10.4. Combination of Assays
Weighted mean potency and its confidence limits are calculated from two or more assay results. Two different types of assay results can be combined.
10.4.1. Variable Selection
1) Confidence Limits are Given: Use this option when the confidence interval of individual assay potencies are known. Potencies, their lower and upper limits and the degrees of freedom should be entered in separate columns and they are selected for analysis by clicking on [Potency], [Low], [High] and [DoF] respectively. These are the compulsory variables. You can optionally select a column containing variances by clicking on [Variance], in which case a Bartlett test of homogeneity of variances will also be performed.
2) Weights are Given: If the confidence limits of potencies are not available, but their weights are, then select this data option. In this case, potencies, their weights and their degrees of freedom should be entered in separate columns and selected by clicking on [Potency], [Weight] and [DoF] respectively. Again, the choice of a column containing variances is optional.
10.4.2. Output Options
If the first data option Confidence Limits are Given is selected, the weights are computed using the confidence limits of each estimated potency:
where the t-statistic has the same degrees of freedom as it was used in calculating the potency.
For both data options the weighted mean potency is defined as:
10.4.2.1. Homogeneity Tests
Homogeneity of Means
The following chi-square statistic is tested:
with n ‑ 1 degrees of freedom. If the probability value displayed is greater than the given significance level (usually 0.05) then the test is said to be satisfactory.
Homogeneity of Variances: Bartlett’s Chi-Square Test
This test statistic is calculated only when the optional [Variance] variable is selected.
where n is the number of assays, N is the total degrees of freedom and:
where ni is the degrees of freedom for each assay. Next, the following term is computed:
and the test statistic is obtained as:
which is approximately chi-square distributed with n ‑ 1 degrees of freedom. If the probability value displayed is greater than the given significance level (usually 0.05) then the test is said to be satisfactory.
10.4.2.2. Combined Potency
European Pharmacopoeia (1997-2008) gives calculations for weighted and unweighted means as two alternatives. The Weighted Mean Potency method (which is more robust) should be used if:
1) Homogeneity of Means test is satisfactory,
2) Bartlett’s Homogeneity of Variance test is satisfactory and
3) g < 1 for each assay.
If one or more of these criteria fail, the Unweighted Mean Potency results can be used.
Weighted Mean Potency
The standard error of weighted mean potency is defined as:
The confidence limits of the weighted mean potency are then:
where the degrees of freedom for the t-statistic is the sum of the number of degrees of freedom for the error mean squares in the individual assays.
Unweighted Mean Potency
This is defined as the arithmetic mean of individual assay potencies:
and its variance is:
The confidence limits of the unweighted mean potency are then:
where the degrees of freedom for the t-statistic is n ‑ 1.
10.4.3. Example
Data is given in European Pharmacopoeia (2008), Table 6.4.1.-I on p. 594.
Open BIOPHARMA6 and select Bioassay → Combine Assays. From the Variable Selection Dialogue select the second data option Weights are Given. Then select C46 LnPotency as [Potency] and columns C42 to C44 respectively as, [Low], [High] and [DoF]. Click [Next] to proceed to Output Options Dialogue. Check both output options and click [Finish] to obtain the following output.
Combination of Assays
Homogeneity Tests
|
|
Chi-Square |
DoF |
Probability |
|
Mean |
4.4274 |
5 |
0.4897 |
|
Variance (Bartlett) |
|
|
|
Combined Potency
|
|
Potency |
Lower 95% |
Upper 95% |
|
Weighted Mean |
9.8085 |
9.7951 |
9.8218 |
|
Unweighted Mean |
9.8088 |
9.8087 |
9.8089 |