Using Blocked Fractional Factorial Designs to Construct Discrete Choice Experiments for Health Care Studies

Abstract

Discrete choice experiments (DCEs) are increasingly used for studying and quantifying subjects preferences in a wide variety of health care applications. They provide a rich source of data to assess real-life decision making processes, which involve trade-offs between desirable characteristics pertaining to health and health care, and identification of key attributes affecting health care.

The choice of the design for a DCE is critical because it determines which attributes’ effects and their interactions are identifiable. We apply blocked fractional factorial designs to construct DCEs and address some identification issues by utilizing the known structure of blocked fractional factorial designs. Our design techniques can be applied to several situations including DCEs where attributes have different number of levels. We demonstrate our design methodology using two health care studies to evaluate (1) asthma patients’ preferences for symptom-based outcome measures, and (2) patient preference for breast screening services.

1. Introduction

There is increasing belief that further development and progress in health care must be accompanied by greater understanding of patients’ preferences [1, 2]. Discrete choice experiments (DCEs) are one of several tools for attaining this objective and are increasingly used to understand and quantify patients’ preferences in a broad variety of health care applications.

A DCE is a survey method for identifying key attributes in health care services and health care outcomes. In a DCE, subjects (e.g., patients) are presented hypothetical scenarios known as choice sets. Each choice set is made up of several options. Each option consists of several attributes and each attribute has one or more levels measuring a different aspect of the outcome of interest. Subjects are asked to select a single option from each choice set. The option chosen is assumed to have the highest utility for the subject, where utility is the benefit that the subject experiences by selecting a particular option. Therefore, the option chosen by the subject implies an implicit trade-off between attributes. For example, a DCE in health care can be used for the quantification of the trade-off between various attributes that characterize preference for a treatment [3]. Section 2 presents an example of a DCE to measure asthma patients’ preferences for symptom-based outcome measures using five attributes each with two levels. This DCE has eight choice sets, where each choice set has two options, and subjects are asked to select a single option from each choice set. A checklist for good research practices in conducting a DCE has been developed by Lancsar and Louviere [4] and Johnson et al. [5].

The design of a DCE is a critical aspect. The design of a DCE refers to the selection of different combinations of attributes and levels to use in the choice sets and options. However, many papers that implement DCEs do not discuss their design choice. For example, De Bekker-Grob et al. [6] reviewed DCEs in health economics and noted that out of the 114 papers in the review, 37% did not report the rationale of the design, while 28% did not report sufficient details on how the choice sets were created. Louviere [7], emphasized the importance of the design of a DCE and stated that “researchers should recognize that the designs chosen for DCEs are at least as, if not more important than the models that one uses to analyze the resulting data.” In particular, it is essential that the researcher justifies the particular choice of the design because it determines which effects are identifiable, i.e. effects that can be estimated unbiasedly from the design. Lancsar and Louviere [4] stated that, “the choice of an appropriate design for a DCE is entirely under the control of the researcher, and identification problems should not occur.” Louviere and Lancsar [8] also suggested “that one first focus on identification because identification cannot be changed once a design is constructed.”

Current methods for constructing DCEs largely aim at estimating the individual effect of each attribute, referred to as a main effect; however, there should also be a focus on estimating the interaction between attributes, referred to as an interaction effect. De Bekker-Grob et al. [6] found that 89% of the designs used in their review focused on estimating main effects and only 5% of the designs could estimate main effects plus two-factor interactions (i.e., the interaction between two attributes). This suggests that some DCE users may not be fully aware that an improperly designed study can waste resources and may not be able to provide important inference on the interaction effects between attributes that can influence subjects’ preferences. Recently, Clark et al. [9], published a paper that updates and builds upon the De Bekker-Grob et al. review (and earlier systematic reviews) by focusing on literature from 2009 – 2012; in this review 54% of the designs focused on the estimation of main effects only, 13% focused on the estimation of main effects plus two-factor interactions, and approximately 30% of the studies did not report which effects could be estimated. Mandeville et al. [10] reviewed the use of DCEs to inform health workforce policy, and found similar results to De Bekker-Grob et al. [6] and Clark et al. [9]. In Mandeville et al. [10], almost 90% of the papers considered in their study used designs that focused on only estimating the main effects. Mandeville et al. [10] state the importance of being able to identify the main effects plus interaction effects, “it is likely to be inaccurate, albeit pragmatic, to assume that the main effects of attributes are not confounded by each other. The inclusion of selected interaction terms in design plans should be encouraged, based on those that are most likely to be conceptually valid.” As a simple illustration, suppose it is known a priori that there is interaction between two attributes: pain severity and pain duration. A subject may not be able to assess pain severity without knowing how long the pain will last, and may not be able to assess pain duration without knowing how severe the pain is during a specified period [5]. Therefore, the identification of the parameters of interest in a study should not be left to chance because understanding the interactive effect between two attributes is key to gaining insight into subjects’ preferences.

DCEs are playing an increasingly important role in public health and are particularly beneficial in health care studies. DCEs “allow the integration of patients’ values on all aspects of care in one measure and can be used to see how subjects trade different health outcomes” [3]. This is useful when deciding on the optimal way to provide a service within limited resources [2]. Some examples of applications to health care studies within the last six years are: (1) women’s preferences for place of delivery in rural Tanzania [11], (2) subject preference for new and existing contraceptive products [12], (3) health worker preferences for community-based health insurance payment mechanisms [13], (4) preferences for working in rural clinics among trainee health professionals in Uganda [14], and (5) preferences and willingness to pay for lifestyle programs for subjects’ with type 2 diabetes [15]. More recently, in the last year, we see a continuing rapid growth of applications of DCEs to study health care problems. They include a DCE to value EQ-5D health states in Australia [16], subjects’ valuation of prescribing nurse in primary care [17], eliciting subjects’ preferences for epilepsy diagnostics [18], subject preferences for prenatal test for Cystic Fibrosis [19], subjects’ preferences for osteoporosis drug treatment [20], and quantifying public preferences for different bowel preparation options prior to screening CT colonography [21]. The above applications are just a very small sample of DCEs seen in the health care setting; there are also many applications of DCEs in economics, transportation, and marketing; see Soetevent and Kooreman [22] and, Kamargianni et al. [23], to name a few.

The aim of our paper is to address some identification problems that can arise from DCEs by constructing designs that utilize the structure of blocked fractional factorial designs (BFFDs). BFFDs are widely used in scientific and industrial experiments and we show how they can be used to construct DCEs. An advantage of our proposed approach is that we know in advance which attributes’ effects and their interactions can be identified. Specifically, we show how a BFFD can be used to construct a DCE by combining attributes and their levels into choice sets such that each choice set is treated as a block and the number of options within each choice set is equal to the block size.

