Table 1.

True risk models and marker distributions for the seven simulation scenarios. The linear logistic regression model logitP(D = 1|T, Y) = Ỹβ₁ + TỸβ₂, with Ỹ = (1, Y), and the classification tree including {T, TY, (1 − T)Y} as predictors are evaluated as working models in all scenarios.

Scenario		True risk model	Marker distribution
The linear logistic working model is correctly specified.	1	logit P(D = 1|T, Y) = 0.3 + 0.2Y1 − 0.2Y2 − 0.2Y3 + T(−0.1 − 2Y1 − 0.7Y2 − 0.1Y3)	Y1, Y2, and Y3 are independent N(0, 1)
	2	logit P(D = 1|T, Y) = 0.3 + 0.2Y1 − 0.2Y2 − 0.2Y3 + T(−0.1 − 2Y1 − 0.7Y2 − 0.1Y3)	Same as Scenario 1 except for 2% of high leverage observations where Y1 ~ Uniform (8, 9)
Link function in the linear logistic working model is incorrectly specified.	3	log{−logP(D = 1|T, Y)} = −0.7 − 0.2Y1 − 0.2Y2 + 0.1Y3 + T(0.1 + 2Y1 − Y2 − 0.3Y3)	Same as Scenario 1
Link function and main effects in the linear logistic working model are incorrectly specified.	4	$log\{-\text{log}P(D=1\mid T,Y)\}=2-1.5Y_1^2-1.5Y_2^2+3Y_1{Y}_2+T(-0.1-Y_1+Y_2)$	Y1, and Y2 are independent Uniform (−1.5, 1.5)
Main effects and interactions are incorrectly specified in the linear logistic working model.	5	$\text{logit}P(D=1\mid T,Y)=-0.1-0.2Y_1+0.2Y_2-0.1Y_3+Y_1^2+T(-0.5-2Y_1-Y_2-0.1Y_3+2Y_1^2)$	Same as Scenario 1
	6	logit P(D = 1|T, Y) = 0.1 − 0.2Y1 + 0.2Y2 − Y1Y2 + T(−0.5 − Y1 + Y2 + 3Y1Y2)
Linear logistic working model is mis-specified for outlying observations.	7	$P(D=1\mid T,Y)=\mathbf{1}\{Y_1<8\}\frac{1}{1+e^{-\eta }}+\mathbf{1}\{Y_1\ge 8\}\left(1-\frac{1}{1+e^{-\eta }}\right)$, where η = 0.3 + 0.2Y1 − 0.2Y2 − 0.2Y3 + T(−0.1 − 2Y1 − 0.7Y2 − 0.1Y2)	Same as Scenario 2