Comparison of Analytical Procedures in Method Transfer and Bridging Experiments

Abstract

Comparison of two analytical procedures is the primary objective of a method transfer or when replacing an old procedure with a new one in a single lab. Guidance for comparing two analytical procedures is provided in USP <1010> based on separate tests for accuracy and precision. Determination of criteria is somewhat problematic for these comparisons because of the interdependence of accuracy and precision. In this paper, a total error approach is proposed that requires a single criterion based on an allowable out-of-specification (OOS) rate at the receiving lab. This approach overcomes the difficulty of allocating acceptance criteria between precision and bias. Computations can be performed with any simulation software. Numerical examples are provided for four experimental designs that are typical in a method transfer study. Finally, recommendations are provided to help the user set criteria that provide an acceptable probability of passing for practical sample sizes.

Graphical Abstract

Appendices

Appendix 1 Example data

Data set 1 (Independent samples)

Lab	Relative potency (%)
Sending	97.3
Sending	104.3
Sending	108.8
Sending	108.5
Sending	100.8
Sending	101
Sending	97.7
Sending	95.9
Sending	107.4
Sending	97.8
Sending	91.4
Sending	89.4
Sending	102.6
Sending	103.2
Sending	97.7
Sending	101.5
Receiving	113
Receiving	101
Receiving	102.3
Receiving	112.7
Receiving	100.1
Receiving	109.3
Receiving	103.4
Receiving	109.1
Receiving	115.1
Receiving	106.8
Receiving	95
Receiving	98.6
Receiving	103.2
Receiving	94.2
Receiving	105.4
Receiving	103.3

Data set 2 (Matched samples)

Lab	Sample type	Relative potency (%)
Sending	1	106.5
Sending	1	104.5
Sending	2	83.2
Sending	2	86.3
Sending	3	86.4
Sending	3	91.6
Sending	4	110.2
Sending	4	117.3
Sending	5	84.5
Sending	5	82.8
Sending	6	101.2
Sending	6	118.3
Sending	7	99.3
Sending	7	89.6
Sending	8	103.1
Sending	8	96.8
Receiving	1	108.4
Receiving	1	111.2
Receiving	2	106.5
Receiving	2	100.7
Receiving	3	103.3
Receiving	3	90.7
Receiving	4	127.8
Receiving	4	120.1
Receiving	5	75.4
Receiving	5	87.8
Receiving	6	93.9
Receiving	6	102.5
Receiving	7	95.7
Receiving	7	90.7
Receiving	8	99.7
Receiving	8	99.2

Data set 3 (Independent samples with ruggedness factor “Assay”)

Assay	Lab	Relative potency (%)
1	Send	105.3
1	Send	103.9
2	Send	102.3
2	Send	103.7
3	Send	97.6
3	Send	97.2
4	Send	94.2
4	Send	93.3
5	Send	92.2
5	Send	94.4
6	Send	94.1
6	Send	93
7	Send	105
7	Send	104.6
8	Send	105.4
8	Send	107.4
9	Receive	101
9	Receive	102
10	Receive	90.3
10	Receive	93.2
11	Receive	100.2
11	Receive	102.5
12	Receive	100.1
12	Receive	98.8
13	Receive	100.8
13	Receive	103.5
14	Receive	108.9
14	Receive	107.9
15	Receive	91.2
15	Receive	90.5
16	Receive	100.7
16	Receive	103.6

Data set 4 (Matched samples with ruggedness factor “Assay”)

Sample type	Lab	Assay	Relative potency (%)
1	Sending	1	101.4
1	Sending	1	101.2
1	Sending	2	105.3
1	Sending	2	109.2
1	Sending	3	104
1	Sending	3	103.9
1	Sending	4	107.9
1	Sending	4	107.7
2	Sending	5	103.3
2	Sending	5	103.1
2	Sending	6	95.7
2	Sending	6	92.8
2	Sending	7	102.5
2	Sending	7	103.7
2	Sending	8	91.8
2	Sending	8	96.9
1	Receiving	9	104.6
1	Receiving	9	106.9
1	Receiving	10	97.5
1	Receiving	10	99.7
1	Receiving	11	96.3
1	Receiving	11	95.2
1	Receiving	12	96.4
1	Receiving	12	98.2
2	Receiving	13	97.8
2	Receiving	13	95.2
2	Receiving	14	95.2
2	Receiving	14	91.7
2	Receiving	15	91.7
2	Receiving	15	85.5
2	Receiving	16	95.3
2	Receiving	16	99.2