In the next section, we present an application of a DCE in the health care setting and discuss how typical design issues can arise in DCEs. Section 3 reviews fundamental concepts from two-level fractional factorial designs (FFDs). Section 4 reviews blocked FFDs when all attributes have two-levels in the context of DCEs. In particular, we show how to use two-level BFFDs to construct DCEs with two-level attributes. In Section 5, we apply BFFDs and construct DCEs with two- and three- level attributes using an asthma study where all attributes have the same number of levels, i.e. symmetric attributes. In Section 6, we extend our approach and construct DCEs when attributes have different number of levels, i.e. asymmetric attributes. In Section 7, we discuss the multinomial logit (MNL) model commonly used for analyzing DCEs and conduct a simulation study to confirm advantages of our proposed approach. Section 8 summarizes our work and briefly discusses alternative ways to construct designs for DCEs.

2. Asthma symptom study

To illustrate the concepts of a DCE, we use McKenzie et al.’s [24] study, which uses a DCE to measure asthma patients’ preferences for symptom-based outcome measures. Five attributes each with three levels were previously selected through existing symptom-based asthma outcome measures and through discussions with the local respiratory medicine department. The five attributes: “daytime cough”, “daytime breathlessness”, “daytime wheeze”, “daytime chest tightness” and “sleep disturbance as a result of night-time asthma symptoms”, and their levels are shown in Table 1. Attributes are denoted as x₁, x₂, x₃, x₄, x₅ and each of the three levels are denoted as 0, 1, 2.

Table 1.

Asthma symptom study attributes and levels; McKenzie et al. [24].

Attributes	Attribute levels	Coded levels
x1: Cough	No cough	0
	Some coughing but no restricted activities	1
	A lot of coughing with restricted activities	2
x2: Breathlessness	No breathlessness	0
	A little breathlessness but no restricted activities	1
	Very breathlessness with restricted activities	2
x3: Wheeze	No wheeze	0
	Some wheezing but no restricted activities	1
	Very wheezy with restricted activities	2
x4: Chest tightness	Chest not tight	0
	A little tightness	1
	Chest very tight	2
x5: Sleep disturbance	No sleep disturbance	0
	Awoke once with cough/breathlessness	1
	Awoke 2–3 times with cough/breathlessness	2

To demonstrate design issues in DCEs, we assume that each attribute has only two levels, and the two levels are coded levels 0 and 1 in Table 1. Once the attributes and their levels have been selected, experimental design theory can be used to combine these attributes and their levels into choice sets and options. Table 2 shows an example of a DCE where each of the five attributes has two levels. This DCE has eight choice sets each with two options. Subject’s are shown these choice sets one at a time in random order and asked to select their preferred option from each choice set. In Section 5, we discuss how this DCE was constructed using a two-level BFFD.

Table 2.

Asthma symptom study. A DCE with eight choice sets each with two options.

We note that the two levels 0 and 1, should be coded as 1 and −1, respectively, for each attribute when constructing a BFFD.

Choice Set	Option	Cough	Breathlessness	Wheeze	Chest tightness	Sleep disturbance
1	1	No cough (0)	Little breathlessness (1)	Some wheezing (1)	A little tightness (1)	No sleep disturbance (0)
	2	Some coughing (1)	No breathlessness (0)	No wheeze (0)	Chest not tight (0)	Awoke once (1)
2	1	No cough (0)	Little breathlessness (1)	Some wheezing (1)	Chest not tight (0)	No sleep disturbance (0)
	2	Some coughing (1)	No breathlessness (0)	No wheeze (0)	A little tightness (1)	Awoke once (1)
3	1	Some coughing (1)	No breathlessness (0)	Some wheezing (1)	Chest not tight (0)	No sleep disturbance (0)
	2	No cough (0)	Little breathlessness (1)	No wheeze (0)	A little tightness (1)	Awoke once (1)
4	1	No cough (0)	Little breathlessness (1)	No wheeze (0)	Chest not tight (0)	Awoke once (1)
	2	Some coughing (1)	No breathlessness (0)	Some wheezing (1)	A little tightness (1)	No sleep disturbance (0)
5	1	No cough (0)	No breathlessness (0)	Some wheezing (1)	A little tightness (1)	Awoke once (1)
	2	Some coughing (1)	Little breathlessness (1)	No wheeze (0)	Chest not tight (0)	No sleep disturbance (0)
6	1	No cough (0)	No breathlessness (0)	Some wheezing (1)	Chest not tight (0)	Awoke once (1)
	2	Some coughing (1)	Little breathlessness (1)	No wheeze (0)	A little tightness (1)	No sleep disturbance (0)
7	1	Some coughing (1)	Little breathlessness (1)	Some wheezing (1)	Chest not tight (0)	Awoke once (1)
	2	No cough (0)	No breathlessness (0)	No wheeze (0)	A little tightness (1)	No sleep disturbance (0)
8	1	Some coughing (1)	Little breathlessness (1)	Some wheezing (1)	A little tightness (1)	Awoke once (1)
	2	No cough (0)	No breathlessness (0)	No wheeze (0)	Chest not tight (0)	No sleep disturbance (0)

Design issues in DCEs can be very specific or more broad ranging and involve several aspects of the design construction. In this particular example, design questions may be on how the options are selected and grouped into choice sets, or on the selection of the number of options in each choice set and the number of choice sets. In this paper, the design questions of interest that we focus on are the number of choice sets, what options to use in each choice set, and how many options per choice set. Throughout we assume the following: (1) the attributes and levels are pre-selected either from focus group studies or by experts in the area; (2) all choice sets have the same number of options; (3) subjects are shown choice sets one at a time and must select an option from each choice set, and their selection is independent from one choice set to another. We further assume that the relative importance of each attribute can be estimated using the MNL model discussed in Section 7.

3. Experimental design: Two-level fractional factorial designs

In this section, we review fundamental concepts from two-level regular FFDs using traditional FFD terminology. An excellent outline and introduction to FFDs in the public health setting can be found in Nair et al. [25], and more generally in Wu and Hamada [26]. Consider a DCE with k two level attributes, then the total number of possible treatment combinations is 2^k. This is known as a full factorial design. However, a full factorial design may require too many treatment combinations for practical purposes. Rather, we may consider a fractional factorial design (FFD). In general, a FFD with k attributes in 2^k–p combinations is said to be a (2^–p)^th fraction of the full 2^k design with p design generators. FFDs have fewer treatment combinations than full factorial designs, but require that one willingly trades off estimating some of the interaction terms [27].

