Mixture amount models for handling constraints in conjoint applications

Mixture amount models for handling constraints in conjoint

applications

Citation for published version (APA):

Dane, G., Timmermans, H. J. P., & Wiley, J. B. (2011). Mixture amount models for handling constraints in conjoint applications. In Second International Choice Modeling Conference, 4–6 July 2011, Leeds, UK (2011) (pp. 1-13)

Document status and date: Published: 01/01/2011

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*This paper is for presentation at the Second International Choice Modeling Conference, July, 2011 Mixture Amount Models for Handling Constraints in Conjoint Applications

Gamze Dane, Urban Planning Group, Eindhoven University of Technology, P.O. Box 513, 5600MB, Eindhoven, The Netherlands, g.z.dane@tue.nl

Harry Timmermans, Urban Planning Group, Eindhoven University of Technology, P.O. Box 513, 5600MB, Eindhoven, The Netherlands, h.j.p.timmermans@tue.nl

James B. Wiley, Departments of Marketing and Statistics, Temple University, Philadelphia, PA 19122, USA, wileyja@temple.edu

Abstract

Conjoint analysis (CA) is concerned with estimating consumer choice behavior for products and services and/or underlying preference structures as a function of tangible specifications such as the cost of a product and duration of a service. Although constraints influence consumers’ preferences by forcing them to make specific trade-off across attributes, and affect what can be offered to the consumers, they have received little attention in the conjoint analysis literature. Such constraints have implications for CA design and the practicality of discrete attribute levels. This paper discusses the principles underlying mixture amount designs and compares the approach to the traditional fractional factorial designs in examples of travel time allocation to four different activities subject to time constraints of individuals.

Keywords: conjoint analysis, mixture amount designs, fractional factorial design, constraints,

time budget

1. Introduction

Stated preference (SP) procedures, such as Conjoint analysis (CA) and Discrete Choice Experimentation (DCE), are used in market research to infer values consumers attach to levels of product/service features. In the SP approach, choice alternatives (profiles) are represented as bundles of attributes and these attributes are typically varied according to a structured plan. For example, plans for studies often are provided by experimental designs, such as Fractional Factorial (FF) and Balanced Incomplete Block (BIBD) designs.

With CA, respondents are asked to rate the planned profiles on a preference scale. The utility function is defined over the attributes and used to infer the parameter estimates. Inferences are based on ratings of product/service profiles, conditional on levels of the experimental

plan. A key principle in interpreting parameters as being utility-like is that respondents must make trade-offs among attributes levels in arriving at their preference judgments (Green et al., 2001). The inferred values for attribute levels facilitate prediction of consumers' preferences for new or modified product/services (Louviere et al, 2000).

Another approach is to have respondents make choices from sets. In a DCE, attribute profiles are placed in choice sets following an experimental design and respondents are asked to choose the choice alternative they like most. The choice of design to create these choice sets depends on the specification of the choice model and the specific purpose of the analysis. Rose et al. (2008) and Hensher et al. (2005) give more detailed reports of choice analysis.

Even though constraints may exist and may influence consumer preferences, in the CA/DCE literature, the construction of choice sets generally does not consider any constraints on which combinations attribute levels may occur in a choice set. In many choice domains, consumer choices are made under time and money budget constraints. It also is the case that firms typically must offer product/services at existing market "price points" and profit and cost considerations constrain the levels of attributes that can be offered at a given price point.

Mixture designs are suited to situations where a given amount of some input (such as expenditure) must be allocated across design attributes. They are used when the objective is to determine the optimal "blend" of attributes, subject to the fixed constraint across attributes. Mixture/amount (MA) designs are appropriate when the objective also is to determine whether the optimal "blend" changes when the amount of the fixed input is varied, e.g., when a different price point is considered (Cornell, 1990; Raghavarao, Wiley, & Chitturi, 2010). Note that with mixture and mixture/amount designs the question of attribute importance cannot be independently answered because change of one attribute requires compensatory change in the remaining attributes (Raghavarao & Wiley, 2009).

