We consider:&lt;BR/&gt;

1. The problem of estimating the parameters of latent variable models such as the Bradley Terry or Thurstone Model by the method of maximum likelihood, given data from a paired comparisons experiment. The parameters of these models can be taken to be weights which are positive and sum to one;&lt;BR/&gt;

2. The problem of determining approximate locally optimal designs for good estimation of these parameters; i.e of determining optimal design weights which are also positive and sum to one

A.C. Y. Kuk

B. Torsney

J. Kiefer

L. L. Thurstone

P. Whittle

R. A. Bradley

S. Mandal

English

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Estimation and Optimal Designing under Latent Variable Models for Paired Comparisons Studies via a Multiplicative Algorithm

We consider:

1. The problem of estimating the parameters of latent variable models such as the Bradley Terry or Thurstone Model by the method of maximum likelihood, given data from a paired comparisons experiment. The parameters of these models can be taken to be weights which are positive and sum to one;

2. The problem of determining approximate locally optimal designs for good estimation of these parameters; i.e of determining optimal design weights which are also positive and sum to one

Torsney, B.

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Estimation and optimal designing under latent variable models for paired comparisons studies via a multiplicative algorithm

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       Torsney, B. (2010) Estimation and optimal designing under latent variable models for paired comparisons studies via a multiplicative algorithm. Contributions to Statistics, 2010 . pp. 213-220. ISSN 1431-1968  http://eprints.gla.ac.uk/47918  Deposited on: 21 December 2010   Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk Estimation and optimal designing under latentvariable models for paired comparisons studiesvia a multiplicative algorithmBernard TorsneyAbstract We consider1. The problem of estimating the parameters of latent variable models such as theBradley Terry or Thurstone Model by the method of maximum likelihood, givendata from a paired comparisons experiment. The parameters of these models canbe taken to be weights which are positive and sum to one.2. The problem of determining approximate locally optimal designs for good esti-mation of these parameters; i.e of determining optimal design weights which arealso positive and sum to one.1 Paired Comparisons1.1 IntroductionWe have two alternative examples of a general problem, namely determining weightsoptimally. Much theory for this problem, e.g. optimality conditions and numericaltechniques have been developed in the optimal design arena. So this can be trans-ported to the estimation problem. We can extend techniques to this case. In section 1we introduce the notion of paired comparisons studies and latent variable models. Insection 2 the parameter estimation problem is outlined with optimality results and ageneral class of multiplicative algorithms outlined in sections 3 and 4 respectively.A specific algorithm is applied to the Bradley Terry log-likelihood in section 5 andlocally optimal designing is considered in section 6.We consider paired comparison experiments in which J treatments or productsare compared in pairs. In a simple form a subject is presented with two treatmentsBernard TorsneyDepartment of Statistics, University of Glasgow, Glasgow G12 8QW,UK e-mail:ben@stats.gla.ac.uk12 Bernard Torsneyand asked to indicate which he/she prefers or considers better. In reality the subjectwill be an expert tester; for example, a food taster in examples arising in food tech-nology. The link with optimal design theory (apart from the fact that a specialiseddesign, paired comparisons, is under consideration) is that, the parameters of latentvariable models for the resultant data are like weights. Hence the theory characteris-ing and the methods developed for finding optimal design weights can be applied tocharacterising and finding the maximum likelihood estimators of these latent vari-able ’weights’.1.2 The DataIn a simple experiment a set of such testers is available and each is presented withone pair from a set of J treatments, say T1,T2, . . . ,TJ . The number of comparisons, ni jof Ti to Tj, we assume has been predetermined. Sufficient summary data comprisesthe set {Oi j : i = 1, . . . ,J; j = 1, . . . ,J; i < j or i > j}, where Oi j is the observedfrequency with which Ti is preferred to Tj. Of