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Chapter 03
The Regression Printout
We have already learned how to run the regression in Microsoft Excel. When you run a regression, you will see lot of output on screen. A sample output is given below;
SUMMARY OUTPUT
Regression Statistics Multiple R (^) 0. R Square (^) 0. Adjusted R Square (^) 0. Standard Error (^) 0. Observations (^35) ANOVA df SS MS F Significance F Regression 1 62.20 62.20 249.96 5.84E- 17 Residual 33 8.21 0. Total 34 70.
Coefficients Standard Error t Stat P-value Intercept (^) - 0.054 0.084 - 0.638 0. X (^) 1.064 0.067 15.810 0.
The purpose of this lesson is to understand the numbers appearing on the screen, the algorithm to compute them and what these numbers stand for.
Consider the regression equation: ๐ฆ๐ = ๐ผ + ๐ฝ๐ฅ๐ + ๐๐
๏ง The first thing in regression analysis is to calculate the estimates of ๐ผ ๐๐๐ ๐ฝ. These estimates can be calculated by the formula we have learned in lecture 2 or by the matrix approach. ๏ง The remaining output visible in this regression output is calculated from these two estimates and the data set. The detail is as follows:
Fitted values or the regression line:
For any value of variable of X-variable, prediction of corresponding value of variable can be obtained by
following formula: ๐ฆ๏ฟฝ = ๐ผ๏ฟฝ + ๐ฝฬ๐ฅ
The vector ๐ฆ๏ฟฝ is the vector of estimates of Y for all values of variable X in the data set. These values formulate a straight line and represent the predictable part of variable Y.
Residuals:
The residuals are calculated as follows:
๐๐ = ๐ฆ๐ โ ๐ฆ๏ฟฝ๐
Estimated or Regression Sum of Square
This is given by the formula ๐ธ๐๐ = โ^ ๐๐=1(๐ฆ๏ฟฝ๐ โ ๐ฆ๏ฟฝ)^2
ESS represents the part of variation in Y that is estimated by the regression, therefore it is called estimated sum of squares.
Residuals Sum of Squares
This is given by the formula ๐
๐๐ = โ ๐๐=1( ๐ฆ๐ โ ๐ฆ๏ฟฝ๐)^2 = โ ๐๐=1๐๐^2
RSS is sum of square of residuals which is difference between true values of Y and its estimates via regression equation. Therefore it represents the part of variation in Y that is estimated by the regression, therefore it is called estimated sum of squares.
Total Sum of Squares
The formula for Total Sum of square is: ๐๐๐ = โ^ ๐๐=1(๐ฆ๐ โ ๐ฆ๏ฟฝ)^2
The formula indicates that TSS is measure of total variation in Y. It can be proven mathematically that TSS=ESS+RSS
R-square
The formula for R โsquare is:
R-square=ESS/TSS
We know that ESS is the portion of variation in Y that is explained using the regression equation, whereas the TSS is measure of total variation in Y. Therefore R-square represent percentage of variation in Y that is explained by the regression.
Note that we have ESS=TSS-RSS
Therefore ๐
โ ๐ ๐๐ข๐๐๐ = ๐๐๐โ๐
๐๐ ๐๐๐ = 1^ โ^
๐
๐๐ ๐๐๐
R-square is called coefficient of determination. This tells us how much part of variation in the dependent variable can be explained by using the variable X. The square root of R-square is called the multiple R.
โ(โ ๐ฅ๐)^2 + ๐ โ ๐ฅ๐^2
And
Therefore we got the regression equation
Further output is calculated as follows:
X Y ๐ฆ๏ฟฝ = ๐ผ๏ฟฝ + ๐ฝฬ ๐ฅ ๐๐^ =^ ๐ฆ๐^ โ^ ๐ฆ๏ฟฝ๐^ ๐ฆ๐^ โ^ ๐ฆ๏ฟฝ
๏ง ESS is sum of squares of the last column which is 22. ๏ง RSS is sum of squares of second last column which is 5.15. ๏ง TSS=RSS+ESS=22.85+5.15=28.
Therefore
๏ง R-square=ESS/TSS=22.85/28=0. ๏ง Multiple R =0.90 and
๏ง Standard Error ๐๏ฟฝ = ๏ฟฝ^ ๐
๐๐ ๐โ๐ =^ ๏ฟฝ
- 15 4 = 1.
The Graph of above mentioned model is as follows:
Table 1 X Y 10 45 21 38 28 56 44 71 50 98 62 81
Exercise 1: For the data in Table 1, calculate the following statistics
- The estimated of parameters for regression equation ๐ฆ = ๐ผ + ๐ฝ๐ฅ + ๐
- The standard error of regression ๐๏ฟฝ
- ESS, TTS and R-square
Exercise 2: Suppose RSS=10.51 and n=
- Calculate ๐๏ฟฝ
- If R-square=0.81, calculate ESS and TSS
Exercise 3: Suppose for some data set, estimated regression equation is ๐ฆ = 2.51 + 0.53๐ฅ + ๐
- Predict the value of y when x=
