designs for first-order interactions in choice experiments ... - CiteSeerX

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DESIGNS FOR FIRST-ORDER INTERACTIONS IN CHOICE EXPERIMENTS WITH BINARY ATTRIBUTES Heiko Großmann1 , Rainer Schwabe2 and Steven G. Gilmour1 1 Queen

Mary, University of London and 2 University of Magdeburg

Abstract: For choice experiments involving pairs of options described by a common set of two-level factors a new method for generating exact designs is presented, which allow the efficient estimation of main effects and first-order interactions. For high efficiencies these designs often require a smaller number of choice sets than available alternatives in the literature. Key words and phrases: Balanced incomplete block design, Hadamard matrix, interactions, optimal design, paired comparisons.

1. Introduction Choice experiments aim at understanding how preferences for goods or services are influenced by the features of competing options and applications in marketing, health economics and other fields abound. A typical choice experiment consists of a series of choice tasks in which attributes of the options are systematically varied. Each task offers a choice set of options and asks the respondent to select, for example, the most attractive alternative. These stated choices are analyzed to estimate utility values which reflect the influence of the attributes. Excellent descriptions of choice experiments, their background, underlying models and analysis can be found in the monographs by Louviere, Hensher and Swait (2000) and Train (2003). The choice sets presented for evaluation are generated according to an experimental design which specifies attribute levels for the options in each set. Originating with the work of Louviere and Woodworth (1983) the optimal and efficient design of choice experiments has attracted considerable attention. Reviews of these developments have been provided by Großmann, Holling and Schwabe (2002) and Louviere, Street and Burgess (2004). In this paper, we are interested in determining efficient designs for experiments with choice sets of size two in

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HEIKO GROSSMANN, RAINER SCHWABE AND STEVEN G. GILMOUR

which the options are characterized by a common set of two-level factors and both main effects and first-order interactions are to be estimated. The corresponding design problem has been considered before by Street, Bunch and Moore (2001) who derived optimal designs under the assumption that within each choice set each option is chosen with the same probability. Yet, these designs are much too large for applications and more practical designs which are also optimal or highly efficient were presented by Street and Burgess (2004). The assumption made in these works that the choice probabilities within the choice sets are equal is equivalent to assuming that the unknown model parameters in the underlying multinomial logit model are equal to zero. In this case, however, the optimal design problem for the choice model is equivalent to the corresponding problem for a linear paired comparison model (see e.g. Großmann et al. (2002)). It follows that optimal designs for this linear model are also optimal for the choice model and vice versa. Optimal designs for estimating first-order interactions and main effects in paired comparison models with twolevel factors which coincide with the designs reported in Street et al. (2001) were derived by van Berkum (1987, pp. 30-31) and corresponding results when the common number of levels is larger than two were presented by Graßhoff, Großmann, Holling and Schwabe (2003). The latter designs are also optimal if only interaction effects are to be estimated, but are again too large for direct use in applications. By using the correspondence between the design problems for the multinomial logit model and linear paired comparisons, here we derive new designs for the case of choices between two options described by binary attributes which are optimal or highly efficient for estimating main effects and first-order interactions and also optimal for estimating interaction effects only. These designs compare favorably with the designs in Street and Burgess (2004) in that for high efficiencies they usually require the same or a considerably smaller number of choice sets. Conversely, for the same number of choice sets they possess the same or a higher efficiency. 2. General Setting Suppose there are K factors each at v = 2 levels that are assumed to drive the preferences for the options in a choice experiment. The options are then