As an example, suppose there is interest to construct a half-FFD to study the five attributes (each with two levels) in the asthma study, i.e., a one-half fraction of the full 2⁵ factorial design. We follow standard practice in FFDs and code the two levels 0 and 1, as 1 and –1 for each attribute. A key issue is how this fraction should be chosen. We first write down all possible 2⁴ level combinations for attributes x₁, x₂, x₃, and x₄. Then introduce attribute x₅ as the design generator. We set the level of attribute x₅ as the product of the levels of attributes x₁x₂x₃x₄, i.e. x₅ = x₁x₂x₃x₄. The main effect x₅ is aliased with a four-factor interaction, x₁x₂x₃x₄, i.e., the main effect is indistinguishable from the four-factor interaction. Additionally, there is aliasing among other effects in this design: each main effect is aliased with a four-factor interaction and each two-factor interaction is aliased with one three-factor interaction. To disentangle these effects, a common and reasonable assumption is that higher-order interactions are assumed to be negligible because they are less likely to be important than lower-order interactions [25, 26]. We say that a main effect or two-factor interaction is clear if none of its aliases are main effects or two-factor interactions [26]. We can estimate clear main effects, or clear two-factor interactions under the assumption that all three-factor interactions and higher are negligible, without having to assume negligibility on other two-factor interactions. A related concept is resolution, which captures the amount of aliasing. This half-FFD has resolution V and all main effects plus all two-factor interactions are clear. To further illustrate the concept of resolution, consider a 2^5–2 FFD with designs generators: x₄ = x₁x₂ and x₅ = x₁x₃. This design is of resolution III, and none of the main effects or two-factor interactions are clear because each main effect is aliased with at least one two-factor interaction. The main effects can only be estimated under the assumption that all aliased two-factor interactions are negligible.

FFDs of resolution III are generally desirable because they have a smaller number of treatment combinations. However, these designs may give misleading results because the main effects are aliased with two-factor interactions, leading to biased estimates of the main effects if two-factor interactions are not negligible. A design of at least resolution IV is desirable because (1) all main effects can be identified as clear; thus, we do not need to assume any two-factor interactions negligible to (clearly) estimate the main effects, and (2) with many resolution IV designs, in addition to being able to identify all main effects as clear, we also can identify select two-factor interactions as clear. By knowing which effects are clear, we know which effects can be identified. Tables of previously constructed FFDs based on the maximum number of clear effects can be found in Appendix 5A.1 – 5A.7 of Wu and Hamada [26]. FFDs are beneficial because the entire aliasing structure of the design is known, which in turn allows us to know which effects are clear and thus identification problems do no exist.

Table 3 provides a summary of key terminology for statistical experimental design for constructing full and fractional factorial designs, as well as terms for blocked fractional factorial designs to be introduced in Section 4.

Table 3.

Statistical Experimental Design Terminology.

Term	Definition/Explanation
k	Number of attributes (factors)
2k	Two-level full factorial design
2k–p	Two-level fractional factorial design (FFD)
Design generator(s)	Determines the fraction of the full factorial design
p	Number of design generators for a FFD
2k–p FFD in 2q blocks	Two-level blocked fractional factorial design (BFFD)
Block generator(s)	Determines the block effect(s)
q	Number of block generators
2q	Number of choice sets (blocks)
2k–p–q	Choice set size, i.e., number of options in each choice set (block size)
Aliasing	Two or more effects cannot be separated
Clear effects	A main effects or two-factor interaction is clear if none of its aliases are main effects or two-factor interactions, or confounded with any block effects
Resolution	Captures the amount of aliasing (Higher resolution = aliasing of higher-order effects)
Confounding	Factorial effect is indistinguishable from block effect

4. Two-level blocked fractional factorial designs for discrete choice experiments

In this section, we show how to construct choice sets and options for DCEs with two-level attributes from previously constructed two-level BFFDs. We consider designs of at least resolution IV to ensure the clear estimation of main effects, as well as possibly select two-factor interactions. Our interest in constructing DCEs using BFFDs comes from the idea that combining attributes and their levels to form choice sets corresponds directly to the use of treatment combinations in BFFDs. BFFDs naturally extend to the area of DCEs, where each choice set is an inherent block [28].

To construct a BFFD we deliberately confound an interaction effect with a block effect. This means that the design is unable to estimate the two effects separately in exchange for higher precision because the differences associated between blocks are eliminated [29]. To block a 2^k–p FFD in 2^q blocks defined by q blocking variables, with blocks of size 2^k–p–q, we need to have two groups of generators: the design generators and the block generators. Any effect, including any aliased effects, associated with these blocking variables, are confounded with the blocks [26]. The concept of a clear effect extends to BFFDs - we say that a main effect or a two-factor interaction is clear in a BFFD if it is not aliased with any other main effects or two-factor interactions, or confounded with any block effects [26]. Hence, if an effect is confounded with a block effect, it cannot be estimated. If an effect is aliased with another effect (not a block effect), it can be estimated if all the aliased effects are negligible. Therefore, it is advantageous to use a BFFD to design a DCE because the entire aliasing and confounding structure of the design is known and it can be predetermined which effects are identifiable.

To use a BFFD to construct the choice sets and options in a DCE we let the number of blocks from a BFFD represent the number of choice sets, and let the size of the block from a BFFD represent the number of options within each choice set, i.e., the number of choice sets in a DCE is 2^q (the number of blocks) and the number of options in the choice sets are 2^k–p–q (the block size). The choice of BFFD depends on the number of attributes k, the desired size of the choice set (i.e., the number of options) and effects to be identified as clear. When using a BFFD to construct the choice sets, the number of options in the choice sets depend on the attribute levels, i.e., the number of options is a power of the attribute levels. Using a two-level BFFD, choice sets can be constructed with either two or four options, i.e., a power of two. To construct a choice set with two options, the necessary block size is 2^k–p–q = 2. To construct a choice set with four options, the necessary block size is 2^k–p–q = 4. If desired, choice sets with more than four options can also be constructed using two-level BFFDs, i.e., 2^k–p–q = 8, 16, . . . etc.

It is advantageous to consider designs of at least resolution IV to ensure all main effects are clear, and selected two-factor interactions are also clear. More commonly, DCEs are often designed for the identification of main effects only. While designs for the identification of main effects plus all two-factor interactions are available, these designs are often too large. In general, the number of clear two-factor interactions in a BFFD depends on the design generators and block generators. Different fractions and generators lead to different designs with different numbers of clear two-factor interactions.

Tables of two-level BFFDs for k = 4, . . ., 9 attributes are provided in Appendix 5B of Wu and Hamada [26]. For two-level attributes, a variety of sizes of choice sets with either two or four options can be constructed from these designs. These tables are organized by total number of treatment combinations: 8, 16, 32, 64, and 128. Within each treatment combination group, BFFDs are displayed according to the number of attributes (k), the number of design generators (p), the number of block generators (q), the design generator, and the block generator. These tables also display clear main effects and clear two-factor interactions. The designs presented in these tables have the maximum number of clear two-factor interactions. In Section 5, we will illustrate how to use these tables to construct DCEs. Furthermore, Xu and Lau [30] and Xu and Mee [31] extended these tables up to 32 attributes in 64 treatment combinations, and 64 attributes in 128 treatment combinations, respectively.