course Oi j +O ji = ni jBradley and El-Helbawy (1976) introduce an example involving 8 coffee types.26 pairwise comparisons were made on each pair, i.e. ni j = 26.So Oi j +O ji = 26 and N = ∑i ∑ j Oi j = 728.The coffees are the eight combinations arising from a 23 factorial structure, thefactors being Brew Strength, Roast Colour, Coffee Brand. We are not exploiting thisstructure and leave them arbitrarily labelled.1.3 Models1.3.1 A General ModelIn the absence of other information the most general model here is to proposeOi j ∼ Bi(ni j,θi j)where, θi j = P(Ti is preferred to Tj).Apart from the constraint Oi j +O ji = ni j, independence between frequencies isan expected assumption. So, apart from the constraint θi j + θ ji = 1, these defineunrelated binomial parameters. The maximum likelihood estimator of θi j is Oi j/ni j(the proportion of times Ti is preferred to Tj in these ni j comparisons), and formalinferences can be based on the asymptotic properties of these.Estimation and optimal designing 31.3.2 Latent Variable ModelsThese are more restricted models in that they impose interrelations between the θi j.Assuming that F(·) is a symmetric distribution function, thenθi j = F(λi−λ j) = F{loge(pi/p j)}where pi = exp(λi). The symmetry of F(·) ensures thatθi j +θ ji = F(λi−λ j)+F(λ j−λi) = 1.The pi or λi can be viewed as indices or quality characteristics, one for each treat-ment. The implication of the model is that the difference in quality between twotreatments has distribution function F(·).Two primary examples of this model are the Bradley Terry and Thurstone mod-els. Respectively these take F(·) to be the Logistic and the Normal distributions.In the Logistic case θi j has the simplistic form: θi j = pi/(pi + p j); see Thurstone(1927), Bradley and Terry (1952), also Kuk (1995).2 Parameter EstimationThe likelihood of the data isL =∏r<∏s[F{loge(pr/ps)}]Ors[F{loge(ps/pr)}]Osr .We focus on the parameters pi and denote the likelihood by L(p).However we cannot estimate these as free parameters. This arises from the fact thatwe only have observations on comparisons between treatments, and is reflected inthe property that θi j is invariant to proportional changes in pi and p j. In consequencethe pi are only unique up to a constant multiple; (likewise the λi up to a constantshift). In keeping with this they are positive as the relationship pi = exp(λi) impliespi > 0. Mathematically speaking θi j and hence L(p) is a homogeneous function ofdegree zero in the pi i.e. L(cp) = L(p), where c is a scalar constant. So L(p) isconstant on rays running out from the origin. It will therefore be maximised alongone specific ray. We can identify this ray by finding a particular optimising p∗.This we can do by imposing a constraint on p. Possible constraints are ∑i pi = 1 or∏i pi = 1, or g(p) = 1 where g(p) is a surface which cuts each ray exactly once. Inthe case J = 2 a suitable g(p) is defined by p2 = h(p1), where h(·) is a decreasingfunction which cuts the two main axes, as in the case of h(p1) = 1− p1 , or hasthese as asymptotes, as in the case of h(p1) = 1/p1. In general a suitable choice ofg(p) is one which is positive and homogeneous of some degree h. Note that otheralternatives are ∑i pi =C or ∏i pi =C, where C is any positive constant; e.g. C = Jor C = 100.4 Bernard TorsneyThe choice of ∏i pi = 1, being equivalent to ∑i ln(pi) = 0, confers on λi = ln(pi)the notion of a main effect. However we will opt for the choice of ∑i pi = 1, whichconveys the notion of pi as a weight. We wish to maximise the likelihood or log-likelihood subject to this constraint and to non-negativity too. This is an example ofthe following general problem:Problem (P)Maximise φ(p) subjec to pi ≥ 0, ∑i pi = 1.We wish to maximise φ(p) with respect to a probability distribution.For the estimation problem we will take φ(p) = ln{L(p)}.There are many examples of this problem arising in various areas of statistics, es-pecially in the area of optimal regression design. We can exploit optimality resultsand algorithms developed in this area. The feasible region is an open but boundedset. Thus there should always be a solution to this problem allowing for the possi-bility of an unbounded maximum, multiple solutions and solutions at vertices (i.e.pt = 1, pi = 0, i 6= t).3 Optimality ConditionsWe assume that φ(·) is differentiable. LetFj = d j− pT d = d j−∑ipidi, where d j = ∂φ/∂ p j.We call Fj the jth vertex directional derivative of φ(·) at p.Note that ∑ j p jFj = 0, so that, in general, some Fj are negative and some are