Transcript Sample Slide Heading Image

Advanced Statistics
for Interventional
Cardiologists
What you will learn
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Introduction
Basics of multivariable statistical modeling
Advanced linear regression methods
Hands-on session: linear regression
Bayesian methods
Logistic regression and generalized linear model
Resampling methods
Meta-analysis
Hands-on session: logistic regression and meta-analysis
Multifactor analysis of variance
Cox proportional hazards analysis
Hands-on session: Cox proportional hazard analysis
Propensity analysis
Most popular statistical packages
Conclusions and take home messages
1st day
2nd day
What you will learn
• Logistic regression
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–
Logistic regression model
Parameter estimates
Odds ratio interpretation
Parameter significance testing
Model checking
Qualitative predictor
Multiple logistic regression
Examples
• Generalized Linear Model
Logistic regression
• How can I predict the impact of left ventricular
ejection fraction (LVEF) on the 12-month risk of
ARC-defined stent thrombosis?
Logistic regression
• How can I predict the impact of left ventricular
ejection fraction (LVEF) on the 12-month risk of
ARC-defined stent thrombosis?
In other words, how can I predict the impact of a
given variable (aka independent) on another
dichotomous, binary variable (aka dependent)
Logistic regression
Simple example
Rained
Rainy
Rainy
Dry
Dry
Frequencies
Level
Count
Probability
Cum Prob
21
0,70000
0,70000
Rainy
9
0,30000
1,00000
Total
30
Dry
2 Levels
The variable “Rained” has two
categories “Rainy” (if precipitation >
0.02) and “Dry” (otherwise). Out of the
30 days in April it was rainy 9 days.
Therefore, if there is no other
information, you predict a 30% chance
of rain for every day.
Let’s investigate if quantitative variables
temperature and barometric pressure can
help in finding a more informative
prediction of the chance/probability of
rain/dry using logistic regression.
Logistic regression
• The goal of the logistic regression is to model the
probability of getting a certain response (eg. “dry”) with
explanatory variables (eg. “temperature”, “pressure”).
• For a binary response, p denotes the probability of the first
response level (eg. “dry”). Then, (1-p) is the probability of
the second response level (“rainy”).
• We could just model the probability by means of ordinary
regression [P (X) = β0 + β1X (linear probability model)]
• This model has a major structural defect. Probabilities fall
between 0 and 1, whereas linear functions take values
over the entire real range. So we need a more complex
model.
Logistic regression
• We model ln (p/(1-p)) instead of just p, and
the linear model is written :
ln(p/(1-p)) = ln(p) – ln(1-p) = β0 + β1*X
• Logistic regression is based on the logit
which transforms a dichotomous dependent
variable into a continuous one
Logistic regression
• An alternative formula for the probability
of getting the first response :
exp( 0  1 x)
p ( x) 
1  exp( 0  1 x)
• Graph
Logistic regression
Quantitative predictors
•
•
Cumulative probability plot on the left shows that the relationship with temperature is very weak.
As the temperature changes from 35 to 75, the probability of dry weather only changes from 0.73
to 0.66 .
A much stronger relationship with pressure is shown. When pressure is 29 inches, the fitted
probability of rain is near 100% (0 probability for Dry at the left of the graph). At 29.8, the
probability of rain drops near zero (near 1 for dry).
Logistic regression
Parameter estimates
The parameter β1 determines the rate of increase of the
S-shaped curve. The sign of b1 indicates whether the
curve ascends or descends.
