Chapter 48

Stability Study Designs

Tsae-Yun Daphne Lin and Chi Wan Chen

48.1 Introduction

The stability of a drug substance or drug product is the capacity of the drug substance or drug product to remain within the established specifications to ensure its identity, strength, quality, and purity during a specified period of time. The U.S. Food and Drug Administration (FDA) requires that the shelf life (also referred to as expiration dating period) must be indicated on the immediate container label for every human drug and biologic on the market.

A good stability study design is the key to a successful stability program. From statistical perspectives, several elements need to be considered in planning a stability study design. First, the stability study should be well designed so the shelf life of the drug product can be estimated with a high degree of accuracy and precision. Second, the stability design should be chosen so that it can reduce bias and identify and control any expected or unexpected source of variations. Third, the statistical method used for analyzing the data collected should reflect the nature of the design and provide a valid statistical inference for the established shelf life.

In many cases, the drug product may have several different strengths packaged in different container sizes because of medical needs. To test every batch under all factor combinations on a real-time, long-term testing schedule can be expensive and time consuming. As an alternative, a reduced design (e.g., bracketing, matrixing) may be considered as an efficient method for reducing the amount of testing needed while still obtaining the necessary stability information. A number of experimental designs are available that can capture the essential information to allow the determination of shelf life while requiring less sampling and testing. Some of these designs require relatively few assumptions about the underlying physical and chemical characteristics, and others require a great many.

Stability requirements for the worldwide registration of pharmaceutical products have changed dramatically in the past few years. A series of guidelines on the design, conduct, and data analysis of stability studies of pharmaceuticals have been published by the ICH (International Conference on Harmonization). ICH Q1A(R2) [1] defines the core stability data package that is sufficient to support the registration of a new drug application in the tripartite regions of the European Union, Japan, and the United States. ICH Q1D [2] provides guidance on reduced designs for stability studies. It outlines the circumstances under which a bracketing or matrixing design can be used. ICH Q1E [3] describes the principles of stability data evaluation and various approaches to statistical analysis of stability data in establishing a retest period for the drug substance or a shelf life for the drug product that would satisfy regulatory requirements.

In this article, the discussion is focused on the statistical aspects of stability study designs in relation to the recent ICH guidelines. The statistical analysis of stability data will not be covered here. In Section 48.2, different types of experimental designs, such as complete factorial design and fractional factorial design, are exemplified. In Section 48.3, several commonly used criteria for design comparison are presented. In Section 48.4, the role of the stability study protocol is emphasized. In Section 48.5, the statistical and regulatory considerations in the selection of stability-study design, in particular, a full design vs. bracketing or matrixing design, are discussed. Finally, conclusions are made in Section 48.6.

48.2 Stability Study Designs

The design of a stability study is intended to establish, based on testing a limited number of batches of a drug substance or product, a retest period or shelf life applicable to all future batches of the drug substance or product manufactured under similar circumstances. Tested batches should therefore be representative in all respects, such as formulation, container and closure, manufacturing process, manufacturing site, and source of drug substance, for the population of all production batches and conform to the quality specification of the drug product. The stability study should be well designed so the shelf life of the product can be estimated with a high degree of accuracy and precision.

A typical stability study consists of several design factors (e.g. batch, strength, container size) and several response variables (e.g., assay, dissolution rate) measured at different time points under a variety of environmental conditions. One can categorize the stability designs as full, bracketing, or matrixing designs by the number of time points or levels of design factors as shown in Table 1. The following sections describe the characteristics, advantages, and disadvantages of various designs.

Table 1: Types of Stability Designs

48.2.1 Full Design

Several different types of experimental design can be applied to the stability study. A full design (also referred to as a complete factorial design) is one in which samples for every combination of all design factors are tested at all time points as recommended in ICH Q1A(R2), for example, a minimum of 0, 3, 6, 9, 12, 18, 24 months and yearly thereafter for long-term testing and 0, 3, and 6 months for accelerated testing.

