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Analysis of Stability Data with Equivalence Testing for Comparing New and Historical Processes Under Various Treatment Conditions Ben Ahlstrom, Rick Burdick, Laura Pack, Leslie Sidor Amgen Colorado, Quality Engineering May 19, 2009 Amgen Corporate Template

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**Agenda Purpose of comparability for stability data**

Problems with the p-value approach Equivalence approach and acceptance criteria methods Example

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**Example Data 2 package types (Bottle, Blister)**

Packaging Data (Chow, Statistical Design and Analysis of Stability Studies, p. 116, Table 5.6) 2 package types (Bottle, Blister) 10 lots (5 for each package type) 6 time points (0 to 18 months)

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**Comparability Analysis for Stability Data**

Purpose Compare the rates of degradation P-value Analysis Steps Fit the regression lines (process*time interaction) Calculate p-value for process*time Compare p-value to =0.05 Draw conclusion about comparability pass (comparable) if p-value > 0.05 fail (not-comparable if p-value < 0.05) I.E.: Evaluate the slopes of the treatment conditions

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**P-value Analysis to Evaluate Comparability for Stability Data**

Bottle vs. Blister: Are the processes comparable? Amgen Corporate Template

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**P-value Approach Hypotheses If p-value < 0.05, reject H0**

H0: slopes are comparable HA: slopes are not comparable If p-value < 0.05, reject H0 If p-value >0.05, fail to reject H0 Does not imply they are comparable, but rather that there isn’t enough evidence to say the slopes are different Amgen Corporate Template

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**P-value Analysis to Evaluate Comparability for Stability Data**

Packaging: Bottle vs. Blister Do we pass or fail the p-value test? Pass: p=0.8453 We compare the slopes using p-values (Pass if p-value > 0.05 and Fail if p-value < 0.05)

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**Problems with P-value Approach**

Reporting a P-value only tells us something about statistical significance. A statistically significant difference in slopes does not necessarily have any practical importance relative to patient safety or efficacy. P-values are non-informative because they do not quantify the difference in slopes in a manner that allows scientific interpretation of practical importance. A p-value approach provides a disincentive to collect more data and learn more about a process. Amgen Corporate Template

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**Equivalence Testing Method**

Fit the model with all historical and new process data (includes different storage conditions, orientations, SKU’s, container types) Compute the difference in slopes for the desired comparison Bottle vs. Blister Compute the 95% one-sided confidence limits around the difference observed over the time frame of interest If the confidence limits are enclosed by the equivalence acceptance criteria, conclude that the historical and new processes are comparable Amgen Corporate Template

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Statistical Model Parameters i and βi are the overall regression parameters for the ith process Random variables aj allow the intercepts to vary for each lot is the time value for process i, lot j, and time k. Model can be extended to more levels

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**Statistical Equivalence Acceptance Criteria (EAC)**

Goal Post is the space of expected historical performance Football = 95% one-sided CLs around difference between slopes over time frame of interest Amgen Corporate Template

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**Methods to Calculate Equivalence Acceptance Criteria (EAC)**

Equivalence Acceptance Criteria (EAC) provide a definition of practical importance The scientific client has the responsibility to determine a definition of practical importance (based on science, safety, specification, reg. commit., etc.) Statistical methods can help establish a starting point for these decisions Three statistical methods include: Method 1: Common cause variability Method 2: Excursion from Product Specification Method 3: Historic Variability of Slope Estimates Amgen Corporate Template

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**3 Statistical Approaches for Defining EAC**

Method 1 Method 2 Method 3 EAC based on common cause variability of the historic process EAC based on product specification EAC based on historic variability of slope estimates -EAC is expressed as average change in response per month -Requires a specification -Requires at least 3 different lots in historic data set -EAC is expressed as change response per month

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**Comparability in Profile Data**

Reference condition Difference between intercepts t = 0 A B-A Total difference between conditions at time T (intercept and slope) Difference in response averages attributed to the difference in slopes B – A = δ B Quality attribute New condition T Time (months) Amgen Corporate Template

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**EAC Method 1: Common Cause Variability**

Criteria is based on historical performance at various conditions Lot to Lot variability Measurement variability Multiplier aligned with other statistical limits used to separate random noise from a true signal Goal Post is the space of expected historical performance Amgen Corporate Template

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**EAC Method 1: Common Cause Variability**

T = Expiry = 18 months Amgen Corporate Template

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**Percent Label Claim, P-value approach vs. Equivalence Test**

