Advanced Statistics for Interventional Cardiologists

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

What you will learn Cox proportional hazards analysis –Checking assumptions –Variable selection methods

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 failure time): days, weeks, years…

Survival analysis 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 Study start- Study end - Lost to f.u. - Withdrawal T Event ?

Survival analysis Assumptions about censoring: 1.non-informative (no info about patient outcome) 2.Patients censored and non censored should have the same chance of failure 3.Chance of censoring independent of failure 4.Censored patients should be representative of those at risk at censoring time 5.Censored patients are supposed to survive to the next time point - issue of patients lost to follow up

Survival analysis ABCDEFABCDEF x Withdrawn Study end Lost x

Survival analysis ABCDEFABCDEF x Withdrawn Study end Lost x ABCDEFABCDEF x Withdrawn Study end Lost x Study end A and F: events B, C, D and E: censored

Survival analysis 0t 1 ∞ 0t 1 ∞ Study end Survival function S(t) = P(T>t) Probability of survival time T at time t S(0) = 1S(∞) = 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

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.

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

Survival analysis Goals of survival analysis: 1.To estimate and interpret survival and/or hazard functions 2.To compare survival functions 0t 1 ∞ 0t 1 ∞ 0t 1 ∞ 3.To assess the relationship of explanatory variables to survival time controlling for covariates -This requires modeling, e.g. using the Cox proportional hazards model

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

Data layout for theory Ordered failure times Number of failures Number of censored Risk set t(0) = 0f(0) = 0c(0) = 0R(t0) = all subjects t(1) = earliest of failure time f(1) = number of failures at t(1) c(1) = number of censored between t(0) and t(1) R(t1) = f all – c1 Risk set allows us to use all information up to time of censorship

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) = Group 2: rate of failures (21/21 / mean time of survival (8) = Hazard Ratio: / = 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

Kaplan Meyer TNMQSurvival function x18/21= x16/17= X14/15= x11/12= X10/11= X6/7= X5/6=0.448

Kaplan Meyer Univariate modeling

Kaplan Meyer

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

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

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

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)

Survival analysis with SPSS

Cosgrave et al, AJC 2005

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

Stratification

IVUS vs. non-IVUS Log Rank: 0.18

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

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

Stratification

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?

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

Cox PH analysis Sir David Cox in 2006

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

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

Cox PH analysis h(t,X) = h 0 (t) e Σβ i X i 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 h 0 (t) e Σβ i X i HR = = exp[Σβ i (X i -X i * )] h 0 (t) e Σβ i X i *

Cox PH analysis h(t,X) = h 0 (t) e Σβ i X i h 0 (t) eΣβiXieΣβiXi Baseline hazard Involves t but not X Not known Exponential Involves X but not t X are assumed to be time-independent If we want Hazard Ratio, h 0 (t) is deleted in the ratio, thus we do not need to calculate it

Cox PH analysis Cosgrave et al, AJC 2005

Cox PH analysis Cosgrave et al, AJC 2005

Cox PH analysis Cosgrave et al, AJC 2005

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

Cox PH analysis Cosgrave et al, AJC 2005 Adjusted Hazard Ratios Unadjusted Hazard Ratios,710,20412,0661,0012,0341,3633,036 Diabetes -,157,198,6331,426,855,5801,259 Stent type BSEWalddfSig.Exp(B)LowerUpper 95,0% CI for Exp(B)

Cox PH analysis Agostoni et al, AJC 2005

Cox PH analysis Agostoni et al, AJC 2005

Cox PH analysis Agostoni et al, AJC 2005

Cox PH analysis Agostoni et al, AJC 2005

Cox PH analysis Agostoni et al, AJC 2005

Cox PH analysis Agostoni et al, AJC 2005

Cox PH analysis Marroquin et al, NEJM Aronud 260 deaths 2.Around 300 MIs 3.Around 500 deaths + MIs

Cox PH analysis Marroquin et al, NEJM 2008

Cox PH analysis

Forward, or Step-Up, Selection This method is often used to provide an initial screening of the candidate variables when a large group of variables exists You begin with no candidate variables in the model Select the most significant variable At each step, select the next most significant candidate variable Stop adding variables when none of the remaining variables are significant Backward, or Step-Down, Selection This method begins with a model in which all variables have been included The user sets the significance level at which variables can enter the model The backward selection model starts with all variables in the model At each step, the variable that is the least significant is removed This process continues until no non-significant variables remain Stepwise regression removes and adds variables to the regression model for the purpose of identifying a useful subset of predictors

Cox PH analysis 1.Age 2.Sex 3.Elective/Urgent 4.Pre PCI 5.Pre CABG 6.CKD 7.CHF 8.DM 9.1/2/3 VD 10.N. Lesions 11.SA/UA/STEMI 12.Therapy 13.Off label DES

Cox PH analysis

Checking PH assumptions Graphical techniques Compare the log-log survival curves over different categories of variables: parallel curves imply PH assumption is ok Compare observed (KM curves) with predicted (using PH model) survival curves: if observed and predicted curves are close, PH assumpiton is ok

Log-Log curves Most commonly used, and relatively easy to perform They can involve subjectivity in the interpretation of the graphs (typically we look for strong indications of non-parallelism) Continuous variables can be a problem (it is not possible to create 2 lines such as in dichotomous variables, however we can create 2 groups by “categorizing” continuous variables, e.g. above and below the mean or the median)

Log-Log curves Cosgrave et al, AJC 2005

If curves are parallel PH asumption is met Log-Log curves Cosgrave et al, AJC 2005

Checking PH assumptions If PH assumption is not met for one variable: Stratify for the variable that does not satisfy the PH assumption and run a Cox analysis into each stratum adjusting for the other variables that meet the PH assumption If the variable can change over time, include time- dependent variable in the model: extended Cox modeling

Extended Cox Model Time-dependent covariates Add interaction term involving time to the model CALL THE STATISTICIAN !

Questions?

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