support.BWS2-package {support.BWS2}

Tools for Case 2 best-worst scaling

Description

The package has three basic functions that support an implementation of Case 2 (profile case) best–worst scaling. The first is to convert an orthogonal main-effect design into questions, the second is to create a dataset suitable for analysis, and the third is to calculate count-based scores. For details, see Aizaki and Fogarty (2019).

Details

The package is under development and thus may be changed substantially in the future.

1) Outline of Case 2 best–worst scaling

Case 2 (profile case) best–worst scaling (BWS) is a question-based survey method to elicit preferences for attribute levels (See Flynn 2010, Flynn et al. 2007 and 2008, Louviere et al. 2015, and Marley et al. 2008 for details of the subsection). A profile (choice set) has three or more attributes and each attribute has two or more levels. The profile is expressed as a combination of attribute levels. Numerous profiles are constructed using experimental designs. Attributes shown in each profile are fixed in all the profiles and a combination of attribute levels in each profile is changed according to the profiles. A profile selected from all the constructed profiles is presented to respondents, who are then asked to choose the best and worst attribute levels in the profile. This question is repeated until all profiles are evaluated. Analyzing the responses enables us to elicit preferences for the attribute levels.

A basic approach to constructing profiles is using an orthogonal main-effect design (OMED). Assume that the profiles have K attributes and each attribute has L_{k} levels. If all the attributes have the same number of levels, L, a L^{K} OMED is used to construct the profiles. Columns of the OMED correspond to attributes, while the rows to profiles. For example, profiles have four attributes and they have three levels: attribute A with levels A1, A2, and A3; attribute B with levels B1, B2, and B3; attribute C with levels C1, C2, and C3; and attribute D with levels D1, D2, and D3. A 3^{4} OMED corresponding to the assumptions is as follows (see the section Examples of the function bws2.dataset() for code to generate the OMED):

1	3	2	3
3	1	2	2
3	3	3	1
2	3	1	2
2	2	2	1
1	1	1	1
1	2	3	2
3	2	1	3
2	1	3	3

Suppose that attributes A, B, C, and D are assigned to the first, second, third, and fourth column of the OMED, respectively, and the values 1, 2, and 3 used in the OMED correspond to the attribute-level values in each attribute: 1 = A1, 2 = A2, and 3 = A3 for attribute A; 1 = B1, 2 = B2, and 3 = B3 for attribute B; 1 = C1, 2 = C2, and 3 = C3 for attribute C; and 1 = D1, 2 = D2, and 3 = D3 for attribute D. Accordingly, the above-mentioned OMED can be transformed into the following:

A1	B3	C2	D3
A3	B1	C2	D2
A3	B3	C3	D1
A2	B3	C1	D2
A2	B2	C2	D1
A1	B1	C1	D1
A1	B2	C3	D2
A3	B2	C1	D3
A2	B1	C3	D3

The resultant OMED consists of nine rows: nine profiles, that is, nine Case 2 BWS questions, are constructed. For example, a profile corresponding to the first row of the OMED comprises A1, B3, C2, and D3. This means that respondents who face the question created from the first row of the OMED are asked to select their best and worst attribute levels from attribute levels A1, B3, C2, and D3, as follows:

Please select your best and worst attribute levels from the following four:

Best	Attribute	Worst
[_]	A1	[_]
[_]	B3	[_]
[_]	C2	[_]
[_]	D3	[_]

There are two approaches for analyzing responses to Case 2 BWS questions: a counting approach and modeling approach. The counting approach calculates scores on the basis of the number of times attribute level i is selected as the best (B_{in}: B score for attribute level i) and the worst (W_{in}: W score for attribute i) among all the questions for respondent n. A (disaggregated) best-minus-worst (BW) score and its standardized variant are defined as

BW_{in} = B_{in} - W_{in},

std.BW_{in} = \frac{BW_{in}}{f_{i}},

where f_{i} is the frequency with which attribute level i appears across all questions.

