An Illustration of Rate Difference Estimation with SAS

in Logistic Regression

Yun Guo

Outline

• Statistical methods for RD

• RD with logistic regression

• SAS realization

• Conclusion

Statistical methods

• Wald/t approximation—no covariate adjust• CMH (weighted estimate)-adjust strata• Logistic Regression-adjust both

Statistical methods

• Logistic regression

,where, x is covariate, s is factor, t is treatment, sub i is indexing subject

(1)

exp( )

1 exp( )i i i

ii i i

x s tp

x s t

(2)

RD with logistic regression

• Odd Ratio through Logistic Regression

Simply subtract among different levels of factor

Say t(t=1) vs. t(t=0),

1 0log( ) log( ) log( ) exp( )1 1t t

p pOR

p p

RD with logistic regression

• Rate through Logistic Regression (say subject i=1)

1

0

1 0

exp( )

1 exp( )

exp( )

1 exp( )

i it

i i

i it

i i

t t

x sp

x s

x sp

x s

RD p p

(3)

Intercept, covariate as well as factor, not gone!!!!

RD with logistic regression

• How about the variance of estimate?

1

0

1 0

exp( )

1 exp( )

exp( )

1 exp( )

var( ) var( )

i it

i i

i it

i i

t t

x sp

x s

x sp

x s

RD p p

(4)

Very messy!!!

RD with logistic regression

Hereafter subscript 1/0 of est p index treatment.

Statistical methods

• Marginal approach (sum over all samples)

Statistical methods

Statistical methods

• To conclude, we have following 4 steps• 1, establish logistic regression • 2, get estimated rates, covariance of estimates,

estimated RD• 3, set up Delta method• 4, get CI

SAS realizationEXAMPLE DATA

SAS realizationPREPARE DATA

SAS realizationSTEP 1: Logistic regression

SAS realizationSTEP 1: Logistic regression

SAS realizationSTEP 1: Logistic regression

COV

EST

SAS realizationSTEP 2: RD estimate

SAS realization

Step 3: Delta method

Step 4: CI

Conclusion

1, When covariates adjust needed, a choice is to use L Regression

2, Other methods could be used too

Thank you!