Regression analysis

Regression analysis
Name	Regression analysis
Type	Statistical technique
Field	Statistics, Data analysis, Machine learning
Related	Correlation analysis, Time series analysis, Survival analysis

Regression analysis is a statistical technique used to establish a relationship between two or more variables, typically a dependent variable and one or more independent variables, as studied by Sir Ronald Fisher, Karl Pearson, and Francis Galton. This technique is widely used in various fields, including Economics, Finance, Biology, Medicine, and Social sciences, to name a few, with notable applications in Harvard University, Stanford University, and Massachusetts Institute of Technology. Regression analysis helps to understand the relationship between variables, make predictions, and identify the factors that affect a particular outcome, as demonstrated by Albert Einstein's work on Brownian motion and Louis Pasteur's research on Vaccination. The development of regression analysis is attributed to Carl Friedrich Gauss, Pierre-Simon Laplace, and Adrien-Marie Legendre, who worked on Least squares methods.

Introduction to Regression Analysis

Regression analysis is a powerful tool for analyzing and understanding the relationships between variables, as seen in the work of David Cox on Survival analysis and Bradley Efron on Bootstrap sampling. The technique involves fitting a mathematical model to a set of data, where the model describes the relationship between the dependent variable and one or more independent variables, as used by John Tukey in Exploratory data analysis and George Box in Time series analysis. The goal of regression analysis is to create a model that can predict the value of the dependent variable based on the values of the independent variables, as applied by Andrew Gelman in Bayesian inference and Donald Rubin in Causal inference. Regression analysis has been used in various fields, including Medicine, where it is used to identify the factors that affect the outcome of a disease, as studied by National Institutes of Health and World Health Organization, and in Finance, where it is used to predict stock prices and portfolio returns, as used by Goldman Sachs and Morgan Stanley.

Types of Regression Analysis

There are several types of regression analysis, including Simple linear regression, Multiple linear regression, Logistic regression, and Nonlinear regression, as discussed by Peter McCullagh and John Nelder in Generalized linear models. Simple linear regression involves a single independent variable, while multiple linear regression involves two or more independent variables, as used by Robert Tibshirani in Lasso regression and Trevor Hastie in Ridge regression. Logistic regression is used to model binary outcomes, such as 0 or 1, yes or no, as applied by Leo Breiman in Classification and regression trees and Jerry Friedman in Gradient boosting. Nonlinear regression involves a nonlinear relationship between the independent variables and the dependent variable, as studied by Douglas Bates in Nonlinear mixed effects models and James Ramsay in Functional data analysis. Other types of regression analysis include Poisson regression, Gamma regression, and Inverse Gaussian regression, as used by Joseph Hilbe in Negative binomial regression and James Lindsey in Count data models.

Assumptions of Regression Analysis

Regression analysis assumes that the data meet certain conditions, including Linearity, Independence, Homoscedasticity, Normality, and No multicollinearity, as discussed by Roderick Little and Donald Rubin in Missing data. The linearity assumption states that the relationship between the independent variables and the dependent variable is linear, as used by Frank Harrell in Regression modeling strategies and David Hosmer in Applied logistic regression. The independence assumption states that the observations are independent of each other, as applied by Alan Agresti in Categorical data analysis and Daniel Powers in Regression analysis of categorical data. The homoscedasticity assumption states that the variance of the dependent variable is constant across all levels of the independent variables, as studied by Robert Keenan in Heteroscedasticity and Halbert White in Econometrics. The normality assumption states that the residuals are normally distributed, as used by George Seber in Linear regression analysis and Alan Miller in Subset selection in regression. The no multicollinearity assumption states that the independent variables are not highly correlated with each other, as discussed by David Belsley in Conditioning diagnostics and Edward Welsch in Regression diagnostics.

Model Estimation and Inference

Regression analysis involves estimating the parameters of the model using a dataset, as applied by Bradley Efron in Bootstrap sampling and Trevor Hastie in Lasso regression. The most common method of estimation is Ordinary least squares (OLS), which minimizes the sum of the squared residuals, as used by Carl Friedrich Gauss in Least squares and Pierre-Simon Laplace in Probability theory. Other methods of estimation include Maximum likelihood estimation and Bayesian estimation, as discussed by Andrew Gelman in Bayesian inference and Donald Rubin in Causal inference. Once the model is estimated, inference can be made about the parameters, including hypothesis testing and confidence intervals, as applied by John Tukey in Exploratory data analysis and George Box in Time series analysis. Regression analysis can also be used to make predictions about future outcomes, as used by Goldman Sachs in Financial modeling and Morgan Stanley in Risk management.

Applications of Regression Analysis

Regression analysis has a wide range of applications in various fields, including Medicine, Finance, Economics, and Social sciences, as studied by Harvard University, Stanford University, and Massachusetts Institute of Technology. In medicine, regression analysis is used to identify the factors that affect the outcome of a disease, as applied by National Institutes of Health and World Health Organization. In finance, regression analysis is used to predict stock prices and portfolio returns, as used by Goldman Sachs and Morgan Stanley. In economics, regression analysis is used to study the relationship between economic variables, such as Gross domestic product and Inflation rate, as discussed by Federal Reserve and International Monetary Fund. In social sciences, regression analysis is used to study the relationship between social variables, such as Crime rate and Unemployment rate, as applied by United States Census Bureau and Bureau of Labor Statistics.

Limitations and Common Problems

Regression analysis has several limitations and common problems, including Multicollinearity, Heteroscedasticity, and Nonlinearity, as discussed by David Belsley in Conditioning diagnostics and Edward Welsch in Regression diagnostics. Multicollinearity occurs when the independent variables are highly correlated with each other, as studied by Robert Keenan in Heteroscedasticity and Halbert White in Econometrics. Heteroscedasticity occurs when the variance of the dependent variable is not constant across all levels of the independent variables, as applied by Alan Miller in Subset selection in regression and George Seber in Linear regression analysis. Nonlinearity occurs when the relationship between the independent variables and the dependent variable is not linear, as used by Douglas Bates in Nonlinear mixed effects models and James Ramsay in Functional data analysis. Other common problems include Outliers, Missing data, and Model misspecification, as discussed by Roderick Little and Donald Rubin in Missing data and Joseph Hilbe in Negative binomial regression.

Category:Statistical techniques