logistic regression ii
DESCRIPTION
Logistic Regression II. Exposure=1. Exposure=0. Disease = 1. Disease = 0. Simple 2x2 Table (courtesy Hosmer and Lemeshow ). Odds Ratio for simple 2x2 Table. (courtesy Hosmer and Lemeshow ). =>55 yrs.TRANSCRIPT
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Logistic Regression II Logistic Regression II
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Simple 2x2 Table Simple 2x2 Table (courtesy (courtesy Hosmer and LemeshowHosmer and Lemeshow))
Exposure=1 Exposure=0
Disease = 1
Disease = 0
1
1
1)/(
e
eEDP
e
eEDP
1)~/(
11
1)/(~
eEDP
eEDP
1
1)~/(~
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e
e
e
ee
e
OR
11
11
1
11
1
1
1
(courtesy (courtesy Hosmer and LemeshowHosmer and Lemeshow))
Odds Ratio for simple 2x2 Table Odds Ratio for simple 2x2 Table
e
e 111 )( ee
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Example 1: CHD and Age Example 1: CHD and Age (2x2)(2x2)
(from Hosmer and Lemeshow) (from Hosmer and Lemeshow)
=>55 yrs <55 years
CHD Present
CHD Absent
21 22
6 51
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Example 1: CHD and Age Example 1: CHD and Age (2x2)(2x2)
(from Hosmer and Lemeshow) (from Hosmer and Lemeshow)
=>55 yrs <55 years
CHD Present
CHD Absent
21 22
6 51
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(younger) unexposed if 0
(older) exposed if 1
))(1
)(log(
1
11
X
XDP
DP
The Logit ModelThe Logit Model
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51226211 )
1
1()
1()
1
1()
1(),(
11
1
e
xe
ex
ex
e
eL
The LikelihoodThe Likelihood
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The Log LikelihoodThe Log Likelihood
1111 loglogloglog
:
eeeee
recall
)1log(510)1log(2222
)1log(60)1log(21)(21
),(log
111
1
ee
ee
L
51226211 )
1
1()
1()
1
1()
1(),(
11
1
e
xe
ex
ex
e
eL
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Derivative(s) of the log Derivative(s) of the log likelihoodlikelihood
1
1
1
1
1
6
1
2121
)]([log
1
1
e
e
e
e
d
Ld
e
e
e
e
d
Ld
1
51
1
2222
)]([log
)1log(510)1log(2222
)1log(60)1log(21)(21
),(log
111
1
ee
ee
L
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Maximize Maximize
51
22
5122
73)1(22
1
7322
01
51
1
2222
e
e
ee
e
e
e
e
e
e
=Odds of disease in the unexposed (<55)
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Maximize Maximize 11
ORx
xe
e
e
e
ee
e
e
226
5121
5122
621
621
6
21
216
)1(2127
01
2721
1
1
1
11
1
1
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Hypothesis TestingHypothesis Testing H H00: : =0=0
2. The Likelihood Ratio test:
1. The Wald test:
)ˆ(error standard asymptotic
0ˆ
Z
2~))](ln(2[))(ln(2
)(
)(ln2
pfullLreducedL
fullL
reducedL
Reduced=reduced model with k parameters; Full=full model with k+p parameters
Null value of beta is 0 (no association)
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Hypothesis TestingHypothesis Testing H H00: : =0=0
2. What is the Likelihood Ratio test here?– Full model = includes age variable– Reduced model = includes only intercept
Maximum likelihood for reduced model ought to be (.43)43x(.57)57
(57 cases/43 controls)…does MLE yield this?…
96.3
221
211
61
511
)2262151
ln(
x
x
Z
1. What is the Wald Test here?
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))(1
)(log(
DP
DP
The Reduced ModelThe Reduced Model
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Likelihood value for reduced modelLikelihood value for reduced model
28.)75ln(.
75.57
43
5743
1004343
01
10043
)(log
)1(57)1(43log43)(log
)1
1()
1()( 5743
e
e
ee
e
e
d
Ld
eeeL
ex
e
eL
= marginal odds of CHD!
305743
5743
101.2)57(.)43(.