CHOICE DESIGNS FOR FIRST-ORDER INTERACTIONS

3

represented by combinations of attribute levels. In what follows, only choices between two options are considered. The first option in each choice set is denoted by s = (s1 , . . . , sK ) and the second one by t = (t1 , . . . , tK ), which are both elements of the set {1, 2}K where the numbers 1 and 2 represent the first and second level of each factor, respectively. Notice that technically speaking the choice sets are ordered pairs. We only consider forced choice experiments which require that one of the options in each choice set must be chosen and do not offer the opportunity to defer choices. Let Y (s, t) be the random variable which equals 1 when s is chosen and 0, if t is selected. The widely used multinomial logit (MNL) model then supposes that for every pair (s, t) the probability of choosing s is given by P (Y (s, t) = 1) = exp[f (s)> β]/(exp[f (s)> β] + exp[f (t)> β]) where f = (f1 , . . . , fp )> is a vector of p known regression functions and β contains the unknown model parameters. The choice probabilities can be rewritten as P (Y (s, t) = 1) = exp[(f (s) − f (t))> β]/(1 + exp[(f (s) − f (t))> β]) from which it is easily seen that for choices between two options the MNL model is equivalent to logistic regression with predictors given by the components of (f (s) − f (t))> . Optimality criteria for designs in the MNL model are typically based on the normalized Fisher information matrix. For an exact design ξN with N pairs P > > (sn , tn ) this matrix is given by M(ξN ; β) = N1 N n=1 Xn (Dn − pn pn )Xn where Xn = (f (sn ), f (tn ))> , Dn = diag(πn , 1 − πn ) is diagonal, pn = (πn , 1 − πn )> and πn = P (Y (sn , tn ) = 1) for every n = 1, . . . , N . Obviously, M(ξN ; β) depends on the unknown vector β via the choice probabilities πn . If πn = 1 − πn = 1/2 for every n or equivalently if β = 0, it follows that M(ξN ; β) = 41 M(ξN ) where M(ξN ) =

1 > NX X

and X = (f (s1 ) − f (t1 ), . . . , f (sN ) − f (tN ))> . The matrix

M(ξN ) can be recognized as the normalized information matrix of ξN in the linear paired comparison model Y˜ (s, t) = (f (s) − f (t))> β˜ + ε

(2.1)

with parameter vector β˜ and uncorrelated homoscedastic errors ε. Since under the assumption β = 0 the matrices M(ξN ; β) and M(ξN ) are proportional, it follows that for every criterion function based on M(ξN ; β) optimal designs in the MNL model can be found by optimizing the same criterion function applied to the normalized information matrix M(ξN ) in model (2.1).

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HEIKO GROSSMANN, RAINER SCHWABE AND STEVEN G. GILMOUR

We now specify the vector of regression functions f for the case where all K factors have v = 2 levels and the model includes both main effects and firstorder interactions. To this end, let g : {1, 2} → {−1, 1} be the function defined by g(1) = 1 and g(2) = −1 which maps the original levels into their effectscoded equivalents. For k = 1, . . . , K set gk = g and for every k, ` ∈ {1, . . . , K} with k < ` define gk,` : {1, 2}2 → {−1, 1} by gk,` (x, y) = g(x)g(y). Finally let f : {1, 2}K → {−1, 1}p be defined by f (x) = (g1 (x1 ), . . . , gK (xK ), g1,2 (x1 , x2 ), . . . , gK−1,K (xK−1 , xK ))>

(2.2)

for every x = (x1 , . . . , xK ) ∈ {1, 2}K where p = K + K(K − 1)/2 is the number of parameters in the corresponding linear paired comparison model (2.1). The first K components of the associated parameter vector β˜ represent the main effects (k)

β1

of level 1 of the k-th factor, k = 1, . . . , K, from which the parameter for the (k)

(k)

level 2 can be obtained as β2 = −β1 . Similarly, the remaining K(K − 1)/2 (k,`) components of β˜ represent the interactions β1,1 of the first levels of the factors k, ` ∈ {1, . . . , K} where k < `. The parameters for the other combinations of (k,`)

(k,`)

(k,`)

(k,`)

(k,`)

the levels of the factors are given by β1,2 = β2,1 = −β1,1 and β2,2 = β1,1 ˜ Thus the parameters are related as in and are not explicitly represented in β. a standard 2K -ANOVA model with identifiability conditions β1

(k)

+ β2

(k,`) β1,1

+

(k,`) β2,2

+

(k,`) β2,1

= 0,

(k,`) β1,2

+

(k,`) β2,2

= 0,

(k,`) β1,1

+

(k,`) β1,2

= 0 and

(k,`) β2,1

(k)

= 0,

= 0.

For technical ease we consider approximate designs which are defined as probability measures on the design region X = {1, 2}K ×{1, 2}K of all pairs (s, t). Every exact design ξN consisting of N pairs (s1 , t1 ), . . . , (sN , tN ) can be identified with the approximate design ξ˜N which assigns equal weight ξ˜N (sn , tn ) = 1/N to each pair (sn , tn ), n = 1, . . . , N . Conversely, every approximate design ξ which assigns only rational weights ξ(s, t) to all pairs (s, t) in its support can be realized as an exact design ξN for some N . For more details about approximate design we refer the reader to Kiefer (1959). The information matrix of an approximate design ξ in the linear paired comparison model (2.1) is defined by X M(ξ) = ξ(s, t)(f (s) − (f (t))(f (s) − (f (t))> . (s,t)∈X

