Statistical Considerations in Design Space Development (Part III of III)

In the final article of a three-part series, the authors discuss how to present a design space and evaluate its graphical representation.

Parts I and II of this article appeared in the July and August 2010 issues, respectively, of Pharmaceutical Technology and discussed experimental design planning and design and analysis in statistical design of experiments (DoE) (1, 2). This article, Part III, covers how to present and evaluate a design space.

Design space is part of the US Food and Drug Administration's quality initiative for the 21st century which seeks to move toward a new paradigm for pharmaceutical assessment as outlined in the International Conference on Harmonization's quality guidelines Q8, Q9, and Q10. The statistics required for design-space development play an important role in ensuring the robustness of this approach.

This article provides concise answers to frequently asked questions (FAQs) related to the statistical aspects of determining a design space as part of quality-by-design (QbD) initiatives. These FAQs reflect the experiences of a diverse group of statisticians who have worked closely with process engineers and scientists in the chemical and pharmaceutical development disciplines, grappling with issues related to the establishment of a design space from a scientific, engineering, and risk-based perspective. Questions 1–7 appeared in Part I of this series (1). Questions 8–22 apeared in Part II (2). The answers provided herein, to Questions 23–29, constitute basic information regarding statistical considerations and concepts to consider when finalizing a design space and will be beneficial to a scientist working to develop and implement a design space in collaboration with a statistician.

Presenting a design space

This section reviews the presentation of the design space, including: tabular display of summary information, and graphical displays such as contour plots, three-dimensional surface plots, overlay plots, and desirability plots. The authors discuss the presentation of the design space based on multifactor equations or a multidimensional rectangular rather than as a system of multifactor equations. Traditionally, one would evaluate the design space before finalizing the presentation. In this article, however, the presentation of the design space is provided first in order to explain the graphics used in the evaluation stage.

Q23: How can a design space be presented?

Table III: Summary of design space for assay and degradate 1.
A: When a design space has been developed from a statistical design of experiments (DoE), there are many effective displays, including tabular and graphical summaries. An example of a tabular summary is presented in Table III* from a two-factor central composite design described in Part II of this article series (2). Table III summarizes the quality characteristic and its specification or requirements, the range of the data realized in the experiment, the regression model, and the root mean square error (RMSE). The RMSE is an estimate of the standard deviation in the data (after model fit) that is not accounted for by the model. If there is an expected amount of variation for the quality characteristic, then this estimate can be used to determine whether the model fits the data appropriately. If the RMSE is too small, then the model may be over-fitting the data; if too large, the model may be missing some terms. The range of the observed data is provided to indicate where the model is applicable. Predictions outside this range (extrapolation) should be confirmed with additional experimentation. This summary is just one possibility for a tabular approach.

Some effective graphical displays include contour plots, three-dimensional surface plots, overlay plots, and desirability plots. Each graph has strengths and weaknesses. It is anticipated that multiple graphs or graph types may be needed to clearly display the design space.