Appendix 1 Data setsPaired data
Donor
DP Cell Viability (%)
Process
Data Set
1
89.9
Pre
1
2
92.3
Pre
1
3
83.9
Pre
1
4
88.9
Pre
1
5
98
Pre
1
1
83.6
Post
1
2
93.5
Post
1
3
82.7
Post
1
4
85.7
Post
1
5
81.5
Post
1
1
89.9
Pre
2
2
92.3
Pre
2
3
83.9
Pre
2
4
88.9
Pre
2
5
98
Pre
2
1
84.7
Post
2
2
90.4
Post
2
3
80.4
Post
2
4
80.7
Post
2
5
90.8
Post
2
Historical pre-change lots
DP Cell Viability (%)
88.1
89.7
94.7
86.1
80.4
94.4
91.2
92.4
90.8
98.1
Appendix 2 Derivation of Eq. (
15)Consider the estimator
$$ }_= }_-[w }_+(1-w) }_]$$
(20)
where
and
$$\beginVar[ }_]=Var( }_)+^Var( }_)+^Var( }_)-2wCov( }_, }_)\\ =\frac_^+_^)}+\frac^(_^+_^)}+\frac^(_^+_^)}-\frac_^}\\ =\frac_^+_^)(h+n)}[^-2w(\frac)+1]\\ =(_^+_^)\times }\\ }}}}} \, }=\frac[^-2w(\frac)+1].\end$$
(22)
Taking the derivative with respect to w and setting equal to zero, the value of w that minimizes the variance is
If \(\rho =0\) then \(^=\frac\) and if \(\rho =1\), \(^=1\) and one ignores the historic data.
We then have
$$\frac}_^}}(_^+_^)}\sim _^$$
(24)
and
$$ }_}\sim N(_-_,}(_^+_^))$$
(25)
which is used to form the exact confidence interval given in Eq. (15).
Appendix 3 Derivation of Eq. (
18)
The approximate confidence interval in Eq. (18) is derived in this section. Although it is tempting to replace \(\rho\) in Eq. (15) with an estimator based on the sample, this does not provide a good approximation because the distribution depends on \(_}^\) in addition to \(_}^\). Re-expressing the variance of \( }_}\) so that the coefficients of the variances are not a function of \(\rho\) yields
$$\beginVar[ }_}]=__^+__^ \, \, }}}}}\\ _=\frac^+2wh}\\ _=\frac^}.\end$$
(26)
Note that \(_\ge _\). In this form, \(Var[ }_}]\) is estimated as
$$_}^=\frac_}^}(_-_)+__}^$$
(27)
and the Satterthwaite approximation is used to derive the confidence interval in Eq. (18). The recommended value for w is the estimated value of \(^\)
$$\widehat^}=\frac_}}$$
(28)
where \(\widehat_}\) is defined in Eq. (19).
留言 (0)