y = a + bx {where a is constant}
This could be extended to a multi-variable concept as follows:
y = a + b1x1 + b2x2 + b3x3 + ..... +bnxn
It should be noted that whether it be for a single variable or for multiple variables, the relationship predicted is always linear.
A graphic explanation of a Regression Analysis
A simple approach to approximate a regression equation for a single variable is to plot the relationship between the variables. The task requires that we first plot the dependent variable against the independent variable. This type of plotting is called the scatter diagram.
Next, identify the straight line that represents the trend through the mid-point of the data, a trend which has the `best fit'. The use of the trend in the regression analysis identifies the relationship between the independent and dependent variables. The relationship, thus identified, is used to predict the various values of the dependent variable, given specific values of the independent variable. This predicted relationship is always in the form of a linear trend.
The table below identifies a set of values for an independent (X) and dependent (Y) variable.
+---------------------------------------------------------------+ ¦ X ¦ 39 ¦ 43 ¦ 21 ¦ 64 ¦ 57 ¦ 47 ¦ 28 ¦ 75 ¦ 34 ¦ 52 ¦ +---+------+-----+-----+-----+-----+-----+-----+-----+-----+----¦ ¦ Y ¦ 68 ¦ 82 ¦ 56 ¦ 86 ¦ 97 ¦ 94 ¦ 77 ¦ 103 ¦ 59 ¦ 79 ¦ +---------------------------------------------------------------+
The scatter plot of the variables is given below:
In case of a multi-variable problem, the regression equation is arrived at in a sequence of multiple linear regression equations, in a stepwise manner. At each step of the sequence, one variable is added to the regression equation. The variable added is the one that makes the greatest reduction in the error sum of squares of the sample data. Equivalently it is the variable that when added, provides the greatest increase in the F value. Variables not having a significant correlation with the dependent variable, are those whose addition does not increase the F value and are not featured in the regression equation.
I. With one independent Variable: The Mathematical Computation of the Regression Coefficients for the case of a single independent variable is given below:
The slope (regression coefficient) for the line of least squares is given by b, where
The intercept of the line is given by a, where
The mathematical formula used for this computation is as follows:
The Residual : The residual is defined as the difference between the actual and predicted values of the dependent variable. The standard error of the estimate is the standard deviation of the residuals. The standard error of the estimate can be calculated as follows:
A Numerical Example: One dependent variable
Let us use the data which produced the above graphical representation of a regression analysis.
