In recent articles on statistical aspects of the ICH Q8 Definition of Design Space, Peterson and colleagues from GlaxoSmithKline (see link below) describe a method to use Bayesian statistics to develop a design space for multiple responses obtained from separate statistical DOE models (one for each response), taking account of uncertainty and ultimately estimating the joint probability of success (for all responses) across the factor space.
Peterson and colleagues also note that:
“Traditionally, multiple response surface optimization has involved two approaches: overlapping mean responses and desirability functions. Tools for such methods are available in statistical packages such as SAS/JMP, Minitab, Design Expert, Statistica, and so forth. However, as shown by Peterson (2004), these two approaches each possess the same two major flaws. One flaw is that they do not account for the model parameter uncertainty. The other flaw is that they do not take into account the correlation structure of the multivariate regression model error vectors.”
It is highly desirable to factor uncertainty into determination of a design space and to quantify the probability of success when operating within the design space. The note linked to below presents example calculations on this topic using results from DynoChem 2008. The approach is applicable when an arbitrary number of responses and factors have to be considered. The probability can be visualised easily in 3-d (two factors) but the method is equally applicable when there are more than two factors.
A technical note may be downloaded here (login required) containing example calculations of design spaces using DynoChem models of i) average responses (overlapping when there is more than one) and ii) estimated probabilities of success, taking account of model parameter uncertainty and error. When a high success probability threshold is used to define the design space (e.g. 0.9), the resulting design space is typically smaller than that obtained when overlapping average responses are used.
Link to GSK paper: http://dx.doi.org/10.1080/10543400802278197