Linear Regression Formula

Linear Regression Formula

Linear regression is the most basic and commonly used predictive analysis. One variable is considered to be an explanatory variable, and the other is considered to be a dependent variable. For example, a modeler might want to relate the weights of individuals to their heights using a linear regression model.

There are several linear regression analyses available to the researcher.

Simple linear regression

  • One dependent variable (interval or ratio)
  • One independent variable (interval or ratio or dichotomous)

Multiple linear regression

  • One dependent variable (interval or ratio)
  • Two or more independent variables (interval or ratio or dichotomous)

Logistic regression

  • One dependent variable (binary)
  • Two or more independent variable(s) (interval or ratio or dichotomous)

Ordinal regression

  • One dependent variable (ordinal)
  • One or more independent variable(s) (nominal or dichotomous)

Multinomial regression

  • One dependent variable (nominal)
  • One or more independent variable(s) (interval or ratio or dichotomous)

Discriminant analysis

  • One dependent variable (nominal)
  • One or more independent variable(s) (interval or ratio)

Formula for linear regression equation is given by:

\[\large y=a+bx\]

a and b are given by the following formulas:

\(\begin{array}{l}\large a \left(intercept\right)=\frac{\sum y \sum x^{2} – \sum x \sum xy} {(\sum x^{2}) – (\sum x)^{2}}\end{array} \)

 

\(\begin{array}{l}\large b\left(slope\right)=\frac{n\sum xy-\left(\sum x\right)\left(\sum y\right)}{n\sum x^{2}-\left(\sum x\right)^{2}}\end{array} \)

Where,
x and y are two variables on the regression line.
b = Slope of the line.
a = y-intercept of the line.
x = Values of the first data set.
y = Values of the second data set.

Solved Examples

Question: Find linear regression equation for the following two sets of data:

  x  2  4  6 8
y 3 7 5  10

Solution:

Construct the following table:

x y x2 xy
2 3 4 6
4 7 16 28
6 5 36 30
8 10 64 80
 
\(\begin{array}{l}\sum x\end{array} \)
= 20		  
\(\begin{array}{l}\sum y\end{array} \)
= 25		  
\(\begin{array}{l}\sum x^{2}\end{array} \)
= 120		  
\(\begin{array}{l}\sum xy\end{array} \)
= 144

\(\begin{array}{l}b\end{array} \)
=
\(\begin{array}{l}\frac{n\sum xy-(\sum x)(\sum y)}{n\sum x^{2}-(\sum x)^{2}}\end{array} \)

\(\begin{array}{l}b\end{array} \)
=
\(\begin{array}{l}\frac{4 \times 144 – 20 \times 25}{4 \times 120 – 400}\end{array} \)

b = 0.95

\(\begin{array}{l}a=\frac{\sum y \sum x^{2} – \sum x \sum xy} {n(\sum x^{2}) – (\sum x)^{2}}\end{array} \)

 

\(\begin{array}{l}a=\frac{25\times 120 – 20\times 144} {4(120) – 400}\end{array} \)

a = 1.5

Linear regression is given by:

y = a + bx

y = 1.5 + 0.95 x

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