+------------------------------------------------+ ¦SL.No¦ y ¦ x ¦ xy ¦ y² ¦ x² ¦ ¦-----+-------+-------+--------+--------+--------¦ ¦ 1 ¦ 68 ¦ 39 ¦ 2652 ¦ 4624 ¦ 1521 ¦ ¦ 2 ¦ 82 ¦ 43 ¦ 3526 ¦ 6724 ¦ 1849 ¦ ¦ 3 ¦ 56 ¦ 21 ¦ 1176 ¦ 3136 ¦ 441 ¦ ¦ 4 ¦ 86 ¦ 64 ¦ 5504 ¦ 7396 ¦ 4096 ¦ ¦ 5 ¦ 97 ¦ 57 ¦ 5529 ¦ 9409 ¦ 3249 ¦ ¦ 6 ¦ 94 ¦ 47 ¦ 4418 ¦ 8836 ¦ 2209 ¦ ¦ 7 ¦ 77 ¦ 28 ¦ 2156 ¦ 5929 ¦ 784 ¦ ¦ 8 ¦ 103 ¦ 75 ¦ 7725 ¦ 10609 ¦ 5625 ¦ ¦ 9 ¦ 59 ¦ 34 ¦ 2006 ¦ 3481 ¦ 1156 ¦ ¦ 10 ¦ 79 ¦ 52 ¦ 4108 ¦ 6241 ¦ 2704 ¦ ¦-----+-------+-------+--------+--------+--------¦ ¦SUM ¦ 801 ¦ 460 ¦ 38800 ¦ 66385 ¦ 23634 ¦ ¦AVG. ¦ 80.1¦ 46 ¦ 3888 ¦ 6638.5 ¦ 2363.4 ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ +------------------------------------------------+
Therefore, the slope is given by:
and the intercept is given by :
Hence the line of best fit is given by :
Y = 43.768553 + 0.789814 X
As an alternate method of deriving the regression equation, a spreadsheet could be used. The line for a single variable regression was derived by using the Lotus spreadsheet. The output from Lotus for the above data set is given below:
+-----------------------------------------+
¦ Regression Output: ¦
¦-----------------------------------------¦
¦ Constant 43.76855 ¦
¦ Std Err of Y Est 9.230407 ¦
¦ R Squared 0.693647 ¦
¦ No. of Observations 10 ¦
¦ Degrees of Freedom 8 ¦
¦ X Coefficient(s) 0.788348 ¦
¦ X Coefficient(s) 0.789814 ¦
+-----------------------------------------+
The Mathematical Computation of the Regression Coefficients in the case of more than one independent variables involves matrix computations. A brief result is given below:
Let X be the data matrix of the predictor (independent) variables. Y is the data vector representing the criterion (dependent) variable and b is the data vector representing the regression coefficients including the constants. The vector of regression coefficients is computed as
b = (X'X)-1X'Y
+------------------------------------+
¦ Y ¦ X0 ¦ X1 ¦ X2 ¦
¦------------+------+-------+--------¦
¦ 4.50 ¦ 1 ¦ 8.00 ¦ 2.00 ¦
¦ 22.50 ¦ 1 ¦ 40.50 ¦ 24.50 ¦
¦ 2.00 ¦ 1 ¦ 4.50 ¦ 0.50 ¦
¦ 0.50 ¦ 1 ¦ 0.50 ¦ 2.00 ¦
¦ 18.00 ¦ 1 ¦ 4.50 ¦ 4.50 ¦
¦ 2.00 ¦ 1 ¦ 7.00 ¦ 8.00 ¦
¦ 32.00 ¦ 1 ¦ 24.50 ¦ 40.50 ¦
¦ 4.50 ¦ 1 ¦ 4.50 ¦ 2.00 ¦
¦ 40.50 ¦ 1 ¦ 32.00 ¦ 24.50 ¦
¦ 2.00 ¦ 1 ¦ 0.50 ¦ 4.50 ¦
¦ ¦ ¦ ¦ ¦
+------------------------------------+
¦ 1 1 1 1 1 1 1 1 1 1 ¦
X' = ¦ 8.00 40.50 4.50 0.50 4.50 7.00 24.50 4.50 32.00 0.50 ¦
¦ 2.00 24.50 0.50 2.00 4.50 8.00 40.50 2.00 24.50 4.50 ¦
X'X = ¦ 10 126.5 113 ¦ ¦ 0.190 -0.00 -0.00 ¦
¦ 126.5 3438.75 2875.25 ¦ (X'X)-1 =¦ -0.00 0.001 -0.00 ¦
¦ 113 2875.25 2957.5 ¦ ¦ -0.00 -0.00 0.001¦
¦ 128.5 ¦
X'Y = ¦ 3152.7 ¦
¦ 2965.5 ¦
¦1.564¦
B = (X'X)-1 X'Y = ¦ .378¦
¦ .574¦
The PC-MDS Command File. The command file defines the various variables, their format and locations, defines missing values for variables and recodes the values of the variables, if desired. For purposes of clarification, the command files are designated as files with an "SPS" extension (i.e., *.SPS). As an example command file, we shall refer to the command file for the hospital data. The file is designated REGRESS.SPS and is listed in the appendix. The name of the PC-MDS command file is specified interactively by the user when each program is run. (Note that the .SPS designation is used for instructional clarity only. The command file may have any name and does not, in reality, require the .SPS extension).