This paper discusses the principles underlying MA designs and illustrates their applicability in an example of travel time allocation to four different activities, subject to time constraints of individuals. In addition, we replicate the MA study by using a fractional factorial design and evaluate results of the two approaches to understand the capabilities of MA design compared to traditional fractional factorial (FF) designs. Reliability and test-retest reliability measures of both designs are compared by using holdout tasks and replications of choice sets in each experiment. Moreover, both designs are compared in the base of utilities.

The paper is organized as follows. In Section 2, we introduce the mixture amount design under different constraints with four attributes. Next, in Section 3, we illustrate the travel

time design using the approach that was outlined in section 2. Then, we illustrate the same example using a fractional factorial design. Finally, we discuss the results of two approaches and conclude the paper with major conclusions and possible lines of further development.

2. Mixture Amount Designs

MA designs can be used to examine the effects of the varying of constraints on consumer preferences and choice behavior. Since MA designs aim to accommodate the allocation of resources such as a budget to the attributes, they are potentially relevant for studying constrained choices in general. Specifically, these designs aim to discover the optimum allocation of resources to the attributes. In standard mixture and MA experiments, simplex lattice designs and simplex-centroid designs are used (Cornell, 1981). The example in this paper is simplex-lattice design.

A {m, n} simplex lattice design for q component is composed of points. Let us consider

linearly independent variables as X1, X2, X3,... . The proportions assumed by each component

are m+1 equally spaced values from 0 to 1, Xi = 0, 1/m, 2/m, ... , 1 for i = 1, 2, ... , q, with the

restriction X1+ X2+...+Xm=1. The restriction can be equal to 100% for unstandardized

variables. There are (m+n-1)!/(n!(m-1)!) design points in the {m, n} simplex-lattice. With a {4, 4} simplex-lattice design there are 35 available design points and a minimum of 10 are required to estimate the parameters of a linear component model with cross products. Discussions of lattice designs in mixture experiments may be found in Cornell (1990).

A simplex lattice {m, m} design will have many more profiles than needed as a simplex lattice design {m, 2} consists of m (m+1)/2 profiles. Therefore, suitable profiles can be chosen from a simplex lattice {m, m} design to estimate parameters and providing opportunities to the respondents to make a choice by comparing the profiles.

A polynomial model can be estimated to fit the data from a mixture experiment. However,

as a result of the restriction X1+ X2+....+Xm=1 on the independent variables, the regression must

be applied without intercept. The linear model typically used in analysis of SP data is given by

∑

If we multiply the termβ as in equation 2 by X1+₀ X2+....+ Xm=1, we obtain

∑

∑

where β_i* =β₀ +β₁ (3) The general second-degree polynomial in q variables is

j i j i q j i i q i i i i q i iX X X X y E < = =

∑

∑∑

∑

Applying the fact that X1+ X2+....+Xm=1, we obtain

∑∑

∑

If responses change with different available budgets allocated to the attributes, this would preclude the use of fractional factorial designs in conjoint preference and choice experiments as the properties of fractional factorial design will violate the constraints. Therefore, we conducted a survey with one mixture design and one fractional factorial design to explore and better understand the capabilities of the mixture amount designs.

2.1. Mixture Design for CA with Four Attributes

Assume a four attribute CA problem. Attributes are denoted as x_j,y_k,z_m,w_n and these

attributes should correspond the constraints 0 ≤ x_j,y_k,z_m,w_n ≤ 1 and xj +yk +zm+wn=

1.00. As it can be seen in Table 1, there are 13 mixtures that correspond to these constraints and allow us to estimate the canonical model.

