Generalized Confidence Interval of SSDL

An upper 100(1-α)th percentile GCI for SSDL can be obtained from the following Monte-Carlo algorithm:

Step 1: Choose a large simulation sample size, say K=10,000. For k equal to 1 through K, carry out the following two steps.

Step 2: Generate LRx1 standard normal random vector Z and central chi-square random variable U with degree of freedom LJ-d-1.

Step 3: For the realized values of Y and S , compute ² R defined in Eq. (6.2.1). _τ,k

The required upper 100(1-α)th percentiles of the distribution of GPQ for SSDL, which is also the upper 100(1-α)th generalized confidence limit for SSDL, is then estimated by the 100(1-α)th sample percentiles of the collection of K=10,000 realizations R , _τ,1

R …….., τ,2 R_τ,10000.

6.4 Statistical Testing Procedure

The upper 100(1-α)% generalized confidence limit for SSDL based on GPQ can be used to test the statistical hypothesis in (6.1.2) for linearity. The null hypothesis in (6.1.2) is rejected and the linearity of a analytical method is concluded at the α significance level if the upper 100(1-α)% generalized confidence limit for SSDL is less than Lδ . ²₀

6.5 Simulation Study

We conducted a simulation study to compare the empirical sizes and powers of the corrected Kroll’s and GPQ-based SSDL methods. Following the specification of the experiment designs for evaluation of linearity, the number of solutions (or dilutions) of different concentrations is set to be 5 or 7 and the number of replications at each concentration is 2, 3, or 4. Throughout the simulation, mean concentration μ is assumed to be 4 and the allowable margin of linearity based on ADL, θ0, is specified 0.05 as

where μ and θ are the mean of the concentrations, and ADL, respectively. It follows that the margin for SSDL for 5 and 7 concentrations are 0.2 and 0.28, respectively. In addition, standard deviation of normal random error is specified as 0.1 and 0.2. For each of 12 combinations, ten thousand (10,000) random samples are generated. For the 5%

nominal significance level, a simulation study with 10,000 random samples implies that 95 percent of the empirical sizes evaluated at the allowable margins will be within 0.0457 and 0.0543 if the proposed methods can adequately control the size at the nominal level of 0.05.

Table 6.5.1 presents the results of the empirical sizes. All of empirical sizes for the corrected Kroll’s is larger than 0.0543. On the other hand, all of empirical sizes of the GPQ method are within the range and showed that it has a better ability for controlling the size at the nominal level than the corrected Kroll’s method. It was introduced in the previous chapters that the poor performance for the corrected Kroll’s method in controlling the size results from the variability of estimators of non-centrality parameters for non-central χ² distribution of the observed ADL being estimated by the square root of residual mean square obtained from best-fitted polynomial model. On the contrary, since one requirement for GPQ is that

p L

R_{μ -μ} is free of nuisance parameter _σ, the GPQ approach can control the size at the nominal level.

Table 6.5.2 presents the results of the empirical powers. For the simulation, the true ADL is specified as 0.005 for both number of solutions of 5 and 7. The results given in Table 6.5.2 also show that the empirical power increases as the numbers of replicates or concentrations increases. Both the methods provide comparable powers except for the one of the GPQ-based SSDL method is 0.6962 when number of solution is 5, number of replicates is 2, and standard deviation of normal random error is 0.2. However, all

Table 6.5.1 Empirical sizes (corrected Kroll’s method vs. GPQ-based SSDL method) No. of

Solution

No. of Replicate

Standard Deviation

Corrected Kroll

GPQ-based SSDL

5 2 0.1 0.0769 0.0535

0.2 0.0734 0.0503

3 0.1 0.0679 0.0523

0.2 0.0643 0.0501

4 0.1 0.0569 0.0476

0.2 0.0596 0.0504

7 2 0.1 0.0670 0.0532

0.2 0.0671 0.0532

3 0.1 0.0573 0.0502

0.2 0.0557 0.0476

4 0.1 0.0563 0.0506

0.2 0.0595 0.0529

empirical powers of the GPQ-based SSDL method for other combinations of parameters are still greater than 90%. In addition, from Table 6.5.1, the corrected Kroll’s method fails to control the size at the nominal level. Therefore, the advantage of power by the corrected Kroll’s method comes at the expense of inflated type I error rates.

Figure 6.5.1 and 6.5.2 present the empirical powers when the standard deviations of normal random error are 0.1 and 0.2, respectively with number of solutions is 5, number of replicates is 3, and the true ADLs are ranged from 0 to 0.08. Figure 6.5.1 shows that when standard deviation is 0.1, the empirical size at ADL=0.05 for the corrected Kroll’s method is 0.0679, while the empirical size of the GPQ-based SSDL method is 0.0521. It shows that the GPQ method can control the size better than the other methods at the nominal level. In addition, the powers reach 0 and 1 at ADL=0.08 and 0.005, respectively for both methods. On the other hand, the power of the GPQ-based SSDL method is quite competitive to the corrected Kroll’s method although it is little lower.

