Partitioning the Sums of Squares

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One useful aspect of regression is that it can divide the variation in Y into two parts:

  • the variation of the predicted scores
  • the variation in the errors of prediction

The variation of Y

  • the sum of squares Y
  • defined as the sum of the squared deviations of Y from the mean of Y

Formula of the Variation of Y

In the population, the formula of The variation of Y

ClipCapIt-140603-230127.PNG
where SSY is the sum of squares Y and 
Y is an individual value of Y, and my is the mean of Y

Example

The mean of Y is 2.06 and SSY is the sum of the values in third column and is equal to 4.597

ClipCapIt-140603-230748.PNG

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

ClipCapIt-140603-230127.PNG


Example

ClipCapIt-140603-230854.PNG
  • The column Y' were computed according to this equation.
  • The column y' contains deviations of Y' from the mean Y'
  • The column y'2 is the square of this column.
  • The column Y-Y' contains the actual scores (Y) minus the predicted scores (Y')
  • The column (Y-Y')2 contains the squares of these errors of prediction


Sum of the squared deviations from the mean

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:
1. the sum of squares predicted (SSY')
  • 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
2.the sum of squares error (SSE)
  • 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

Example

ClipCapIt-140603-231354.PNG

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.
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.
SSY is the total variation
SSY' is the variation explained
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


r2

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 the proportion of variation explained is 1
  • if r = 0, then the proportion explained is 0;
  • if r = 0.4, then 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

Example

ClipCapIt-140603-231640.PNG
  • the first term is the variance total
  • the second term is the variance of Y'
  • 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.

Summary Table

It is often convenient to summarize the partitioning of the data in a table.

  • The degrees of freedom column (df) shows the degrees of freedom for each source of variation.
  • The degrees of freedom for the sum of squares explained is equal to the number of predictor variables.
  • This will always be 1 in simple regression.
  • The error degrees of freedom is equal to the total number of observations minus 2.
  • In this example, it is 5 - 2 = 3.
  • The total degrees of freedom is the total number of observations minus 1.
Source Sum of Squares df Mean Square
Explained 1.806 1 1.806
Error 2.791 3 0.930
Total 4.597 4

Quiz

1

If these data are converted to deviation scores, the last value (15) would have a value of

Y
 2
 9
11
13
15

Answer >>

15

To compute a deviation score you subtract the mean. 15 - 10 is 5.


2

Compute the sum of squares Y.

Y
 2
 9
11
13
15

Answer >>

100

To compute SSY, first compute the deviation scores (y) by subtracting the mean (10) from each number. Then square these values and add them together: (-8)2 + (-1)2 + 12 + 32 + 52 equals to 100


3

If SSY is 25.5 and SSY' is 18.3, what is SSE?

Y
 2
 9
11
13
15

Answer >>

7.2

SSY is SSY' + SSE; SSE is SSY - SSY' 25.5 - 18.3 equals to 7.2


4

The larger ________ is, the larger the proportion of variation explained is.

SSY
SSY'
SSE
Y

Answer >>

False

Proportion of variation explained is SSY'/SSY, so as SSY' increases, so does the proportion of variation explained.


5

The proportion of variation explained is 0.3. If SSY is 20, what is SSY'?

Answer >>

6

Proportion explained is SSY'/SSY; SSY' is (.3)(20) equals to 6


6

If r is .84, what proportion of variation is explained?

Answer >>

0.71

r2 is the proportion of variation explained. (.84)2 is .71