Correlation and Regression - CMA

Regression Equations

We use linear regression models to research and estimate the linear or straight line, relationships.  The linear relationship between two variables summarizes the amount of change in one variable that is associated with the change in another variable or variables. Regression equations are super fun to learn and it is very informative.

Introduction

Linear regression models the relationship between the two variables. It is used to test the statistical significance which can be used to test whether the observed linear relationship could have emerged. if it fits a linear equation to observed data called a regression equation.

In statistics, we use a regression equation to come up with an equation-like model. This equation like model helps to represent the pattern and patterns present in data.

Regression Variables

In regression models, we have independent variables which are of two types’ dependent and independent variable. We can show the relationship between two variables on the graph.

We use regression equations for the prediction of values of the independent variable. The dependent variable is an outcome variable. Independent variable for the gross data is the predictor variable.

Regression Equations

A regression equation can be defined as a statistical model, used to determine the specific relationship between the predictor variable and the outcome variable. A model regression equation allows predicting outcome with a very small error.

Yi=b0 +b1xi +e

In this equation, Yi represents an outcome variable and Xi represents its corresponding predictor variable. It is an equation which contains numerical relationships between the predictor and the outcome.

Here the term b0 is the intercept for the model if the predictor is of zero value. We can consider it as a baseline or control point. The term bi represents the numerical relationship between the predictor variable and the outcome of the term.

Least Square Method 

Suppose there is a researcher who decides that the variable X is an independent variable and it has some influence on dependent variable Y. it does not imply that Y is directly caused by X.

Researchers consider some reasons for X to be an independent variable. X can occur before Y, from which the researcher conclude X influence Y.

If there are n observations on each X and Y. All these can be plotted in a scatter diagram. All the independent variables on the horizontal axis and all the dependent variable on the Vertical axis.

Regression Equations

 

By using the scatter diagram, the researchers make an observation of scattering of points. According to the observation, they decide whether it will form a straight line or not by connecting the two variables. It provides a ready way to estimate the regression line.

Although, it is not a systematic approach since the same data can show a different line, which makes a different judgment. It concerns to have a straight line between the variables.

So, in order to provide a systematic estimate of line, various statisticians have devised some procedure which gives an estimate of the line, which may fit the points better than other possible lines. Some call this the least square criterion and the regression line from this method is known as the least squares regression lines.

Solved Question on Regression Equations

Q. Consider the following set of points :

{(-2,-1),(1,1),(3,2)}

Find the least square regression line for given data points.

Solution:

x y x y x 2
-2 -1 2 4
1 1 1 1
3 2 6 9
Σx = 2 Σy = 2 Σxy = 9 Σx= 14

a = (nΣx y – ΣxΣy) / (nΣx2 – (Σx)2)

(3*9 – 2*2) / (3*14 – 22) = 23/38

b = (1/n)(Σy – a Σx)

(1/3)(2 – (23/38)*2) = 5/19

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