Appendix 2 Details of GCI

One constructs a GCI on a parameter such as π_Receive in Eq. (7) by replacing the unknown parameters with a generalized pivotal quantity (GPQ) based on the statistics used in the estimators. The two GPQs shown in Eq. (12) are based on the statistics ${\overline{Y}}_{\boldsymbol{Receive}}-{\overline{Y}}_{\boldsymbol{Send}},{S}_{\boldsymbol{Receive}}^2,$ and ${S}_{\boldsymbol{Send}}^2$. Recall that ${S}_{\boldsymbol{Receive}}^2$ and ${S}_{\boldsymbol{Send}}^2$ generally underestimate ${\sigma}_{\boldsymbol{Receive}}^2$ and ${\sigma}_{\boldsymbol{Send}}^2$, respectively, due to the relatively small size of the transfer study. That is $E\left({S}_{\boldsymbol{Receive}}^2\right)={\sigma}_{{\boldsymbol{Receive}}^{\ast}}^2\le {\sigma}_{\boldsymbol{Receive}}^2$ and $E\left({S}_{\boldsymbol{Receive}}^2\right)={\sigma}_{{\boldsymbol{Send}}^{\ast}}^2\le {\sigma}_{\boldsymbol{Send}}^2$. Under the assumption of independent normal errors, these statistics have the following distributional properties

$${\displaystyle \begin{array}{l}\frac{{\overline{Y}}_{\mathrm{Receive}}-{\overline{Y}}_{\mathrm{Send}}-\left({\mu}_{\mathrm{Receive}}-{\mu}_{\mathrm{Send}}\right)}{\sqrt{\frac{\sigma_{\mathrm{Receive}\ast}^2}{n_{\mathrm{Receive}}}+\frac{\sigma_{\mathrm{Send}\ast}^2}{n_{\mathrm{Send}}}}}=Z\\ {}\frac{d{f}_{\mathrm{Receive}}\times {S}_{\mathrm{Receive}}^2}{\sigma_{\mathrm{Receive}\ast}^2}={W}_{\mathrm{Receive}}\\ {}\frac{d{f}_{\mathrm{Send}}\times {S}_{\mathrm{Send}}^2}{\sigma_{\mathrm{Send}\ast}^2}={W}_{\mathrm{Send}}\end{array}}$$

(20)

where W_Receive is a chi-squared random variable with n_Receive − 1 degrees of freedom, W_Send is a chi-squared random variable with n_Send − 1 degrees of freedom, and Z is a standard normal random variable with mean 0 and variance 1. Invert the pivotal quantities in (20) and replace the statistics with their realized values to obtain