As an example for a command file for a regression analysis we shall refer to the file REGRESS.SPS:
TITLE BMD02R STEPWISE REGRESSION TEST DATA
FILE NAME 'REGRESS.DAT'
DATA LIST V1 TO V6
6 (F6.2,F6.0,2F6.2,2F6.0)
1) Stepwise input of independent variables
VARIABLE LABELS
V1 'AGE'
V2 'WEIGHT'
V3 'VARIABLE 3'
V4 'HEIGHT'
V5 'STATUS'
V6 'DEPENDENT'
The Data File: The data file contains the data in the format described in the Command File. The data files are usually named with a "DAT" extension (i.e., *.DAT). The example data file for the hospital data is called REGRESS.DAT and is given in the appendix. The data file is specified in line 2 of the command file (the FILENAME command). As an example for a data file for a regression analysis we shall refer to the file REGRESS.DAT given below.
00025 00025 02500 00150 00034 00064 01300 00021 02100 00087 00036 00065 00350 00022 02200 00043 00041 00082 00175 00009 00130 00180 00015 00023 00300 00023 02300 00200 00033 00064 00200 00010 00060 00330 00013 00016 00550 00007 00140 00340 00016 00012 00600 00006 00080 00500 00011 00027 00130 00008 00270 00150 00019 00048 00500 00018 00360 00180 00027 00050 ----------------------------------- ...........CONTINUES........... ----------------------------------- 00500 00022 02200 00120 00039 00100 00100 00015 00500 00080 00029 00050 01700 00009 00300 01300 00010 00080 00500 00030 03500 00090 00058 00065 00130 00010 00130 00900 00010 00025
The Output File: The output file must be specified by the user while running each of PC-MDS programs. The output file is the file to which the Regression Analysis is printed. A common convention is to name the file with a "PRN" extension to signify a print file. (i.e.*.PRN) For the regression analysis, the output file contains the following output.
HOW TO RUN THE REGRESS PROGRAM
STEP 1: Enter the EDITOR (a word processor or program editor that produces ASCII files will suffice), and prepare the command file and the data file.
STEP 2: Load the REGRESS program. The program is loaded by simply typing REGRESS and then pressing the [ENTER] key.
A> REGRESS [ENTER]
STEP 3: After the initial logo identifying the program, a message will appear on the screen requesting the location and name of the command file.
ENTER THE NAME OF THE PC-MDS COMMAND FILE USE THE FORM: DRV:FILENAME.EXT (e.g. B:STAT.SPS)
A:REGRESS.SPS |
RESPOND with the location and name of the command file:
A:REGRESS.SPS [ENTER]
(Assumes the REGRESS.SPS file is in the A: drive). If the specifications of the command file name was not acceptable, then a message will ask you to re-enter the command file name.
STEP 4: If the command file name was specified correctly, the next menu item will pop up asking you to specify the location and name of the output file.
|
ENTER THE NAME OF THE FILE TO SAVE OUTPUT USE THE FORM: DRV:FILENAME.EXT(e.g. B:STAT.PRN)
A:REGRESS.PRN |
Enter the name of the output file:
A:REGRESS.PRN [ENTER]
(Assumes you want the output file REGRESS.PRN written to the A: drive). If a file already exists with the same name, then the message will appear on screen:
|
THIS OUTPUT FILE NAME ALREADY EXISTS! DO YOU WANT TO OVERWRITE IT? (Y/N) Y |
STEP 5: The program then asks the type of data being studied.
|
|
|
|
The program next reads the first line of data, displays the input format for reading the data, and lists the values for the first data case. If the data is read incorrectly, you may re-specify the format statement. After you indicate that the data was read correctly, the program proceeds with the regression analysis of the data.
+---------------------------------------------------------------------------+ ¦ STMT# #VARIABLES FORMAT STATEMENT AND DATA ¦ ¦---------------------------------------------------------------------------¦ ¦ 1 6 ¦ ¦ (F6.2,F6.0,2F6.2,2F6.0) ¦ +---------------------------------------------------------------------------+ +---------------------------------------------------------------------------+ ¦ 2.500000e+000 2.500000e+001 2.500000e+001 1.500000e+000 3.400000e+001 ¦ ¦ 6.400000e+001 ¦ ¦ ¦ ¦ ¦ +---------------------------------------------------------------------------+
|
WAS THE DATA READ CORRECTLY? Y
|
STEP 6: Once the output file name is correctly entered, the initial computations required for reading the command file take place. Initial error messages associated with the command file, if any, will be displayed:
If errors are found, the program aborts. It is recommended that the user makes a note of the errors. The user must edit the Command file to correct the errors. The Regression program may then be rerun.
