logistic regression analysis of matched data

Logistic Regression Logistic Regression Analysis of Analysis of

Matched DataMatched Data

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

]

Logit form:

Special Case: No Interaction, I.e., all = 0.

OR = exp [ ] = e

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)

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

logit P(X) = + E + 1iDii=1

62

+ 21GALL

logit P(X) = + E + 1iDii=1

62

+ 21GALL

logit P(X) = + E + 1iDii=1

62

+ 21GALL

OR(adj) = exp []

exp

OR(adj) = exp [2.209] = 9.11

logit P(X) = + E + 1iDii=1

62

+ 21GALL

OR(adj) = exp [2.209] = 9.11

logit P(X) = + E + 1iDii=1

62

+ 21GALL

where the Di denote 62 dummy variables for the 63 matched sets

logit P(X) = + E + 1iDii=1

62

OR(adj) = exp [2.209] = 9.11

logit P(X) = + E + 1iDii=1

62

+ 21GALL

H0: OR(adj) = 1 = 0

OR(adj) = exp [2.209] = 9.11

H0: OR(adj) = 1 = 0

logit P(X) = + E + 1iDii=1

62

+ 21GALL

OR(adj) = exp [2.209] = 9.11

H0: OR(adj) = 1 = 0

OR(adj) = exp [2.209] = 9.11

= (2.76, 30.10)

logit P(X) = + E + 1iDii=1

62

+ 21GALL

exp [ 1.96s]

OR(adj) = exp [2.209] = 9.11

(2.76, 30.10)

(2.76, 30.10)

INTERACTION MODEL: 63 matched pairs

Note: Previous model was a no interaction model

OR(adj.) = exp [ + 1GALL ]

Logit P(X) = + E + 1iDi + 21GALL + 1EGALL62

95% CI for OR involving interaction?

e.g., What is the 95% CI for

?OR(adj.) = exp [ + 1GALL ]

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

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

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)