As mentioned, ICH has recently issued a series of guidelines on the design and conduct of stability studies. ICH Q1A(R2) recommends that the drug substance or product should be stored at the long-term condition (e.g., 25°C ± 2°C/60% RH ± 5% RH or 30°C ± 2°C/65% RH ± 5% RH) that is reflective of the storage condition intended for the container label. Unless the drug substance or product is destined for freezer storage, it should also be stored at the accelerated condition for 6 months and tested minimally at 0, 3, and 6 months. The guideline also states that if long-term studies are conducted at 25°C ± 2°C/60% RH ± 5% RH and “significant change” occurs at any time during 6 months’ testing at the accelerated storage condition, additional testing at the intermediate storage condition should be conducted and evaluated.

A product may be available in several strengths, and each strength may be packaged in more than one container closure system and several container sizes. In this case, the resources needed for stability testing are considerable. Table 2 illustrates an example of a stability protocol using a full design for a drug product manufactured in three strengths and packaged in three container sizes and four container closure systems. This example shows that it will require 1188 test samples for long-term and accelerated stability testing.

Table 2: An Example of a Stability Protocol Using a Full Design According to the ICH Q1A(R2) Guideline

^* The long-term condition could be 25°C ± 2°C/60% RH ± 5% RH or 30°C ± 2°C/65% RH ± 5%RH

^** If long-term studies are conducted at 25°C ± 2°C/60% RH ± 5% RH and “significant change” occurs at any time during 6 months’ testing at the accelerated storage condition, then the intermediate testing is needed.

In addition, if long-term studies are conducted at 25°C ± 2°C/60% RH ± 5% RH and “significant change” occurs at any time during 6 months’ testing at the accelerated storage condition, the intermediate testing is needed and the number of test samples will be increased to 1620.

Table 3 shows an example of a simple full design. This example describes a protocol for long-term stability testing (25°C/60%RH) of a drug product manufactured in three strengths (25, 50, and 100 mg) and packaged in three container sizes (10, 50, and 100 ml). Samples for every combination of all design factors (i.e., strength, batch, and container size) are tested at all time points. Hence, this example is a complete factorial design, and the total number of samples tested is N = 3 × 3 × 3 × 8 = 216.

Table 3: Example of a Full Stability Study Design

As shown in the above examples, a full design involves complete testing of all factor combinations at all time points, which can be costly. The pharmaceutical industry would like to apply a reduced design so that not every factor combination will be tested at every time point. ICH Q1D listed some principles for situations in which a reduced design can be applied. However, before a reduced design is considered, certain assumptions should be assessed and justified. The potential risk should be considered for establishing a shorter retest period or shelf life than could be derived from a full design because of the reduced amount of data collected.

48.2.2 Reduced Design

As defined in ICH Q1D Guideline, a reduced design (also referred to as a fractional factorial design) is one in which samples for every factor combination are not all tested at all time points. Any subset of a full design is considered a reduced design. Bracketing and matrixing designs are two most commonly used reduced designs.

In 1989, Wright [4] proposed the use of factorial designs in stability studies. Nakagaki [5] used the terminologies matrix and bracket in his 1991 presentation. In 1992, Nordbrock [6], Helboe [7], and Carstensen et al. [8] published articles discussing several methods for reducing the number of samples tested from the chemistry and economic aspects of stability studies. Nordbrock investigated various types of fractional factorial designs and compared these designs based on the power of detecting a significant difference between slopes. He concluded that the design that gives acceptable performance and has the smallest sample size can be chosen based on power. Lin [9] investigated the applicability of matrixing and bracketing approaches to stability studies. She concluded that the complete factorial design is the best design when the precision of shelf life estimation is the major concern but indicated that matrixing designs could be useful for drug products with less variability among different strengths and container sizes. With regard to the statistical analysis of data from complex stability studies, Fairweather et al. [10], Ahn et al. [11], Chen et al. [12], and Yoshioka et al. [13] have proposed different procedures for testing and classifying stability data with multiple design factors.

Bracketing Design versus Matrixing Design ICH Q1D states that a reduced design can be a suitable alternative to a full design when multiple design factors are involved in the product being evaluated. However, the application of a reduced design has to be carefully assessed, taking into consideration any risk to the ability of estimating an accurate and precise shelf life or the consequence of accepting a shorter-than-desired shelf life. Bracketing and matrixing designs are the two most commonly used reduced designs. The reduced stability testing in a bracketing or matrixing design should be capable of achieving an acceptable degree of precision in shelf life estimation without losing much information.