If there were no errors then the option to SPECIFY THE DEPENDENT VARIABLE will appear on screen. Specify the dependent variable and press enter.
+-------------------------------------------------+ ¦ THE DEPENDENT VARIABLE IS THE Y VARIABLE IN THE¦ ¦ FORMULA: Y = A + B1X1 + ... + BkXk ¦ ¦ YOU ARE PREDICTING THE DEPENDENT VARIABLE, Y. ¦ ¦ ¦ ¦ +--------------------------------------------+ ¦ ¦ ¦ ENTER THE DEPENDENT VARIABLE FOR REGRESSION¦ ¦ ¦ +--------------------------------------------+ ¦ ¦ 6 ¦ +-------------------------------------------------+
STEP 7: The following message for choosing the independent variables will appear.
+-----------------------------------------------------+ ¦ REGRESSION ANALYSIS PROGRAM OPTIONS: ¦ ¦ ¦ ¦ 6 VARIABLES HAVE BEEN DECLARED. ¦ ¦ SELECT THE APPROPRIATE OPTION: ¦ ¦ ¦ ¦ (1) SPECIFY THE VARIABLES FOR ANALYSIS ¦ ¦ (VARIABLES ARE SPECIFIED BY SEQUENCE NUMBER) ¦ ¦ (2) VIEW A LIST OF VARIABLE NUMBERS ¦ ¦ (3) QUIT PROGRAM ¦ ¦ ¦ ¦ YOUR CHOICE : 2 ¦ +-----------------------------------------------------+ +-----------------------------------------------------+ ¦SEQ# NAME VARIABLE LABEL ¦ +-----------------------------------------------------+ +-----------------------------------------------------+ ¦ 1 V1 VARIABLE 1 ¦ ¦ 2 V2 VARIABLE 2 ¦ ¦ 3 V3 VARIABLE 3 ¦ ¦ 4 V4 VARIABLE 4 ¦ ¦¦ +-----------------------------------------------------+
Option 2 was selected to VIEW THE VARIABLE LIST. Option 1 is then selected to specify the variables that are to be included in the analysis.
STEP 8: The option to SPECIFY THE INDEPENDENT VARIABLES will give the following message. Enter the independent variables you want to study and press enter. The variables selected are listed.
+-----------------------------------------------------+ ¦ REGRESSION ANALYSIS VARIABLES SPECIFICATION: ¦ ¦ ¦ ¦ ENTER VARIABLES ONE AT A TIME. ¦ ¦ A blank space must follow each variable number. ¦ ¦ The dash (-) may be used to simplify statements. ¦ ¦ PRESS ENTER to quit this menu ¦ ¦ For example, ¦ ¦ 1 2 3 4 5 and 1 - 5 are equivalent statements. ¦ ¦ ¦ ¦ 1 - 5 ¦ +-----------------------------------------------------+ +---------------------------------------------------------------------------+ ¦ SELECTED VARIABLES ¦ +---------------------------------------------------------------------------+ +---------------------------------------------------------------------------+ ¦ 1 V1 2 V2 3 V3 4 V4 5 V5 ¦ ¦ ¦ ¦ VARIABLES CORRECT? Y ¦ +---------------------------------------------------------------------------+
Once the command file has been read, the program prompts for the print option. You may specify that the correlation matrix be saved as a separate file.
