The supplementary material contains the proofs and derivations for equations (

Systems with both quantitative and qualitative responses are widely encountered in many applications. Design of experiment methods are needed when experiments are conducted to study such systems. Classic experimental design methods are unsuitable here because they often focus on one type of response. In this paper, we develop a Bayesian

In many applications, both quantitative and qualitative responses are often collected for evaluating the quality of the system. Often, the two types of responses are mutually dependent. We call such a system with both types of quality responses quantitative-qualitative system. Such systems are widely encountered in practice [

This work is motivated by a study of an etching process in a wafer manufacturing process. In the production of silicon wafers, the silicon ingot is sliced into wafers in fine geometry parameters. Inevitably, this step leaves scratches on the wafers’ surface. An etching process is used to improve the surface finish, during which the wafers are submerged in the container of etchant for chemical reaction. The quality of the wafers after etching is measured by two response variables: the total thickness variation of the wafer (TTV) and the binary judgment that whether the wafer has cloudy stains in its appearance. The two responses measure the quality from different but connected aspects. There is a hidden dependency between the continuous TTV and binary judgment of stains. To improve the etching quality, expensive experiments are to be carried out to reveal this hidden dependency and to model the quality-process relationship. Therefore, an ideal experimental design for continuous and binary responses should be able to extract such useful information with economic run size.

The classic design of experiments methods mainly focus on experiments with a single continuous response. There have been various methods developed for a single discrete response too, including [

Denote by

(a) Observations from design for

In this example, there is a strong dependency between the two responses since the true underlying models of

Such experiments call for new experimental design methods to account for both continuous and binary responses. Note that under the experimental design framework, the linear model is often considered for modeling the continuous response, and the generalized linear model (GLM) is often considered for modeling the qualitative response. A joint model must be developed to incorporate both types of responses. Compared to the classic design methods for linear models or GLMs, design for the joint model is more challenging due to the following aspects. First, the design criterion for the joint model is more complicated, as the joint model is more complicated than the separate ones. Second, experimental design for the GLM itself is more difficult than that for the linear model, which is naturally inherited by the design for the joint model. Third, efficient design construction algorithms are needed to handle the complexity of the design criterion based on the joint model. [

In this paper, we choose the most commonly used

The rest of the paper is organized as follows. Section

We first review the general Bayesian QQ model introduced in [

The association between the two responses

Denote

When lacking domain knowledge or proper historical data, experimenters often favor the frequentist approach as no priors need to be specified. The frequentist approach can be seen as the Bayesian approach using noninformative priors. In this section, we derive the optimal design criterion and the regularity conditions for noninformative priors.

Assume the non-informative priors

Using the posterior distributions (

To construct the optimal design, we consider maximizing the approximated

The proof of Theorem

The matrices

Let

The if-and-only-if condition for

Proposition

Proposition

When prior information for parameters

For the parameters

For the parameters

The proof of Proposition

For the same argument as in Section

Note that the criterion in (

The criterion

Note that the conjugate prior

The design criterion

The setting of

To specify different prior distributions for

In computation, we construct

In this work, we focus on the construction of optimal design based on factorial design, which is suited for the prior distribution introduced in Section

For a fixed

The point-wise exchange algorithm finds the optimal design from a candidate set. Here the candidate set is chosen to be the full factorial design without replicates. For now, we develop the method for 2- and 3-level factors, but it can be generalized to factors of more levels. Use previous notation that

Next we construct the initial design of size

To simplify the notation for

One key of the point-wise exchange algorithm is to compute

Given the initial design, we can iteratively exchange the current design points with candidate design points to improve the objective

Exchange-Point Algorithm for Local

Generate the candidate design set from full factorial design. Filter out the points with probabilities

Generate the initial design. Based on the initial design

Compute the deletion function

Find

Repeat

The Algorithm

Based on Algorithm

Algorithm for Global

If

For each

For each point in the candidate set, count its frequency of being selected in the local optimal designs. The continuous optimal design is formed by the normalized frequency as a discrete distribution.

To obtain a discrete optimal design, sample

In Step 1 of generating

In this section, we use two examples to demonstrate the proposed Bayesian

In this artificial experiment, there are three 2-level factors

First order effects:

An example of

Effect | Effect | Effect | Effect | ||||

Intercept | −0.0153 | −0.6067 | 0.7212 | 0.0080 | |||

−0.1682 | 0.0010 | 0.1349 | 0.0283 | ||||

0.0594 | −0.1719 | 0.1492 | −0.1468 | ||||

0.0553 | −0.0634 | −0.2629 | −0.0660 | ||||

−0.1054 | −0.0857 | −0.0807 | −0.1198 | ||||

−0.0292 | −0.1336 |

We set the experimental run size to be

To evaluate the performance of the proposed design in comparison with alternative designs, we consider the efficiency between two designs [

Design efficiency between the proposed local design

0 | 1.08 | 1.11 | 1.05 |

0.3 | 1.10 | 1.14 | 1.07 |

Next, we focus on the comparison of

Artificial example: the efficiency between each local

Based on Algorithm

To compare the performances of the global designs, the design efficiencies in (

Artificial example: global QQ optimal designs for (a)

Artificial example: the efficiency between each global QQ optimal design and combined design under different

In the etching process described in Section

Because of the newly developed process, the historical data on similar processes are not directly applicable to this experiment. Based on some exploratory analysis, we set

We generate a maximin Latin hypercube design of

Fractional factorial design [

Etching experiment: (a) the global Bayesian QQ

Etching experiment: histograms of the efficiencies (a) efficiencies between local designs

In this paper, we propose the Bayesian

Other than the conjugate prior

In this paper, we focus on optimal designs for one quantitative response and one qualitative response. The proposed method can also be generalized to accommodate the QQ models with multiple quantitative responses

The point-exchange algorithm is to construct the exact discrete optimal designs, which are different from the theoretical continuous optimal designs. As described in Sections

The proposed design is not restricted to the logit model for the binary response. For example, if the probit model is used, the Bayesian

It is worth pointing out that the design criterion in the work is based on the QQ model constructed by the joint model of