Note that for an exact design ξN the normalized information matrix M(ξN )

CHOICE DESIGNS FOR FIRST-ORDER INTERACTIONS

5

coincides with the information matrix M(ξ˜N ) of the corresponding approximate design ξ˜N . Optimality criteria for approximate designs ξ are usually functionals of M(ξ). For example, an approximate design ξ ∗ is D-optimal if it maximizes the determinant of the information matrix, that is, if det M(ξ ∗ ) ≥ det M(ξ) for every approximate design ξ. As in most works about optimal designs for choice experiments, in the following mainly D-optimality is considered. 3. Optimality Results We consider the linear paired comparison model (2.1) with the vector of regression functions f specified by (2.2). The design region X can be partitioned into disjoint sets such that the pairs in each set differ only in some of the factors. These sets play an important role in the construction of optimal designs. More precisely, for d = 0, . . . , K let Xd = {(s, t) ∈ X : |{k : sk 6= tk }| = d} be the set of Nd = 2K K!/[d!(K − d)!] pairs which vary in exactly d attributes and denote by ξ¯d the uniform approximate design which gives equal weight ξ¯d (s, t) = 1/Nd to each pair in Xd and weight zero to all remaining pairs in X . Following Graßhoff et al. (2003) we refer to d as the comparison depth. Obviously, for every pair (s, t) ∈ Xd the comparison depth d is equal to the Hamming distance of s and t. It was shown by van Berkum (1987, p. 30) that the information matrix of ξ¯d is equal to M(ξ¯d ) =

4d K IK

0

0

8d(K−d) K(K−1) IK(K−1)/2

! (3.1)

where Im is the identity matrix of order m for every m. Moreover, he proved the D-optimality of the design ξ ∗ = w∗ ξ¯d∗ + (1 − w∗ )ξ¯d∗ +1 where d∗ = (K + 1)/2, w∗ = 1 for K odd and d∗ = K/2, w∗ = (d∗ + 1)/(K + 1) for K even. The information matrix of ξ ∗ is then equal to M(ξ ∗ ) = 4d∗ /KIp if K is odd and M(ξ ∗ ) = 4(d∗ + 1)/(K + 1)Ip if K is even, with p denoting the number of parameters. The values of d∗ and w∗ for K = 3, . . . , 10 are shown in Table 3.1. Thus if K is odd, considering only pairs with a single comparison depth is sufficient to generate a D-optimal design, whereas two comparison depths are needed if K is even. It is easy to see that for even K the uniform design on the union XK/2 ∪ XK/2+1 can be represented as ξ ∗ = w∗ ξ¯K/2 + (1 − w∗ )ξ¯K/2+1 and is hence D-optimal (Graßhoff et al. (2003)). Also, it can be easily verified by means of the general equivalence theorem (Kiefer, 1974) that the design ξ ∗ is Φq -optimal for

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HEIKO GROSSMANN, RAINER SCHWABE AND STEVEN G. GILMOUR

every qth matrix mean under the present parameterization. This includes the Acriterion for minimizing the average variances of the parameter estimates (q = 1), as has been mentioned by Street et al. (2001), and the E-criterion for minimizing the maximum eigenvalue of the corresponding covariance matrix (q = ∞). Moreover, ξ ∗ is also M V -optimal, i.e. it minimizes the maximum variance for the single parameter estimates. We further mention that the designs ξ¯d∗ are optimal with regard to every optimality criterion which is invariant with regard to both permutations of the factors and permutations of the levels, when only the interaction effects in the model are to be estimated (Graßhoff et al. (2003)). An interesting question is how well these designs perform for estimating main effects and interactions when K is even. More generally, for arbitrary K we consider the D-efficiency eff D (ξ¯d ) = (det M(ξ¯d )/ det M(ξ ∗ ))1/p of the uniform design ξ¯d . From (3.1) it follows that 2d eff D (ξ¯d ) = cK