We note that FFDs have been used for constructing DCEs, see Street and Burgess [32] and more recently Bush [33], among others. These designs are based on a starting design, which is either a full factorial or fractional factorial design, whose entries represent the first option in each choice set. Generators are then added component wise to the starting design to form the remaining options in each choice set. Our method differs from these methods because we use a BFFD to construct a DCE. The main advantage of our designs is that the aliasing structure and which effects are clear are known. A drawback of using BFFDs to construct DCEs is that the resulting DCEs are limited by the number of options in each choice set, i.e., the number of options must be a power of the number of attribute levels.

5. Applications

We revisit the asthma study and demonstrate how BFFDs can be used to construct DCEs. In particular, we construct DCEs with two-level symmetric attributes when interest is restricted to the identification of main effects or the identification of main effects plus select two-factor interactions. We also discuss how this methodology can be extended to DCEs with three-level symmetric attributes.

5.1. DCEs with two-level symmetric attributes

Consider the five attributes each with two levels in the asthma study. A full factorial design would require 2⁵ = 32 treatment combinations, which may be prohibitive to run. Rather, a one-half fraction of the full factorial design may be more desirable. We set p = 1 and the 2^5–1 FFD has 16 treatment combinations. Here 5 represents the number of attributes and $2^{-1}=\frac{1}{2}$ represents the fraction of the full design. This design is called a one-half fraction of the full factorial 2⁵ design with one design generator.

Suppose now we wish to construct this 2^5–1 FFD in eight blocks (2^q = 2³; i.e., we need three block generators) each of size two (2^k–p–q = 2^5–1–3). This design has one design generator x₅ = x₁x₂x₃ and is of resolution IV, three block generators: b₁ = x₁x₄, b₂ = x₂x₄, and b₃ = x₃x₄, and the block effects are confounded with these two-factor interaction effects. Table 2 shows this 16 (2^5–1) treatment combination BFFD in eight blocks (i.e., eight choice sets) where each block is of size two (i.e., two options in each choice set). A subject would then be asked to select one of the two options in each of the eight choice sets. From the BFFD, all five main effects are clear.

In addition, BFFDs can be used to construct DCEs for the identification of clear main effects plus select clear two-factor interactions. This is beneficial because if it is known a priori that there is interaction between certain attributes, then it is important that the DCE is able to identify these interactions as clear. In the asthma study, if we know a priori that the attributes chest tightness (x₄) and sleep disturbance (x₅) interact with the three attributes: coughing (x₁), breathlessness (x₂), and wheezing (x₃), then it is important to consider a design for the DCE that can identify these interactions. For example, consider a DCE with four choice sets (i.e., four blocks) each with four options per choice set. Then, we begin with a 2^5–1 FFD in 2² = 4 blocks each of size four (2^5–1–2). This design has one design generator: x₅ = x₁x₂x₃x₄ and is of resolution V. Then define two block generators: b₁ = x₁x₂ and b₂ = x₁x₃. With this design, we can identify all five main effects as clear plus the following seven two-factor interactions: x₁x₄, x₁x₅, x₂x₄, x₂x₅, x₃x₄, x₃x₅, x₄x₅, i.e., the interactions between the attributes chest tightness (x₄) and sleep disturbance (x₅) and the three attributes: coughing (x₁), breathlessness (x₂), and wheezing (x₃) This design can also be found in Appendix 5B.2 of Wu and Hamada [26].

To further illustrate how to use Appendix 5B in Wu and Hamada [26], Table 4 presents select two-level BFFDs to construct DCEs with choice sets of size four and k ≤ 9 attributes. Table 4 is arranged by five sizes of choice experiments, each with four options: 2 choice sets, 4 choice sets, 8 choice sets, 16 choice sets, and 32 choice sets, and displays the following: k - the number of attributes, p - the number of treatment (or design) generators, q - the number of block generators, the design generators, the block generators, and the clear effects — main effects and/or two-factor interactions. BFFDs with k > 9 attributes are available in [30, 31].

Table 4.

DCEs with two-level attributes and choice sets with four options.

k	p	q	Design generators	Block generators	Clear effects
2 choice sets
3	0	1	—–	b1 = x1x2x3	all 3 ME’s plus all 2fi’s
4	1	1	x4 = x1x2x3	b1 = x1x2	all 4 ME’s
5	2	1	x4 = x1x2, x5 = x1x3	b1 = x2x3	none
6	3	1	x4 = x1x2, x5 = x1x3, x6 = x2x3	b1 = x1x2x3	none
4 choice sets
4	0	2	—–	b1 = x1x3x4, b2 = x2x3x4	all 4 ME’s plus all 2fi’s except: x1x2
5	1	2	x5 = x1x2x3x4	b1 = x1x2, b2 = x1x3	all 5 ME’s, x1x4, x1x5, x2x4, x2x5, x3x4, x3x5, x4x5
6	2	2	x5 = x1x2x3, x6 = x1x2x4	b1 = x1x3x4, b2 = x2x3x4	all 6 ME’s
7	3	2	x5 = x1x2x3, x6 = x1x2x4, x7 = x1x3x4	b1 = x1x2, b2 = x1x3	all 7 ME’s
8	4	2	x5 = x1x2x3, x6 = x1x2x4, x7 = x1x3x4, x8 = x2x3x4	b1 = x1x2, b2 = x1x3	all 8 ME’s
9	5	2	x5 = x1x2, x6 = x1x3, x7 = x1x4, x8 = x2x3x4, x9 = x1x2x3x4	b1 = x2x3, b2 = x2x4	none
8 choice sets
5	0	3	—–	b1 = x1x3x5, b2 = x2x3x5, b3 = x1x2x3x4	all 5 ME’s plus all 2fi’s except: x1x2, x3x4
6	1	3	x6 = x1x2x3x4x5	b1 = x1x3x5, b2 = x2x3x5, b3 = x1x4x5	all 6 ME’s plus all 2fi’s except: x1x2, x3x4, x5x6
7	2	3	x6 = x1x2x3, x7 = x1x2x4x5	b1 = x2x3x4, b2 = x2x3x5, b3 = x1x3x4x5	all 7 ME’s, xixj (i = 1, 2, 3; j = 4, 5, 7), x4x6, x5x6, x6x7
8	3	3	x6 = x1x2x3, x7 = x1x2x4, x8 = x1x3x4x5	b1 = x1x3, b2 = x2x3, b3 = x1x4	all 8 ME’s, xixj (i = 1, 2, 3, 4, 6, 7; j = 5, 8), x5x8
9	4	3	x6 = x1x2x3, x7 = x1x2x4, x8 = x1x3x4, x9 = x2x3x4x5	b1 = x1x2, b2 = x1x3, b3 = x1x4	all 9 ME’s, xixj (i = 1, 2, 3, 4, 6, 7, 8; j = 5, 9), x5x9
16 choice sets
6	0	4	—–	b1 = x1x3x6, b2 = x1x2 x3x4, b3 = x3x4x5x6, b4 = x1x2x3x4x5x6	all 6 ME’s plus all 2fi’s except: x1x2, x3x4, x5x6
7	1	4	x7 = x1x2x3x4x5	b1 = x1x2, b2 = x3x4, b3 = x1x3x5, b4 = x1x6	all 7 ME’s plus all 2fi’s except: x1x2, x1x6, x2x6, x3x4, x5x7
8	2	4	x7 = x1x2x3x4, x8 = x1x2x5x6	b1 = x1x3, b2 = x1x4, b3 = x2x5, b4 = x2x6	all 8 ME’s, xixj (i = 1, 2, 3, 4, 5, 6, j = 7, 8), x1x2, x1x5, x1x6, x2x3, x2x4, x3x5, x3x6, x4x5, x4x6
9	3	4	x7 = x1x2x3, x8 = x1x2x4x5, x9 = x1x3x4x6	b1 = x1x2, b2 = x1x3, b3 = x1x4, b4 = x5x6	all 9 ME’s, xixj (i = 1, 2, 3, 4, j = 5, 6, 8, 9), x5x7, x5x8, x5x9, x6x7, x6x8, x6x9, x7x8, x7x9
32 choice sets
7	0	5	—–	b1 = x1x2x3x4x6, b2 = x1x2x3x4x7, b3 = x1x2x4x5x6x7, b4 = x1x3x4x5x6x7, b5 = x2x3x4x5x6x7	all 7 ME’s plus all 2fi’s except: x1x2, x1x3, x2x3, x4x5, x6x7
8	1	5	x8 = x1x2x3x4x5x6	b1 = x1x2, b2 = x1x3, b3 = x4x5, b4 = x4x6, b5 = x1x4x7	all 8 ME’s plus all 2fi’s except: x1x2, x1x3, x2x3, x4x5, x4x6, x5x6, x7x8
9	2	5	x8 = x1x2x3x4x5, x9 = x1x2x3x6x7	b1 = x1x2x3, b2 = x1x4, b3 = x2x5, b4 = x1x6, b5 = x2x7	all 9 ME’s plus all 2fi’s except: x1x4, x1x6, x2x5, x2x7, x3x8, x3x9, x4x6, x5x7, x8x9