P(tem p) 
exp(1.34  0.0086* tem p)
1  exp(1.34  0.0086* tem p)
P( pressure) 
Rained By Pressur
Rained By Temp
Converged by Gradient
Converged by Gradient
Whole-Model Test
Whole-Model Test
Model
exp(405 13.82* pressure)
1  exp(405 13.82* pressure)
-LogLikelihood
Difference
DF
ChiSquare
Prob>ChiSq
1
0,026668
0,8703
0,013334
Model
-LogLikelihood
Difference
Full
18,312595
Full
12,075078
Reduced
18,325929
Reduced
18,325929
RSquare (U)
0,0007
Observations (or Sum Wgts)
RSquare (U)
30
1
ChiSquare
12,5017
Prob>ChiSq
0,0004
0,3411
Observations (or Sum Wgts)
Parameter Estimates
Term
DF
6,250851
30
Parameter Estimates
Estimate
Std Error
ChiSquare
Prob>ChiSq
Term
Estimate
Std Error
ChiSquare
Prob>ChiSq
Intercept
1,34073823
3,0620868
0,19
0,6615
Intercept
-405,36267
169,29517
5,73
0,0166
Temp
-0,0086266
0,0529847
0,03
0,8707
Pressur
13,8233881
5,7651316
5,75
0,0165
Logistic regression
Odds ratio interpretation
• A very popular interpretation of the logistic regression
model uses the odds and the odds ratio. For the logit
model, the odds of response (eg. “dry”) are
p( x)
 exp( 0  1 x)  e  0 (e 1 ) x
1  p ( x)
• This exponential relationship provides an interpretation
1
for β1: the odds increase multiplicatively by e for
every unit increase in x. That is, the odds at level x+1
equal the odds at x multiplied by e 1 .

When β1 = 0, e 1 = 1 and the odds do not change as x
changes.
• Odds ratio associated with unit change of x :
oddsx 1 px 1 /(1  px 1 )

 e 1  exp(1 )
oddsx
px /(1  px )
Logistic regression
Odds ratio interpretation
Odds ratio interpretation of the β1 parameter.
Odds ratio associated with one unit increase in temperature =
exp(-0.0086) = 0.99 (each increase with one degree of the
temperature results in 1% decrease in the odds of having a “dry” day).
Odds ratio associated with one unit increase in pressure =
exp(13.82) = 1 004 500
Rained By Pressur
Rained By Temp
Converged by Gradient
Converged by Gradient
Whole-Model Test
Whole-Model Test
Model
-LogLikelihood
Difference
DF
ChiSquare
Prob>ChiSq
1
0,026668
0,8703
0,013334
Model
-LogLikelihood
Difference
Full
18,312595
Full
12,075078
Reduced
18,325929
Reduced
18,325929
RSquare (U)
0,0007
Observations (or Sum Wgts)
RSquare (U)
30
1
ChiSquare
12,5017
Prob>ChiSq
0,0004
0,3411
Observations (or Sum Wgts)
Parameter Estimates
Term
DF
6,250851
30
Parameter Estimates
Estimate
Std Error
ChiSquare
Prob>ChiSq
Term
Estimate
Std Error
ChiSquare
Prob>ChiSq
Intercept
1,34073823
3,0620868
0,19
0,6615
Intercept
-405,36267
169,29517
5,73
0,0166
Temp
-0,0086266
0,0529847
0,03
0,8707
Pressur
13,8233881
5,7651316
5,75
0,0165
Logistic regression
Significance testing
Is the effect of X on the binary response significant ?
Is the probability of the response independent of X ?
→ Hypothesis test H0: β=0 versus H1: β≠0
Wald statistic :
b




Std
.
Error


2
has a chi-squared distribution with df=1 for large samples
It is also possible to construct confidence intervals to evaluate
the significance of the effects.
Likelihood-ratio test compares the maximum log-likelihood for the
simple model when β1=0 (L0) to the maximum log-likelihood for
the full model with unrestricted β1 (L1). The test statistic -2(L0-L1)
has a chi-squared distribution with df=1. Likelihood ratio test is
more reliable for small sample sizes.
Logistic regression
Significance testing
What are your conclusions about the significance of
the effect of temperature and pressure on the
probability of a “dry” day ? Motivate with Wald statistic
and with whole-model Likelihood ratio test. Compare
with ANOVA table for continuous responses.
