Statistical Methods in Medical Research 4ed

y
Fig. 7.6 A three-dimensional representation of a bivariate normal distribution, the vertical dimension
representing probability density (reprinted from Yule and Kendall, 1950, by permission of the
authors and publishers).
Errors of prediction
The best estimate of the mean value of y at a given value of x, say x0, is given by
the regression equation
The sampling variance of Y is
7.4 Sampling errors in regression and correlation 201
Y ˆ a ‡ bx0 ˆ y ‡ b…x0 x†: …7:20†
var…Y† ˆvar…y†‡…x0 x† 2 var…b†
2 1
ˆ s
n ‡ …x0 x† 2
" #
P 2 ,
…x x†
which may be estimated by
var…Y† ˆs 2 1
0
n ‡ …x0 x† 2
" #
P 2 :
…x x†
Again the t distribution on n 2 DF is required. For instance, 95% confidence
limits for the predicted mean are
1
Y tn 2, 0 05s0
n ‡ …x0 x† 2
s"
P
…x
#
2 :
x†
…7:21†
x