Table 1. Four Attribute Mixture Design

Mixture Attribute 1 Attribute 2 Attribute 3 Attribute 4

1 1 0 0 0 2 0 1 0 0 3 0 0 1 0 4 0 0 0 1 5 1/2 1/4 1/4 0 6 0 1/2 1/4 1/4 7 1/4 0 1/2 1/4 8 1/4 1/4 0 1/2 9 1/4 1/2 1/4 0 10 0 1/4 1/2 1/4 11 1/4 0 1/4 1/2 12 1/2 1/4 0 1/4 13 1/4 1/4 1/4 1/4

Table 2. Choice sets for Four Attributes and Four Amounts

Choice Set Profile a Profile b Profile c Profile d Base 1 1 a (1) a₂(2) a₃(4) a₄(10) No Choice 2 1 a (2) a₂(3) a₃(5) a₄(11) No Choice 3 1 a (3) a₂(4) a₃(6) a₄(12) No Choice 4 1 a (4) a₂(5) a₃(7) a₄(13) No Choice 5 1 a (5) a₂(6) a₃(8) a₄(1) No Choice 6 1 a (6) a₂(7) a₃(9) a₄(2) No Choice 7 1 a (7) a₂(8) a₃(10) a₄(3) No Choice 8 1 a (8) a₂(9) a₃(11) a₄(4) No Choice 9 1 a (9) a₂(10) a₃(12) a₄(5) No Choice 10 1 a (10) a₂(11) a₃(13) a₄(6) No Choice 11 1 a (11) a₂(12) a₃(1) a₄(7) No Choice 12 1 a (12) a₂(13) a₃(2) a₄(8) No Choice 13 1 a (13) a₂(1) a₃(3) a₄(9) No Choice

We assume that there are four amounts (a1,a2,a3,a4) to be allocated to the attributes

according to Table 1. Therefore, there are 52 profiles that respondents should choose from. Choice sets are organized according to the balanced incomplete block design as shown in Table 2. As all amounts are shown in each choice set, amounts should not be dominated by

others. Thus, amounts are restricted as ½ max (a₁,a_2,a₃,a₄) ≤ min(a₁,a_2,a₃,a₄). Using

responses uijkmnfor profile (xj,yk,zm,wn) and amount a , we can fit the model: i

n m n k m k n j m j k j m k j m i k i j i i i ijkmn w z w y z y w x z x y x z y x z a y a x a a a u E 34 24 23 14 13 12 3 2 1 3 2 1 2 ) ( δ δ δ δ δ δ δ δ δ γ γ γ β α + + + + + + + + + + + + + = (6)

In equation (6), the term w_n, a_iw_n is not used becausex_j +y_k +z_m +w_n =1.00.

3. Experiment

In this paper, we compare two design approaches for conjoint analysis which are mixture amount designs and traditional fractional factorial designs. Before discussing the results, we will first outline the creation of these two designs in this section.

3.1. Experiment with Mixture Amount Design

For this CA experiment, there are four attributes, representing travel times to activities. These activities are education, grocery shopping, cultural activities and sports. The average base values for the travel times of these activities are assumed to be 10 minutes, 5 minutes, 15 minutes and 10 minutes since total travel time for a day is generally approximately 40 minutes. We assumed four discretionary time budgets, set as 10, 14, 18, and 20 minutes that can be allocated to these travel times to evaluate travel time sensitivity to activities. A total of 13 profiles were created by using the mixture design for four attributes as shown in Table 1 for each discretionary time budget. There are four discretionary amounts to be allocated; therefore there are 52 profiles in total. These 52 mixture profiles are listed in Table 3. A balanced incomplete block design was used to create the choice sets, shown in Table 2.

In addition, two holdout choice sets were created and used in the experiment to measure validity. These holdout profiles were not used for estimating utilities. However, they were used to test how well the estimated part-worth utilities can predict responses to the hold-outs not used in utility estimation. Moreover, some of the choice sets are asked twice to assess test-retest reliability.

