Transcript LAB6&7

Labs 6 & 7
Case-Control Analysis
----Logistic Regression
Henian Chen, M.D., Ph.D.
Applied Epidemiologic Analysis P8400
Fall 2002
Data Files
Today we will use the case-control study data of
esophageal cancer.
If you use “infile” statement to read the ‘case-control978. dat’
file,Please make sure that you corrected the miscoded
values and the two abnormally high values for alcohol.
I corrected case-control978.dbf, case-control978.wk3, and
case-control978.txt. You are welcome to use one of them.
proc import datafile='a:case-control978.txt' out=case_control978 dbms=tab replace;
getnames=yes; run;
proc import datafile='a:case-control978.wk3' out= case_control978 dbms=wk3 replace;
getnames=yes; run;
proc import datafile='a:case-control978.dbf' out= case_control978 dbms=dbf replace;
run;
Applied Epidemiologic Analysis P8400
Fall 2002
Logistic Regression Model
A regression model in which the dependent variable is
binary (yes, no). A form of the generalized linear model
in which the link function is the logit, and the
regression parameters are expressed as log odds
associated with unit increase in the predictors.
For ordinal response outcomes (no pain, slight pain,
substantial pain), we can model the cumulative logits
by performing ordered logistic regression using the
proportional odds model
For nominal outcomes (Democrate, Republicans,
Independents), we can model the generalized logits by
performing logistic analysis using the log-linear model
Applied Epidemiologic Analysis P8400
Fall 2002
Logistic Regression for Intercept only
SAS Program
proc logistic data=case_control978 descending;
model status=;
run;
* Descending: to get the probability and OR for dependent variable=1
SAS Output
The LOGISTIC Procedure
Model Information
Data Set
Response Variable
Number of Response Levels
Number of Observations
Model
Optimization Technique
WORK.CASE_CONTROL978
status
2
978
binary logit
Fisher's scoring
Applied Epidemiologic Analysis P8400
Fall 2002
Logistic Regression for Intercept only
SAS Output
Response Profile
Ordered
Value
1
2
Total
Frequency
200
778
status
1
0
Probability modeled is status=1.
Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.
-2 Log L = 990.8635
Analysis of Maximum Likelihood Estimates
Parameter
DF
Estimate
Standard
Error
Intercept
1
-1.3584
0.0793
Wald
Chi-Square
Pr > ChiSq
293.5837
<.0001
Applied Epidemiologic Analysis P8400
Fall 2002
Logistic Regression for Intercept only
1. Calculate the log odds
In our model, intercept (α) = -1.3584,
-1.3584 is the log odds of cancer for total sample
2. Take the antilog to get the odds
Odds=exp(-1.3584)=0.2571
3. Divide the odds by (1+odds) to get the P
(P means probability in cohort or population,
in case-control study P means proportion)
P = 0.2571/(1+0.2571)=0.2045
= 200/(200+778)
P is related to α in Logistic Model
Applied Epidemiologic Analysis P8400
Fall 2002
Logistic Regression for Dichotomous Predictor
Alcohol Consumption (alcgrp): 0=0-39 gm/day; 1=40+ gm/day
SAS Program
proc logistic data=case_control978 descending;
model status=alcgrp;
run;
SAS Output
Model Fit Statistics
Criterion
-2 Log L
Intercept Only
990.863
Intercept and Covariates
901.036
Likelihood Ratio Test
G = 990.863 – 901.036 = 89.827
df = 1
The model with variable ‘alcgrp’ is significantly.
Applied Epidemiologic Analysis P8400
Fall 2002
Logistic Regression for Dichotomous Predictor
SAS Output
Analysis of Maximum Likelihood Estimates
Parameter
DF
Estimate
Standard
Error
Intercept
alcgrp
1
1
-2.5911
1.7641
0.1925
0.2132
Wald
Chi-Square
Pr > ChiSq
181.1314
68.4372
<.0001
<.0001
Odds Ratio Estimates
Effect
Point
Estimate
alcgrp
5.836
95% Wald
Confidence Limits
3.843
8.864
OR = exp(β) = exp(1.7641) = 5.836
Heavy drinkers (alcgrp=1) are about 6 times more
likely to get cancer than light drinkers (alcgrp=0).
OR is not related to α in Logistic Model
Applied Epidemiologic Analysis P8400
Fall 2002
Logistic Regression for Dichotomous Predictor
1. Calculate the log odds
Light drinkers (alcgrp=0), log odds=-2.5911
Heavy drinkers (alcgrp=1),
log odds=-2.5911+1.7641=-0.827
2. Take the antilog to get the odds
Light drinkers, Odds=exp(-2.5911)=0.0749
Heavy drinkers, Odds=exp(-0.827)=0.4374
3. Divide the odds by (1+odds) to get the P(x)
Light drinkers, P(x)=0.0749/(1+0.0749)=0.0697
Heavy drinkers, P(x)=0.4374/(1+0.4374)=0.3043
Applied Epidemiologic Analysis P8400
Fall 2002
Logistic Regression for Ordinal Predictor
Alcohol Consumption (alcgrp4): 0=0-39 gm/day; 1=40-79 gm/day
2=80-119 gm/day; 3=120+ gm/day
SAS Program
proc logistic data=case_control978 descending;
model status=alcgrp4;
run;
SAS Output
Model Fit Statistics
Criterion
-2 Log L
Intercept Only
990.863
Intercept and Covariates
846.467
Likelihood Ratio Test
G = 990.863 – 846.467 = 144.396
df = 1
The model with variable ‘alcgrp4’ is significantly.