The modeling approach uses discrete choice models to analyze responses. When using the modeling approach, a model type must be selected according to the assumption for respondents' choice behavior in Case 2 BWS questions and then a dataset must be formatted as per the selected model. There are three standard models: paired, marginal, and marginal sequential models. Although the three models commonly assume that the respondents derive utility for each attribute level shown in the profile, the assumption for how they select the best and worst attribute levels from the set of attribute levels in the profile differs among the three models.

The number of possible pairs in which attribute level i is selected as the best and attribute level j is selected as the worst (i \neq j) from K attribute levels is K \times (K - 1). The paired model assumes that respondents select attribute level i as the best and attribute level j as the worst because the difference in utility between i and j represents the greatest utility difference among K \times (K - 1) utility differences. Consider the example profile mentioned above. It contains four attribute levels: A1, B3, C2, and D3. The number of possible pairs is 12 (= 4 \times (4 - 1)). There are 12 possible pairs of the best and worst attribute levels (in each pair, the former is the best and the latter is the worst): (A1, B3), (A1, C2), (A1, D3), (B3, A1), (B3, C2), (B3, D3), (C2, A1), (C2, B3), (C2, D3), (D3, A1), (D3, B3), and (D3, C2). If a respondent selects A1 as the best attribute level and C2 as the worst, the paired model assumes that the respondent calculates 12 utility differences as per the 12 above-mentioned pairs and that the difference in utility between A1 and C2 is the maximum among the 12 utility differences.

The marginal model assumes that there are K possible best attribute levels and K possible worst attribute levels in a profile, that attribute level i is selected as the best from K possible best attribute levels in the profile, and that attribute level j is selected as the worst from K possible worst attribute levels. This is because the utility for attribute level i is the maximum among the utilities for K attribute levels and that for attribute level j is the minimum. Following the above example, the marginal model assumes that there are four possible best attribute levels and four possible worst attribute levels in the profile and interprets the respondent's choice behavior as follows: utility for A1 is the maximum among the four utilities for A1, B3, C2, and D3 and that for C2 is the minimum among the four.

The assumption of the marginal model that the worst attribute level is selected from K attribute levels would not be appropriate because the best attribute level in a profile must differ from the worst one in the profile. Thus, the marginal sequential model assumes that respondents select attribute level i as the best from K attribute levels in the profile and then attribute level j as the worst from the remaining K - 1 attribute levels. Following the above example, under the marginal sequential model assumption, there are four possible best attribute levels and three possible worst attribute levels in the profile. The model considers that the respondent selects A1 as the best from the four possible attribute levels because the utility for A1 is the highest among the utilities for A1, B3, C2, and D3, but selects C2 as the worst from three possible worst levels, B3, C2, and D3, because the utility for C2 is the least among the three.

The three models generally assume that the utility for attribute level i selected as the worst is the negative of the one selected as the best. Under these assumptions, and given the assumption for the stochastic component of utility, the probability of selecting attribute level i as the best and attribute level j as the worst can be expressed as a conditional logit model.

2) Role of the package and other packages needed to complete implementing Case 2 BWS

The package support.BWS2 provides functions to convert an OMED into a series of Case 2 BWS questions, create a dataset for the analysis from the OMED and the responses to the questions, and calculate BWS scores. Other packages are needed to complete implementing Case 2 BWS with R: a package to construct OMEDs and another to analyze the responses on the basis of the modeling approach. For example, the oa.design() function in DoE.base (Groemping 2018) can construct OMEDs, while the functions clogit() in survival (Therneau 2016), mlogit() in mlogit (Croissant 2013), and gmnl() in gmnl (Sarrias and Daziano 2017) can fit the conditional logit model. The latter two functions are also used to fit advanced discrete choice models such as a mixed (random parameters) logit model. Refer to the task views about experimental designs (Groemping 2016) and econometrics (Zeileis 2017) on CRAN for details on packages for experimental designs and discrete choice models in R.

Acknowledgments

I would like to thank Professor Kazuo Sato for his kind support. This work was supported by JSPS KAKENHI Grant Numbers 25450341, 16K07886, and 20K06251.

Author(s)

Hideo Aizaki

References

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