I am trying to simulate a selection model for a variable $Y$ dependent on covariate vector $X$, so that two groups/sub-sets $S=(0,1)$ of observations on $Y$ are created, which have a fixed difference in expectations (means) $\theta$ (i.e., selection bias) $$E(Y|S=1)-E(Y|S=0)= \theta$$
I am looking for solutions to approach this problem, where in principle I could also simulate two variables $Y_1$ and $Y_2$ and overlay these as a mixture distribution $Y$. However, note that the selection mechanism $S$ should be conditionally independent of $Y$ given $X$ (ignorable selection) $$P(S|Y,X) = P(S|X),$$ but not be marginally independent $$P(S|Y) \ne P(S).$$
Whereas it is easy to parameterize a model for $P(S|X)$, and also relate $E(Y|X)$, it seems very difficult to me to specify this model to achieve a fixed $\theta$ and keep variances of $Y$ across sub-sets approximately constant (the latter is not absolutely necessary, but desirable). If the mixture approach is chosen, I do not know how to overlay distributions in dependence of $X$. Thanks for suggestions!