Rained By Pressur
Rained By Temp
Converged by Gradient
Converged by Gradient
Whole-Model Test
Whole-Model Test
Model
-LogLikelihood
Difference
DF
ChiSquare
Prob>ChiSq
1
0,026668
0,8703
0,013334
Model
-LogLikelihood
Difference
Full
18,312595
Full
12,075078
Reduced
18,325929
Reduced
18,325929
RSquare (U)
0,0007
Observations (or Sum Wgts)
RSquare (U)
30
1
ChiSquare
12,5017
Prob>ChiSq
0,0004
0,3411
Observations (or Sum Wgts)
Parameter Estimates
Term
DF
6,250851
30
Parameter Estimates
Estimate
Std Error
ChiSquare
Prob>ChiSq
Term
Estimate
Std Error
ChiSquare
Prob>ChiSq
Intercept
1,34073823
3,0620868
0,19
0,6615
Intercept
-405,36267
169,29517
5,73
0,0166
Temp
-0,0086266
0,0529847
0,03
0,8707
Pressur
13,8233881
5,7651316
5,75
0,0165
Logistic Regression
Model checking
• Let’s find out if a particular model provides a
good fit to the observed outcomes.
• Fitted logistic regression models provide
predicted probabilities that Y=1. At each setting
of the explanatory variables, one can multiply
this predicted probability by the number of
subjects to obtain a fitted count.
• The test of the null hypothesis that the model
holds compares the fitted and observed counts
using a Pearson χ2 or likelihood-ratio G2 test
statistic.
Logistic Regression
Model checking
• For a fixed number of settings, when most fitted counts
equal at least 5, χ2 and G2 have approximate chi-squared
distributions with df equal to the number of settings of
explanatory variables minus the number of model
parameters.
• Large χ2 and G2 provide evidence of lack of fit and the pvalue is the right-tailed probability above the observed
value.
• When the fit is poor, residuals and other diagnostic
measures are used to describe the influence of individual
observations on the model fit.
Logistic Regression
Model checking
So for grouped observed counts and fitted values
we can calculate lack of fit statistics
with following formulas:
(observed  fitted )
 
fitted
2
2
 observed 

G  2 (observed) log
 fitted 
2
Logistic Regression
Model checking
• We can also detect lack of fit by using the
likelihood-ratio test to compare a working model
to more complex ones.
• This approach is more useful from a scientific
perspective. A large goodness-of-fit statistic
simply indicates there is some lack of fit.
Comparing a model to a more complex model
indicates whether lack of fit exists of a particular
type.
Logistic Regression
Model checking
•
The deviance is the likelihood-ratio statistic for comparing model M
to the Saturated model (= most complex model with separate
parameter at each explanatory setting = perfect fit).
Deviance = -2(Lm- Ls)
•
•
•
The deviance, which has the same form as the G2 likelihood-ratio
goodness-of-fit statistic, follows a chi-square distribution and is
used to test the model fit.
In testing whether M fits, we test whether all parameters that are in
the saturated model but not in M equal zero.
The difference in the deviance of two models is used to compare
the fit of any two models. The statistic is large when M0 fits poorly
compared to M1.
Logistic regression
Model checking
• Rsquare or Uncertainty coefficient U.
The ratio of the reduction of the negative LogLikelihood of the
working model to the negative LogLikelihood of the reduced model
(the proportion of the uncertainty explained by the model).
• Lack-of-Fit test is the opposite of the whole-model test.
Where the whole-model tests whether anything you have in your
model is significant, the lack-of-fit tests whether anything you left out
of your model is significant. Lack-of-fit compares the fitted model
with the saturated model using the same terms. If the lack-of-fit test
is significant, then add more effects to the model using higher orders
of terms already in the model.
Logistic Regression
Model checking
Rained By Pressur
Rained By Temp
Converged by Gradient
Converged by Gradient
Whole-Model Test
Whole-Model Test
Model
-LogLikelihood
Difference
DF
ChiSquare
Prob>ChiSq
1
0,026668
0,8703
0,013334
Model
-LogLikelihood
Difference
DF
6,250851
Full
18,312595
Full
12,075078
Reduced
18,325929
Reduced
18,325929
RSquare (U)
0,0007
Observations (or Sum Wgts)
RSquare (U)
30
12,5017
Prob>ChiSq
0,0004
0,3411
Observations (or Sum Wgts)
Parameter Estimates
Term
1
ChiSquare
30
Parameter Estimates
Estimate
Std Error
ChiSquare
Prob>ChiSq
Term
Estimate
Std Error
ChiSquare
Prob>ChiSq
Intercept
1,34073823
3,0620868
0,19
0,6615
Intercept
-405,36267
169,29517
5,73
0,0166
Temp
-0,0086266
0,0529847
0,03
0,8707
Pressur
13,8233881
5,7651316
5,75
0,0165
Lack of Fit
Lack of Fit
Source
DF
-LogLikelihood
Lack of Fit
19
10,788654
Pure Error
9
Total Error
28
ChiSquare
Source
DF
21,57731
-LogLikelihood
ChiSquare
Lack of Fit
22
9,825737
19,65147
7,523941
Prob>ChiSq
Pure Error
6
2,249341
Prob>ChiSq
18,312595
0,3058
Total Error
28
12,075078
0,6048
What are your conclusions about the goodness of fit of the model ?
