Look into the built-in function Composition.

Good question!

1. I was trying to do a minimal and quick to execute example -- the code of the post is included in the package ROCFunctions.m.

2. I am aware that there are better alternatives of LinearModelFit, but I plan to discuss those in the MathematicaVsR project: "Regression with ROC". (As the name hints, it is not just about Linear regression.)

3. Another point is that although the Linear regression is not a good method for this classification, nevertheless using ROC it can be made to give better results than, say, the built-in "NeuralNetwork" method. See the last section of "Linear regression with ROC.md".

Two years ago submitted more than a few answers to this Community question

"Regression frustration"

one of them with a notebook that had/has ROC implementations. While preparing for WTC-2016, I decided that it is good idea to make a package with ROC functionalities. (Making an Association-centric implementation.)

For a more elaborated example of Linear regression with ROC see this

"Regression with ROC"

project which is part of

MathematicaVsR at GitHub.