10.5. Combination of Assays
Weighted and unweighted, geometric / arithmetic mean potencies and their confidence limits are calculated from two or more assay results. Semiweighted means can be computed according to European Pharmacopoeia (19972017) and United States Pharmacopoeia (2009).
It is also possible to run multiple assays and to combine their results in one go. See section 10.0.4. Multiple Assays with Combination.
10.5.1. Combined Potency Variable Selection
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. All four variables are compulsory.
10.5.2. Combined Potency Output Options
Usually, arithmetic means are calculated for Slope Ratio Method and geometric means for Parallel Line Method, Quantal Response Method and FourParameter Logistic Model. If the geometric means are to be computed, all potencies and their confidence limits are logged first. The results are antilogged just before they are displayed.
Weights are computed using confidence limits of each estimated potency:
where the tstatistic has the same degrees of freedom as it was used in calculating the potency.
10.5.2.1. Combined Potency Homogeneity Tests
Homogeneity of Means
The following chisquare statistic is calculated:
with n – 1 degrees of freedom and n is the number of valid assays combined. 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 ChiSquare Test
The variance of each potency to be combined is:
The chisquare statistic is computed as:
where:
n_{i} is the degrees of freedom for each assay and N is the sum of n_{i}. Next, the following term is computed:
and the test statistic is obtained as:
which is approximately chisquare 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.
10.5.2.2. Combined Potency EP
European Pharmacopoeia (19972017) describes calculations for weighted, semiweighted and unweighted means. 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 EP
The weighted mean potency is defined as:
Standard error of weighted mean potency is defined as:
Confidence limits are:
where the degrees of freedom for the tstatistic is the sum of the number of degrees of freedom for the error mean squares in the individual assays (n_{i}).
Semi Weighted Mean Potency EP
When the chisquare test for combined mean potency has a value p ≥ α, semi weighted mean potency can be used. The variance of the heterogeneity between assays is calculates as:
where:
A semi weight is then defined as:
The semi weighted mean potency and its confidence interval is then calculated as in the Weighted Mean Potency.
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 tstatistic is n – 1.
10.5.2.3. Combined Potency USP
Weighted Mean Potency USP
USP <111> (US Pharmacopoeia, 2009) computes the weighted mean potency as shown above for EP, however it differs in the definition of its confidence interval.
USP adds a correction term to the standard error of weighted mean potency as follows:
where:
The confidence limits of the weighted mean potency are:
where the degrees of freedom for the tstatistic is:
Semi Weighted Mean Potency USP
When the chisquare test for combined mean potency has a value p ≥ α, USP advises use of semi weighted mean potency. The variance of the heterogeneity between assays is calculates as:
where the variance is:
If ν < 0, then it is recalculated by omitting the term following the minus sign. A semi weight is then defined as:
Weighted Mean Potency
This is the same as in EP.
10.5.2.4. Plot of Potencies
Potency data and confidence limits are plotted.
10.5.3. Combined Potency 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. Then select C41 Potency as [Potency], column C42 LL as [Low], C43 UL as [High] and C44 as [DoF]. Click [Next] to proceed to Output Options Dialogue. Check all output options and click [Finish] to obtain the following output.
Combination of Assays
Data
Assay 
Potency 
Lower 95% 
Upper 95% 
DoF 
1 
18367.0000 
17755.0000 
19002.0000 
20 
2 
18003.0000 
17415.0000 
18610.0000 
20 
3 
18064.0000 
17319.0000 
18838.0000 
20 
4 
17832.0000 
17253.0000 
18429.0000 
20 
5 
18635.0000 
17959.0000 
19339.0000 
20 
6 
18269.0000 
17722.0000 
18834.0000 
20 
Homogeneity Tests

ChiSquare 
DoF 
Probability 
Mean 
4.4195 
5 
0.4907 
Variance 
2.5984 
5 
0.7616 
Combined Potency EP
Geometric Mean 
Potency 
Lower 95% 
Upper 95% 
Weighted EP 
18186.8537 
17946.0755 
18430.8624 
Semiweighted EP 
18187.7481 
17926.1824 
18453.1304 
Unweighted 
18193.1071 
17894.1048 
18497.1056 
Combined Potency USP
Geometric Mean 
Potency 
Lower 95% 
Upper 95% 
Weighted USP 
18186.8537 
17923.4957 
18454.0813 
Semiweighted USP 
18190.0887 
17855.8964 
18530.5359 
Unweighted 
18193.1071 
17894.1048 
18497.1056 