Note: ME’s (main effects) and 2fi’s (two-factor interactions).

To demonstrate how to use Table 4 suppose that we have k = 9 attributes each with two levels and we want choice sets of size four. There are four BFFDs to choose from depending on the problem of interest. Consider a 2^9–5 FFD in 2² blocks (k = 9, p = 5, q = 2). This design has four choice sets each with four options and design generators: x₅ = x₁x₂, x₆ = x₁x₃, x₇ = x₁x₄, x₈ = x₂x₃x₄, x₉ = x₁x₂x₃x₄, and block generators: b₁ = x₂x₃ and b₂ = x₂x₄. With this particular design none of the main effects are clear and we can only estimate the main effects under the assumption that all two-factor interactions aliased with the main effects are assumed to be negligible. Alternatively, for each main effect to be clear, we have three designs to consider. We may either construct a DCE with 8 choice sets, 16 choice sets, or 32 choice sets. In addition, these designs have an added benefit of clearly estimating some two-factor interactions. The 8, 16, and 32 choice sets each have 15, 24, and 27 clear two-factor interactions, respectively.

Similarly, for choice sets of size two, the supplementary material for this paper presents select two-level blocked fractional factorial designs from Wu and Hamada [26] for k ≤ 9 attributes. We note that for choice sets of size two only the main effects are clear.

5.2. DCEs with three-level symmetric attributes

A similar procedure can be used to construct DCEs with three-level symmetric attributes. Appendix 6B of Wu and Hamada [26] gives efficient blocking schemes for three-level attributes in 9-, 27-, and 81- treatment combinations. Similar to two-level BFFDs, the choice of which three-level BFFD to use depends on the given number of attributes k, the desired number of options within each choice set and which effects are to be identified. For three-level attributes, choice sets can be constructed with options that are a power of three. To construct a choice set with three options, the necessary block size is: 3^k–p–q = 3. If desired, choice sets with more than three options can also be constructed using three-level BFFDs, i.e., 3^k–p–q = 3, 9, . . . etc.

To demonstrate the use of three-level BFFDs, we revisit the asthma study presented in Table 1 with five attributes each with three levels. Two BFFDs from Appendix 6B.1 of Wu and Hamada [26] can be used to construct a DCE with nine choice sets each with three options, i.e., a 3^5–2 BFFD in 3² blocks of size 3^5–2–2. The first design has design generators x₄ = x₁x₂ and $x_5=x_1{x}_2^2{x}_3$ and block generators b₁ = x₁x₃ and b₂ = x₂x₃. This design can identify two clear main effects: x₃ and x₅, plus one clear two-factor interaction: x₃x₅. The second design has design generators x₄ = x₁x₂ and x₅ = x₁x₃ and block generators $b_1=x_1{x}_2^2$ and b₂ = x₁x₂x₃. This design can identify four clear two-factor interactions x₂x₃, $x_2{x}_5^2,x_3{x}_4^2$ and x₄x₅. Alternatively, a DCE with 27 choice sets each with three options can be constructed using a 3^5–1 BFFD in 3³ blocks of size 3^5–1–3. This design has design generator x₅ = x₁x₂x₃x₄ and block generators: b₁ = x₁x₂, b₂ = x₁x₃, b₃ = x₁x₄. With this design, we can identify all five main effects plus 10 two-factor interactions as clear.

In general, for a DCE with three-level attributes, and for all main effects to be identified as clear, a DCE with nine choice sets (each with three options) can be constructed for k = 3 and k = 4 attributes, and a DCE with 27 choice sets (each with three options) can be constructed for k = 5, . . ., 9 attributes. The supplementary material for this paper presents select three-level BFFDs from Wu and Hamada [26] to construct DCEs with choice sets of size three and k ≤ 9 attributes. Additional DCEs with nine choice sets each with three options can be constructed for k = 6–12 attributes for the estimation of main effects, assuming all interactions negligible.

6. Extensions

In the previous section we focused on constructing DCEs with symmetric attributes. We now discuss how this methodology can be extended for constructing DCEs with asymmetric attributes for the identification of main effects. Mixed-level orthogonal arrays (Wu and Hamada, [26]; Chapter 8) can be used for this purpose because of their run size economy and great flexibility. In general, orthogonal arrays are popular experimental designs for identifying important factors. In this section, we give a condensed introduction on how to use previously constructed mixed-level orthogonal arrays to construct DCEs.