The terms of bracketing and matrixing are defined in ICH Q1D as follows:

As defined above, bracketing and matrixing are two different approaches to designing a stability study. Each design has its own assumptions, advantages, and disadvantages. The applicability of either design to a stability study generally depends on the manufacturing process, stage of development, assessment of supportive stability data, and other factors as described in ICH Q1D. The following additional points can be considered regarding the use of a bracketing or matrixing design in a stability study.

In a bracketing design, samples of a given batch for a selected extreme of a factor are analyzed at all time points. Therefore, it is easier to assess the stability pattern in a bracketing study than in a matrixing study, in which samples of a given batch are often tested at fewer time points. If all selected strengths or container sizes tested show the same trend, it can be concluded with a high degree of certainty that the stability of the remaining strengths or container sizes is represented, or bracketed, by the selected extremes.

In a matrixing design, the samples to be tested are selected across all factor combinations. This procedure may be less sensitive to assessing the stability pattern than bracketing because of the reduced number of time points. Therefore, a matrixing design is more appropriate for confirming a prediction or available stability information, and is more effective in the following situations: (1) in the later stages of drug development when sufficient supporting data are available, (2) for stability testing of production batches, and (3) for annual stability batches. One of the advantages of matrixing over bracketing is that all strengths and container sizes are included in the stability testing.

The general applicability of bracketing and matrixing has been discussed in the literature [e.g., References 9,14–21]. From statistical and regulatory perspectives, the applicability of bracketing and matrixing depends on the type of drug product, type of submission, type of factors, data variability, and product stability. The factors that may be bracketed or matrixed in a stability study are outlined in ICH Q1D. This ICH guideline also briefly discusses several conditions that need to be considered when applying these types of design. A bracketing or matrixing design may be preferred to a full design for the purpose of reducing the number of samples tested, and consequently the cost. However, the ability to adequately predict the product shelf life by these types of reduced designs should be carefully considered.

In general, a matrixing design is applicable if the supporting data indicate predictable product stability. Matrixing is appropriate when the supporting data exhibit only small variability. However, where the supporting data exhibit moderate variability, a matrixing design should be statistically justified. If the supportive data show large variability, a matrixing design should not be applied.

A statistical justification could be based on an evaluation of the proposed matrixing design with respect to its power to detect differences among factors in the degradation rates or its precision in shelf life estimation.

If a matrixing design is considered applicable, the degree of reduction that can be made from a full design depends on the number of factor combinations being evaluated. The more factors associated with a product and the more levels in each factor, the larger the degree of reduction that can be considered. However, any reduced design should have the ability to adequately predict the product shelf life.

An example of a bracketing design is given in Table 4. Similar to the full design in Table 3, this example is based on a product available in three strengths and three container sizes. In this example, it should be demonstrated that the 10-ml and 100-ml container sizes truly represent the extremes of the container sizes. In addition, the 25-mg and 100-mg strengths should also represent the extremes of the strengths. The batches for each selected combination should be tested at each time point as in a full design. The total number of samples tested will be N = 2 × 2 × 3 × 8 = 96.

Table 4: Example of a Bracketing Design

An example of a matrixing-on-time-points design is given in Table 5. The description of the drug product is similar to the previous examples. Three time codes are used in two different designs. As an example of a complete one-third design, the time points for batch B in strength 100 mg and container size 10 ml are 0, 9, 24, and 36 months. The total number of samples tested for this complete one-third design will be N = 3 × 3 × 3 × 4 = 108.

Table 5: Example of a Matrixing-On-Time-Points Design

Table 6 illustrates an example of matrixing on both time points and factors. Similar to the previous example, three time codes are used. As an example, batch B in strength 100 mg and container 10 ml is tested at 0, 6, 9, 18, 24, and 36 months. The total number of samples tested for this example will be N = 3 × 3 × 2 × 6 = 108.

Table 6: Example of Matrixing On-Time-Points and Factors

48.2.3 Other Fractional Factorial Designs

Fractional factorial designs other than those discussed above may be applied. As bracketing and matrixing designs are based on different principles, the use of bracketing and matrixing in one design should be carefully considered. If this type of design is applied, scientific justifications should be provided.

48.3 Criteria for Design Comparison

Several statistical criteria have been proposed for comparing designs and choosing the appropriate design for a particular stability study. Most of the criteria are similar in principles but different in procedures. This section discusses several commonly used criteria for design comparison. As the statistical method used for analyzing the data collected should reflect the nature of the design, the availability of a statistical method that provides a valid statistical inference for the established shelf life should be considered when planning a study design.

