Statistical analysis and Extrapolation of Stability Data

This article is to explain the statistical evaluation and extrapolation of Stability data. We are followed ICH Q1E Guidance.

Software is downloaded from http://www.r-project.org/.

Our program is stab for R-Project.

Methods:: This package includes two steps.

In the first step, Decision Tree for Data Evaluation follows "Appendix A" of ICH guideline "Q1E Evaluation for Stability Data" that assists users evaluating stability data and guide users to consider doing an extrapolation for a proposed retest period or shelf life.

In the second step, Statistical Approaches to Stability Data Analysis is conducted for two different situations.

First one is for a single batch. This approach estimates the retest period or shelf life for a single batch of drug product. The relationship between residuals and time is assumed to be linear. Two-sided 95 % confidence intervals of the regression line for residuals (% relative to the original amount) of a drug product intersect with upper and lower acceptance criteria of label claimed. Then, the shortest one is the shelf life.

When there are multiple batches (e.g. 3 batched) available, analysis of covariance (ANCOVA) is first employed to test the difference in slopes and intercepts of the regression lines with different factors (packages, dosage forms, etc.). Then, based on the statistical results, there can be three possibilities.

(1). slope (P>=0.25) and intercept (P>=0.25): the tests for equality of slopes and equality of intercepts are all no differences. The data from all batches then should be combined. Then, a single retest period or shelf life is estimated from the combined data.

(2). slope (P>=0.25) and intercept (P<0.25): the test rejects the hypothesis of equality of intercepts but fails to reject the hypothesis with that all slopes are equal. The data should be combined to estimate the common slope. The retest periods or shelf lives for individual batches can be estimated. Then, the shortest estimate among batches should be chosen as the shelf life for all batches.

(3). [slope (P<0.25) and intercept (P>=0.25)] or [slope (P<0.25) and intercept (P<0.25)]: the result in this scenario shows that the test rejects the hypothesis of equality of all slopes. It is not appropriate to combine the data from all batches in this situation. The retest periods or shelf lives for individual batches is estimated. Then, the shortest estimate among batches should be chosen as the shelf life for all batches.

Example for Multiple batches pooling study with ANCOVA.

Step 1:

a) significant change in accelerated Studies Yes/no

Selection: No

b) Long term data : (1) show little or no variability or (2) little or no change over time

Selection: No to(1) or (2)

c) Long term data amenable to staistical analysis and Statistical analysis performed

Slection: yes to both

------------------ stab for R v0.1.3 -------------------

developed by Hsin-ya Lee and Yung-jin Lee, 2007-2010.

generated on Tue Apr 17 19:54:53 2012

<< --- List of input data --- >>

batch time assay (%)

1 1 0 100.12

2 1 3 99.99

3 1 6 98.99

4 1 9 99.67

5 1 12 99.87

6 1 18 99.67

7 2 0 99.89

8 2 3 100.22

9 2 6 99.89

10 2 9 99.67

11 2 12 99.56

12 2 18 99.12

13 3 0 99.85

14 3 3 99.69

15 3 6 99.68

16 3 9 100.20

17 3 12 99.98

18 3 18 99.83

Analysis settings for multiple batches:

---------------------------------------------

The lower acceptance limit is set to 98.5 %.

<<Output: ANCOVA model: batch vs. time vs. assay (%)>>

Analysis of Variance Table

Response: assay

Df Sum Sq Mean Sq F value Pr(>F)

batch 2 0.09013 0.045067 0.5120 0.6118

time 1 0.23317 0.233166 2.6491 0.1296

batch:time 2 0.39117 0.195583 2.2222 0.1510

Residuals 12 1.05618 0.088015

Type P values

1 Intercept 0.6117985

2 Slope 0.1510063

--------------------------

at a sig. level of 0.25.

--------------------------------------------------------------------------

<< ANCOVA Output: Testing for poolability of batches >>

--------------------------------------------------------------------------

The test rejects the hypothesis of equality of slopes (there is a

significant difference in slopes and intercepts among batches).

<<Model #3: one-sided lower LC analysis>>

separate intercepts and separate slopes between batches.

------------------------------------------------------------------------

<<linear regression model: Assay (%) vs. time>>

Call:

lm(formula = assay ~ batch * time, data = ANCOVAdata)

Coefficients:

(Intercept) batch2 batch3 time batch2:time batch3:time

99.83414 0.30571 -0.03143 -0.01448 -0.03738 0.02310

Analysis of Variance Table

Response: assay

Df Sum Sq Mean Sq F value Pr(>F)