)75.1
1()
75.1
75.()28.(
xx
xL
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Likelihood value of full modelLikelihood value of full model
265122621
51226211
1043.2)43.1
1()
43.1
43.()
5.4
1()
5.4
5.3(
)
5122
1
1()
5122
1
5122
()
621
1
1()
621
1
621
()(
xxxx
xxxL
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Finally the LR…Finally the LR…
2
2630
)96.3(7.18
7.1896.1177.136)]1043.2ln(2[)101.2ln(2
)(
)(ln2
xx
fullL
reducedL
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Example 2: Example 2: >2 exposure levels>2 exposure levels*(dummy coding) *(dummy coding)
CHD status
White Black Hispanic Other
Present 5 20 15 10
Absent 20 10 10 10
(From Hosmer and Lemeshow)
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SAS CODESAS CODEdata race;
input chd race_2 race_3 race_4 number;datalines;
0 0 0 0 201 0 0 0 50 1 0 0 101 1 0 0 200 0 1 0 101 0 1 0 150 0 0 1 101 0 0 1 10end;run;
proc logistic data=race descending;weight number;model chd = race_2 race_3 race_4;
run;
Note the use of “dummy variables.”
“Baseline” category is white here.
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What’s the likelihood here?What’s the likelihood here?
10101015
1020205
)1
1()
1()
1
1()
1( x
)1
1()
1()
1
1()
1()(
otherwhiteotherwhite
otherwhite
hispwhitehispwhite
hispwhite
blackwhiteblackwhite
blackwhite
whitewhite
white
ex
e
e
ex
e
e
ex
e
ex
ex
e
eL
β
In this case there is more than one unknown beta
(regression coefficient)—so this symbol represents a vector of beta coefficients.
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SAS OUTPUT – model fitSAS OUTPUT – model fit
Intercept Intercept and Criterion Only Covariates AIC 140.629 132.587 SC 140.709 132.905 -2 Log L 138.629 124.587 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 14.0420 3 0.0028 Score 13.3333 3 0.0040 Wald 11.7715 3 0.0082
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SAS OUTPUT – regression SAS OUTPUT – regression coefficientscoefficients
Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -1.3863 0.5000 7.6871 0.0056 race_2 1 2.0794 0.6325 10.8100 0.0010 race_3 1 1.7917 0.6455 7.7048 0.0055 race_4 1 1.3863 0.6708 4.2706 0.0388
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SAS output – OR estimatesSAS output – OR estimates The LOGISTIC Procedure Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits race_2 8.000 2.316 27.633 race_3 6.000 1.693 21.261 race_4 4.000 1.074 14.895
Interpretation:
8x increase in odds of CHD for black vs. white
6x increase in odds of CHD for hispanic vs. white
4x increase in odds of CHD for other vs. white
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Example 3: Prostrate Cancer Study Example 3: Prostrate Cancer Study (same data as from lab 3)(same data as from lab 3)
Question: Does PSA level predict tumor penetration into the prostatic capsule (yes/no)? (this is a bad outcome, meaning tumor has spread).
Is this association confounded by race?
Does race modify this association (interaction)?
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1.1. What’s the relationship What’s the relationship between PSA (continuous between PSA (continuous variable) and capsule variable) and capsule penetration (binary)?penetration (binary)?
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Capsule (yes/no) vs. PSA (mg/ml)Capsule (yes/no) vs. PSA (mg/ml)psa vs. capsule
capsule
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
psa0 10 20 30 40 50 60 70 80 90 100 110 120 130 140
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Mean PSA per quintile vs. proportion capsule=yes S-shaped?
proportion with
capsule=yes
0.180.200.220.240.260.280.300.320.340.360.380.400.420.440.460.480.500.520.540.560.580.600.620.640.660.680.70
PSA (mg/ml)0 10 20 30 40 50
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logit plot of psa predicting capsule, by quintiles linear in the logit?
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logit plot of psa predicting capsule, by QUARTILE linear in the logit?
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logit plot of psa predicting capsule, by decile linear in the logit?
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model: capsule = psamodel: capsule = psa
Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 49.1277 1 <.0001 Score 41.7430 1 <.0001 Wald 29.4230 1 <.0001 Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -1.1137 0.1616 47.5168 <.0001 psa 1 0.0502 0.00925 29.4230 <.0001
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Model: capsule = psa raceModel: capsule = psa race Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -0.4992 0.4581 1.1878 0.2758 psa 1 0.0512 0.00949 29.0371 <.0001 race 1 -0.5788 0.4187 1.9111 0.1668
No indication of confounding by race since the regression coefficient is not changed in magnitude.
