Geoscience Reference
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Fig. 4.2 Example of practical use of confidence belts for a regression line. Straight line was fitted
to data with more than 30 % forsterite; only one of these falls outside the 95-% belt. All
observation data with %FO
30 % fall below the 95-% confidence belt for data not used for
estimation (Source: Agterberg 1974 , Fig. 38)
<
p
2 F 2, n
R p ; The purpose of this belt which is wider than the
preceding one is to define a confidence region for the entire calculated
regression line.
Y i
2.
ð
2
Þ
s res
R p ; This represents a confidence belt for the results of data
which have been used to estimate Y i . It allows testing of the residuals for
normality. Like belt (4), belt (3) is considerably wider than belts (1) and (2),
although all belts are hyperbolas.
1
Y i
3.
tn
ð
2
Þ
s res
p
1
Y i
4.
tn
ð
2
Þ
s res
þ
R i
; This belt is for “new” observations that have not been
used to estimate Y i .
The techniques for linear regression discussed in this section can be extended to
(a) testing additional terms ( x 2 , x 3 , etc.) in curvilinear regression for statistical
significance, (b) testing other explanatory variables, in addition to x in applications
of the general linear model, and (c) additional terms consisting of powers and cross-
products of powers in 2-D or 3-D trend analysis. Various generalizations of these
types will be discussed later. Use of two of the preceding confidences belts will be
illustrated in the following example.
One method of determining the forsterite (FO) content of olivine consists
of measuring the d 174 cell edge. In Agterberg ( 1964 ), a linear regression equation
was fitted to 20 measurements with %FO
30 in an experiment for which the
olivine crystals were chemically analyzed. The XY -plot for this experiment
(Fig. 4.2 ) showed that these 20 data closely follow a straight line with equation:
Y i ¼
>
X . Figure 4.2 shows the 95 % confidence belt for these data.
Only one of the 20 points falls slightly outside this Type (3) belt. This indicates that
4, 145
3, 970
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