$${\displaystyle \begin{array}{l}{\sigma}_{\mathrm{Receive}\ast \mathrm{GPQ}}^2=\frac{d{f}_{\mathrm{Receive}}\times {S}_{\mathrm{Receive}}^2}{W_{\mathrm{Receive}}}\\ {}{\sigma}_{\mathrm{Send}\ast \mathrm{GPQ}}^2=\frac{d{f}_{\mathrm{Send}}\times {S}_{\mathrm{Send}}^2}{W_{\mathrm{Send}}}\\ {}{\mu}_{\mathrm{Receive}\mathrm{GPQ}}={\mu}_{\mathrm{Send}}+\left({\overline{Y}}_{\mathrm{Receive}}-{\overline{Y}}_{\mathrm{Send}}\right)-Z\times \sqrt{\frac{\sigma_{\mathrm{Receive}\ast \mathrm{GPQ}}^2}{n_{\mathrm{Receive}}}+\frac{\sigma_{\mathrm{Send}\ast \mathrm{GPQ}}^2}{n_{\mathrm{Send}}}}\\ {}={\mu}_{\mathrm{Send}}+\left({\overline{Y}}_{\mathrm{Receive}}-{\overline{Y}}_{\mathrm{Send}}\right)-Z\times \sqrt{\frac{d{f}_{\mathrm{Receive}}\times {S}_{\mathrm{Receive}}^2}{n_{\mathrm{Receive}}\times {W}_{\mathrm{Receive}}}+\frac{d{f}_{\mathrm{Send}}\times {S}_{\mathrm{Send}}^2}{n_{\mathrm{Send}}\times {W}_{\mathrm{Send}}}}\\ {}{\sigma}_{\mathrm{Receive}\mathrm{GPQ}}^2={\sigma}_{\mathrm{Send}}^2\times \frac{\sigma_{\mathrm{Receive}\ast \mathrm{GPQ}}^2}{\sigma_{\mathrm{Send}\ast \mathrm{GPQ}}^2}\\ {}={\sigma}_{\mathrm{Send}}^2\times \left(\frac{S_{\mathrm{Receive}}^2}{S_{\mathrm{Send}}^2}\right)\times \left(\frac{d{f}_{\mathrm{Receive}}\times {W}_{\mathrm{Send}}}{d{f}_{\mathrm{Send}}\times {W}_{\mathrm{Receive}}}\right)\end{array}}$$

(21)

The last two terms in Eq. (21) correspond to the GPQs in Eq. (12) of the text. To form the generalized pivotal quantity for π_Receive, simply place μ_ReceiveGPQ and ${\sigma}_{\boldsymbol{ReceiveGPQ}}^2$ from Eq. (21) into Eq. (11). More detail on this process is provided in Appendix B.2 of Burdick et al. [9].

Appendix 3 SAS IML code for example

proc iml;

start;

*Input values;

gpqiterations=100000;

type1error=0.05;

Nsend=16;

Nrec=16;

dfsend=15;

dfrec=15;

Truemeansend=102;

Trueanalvarsend=35;

Truelotsvar=10;

USL=125;

LSL=75;

*Results from the transfer study;

meanrec=104.53;

meansend=100.33;

varrec=38.27;

varsend=31.19;

Q=j(gpqiterations,1,0);

Seed1=53876321;

Seed2=29325845;

Seed3=85894525;

*GCI interval;

do c=1 to gpqiterations;

WS=2*RANGAM(Seed1,(nsend-1)/2);

WR=2*RANGAM(Seed2,(nrec-1)/2);

trueanalsendgpq=(dfsend)*varsend/WS;

trueanalrecgpq=(dfrec)*varrec/WR;

varrecgpq=trueanalvarsend*(trueanalrecgpq/trueanalsendgpq);

ZR=rannor(Seed3);

meanrecgpq=(meanrec-meansend)-ZR*sqrt(trueanalrecgpq/nrec+trueanalsendgpq/nsend)

+truemeansend;

OOSRecGPQ=1-probnorm((USL-meanrecgpq)/sqrt(varrecgpq+truelotsvar))+probnorm((LSL-meanrecgpq)/sqrt(varrecgpq+truelotsvar));

Q(|C,1|)= OOSRecGPQ;

end;

*Sort for OOSRecGPQ;

finalesgci1=j(gpqiterations,1,0);

x1=Q(|,1|);

b1=x1;

x1[rank(x1),]=b1;

finalesgci1(|,1|)=x1;

ubOOSgci=finalesgci1(|gpqiterations*(1-type1error),|);

meanrecest=truemeansend+(meanrec-meansend);

varrecest=trueanalvarsend*varrec/varsend;

OOSpointest=1-probnorm((USL-meanrecest)/sqrt(varrecest+truelotsvar))

+probnorm((LSL-meanrecest)/sqrt(varrecest+truelotsvar));

print OOSpointest ubOOSgci;

finish;