Logistic Regression Analysis of Matched Data
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Transcript of Logistic Regression Analysis of Matched Data
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Logistic Regression Logistic Regression Analysis of Analysis of
Matched DataMatched Data
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THE GENERAL LOGISTIC MODEL
Logit form of logistic model:
Logit P(X) = + iXi
Pr(D=1| X1,..., Xp) = P(X ) = 1
1 + exp[ + iXii=1
p
]
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Logit form:
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Special Case: No Interaction, I.e., all = 0.
OR = exp [ ] = e
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EVW LOGISTIC MODEL FOR MATCHED DATA
Logit P(X) = + E + 1iV1i + 2iV2i + EkWk
E = (0, 1) exposure
V1i’s denote dummy variables used to identify matching strata
V2j’s denote potential confounders other than matching variables
Wk’s denote potential effect modifiers
(usually other than matching variables)
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Adjusted OR Comparing E=1 vs. E=0 Controlling for the V’s and W’s
Special Case: No Interaction, I.e., all = 0.
OR = exp [ + kWkk
]
OR = exp [ ] = e
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logit P(X) = + E + 1iDii=1
62
+ 21GALL
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logit P(X) = + E + 1iDii=1
62
+ 21GALL
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logit P(X) = + E + 1iDii=1
62
+ 21GALL
OR(adj) = exp []
exp
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OR(adj) = exp [2.209] = 9.11
logit P(X) = + E + 1iDii=1
62
+ 21GALL
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OR(adj) = exp [2.209] = 9.11
logit P(X) = + E + 1iDii=1
62
+ 21GALL
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where the Di denote 62 dummy variables for the 63 matched sets
logit P(X) = + E + 1iDii=1
62
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OR(adj) = exp [2.209] = 9.11
logit P(X) = + E + 1iDii=1
62
+ 21GALL
H0: OR(adj) = 1 = 0
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OR(adj) = exp [2.209] = 9.11
H0: OR(adj) = 1 = 0
logit P(X) = + E + 1iDii=1
62
+ 21GALL
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OR(adj) = exp [2.209] = 9.11
H0: OR(adj) = 1 = 0
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OR(adj) = exp [2.209] = 9.11
= (2.76, 30.10)
logit P(X) = + E + 1iDii=1
62
+ 21GALL
exp [ 1.96s]
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OR(adj) = exp [2.209] = 9.11
(2.76, 30.10)
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(2.76, 30.10)
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INTERACTION MODEL: 63 matched pairs
Note: Previous model was a no interaction model
OR(adj.) = exp [ + 1GALL ]
Logit P(X) = + E + 1iDi + 21GALL + 1EGALL62
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95% CI for OR involving interaction?
e.g., What is the 95% CI for
?OR(adj.) = exp [ + 1GALL ]
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GENERAL 100(1) CI FORMULA IN A LOGISTIC MODEL FOR MATCHED DATA
100(1 - )% CI for OR (adj.):
exp [ L Z1 -
2
Var (L) ]
OR(adj.) = exp [ L ] where L = + kWkk
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VARIANCE FORMULA
Var (L) = Var () + Wk2Var (
k)
k
+ 2 Wk2Cov (,
k)
k
+ 2 Wk2Cov (
k,k)
k
?
k °
OR(adj.) = exp [ L ] where L = + kWkk
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Example: 95% CI formula
exp [ L 1.96 Var (L) ]
OR(adj.) = exp [ + 1GALL ]
L = exp [ + 1GALL ]
Var (L) = Var () + (GALL)2Var (1)
+ 2(GALL)2Cov (,1)