CHAPTER 9 ELEMENTARY REGRESSION The "Elementary Regression" module of SPPC is provided for those who may need to conduct a simple regression analysis by entering data from the keyboard. It saves the trouble of creating a raw data file and then using the multiple regression module. 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. DATA REQUIREMENTS The elementary regression module permits data entry only from the keyboard. However, you may elect to enter either raw or summary data. If you elect to use summary data input, you will need to have the mean and standard deviation for all variables as well as the sample size and the correlations among all 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 understand how that works. Suppose, for example, that you wanted 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