Partitioning the Sums of Squares

Prerequisites
Introduction to Linear Regression

Learning Objectives

  1. Compute the sum of squares Y
  2. Convert raw scores to deviation scores
  3. Compute predicted scores from a regression equation
  4. Partition sum of squares Y into sum of squares predicted and sum of squares error
  5. Define r2 in terms of sum of squares explained and sum of squares Y.

One useful aspect of regression is that it can divide the variation in Y into two parts: the variation of the predicted scores and the variation in the errors of prediction. The variation of Y is called the sum of squares Y and is defined as the sum of the squared deviations of Y from the mean of Y. In the population, the formula is

where SSY is the sum of squares Y, Y is an individual value of Y, and my is the mean of Y. A simple example is given in Table 1. The mean of Y is 2.06 and SSY is the sum of the values in third column and is equal to 4.597.

Table 1. Example of SSY.
Y
Y-my
(Y-my)2
1.00
2.00
1.30
3.75
2.25

-1.06
-0.06
-0.76
1.69
0.19

1.1236
0.0036
0.5776
2.8561
0.0361

When computed in a sample, you should use the sample mean, M, in place of the population mean.

It is sometimes convenient to use formulas that use deviation scores rather than raw scores. Deviation scores are simply deviations from the mean. By convention, small letters rather than capitals are used for deviation scores. Therefore the score, y, indicates the difference between Y and the mean of Y. Table 2 shows the use of this notation. The numbers are the same as in Table 1.

Table 2. Example of SSY using Deviation Scores.
Y
y
y2
1.00
2.00
1.30
3.75
2.25

-1.06
-0.06
-0.76
1.69
0.19

1.1236
0.0036
0.5776
2.8561
0.0361

The data in Table 3 are reproduced from the introductory section. The column X, has the values of the predictor variable and the column Y has the criterion variable. The third column, y, contains the the differences between the column Y and the mean of Y.

Table 3. Example data.
X
Y
y
y2
Y'
y'
y'2
Y-Y'
(Y-Y')2
 

1.00
2.00
3.00
4.00
5.00

1.00
2.00
1.30
3.75
2.25
-1.06
-0.06
-0.76
1.69
0.19
1.1236
0.0036
0.5776
2.8561
0.0361

1.210
1.635
2.060
2.485
2.910

-0.850
-0.425
0.000
0.425
0.850
0.7225
0.1806
0.0000
0.1806
0.7225

-0.210
0.365
-0.760
1.265
-0.660

0.044
0.133
0.578
1.600
0.436

sum
15.00
10.30
0.00
4.597
10.300
0.000
1.806
0.000
2.791


The fourth column, y2, is simply the square of the y column. The column Y' contains the predicted values of Y. In the introductory section it was shown that the equation for the regression line for these data is

Y' = 0.425X + 0.785.

The values of Y' were computed according to this equation. The column y' contains deviations of Y' from the mean Y' and y'2 is the square of this column. The next-to-last column, Y-Y' contains the actual scores (Y) minus the predicted scores (Y'). The last column contains the squares of these errors of prediction.

We are now in position to see how the SSY is partitioned. Recall that SSY is the sum of the squared deviations from the mean. It is therefore the sum of the y2 column and is equal to 4.597. SSY can be partitioned into two parts: the sum of squares predicted (SSY') and the sum of squares error (SSE). The sum of squares predicted is the sum of the squared deviations of the predicted scores from the mean predicted score. In other words, it is the sum of the y'2 column and is equal to 1.806. The sum of squares error is the sum of the squared errors of prediction. It is there fore the sum of the (Y-Y')2 column and is equal to 2.791. This can be summed up as:

SSY = SSY' + SSE
4.597 = 1.806 + 2.791

There are several other notable features about Table 3. First, notice that the sum of y and the sum of y' are both zero. This will always be the case because these variables were created by subtracting their respective means from each value. Also notice that the mean of Y-Y' is 0. This indicates that although some Y's are higher than there respective Y's and some are lower, the average difference is zero.

The SSY is the total variation, SSY' is the variation explained, and the SSE is the variation unexplained. Therefore, the proportion of variation explained can be computed as:

Proportion explained = SSY'/SSY

Similarly, the proportion not explained is:

Proportion not explained = SSE/SSY

There is an important relationship between the proportion of variation explained and Pearson's correlation: r2 is the proportion of variation explained. Therefore, if r = 1, then, naturally, the proportion of variation explained is 1; if r = 0, then the proportion explained is 0. One last example: for r = 0.4, the proportion of variation explained is 0.16.

Since the variance is computed by dividing the variation by N (for a population) or N-1 (for a sample), the relationships spelled out above in terms of variation also hold for variance. For example,

where the first term is the variance total, the second term is the variance of Y' and the last term is the variance of the errors of prediction (Y-Y'). Similarly, r2 is the proportion of variance explained as well as the proportion of variation explained.