2(K − d) K −1

 K−1 K+1

where cK = K +1 for K odd and cK = K(K +2)/(K +1) for K even, respectively. If K is even and d = d∗ the expression simplifies to eff D (ξ¯d∗ ) = (K + 1)(K + 2)−1 (K/(K − 1))(K−1)/(K+1) which converges quickly to 1. Table 3.1 presents the D-efficiencies of the uniform designs with comparison depths d∗ − 1, d∗ and d∗ + 1 for K = 3, . . . , 10. For even numbers of factors the table illustrates that ξ¯d∗ is nearly optimal with eff D (ξ¯d∗ ) > 0.99 if K > 2. Moreover, uniform designs with comparison depth d∗ − 1 or d∗ + 1 are often highly efficient. In fact, the corresponding D-efficiencies also converge to 1 as K grows larger. These designs can be attractive when they can be realized as exact designs with small numbers of pairs. The efficiency value of eff D (ξ¯3 ) = 0 for K = 3 reflects the well-known fact that the interaction effects cannot be estimated with designs which use only pairs that differ in all K attributes (e.g. Graßhoff et al. (2003)). 4. Efficient exact designs The uniform designs ξ¯d∗ perform very well in terms of the D-optimality criterion but the number of pairs Nd∗ they require is usually too large for practical applications. However, if an exact design with a practical number of pairs can be found whose normalized information matrix coincides with M(ξ¯d∗ ), this design

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CHOICE DESIGNS FOR FIRST-ORDER INTERACTIONS

Table 3.1: D-efficiencies of uniform designs for estimating main effects and interactions K

3

4

5

6

7

8

9

10

d∗ w∗ eff D (ξ¯d∗ ) eff D (ξ¯d∗ −1 ) eff D (ξ¯d∗ +1 )

2 1 1.0000 0.7071 0.0000

2 3/5 0.9903 0.6315 0.9801

3 1 1.0000 0.8736 0.8399

3 4/7 0.9967 0.8161 0.9948

4 1 1.0000 0.9306 0.9222

4 5/9 0.9985 0.8908 0.9979

5 1 1.0000 0.9564 0.9533

5 6/11 0.9992 0.9279 0.9989

will be as efficient as ξ¯d∗ . The corresponding statement is also true for the designs ξ¯d . Since these designs are often also very efficient, in the following we consider the slightly more general problem of constructing exact designs for which the normalized information matrix is equal to M(ξ¯d ) where 1 ≤ d < K and present a new method for generating exact designs with this property. The designs are generated using fractional factorials, balanced incomplete block designs and Hadamard matrices as building blocks. We consider again model (2.1) with K two-level factors and f defined by (2.2). For d ∈ {1, . . . , K −1} let n be the smallest number for which a Hadamard matrix of order not less than d exists and let H be such a matrix. In other words, H is an n×n matrix with elements in {−1, 1} and H> H = nIn . For every d up to 424 the number n is either equal to 1, 2 or the smallest multiple of 4 which is greater than or equal to d (Hedayat, Sloane and Stufken (1999), p. 147). Furthermore, let F be an m × (K − d) regular two-level fractional factorial of resolution III or higher for K −d factors (or the 2K−d full factorial if K −d ≤ 2) in which the levels of each factor are coded as 1 and 2, respectively. Finally, suppose there exists a balanced incomplete block design BIBD(K, b, r, d, λ) with parameters K (treatments), b (blocks), r (replication), d (block size) and λ (index). We assume that this design is given in the form of a d×b matrix B in which the treatments are represented by the numbers 1, . . . , K. Notice that only the parameters representing the number of treatments and block size are required to have specific values and that for every K and d any BIBD(K, b, r, d, λ) is completely determined by choosing a value for one of the remaining parameters b, r or λ. We construct an exact design ξN,d with N = bmn pairs in Xd .

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HEIKO GROSSMANN, RAINER SCHWABE AND STEVEN G. GILMOUR

The construction is divided into three stages which are described here in algorithmic form. In what follows for simplicity of notation the element in row i and column j of every matrix C is denoted by Ci,j . Step 1 Select an n × d matrix A consisting of d columns of H. Let L be the n × d matrix with Li,j = 1 if Ai,j = 1 and Li,j = 2 if Ai,j = −1 for every i and j. Similarly, let R be the n × d matrix with Ri,j = 2 if Ai,j = 1 and Ri,j = 1 if Ai,j = −1. Step 2 For j = 1, . . . , b let c1,j < . . . < cK−d,j be the elements in {1, . . . , K} \ {B1,j , . . . , Bd,j } and let σj be the permutation of the numbers 1, . . . , K defined by σj (k) = i if k = ci,j and σj (k) = K − d + i if k = Bi,j . Step 3 For g = 1, . . . , b, h = 1, . . . , m, i = 1, . . . , n and k = 1, . . . , K set sα(g,h,i),k = tα(g,h,i),k = Fh,σg (k)

