Transcript Parametric Regression Models - University of Minnesota

Applications - SAS
• Parametric Regression in SAS
– PROC LIFEREG
– PROC GENMOD
– PROC LOGISTIC
Reference: SAS ver. 8.0 SAS/STAT User’s Guide,
SAS Institute, Inc., Cary, NC
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PubH8420: Parametric Regression Models
Slide 1
Applications – PROC LIFEREG
• Mathematical Model
y  X  
where y is a vector of response values,
(often the log of the failure times)
X is a matrix of covariates variables
(usually including an intercept term),
β is a vector of unknown regression parameters
σ is an unknown scale parameter, and
ε is a vector of errors
(assumed to come from any known distribution)
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PubH8420: Parametric Regression Models
Slide 2
Applications – PROC LIFEREG
• Log Likelihood
– if all the responses are observed
L   log(
f ( wi )

),
where
wi 
1

( yi  xi )
– If some of the responses are right censored,
L   log(
2 December 2004
f ( wi )

)   log( F ( wi ))
PubH8420: Parametric Regression Models
Slide 3
Applications – PROC LIFEREG
• Model & Estimation
– Accelerated Failure Time (Life) Model
if y  log(T )
• The effect of independent variables on an event time distribution is
multiplicative on the event time
• The effect of the covariates : change the scale of a baseline
distribution of failure times, not the location
– Estimation : MLE using a Newton-Raphson
algorithm
– Standard Errors of the parameter estimates : the
inverse of the observed information matrix
– Test : Normal based Test (e.g. chi-sq test, LRT)
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PubH8420: Parametric Regression Models
Slide 4
Applications – PROC LIFEREG
• Kidney Transplant Data
PROC FORMAT;
VALUE female 0='Male' 1='Female';
VALUE algfmt 0='Non-ALG' 1='ALG';
RUN
DATA kidney;
INFILE "surd01.dat";
INPUT id 1-4 age 5-6 sex 7 Alg 22
duration 25-27 status 28;
lntime = log(duration);
FORMAT sex female. Alg algfmt.;
RUN;
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PubH8420: Parametric Regression Models
Slide 5
Applications – PROC LIFEREG
• Exponential Regression
TITLE1 "Kidney Transplants Data";
PROC LIFEREG DATA=kidney;
CLASS ALG;
MODEL DURATION*STATUS(0)= ALG/
DIST=EXPONENTIAL;
OUTPUT OUT=out CDF=prob;
TITLE2 "Simple Exponential
Regression”;
RUN;
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PubH8420: Parametric Regression Models
Slide 6
Applications – PROC LIFEREG
Output
Kidney Transplants Data
Simple Exponential Regression
1
The LIFEREG Procedure
Model Information
Data Set
Dependent Variable
Censoring Variable
Censoring Value(s)
Number of Observations
Noncensored Values
Right Censored Values
Left Censored Values
Interval Censored Values
Name of Distribution
Log Likelihood
WORK.KIDNEY
Log(duration)
status
0
469
192
277
0
0
Exponential
-645.2158149
Algorithm converged.
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PubH8420: Parametric Regression Models
Slide 7
Applications – PROC LIFEREG
Output Continued
Type III Analysis of Effects
Effect
ALG
Wald
Chi-Square
6.7769
DF
1
Pr > ChiSq
0.0092
Analysis of Parameter Estimates
Parameter
Intercept
Alg
ALG
Alg Non-ALG
Scale
Weibull Shape
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Standard
DF Estimate Error
1
1
0
0
0
4.2155
0.4254
0.0000
1.0000
1.0000
0.1400
0.1634
0.0000
0.0000
0.0000
95% Confidence ChiLimits
Square
3.9410
0.1051
0.0000
1.0000
1.0000
4.4899
0.7456
0.0000
1.0000
1.0000
906.28
6.78
.
PubH8420: Parametric Regression Models
Slide 8
Applications – PROC LIFEREG
• Interpretation (Risk = λ exp(xβ) )
– λ = Exp(-β0) = exp(-4.215) = 0.015
– β1 = coefficient for ALG = 0.425