Table 3. Mixture profiles for four different travel time budgets (minutes) Mixture Total Travel Time Travel Time to Education Travel Time to Grocery Shopping Travel Time to Cultural Activities Travel Time to Sports A llo ca tio n 1 1 50 20 5 15 10 2 50 10 15 15 10 3 50 10 5 25 10 4 50 10 5 15 20 5 50 15 7.5 17.5 10 6 50 10 10 17.5 12.5 7 50 12.5 5 20 12.5 8 50 12.5 7.5 15 15 9 50 12.5 10 17.5 10 10 50 10 7.5 20 12.5 11 50 12.5 5 17.5 15 12 50 15 7.5 15 12.5 13 50 12.5 7.5 17.5 12.5 A llo ca tio n 2 14 54 24 5 15 10 15 54 10 19 15 10 16 54 10 5 29 10 17 54 10 5 15 24 18 54 17 8.5 18.5 10 19 54 10 12 18.5 13.5 20 54 13.5 5 22 13.5 21 54 13.5 8.5 15 17 22 54 13.5 12 18.5 10 23 54 10 8.5 22 13.5 24 54 13.5 5 18.5 17 25 54 17 8.5 15 13.5 26 54 13.5 8.5 18.5 13.5 A llo ca tio n 3 27 58 28 5 15 10 28 58 10 23 15 10 29 58 10 5 33 10 30 58 10 5 15 28 31 58 19 9.5 19.5 10 32 58 10 14 19.5 14.5 33 58 14.5 5 24 14.5 34 58 14.5 9.5 15 19 35 58 14.5 14 19.5 10 36 58 10 9.5 24 14.5 37 58 14.5 5 19.5 19 38 58 19 9.5 15 14.5 39 58 14.5 9.5 19.5 14.5 A llo ca tio n 4 40 60 30 5 15 10 41 60 10 25 15 10 42 60 10 5 35 10 43 60 10 5 15 30 44 60 20 10 20 10 45 60 10 15 20 15 46 60 15 5 25 15 47 60 15 10 15 20 48 60 15 15 20 10 49 60 10 10 25 15 50 60 15 5 20 20 51 60 20 10 15 15 52 60 15 10 20 15

3.2. Experiment with Fractional Factorial Design

In the mixture amount application, each attribute has four different values in one allocation task due to the proportions (1, ½, ¼, 0) underlying the design (see Table 1). These four values for each attribute were taken as standard values for the attribute levels in the construction of the fractional factorial design. There are four attributes and each attribute has four different

values. In addition, there are four different allocations. Therefore, a 45 FF design (5 attributes

and 4 levels) with 64 profiles was constructed. The first four columns were used for the attributes while the last column was used for allocating 16 profiles to 4 different allocations. The allocation tasks and attribute levels were coded as shown in Table 4. Choice sets were created randomly from the 64 profiles. Each choice set has two options. Therefore, there are 32 choice sets.

In this fractional factorial design, the sum of variables x_j,y_k,z_m,w_nis not equal to 1.

However, this design uses the same proportions as MA and therefore allows us to fit the FF data to model of MA.

n m n k m k n j m j k j m k j m i k i j i i i ijkmn w z w y z y w x z x y x z y x z a y a x a a a u E 34 24 23 14 13 12 3 2 1 3 2 1 2 ) ( δ δ δ δ δ δ δ δ δ γ γ γ β α + + + + + + + + + + + + + = (7)

Table 4. Coding for Allocation Tasks and Attribute Values

Coding For Allocation Allocation Task Coding For Attributes Travel Time to Education Travel Time to Grocery Shopping Travel Time to Cultural Activities Travel Time to Sports 0 A llo ca tio n 1 0 10.0 5.0 15.0 10.0 1 12.5 7.5 17.5 12.0 2 15.0 10.0 20.0 15.0 3 20.0 15.0 25.0 20.0 1 A llo ca tio n 2 0 10.0 5.0 15.0 10.0 1 13.5 8.5 18.5 13.5 2 17.0 12.0 22.0 17.0 3 24.0 19.0 29.0 24.0 2 A llo ca tio n 3 0 10.0 5.0 15.0 10.0 1 14.5 9.5 18.5 14.5 2 19.0 14.0 22.0 19.0 3 28.0 23.0 29.0 28.0 3 A llo ca ti on 4 0 10.0 5.0 15.0 10.0 1 15.0 10.0 20.0 15.0 2 20.0 15.0 25.0 20.0 3 30.0 25.0 35.0 30.0

4. Data

The data were collected in February, 2011 in Eindhoven, the Netherlands. The questionnaire was divided into two parts. Consequently, two respondents were required for one completion of the designs. Each respondent had to respond to both the MA experiment and the FF experiment choice sets. There were 105 respondents for the survey. However; 64 of them finished the MA choice sets, while 62 finished FF choice sets. As a result, 32 completed MA and 31 completed FF choice sets could be used for the analysis. This will impact the significance of the results, but the insignificance of the results does not influence our objective of discussing and illustrating the rationale for MA designs, their analysis, and empirical approaches for their evaluations.