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Model: Model: capsule = psa race psa*racecapsule = psa race psa*race
Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -1.2858 0.6247 4.2360 0.0396 psa 1 0.0608 0.0280 11.6952 0.0006 race 1 0.0954 0.5421 0.0310 0.8603
psa*race 1 -0.0349 0.0193 3.2822 0.0700
Evidence of effect modification by race (p=.07).
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---------------------------- race=0 ----------------------------
Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -1.1904 0.1793 44.0820 <.0001 psa 1 0.0608 0.0117 26.9250 <.0001 ---------------------------- race=1 ---------------------------- Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -1.0950 0.5116 4.5812 0.0323 psa 1 0.0259 0.0153 2.8570 0.0910
STRATIFIED BY RACE:
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How to calculate ORs from How to calculate ORs from model with interaction termmodel with interaction term
Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -1.2858 0.6247 4.2360 0.0396 psa 1 0.0608 0.0280 11.6952 0.0006 race 1 0.0954 0.5421 0.0310 0.8603
psa*race 1 -0.0349 0.0193 3.2822 0.0700
Increased odds for every 5 mg/ml increase in PSA:
If white (race=0):
If black (race=1):
36.1)0608.*5( e
14.1))0349.0608*(.5( e
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How to calculate ORs from How to calculate ORs from model with interaction termmodel with interaction term
Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -1.2858 0.6247 4.2360 0.0396 psa 1 0.0608 0.0280 11.6952 0.0006 race 1 0.0954 0.5421 0.0310 0.8603
psa*race 1 -0.0349 0.0193 3.2822 0.0700
Increased odds for every 5 mg/ml increase in PSA:
If white (race=0):
If black (race=1):
36.1)0608.*5( e
14.1))0349.0608*(.5( e
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ORs for increasing psa at ORs for increasing psa at different levels of race.different levels of race.
30.1e
:menblack among level psain increase mg/ml 10 afor OR
14.1e
:menblack among level psain increase mg/ml 5 afor
82.1e
:men whiteamong level psain increase mg/ml 10 afor OR
36.1e
:men whiteamong level psain increase mg/ml 5 afor
)0349.0608(.100)(*0349.*(1)0954.*(0)0608.
)1*01(*0349.*(1)0954.*(5)0608.
5*0349.5*0608.)1*0(*0349.*(1)0954.*(0)0608.
1)*5(*0349.*(1)0954.*(5)0608.
10*0608.0)(*0349.*(0)0954.*(0)0608.
0)(*0349.*(0)0954.*(10)0608.
5*0608.0)(*0349.*(0)0954.*(0)0608.
0)(*0349.*(0)0954.*(5)0608.
ee
ee
OR
ee
ee
OR
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ORs for increasing psa at ORs for increasing psa at different levels of race.different levels of race.
30.1e
:menblack among level psain increase mg/ml 10 afor OR
14.1e
:menblack among level psain increase mg/ml 5 afor
82.1e
:men whiteamong level psain increase mg/ml 10 afor OR
36.1e
:men whiteamong level psain increase mg/ml 5 afor
)0349.0608(.100)(*0349.*(1)0954.*(0)0608.
)1*01(*0349.*(1)0954.*(5)0608.
5*0349.5*0608.)1*0(*0349.*(1)0954.*(0)0608.
1)*5(*0349.*(1)0954.*(5)0608.
10*0608.0)(*0349.*(0)0954.*(0)0608.
0)(*0349.*(0)0954.*(10)0608.
5*0608.0)(*0349.*(0)0954.*(0)0608.
0)(*0349.*(0)0954.*(5)0608.
ee
ee
OR
ee
ee
OR
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OR for being black (vs. white), at OR for being black (vs. white), at different levels of psa.different levels of psa.
10.1e
:mg/ml 0psamen with among white)(vs.black beingfor
06.1e
:mg/ml 1psamen with among white)(vs.black beingfor
19.0e
:mg/ml 05psamen with among white)(vs.black beingfor
034.0e
:mg/ml 100psamen with among white)(vs.black beingfor
*(1)0954.0)(*0349.*(0)0954.*(0)0608.
0)(*0349.*(1)0954.*(0)0608.
