Suppose we have an observed data matrix $X$ of length $N$ with $2$ column predictors. If I wanted to generate continuous response data from this, we might do
$$ Y^{cont} = X\beta + N(0,1) $$
or in R
Y = beta1*X[,1] + beta2*X[,2] + rnorm(N, 0, 1)
If instead I wanted to generate binary response data, is it valid to do
$$ Y^{bin} = Bin[\sigma(X\beta + N(0,1))] $$
or in R
Y = rbinom(N, 1, sigmoid(beta1*X[,1] + beta2*X[,2] + rnorm(N, 0, 1)))
where
sigmoid = function(x) 1/(1+exp(-x))
is the sigmoid function? How can I effectively add noise to data in order to generate binary data?