Logistic Regression
Residuals
• Goodness-of-fit statistics such as χ2 and G2 are indicators
of overall quality of fit. Additional diagnostics are
necessary to describe the nature of any lack of fit.
Pearson residuals comparing observed and fitted counts
are useful for this purpose.
yi  ni pi
ei 
[ni pi (1  pi )]
• Each residual divides the difference between an
observed count and its fitted value by the estimated
binomial standard deviation of the observed count.
Logistic Regression
Residuals
• The Pearson statistic for testing the model fit :
   ei
2
2
• The Pearson residual has an approximate normal
distribution around zero, when the binomial index ni is
large. Pearson residuals are treated like standard normal
deviates, with values larger than 2 indicating possible
lack of fit.
• Residuals have limited meaning when the fitted values
are very small.
Logistic Regression
Diagnostic Measures of Influence
• An influential point is an observation which changes
much the estimated parameters when removed from
the sample.
• The measures are algebraically related to an
observation’s leverage h. The greater an
observation’s leverage the greater its potential
influence.
• Adjusted Pearson residual :
e
i
1  hi
• Formulas of the measures are rather complex and not
reproduced here.
What you will learn
• Logistic regression
–
–
–
–
–
–
–
–
Logistic regression model
Parameter estimates
Odds ratio interpretation
Parameter significance testing
Model checking
Qualitative predictor
Multiple logistic regression
Examples
• Generalized Linear Model
Logistic Regression
Qualitative Predictors
• Like ordinary regression, logistic regression extends to
models incorporating multiple explanatory variables,
some of them can be qualitative.
• We will use dummy variables for including qualitative
predictors, called factors, in the model.
• Let us have a look at a simple example, using the fitness
data.
• We evaluate the effect of gender on the binary response
variable Oxy_H_L having two levels: High (>= 50) and
Low (< 50).
Logistic Regression
Qualitative Predictors – Fitness Example
To evaluate the relation between Sex
and Oxy_H_L we can use the
crosstable analysis methods.
Females have 43.75 % H and Males
have only 6.67 % H. The risk for a
female to have high oxygen uptake is
6.56 larger than for a male (relative
risk), suggesting an effect of sex.
We can also calculate the ratio of the
odds. The odds-ratio, dividing the
odds for females by the odds for
males, is 10.88.
The Pearson X2 (df=1) of 5.56
confirms the dependence
(p=0.0184).
Logistic Regression
Qualitative Predictors
We can use the logits and the logistic regression model to
evaluate the effect of sex on the binary response. Sex is
incorporated in the model with the dummy variable (“F”=1
and “M”=0).
 P(Oxy  High) 
 = -2,638 + 2,387 . Sex
ln
Predicted Logit(Oxy_H_L) = 
 P(Oxy  Low) 
Interpretation:
Odds Ratio for Males : exp (-2,638) = 0,0715
Odds Ratio for Females : exp (2,387) = 10,88
Multiple Logistic Regression
• The logistic regression model, like ordinary regression
models, generalizes to allow for several explanatory
variables. The predictors can be quantitative, qualitative,
or of both types.
• The model equation is :
Logit(Π) = α + β1x1 + β2x2 + … + βkxk
• βi refers to the effect of Xi on the log odds that Y = 1,
controlling the other X’s. For instance, exp(βi) is the
multiplicative effect on the odds of a 1-unit increase in Xi,
at fixed levels of the other Xs.