+-------------------------------------------------+ ¦ THE CORRELATION MATRIX MAY BE SAVED TO BE USED ¦ ¦ IN FUTURE ANALYSES. (REDUCES PROCESSING TIME) ¦ ¦ ¦ ¦ PRESS ENTER TO CONTINUE WITHOUT SAVING ¦ ¦ +--------------------------------------------+ ¦ ¦ ¦ ENTER THE NAME OF THE CORRELATION MATRIX. ¦ ¦ ¦ +--------------------------------------------+ ¦ ¦ ¦ +-------------------------------------------------+
STEP 9: The following subproblem menu will next appear:
+----------------------------------------------------+ ¦ SUBPROBLEM MENU ¦ ¦ SELECT ITEM TO ENTER A NEW VALUE ¦ ¦ ¦ ¦ ITEM DESCRIPTION CURRENT VALUE ¦ ¦ --------------------------- --------------- ¦ ¦ 1 No changes, Start Analysis ¦ ¦ 2 Max Number of steps 12 ¦ ¦ 3 F Value for inclusion .010000 ¦ ¦ 4 F Value for deletion .005000 ¦ ¦ 5 Tolerance Level .001000 ¦ ¦ 6 Control Delete Option: YES ¦ ¦ 7 PRINT Residuals Option: NO ¦ ¦ 8 PLOT Residuals vs. X Variables: ¦ ¦ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ¦ ¦ ¦ ¦ ENTER CHOICE: ¦ +----------------------------------------------------+
NOTE: Tolerance level is also known as the Significance Level
STEP 10: The program begins computations. Once the statistical analysis is complete, the program prompts the user with an options menu. The QUIT PROGRAM option should be used to exit the regression program.
+-----------------------------------------------------+ ¦ STEPWISE REGRESSION ANALYSIS COMPLETE: ¦ ¦ ¦ ¦ PROGRAM OPTIONS: ¦ ¦ SELECT THE APPROPRIATE OPTION: ¦ ¦ ¦ ¦ (0) QUIT PROGRAM ¦ ¦ (1) START A NEW ANALYSIS (NEW COMMAND FILE) ¦ ¦ (2) FURTHER ANALYSIS WITH CURRENT DATA ¦ ¦ ¦ ¦ YOUR CHOICE : ¦ +-----------------------------------------------------+
When the analysis is complete, the output will be in the REGRESS.PRN file. Run the EDITOR or a word processing program to read the output file. The output file may be printed from the editor. A printed copy of the Regression Data and example follows.
PC-MDS
STEPWISE REGRESSION ANALYSIS
ANALYSIS TITLE STEPWISE REGRESSION
INPUT DATA FILE REGRESS.DAT
OUTPUT PRINT FILE TEST.PRN
NO. OF VARIABLES 6
DATA TREATED AS HAVING NO MISSING VALUES
DATA FOR RECORD: 1
.25E+01 .25E+02 .25E+02 .15E+01 .34E+02 .64E+02
DATA FOR RECORD: 68
.13E+01 .10E+02 .13E+01 .90E+01 .10E+02 .25E+02
VARIABLE MEAN STAND. DEV. MINIMUM MAXIMUM
V1 6.9956 6.47375 .25000 30.00000
V2 15.2500 9.35753 2.00000 38.00000
V3 10.4251 11.62703 .40000 38.00000
V4 3.0996 5.99713 .01000 48.00000
V5 25.3971 12.47940 5.00000 58.00000
V6 56.7941 43.55013 7.00000 208.00000
VARIANCE-COVARIANCE MATRIX
V1 .41909E+02
V2 -.10690E+02 .87563E+02
V3 .38646E+00 .94438E+02 .13519E+03
V4 .99188E+01 .56522E+01 .87765E+01 .35966E+02
V5 -.15809E+02 .10266E+03 .10910E+03-.10510E+02 .15574E+03
V6 .23134E+02 .30549E+03 .39803E+03 .89470E+02 .35064E+03 .18966E+04
1 2 3 4 5 6
Variance-Covariance Matrix : It is the matrix of the variance of the independent variables.
CORRELATION MATRIX
V1 .10000E+01
V2 -.17646E+00 .10000E+01
V3 .51343E-02 .86799E+00 .10000E+01
V4 .25548E+00 .10072E+00 .12587E+00 .10000E+01
V5 -.19569E+00 .87912E+00 .75194E+00-.14044E+00 .10000E+01
V6 .82056E-01 .74962E+00 .78606E+00 .34257E+00 .64517E+00 .10000E+01
1 2 3 4 5 6
Note: Correlation Matrix = It is the Matrix that gives the correlation between the dependent
and independent variables. This table is also useful for studying multi-collinearity,
(correlation between the independent variables).
