What is a Regression Analysis?

Regression analysis is a statistical method employed in statistical modeling to scrutinize the relationship between a dependent response variable and one or more independent predictor variables. It provides an insight into the manner in which the distinctive value of the dependent variable (also referred to as the criterion variable) changes with a change in any one of the independent variables, provided the values of the other independent variables are not altered. Regression analysis not only interprets how different variables interact with one another but also offers cognizance of which variables are salient and which can be ignored.

How does a Regression Analysis Work?

Regression analysis is typically used to determine the average value of the dependent variable for fixed values of the independent variables. It is also used, albeit less frequently, to determine a quantile or any other location parameter of the conditional distribution of the dependent variable for fixed values of the independent variables. The primary objective of these endeavors is to estimate the regression function, which is a function of the independent variables. In simpler terms, regression analysis answers the following two questions:

1. Are the independent predictor variables competent in predicting dependent criterion variables?
2. Which independent variables are the most essential predictors of dependent variables?

Regression analysis also mandates the use of a probability distribution to characterize the variation of the dependent variable around the prediction of the regression function. Then, there is also the vastly similar approach known as Necessary Condition Analysis (NCA) that is used to derive the maximum value of the dependent variable for a fixed value of the independent variable. The goal of NCA is to determine a value of the independent variable that is essential but insufficient for a fixed value of the dependent variable.

Uses of Regression Analysis

There are three principal uses of regression analysis:

1. To calculate the strength of independent variables or predictors.
2. To forecast an outcome.
3. To forecast a trend.

When used as a tool for prediction and forecasting, regression analysis has applications that are analogous to artificial intelligence. There are situations where regression analysis can also be used, with due caution, to theorize a causal connection between dependent and independent variables. There are several methods of performing regression analysis; however all methods can be broadly classified into two categories: Parametric: Parametric regressions are those regressions where the regression functions can be expressed in terms of a limited number of unknown parameters obtainable from the available data. Examples include linear regression and least squares regression. Nonparametric: Nonparametric regressions are regressions in which the regression function can remain in a defined set of possibly infinite-dimensional functions. The structure of the data generating process and its correlation with the regression approach can greatly influence the performance of regression analysis procedures. Since such process structures remain typically unascertained, assumptions are often made by regression analysis procedures regarding those processes. However, the availability of a sufficient quantity of data makes it possible to put these assumptions to the test. Although moderately violated assumptions do not significantly affect usability of regression models as tools for prediction, optimal performance cannot be guaranteed in such situations. Moreover, applications that have small effects of causality based on observational data can cause regression procedures to project anomalous outcomes. Often, regression may take on the specific role of appraising dependent variables instead of the the discrete response variables used in classification. This is known as metric regression.