By David G. Kleinbaum
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Extra info for Logistic Regression: A Self-Learning Text (2nd Edition)
State the form of the estimated logistic model obtained from fitting the model to the data set. 24. State the estimated logistic model in logit form. 25. Assuming the study design used was a follow-up design, compute the estimated risk for a 40-year-old male (SEX=1) smoker (SMK=1) with CHOL=200 and OCC=1. ) 26. Again assuming a follow-up study, compute the estimated risk for a 40year-old male nonsmoker with CHOL=200 and OCC=1. ) 27. Compute and interpret the estimated risk ratio that compares the risk of a 40-year-old male smoker to a 40-year-old male nonsmoker, both of whom have CHOL=200 and OCC=1.
Starting with the general formula for the ROR, we then substitute the values for the X1 and X0 variables in the formula. = eβ1 (1−0)+β2 (40− 40)+β3 (0−0) We then obtain ROR equals e to the ␤1 times (1 Ϫ 0) plus ␤2 times (40 Ϫ 40) plus ␤3 times (0 Ϫ 0). = eβ1 +0+0 The last two terms reduce to 0, = eβ1 coefficient of CAT in ( ) logit P X = α + β1CAT + β2 AGE + β3ECG ROR X1 , X 0 = eβ1 (1) CAT = 1, AGE = 40, ECG = 0 (0) CAT = 0, AGE = 40, ECG = 0 ROR X1 , X 0 = eβ1 = an “adjusted” OR AGE and ECG: • • • fixed same control variables so that our final expression for the odds ratio is e to the ␤1, where ␤1 is the coefficient of the variable CAT.
The second expression shows that when CAT=0, but AGE and ECG remain fixed at 40 and 0, respectively, the logit reduces to ␣ ϩ 40 ␤2. If we subtract the logit for CAT=0 from the logit for CAT=1, after a little arithmetic, we find that the difference is ␤1, the coefficient of the variable CAT. Thus, letting the symbol ᭝ denote change, we see that ␤1 represents the change in the logit that would result from a unit change in CAT, when the other variables are fixed. An equivalent explanation is that ␤1 represents the change in the log odds that would result from a one unit change in the variable CAT when the other variables are fixed.