Slope Bottle Slope Blister 0.8453 NA Slope difference over 18 months Goal Post +/ Result PASS Key Point Slope estimates are the same for both approaches Equivalence graph 0.2722 Difference in Slopes Amgen Corporate Template

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**EAC Method 2: Product Specification**

Maximum allowable difference in slopes where new and historic have < p% excursion rate at expiry Typically p=0.01, 0.025, 0.05 Use historic data Relates comparability to specification Typically p=.01, .025, .05 Amgen Corporate Template

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**EAC Method 2: Product Specification**

Spec (LSL) K bHist bNew E (Expiry) Mean of historical at expiry Response Time (months) Pth lower percentile centered at historic mean where P is probability of excursion centered at new mean Acceptable difference in slopes is q = K/E. Amgen Corporate Template

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**EAC Method 2: Product Specification**

K is unknown, so replace term in brackets with lower one-sided (1-P)*100% individual confidence bound based on historical (prediction bound) Assume Lower Spec Limit (LSL) = 95 Expiry = 18 months Amgen Corporate Template

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**EAC Method 3: Historic Slope Variability**

Use historical data for calculation Historical dataset provides nH independent estimates of the common slope β EAC based on 99.5th percentile of distribution of difference in slopes from same lot. If observed slope difference is consistent with this variability, equivalence is demonstrated. Amgen Corporate Template

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**EAC Method 3: Historic Slope Variability**

^ ^ ^ Amgen Corporate Template

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**EAC Method 3: Historic Slope Variability**

θ3 is the 99.5th percentile of the distribution of 2.576 is the 99.5th percentile of the standard normal distribution U is a 95% upper bound on the standard error for an estimate of β based on a single lot

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**Comparison of Equivalence Acceptance Criteria**

Hard for a client to know what a difference in slopes of, say, 0.1 % looks like in a table Once client sees graph, they can get a feel for what a difference in slope means Can visualize what the possible range of regression lines could be to still claim equivalence

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**Comparison of Equivalence Acceptance Criteria**

Based only on historical data Graph is created before data for the new process is collected Amgen Corporate Template

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Results by Method HA: Show δ is less than some amount deemed practically important Equivalence is demonstrated by computing two one-sided tests (TOST) If the 95% lower one-sided confidence bound on δ is greater than -θ and the 95% upper one-sided confidence bound is less than θ, then equivalence is demonstrated

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**P-value Approach vs. Equivalence Approach**

Ho: slopes are comparable HA: slopes are not comparable P-value Equivalence Approach Ho: slopes are not comparable HA: slopes are comparable Equivalence acceptance criteria set a priori Based on interval estimates of slope difference using mixed regression model with random lots Statistical convention is to have research objective in HA Amgen Corporate Template

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**Move to Equivalence Testing for Comparability**

Summary P-value approach to comparability has numerous issues High p-values do NOT prove equivalence High p-values only indicate that there is NOT enough evidence to conclude slopes are different At times, leads to ad hoc analysis requests when p-value is small P-values sensitive to sample size Goal posts allow you to state equivalence Industry is moving in the direction of equivalence tests Can be extended to accelerated studies Move to Equivalence Testing for Comparability Amgen Corporate Template

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References Limenati, G. B., Ringo, M. C., Ye, F., Bergquist, M. L., and McSorley, E. O. (2005). Beyond the t-test: Statistical equivalence testing. Analytical Chemistry, June 2005, pages 1A-6A. Chambers, D. , Kelly, G., Limentani, G., Lister, A., Lung, K. R., and Warner, E. (2005) Analytical method equivalency: An acceptable analytical practice. Pharmaceutical Technology, Sept 2005, pages Richter, S. , and Richter, C. (2002). A method for determining equivalence in industrial applications. Quality Engineering, 14(3), pages Park, D. J. and Burdick, R. K. (2004). Confidence Intervals on Total Variance in a Regression Model with an Unbalanced Onefold Nested Error Structure, Communications in Statistics, Theory and Methods, 33, No. 11, pages

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Back up slides

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**Back up slides EAC Method 2 Equal Difference Assumption:**

This assumption may not always hold The p-value for the interaction between time, process, and temperature tests this assumption

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**Comparison of Equivalence Acceptance Criteria**

Plot regression line for historical process At time=0 the value is Calculate Plot 2 additional lines Value at time=0 is Values at time=T are Amgen Corporate Template

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