5. Results

5.1. Estimation of Models

Due to the small size data sample, none of the variables is significant except 2

i

a in the MA

estimation and ykwnin the FF estimation. Therefore, only the effects of coefficients will be

interpreted in this section.

Table 5. Estimation results of MA data Table 6. Estimation Results of FF data Variable Coefficient P[|Z|>z] Variable Coefficient P[|Z|>z]

i a 0.16132358 0.5182 ai 0.01505120 0.8342 2 i a -0.00831388 0.0309 a_i2 -0.00400645 0.1102 j ix a 0.00680499 0.2137 j ix a -0.00171593 0.5446 k iy a 0.01029499 0.1087 aiyk 0.00283746 0.5240 m iz a 0.00637031 0.2463 aizm 0.00363208 0.3681 j x 0.22905949 0.6355 xj 0.23217947 0.0876 k y -0.12205970 0.7521 k y -0.21471728 0.1466 m z -0.10652198 0.7473 m z 0.14931736 0.1346 k jy x 0.01946393 0.2910 k jy x 0.00885300 0.1144 m jz x -0.01008756 0.3581 xjzm -0.00121535 0.8551 n jw x -0.01773007 0.3257 n jw x 0.01245900 0.0598 m kz y -0.02524887 0.1802 ykzm 0.00038963 0.9341 n kw y -0.00283175 0.7955 ykwn 0.01074977 0.0106 n mw z 0.03233220 0.0862 zmwn 0.00270175 0.3538

The estimation of MA data results are shown in Table 5. The quadratic term a_i2 has a negative sign which means the utility increases at a decreasing rate with increasing amounts. The amounts have a positive but small effect on travel time to education (x), grocery shopping (y) and cultural activities (z). In addition, travel time to grocery shopping and travel time to cultural activities have negative effects while travel time to education has a positive effect. This suggests that people are willing to spend more time on travel to education and less time on travel to the other activities, when faced with fixed travel time budgets. The interactions between travel time to education and cultural activities, travel time to education and sports (w), travel time to grocery shopping and cultural activities, travel time to grocery shopping and sports have negative effects while the interactions between travel time to education and grocery shopping, travel time to cultural activities and sports have a positive effect. This shows us that the combination of travel time to a committed activity and a leisure activity decreases utility. However, combination of travel time to committed activities such as education and grocery shopping increases utility. Moreover, combination of travel to leisure activities such as cultural activities and sports increases the utility as well.

The estimation of FF data results are shown in Table 6. The quadratic term a has a i2

negative sign, indicating that utility increases at a decreasing rate with increasing amounts as in MA results. The amounts have small effects on travel time to education, grocery shopping and cultural activities. Moreover, increasing amount for travel time to education has a negative effect which means that increasing amounts of time for traveling to education decreases the utility that individuals derive. Travel time to grocery shopping has a negative effect while travel time to education and travel time to cultural activities have a positive effect. This suggests that people prefer to spend more time on travel to education and cultural activities and less time to travel for grocery shopping. The interactions between travel time to education and cultural activities shows that this combination decreases utility. The interactions between travel time to education and grocery shopping, travel time to education and sports, travel time to grocery shopping and cultural activities, travel time to grocery shopping and sports, travel time to cultural activities and sports have positive effects. The combination of travel time to education and cultural activities decreases the utility.

Comparing the overall results of the two design approaches clearly indicates that estimated effects differ for most attributes. The effects of time budget is stronger articulated in the MA experiment compared to the FF experiment. Moreover, the negative signs of the travel time

allocation variables in the MA experiment may suggest that this approach better picks up the trade-offs between time allocations under budget constraints.

