---
title: "Logistic Regression"
author: "Prof. Eric A. Suess"
date: "1/29/2024"
format: 
  revealjs:
    self-contained: true
---

## Introduction

Lantz Chapter 6 is about Regression Methods for Predicting Numeric Data.  The author presents the basics of Simple Linear Regression and Multiple Linear Regression.

Regression Methods can also be used for Classification.

- Logistic Regression 
- CART

## Why not linear regression?

The author applies linear regression to the launch data.
In this data set the dependent variable is **distress_ct**.
This variable has only 3 categories. 

And making predictions less than 0 or greater than 3 does not
make much sense.

**Logistics Regression** or **Multinomial Regression** would make more sense.

These are **Generalized Linear Models** (GLMs).

## An excellent introduction to Logistic Regression

Here is a link to an R-bloggers post [How to perform a Logistic Regression in R](http://www.r-bloggers.com/how-to-perform-a-logistic-regression-in-r/).

The author of the post creates training and test data sets.  And introduces
the use of the **ROC** to evaluate the fitted model.

## Logistic Regression

A logistic regression model, models a **binary dependent variable**

$Y = 1$  or Yes 

or

$Y = 0$  or No

where $P(Y = 1 | X)$ is modeled in terms of the predictors $X$.

## Logistic Regression

Try

$P(Y = 1 | X) = \beta_0 + \beta_1 X$

but all probabilities need to be between 0 and 1.

What is used is the **logit** function, to keep the values of the 
probabilities between 0 and 1.

$P(Y = 1 | X) = \frac{e^{\beta_0 + \beta_1 X}}{1+e^{\beta_0 + \beta_1 X}}$

## Logistic Regression

So it turns out that the **log odds** are linear

$log\left(\frac{P(Y=1 |X)}{1-P(Y=1|X)}\right) = \beta_0 + \beta_1 X$

This gives a nonlinear model that is estimated using MLEs by numerical methods.

## Multiple Logistic Regression

**Multiple Logistic Regression** can be used when there is more 
than one predictor variable.

Categorical or Numeric variables can be used as predictors.

## Evaluations

The **AIC** is used to compare models.

The **ROC** curve is used to compare models. 

The Area under the ROC is commonly used to evaluate and compare models.

## Logistic Regression

Try Logistic Regression with the **launch** data and the **credit** data.

## CART

Try CART with the **credit** data.  

This will be using the "C" in CART.

## An excellent introduction to Generalized Linear Models (GMLs)

Here is a link to a Quick-R post [Generalized Linear Models](http://www.statmethods.net/advstats/glm.html).