Multiple Logistic Regression
example: fictitious trial
• 12 months duration
– Monthly visits
– Baseline at month 0
– Final evaluation at month 12
• Placebo versus Drug
• Primary Objective
– To show that there are 20% more responders on drug compared
to placebo after 12 months of treatment.
• Primary Efficacy Variable = Disease Activity Scale (DAS)
– DAS with range 0 – 140
– DAS at study entry minimum 50
– Responder defined as a 20 point decrease from baseline DAS to
month 12
Logistic regression
fictitious trial: key variables
Variable
Description
trt
sex
age
grade
duration
surgery
DAS.bl
DAS.12
DAS.12.cfb
DAS.wd
DAS.12.cfb
time.wd
res20.12
res20.wd
time.res
Randomized treatment (0 = placebo, 1 = drug)
Patient’s sex (0 = female, 1 =male)
Patient’s age at baseline (years)
Disease grade (1 good – 4 very bad)
Disease duration at baseline in years
prior surgery for disease
DAS at baseline
DAS at month 12
Change from baseline at month 12 in DAS
DAS at withdrawal
Change from baseline at withdrawal in DAS
Time to withdrawal in days
Responder at month 12 (0 = No, 1 = Yes)
Responder at withdrawal (0 = No, 1 = Yes)
Time to first response in days
Logistic regression
fictitious trial: primary efficacy analysis
• To show that there are 20% more responders
on drug compared to placebo after 12 months
of treatment.
• Responder defined as a 20 point decrease
from baseline DAS to month 12
• Possible analysis
– Contingency table analysis – X2 test
– Logistic regression adjusting for prognostic factors
Logistic regression
fictitious trial: contingency table
Responder? Placebo
Drug
Total
No
83 (74.8%)
45 (42.1%)
128 (58.7%)
Yes
28 (25.2%)
62 (57.9%)
90 (41.3%)
Total
111 (100%)
107 (100%)
218 (100%)
X2 test p < 0.001 → treatment and response are dependent.
Logistic regression
fictitious trial: odds ratio
• The odds of being a responder on drug are 62/45 (1.378)
• The odds of being a responder on placebo are 28/83 (0.337)
• The odds ratio is 1.378 divided by 0.337 which equals 4.08
Patients are more likely to respond on drug compared to placebo.
• 95% confidence interval for odds ratio is (2.319, 7.342)
Logistic regression
Model 1: logit(res20.12) = trt
Parameter
Estimate
Odds ratio
95% conf. int.
Intercept
trt
-1.087
1.407
0.337
4.084
(0.216, 0.511)
(2.318, 7.342)
• Odds ratio is the exponential of the estimate.
• Results are identical to previous slide.
Logistic regression
Model 2: logit(res20.12) = trt + duration
Parameter
Estimate
Odds ratio
95% conf. int.
Intercept
trt
duration
-0.698
1.408
-0.112
0.497
4.088
0.894
(0.200, 1.211)
(2.317, 7.361)
(0.708, 1.123)
• Adjustment for duration does not seem to have much influence on
treatment effect.
• Odds ratio of duration is close to 1 and confidence interval does
not include 1 so no real influence on the response.
Logistic regression
Model 3: logit(res20.12) = trt + duration + age + sex
Parameter
Estimate
Odds ratio
95% conf. int.
Intercept
trt
duration
age
sex
-0.738
1.412
-0.119
-0.001
0.215
0.478
4.102
0.888
0.999
1.240
(0.063, 3.469)
(2.321, 7.405)
(0.702, 1.119)
(0.961, 1.039)
(0.694, 2.223)
• Duration, age and sex do not seem to have an influence on
treatment effect.
Logistic regression
Model 4: logit(res20.12) = trt + surgery
Parameter
Estimate
Odds ratio
95% conf. int.
Intercept
trt
surgery
0.263
1.341
-2.669
1.300
3.823
0.069
(0.740, 2.308)
(1.925, 7.849)
(0.033, 1.137)
• Surgery does seem to have an influence on treatment effect.