The similar results are observed in Figure 6.5.2 when standard deviation of normal random error is 0.2. The empirical sizes for the corrected Kroll’s and the GPQ-based SSDL methods at ADL=0.05 are 0.0827 and 0.0494, respectively. In addition, the powers for both methods when the standard deviation is 0.2 are lower than those when the standard deviation is 0.1.

6.6 Numerical Example

The same example of calcium used in Chapter 5 from Example 2 of CLSI guideline EP6-A (Tholen et al., 2003) is used to illustrate the proposed testing procedures. In this example, the allowable margin of percent bound for ADL is set as 0.05. As indicated in EP6-A (ICH Expert Working Group, 1995), the criteria of μ -μ_{Pi Li} for claiming

Table 6.5.2 Empirical powers with the true ADL=0.005 (corrected Kroll’s method vs.

GPQ-based SSDL method)

No. of

Solution No. of

Replicate Standard

Deviation Corrected

Kroll GPQ

5 2 0.1 1.0000 0.9994

0.2 0.9261 0.6962

3 0.1 1.0000 1.0000

0.2 0.9454 0.9256

4 0.1 1.0000 1.0000

0.2 0.9828 0.9781

7 2 0.1 1.0000 1.0000

0.2 0.9327 0.9078

3 0.1 1.0000 1.0000

0.2 0.9901 0.9873

4 0.1 1.0000 1.0000

0.2 0.9980 0.9972

0.00 0.02 0.04 0.06 0.08

0.00.20.40.60.81.0

ADL

power

Corrected Kroll GPQ-based SSDL

Figure 6.5.1 The empirical powers when standard deviation of normal random error is 0.1, number of solutions is 5, and number of replicates is 3 (corrected Kroll’s method vs. GPQ-based SSDL method)

0.00 0.02 0.04 0.06 0.08

0.00.20.40.60.81.0

ADL

power

Corrected Kroll GPQ-based SSDL

Figure 6.5.2 The empirical powers when standard deviation of normal random error is 0.2, number of solutions is 5, and number of replicates is 3 (corrected Kroll’s method vs. GPQ-based SSDL method)

linearity is set as 0.2 mg/dL, the allowable limit of SSDL is set as 0.2 which is calculated by square of 0.2 mg/dL multiplying 5 concentrations. Table 6.6.1 presents the results of the two testing procedures. According to the decision rule of the corrected Kroll method, the analytical method can be concluded linear at the 5% significance level. On the other hand, the 95% upper limit confidence limit for SSDL of the GPQ methods is 0.2664, respectively. As a result, the GPQ-based SSDL method can not conclude that the analytical procedure is linear at the 5% significance level. The results presented above show the consistent results with the simulation results in Section 6.5 which the GPQ-based SSDL method is more conservative than the corrected Kroll’s method. However, as demonstrated by the simulation, the GPQ-based SSDL method is the procedure that can adequately control the size at the nominal level.

6.7 Summary

The ADL is an aggregate criterion proposed by Kroll et al. (Kroll, 2000) for evaluating the linearity in assay validation. In this chapter, we propose an alternative criterion of SSDL based on the GPQ approach to assess the linearity. Simulation results show that the GPQ-based SSDL method not only can adequately control the type I error rate at the nominal level better than the corrected Kroll’s method but also keep a competitive performance of the power. The reason for the poor performance corrected Kroll’s method in controlling the size at the nominal level is the variability of estimators of non-central parameters for non-central χ distribution of the observed ADL being ² estimated by the square root of residual mean square obtained from best-fitted polynomial model. Therefore, we can conclude the proposed statistical hypothesis based on the aggregate criteria SSDL in conjunction with the testing procedure derived from

Table 6.6.1 Results of the linearity evaluation for the example of calcium by the corrected Kroll’s and GPQ-based SSDL methods

Method

Sample Statistic /

Critical Value or Allowable Bound Conclusion Sample ADL 0.0146

Corrected Kroll

Critical Value 0.0437 Linear Upper 95% C.L. 0.2664

GPQ-based SSDL

Allowable Upper Bound 0.2 Nonlinear 95% C.L. : Upper 95% Confidence limit. of SSDL

the GPQ method for evaluating the linearity in assay validation is better than the corrected Kroll’s method.

Chapter 7

Alternative Criterion - Sum of Squares of the Deviation from Linearity Related to the Variation (CVDL)

The SSDL we introduced in Chapter 7 is based on the un-scaled deviations from linearity while ADL is based on the deviations from linearity scaled by the population average of concentrations of all solutions of the assay. Both ADL and SSDL do not take the experimental variability into consideration. As the repeatability is also the important characteristic which stands for reliability of a assay method, Wu (Wu, 2008) propose the coefficient of variation of the deviations from linearity (CVDL) as an alternative measure which can be used to evaluate the linearity and repeatability simultaneously.