if σg (k) ∈ {1, . . . , K − d}

sα(g,h,i),k = Li,σg (k)−K+d

if σg (k) ∈ {K − d + 1, . . . , K}

tα(g,h,i),k = Ri,σg (k)−K+d

if σg (k) ∈ {K − d + 1, . . . , K}

where α(g, h, i) = (g − 1)(mn) + (h − 1)n + i. Finally, the exact design ξN,d is defined by the pairs (sα(g,h,i) , tα(g,h,i) ), g = 1, . . . , b, h = 1, . . . , m, i = 1, . . . , n, where sα(g,h,i) = (sα(g,h,i),1 , . . . , sα(g,h,i),K ) and tα(g,h,i) = (tα(g,h,i),1 , . . . , tα(g,h,i),K ). To illustrate the construction of ξN,d we consider K = 3 factors and the comparison depth d = d∗ = 2. As the building blocks we choose ! ! ! 1 1 1 1 1 2 H= , F= , and B = 1 −1 2 2 3 3 so that the design will consist of N = 3 × 2 × 2 = 12 pairs. Since d = n = 2, in the first step we set A = H, from which the matrices ! ! 1 1 2 2 L= and R = 1 2 2 1 are obtained. In the second step, the matrix B gives rise to the three permutations σ1 with σ1 (1) = 2, σ1 (2) = 3, σ1 (3) = 1, σ2 with σ2 (1) = 2, σ2 (2) = 1, σ2 (3) = 3 and σ3 with σ3 (1) = 1, σ3 (2) = 2, σ3 (3) = 3. The final design resulting

CHOICE DESIGNS FOR FIRST-ORDER INTERACTIONS

9

Table 4.2: Exact design for three factors and comparison depth two s

t

g

h

i

α

sα,1

sα,2

sα,3

tα,1

tα,2

tα,3

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

1 1 2 2 1 1 2 2 1 1 2 2

1 2 1 2 1 2 1 2 1 2 1 2

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

1 1 1 1 1 1 1 1 1 1 2 2

1 2 1 2 1 1 2 2 1 1 1 1

1 1 2 2 1 2 1 2 1 2 1 2

2 2 2 2 2 2 2 2 1 1 2 2

2 1 2 1 1 1 2 2 2 2 2 2

1 1 2 2 2 1 2 1 2 1 2 1

from the third step is shown in Table 4.2, where for convenience the values of g, h, i and α = α(g, h, i) are also given. In the table, factor settings that are held constant within a pair are printed in boldface. By looking at the example the structure of the designs ξN,d can be grasped more easily. Each block of the balanced incomplete block design B specifies d factors k1 , . . . , kd in which mn pairs of the design differ. For a given column of B these mn pairs are obtained as follows. First, the settings for the factors k1 , . . . , kd of s and t in n pairs are specified by the d columns of L and R, respectively, which are both derived from the Hadamard matrix H. The settings of the K − d factors which have the same level in s and t are given by a single row of the (fractional) factorial F. More precisely, in each of the n pairs the setting for the first constant factor of s and t is defined by the first component of the same fixed row of F, the setting for the second constant factor by the second component of the given row of F and so forth. Second, repeating this procedure for all m rows of F generates all mn pairs. The following theorem implies that the design ξN,d performs as well as the uniform design ξ¯d in terms of every optimality criterion which is based on the