The optimality criteria have been applied by several authors (Nordbrock [6], Ju and Chow [22]) to the selection of stability design. These optimality criteria are statistical efficiency criteria and have been developed and widely used (Kiefer [23] and Fedorov [24]) in other scientific areas for many years. Hundreds of journal articles on these optimality criteria have been published. The basic principle of this approach is described below.

Let Y denote the result of a test attribute, say assay. The following model can be used to describe Y:

where X is the design matrix, β is the coefficient vector, and is the residuals vector. The least squares solution of this matrix equation with respect to the model coefficient vector is β = (X′X)⁻¹ X′Y, and X′X is called the information matrix because its determinant is a measure of the information content in the design.

Several different optimality criteria are used for design comparison, and, among them, the D-optimality and A-optimality are the most commonly used. The basic principle behind these criteria is to choose a design that is optimal with respect to the precision of the parameter estimators. For example, say X₁ and X₂ are the design matrices for two different fractional factorial designs. A D-optimality criterion is that if Det(X′₁ X₁) > Det(X′₂ X₂), then design X₁ is said to be preferred to design’ X₂. An A-optimality criterion is that if Trace (X′₁ X₁) > Trace (X′₂ X₂), then design X₁ is said to be preferred to design X₂. The D-optimal design is used most often in experimental designs.

If the D-optimality concept is applied directly to stability studies, then the selection of observations (i.e., time points) that give the minimum variance for the slope is to place one-half at the beginning of the study and one-half at the end. However, in reality, the linearity of the test-attribute-vs.-time may not hold for many chemical and physical characteristics of the drug product. Hence, the statistically designed stability studies should not only apply this basic optimality principle but also include several intermediate time points (e.g., 3 and 6 months) as a check for linearity. However, depending on the drug product, manufacturing process, marketing strategy, and other factors, different designs may be chosen, and no single design is optimal in all cases. As analyses are typically done at several different times (e.g., at the time a registration application for the new drug product is filed, or yearly for a marketed product), the choice of design should also take into account that the analysis will be done after additional data are collected.

Nordbrock [6] proposed choosing a design based on the power for detecting a significant difference among slopes. The method he proposed consists of three steps. The first step is to list slopes that must be compared [i.e., to list factors and factor interactions (for slopes) that may affect stability]. The second step is to list some alternative designs with reduced sample sizes. Some general experimental design considerations are used to derive these alternative designs. The third step is to compare statistical properties of the designs based on the power of contracts of interest. He then suggested that “among designs with the same sample size, the design with the highest power is the best design. One way to select a design for a particular study is to choose the desired power, and then, from designs having at least the desired power, the design with the smallest sample size is the best design” [6].

The primary objective of a stability study is to establish a shelf life for the drug product and not to examine the effect of factors used in the experiment. Thus, the design chosen based on the power of contracts of interest as proposed by Nordbrock may not be the best choice. As an alternative, Ju and Chow [22] proposed a design criterion based on the precision of the shelf life estimation. Mathematically, this criterion of choosing a design is based on the following comparison:

Design X_A is considered to be better than design X_B if

where x(t) is the design matrix X at time t, and t is chosen based on the true shelf life. This article applied the shelf life estimation procedure developed by Shao and Chow [25] so that a random effect model was used. This criterion also takes into account the batch variability; hence, it could be applied to the balanced cases and could be extended to unbalanced cases.

Based on the above criterion and comparison results, Ju and Chow then proposed that “For a fixed sample size, the design with the best precision for shelf life estimation is the best design. For a fixed desired precision of shelf life estimation, the design with the smallest sample size is the best design” [22].

Murphy [26] introduced the uniform matrix design for drug stability studies and compared it with standard matrix designs. The strategy of the uniform matrix design is to delete certain times (e.g., 3, 6. 9, and 18 months); therefore, testing is done only at 12, 24, and 36 months. This design has the advantage of simplifying the data entry of the study design and eliminating time points that add little to reducing the variability of the slope of the regression line. The disadvantage is that, if major problems exist with the stability, no early warning occurs because early testing is not done. Furthermore, it may not be possible to determine whether the linear model is appropriate.

