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Multiple regression occurs when there are more than one predictor variable is present in the regression equation. For example:
y = a0 + a1x1 + a2x2
Here a0, a1 and a2 we take (x1;, x2;, Y;), i = 1,2, …… , n as observed data. Note must be made upon the values of x that are errorless and values of y that are random.
So when,
n
s =∑[yi – (a0 + a1x1i + a2x2i)]2
i=1
Here you will find sum of squares of errors. To proceed in minimizing S,
αS/αa0 = 0, αS/αa1 = 0 and αS/αa2 = 0
From here on we come to three equations:
∑yi = n a0 + a1∑x1i + a2 ∑x2i
∑xiyi = a0∑x1i + a1∑x1i2+ a2∑x1i x2i
∑x2iyi = a0∑x2i + a1 ∑x1i x2i + a2∑x22i
This is how we’ll come to least square estimates of a0, a1 and a2. There are few things one must remember here;this is specifically a regression plane and by any chance value of a2 disappears then the regression line will represent y upon x.
Let us consider this data chart,
| x1 | 2 | 4 | 5 | 6 | 3 | 1 |
| x2 | 1 | 2 | 1 | 3 | 5 | 2 |
| y | 14 | 16 | 17 | 20 | 18 | 12 |
The first thing is to fit these data into a regression plane, for example y. n = 6.
Then, y= a0 + a1 x1 + a2 x2.
| x1 | x2 | x12 | x22 | x1x2 | x1y | x2y | y | |
| 2
4 5 6 3 1 |
1
2 1 3 5 2 |
1
16 25 36 9 1 |
1
4 1 9 25 4 |
2
8 5 18 15 2 |
28
64 85 120 54 12 |
14
32 17 60 90 24 |
14
16 17 20 18 12 |
|
| ∑ | 21 | 14 | 91 | 44 | 50 | 363 | 237 | 97 |
So when placing data in the above mentioned normal equations, we get:
97 = 6a0 + 21 a1 + 14a2
363 = 21 a0 + 91 a1 +50 a2
237 = 14 a0 + 50 a1 + 44 a2
So at last we come to:
a0= 9.7, a1= 1.3, a2= 0.83
Links of Previous Main Topic:-
- Introduction to statistics
- Knowledge of central tendency or location
- Definition of dispersion
- Moments
- Bivariate distribution
- Coefficient of correlation
- Regression of equations
- Rank correlation
- Correlation of bivariate frequency distribution
Links of Next Statistics Topics:-
- Theorem of total probability addition theorem
- Random variable
- Binomial distribution
- What is sampling
- Estimation
- Statistical hypothesis and related terms
- Analysis of variance introduction
- Definition of stochastic process
- Introduction operations research
- Introduction and mathematical formulation in transportation problems
- Introduction and mathematical formulation
- Queuing theory introduction
- Inventory control introduction
- Simulation introduction
- Time calculations in network
- Introduction of game theory