N-MATRIX
V1 68
V2 68 68
V3 68 68 68
V4 68 68 68 68
V5 68 68 68 68 68
V6 68 68 68 68 68 68
1 2 3 4 5 6
Note: N-Matrix = It is the matrix of the number of counts of for each independent variable.
In this case, for the data set REGRESS.DAT, because there are 68 sets of values for each variable,
each value in the matrix is 68.
SUB-PROBLEM 1
DEPENDENT VARIABLE V6
MAXIMUM NUMBER OF STEPS 12
F-LEVEL FOR INCLUSION .010000
F-LEVEL FOR DELETION .005000
TOLERANCE LEVEL .001000
**********************************************************************
STEP NUMBER VARIABLE ACTION R-SQUARED
1 V3 ENTERED .6179
**********************************************************************
STEP NUMBER 1
VARIABLE ENTERED V3
MULTIPLE R .7861
STD. ERROR OF EST. 27.1238
ANALYSIS OF VARIANCE
DF1 SUM OF SQUARES MEAN SQUARE2 F-RATIO3 F-PROB4.
REGRESSION 1 78516.9400 78516.9400 106.7242 .0000
RESIDUAL 66 48556.1800 735.6997
VARIABLES IN EQUATION . VARIABLES NOT IN EQUATION
STD5. UNSTD6. . PARTIAL
VAR COEFF. COEFF STD. ERROR F TO REMOVE . VAR. CORR. TOLERANCE F TO ENTER
(CONST7 26.09981) .
V3 .78606 2.94426 .28500 106.7242 (2) . V1 .12622 1.0000 1.0523 (2)
. V2 .21932 .2466 3.2847 (2)
. V4 .39729 .9842 12.1821 (2)
. V5 .13276 .4346 1.1662 (2)
******************************************************************************
STEP NUMBER VARIABLE ACTION R-SQUARED8
2 V4 ENTERED .6782
******************************************************************************
1. The Degree of Freedom for the Regression Model also called the explained model is given by k,
where k = number of independent variables in the regression equation.
For the Residual, the error unexplained by the regression model, the Degree of Freedom is
given by (n-k-1), where n = number of counts of the independent variable in the data set.
2. Mean Square = (Sum of Squares)/(DF)
3. F Ratio = (Mean Square of the Regression)/(Mean Square of the Residual)
4. F-Prob = Level of significance corresponding to the F Value
5. Std. Coeff = Standardized Coefficient of Regression for the independent variable.
6. Unstd. Coeff = Unstandardized Coefficient of Regression for the independent variable.
7. Const = The Intercept of the Regression Equation
8. The R-Squared Value tells the percentage of the changes in the dependent variable that
can be explained by the regression equation.
STEP NUMBER 2
VARIABLE ENTERED V4
MULTIPLE R .8235
STD. ERROR OF EST. 25.0821
ANALYSIS OF VARIANCE
DF SUM OF SQUARES MEAN SQUARE F-RATIO F-PROB.
REGRESSION 2 86180.8500 43090.4300 68.4941 .0000
RESIDUAL 65 40892.2700 629.1118
.
VARIABLES IN EQUATION . VARIABLES NOT IN EQUATION
STD. UNSTD. . PARTIAL
VAR COEFF. COEFF STD. ERROR F TO REMOVE . VAR. CORR. TOLERANCE F TO ENTER
(CONST 21.74448) .
V3 .75490 2.82755 .26566 113.2843 (2) . V1 .02724 .9340 .0475 (2)
V4 .24755 1.79768 .51505 12.1821 (2) . V2 .24653 .2465 4.1414 (2)
. V5 .32179 .3784 7.3925 (2)
******************************************************************************
STEP NUMBER VARIABLE ACTION R-SQUARED
3 V5 ENTERED .7115
******************************************************************************
STEP NUMBER 3
VARIABLE ENTERED V5
MULTIPLE R .8435
STD. ERROR OF EST. 23.9328
ANALYSIS OF VARIANCE
DF SUM OF SQUARES MEAN SQUARE F-RATIO F-PROB.