To use a mixed-level orthogonal array to construct the choice sets in a DCE with asymmetric attributes we modify traditional mixed-level orthogonal arrays to include a column to represent the blocks and use the following notation: $OA(N,B^1{l}_1^{n_1}{l}_2^{n_2}\dots l_j^{n_j})$, where N is the total number of treatment combinations in the design, B¹ is the total number of blocks/choice sets, and $l_1^{n_1},\dots ,l_j^{n_j}$ are the levels of the k = n₁ + ⋯ + n_j attributes. With this modified orthogonal array, the block effects are orthogonal to the main effects when one of the columns in the mixed-level orthogonal array is used to determine the choice sets. This method is general and works as long as such an orthogonal array exists. There are various methods for constructing mixed level orthogonal arrays, see Dey [34], Wang and Wu [35], Hedayat et al. [36], Xu [37], and Wu and Hamada [26]. More specifically, Xu [37] provides an algorithm for constructing mixed-level orthogonal arrays with source C code (http://www.stat.ucla.edu/~hqxu/pub/woa.c); Kuhfeld and Tobias [38] provide a SAS macro to generate thousands of orthogonal, and mixed-level orthogonal, arrays provided by Hedayat et al. [36]; and Neil Sloane provides a library of over 200 orthogonal arrays at http://neilsloane.com/oadir/index.html.

To demonstrate the use of mixed-level orthogonal arrays for constructing choice sets and options for a DCE with asymmetric attributes, we use the study by Gerard et al. [39] on women’s preferences regarding invitation to re-attend a breast screening service. The goal of the study was to determine women’s preferences for breast screening based on various attributes which may influence a woman’s decision to participate. There were 10 attributes in the study, nine of which were related to the different aspects of the screening process and one was related to the screening outcome. These selected attributes and their levels are shown in Table 5 and they represent features that may influence a women’s decision that could also be reconfigured by the provider. The attributes are denoted by x₁, x₂, . . ., x₁₀: five attributes have four levels, denoted by 0, 1, 2 and 3; and five attributes have two levels, denoted by 0 and 1. The fifth attribute, level of accuracy of the screening test (x₅) measures the sensitivity of screening, which is the true cancer detection rate.

Table 5.

Breast screening study attributes and levels; Gerard et al. [39].

Attributes	Attribute levels	Coded levels
x1: Method of invitation	Personal letter from local service	0
	Personal letter from GP	1
	Media campaign	2
	Recommendation from family and friends	3
x2: Time spent traveling	Not more than 20 min	0
	Between 20 and 40 min	1
	Between 40 and 60 min	2
	Between 1 and 2 hr	3
x3: Time spent	20 min	0
	30 min	1
	40 min	2
	50 min	3
x4: Time to notification of results	8 working days	0
	10 working days	1
	12 working days	2
	14 working days	3
x5: Level of accuracy of the screening test	70%	0
	80%	1
	90%	2
	100%	3
x6: Info. included with invitation	Sheet about procedure	0
	No info. sheet	1
x7: Time to wait for an appointment	1 week	0
	4 weeks	1
x8: Choice of appointment times	Usual office hours	0
	Usual office hours, one evening per week, Saturday morning	1
x9: Staff mannerism at the screening service	Welcoming manner	0
	Reserved manner	1
x10: Attention paid to privacy	Private changing area	0
	Open changing area	1

Our approach can be used to construct a DCE for this study such that each choice set has four options. A mixed level orthogonal array with 32 treatment combinations in eight blocks can be generated using the algorithm in Xu [37] or SAS macro in Kuhfeld and Tobias [38]. This design is OA(32, 8¹4⁵2⁵), where N = 32 is the number of total treatment combinations in B = 8 blocks each of size 4 with l₁ = 4 and n₁ = 5, representing the five four-level attributes, and l₂ = 2 and n₂ = 5, representing the five two-level attributes. This orthogonal array is shown in Table 6. The first column defines the eight blocks (coded as 0, 1, 2, 3, 4, 5, 6, 7) or choice sets each of size four. The next five columns represent the five four-level attributes. The last five columns represent the five two-level attributes. For this mixed level orthogonal array, we would present subjects with eight choice sets each of size four and elicit their response on each choice set. This design can estimate all 10 main effects, assuming all interactions negligible.

Table 6.

Mixed-level orthogonal array for the breast screening study: OA(32, 8¹4⁵2⁵).

Block	x1	x2	x3	x4	x5	x6	x7	x8	x9	x10
0	0	3	1	3	3	0	1	1	0	0
0	1	2	2	1	0	1	1	0	1	1
0	2	1	0	2	1	0	0	0	0	0
0	3	0	3	0	2	1	0	1	1	1
1	0	2	0	0	3	0	0	0	1	1
1	1	1	2	3	2	1	1	0	0	0
1	2	0	1	1	1	0	1	1	1	1
1	3	3	3	2	0	1	0	1	0	0
2	0	3	2	2	1	1	1	1	1	1
2	1	0	0	1	0	0	0	1	0	0
2	2	2	3	3	3	1	0	0	0	1
2	3	1	1	0	2	0	1	0	1	0
3	0	2	3	1	2	0	1	1	0	0
3	1	1	1	2	3	1	0	1	1	1
3	2	3	0	0	0	1	1	0	1	0
3	3	0	2	3	1	0	0	0	0	1
4	0	1	2	0	0	0	0	1	0	1
4	1	0	3	2	3	0	1	0	1	0
4	2	3	1	1	2	1	0	0	0	1
4	3	2	0	3	1	1	1	1	1	0
5	0	1	3	1	1	1	0	0	1	0
5	1	3	0	3	2	0	0	1	1	1
5	2	0	2	0	3	1	1	1	0	0
5	3	2	1	2	0	0	1	0	0	1
6	0	0	0	2	2	1	1	0	0	1
6	1	2	1	0	1	1	0	1	0	0
6	2	1	3	3	0	0	1	1	1	1
6	3	3	2	1	3	0	0	0	1	0
7	0	0	1	3	0	1	0	0	1	0
7	1	3	3	0	1	0	1	0	0	1
7	2	2	2	2	2	0	0	1	1	0
7	3	1	0	1	3	1	1	1	0	1

7. The multinomial logit model and simulation

The multinomial logit (MNL) model is commonly used for analyzing DCEs. Using our proposed approach described in the previous sections, we construct three DCEs using three BFFDs and perform a simulation study to investigate the effects of identification when estimating the parameters in the MNL model. We first begin by describing the MNL model.

7.1. The multinomial logit model

The responses from the subjects in a DCE are analyzed using economic and random utility theory. For example, if a subject attaches a higher utility to option 1 than option 2, then a rational subject will select option 1, where the utility is the benefit that the subject experiences by selecting a particular option. Following Kessels et al. [40], in developing DCEs for the MNL model, we define the utility as the sum of two parts: (1) an explainable systematic component based on the observed attributes; and (2) a non-explainable random component that captures other attributes that may be relevant, but not specified. The utility for a subject that chooses option j in choice set s is modeled as

$$U_{js}={\mathit{x}}_{js}^{\prime }\mathit{\beta }+{\epsilon }_{js},$$ (1)

where x_js is a k* × 1 vector containing the model expansion of the attribute levels of option j in choice set s; k* is the number of parameters to be estimated; β is a k*× 1 vector of model parameters representing the effect of the attribute levels on the utility; and ε_js is an error term following an independent identically distributed extreme value type 1 distribution. In the MNL model, each parameter (β) represents the effect of an attribute or its interaction with other attributes.

