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Unformatted text preview: p is the number of independent variables in the regression. Note that the
Adjusted R Square will always be smaller than R Square. ˆ
The Standard Error, often written as s (or s), is an estimate of s , the standard
deviation of error term of the regression. It is computed as SSE / (n  p  1) , where SSE stands for the Sum of Squared Errors and will be explained in section “The
ANOVA Table”. 3. The ANOVA Table The ANOVA (ANalysis Of VAriance) table breaks down the variation in the dependent
variable into different components related to the regression model. The total variation of
the dependent variable around its mean is known as the Total Sum of Squares (SST),
and is broken down into the variation explained by the regression model, known as the
Regression (or Explained) Sum of Squares (SSR), and the variation explained by the
residuals (or the variation unexplained by the model), known as the Residual (or Error)
Sum of Squares (SSE). The following equation relates the three sums of squares
mentioned above:
VER. 2/2/2012. © P. KOLM 4 SST = SSR + SSE The sums of squares are computed using the following formulas:
SST = å (yi  y )2
i ˆ
SSR = å (yi  y )2
i ˆ
SSE = å (yi  yi )2
i ˆ
where yi stands for the predicted value for the ith observation of the dependent
variable y, and y stands for the sample mean. SSR, SSE and SST are displayed in the
third column of the ANOVA table:
Sum of Squares
ANOVA
df
Regression
Residual
Total 2
174
176 SS
19.3365618
45.3096585
64.6462203 MS
F
Significance F
9.6682809 37.1285270
0.0000000
0.2604003 SSR
SSE
SST Column “df” in the ANOVA table shows the degrees of freedom of each component.
Intuitively, the degrees of freedom indicates the number of “independent pieces of
information” that go into the estimate of a parameter. They are computed as follows:
ANOVA
df
Regression
Residual
Total 2
174
176 SS
19.3365618
45.3096585
64.6462203 MS
F
Significance F
9.6682809 37.1285270
0.0000000
0.2604003 =p
= n  p 1
= n 1 Degree of Freedom Column “MS” shows the mean square of each component. They represent the “average”
variation explained by the regression and the error component. They are computed as
follows:
ANOVA
df
Regression
Residual
Total 2
174
176 SS
19.3365618
45.3096585
64.6462203 MS
F
Significance F
9.6682809 37.1285270
0.0000000
0.2604003 = SSR / dfSSR
= SSE / dfSSE Finally, the F statistic indicates the overall level of significance of the regression model
Mean Square
(explained in more detail in section “Testing Overall Significance of the Regressors” VER. 2/2/2012. © P. KOLM 5 below). It is computed as MSSSR / MSSSE . Column “Significance F” shows the pvalue
associated with the F statistic according to the F (dfSSR , dfSSE ) distribution. 4. The Graphical Output from Regression There are three t...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.
 Fall '14

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