– RR(ALG=1:ALG=0) = exp(β1) = 0.654
• the risk of ALG group = λ exp(β1)
= 0.015*0.654 = 0.0096
• the risk of Non-ALG group
= λexp(0) = 0.015
• Testing & Conclusion
– Using ALG decreased the risk 34.6%
– Significant effect
(  2  6.78, pValue  0.0092 )
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PubH8420: Parametric Regression Models
Slide 9
Applications – PROC LIFEREG
Estimated CDF of Residuals Vs. Observed Duration
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PubH8420: Parametric Regression Models
Slide 10
Applications – PROC LIFEREG
• Multiple Regression
PROC LIFEREG DATA=kidney;
CLASS ALG;
MODEL DURATION*STATUS(0)= AGE ALG/
DIST=EXPONENTIAL;
OUTPUT OUT=out QUANTILES=.5
STD=STD
P=MED_DURATION;
RUN;
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PubH8420: Parametric Regression Models
Slide 11
Applications – PROC LIFEREG
• Estimation Comparison
Exponential Regression
Parameter
Hazards
Ratio
age
1.022
1.010
1.034
1.017
1.006
1.029
ALG
0.651
0.473
0.897
0.577
0.417
0.798
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95% Confidence
Limits
Cox Regression
Hazards
Ratio
PubH8420: Parametric Regression Models
95% Confidence
Limits
Slide 12
Applications – PROC LIFEREG
• Predicted Values and Confidence Intervals
DATA out1;
SET out;
ltime=log(med_duration);
stde=std/med_duration;
upper=exp(ltime+1.64*stde);
lower=exp(ltime-1.64*stde);
RUN;
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PubH8420: Parametric Regression Models
Slide 13
Applications – PROC LIFEREG
Median Predicted Values Vs. AGE by the Use of ALG
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PubH8420: Parametric Regression Models
Slide 14
Applications – PROC LIFEREG
• Other supported distributions
–
–
–
–
Generalized Gamma
Loglogistic
Lognormal
Weibull
• Some relations among the distributions:
The Weibull with Scale=1 : exponential distribution
The gamma with Shape=1 : Weibull distribution.
The gamma with Shape=0 : lognormal distribution.
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PubH8420: Parametric Regression Models
Slide 15
Applications – PROC GENMOD
• Piecewise exponential distribution
(Poisson Regression)
TITLE1 "Kidney Transplants Data";
PROC GENMOD DATA=kidney;
CLASS ALG;
MODEL STATUS = AGE ALG/
DIST=POISSON
LINK=log OFFSET=lntime type3;
TITLE2 "Multiple Piecewise
Exponential Regression";
RUN;
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PubH8420: Parametric Regression Models
Slide 16
Applications – PROC LOGISTIC
• Dichotomized data
DATA kidney1;
SET kidney;
DO month=1 TO duration;
IF month=duration AND status=1
THEN fail=1;
ELSE fail=0;
OUTPUT;
END;
RUN;
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PubH8420: Parametric Regression Models
Slide 17
Applications – PROC LOGISTIC
• LOGISTIC REGRESSION with LOGIT LINK
PROC LOGISTIC DATA=kidney1;
CLASS month fail/
PARAM=reference REF=first;
MODEL fail=age ALG;
RUN;
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PubH8420: Parametric Regression Models
Slide 18
Applications – PROC LOGISTIC
• LOGISTIC REGRESSION
with CLOGLOG LINK
PROC LOGISTIC DATA=kidney1 ;
CLASS month fail/
PARAM=reference REF=first;
MODEL fail=age ALG/
LINK=CLOGLOG;
RUN;
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PubH8420: Parametric Regression Models
Slide 19
Applications - SAS
• Comparison of Parameter Estimates
– Hazards Ratio in Log Scale
PHREG
LIFEREG
GENMOD
Method
Cox Reg.
Exp. Reg
( -β )
Piecewise
Exp. Reg
LOGIT
CLOGLOG
AGE
0.0168
0.0216
0.0216
0.0219
0.0217
ALG
-0.549
-0.429
-0.429
-0.4346
-0.431
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PubH8420: Parametric Regression Models
LOGISTIC
Slide 20