Statistical Methods in Medical Research 4ed

192 Regression and correlation
on substituting for b from (7.3).
Finally, it can be shown that an unbiased estimator of s2 is
s 2 0 ˆ
P 2
…y Y†
, …7:6†
n 2
the residual sum of squares, P …y Y† 2 , being obtainable from (7.5). The divisor
n 2 is often referred to as the residual degrees of freedom, s2 0 as the residual
mean square, and s0 as the standard deviation about regression.
The quantities a and b are called the regression coefficients; the term is often
used particularly for b, the slope of the regression line.
The expression in the numerator of (7.3) is the sum of products of deviations of x and y
about their means. A short-cut formula analogous to (2.3) is useful for computational
work:
P …xi x†…yi y† ˆ P xiyi
… P xi†… P yi†
: …7:7†
n
Note that, whereas a sum of squares about the mean must be positive or zero, a
sum of products of deviations about the mean may be negative, in which case,
from (7.3), b will also be negative.
The above theory is illustrated in the following example, which will also be
used later in the chapter after further points have been considered. Although the
calculations necessary in a simple linear regression are feasible using a scientific
calculator, one would usually use either a statistical package on a computer or a
calculator with keys for fitting a regression, and the actual calculations would
not be a concern.
Example 7.1
Table 7.1 gives the values for 32 babies of x, the birth weight, and y, the increase in weight
between the 70th and 100th day of life expressed as a percentage of the birth weight. A
scatter diagram is shown in Fig. 7.4 which suggests an association between the two
variables in a negative direction. This seems quite plausible: when the birth weight is
low the subsequent rate of growth, relative to the birth weight, would be expected to be
high, and vice versa. The trend seems reasonably linear.
From Table 7.1 we proceed as follows:
n ˆ 32 P P
x ˆ 3576 y ˆ 2281
x ˆ 3576=32 y ˆ 2281=32
P 2 x ˆ 409 880
ˆ 111 75: ˆ 71 28:
P
xy ˆ 246 032
P 2 y ˆ 179 761
2
P
x =n ˆ 399 618 00
P
x
P
y =n ˆ 254 901 75
2
P
y =n ˆ 162 592 53
P
…x
2
x† ˆ 10 262 00
P
…x x†…y y† ˆ 8869 75
P
…y
2
y† ˆ 17 168 47: