Search in book...
Toggle Font Controls
Create new playlist

Name your new playlist

Playlist description (optional)
Sign In

Email address

Password

Forgot Password?

or

Continue with Facebook

Continue with Google
Sign Up

Full Name

Email address

Confirm Email Address

Password

or

Continue with Facebook

Continue with Google

Appendix 4.1: Formulas for Regression Calculations

When performing regression calculations by hand, there are other formulas that can make the task easier and are mathematically equivalent to the ones presented in the chapter. These, however, make it more difficult to see the logic behind the formulas and to understand what the results actually mean.

When using these formulas, it helps to set up a table with the columns shown in Table 4.7, which has the Triple A Construction Company data that were used earlier in the chapter. The sample size (n) is 6. The totals for all columns are shown, and the averages for X and Y are calculated. Once this is done, we can use the following formulas for computations in a simple linear regression model (one independent variable). The simple linear regression equation is again given as

\hat{Y} = b_{0} + b_{1} X

$\hat{Y} = b_{0} + b_{1} X$

Slope of regression equation:

\begin{array}{l} b_{1} & = & \frac{Σ X Y - n \bar{X Y}}{Σ X^{2} - n {\bar{X}}^{2}} \\ b_{1} & = & \frac{180.5 - 6 (4) (7)}{106 - 6 (4^{2})} = 1.25 \end{array}

$\begin{array}{l} b_{1} & = & \frac{Σ X Y - n \bar{X Y}}{Σ X^{2} - n {\bar{X}}^{2}} \\ b_{1} & = & \frac{180.5 - 6 (4) (7)}{106 - 6 (4^{2})} = 1.25 \end{array}$

Intercept of regression equation:

\begin{array}{l} b_{0} & = & \bar{Y} - b_{1} \bar{X} \\ b_{0} & = & 7 - 1.25 (4) = 2 \end{array}

$\begin{array}{l} b_{0} & = & \bar{Y} - b_{1} \bar{X} \\ b_{0} & = & 7 - 1.25 (4) = 2 \end{array}$

Sum of squares of the error:

\begin{array}{l} SSE & = & Σ Y^{2} - b_{0} Σ Y - b_{1} Σ X Y \\ SSE & = & 316.5 - 2 (42) - 1.25 (180.5) = 6.875 \end{array}

$\begin{array}{l} SSE & = & Σ Y^{2} - b_{0} Σ Y - b_{1} Σ X Y \\ SSE & = & 316.5 - 2 (42) - 1.25 (180.5) = 6.875 \end{array}$

Estimate of the error variance:

\begin{array}{l} s^{2} & = & MSE & = & \frac{SSE}{n - 2} \\ s^{2} & = & \frac{6.875}{6 - 2} & = & 1.71875 \end{array}

$\begin{array}{l} s^{2} & = & MSE & = & \frac{SSE}{n - 2} \\ s^{2} & = & \frac{6.875}{6 - 2} & = & 1.71875 \end{array}$

Estimate of the error standard deviation:

\begin{array}{l} s & = & \sqrt{MSE} \\ s & = & \sqrt{1.71875} = 1.311 \end{array}

$\begin{array}{l} s & = & \sqrt{MSE} \\ s & = & \sqrt{1.71875} = 1.311 \end{array}$

Table 4.7 Preliminary Calculations for Triple A Construction

Y	X	Y²	X²	XY
6	3	$6^{2} = 36$ $6^{2} = 36$	$3^{2} = 9$ $3^{2} = 9$	$3 (6) = 18$ $3 (6) = 18$
8	4	$8^{2} = 64$ $8^{2} = 64$	$4^{2} = 16$ $4^{2} = 16$	$4 (8) = 32$ $4 (8) = 32$
9	6	$9^{2} = 81$ $9^{2} = 81$	$6^{2} = 36$ $6^{2} = 36$	$6 (9) = 54$ $6 (9) = 54$
5	4	$5^{2} = 25$ $5^{2} = 25$	$4^{2} = 16$ $4^{2} = 16$	$4 (5) = 20$ $4 (5) = 20$
4.5	2	${4.5}^{2} = 20.25$ ${4.5}^{2} = 20.25$	$2^{2} = 4$ $2^{2} = 4$	$2 (4.5) = 9$ $2 (4.5) = 9$
9.5	5	${9.5}^{2} = 90.25$ ${9.5}^{2} = 90.25$	$5^{2} = 25$ $5^{2} = 25$	$5 (9.5) = 47.5$ $5 (9.5) = 47.5$
$\begin{array}{l} \sum Y & = & 42 \\ \bar{Y} & = & 42 / 6 = 7 \end{array}$ $\begin{array}{l} \sum Y & = & 42 \\ \bar{Y} & = & 42 / 6 = 7 \end{array}$	$\begin{array}{l} \sum X & = & 24 \\ \bar{X} & = & 24 / 6 = 4 \end{array}$ $\begin{array}{l} \sum X & = & 24 \\ \bar{X} & = & 24 / 6 = 4 \end{array}$	$\sum Y^{2} = 316.5$ $\sum Y^{2} = 316.5$	$\sum X^{2} = 106$ $\sum X^{2} = 106$	$\sum X Y = 180.5$ $\sum X Y = 180.5$

Coefficient of determination:

\begin{array}{l} r^{2} & = & 1 - \frac{SSE}{Σ Y^{2} - n {\bar{Y}}^{2}} \\ r^{2} & = & 1 - \frac{6.875}{316.5 - 6 (7^{2})} = 0.6944 \end{array}

$\begin{array}{l} r^{2} & = & 1 - \frac{SSE}{Σ Y^{2} - n {\bar{Y}}^{2}} \\ r^{2} & = & 1 - \frac{6.875}{316.5 - 6 (7^{2})} = 0.6944 \end{array}$

This formula for the correlation coefficient automatically determines the sign of r. This could also be found by taking the square root of $r^{2}$ $r^{2}$ and giving it the same sign as the slope:

\begin{array}{l} r & = & \frac{n \sum X Y - \sum X \sum Y}{\sqrt{[n \sum X^{2} - {(\sum X)}^{2}] [n \sum Y^{2} - {(\sum Y)}^{2}]}} \\ r & = & \frac{6 (180.5) - (24) (42)}{\sqrt{[6 (106) - 24^{2}] [6 (316.5) - 42^{2}]}} = 0.833 \end{array}

$\begin{array}{l} r & = & \frac{n \sum X Y - \sum X \sum Y}{\sqrt{[n \sum X^{2} - {(\sum X)}^{2}] [n \sum Y^{2} - {(\sum Y)}^{2}]}} \\ r & = & \frac{6 (180.5) - (24) (42)}{\sqrt{[6 (106) - 24^{2}] [6 (316.5) - 42^{2}]}} = 0.833 \end{array}$

..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.

Table of Contents for Appendix 4.1: Formulas for Regression Calculations

Create new playlist

Sign In

Sign Up

Table of Contents for
Appendix 4.1: Formulas for Regression Calculations