Applied Epidemiologic Analysis P8400
Fall 2002
Logistic Regression for Ordinal Predictor
SAS Output
Analysis of Maximum Likelihood Estimates
Parameter
DF
Estimate
Standard
Error
Intercept
alcgrp4
1
1
-2.4866
1.0453
0.1459
0.0934
Wald
Chi-Square
Pr > ChiSq
290.4172
125.2007
<.0001
<.0001
Odds Ratio Estimates
Effect
alcgrp4
Point
Estimate
2.844
95% Wald
Confidence Limits
2.368
3.416
OR = exp(1.0453) = 2.844. Men with alcgrp4=1 are about 3 times more
likely to get cancer than men with alcgrp4=0. This OR is also for
alcgrp4= 1 vs. alcgrp4=2; or alcgrp4=2 vs. alcgrp4=3.
OR = exp[(3-1)*1.0453] = exp(2.0906) = 8.090 for alcgrp4=1 vs. alcgrp4=3
OR = exp[(3-0)*1.0453] = exp(3.1359) = 23.009 for alcgrp4=0 vs. alcgrp4=3
Applied Epidemiologic Analysis P8400
Fall 2002
OR=exp(βx) is a special case when
1. X is a binary variable
2. No interactions between X and other
variables
If X is not a binary variable
OR=exp[βx(X*-X**)]
If X is not a binary variable, and there is a
interaction between X and W,
OR=exp[(X*-X**)(βx+ βxwW)]
Applied Epidemiologic Analysis P8400
Fall 2002
Logistic Regression for Continuous Predictor
Alcohol Consumption (alcohol): daily consumption in grams
SAS Program
proc logistic data=case_control978 descending;
model status=alcohol;
run;
SAS Output
Analysis of Maximum Likelihood Estimates
Parameter
DF
Estimate
Standard
Error
Intercept
alcohol
1
1
-2.9741
0.0261
0.1807
0.00232
Wald
Chi-Square
Pr > ChiSq
270.9266
126.4179
<.0001
<.0001
Odds Ratio Estimates
Effect
Point
Estimate
alcohol
1.026
95% Wald
Confidence Limits
1.022
Applied Epidemiologic Analysis P8400
Fall 2002
1.031
Logistic Regression for Continuous Predictor
OR = exp(0.0261) = 1.026.
The odds of cancer increase by a factor of 1.026
for each unit in alcohol consumption
OR = exp[40*(0.0261)] = exp(1.044) = 2.8406
for a 40-grams increase in alcohol consumption
per day
OR = exp[120*(0.0261)] = 22.825
for a man who drinks 160 grams per day compare
with a man who is similar in other respects
but drinks 40 grams per day.
Applied Epidemiologic Analysis P8400
Fall 2002
Interaction in Logistic Regression
model status = α + β1 alcgrp + β2 tobgrp
β1 : the effect of alcohol on cancer, controlling for tobacco (i.e., the
same OR across levels of tobacco)
β2 :the effect of tobacco on cancer, controlling for alcohol (i.e., the
same OR across levels of alcohol)
model status = α + β1 alcgrp + β2 tobgrp + β3 alcgrp*tobgrp
β1 : the effect of alcohol on cancer among non-smokers (tobgrp=0)
β2 :the effect of tobacco on cancer among non-drinkers (alcgrp=0)
β3 : interaction between smokers and drinkers
Applied Epidemiologic Analysis P8400
Fall 2002
Interaction in Logistic Regression
model status = -3.33 + 2.28 (alcgrp) + 1.38 (tobgrp) –0.98 (alcgrp*tobgrp)
A: alcgrp=0 & tobgrp=0
B: alcgrp=1 & tobgrp=0
C: alcgrp=0 & tobgrp=1
D: alcgrp=1 & tobgrp=1
Log odds
2.28*0 + 1.38*0 – 0.98*0*0 = 0.00
2.28*1 + 1.38*0 – 0.98*1*0 = 2.28
2.28*0 + 1.38*1 – 0.98*0*1 = 1.38
2.28*1 + 1.38*1 – 0.98*1*1 = 2.68
A vs. B
A vs. C
A vs. D
B vs. D
C vs. D
Odds Ratio
9.78 = 9.78/1.00
3.97 = 3.97/1.00
14.59 = 14.59/1.00
1.49 = 14.59/9.78
3.68 = 14.59/3.97
Applied Epidemiologic Analysis P8400
Fall 2002
odds
1.00
9.78
3.97
14.59