Murphy used five efficacy measures (design moment, D-efficiency, uncertainty, G-efficiency, and statistical power) to compare uniform matrix designs with other matrix designs. Three of the criteria (moment, D-efficiency, and power) are well known and widely used. Uncertainty is a measure of the reliability of the shelf life estimates, which is dependent on the reliability of the estimated slopes. Another measure of uncertainty is the width of a confidence interval on the fitted slope, which, for a fixed amount of residual variation, is dependent only on the number and spacing of the data points. G-efficiency is a relatively new concept that is defined as follows:

where

n = number of points in the design,

p = number of parameters,

σ_max = maximum standard error of prediction over the design.

Based on the comparisons of five different criteria, Murphy stated that “the uniform matrix designs provide superior statistical properties with the same or fewer design points than standard matrix design” [26].

Pong and Raghavarao [27] compared the power for detecting a significant difference between slopes and the mean square error to evaluate the precision of the estimated shelf life of bracketing and matrixing designs. They found the power of both designs to be similar. Based on the conditions under study, they concluded that bracketing appears to be a better design than matrixing in terms of the precision of the estimated shelf life.

Evidently, the above-mentioned criteria are based on different procedures. Thus, the final choice of design might be different with different criteria. In general, in addition to chemical and physical characteristics of the drug product, regulatory and statistical aspects need to be considered. Ideally, one would like to choose a design that is optimal with respect to the precision of the shelf life estimation and the power of detecting meaningful effects.

48.4 Stability Protocol

A good stability study design is the key to a successful stability program. The program should start with a stability protocol that specifies clearly the study objective, the study design, batch and packaging information, specifications, time points, storage conditions, sampling plan, statistical analysis method, and other relevant information. The protocol should be well designed and followed rigorously, and data collection should be complete and in accordance with the protocol. The planned statistical analysis should be described in the protocol to avoid the appearance of choosing an approach to produce the most desirable outcome at the time of data analysis. Any departure from the design makes it difficult to interpret the resulting data. Any changes made to the design or analysis plan without modification to the protocol or after examination of the data collected should be clearly identified.

48.5 Basic Design Considerations

48.5.1 Impact of Design Factors on Shelf Life Estimation

A drug product may be available in different strengths and different container sizes. In such a case, stability designs for long-term stability studies will involve the following design factors: strength, container size, and batch. As mentioned by Nordbrock [6] and Chow and Liu [28], it is of interest to examine several hypotheses, such as the following:

Hence, a need exists to investigate the impact of main effects (e.g., strength, container size, batch) and interaction effects on the stability of the drug product under long-term testing. In constructing a stability design, one needs to consider the extent to which it is acceptable to pool data from different design factors. For example, if a statistically significant interaction effect occurs between container size and strength, then a separate shelf life should be estimated for each combination of container size and strength. On the other hand, if no statistically significant interaction effect occurs (i.e., the degradation rates for all container sizes, strengths, and batches are the same), it will be acceptable to pool all results and calculate one single shelf life for all container sizes, strengths, and batches produced under the same circumstances. A design should be adequately planned such that it is capable of detecting the possible significant interaction effects and main effects.

A full design can provide not only valid statistical tests for the main effects of design factors under study but also better precision in the estimates for interactions. Hence, the precision of the estimated drug shelf life for a full design is better than a reduced design. A reduced design is preferred to a full design for the purpose of reducing the number of test samples and, consequently, the cost. However, it has the following disadvantages: (1) One may not be able to evaluate some interaction effects for certain designs. For example, for a 2^4–1 fractional factorial design, two-factor effects are confounded with each other; hence, one may not be able to determine whether the data should be pooled. (2) If interactions between two factors exist, such as strength by container size, the data cannot be pooled to establish a single shelf life. It is recommended that a separate shelf life for each combination of strength and container size should be established. However, no shelf life estimation for the missing factor combinations can be assessed. (3) If many missing factor combinations exist, there may not be sufficient precision for the estimated shelf life.

It is generally impossible to test the assumption that the higher-order terms are negligible. Hence, if the design does not permit the estimation of interactions or main effects, it should be used only when it is reasonable to assume that these interactions are very small. This assumption must be made on the basis of theoretical considerations of the formulation, manufacturing process, chemical and physical characteristics, and supporting data from other studies.

