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Linear regression is used to find out the relationship between two variables by applying a linear equation to observed data. It must be noted that one variable is an independent variable and the other variable is a dependent variable. In simpler terms, linear regression is used to examine two things at a time. For example, an increase in height of a person also results in an increase in his/her weight. Thus, the height of a person is linearly related to weight. It is not necessary that one variable is dependent on the other but there is always some relationship between the two variables. To find the relationship between two variables the scatter plot is used. In situations when there is no relationship between the variables, the scatter plot does not indicate any value.
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Key Takeaways: Linear Regression, Simple Linear Regression, Multiple Linear Regression, Variables, Correlation Coefficient
Linear Regression Equation
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The correlation coefficient shows the extent of the relationship between two variables. It shows the association of the observed data of two variables. Coefficient range is between -1 to +1.
A linear regression line equation is written in the form of:
Y = a + bX
(X = independent variable and it is plotted along the x-axis)
(Y = dependent variable and it is plotted along the y-axis)
The slope of the line is b, and a is the intercept (the value of y when x = 0).
Check Important Formula for Coefficient of Determinant
Linear Regression Formula
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The linear regression equation is somehow similar to the slope formula. It is given by Y = a + bX.
The formula used in linear regression are as follows:
Formula for Linear Regression
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Simple Linear Regression
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Simple linear regression is known as the situation of a single scale predictable variable x and single scale response variable y. The equation is: y = a + bx
The expansion to multiple and vector-valued predictor variables is known as multiple linear regression. The equation is: Y = a + bX
Linear Regression Line
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Least-square method is used to fit the regression line in the XY plot. It provides the best-fitting line for the data by reducing the sum of the squares of vertical deviations from every data point to the line.
Linear Regression Line
Linear regression establishes the straight line known as the least-squares regression line or LSRL. It is the perfect way to present observations in a bivariate analysis of a data set.
If Y is a dependent variable, and X is an independent variable, then the population regression line is given by;
Y = BO + B1X
Here, BO is a constant, and B1 is the regression coefficient.
If a random sample of observations is given, then the regression line is expressed by;
Å· = bO + b1x
where bO is a constant, b1 is the regression coefficient, x is the independent variable, and Å· is the predicted value of the dependent variable.
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Properties of Linear Regression
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The properties of linear regression are mentioned below:
- The line decreases the aggregate of the squared difference between observed and predicted values.
- The regression line intersects the mean of X and Y variable values.
- In linear regression, the regression constant (BO) is equal to y-intercept.
- The slope of the regression line is the regression coefficient (B1). It is equal to the average change in the dependent variable (Y) and independent variable (X).
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Things to Remember
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- Linear regression is used to express the relation between an independent and dependent variable.
- The extent of relationship between two variables is expressed by correlation coefficient.
- The equation for the linear regression line is given by; Y = a + bX
- Least square method helps to plot the linear regression line in an XY plot.
- In linear regression, regression constant (BO) is equal to the y-intercept and regression coefficient (B1) is given by the slope of the regression line.
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Sample Questions
Ques. What is the range of the correlation coefficient? (1 Mark)
Ans. The range of the correlation coefficient is -1 to +1.
Ques. Name different types of linear regression? (2 Marks)
Ans. Types of linear regression as are follows:
- Simple linear regression
- Multiple linear regression
- Logistic regression
- Ordinal regression
- Multinomial regression
- Discriminant regression
Ques. In which sector regression lines are used? (2 Marks)
Ans. Regression lines are used most importantly in the finance sector for various financial analytics such as stock price, stock value etc.
Ques. What is the main difference between single and multiple regressions? (2 Marks)
Ans. The major difference between single linear regression and multiple linear regression is that the former one contains one dependent variable and the latter one has more than one dependent variable.
Ques. Find a linear regression equation for the following two sets of data: (5 Marks)
Ans. First construct the table
| x | y | x2 | xy |
|---|---|---|---|
| 2 | 3 | 4 | 6 |
| 4 | 7 | 16 | 28 |
| 6 | 5 | 36 | 30 |
| 8 | 10 | 64 | 80 |
| Σx = 20 | Σy = 25 | Σx2 = 120 | Σxy = 144 |
\(a = \frac{(?Y)(?X^{2})?(?X)(?XY)}{n(?x^{2})?(?x)^{2}}\)
Now we put the values in equation;
\(a = \frac{(25) × (120) ? (20) × (144)}{(4) × (120) ? (400)}\)
\(a = \frac{120}{80}\)
\(a = 1.5\)
Now we find the value of B;
\(b = \frac{(4) × (144) ? (20) × (25)}{(4) × (120) ? (400)}\)
\(b = \frac{76}{80}\)
\(b = 0.95\)
Thus, the value of a = 1.5 and b = 0.95
Therefore, the linear regression equation is y = 1.5 + 0.95x
Ques. State a few of the real life examples of linear regression? (2 Marks)
Ans. The effect of height on weight, effect of drugs on blood pressure are some of the real life examples of linear regression.
Ques. Find the least square regression line for the following set of data
{(-1 , 0),(0 , 2),(1 , 4),(2 , 5)}. (5 Marks)
Ans. The table is as follows:
| x | y | xy | x2 |
|---|---|---|---|
| -1 | 0 | 0 | 1 |
| 0 | 2 | 0 | 0 |
| 1 | 4 | 4 | 1 |
| 2 | 5 | 10 | 4 |
| Σx = 2 | Σy = 11 | Σxy = 14 | Σx2 = 6 |
\(a = \frac{(n \sum x y - \sum x \sum y) }{ (n \sum x^{2} - ( \sum x)^{2})} = \frac{(414 - 211) }{ (46 - 22)} = \frac{17}{10} = 1.7\)
\(b = (\frac{1}{n})(?y - a ?x) = (\frac{1}{4})(11 - 1.72) = 1.9\)
Ques. What is the correlation coefficient? (1 Mark)
Ans. Correlation coefficient depicts the extent of the relationship between two variables.
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