Advanced Statistics for Interventional Cardiologists

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Advanced Statistics for Interventional
Cardiologists
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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

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What you will learn

• Cox proportional hazards analysis
• Checking assumptions
• Variable selection methods

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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

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Survival analysis

• KEY PROBLEM
• Censored data
• We dont 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

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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

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Survival analysis
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Survival analysis
A and F events B, C, D and E censored
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Survival analysis
Survival function S(t) P(Tgtt) Probability of
survival time T at time t S(0) 1 S(8) 0 S(t)
is not increasing as t increases It is a
probability thus 0S(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
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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
0h(t)lt8 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.
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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
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Survival analysis

• Goals of survival analysis
• To estimate and interpret survival 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

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Data layout for the CPU
MACE event (1,0) TimeMACE time to event
(days) Sex, Age, Typesten, explanatory
variables
Cosgrave et al, AJC 2005
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Data layout for theory
Ordered failure times Number of failures Number of censored Risk set
t(0) 0 f(0) 0 c(0) 0 R(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) f1 ------------ all c1
Risk set allows us to use all information up to
time of censorship
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Hazard ratio
Example (2 cohorts of patients with max follow up
35 weeks) Group 1 (n21) failures9 (censored
12), time to failure17 weeks Group 2 (n21)
failures21(censored 0), time to failure8
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 HR1
gt no relationship HR5 gt hazard of the
exposed is 5 times the one of unexposed HR0.5 gt
the hazard of the exposed is half that of the
unexposed
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Kaplan Meyer
T N M Q Survival function
0 21 0 0 1
6 21 3 1 1x18/210.857
7 17 1 1 0.857x16/170.807
10 15 1 2 0.807X14/150.753
13 12 1 0 0.753x11/120.69
16 11 1 3 0.69X10/110.623
22 7 1 0 0.623X6/70.538
23 6 1 5 0.538X5/60.448
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Kaplan Meyer
Univariate modeling
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Kaplan Meyer
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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

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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

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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
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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)

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Survival analysis with SPSS
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Survival analysis with SPSS
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Survival analysis with SPSS
Cosgrave et al, AJC 2005
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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 create subgroups strata according to
  these factors
• Multivariable modeling

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Stratification
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Stratification
IVUS vs. non-IVUS Log Rank 0.18
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Stratification
Distal vs. Non-distal LM Log Rank 0.02
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Stratification
IVUS in 54 of non-distal LM IVUS in 31 of
distal LM
P0.08
Log Rank 0.69
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Stratification
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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?

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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

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Cox PH analysis
Sir David Cox in 2006
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Cox PH analysis

• Problem
• Cant use ordinary linear regression because how
  do we account for the censored data?
• Cant 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

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Cox PH analysis
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Cox PH analysis
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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

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Cox PH analysis

• h(t,X) h0(t) eSßiXi
• 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) eSßiXi
• HR expSßi(Xi-Xi)
• h0(t) eSßiXi

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Cox PH analysis
h(t,X) h0(t) eSßiXi
h0(t) eSßiXi
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, h0(t) is deleted in the
ratio, thus we do not need to calculate it
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Cox PH analysis
Cosgrave et al, AJC 2005
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Cox PH analysis
Cosgrave et al, AJC 2005
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Cox PH analysis
Cosgrave et al, AJC 2005
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Cox PH analysis
Cosgrave et al, AJC 2005
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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
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Cox PH analysis
Agostoni et al, AJC 2005
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Cox PH analysis
Agostoni et al, AJC 2005
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Cox PH analysis
Agostoni et al, AJC 2005
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Cox PH analysis
Agostoni et al, AJC 2005
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Cox PH analysis
Agostoni et al, AJC 2005
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Cox PH analysis
Agostoni et al, AJC 2005
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Cox PH analysis

1. Aronud 260 deaths
2. Around 300 MIs
3. Around 500 deaths MIs

Marroquin et al, NEJM 2008
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Cox PH analysis
Marroquin et al, NEJM 2008
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Cox PH analysis
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Stepwise regression removes and adds variables to
the regression model for the purpose of
identifying a useful subset of predictors

• 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

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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

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Cox PH analysis
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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

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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)

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

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Extended Cox Model

• Time-dependent covariates
• Add interaction term involving time to the model

CALL THE STATISTICIAN !
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Questions?
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For further slides on these topics please feel
free to visit the metcardio.org
websitehttp//www.metcardio.org/slides.html