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HEIKO GROSSMANN, RAINER SCHWABE AND STEVEN G. GILMOUR

information matrix. In particular, with regard to the D-optimality criterion ξN,d and ξ¯d possess the same efficiency. Theorem 1 The normalized information matrix M(ξN,d ) of the exact design ξN,d and the information matrix M(ξ¯d ) of the approximate design ξ¯d are equal. The proof is given in the appendix. Of course, the theorem applies if d = d∗ . Table 4.3 presents the building blocks needed for constructing exact designs ξN,d∗ for up to K = 10 factors. The table specifies the order of the Hadamard matrix H, the balanced incomplete block design and the factorial part F and lists the number of pairs N required by each design. The resolution of each fractional factorial is indicated by a subscript. Since they use pairs with comparison depth d∗ , the designs possess the D-efficiencies shown in the fourth row of Table 3.1. Notice that for given K and d∗ there are usually several Hadamard matrices, fractional factorials and balanced incomplete block designs satisfying the specifications in Table 4.3, which all give rise to exact designs with the same efficiency. Street and Burgess (2004) provide designs for K = 3, . . . , 8 factors generated by a different method of construction in their Table 6. For each value of K the table presents the numbers of pairs and D-efficiencies of different designs. For odd K = 3, 5, 7 the D-optimal designs require 12, 160 and 224 pairs, respectively, whereas 80 and 224 pairs are needed for K = 4 and K = 6. For K = 8 the design with the highest D-efficiency of 0.9997 consists of 1120 pairs. In addition, several smaller designs are given. For example, for K = 4 there is a design with 48 pairs and a D-efficiency of 0.9903 and for K = 5 a design with 80 pairs and a Defficiency of 0.9649. Table 4.3 shows that for odd K the D-optimal designs ξN,d∗ are of the same or a considerably smaller size. For K = 7, the smallest design given by Street and Burgess (2004) with an efficiency close to the optimum still requires 160 pairs. Although not optimal, for even K the designs in Table 4.3 are highly efficient with D-efficiencies of 0.9903, 0.9967 and 0.9985 for K = 4, 6, 8, respectively. For K = 6 and K = 8 designs with a comparable performance in Table 6 of Street and Burgess (2004) require 176 and 960 pairs. It seems that for D-efficiencies close to 1 the difference between the numbers of pairs needed by the designs ξN,d∗ and those of Street and Burgess grows larger with K. Unfortunately, that paper does not contain the exact information necessary for making comparisons for larger

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CHOICE DESIGNS FOR FIRST-ORDER INTERACTIONS

Table 4.3: Building blocks for efficient exact designs K

d∗

Hadamard matrix

BIBD(K, b, r, d∗ , λ)

Factorial part

Pairs

2 3 4 5 6 7 8 9 10

1 2 2 3 3 4 4 5 5

1 2 2 4 4 4 4 8 8

BIBD(2, 2, 1, 1, 0) BIBD(3, 3, 2, 2, 1) BIBD(4, 6, 3, 2, 1) BIBD(5, 10, 6, 3, 3) BIBD(6, 10, 5, 3, 2) BIBD(7, 7, 4, 4, 2) BIBD(8, 14, 7, 4, 3) BIBD(9, 18, 10, 5, 5) BIBD(10, 18, 9, 5, 4)

21 21 22 22

4 12 48 160 160 112 448 1152 1152

23−1 III 23−1 III 24−1 IV 24−1 IV 25−2 III

values of K. It should be noted, however, that for lower efficiencies the methods of Street and Burgess (2004) are often capable of generating designs which are smaller than the exact designs in Table 4.3, as is illustrated for K = 5 by their design with 80 pairs. In such cases designs ξN,d with d 6= d∗ may represent viable alternatives. For example, the design ξ40,4 for K = 5 derived from a Hadamard matrix of order 4, the BIBD(5, 5, 4, 4, 3) and the 21 full factorial requires only 40 pairs and still possesses a reasonable efficiency of eff D (ξ40,4 ) = 0.8399. 5. Concluding remarks We have considered the problem of finding exact designs for choice experiments with choice sets of size two which allow the efficient estimation of main effects and first-order interactions. When it is assumed that the choice probabilities for the options in each set are equal, this is equivalent to the corresponding problem for a linear paired comparison model. It appears that for high efficiencies the designs derived herein require fewer pairs than alternative designs in the literature. A similar comparison for main-effects models has been presented by Großmann, Holling, Graßhoff and Schwabe (2007). The proposed method for constructing designs can easily be implemented in a computer program and the building blocks—Hadamard matrices, fractional factorials and balanced incomplete block designs—are available from many sources.