REGRESSION 3 90415.1500 30138.3800 52.6177 .0000
RESIDUAL 64 36657.9700 572.7808
.
VARIABLES IN EQUATION . VARIABLES NOT IN EQUATION
STD. UNSTD. . PARTIAL
VAR COEFF. COEFF STD. ERROR F TO REMOVE . VAR. CORR. TOLERANCE F TO ENTER
.
(CONST 2.91075) .
V3 .52285 1.95840 .40797 23.0429 (2) . V1 .11111 .8832 .7875 (2)
V4 .31843 2.31239 .52665 19.2786 (2) . V2 .01574 .1134 .0156 (2)
V5 .29673 1.03553 .38086 7.3925 (2) .
******************************************************************************
STEP NUMBER VARIABLE ACTION R-SQUARED
4 V1 ENTERED .7151
******************************************************************************
STEP NUMBER 4
VARIABLE ENTERED V1
MULTIPLE R .8456
STD. ERROR OF EST. 23.9727
ANALYSIS OF VARIANCE
DF SUM OF SQUARES MEAN SQUARE F-RATIO F-PROB.
REGRESSION 4 90867.7400 22716.9400 39.5291 .0000
RESIDUAL 63 36205.3800 574.6885
.
VARIABLES IN EQUATION . VARIABLES NOT IN EQUATION
STD. UNSTD. . PARTIAL
VAR COEFF. COEFF STD. ERROR F TO REMOVE . VAR. CORR. TOLERANCE F TO ENTER .
(CONST -1.25280) .
V1 .06350 .42721 .48139 .7875 (2) . V2 .05241 .1029 .1708 (2)
V3 .50640 1.89677 .41451 20.9389 (2) .
V4 .30755 2.23335 .53500 17.4266 (2) .
V5 .32000 1.11674 .39232 8.1027 (2) .
******************************************************************************
STEP NUMBER VARIABLE ACTION R-SQUARED
5 V2 ENTERED .7159
******************************************************************************
STEP NUMBER 5
VARIABLE ENTERED V2
MULTIPLE R .8461
STD. ERROR OF EST. 24.1320
ANALYSIS OF VARIANCE
DF SUM OF SQUARES MEAN SQUARE F-RATIO F-PROB.
REGRESSION 5 90967.1800 18193.4400 31.2412 .0000
RESIDUAL 62 36105.9400 582.3538
VARIABLES IN EQUATION . VARIABLES NOT IN EQUATION
STD. UNSTD. . PARTIAL
VAR COEFF. COEFF STD. ERROR F TO REMOVE . VAR. CORR. TOLERANCE F TO ENTER .
(CONST -1.84213) .
V1 .07303 .49129 .50880 .9323 (2) .
V2 .08720 .40584 .98214 .1708 (2) .
V3 .46858 1.75512 .54002 10.5633 (2) .
V4 .29432 2.13728 .58658 13.2759 (2) .
V5 .27179 .94847 .56726 2.7956 (2) .
F-LEVEL OR TOLERANCE INSUFFICIENT FOR FURTHER COMPUTATION
SUMMARY TABLE
STEP VARIABLE MULTIPLE INCREASE F VALUE TO SUM OF SQ.
NUMBER ENTERED REMOVED R RSQ IN RSQ1 ENTER OR REMOVE ADDED
1 V3 .7861 .6179 .6179 106.7242 78516.9400
2 V4 .8235 .6782 .0603 12.1821 7663.9170
3 V5 .8435 .7115 .0333 7.3925 4234.2900
4 V1 .8456 .7151 .0036 .7875 452.6008
5 V2 .8461 .7159 .0008 .1708 99.4410
COMPLETION OF STEPWISE REGRESSION ANALYSIS
1. A variable is added as long as its addition contributes a positive increase in the
R-Square value of the model; i.e. as long as it meets the significant level of the test.