Thus, to achieve a better precision of the estimated shelf life, a design should be chosen so as to avoid possible confounding or interaction effects. Once the design is chosen, statistical analysis should reflect the nature of the design selected.

48.5.2 Sample Size and Sampling Considerations

The total number of samples tested in a stability study should be sufficient to establish the stability characteristics of the drug product and to estimate the shelf life of the drug product with an acceptable degree of precision.

The total number of test samples needed in a stability study generally depends on the objective and design of the study. For example, for a drug product available in a single strength and a single container size, the choice of design is limited (i.e., the design should have three batches of the product and samples should be tested every 3 months over the first year, every 6 months over the second year, and then annually). For drug products involving several design factors, several different types of design, such as, full, bracketing, and matrixing designs, can be chosen. In general, the number of design factors planned in the study, the number of batches, data variability within or across design factors, and the expected shelf life for the product all need to be considered when choosing a design. In addition, available stability information, such as the variability in the manufacturing process and the analytical procedures, also needs to be evaluated.

The estimation of shelf life of a drug product is based on testing a limited number of batches of a drug product. Therefore, tested batches should be representative in all respects as discussed above.

48.5.3 Other Issues

The purpose of selecting an appropriate stability design is to improve the accuracy and precision of the established shelf life of a drug product. Background information, such as regulatory requirements, manufacturing process, proposed specifications, and developmental study results are helpful in the design of a stability study.

The choice of stability study design should reflect the formulation and manufacturing process. For example, the study design and associated statistical analysis for a product available in three strengths made from a common granulation will be different from those made from different granulations with different formulations.

The stability study design should be capable of avoiding bias and achieving minimum variability. Therefore, the design should take into consideration variations from different sources. The sources of variations may include individual dosage units, containers within a batch, batches, analytical procedures, analysts, laboratories, and manufacturing sites. In addition, missing values should be avoided, and the reason for the missing values should be documented.

When choosing a matrixing design, one cannot rely on the assumption that the shelf life of the drug product is the same for all design factors. If the statistical results show that a significant difference exists among batches and container sizes, one cannot rely on certain combinations of batch and container size or on the statistical model to provide reliable information on the missing combinations. The shelf life must be calculated for the observed combinations of batch and container size. The shortest observed shelf life is then assigned to all container sizes.

48.6 Conclusions

A good stability study design is the key to a successful stability program. The number of design factors planned in the study, the number of batches, data variability within or across design factors, and the expected shelf life for the product all need to be considered when choosing a design. The recently published ICH guidelines, such as Q1A(R2), Q1D, and Q1E, should be perused, and the recommendations therein should be followed.

A reduced design, such as bracketing and matrixing, can be a suitable alternative to a full design when multiple design factors are involved in the product being evaluated. However, the application of a reduced design has to be carefully assessed, taking into consideration any risk to the ability of estimating an accurate and precise shelf life or the consequence of accepting a shorter-than-desired shelf life. When the appropriate study design is chosen, the total number of samples tested in a stability study should be sufficient to establish the stability characteristics of the drug product and to estimate the shelf life of the drug product with an acceptable degree of precision. To achieve better precision of the estimated shelf life, a design should be chosen to avoid possible confounding or interaction effects. Once the design is chosen, the statistical method used for analysis of the data collected should reflect the nature of the design and provide a valid statistical inference for the established shelf life.

Acknowledgments. The authors would like to thank the FDA CDER Office of Biostatistics Stability Working Group for support and discussion on the development of this manuscript.

References

[1] Guidance for Industry: ICH Q1A(R2) Stability Testing of New Drug Substances and Products. Food and Drug Administration, Center for Drug Evaluation and Research and Center for Biologics Evaluation and Research, January, 2003.

[2] Guidance for Industry: ICH Q1D Bracketing and Matrixing Designs for Stability Testing of New Drug Substances and Products. Food and Drug Administration, Center for Drug Evaluation and Research and Center for Biologics Evaluation and Research, January, 2003.

[3] Draft ICH Consensus Guideline Q1E Stability Data Evaluation. Food and Drug Administration, Center for Drug Evaluation and Research and Center for Biologics Evaluation and Research, January 2003.

[4] J. Wright. Use of factorial designs in stability testing. In Proceedings of Stability Guidelines for Testing Pharmaceutical Products: Issues and Alternatives. AAPS Meeting, December 1989.

