Introductory Biostatistics

8.1.6 Analysis-of-Variance Approach

The variation inY is conventionally measured in terms of the deviations
ðYi
YÞ; the total variation, denoted by SST, is the sum of squared deviations:

SST¼

X

ðYi
YÞ^2

For example, SST¼0 when all observations are the same. SST is the numera-
tor of the sample variance ofY. The larger the SST, the greater the variation
amongYvalues.
When we use the regression approach, the variation inYis decomposed into
two components:

Yi
Y¼ðYi
YY^iÞþðYY^i
YÞ

1. The first term reflects thevariation around the regression line; the part
   that cannot be explained by the regression itself with the sum of squared
   deviations:

SSE¼

X

ðYi
YY^iÞ^2

called theerror sum of squares.

2. The di¤erence between the two sums of squares,

SSR¼SST
SSE

¼

X

ðYY^i
YÞ^2

is called theregression sum of squares. SSR may be considered a measure

of the variation inYassociated with the regression line. In fact, we can

express the coe‰cient of determination as

r^2 ¼

SSR

SST

Corresponding to the partitioning of the total sum of squares SST, there is
partitioning of the associated degrees of freedom (df). We haven
1 degrees of
freedom associated with SST, the denominator of the sample variance ofY;
SSR has 1 degree of freedom representing the slope, the remainingn
2 are
associated with SSE. These results lead to the usual presentation of regression
analysis by most computer programs:

1. Theerror mean square,

MSE¼

SSE

n
 2

292 CORRELATION AND REGRESSION