batch 2 0.09013 0.045067 0.5120 0.6118

time 1 0.23317 0.233166 2.6491 0.1296

batch:time 2 0.39117 0.195583 2.2222 0.1510

Residuals 12 1.05618 0.088015

**************************************************************************

<< Output >>

--------------------------------------------------------------------------

<<Summary: linear regression model>>

--- Batch#: 1 ---

Y = 99.83414 +( -0.01447619 ) X

Time Observed assay(%) Calculated assay(%) Residuals

1 0 100.12 99.83414 -0.28585714

2 3 99.99 99.79071 -0.19928571

3 6 98.99 99.74729 0.75728571

4 9 99.67 99.70386 0.03385714

5 12 99.87 99.66043 -0.20957143

6 18 99.67 99.57357 -0.09642857

--- Batch#: 2 ---

Y = 100.1399 +( -0.05185714 ) X

Time Observed assay(%) Calculated assay(%) Residuals

1 0 99.89 100.13986 0.249857143

2 3 100.22 99.98429 -0.235714286

3 6 99.89 99.82871 -0.061285714

4 9 99.67 99.67314 0.003142857

5 12 99.56 99.51757 -0.042428571

6 18 99.12 99.20643 0.086428571

--- Batch#: 3 ---

Y = 99.80271 +( 0.008619048 ) X

Time Observed assay(%) Calculated assay(%) Residuals

1 0 99.85 99.80271 -0.04728571

2 3 99.69 99.82857 0.13857143

3 6 99.68 99.85443 0.17442857

4 9 100.20 99.88029 -0.31971429

5 12 99.98 99.90614 -0.07385714

6 18 99.83 99.95786 0.12785714

**************************************************************************

<< Summary and plots >>

--------------------------------------------------------------------------

One-sided lower LC analysis

batch# shelf life*

1 1 22.65409

2 2 23.23252

3 3 69.31091

-------------------------

*: estimated shelf life

time fit Lower

1 0 99.83414 99.20063

2 1 99.81967 99.23612

3 2 99.80519 99.26869

4 3 99.79071 99.29753

5 4 99.77624 99.32154

6 5 99.76176 99.33942

7 6 99.74729 99.34966

8 7 99.73281 99.35079

9 8 99.71833 99.34165

10 9 99.70386 99.32184

11 10 99.68938 99.29176

12 11 99.67490 99.25256

13 12 99.66043 99.20573

14 13 99.64595 99.15276

15 14 99.63148 99.09498

16 15 99.61700 99.03345

17 16 99.60252 98.96901

18 17 99.58805 98.90230

19 18 99.57357 98.83379

20 19 99.55910 98.76385

21 20 99.54462 98.69277

22 21 99.53014 98.62075

23 22 99.51567 98.54796

24 23 99.50119 98.47454

25 24 99.48671 98.40058

26 25 99.47224 98.32617

27 26 99.45776 98.25138

28 27 99.44329 98.17626

29 28 99.42881 98.10086

30 29 99.41433 98.02521

31 30 99.39986 97.94935

32 31 99.38538 97.87329

33 32 99.37090 97.79707

34 33 99.35643 97.72071

35 34 99.34195 97.64421

36 35 99.32748 97.56759

37 36 99.31300 97.49087

time fit Lower

1 0 100.13986 99.87494

2 1 100.08800 99.84398

3 2 100.03614 99.81179

4 3 99.98429 99.77805

5 4 99.93243 99.74229

6 5 99.88057 99.70396

7 6 99.82871 99.66244

8 7 99.77686 99.61711

9 8 99.72500 99.56748

10 9 99.67314 99.51339

11 10 99.62129 99.45501

12 11 99.56943 99.39282

13 12 99.51757 99.32743

14 13 99.46571 99.25948

15 14 99.41386 99.18951

16 15 99.36200 99.11798

17 16 99.31014 99.04522

18 17 99.25829 98.97152

19 18 99.20643 98.89707

20 19 99.15457 98.82202

21 20 99.10271 98.74649

22 21 99.05086 98.67057

23 22 98.99900 98.59433

24 23 98.94714 98.51782

25 24 98.89529 98.44109

26 25 98.84343 98.36417

27 26 98.79157 98.28709

28 27 98.73971 98.20988

29 28 98.68786 98.13254

30 29 98.63600 98.05510

31 30 98.58414 97.97758

32 31 98.53229 97.89997

33 32 98.48043 97.82229

34 33 98.42857 97.74455

35 34 98.37671 97.66676

36 35 98.32486 97.58892

37 36 98.27300 97.51103

time fit Lower

1 0 99.80271 99.49578

2 1 99.81133 99.52861

3 2 99.81995 99.56002

4 3 99.82857 99.58962

5 4 99.83719 99.61689

6 5 99.84581 99.64119

7 6 99.85443 99.66178

8 7 99.86305 99.67796

9 8 99.87167 99.68917

10 9 99.88029 99.69520

11 10 99.88890 99.69626

12 11 99.89752 99.69290

13 12 99.90614 99.68584

14 13 99.91476 99.67581

15 14 99.92338 99.66345

16 15 99.93200 99.64927

17 16 99.94062 99.63368

18 17 99.94924 99.61699

19 18 99.95786 99.59943

20 19 99.96648 99.58118

21 20 99.97510 99.56238

22 21 99.98371 99.54312

23 22 99.99233 99.52348

24 23 100.00095 99.50354

25 24 100.00957 99.48334

26 25 100.01819 99.46293

27 26 100.02681 99.44232

28 27 100.03543 99.42156

29 28 100.04405 99.40066

30 29 100.05267 99.37964

31 30 100.06129 99.35852

32 31 100.06990 99.33730

33 32 100.07852 99.31601

34 33 100.08714 99.29464

35 34 100.09576 99.27321

36 35 100.10438 99.25172

37 36 100.11300 99.23018

------------------------------------------------------------------

Drug product with lower acceptance limit of 98.5 % of label claim

Shelf life = 22 (months/weeks)

******************************************************************

Note: there are many situations arise during the data analysis. I was given only one of them.