Under the MNL model, the probability p_js that a subject selects option j in choice set s is

$$p_{js}=\frac{e^{({\mathit{x}}_{js}^{\prime }\mathit{\beta })}}{{\displaystyle \sum _{r=1}^J}{e}^{({\mathit{x}}_{rs}^{\prime }\mathit{\beta })}},$$ (2)

where β is estimated using maximum likelihood estimation. We note that it is assumed in the MNL model that β is the same for every subject and that subject’s preferences for the attribute levels are homogeneous across the population [40]. Additionally, we assume all subjects are given the same choice sets, and the choice of the option in each choice set is independent because the error terms are assumed to be independent.

7.2. Case study

To illustrate the advantage of BFFDs to identify main effects and two-factor interactions, we perform a simulation study and construct three DCEs using three different BFFDs, each with a different number of clear main effects and two-factor interactions. We call them DCE I, II, and III. Each DCE has five two-level attributes and four choice sets each with four options (2^5–1 FFD in 2² blocks). We analyze the results of each DCE using the MNL model and compare their parameter estimates. Consider a true model, with five main effects and three two-factor interactions: x₁x₄, x₂x₅, and x₄x₅,

$$\mu =-0.5x_1+0.5x_2-0.5x_3+0.5x_4-0.5x_5+0.5x_1{x}_4-0.5x_2{x}_5+0.5x_4{x}_5.$$ (3)

We then computed the MNL probability (2) for each option within each choice set. Using these probabilities, we simulated a response according to the multinomial distribution for each of the three DCEs. We replicated each DCE 500 times to represent 500 subjects.

Table 7 provides an example of a single replicate within the simulation for DCE I. The first column, choice set, indicates the choice set number. The second column gives the option number within each choice set, each choice set has four options. The next eight columns represent the BFFD, which contains the five main effects plus three two-factor interactions from the true model (3). The second to last column contains the probability (2) of each option. The final column indicates the choice selected, i.e., 1 represents the selected option. Using the simulated responses (the choice column in Table 7) from each of the three DCEs, we fit a model containing all five main effects plus all clear two-factor interactions from each BFFD. We estimated the parameter values for each DCE using the “mlogit” package in R [41]. Table 8 presents the parameter estimates and the standard errors for the estimates for each DCE.

Table 7.

Simulated DCE I.

Choice Set	Option	x1	x2	x3	x4	x5	x1x4	x2x5	x4x5	Probability	Choice
1	1	−1	1	1	−1	1	1	1	−1	0.060	0
1	2	−1	1	1	1	−1	−1	−1	−1	0.440	0
1	3	1	−1	−1	−1	−1	−1	1	1	0.060	0
1	4	1	−1	−1	1	1	1	−1	1	0.440	1
2	1	−1	1	−1	−1	−1	1	−1	1	0.865	1
2	2	−1	1	−1	1	1	−1	1	1	0.117	0
2	3	1	−1	1	−1	1	−1	−1	−1	0.002	0
2	4	1	−1	1	1	−1	1	1	−1	0.016	0
3	1	−1	−1	1	−1	−1	1	1	1	0.105	0
3	2	−1	−1	1	1	1	−1	−1	1	0.105	0
3	3	1	1	−1	−1	1	−1	1	−1	0.014	0
3	4	1	1	−1	1	−1	1	−1	−1	0.776	1
4	1	−1	−1	−1	−1	1	1	−1	−1	0.250	0
4	2	−1	−1	−1	1	−1	−1	1	−1	0.250	0
4	3	1	1	1	−1	−1	−1	−1	1	0.250	0
4	4	1	1	1	1	1	1	1	1	0.250	1

Table 8.

Estimates and standard errors for the simulation.

Effect	DCE I	DCE II	DCE III
x1	−0.492 (0.075)	−0.532 (0.042)	0.010 (0.034)
x2	0.586 (0.075)	0.572 (0.037)	0.511 (0.036)
x3	−0.575 (0.075)	−0.570 (0.037)	−0.514 (0.032)
x4	0.462 (0.075)	0.566 (0.053)	0.495 (0.034)
x5	−0.436 (0.075)	−0.552 (0.042)	−0.034 (0.034)
x1x2	–	–	–
x1x3	–	–	–
x1x4	0.406 (0.075)	0.544 (0.036)	–
x1x5	0.055 (0.075)	–	–
x2x3	–	–	–
x2x4	0.023 (0.075)	−0.028 (0.035)	−0.018 (0.032)
x2x5	−0.526 (0.075)	–	−0.491 (0.032)
x3x4	0.038 (0.075)	0.046 (0.035)	0.001 (0.028)
x3x5	−0.068 (0.075)	–	−0.014 (0.030)
x4x5	0.516 (0.075)	0.551 (0.036)	–

DCE I has design generator: x₅ = x₁x₂x₃x₄ and block generators: b₁ = x₁x₂, b₂ = x₁x₃. The two block generators define three block effects, x₁x₂, x₁x₃ and (x₁x₂)(x₁x₃) = x₂x₃. These three two-factor interactions are confounded with block effects and they cannot be estimated. For this design, all five main effects are clear, plus seven two-factor interactions: x₁x₄, x₁x₅, x₂x₄, x₂x₅, x₃x₄, x₃x₅, x₄x₅. Comparing the parameter estimates for DCE I in Table 8 with the true model (3), we see that all five main effects are consistent with the coefficients in (3). Furthermore, the two-factor interactions: x₁x₄, x₂x₅, x₄x₅, which are included in the true model (3), are also consistent because they are clear in the BFFD.

DCE II has design generator: x₅ = x₁x₂x₃ and block generators: b₁ = x₁x₂, b₂ = x₁x₃. With this design, each block effect is confounded with two two-factor interactions, b₁ = x₁x₂ = x₃x₅, b₂ = x₁x₃ = x₂x₅, b₃ = b₁b₂ = x₂x₃ = x₁x₅. These effects cannot be estimated. For this design, all five main effects are clear, plus four two-factor interactions: x₁x₄, x₂x₄, x₃x₄, x₄x₅. Comparing the parameter estimates for DCE II in Table 8 with the true model, again we see that all five main effects are consistent with the coefficients in the true model. In DCE II, only two of the three two-factor interactions included in the true model are estimated: x₁x₄, x₄x₅; these two two-factor interactions are also consistent with the true model because they are clear in the BFFD.

