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║ ███ ▄ ███ ║  CHAPTER 4
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║ ▄▄▄▄▄▄▄▄▄ ║  ELEMENTARY REGRESSION
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  The "Elementary Regression" module of FREESTAT is provided for
those who may need to conduct a simple regression analysis by
entering data from the keyboard.  Thus, it can be a very handy
labor saver for those quick and dirty jobs.  In addition to the
simple regression procedures, you may also conduct multiple
regression using two independent variables.


                    VARIABLE TRANSFORMATIONS

  The regression procedures will also permit data transformations
if you elect to use raw data input.  The permissible
transformations are:

     A  X = X + a         a = Numerical constant
     B  X = X * b         b = Numerical constant
     C  X = 1/X           Reciprocal transformation
     D  X = SQRT(X)       Square root transformation
     E  X = LN(X)         Naperian logarithm
     F  X = LN10(X)       Common logarithm
     G  X = ARCSIN(X)     Arcsin transformation
     H  X = LOGIT(X)      Logit transformation
     I  X = PROBIT(X)     Probit transformation
     J  X = X^y           Power of y = Numerical constant  
     K  Cancel transformations

  When you choose the option to transform a variable, the above
transformations will be presented to you for selection.  The
transformations will be carried out in the order in which you
choose them.  It is very important that you understand how that
works. 

  Suppose, for example, that you want to transform X such that X
= 14 + 3 * X.  You would first choose Option B and then enter the
value of 3.  You would then choose Option A and then enter the
value of 14.  The resulting transformation would then be:

     X = (X * 3) + 14

  Now consider what would happen if you chose Option A first and
then Option B.  If you chose Option A and then entered 14 you
would have X = X + 14.  If you then chose Option B and entered
the value of 3, you would have the transformation,

     X = (X + 14) * 3

which would be incorrect.  Just remember, all transformations are
executed in the order they are selected.


                      BIVARIATE REGRESSION

  The bivariate regression procedure enables you to conduct
simple linear regression analyses of the form:

  Simple linear regression -- Y = a + bX + e
  Geometric regression --     Y = a * X^b, and
  Exponential regression --   Y = a * b^X

  The following is a sample of the output generated by the
bivariate regression procedure using summary data and simple
linear regression.

Simple Linear Regression
 Mean of Y = 34.90000
 Mean of X = 18.30000
 SD of Y   = 12.80000
 SD of X   = 8.30000 
 r         = 0.64000 
 N         = 300     
 Sum y^2   = 48988.16000
 Sum x^2   = 20598.11000
 Sum xy    = 20330.08640

Regression of Y on X
 Slope            b = 0.98699
 Correlation      r = 0.64000
 Explained SS   ESS = 20065.55034
 Residual SS    RSS = 28922.60966
 Std error       SE = 9.85169
            F-ratio = 206.74255
                 p <= 0.00000

    95% Confidence Intervals
    15.72330 <= a <= 17.95294
     0.85245 <= b <= 1.12153
     0.56792 <= r <= 0.70234

Regression of X on Y
 Intercept        a = 3.81650
 Slope            b = 0.41500
 Correlation      r = 0.64000
 Explained SS   ESS = 8436.98586
 Residual SS    RSS = 12161.12414
 Std error       SE = 6.38820
            F-ratio = 206.74255
                 p <= 0.00000

    95% Confidence Intervals
     3.09361 <= a <= 4.53939
     0.32776 <= b <= 0.50224
     0.56792 <= r <= 0.70234

Estimated Values
For the model, Y' = a + bX
 X  = 27.00000
 Y' = 43.48680
       95% Interval for Mean
    41.87035 <= Y' <= 45.10324
95% Interval for Individual Value
    24.10995 <= Y' <= 62.86364

For the model, Y' = a + bX
 X  = 11.00000
 Y' = 27.69499
       95% Interval for Mean
    26.20924 <= Y' <= 29.18073
95% Interval for Individual Value
     8.32861 <= Y' <= 47.06137


                       MULTIPLE REGRESSION

  As indicated above, the multiple regression procedure allows
you to enter two independent variables and one dependent
variable.  Once you have obtained your fitted model, you may
elect to produce estimated values of your dependent variable, Y',
for values of X1 and X2 that you wish to enter.

  The following is a sample of the output generated by the
multiple regression procedure.

Summary Data
 Mean of Y  = 25.00000
 Mean of X1 = 47.00000
 Mean of X2 = 83.00000
 SD of Y    = 11.00000
 SD of X1   = 21.00000
 SD of X2   = 28.00000

 N          = 290
 Sum y^2    = 34969.00000
 Sum x1^2   = 127449.00000
 Sum x2^2   = 226576.00000

 Sum yx1    = 22698.06000
 Sum yx2    = 37385.04000
 Sum x1x2   = 28888.44000

 Multiple R = 0.50069
 R-Square   = 0.25069
 r(y,x1)    = 0.34000
 r(y,x1)^2  = 0.11560
 r(y,x2)    = 0.42000
 r(y,x2)^2  = 0.17640
 r(x1,x2)   = 0.17000
 r(x1,x2)^2 = 0.02890
 F = 96.35516    p <= 0.0000

Semi-partial & partial correlations
 sr(y,x1)    = 0.27257
 sr(y,x1)^2  = 0.07429
 sr(y,x2)    = 0.36755
 sr(y,x2)^2  = 0.13509
 pr(y,x1)    = 0.30034
 pr(y,x1)^2  = 0.09021
 pr(y,x2)    = 0.39083
 pr(y,x2)^2  = 0.15275

Raw Score Model
      a = 6.02875
     b1 = 0.14488
     b2 = 0.14653

Standardized Model
 Beta 1 = 0.27659
 Beta 2 = 0.37298

95 Percent Confidence Intervals
   0.40877 <=     R    <= 0.58256
   0.23379 <= r( y,x1) <= 0.43819
   0.32013 <= r( y,x2) <= 0.51064
   0.05571 <= r(x1,x2) <= 0.27989
   0.16209 <=    sr1   <= 0.37629
   0.26315 <=    sr2   <= 0.46344
   0.19140 <=    pr1   <= 0.40197
   0.28827 <=    pr2   <= 0.48453

 ssy        = 34969.00000
 ssy'       = 8766.48451
 ssx1       = 2597.95291
 ssx2       = 4724.06811
 sse        = 26202.51549

 Std Err y  = 9.55500
     SE b1  = 0.05630
     SE b2  = 0.05433

   4.92901 <=    a1    <= 7.12848
   0.08708 <=    b1    <= 0.20269
   0.10469 <=    b2    <= 0.18836

Estimated Values
For the model Y'= a + b1X1 +b2X2
X1 = 37.00000
X2 = 44.00000
Y' = 17.83660

For the model Y'= a + b1X1 +b2X2
X1 = 57.00000
X2 = 69.00000
Y' = 24.39744


                         END OF CHAPTER
