## BLOG

November 19, 2015
The Three Most Commonly Used Regression Models
By: Billy Hou

Regression analysis is a statistical process for estimating the relationships among variables. It includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables (predictors). More specifically, regression analysis helps one understand how the typical value of the dependent variable changes when any one of the independent variables is varied, while the other independent variables are held fixed.

Regression analysis is widely used in the business world for prediction and forecasting and nowaday overlaps with machine learning. We often use it in the predictive stage of the four levels of analytics. Practical applications include demand forecasting and price optimization that require understanding of the relationship between independent and dependent variables.

There are hundreds of types of regression techniques developed since 1805, when the first regression technique, method of least squares, was published by Adrien-Marie Legendre. How do you decide which one to use? We review the top three most used regression techniques to help you decide which to use depending on your context.

Linear Regression

Linear Regression or Linear Probability (LP) is the oldest type of regression technique and used extensively in practical applications such as forecasting, predicting dependent variables with one or more independent variable, and inference, quantifying the strength of the relationship between the dependent variable and one or more independent variable.

Usage of linear regression is limited by four principle assumptions, linearity, homoscedasticity, independence, and normality. It does not capture highly non-linear or chaotic patterns and highly correlated independent variables will cause the model to collapse. Practitioners will need to use factor reduction techniques or other constrained regression techniques to address these issues.

Logistic Regression

Logistic regression is a regression model that predicts a binary dependent variable. This technique is very useful in social sciences, medical research, and fraud detection. Unlike linear regression, logistic regression does not require assumptions such as linearity (between dependent variable and independent variable), normality, and homoscedasticity. However, some assumptions still apply. Logistic regression requires a dependent variable to be binary or ordinal, error terms need to be independent, linearity of independent variables and log odds, and at least 30 observations for each parameter to be estimated.

Ridge Regression

Ridge regression is a modeling technique for analyzing multiple regression data that suffer from multi-colinearity. It is like the least squares method but shrinks the estimated coefficients towards zero. One major benefit is that ridge regression does not over fit when compared to linear regression, see graph below for example. In short, it is a more robust version of linear regression. It has the same assumptions as the linear regression model except normality is not a required assumption.

A major drawback of ridge regression is that it cannot zero out coefficients; thus you either end up including all the coefficients in the model, or none of them. The least absolute shrinkage and selection operator, or LASSO, is a technique that addresses this issue. As with ridge regression, it assumes the covariates are standardized. A practitioner must consider using a factor selection and reduction method before starting to use ridge regression or consider to use LASSO regression instead of ridge regression.

Regression analysis is at the core of forecasting and machine learning. However, there are hundreds of regression techniques and it is up to the practitioners to select the correct technique based on their context. They must understand the advantages and disadvantages of each technique to make the correct choice.

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