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HEIKO GROSSMANN, RAINER SCHWABE AND STEVEN G. GILMOUR

Acknowledgement The research of Großmann and Gilmour was funded by the UK Engineering and Physical Sciences Research Council (EPSRC) under grant EP/C54171/1. Schwabe acknowledges support from the Deutsche Forschungsgemeinschaft (DFG) under grant HO 1286/2-3. Appendix A. Proof of Theorem 1 Let the d × b matrix B representing the balanced incomplete block design, the m×(K −d) (fractional) factorial F and the n×n Hadamard matrix be defined as in Section 4. Also, let A, L and R be the n × d matrices in Step 1 of the ˜ = (F ⊗ 1n , 1m ⊗ L) and R ˜ = (F ⊗ 1n , 1m ⊗ R), where construction of ξN,d . Set L the symbol ‘⊗’ denotes the Kronecker product and where for every u, as usual, 1u is a column vector of length u with all elements equal to 1. The N = bmn options sα(g,h,i) and tα(g,h,i) in Step 3 are the rows of the matrices     ˜ 1 ˜ 1 LP RP  .     ..  and  ...  ,     ˜ b ˜ b LP RP respectively, where P1 , . . . , Pb are the K ×K permutation matrices corresponding to σ1 , . . . , σb in Step 2 of the construction. Recall that the number of parameters in model (2.1) is equal to p = K + K(K − 1)/2 and consider the p × p matrix ˜ = (f (L) ˜ − f (R)) ˜ > (f (L) ˜ − f (R)) ˜ where f is defined by (2.2) and applied to each M ˜ and R, ˜ respectively. It follows that the normalized information matrix row of L of ξN,d is equal to M(ξN,d ) =

b 1 X ˜> ˜ ˜ diag(P> g , Pg )M diag(Pg , Pg ) N

(A.1)

g=1

˜ g , g = 1, . . . , b, is a permutation matrix corresponding to a permutation where P σ ˜g of the numbers K + 1, . . . , p induced by σg . Note that we do not need to know the exact form of any of the permutation matrices. For every u and every v, in the following 0u×v denotes a u × v zero matrix. It is now shown that ˜ = 4mn diag(0(K−d)×(K−d) , Id , G1 , . . . , GK−d , 0(d(d−1)/2)×(d(d−1)/2) ) M

(A.2)

where Gj = diag(0(K−d−j)×(K−d−j) , Id ), j = 1, . . . , K − d, is a (K − j) × (K − j) diagonal matrix, in particular GK−d = Id . In order to prove this equality, notice

13

CHOICE DESIGNS FOR FIRST-ORDER INTERACTIONS

˜ − f (R) ˜ = (0mn×(K−d) , 21m ⊗ A, C), where that f (L)  (1)  (1) C1 . . . CK−1  .  ..  . C= .  .  (m) (m) C1 . . . CK−1 (h)

with n × (K − j) matrices Cj . For h = 1, . . . , m and j = 1, . . . , K − d these (h)

are given by Cj

= (0n×(K−d−j) , 2g(Fh,j )A) where g : {1, 2} → {−1, 1} with

g(1) = 1 and g(2) = −1 is the function considered in Section 2 and, as before, Fh,j denotes the element in row h and column j of F. For h = 1, . . . , m and (h)

j = K − d + 1, . . . , K − 1 the matrix Cj for these h and j it holds that

(h) Cj

does not depend on h. In fact,

= 0n×(K−j) which can be seen as follows.

For every j = K − d + 1, . . . , K − 1 and h = 1, . . . , m it is easy to verify that (h)

Cj

= Uj − Vj is the difference of two matrices Uj and Vj which depend on

the columns j − K + d, . . . , d of A. More precisely, if these columns are denoted by a(j−K+d) , . . . , a(d) , then Uj = (a(j−K+d) ∗ a(j−K+d+1) , . . . , a(j−K+d) ∗ a(d) ) and Vj = ((−a(j−K+d) ) ∗ (−a(j−K+d+1) ), . . . , (−a(j−K+d) ) ∗ (−a(d) )) where the symbol ‘∗’ represents the elementwise product of the columns. Since obviously Uj = Vj , the result follows. ˜ − f (R) ˜ = (0mn×(K−d) , 21m ⊗ A, C) and (1m ⊗ A)> (1m ⊗ A) = Since f (L) P > (h) mnId , to prove (A.2), it is sufficient to show that m h=1 A Cj = 0d×(K−j) for j = 1, . . . , K − d and that m X

( (h) (h) (Cj )> Cj 0

=

h=1

for j, j 0 = 1, . . . , K − d. Now,

4mnGj

if j = j 0

0(K−j)×(K−j)

if j 6= j 0

(h) h=1 Cj

Pm

= (0n×(K−d−j) , 2A

Pm

h=1 g(Fh,j ))

=

0n×(K−j) for every j = 1, . . . , K − d since in each column of F the levels 1 and 2 occur an equal number of times, which shows that the first condition is fulfilled. Furthermore, for j, j 0 = 1, . . . , K − d it holds that m X h=1

(h) (h) (Cj )> Cj 0

=

0(K−d−j)×(K−d−j 0 ) 0d×(K−d−j 0 )

0(K−d−j)×d Pm > 4A A h=1 g(Fh,j )g(Fh,j 0 )