• We can conclude that some of the difference in response at month
12 can be put down to prior surgery.
Multiple logistic regression
Model comparison
• One can use the likelihood-ratio method to test
hypotheses about parameters in logistic
regression models. Compare the maximized loglikelihood L1 for the full model to the maximized
log-likelihood L0 for the simpler model with those
parameters equal 0, using test statistic -2(L0 – L1)
• More generally, one can compare maximized loglikelihoods for any pair of models and select the
most parsimonious model.
Multiple Regression
Backward elimination
• Backward elimination of predictors, starting with
a complex model and successively taking out
terms is often used to find a good model.
• At each stage, we might eliminate the term in the
model that has the largest p-value when we test
that its parameters equal zero. We first test the
highest-order terms for each variable. We do not
remove a main effect term if the model contains
higher-order-interactions involving that term.
What you will learn
• Logistic regression
–
–
–
–
–
–
–
–
Logistic regression model
Parameter estimates
Odds ratio interpretation
Parameter significance testing
Model checking
Qualitative predictor
Multiple logistic regression
Examples
• Generalized Linear Model
Logistic regression
Logistic regression
Sangiorgi et al, AHJ 2008
Other examples
Fajadet et al, Circulation 2006
Other examples
Moses et al, NEJM 2003
Other examples
Corbett et al, EHJ 2006
Other examples
Corbett et al, EHJ 2006
Other examples
What you will learn
• Logistic regression
–
–
–
–
–
–
–
–
Logistic regression model
Parameter estimates
Odds ratio interpretation
Parameter significance testing
Model checking
Qualitative predictor
Multiple logistic regression
Examples
• Generalized Linear Model
Generalized Linear Models
• The generalized linear model (GLM) is a flexible
generalization of ordinary regression.
• The GLM is a broad class of models that
includes ordinary regression and ANOVA models
for continuous response variables as well as
models for categorical responses.
• Logistic regression is one type of GLM.
Generalized Linear Models
• All generalized linear models have three components :
– Random component identifies the response variable and
assumes a probability distribution for it
– Systematic component specifies the explanatory variables
used as predictors in the model (linear predictor).
– Link describes the functional relationship between the
systematic component and the expected value (mean) of
the random component.
• The GLM relates a function of that mean to the explanatory
variables through a prediction equation having linear form.
• The model formula states that: g(µ) = α + β1x1 + … + βkxk
Generalized Linear Models
• Through differing link functions, GLM
corresponds to other well known models
Distribution
Name
Normal
Exponential
Identity
Inverse
Gamma
Inverse
Gaussian
Inverse
squared
Poisson
Log
Binomial
Logit
Link Function
Mean
Function
Generalized linear Models
Normal GLM
• Ordinary regression and ANOVA models are special
cases of GLMs, assuming a normal distribution of the
random component, modelling the mean directly.
• A GLM generalizes ordinary regression in two ways:
– allows the random component to have a distribution other than
normal
– allows modelling some function of the mean
• A traditional way of analyzing non-normal data
transforms the response, so it becomes normal with
constant variance
• With GLMs it is unnecessary to transform data so that
normal-theory methods apply. This is because the GLM
fitting process utilizes maximum likelihood methods.
Logistic Regression with SPSS
Variables in the Equation
B
Step
a
1
TOTLENGT
Constant
,011
-2,031
S.E.
,003
,195
a. Variable(s) entered on step 1: TOTLENGT.
Wald
16,821
108,924
df
1
1
Sig .
,000
,000
Exp(B)
1,011
,131
95,0% C.I.for EXP(B)
Lower
Upper
1,006
1,016
Questions?
Take-home messages
• Logistic regression is used to analyze the effect of
explanatory variables on the probability of an outcome of
a binary response variable.
• Logistic regression is based on the logit (ln(p/(1-p))
which transforms the dichotomous dependent
variable into a continuous one.
• A βx parameter estimate is the odds ratio associated
with a unit increase of the independent X variable.
• Likelihood-ratio statistic is used to compare models.
• Examine Pearson residuals to evaluate lack of fit.
• A flexible extension of the Regression and ANOVA model
is the Generalized Linear Model.
And now a brief break…
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