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1 Parallelism in practice USP Bioassay Workshop August 2010 Ann Yellowlees Kelly Fleetwood Quantics Consulting Limited

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2 Contents What is parallelism? Approaches to assessing parallelism Significance Equivalence Experience Discussion

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3 Setting the scene: Relative Potency RP: ratio of concentrations of reference and sample materials required to achieve the same effect RP = C ref / C samp

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4 Parallelism One curve is a horizontal shift of the other These are ‘parallel’ or ‘similar’ curves Finney: A prerequisite of all dilution assays

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5 Real data: continuous response

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6 Linear model (4 concentrations) Parallel when the slopes β equal Linear: Y = a + β log (C)

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7 Four parameter logistic model 4PL: Y = γ + (δ - γ) / [1 + exp (β log (C) – α)] Parallel when asymptotes γ, δ slope β equal

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8 Five parameter logistic model Parallel when asymptotes γ, δ slope β asymmetry φ equal 5PL: Y = γ + (δ - γ) / [1 + exp (β log (C) – α) ] φ

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9 Tests for parallelism Approach 1 Is there evidence that the reference and test curves ARE NOT parallel? Compare unrestricted vs restricted models Test loss of fit when model restricted to parallel ‘p value’ approaches 1.Traditional F test approach as preferred by European Pharmacopoeia 2.Chi-squared test approach as recommended by Gottschalk & Dunn (2005)

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10 Approach 2 Is there evidence that the reference and test curves ARE parallel? Equivalence test approach as recommended in the draft USP guidance (Hauck et al 2005) Fit model allowing non-parallel curves Confidence intervals on differences between parameters Pharmacopoeial disharmony exists!! (existed?)

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11 In practice... Four example data sets Data set 1: 60 RP assays (96 well plates, OD: continuous) Data set 2: 15 RP assays (96 well plates, OD : continuous) Data set 3: 12 RP assays (96 well plates, OD : continuous) Data set 4: 60 RP assays (in vivo, survival at day x: binary*) * treated as such for this purpose; wasteful of data

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12 In practice... We have applied the proposed methods in the context of individual assay ‘pass/fail’ (suitability): Data set 1 Compare 2 ‘significance’ approaches Compare ‘equivalence’ with ‘significance’ Data sets 2, 3 Compare 2 ‘significance’ approaches Data set 4 Compare ‘F test’ (EP) with ‘equivalence’ (USP)

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13 Data set 1 60 RP assays 8 dilutions 2 independent wells per dilution 4PL a good fit (vs 5PL) NB precision

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14 Data set 1: F test and chi-squared test F test: straightforward Chi-squared test: need to establish mean-variance relationship

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15 Data set 1: F test and chi-squared test F test: 12/60 = 20% of assays have p < 0.05 Evidence of dissimilarity? – OR – Precise assay? Chi-squared test: 58/60 = 97% of assays have p < 0.05! Intra-assay variability is low differences between parallel and non-parallel model are exaggerated

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16 Data set 1: Comparison of approaches to parallelism

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Some evidence of ‘hook’ in model Residual SS inflated 17

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18 Data set 1: Comparison of approaches to parallelism

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19 Data set 1: F test and chi-squared test RSS parallel = 159 RSS non-parallel = 112 RSS p – RSS np = 47 Pr( 2 3 >47) < 0.01 F test: P = 0.03

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20 Data set 1: F test and chi-squared test RSS parallel = 100.2 RSS non-parallel = 99.0 RSS p – RSS np = 1.2 Pr( 2 3 >1.2) = 0.75

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Data set 1: USP methodology Prove parallel Lower asymptote:

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22 Data set 1: USP methodology Upper asymptote:

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23 Data set 1: USP methodology Scale:

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24 Data set 1: USP methodology Criteria for 90% CI on difference between parameter values: Lower asymptotes: (-0.235, 0.235) Upper asymptotes: (-0.213, 0.213) Scales: (-0.187, 0.187) Applying the criteria: 3/60 = 5% of assays fail the parallelism criteria No assay fails more than one criterion

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25 Data set 1: Comparison of approaches to parallelism

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This plate ‘fails’ all 3 tests USP: Lower asymptote 26

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27 Data set 1: Comparison of approaches to parallelism Equivalence test: scales not equivalent F test p-value = 0.60 Chi-squared test p-value < 0.001

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28 Data set 2: Comparison of approaches to parallelism

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29 Data set 3: Comparison of approaches to parallelism

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30 In practice... Data set 4: Compare ‘F test’ with ‘equivalence’ Methodology for Chi-squared test not developed for binary data

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31 Data set 4 60 RP assays 4 dilutions 15 animals per dilution

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32 Data set 4: Comparison of approaches to parallelism

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33 Data set 4: Comparison of approaches to parallelism

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34 Broadly... F test Fail (?wrongly?) when very precise assay Pass (?wrongly?) when noisy Linear case: p value can be adjusted to match equivalence Chi-squared Fail when very precise assay (even if difference is small) If model fits badly – weighting inflates RSS (e.g. hook) 2 further data sets supported this USP Limits are set such that the extreme 5% will fail They do! Regardless of precision, model fit etc

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35 Stepping back… What are we trying to do? Produce a biologic to a controlled standard that can be used in clinical practice For a batch we need to know its potency With appropriate precision In order to calculate clinical dose

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36 Some thoughts 1.Establish a valid assay Use all development assay results unless a physical reason exists to exclude them Statistical methodology can be used to flag possible outliers for investigation USP applies this to individual data points Parallelism / similarity Are the parameter differences fundamentally zero? Or is there a consistent slope difference (e.g)? Equivalence approach + judgment for acceptable margin

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37 Some thoughts 2. Set number of replicates to provide required precision Combine RP values plus confidence intervals for reportable value 3.Per assay, use all results unless physical reason not to (They are part of the continuum of assays) Flag for investigation using statistical techniques Reference behaviour Parallelism 4. Monitor performance over time (SPC) Reference stability Parallelism

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38 Which parallelism test? Our view: Chi squared test requires too many complex decisions and is very sensitive to the model F test not generally applicable to the assay validation stage Does not allow examination of the individual parameters Does not lend itself to judgment about ‘How parallel is parallel?’ The equivalence test approach fits in all three contexts With adjustment of the tolerance limits as appropriate

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39 Thank you USP the invitation Clients use of data BioOutsource: www.biooutsource.comwww.biooutsource.com Other clients who prefer to remain anonymous Quantics staff analysis and graphics Kelly Fleetwood (R), Catriona Keerie (SAS)

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