5.2. Holdout Profile Results

Two holdout choice sets were included to the questionnaire to compare the ability of the estimated models using the two different approaches to predict the holdout profiles. As the models for two approaches are the same, we used the parameters of each design to predict the MA holdouts, because the MA experiment has constrained sets which are more realistic to the travelers. Although, it appears from Table 7 and 8 that the designs do not predict the holdouts very accurately, in fact chi-square results indicate that in none of the approaches do the predicted proportions differ significantly from the observed proportions. The insignificance of model parameters in the respective models precludes comparison between models. This will have to await a larger sample.

Table 7. Mixture Amount Design Holdout Choice Set 1 Predictions with MA and FF parameters MA Holdout

Choice Set 1 Observed

Predicted with MA parameters Predicted with FF parameters Profile 1 28.1 35.9 19.4 Profile 2 21.9 31.9 18.8 Profile 3 32.8 16.9 33.8 Profile 4 9.4 4.5 23.9 Profile 5 7.8 10.8 4.2 χ2= 2.08 (p=.72) χ2= 1.31(p = .86)

Table 8. Mixture Amount Design Holdout Choice Set 2 Predictions with MA and FF parameters MA Holdout

Choice Set 2 Observed

Predicted with MA parameters Predicted with FF parameters Profile 1 39.1 48.7 23.5 Profile 2 17.2 16.4 30.9 Profile 3 6.3 13.2 11.5 Profile 4 29.7 11.1 30.0 Profile 5 7.8 10.7 4.1 χ2= 1.81 (p=.77) χ2= 1.85(p = .76) 5.3. Repeated Choice Set Results

Four repeated choice sets were used in the questionnaire to compare the consistency of responses. As it can be seen from Table 9, the responses in the FF experiment are more consistent compared to those in the MA experiment. Three choice options were used in the

FF design while 5 choice options were used in the MA experiment, so the base error rate differs between the experiments. The larger chi-square results for the MA experiment indicates that the "hit rates" for the MA design differed more from their expected rate of .20 than those of the FF design differed from their expected rate of .33. Moreover, attribute values are closer in MA choice options, and therefore may be more difficult to discriminate.

Table 9. Comparison of Repeated Choice Sets

Repeated Choice Sets MA Experiment FF Experiment Repeated Choice Set 1 65.63% 90.63% Repeated Choice Set 2 59.38% 65.63% Repeated Choice Set 3 43.75% 80.65% Repeated Choice Set 4 59.38% 58.06% χ2= 30.65 (d.f.=3) χ2= 22.25(d.f.=3) 6. Conclusion

In this paper, we have focused the attention on the potential applicability of mixture-amount models in time use and transportation studies to examine consumer choice under budget constraints. To explore their applicability empirically, we conducted both a mixture amount experiment and fractional factorial experiment using travel time allocation to four different activities as an example. Mixture amount designs provide a framework for conducting conjoint analysis experiments subject to constraints. Moreover, the effect of changing amount and attribute values can be investigated in this framework. Thus, these designs allow understanding trade-offs between attributes and between attributes and constrained budgets. In addition, this framework allows researchers to examine the optimal allocation of money or time, subject to corresponding budget constraints (Raghavarao & Wiley, 2009). The value of these mixture-amount models concerns the fact that respondents are shown constrained attribute values. This is impossible using fractional factorial design and therefore it may be argued that traditional conjoint studies using such fractional factorial designs may not generate preference or choice responses necessary to estimate utility functions under budget constraints.

The results of some first analyses reported in this paper of a comparison between the two approaches suggest that indeed the utility estimates between the two approaches are significantly different. Supportive of the potentially higher validity of the mixture amount model are the results that the effects of the budget stand out more clearly and that the estimated effects for the attributes are more in line with constraints. On the other hand,

mixture amount experiments have added complexity. Processing the information may be more demanding, which in turn may reduce the reliability of the response patterns.

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