DCE III has design generator: x₅ = x₁x₄ and block generators: b₁ = x₁x₂, b₂ = x₁x₃. With this design, three two-factor interactions (x₁x₂, x₁x₃ and x₂x₃) are confounded with block effects and cannot be estimated. Only two of the five main effects are clear (x₂ and x₃) and four two-factor interactions are clear: x₂x₄, x₂x₅, x₃x₄, x₃x₅. Three main effects are aliased with two-factor interactions: x₁ = x₄x₅, x₄ = x₁x₅, x₅ = x₁x₄, and these main effects can only be estimated if the aliased two-factor interactions are negligible. Comparing the parameter estimates for DCE III in Table 8 with the true model, the two clear main effects: x₂ and x₃, are consistent with the true model. Since x₁ and x₅ are aliased with two-factor interactions included in the true model (3), the estimates of these main effects are biased by their aliased two-factor interactions. Whereas, x₄ is also aliased with a two-factor interaction x₁x₅, however, this two-factor interaction is not included in the true model; therefore, we can estimate x₄ unbiasedly and its estimate in Table 8 is consistent with the coefficient in (3). The two-factor interaction x₂x₅ is clear and its estimate is also consistent with the true model.

This simulation study shows the importance of the choice of the design for a DCE because it determines which attributes and their interactions are identifiable. Among the three DCEs considered, the BFFD used to construct DCE I is the most appropriate because this design is capable of identifying all of the parameters included in the true model (3).

8. Discussion

DCEs are increasingly used to understand and quantify subjects’ preferences for a diverse range of health care applications. Much of the current literature for constructing DCEs does not seem to focus on identification issues. Using two health care studies, we reviewed fundamentals from two-level FFDs and used blocking ideas from them to construct DCEs for identifying all main effects, and in some situations for identifying all main effects plus select two-factor interactions. We also showed how our approach to construct DCEs with two-level attributes can be extended to find DCEs with three-level attributes and for DCEs with asymmetric attributes.

The designs we constructed are optimal for estimating the parameters in the MNL model under the assumption that all options are equally attractive [33]. They are referred to as locally optimal (or D-optimal) designs in the literature. The information matrix for a locally optimal design depends on the the parameters in the model and a best guest for the nominal values of the parameters is required before the locally optimal design can be found. Typically the nominal values are obtained from pilot studies, expert opinions or from similar studies. It is often assumed that the utility for each option is the same; see for example, Burgess and Street [43, 44]; Bush [33]; Grasshoff et al. [45, 46]; Grossmann et al. [47]; Street and Burgess [48, 32]; Street, Burgess, and Louviere [49]; Street, Bunch, and Moore [50]. A potential problem with this approach is that if the nominal values are mis-specified, the locally optimal design may be potentially inefficient.

Alternative design approaches for constructing DCEs include a Bayesian approach, a maximin approach or a sequential approach. A Bayesian approach requires a careful elicitation of information on the unknown parameters into a prior distribution for averaging out the criterion values over possible values for the nominal parameters, see for example, Sandor and Wedel [51] and Kessels et al. [40]. A maximin design approach can be used in a variety of ways to overcome model uncertainty. For maximin optimal designs, one first computes a locally optimal design for a range of values for the parameters, and then determines the set of parameter values that produce the smallest determinant. The maximin optimal design then maximizes the smallest determinant among all possible designs. Minimax optimal designs are similar in spirit to maximin optimal designs with a slightly different formulation to minimize the maximum inefficiency. Examples of recent work using a Bayeisan approach or maximin approach to find optimal designs for nonlinear models are [52, 53]. We note that these two approaches require more assumptions than a single best guess for the model parameters to construct optimal designs. While sequential approaches, build and implement the design in a stepwise manner. Information gained from the previous stage is incorporated into the next step for finding an efficient design. For instance, it is common that after a locally optimal design is implemented, data from the design is used to estimate the model parameters and they then serve as nominal values in the next step in determining the locally optimal design. When feasible, it is recommended to implement a sequential approach, which uses the most up-to-date information gained from the previous steps to construct an efficient design.

The question then becomes, are our locally optimal designs robust to model mis-specifications? To answer this question, Kessels et al. [40] conducted simulations to compare nine Bayesian optimal designs to a locally optimal design constructed from a blocked 2^6–2 FFD. The prior distribution they used was a multivariate normal, $N({\mathit{\beta }}_0,{\sigma }_0^2\mathbf{I})$, with prior mean β₀ and prior variance-covariance matrix ${\sigma }_0^2\mathbf{I}$, where I denotes an identity matrix. Kessels et al. [40] found that the efficiency of the locally optimal design depends on β₀ and ${\sigma }_0^2$. When σ₀ is small and β₀ is close to the zero vector, locally optimal designs are nearly as efficient as Bayesian optimal designs. The efficiency of the locally optimal design decreases as σ₀ increases or β₀ deviates farther from the zero vector. They concluded that Bayesian optimal designs appear to be more robust than locally optimal designs; however, the construction of Bayesian optimal designs is computationally intensive and becomes a very challenging task when the numbers of attributes and choice sets are large.

In contrast, DCEs constructed from BFFDs are locally optimal for main effect models and models with all main effects plus any clear two-factor interactions. Our method constructs locally optimal DCEs by using optimal BFFDs that are already tabulated for 128 treatment combinations up to 64 attributes with block sizes of 2, 4, ..., 64 for two-level attributes [31]; and 81 treatment combinations up to 39 attributes with block sizes of 3, 9, 27 for three-level attributes [30]. In addition, Kuhfeld and Tobias [38] provide a SAS macro to produce more than 115,000 orthogonal arrays with mixed levels and up to 513 treatment combinations. As illustrated in Section 6, these orthogonal arrays can be used as BFFDs, which are locally optimal for identifying main effects in a DCE with asymmetric attributes. The optimality follows from the fact that an orthogonal array (of strength two), i.e., of resolution III, is universally optimal for the estimation of main effects [42] and the block effects are orthogonal to the main effects. Therefore, given the readily available designs for constructing optimal BFFDs, our method provides an attractive option for practitioners to implement DCEs to ensure identification. The BFFDs that are considered in this paper are chosen to maximize the number of clear main effects and two-factor interactions. An alternative approach is to choose BFFDs according to the minimum aberration criterion, which aims to minimize the contamination of non-negligible interactions on the estimation of main effects. We will continue the research on the use of minimum aberration BFFDs for constructing DCEs.

In summary, our proposed design strategy uses ideas from BFFDs to construct various types of DCEs so that it is known in advance which attribute effects and their interactions can be identified. This can be a critical factor in determining whether DCEs are used successfully in health care applications or not. Using two health care studies, we demonstrated how we took advantage of the properties of BFFDs and constructed choice sets and options. Generally, our proposed designs are easy to construct and for many practical scenarios are already available from the literature. Through the bundling of attributes, DCEs can generate a rich source of data to assess real-life decision making processes that involve trade-offs among desirable characteristics pertaining to health and health care [54, 11].