[5] P. Nakagaki, AAPS Annual Meeting, 1990.

[6] E. Nordbrock, Statistical comparison of stability study designs. J, Biopharm. Stat. 1992; 2: 91–113.

[7] P. Helboe, New designs for stability testing programs: matrix or factorial designs. Authorities viewpoint on the predictive value of such studies. Drug Info J. 1992; 26: 629–634.

[8] J. T. Carstensen, M. Franchini, and K. Ertel, Statistical approaches to stability protocol design. J. Pharm. Sci. 1992; 81(3): 303–308.

[9] T. D. Lin, Applicability of matrix and bracket approach to stability study design. In:Proceedings of the American Statistical Association Biopharmaceutical Section, 1994: 142–147.

[10] W. Fairweather, T. D. Lin, and R. Kelly, Regulatory, design, and analysis aspects of complex stability studies. J. Pharm. Sci. 1995; 84(11): 1322–1326.

[11] H. Ahn, J. Chen, and T. D. Lin, A two-way analysis of covariance model for classification of stability data. Biometr. J. 1997; 39(5): 559–576.

[12] J. Chen, H. Ahn, and Y. Tsong, Shelf life estimation for multi-factor stability studies. Drug Info J. 1997; 31(2): 573–587.

[13] S. Yoshioka, Y. Aso, and S. Kojima, Assessment of shelf-life equivalence of pharmaceutical products. Chem. Pharm. Bull. 1997; 45: 1482–1484.

[14] T. D. Lin and W. R. Fairweather, Statistical design (bracketing and matrixing) and analysis of stability data for the US market. Proceedings from IBC Stability Testing Conference, London, 1997.

[15] T. D. Lin, Statistical considerations in bracketing and matrixing. Proceedings from IBC Bracketing and Matrixing Conference, London, 1999.

[16] T. D. Lin, Study design, matrixing and bracketing. AAPS Workshop on Stability Practices in the Pharmaceutical Industry — Current Issues, Arlington, Virgina, 1999.

[17] W. P. Fairweather and T. D. Lin, Statistical and regulatory aspects of drug stability studies: an FDA perspective. In: D. Mazzo, ed. International Stability Testing. Interpharm Press, Inc., 1999, pp. 107–132.

[18] C. Chen, US FDA’s perspective of matrixing and bracketing. In: Proceedings from EFPIA Symposium: Advanced Topics in Pharmaceutical Stability Testing Building on the ICH Stability Guideline, EFPIA, Brussels, Belgium, 1996.

[19] D. Chambers, Matrixing/Bracketing. US industry views. In: Proceedings from EFPIA Symposium: Advanced Topics in Pharmaceutical Stability Testing Building on the ICH Stability Guideline, EFPIA, Brussels, Belgium, 1996.

[20] P. Helboe, Matrixing and bracketing designs for stability studies: an overview from the European perspective. In: D. Mazzo, ed. International Stability Testing. Interpharm Press, Inc., 1999, pp. 135–160.

[21] S. Yoshioka, Current application in Japan of the ICH stability guidelines: does Japanese registration require more than others do? In: D. Mazzo, ed. International Stability Testing. Interpharm Press, Inc., 1999, pp. 255–264.

[22] H. L. Ju and S. C. Chow, On stability designs in drug shelf life estimation. J. Biopharm. Stat. 1995; 5(2): 201–214.

[23] J. Kiefer, Optimal experimental designs. J. Royal Stat. Soc. Series, B 1959; 21: 272–319.

[24] V. V. Fedorov, Theory of Optimal Experiments. W. J. Studden and E. M. Klimko, eds. New York: Academic Press, 1972.

[25] J. Shao and S. C. Chow, Statistical inference in stability analysis. Biometrics 1994; 50(3): 753–763.

[26] J. R. Murphy, Uniform matrix stability study designs. J. Biopharm. Stat. 1996; 6(4): 477–494.

[27] A. Pong and D. Raghavarao, Comparison of bracketing and matrixing designs for a two-year stability study. J. Pharm. Stat. 2000; 10(2): 217–228.

[28] S. C. Chow and J. P. Liu, eds. Statistical Designs and Analysis in Pharmaceutical Science. New York: Marcel Dekker, 1995.