!

and because F is of resolution III or higher (or the 2K−d full factorial for K − d ≤ P 0 2) the sum m h=1 g(Fh,j )g(Fh,j 0 ) is equal to m if j = j and 0 otherwise. In view

14

HEIKO GROSSMANN, RAINER SCHWABE AND STEVEN G. GILMOUR

of A> A = nId this finally proves (A.2). From (A.1) it then follows, that M(ξN,d ) is a diagonal matrix. Next, the first K diagonal elements of M(ξN,d ) are derived. Let N be the K × b incidence matrix with elements in {0, 1} and columns n(1) , . . . , n(b) corresponding to the BIBD(K, b, r, d, λ) represented by B. The definition of σg in Step 2 of the construction implies diag(0(K−d)×(K−d) , Id ) = Pg diag(n(g) )P> g for every g = 1, . . . , b so that b X

P> g diag(0(K−d)×(K−d) , Id )Pg =

g=1

b X

diag(n(g) ) = rIK .

g=1

From this and (A.2) it then follows that the first K diagonal elements of M(ξN,d ) are equal to 4r/b. Furthermore r/b = d/K, which shows that the first K diagonal elements of M(ξN,d ) coincide with the corresponding elements of M(ξ¯d ) in (3.1). We now derive the remaining diagonal elements of M(ξN,d ). To this end consider the K × K matrix Λ=

0(K−d)×(K−d) 1K−d 1> d 1d 1> K−d

0d×d

! .

The elements Λj,j+1 , . . . , Λj,K in row j of Λ are equal to the diagonal elements of Gj for j = 1, . . . , K − d and are the diagonal elements of 0(K−j)×(K−j) for j = K − d + 1, . . . , K − 1. Moreover, the diagonal elements of b X

˜ > diag(G1 , . . . , GK−d , 0(d−1)×(d−1) , 0(d−2)×(d−2) , . . . , 01×1 )P ˜g P g

(A.3)

g=1

correspond to elements in the upper triangular part (excluding the diagonal) of ˜ = Pb P> ΛPg . From the definition of the permutation σg , g = 1, . . . , b, it Λ g=1

g

˜ k,k0 where k, k 0 ∈ {1, . . . , K} with k < k 0 is equal to the follows that the element Λ number of blocks which contain k but not k 0 . In order to calculate this number we note that the number of blocks containing both k and k 0 equals the index λ of the balanced incomplete block design, and that the number of blocks containing neither k nor k 0 is given by the index b − 2r + λ of the complement design in which each of the original blocks is replaced by its complement with regard to ˜ k,k0 = b − λ − (b − 2r + λ) = 2(r − λ) for all the set {1, . . . , K}. It follows that Λ k < k 0 . Since this value does not depend on k or k 0 it follows that all diagonal

CHOICE DESIGNS FOR FIRST-ORDER INTERACTIONS

15

elements of the matrix in (A.3) are equal to 2(r − λ). The well-known equalities Kr = bd and λ(K − 1) = r(d − 1) for balanced incomplete block designs imply r − λ = bd(K − d)/(K(K − 1)). Thus, taking into account the factor 4mn in (A.2) it finally follows that the elements in positions K + 1, . . . , p of the diagonal of M(ξN,d ) are equal to the corresponding elements of M(ξ¯d ) which completes the proof.

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Louviere, J. J. and Woodworth, G. (1983). Design and analysis of simulated consumer choice or allocation experiments: An approach based on aggregate data. Journal of Marketing Research 20, 350-367. Street, D. J., Bunch, D. S. and Moore, B. S. (2001). Optimal designs for 2k paired comparison experiments. Communications in Statistics—Theory and Methods 30, 2149-2171. Street, D. J. and Burgess, L. (2004). Optimal and near-optimal pairs for the estimation of effects in 2-level choice experiments. Journal of Statistical Planning and Inference 118, 185-199. Train, K. E. (2003). Discrete Choice Methods with Simulation. Cambridge University Press, Cambridge. van Berkum, E. E. M. (1987). Optimal Paired Comparison Designs for Factorial Experiments, CWI Tract 31. Centrum voor Wiskunde en Informatica, Amsterdam.

School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom. E-mail: ([email protected]) Institut f¨ ur Mathematische Stochastik, Otto-von-Guericke-Universit¨at Magdeburg, PF 4120, D-39016 Magdeburg, Germany. E-mail: ([email protected]) School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom. E-mail: ([email protected])