Statistical Methods in Medical Research 4ed

202 Regression and correlation
y , Increase in weight between 70 and 100 days, as % of x
120
100
80
60
40
20
0
80 90 100 110 120 130 140 150
x , Birth weight (oz)
Fig. 7.7 Confidence limits (95%) for the predicted mean value of y for specified values, x0, ofx (data
as in Fig. 7.4).
In (7.21) the width of the confidence interval increases with …x0 x† 2 , and is
therefore a minimum when x0 ˆ x. Figure 7.7 shows the limits for various values
of x0 in Example 7.1. The reason for the increase in the width of the interval is
that slight sampling errors in b will have a greater effect for values of x0 distant
from x than for those near x. The regression line can be thought of as a rod
which is free to move up and down (corresponding to the error in y) and to
pivot about a central point (corresponding to the error in b). Points near the
ends of the rod will then be subject to greater oscillations than those near
the centre.
A different prediction problem is that of estimating an individual value y0 of
y corresponding to a given x0. The best single estimate is again the value given by
the regression equation, but the limits of error are different from those in the
previous case. Two slightly different problems can be distinguished.
1 A single prediction is required. The appropriate limits are not strictly confidence
limits because the quantity to be estimated is a value taken by a
random variable, not a parameter of a distribution. However, writing
y0 ˆ Y ‡ e,
where e is the deviation of y0 from the predicted value Y, we have