1)(*0349.*(1)0954.0)(*0349.*(0)0954.*(1)0608.
1)(*0349.*(1)0954.*(1)0608.
50)*1(*0349.*(1)0954.0)(*0349.*(0)0954.*(50)0608.
50)*1(*0349.*(1)0954.*(50)0608.
100)*1(*0349.*(1)0954.100)*0(*0349.*(0)0954.*(100)0608.
100)*1(*0349.*(1)0954.*(100)0608.
ee
OR
ee
OR
ee
OR
ee
OR
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PredictionsPredictionsThe model:
What’s the predicted probability for a white man with psa level of 10 mg/ml?
)*(0349.)(0954.)(0608.2858.1 1)(capsulelogit racepsaracepsa
34.51.1
51.
1
110)psa1/white,P(capsule
11)P(capsule
1)P(capsule-1
1)P(capsule
)*(0349.)(0954.)(0608.2858.1 )1)P(capsule-1
1)P(capsuleln(
)10(0608.2858.1
)10(0608.2858.1
)0(0349.)0(0954.)10(0608.2858.1
)0(0349.)0(0954.)10(0608.2858.1
)*(0349.)(0954.)(0608.2858.1
)*(0349.)(0954.)(0608.2858.1
)*(0349.)(0954.)(0608.2858.1
e
e
e
e
e
e
e
racepsaracepsa
e
racepsaracepsa
racepsaracepsa
racepsaracepsa
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PredictionsPredictionsThe model:
What’s the predicted probability for a black man with psa level of 10 mg/ml?
)*(0349.)(0954.)(0608.2858.1 1)(capsulelogit racepsaracepsa
28.39.1
39.
110)psa1/black,P(capsule
11)P(capsule
)10(0349.)1(0954.)10(0608.2858.1
)10(0349.)1(0954.)10(0608.2858.1
)*(0349.)(0954.)(0608.2858.1
)*(0349.)(0954.)(0608.2858.1
e
e
e
eracepsaracepsa
racepsaracepsa
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PredictionsPredictionsThe model:
What’s the predicted probability for a white man with psa level of 0 mg/ml (reference group)?
)*(0349.)(0954.)(0608.2858.1 1)(capsulelogit racepsaracepsa
22.28.1
28.
110)psa1/black,P(capsule
11)P(capsule
2858.1
2858.1
)*(0349.)(0954.)(0608.2858.1
)*(0349.)(0954.)(0608.2858.1
e
e
e
eracepsaracepsa
racepsaracepsa
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PredictionsPredictionsThe model:
What’s the predicted probability for a black man with psa level of 0 mg/ml?
)*(0349.)(0954.)(0608.2858.1 1)(capsulelogit racepsaracepsa
23.30.1
30.
110)psa1/black,P(capsule
11)P(capsule
)1(0954.2858.1
)1(0954.2858.1
)*(0349.)(0954.)(0608.2858.1
)*(0349.)(0954.)(0608.2858.1
e
e
e
eracepsaracepsa
racepsaracepsa
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Diagnostics: ResidualsDiagnostics: Residuals
What’s a residual in the context of logistic regression?
Residual=observed-predicted
For logistic regression:
residual= 1 – predicted probability
OR residual = 0 – predicted probability
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Diagnostics: ResidualsDiagnostics: Residuals
88.22.1Residual
22.22.0Residual
What’s the residual for a white man with psa level of 0 mg/ml who has capsule penetration?
What’s the residual for a white man with psa level of 0 mg/ml who does not have capsule penetration?
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In SAS…recall model with psa In SAS…recall model with psa and gleason…and gleason…
proc logistic data = hrp261.psa;
model capsule (event="1") = psa gleason;
output out=MyOutdata l=MyLowerCI
p=Mypredicted u=MyUpperCI resdev=Myresiduals;
run;
proc gplot data = MyOutdata;
plot Myresiduals*predictor;
run;
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Residual*psaResidual*psaDevi ance Res i dual
- 3
- 2
- 1
0
1
2
3
psa
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140
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Estimated prob*gleasonEstimated prob*gleasonEs t i mat ed Pr obabi l i t y
0. 0
0. 1
0. 2
0. 3
0. 4
0. 5
0. 6
0. 7
0. 8
0. 9
1. 0
gl eason
0 1 2 3 4 5 6 7 8 9