Regression analysis would be inappropriate when there are only ten cases and is
performed here for illustrative purposes only. Most statistical analysis programmes
handle the tedious calculations for regression and correlation and the researcher would
seldom have need to compute sums of squares and regression weights from raw data.
However, it is important that you understand how these quantities are derived because
sums of squares and mean squares appear in the regression and ANOVA output of many
propriety statistical packages.
Significance tests and confidence intervals for the slope of the line as well as
confidence intervals for the mean value of the response variable, for a given value of
xi, and a prediction interval can be calculated once the regression model has been
determined.
1 Computation of Sums of Squares and Cross Product Sums of SquaresThe total corrected sums of squares for Y is given by:
Total
correct
ed
Sums of
Squares
for Y—
8.3The corrected sums of squares is the same as the sum of squared deviations about the
mean (recall the idea of least squares regression to sum the squared deviations about the
regression line, which goes through the mean of Y).
A correction term, is subtracted from the total sums of squares to give the
corrected total sums of squares. This is done to correct for the fact that the original values
of the variable Y were not expressed as deviations from the mean of Y. To calculate the
corrected sums of squares the following values are required: ΣY^2 and ΣY. The terms ΣX^2
and ΣX will also be required for computing the corrected sums of squares for X (SSXX),
and ΣXY will be needed to compute the corrected sums of squares for cross products XY
(SSXY). These values are most easily computed if a table is set out as follows:
VARIABLES
Y Y^2 X X^2 XY
110 12100 5 25 550
133 17689 10 100 1330
109 11881 5 25 545
114 12996 3 9 342
128 16384 8 64 1024
109 11881 5 25 545
119 14161 8 64 952
Inferences involving continuous data 261