
Confidence in the coefficients
The variances on the linear coefficients of the regression of the first kind are
usually estimated using the following relations, which assume that the dis-
persion around the line is Gaussian with a constant standard deviation and is
the result of the measurement errors:
s
n
¼
1
s
x
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s
2
y
ns
xy
N 2
s
and s
a
¼ s
n
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
N
X
N
i ¼1
x
2
i
v
u
u
t
ð7:17Þ
If T(P;) is the significance limit of the two-sided Student distribution for a
probability, P , for degree of freedom, , then the confide nce levels on the
coefficients are:
I
a
¼ s
a
TðP; N 2Þð7:18Þ
I
n
¼ s
n
TðP; N 2Þð7:19Þ
This means that with a probability, P, the coefficient, a, lies in the in terval
½a I
a
; a þ I
a
and the same for n.
The estimate of the variance around the regression line at abscissa x is:
s
y
ðxÞ¼s
n
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðN 1Þ
N
s
2
x
þðx
xxÞ
2
r
ð7:20Þ
and the confidence interval in the estimate of y using the regression line for any
x is:
I
y
ðxÞ¼s
y
ðxÞTðP ; N 2Þð7:21Þ
The values of the two-sided Student distribution are given in Table 7.6. The
relation (Equation 7.17) can be used to obtain the confidence intervals for a
0
and n
0
in the second kind regression if the roles of x and y are permuted.
Table 7.6 Two-sided confidence limits TðP; N 2Þ for a Student distribution
TðP; N 2Þ for probability P ¼
N 2 0.8 0.9 0.95 0.9 9 0.995 0.999
1 3.078 6.3138 12.706 63.657 127.32 636.619
2 1.886 2.9200 4.3027 9.9248 14.089 31.598
3 1.638 2.3534 3.1825 5.8409 7.4533 12.924
4 1.533 2.1318 2.7764 4.6041 5.5976 8.610
5 1.476 2.0150 2.5706 4.0321 4.7733 6.869
6 1.440 1.9432 2.4469 3.7074 4.3168 5.959
7 1.415 1.8946 2.3646 3.4995 4.0293 5.408
8 1.397 1.8595 2.3060 3.3554 3.8325 5.041
9 1.383 1.8331 2.2622 3.2498 3.6897 4.781
10 1.372 1.8125 2.2281 3.1693 3.5814 4.5787
Common Methods and Techniques 149
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