risk at censoring time • Censored patients are supposed to survive to the next time point • - issue of patients lost to follow up

7. Survival analysis 2 4 6 8 10 12 A B C D E F x Study end Withdrawn Lost x

8. Survival analysis 2 4 6 8 10 12 2 4 6 8 10 12 A B C D E F A B C D E F x x Study end Study end Withdrawn Withdrawn Study end Lost Lost x x A and F: eventsB, C, D and E: censored

9. Survival analysis 1 ∞ 0 t 1 ∞ Study end 0 t Survival function S(t) = P(T>t) Probability of survival time T at time t S(0) = 1 S(∞) = 0 S(t) is not increasing as t increases It is a probability thus 0≤S(t)≤1 Theoretical S(t) is curvilinear In practice (Kaplan Meyer, Cox) S(t) is a step-function We want to study how S(t) goes down

10. Survival analysis Hazard function h(t) Instantaneous failure rate - The event rate at time t conditional on survival until time t or later - Instantaneous potential for failure per unit of time given survival up to time t - It is a rate thus 0≤h(t)<∞ If I am driving 55 Km/h, this does not mean that I will do 55 Km in the next hour, but I have the potential to do so. If I change my instantaneous speed I can change also the potential kilometers I can do in a fixed time.

11. Survival analysis There is a mathematical relationship between Survival function S(t) and Hazard function h(t) S(t) = e-h(t)*t In practice, the higher the hazard rate the lower the survival probability

12. Survival analysis 1 1 ∞ 0 t 1 ∞ 0 t ∞ 0 t • Goals of survival analysis: • To estimate and interpretsurvival and/or hazard functions • To compare survival functions • To assess the relationship of explanatory variables to survival time controlling for covariates • This requires modeling, e.g. using the Cox proportional hazards model

13. Data layout for the CPU MACE: event (1,0) TimeMACE: time to event (days) Sex, Age, Typesten, …: explanatory variables Cosgrave et al, AJC 2005

14. Data layout for theory Risk set allows us to use all information up to time of censorship

15. Hazard ratio Example (2 cohorts of patients with max follow up 35 weeks): Group 1 (n=21): failures=9 (censored 12), time to failure=17 weeks Group 2 (n=21): failures=21(censored 0), time to failure=8 weeks Hazard: Group 1: rate of failures (9/21) / mean time of survival (17) = 0.025 Group 2: rate of failures (21/21 / mean time of survival (8) = 0.125 Hazard Ratio: 0.125 / 0.025 = 5 (this is a “cumulative ratio”, we can also calculate istantaneous ratios) Interpretation of the Hazard Ratio: similar to the Odds Ratio HR=1 => no relationship HR=5 => hazard of the exposed is 5 times the one of unexposed HR=0.5 => the hazard of the exposed is half that of the unexposed

16. Kaplan Meyer

17. Kaplan Meyer Univariate modeling

18. Kaplan Meyer

19. Any survival curve has a ladder trend, with many steps Each step occurs when an event occurs, and the height of the step depends on the number of events and of censored data at each specific time Impact of a few changes in events

20. Any survival curve has a ladder trend, with many steps Each step occurs when an event occurs, and the height of the step depends on the number of events and of censored data at each specific time Impact of a few changes in events

21. Kaplan-Meier and log-rank Comparison between survival curves is usually performed with the non-parametric Mantel-Haenzel-Cox test (log-rank test) TAPAS 1 year, Lancet 2008

22. Log-rank test Are the K-M curves statistically equivalent? • Chi-square test • Overall comparison of KM curves • Observed versus Expected counts • Categories defined by ordered failure times (O-E)2 Log rank statistic = Var(O-E) Censorship plays a role in the subjects at risk for every time point when O-E is computed (i.e. when an event occurs)

23. Survival analysis with SPSS

24. Survival analysis with SPSS

25. Survival analysis with SPSS Cosgrave et al, AJC 2005

26. K-M curves and log rank test are appropriate if the comparison comes from randomized allocation (univariate analysis)… How do we deal with registry/observational data? It is possible to adjust for other relevant factors which may be heterogeneously distributed across groups We can create subgroups – strata – according to these factors Multivariable modeling Hypothesis testing for survival

27. Stratification

28. Stratification IVUS vs. non-IVUS Log Rank: 0.18

29. Stratification Distal vs. Non-distal LM Log Rank: 0.02

30. Stratification IVUS in 54% of non-distal LM IVUS in 31% of distal LM P=0.08 Log Rank: 0.69

31. Stratification

32. Hypothesis testing for survival • K-M curves and log rank test allow for comparisons based on one grouping factor (predictor) at a time • How can we account for multiple factors simultaneously for each subject in a time to event study? • How can we estimate adjusted survival-predictor relationships in the presence of potential confounding?

33. Hypothesis testing for survival • K-M curves and log rank test are appropriate if the comparison comes from randomized allocation (univariate analysis)… • How do we deal with registry/observational data? • It is possible to adjust for other relevant factors which may be heterogeneously distributed across groups • We can use Cox Proportional Hazards (PH) analysis

34. Cox PH analysis Sir David Cox in 2006

35. Cox PH analysis • Problem • Can’t use ordinary linear regression because how do we account for the censored data? • Can’t use logistic regression without ignoring the time component • with a continuous outcome variable we use linear regression • with a dichotomous (binary) outcome variable we use logistic regression • where the time to an event is the outcome of interest, Cox regression is the most popular regression technique

36. Cox PH analysis

37. Cox PH analysis

38. Cox PH analysis • Allows for prognostic factors • Explore the relationship between survival and explanatory variables • Multivarible modeling • Models and compares the hazards and their magitude for different groups/factors • Important assumption: • Survival curves must have proportional hazards (i.e. risk of an event at different time points) • It assumes the ratio of time-specific outcome (event) risks (hazard) of two groups remains about the same over time • This ratio is called the hazards ratio

39. Cox PH analysis h(t,X) = h0(t) eΣβiXi • Cox PH analysis models the effect of covariates on the hazard rate but leaves the baseline hazard rate unspecified • Does NOT assume knowledge of absolute risk • Estimates relative rather than absolute risk h0(t) eΣβiXi HR = = exp[Σβi(Xi-Xi*)] h0(t) eΣβiXi*

40. Cox PH analysis h(t,X) = h0(t) eΣβiXi If we want Hazard Ratio, h0(t) is deleted in the ratio, thus we do not need to calculate it

41. Cox PH analysis Cosgrave et al, AJC 2005

42. Cox PH analysis Cosgrave et al, AJC 2005

43. Cox PH analysis Cosgrave et al, AJC 2005

44. Cox PH analysis Diabetes Stent Type Diabets*Stent Type Cosgrave et al, AJC 2005

45. Cox PH analysis Adjusted Hazard Ratios Unadjusted Hazard Ratios 95,0% CI for Exp(B) B SE Wald df Sig. Exp(B) Lower Upper Stent type -,157 ,198 ,633 1 ,426 ,855 ,580 1,259 Diabetes ,710 ,204 12,066 1 ,001 2,034 1,363 3,036 Cosgrave et al, AJC 2005

46. Cox PH analysis Agostoni et al, AJC 2005

47. Cox PH analysis Agostoni et al, AJC 2005

48. Cox PH analysis Agostoni et al, AJC 2005

49. Cox PH analysis Agostoni et al, AJC 2005

50. Cox PH analysis Agostoni et al, AJC 2005