posi-tive.Given φ(·) is differentiable at p∗, then a necessary condition for φ(p∗) to be alocal maximum of φ(·) in the feasible region of Problem (P) isF∗j = 0 for p∗j > 0,F∗j ≥ 0 for p∗j = 0.If φ(·) is concave on its feasible region, then these first order stationarity conditionsare both necessary and sufficient. This is the general equivalence theorem in optimaldesign. See Whittle (1973), Kiefer (1974). In fact the second condition is redundantfor this estimation problem, while, given homogeneity of degree zero of L(p), thefirst reduces to standard first order conditions: d∗j = 0.Estimation and optimal designing 54 Algorithms4.1 Multiplicative AlgorithmProblem (P) has a distinct set of constraints, namely the variables p1, p2, . . . , pJmust be nonnegative and sum to 1. Let f (d,δ ) be a function satisfying (for δ > 0):• f (d,δ )> 0,• ∂ f (d,δ )∂d > 0 (for δ > 0),• f (d,0) = constant(e.g. f (d,δ ) = Φ(δd) or f (d,δ ) = dδ (if d > 0.))An iteration which neatly submits to these and has some suitable properties is themultiplicative algorithm:p(r+1)j =p(r)j f (d(r)j )∑i p(r)i f (d(r)i )where d(r)j =∂φ∂ p j∣∣∣∣p= p(r) , while f (d) is positive and strictly increasing in d andmay depend on one or more free parameters.4.2 Properties of the AlgorithmUnder the conditions imposed on f (·, ·), the above iterations possess the follow-ing properties which are considered in more detail in Torsney (1988), Torsney andAlahmadi (1992) and Mandal and Torsney (2000):1. p(r) is always feasible.2. Fφ{p(r), p(r+1)} ≥ 0, with equality when the d j’s corresponding to nonzero p j’shave a common value d(= ∑i pidi ), in which case p(r) = p(r+1).So an iterate p(r) is a fixed point of the iteration if derivatives d(r)j correspondingto nonzero p(r)j are equal; i.e. if corresponding vertex directional derivatives F(r)jare zero.3. If δ = 0 there is no change in p(r), given f (d,δ ) = constant4. So the algorithm should be monotonic for small positive δ .5 Fitting Bradley Terry ModelsOur criterion isφ(p) = ln{L(p)}.6 Bernard TorsneySince L(p) is a homogeneous function of degree zero ∑i pidi = 0. In fact d j = Fj. Sothere are always positive and negative d j unless all are zero. We require a functionf (d,δ ) which is defined for positive and negative d, where we take d to representa partial derivative. Noting that all p∗j must be positive a suitable choice should begoverned by the fact that at the optimum d∗j = 0, j = 1,2, . . . ,J.We opt for f (d,δ ) = Φ(δd), so that iterations prove to bep(r+1)j =p(r)j Φ(δd(r)j )∑i p(r)i Φ(δd(r)i )Coffee Example.In this case J = 8 coffee types were compared yielding a total of N = 728 obser-vations; i.e. ∑∑Oi j = 728. A suitable δ is δ = 1/N. In effect we are standardisingthe sample size to 1, through replacing observed by relative frequencies in the log-likelihood, and then taking δ = 1.Torsney (2004) reported the following results. Starting from p(0)j = 1/J, the numbersof iterations needed to achieve max |d j| = max |Fj| ≤ 10−n, for n = 0,1, . . . ,7 re-spectively are 17, 21, 25, 32, 38, 45, 51, 59. The optimal p∗ is (0.190257, 0.122731,0.155456, 0.106993, 0.091339, 0.149406, 0.080953, 0.102865).Iterations were monotonic.6 Local Optimal DesigningWe have not introduced any design variables. However we can pose the question:how many comparisons ni j there should be between Ti and Tj? This of course is anexact design problem. The easier approximate design problem poses the question:what proportion λi j of such comparisons there should be?This depends on our model. We focus on the Bradley Terry Model. The param-eters are now p1, p2, . . . , pJ . We wish good estimation of these. The informationmatrix isM(λ ) =∑∑i< jλi jwi jvi jvTi jwhere vi j = (ei− e j),ei being the ith unit vector wi j = 1/(pi+ p j)2.We note the following properties:1. M(λ ) has the form of the information matrix of a weighted linear model withweights wi j. This happens with a wide range of generalised linear models.2. M(λ ) depends on the pi’s (but only through the wi j’s).We need provisional values for them. A conventional choice is p j = 1/J.However we have maximum likelihood estimates This does not seem to havebeen considered in the literature before.3. M(λ ) is singular. This is another manifestation of the fact that we only have ob-servations on comparisons between treatments. We can only estimate differencesEstimation and optimal designing 7between treatments. This has implications for choice of design criteria. We mustrestrict consideration to good estimation of such differences (or other contrasts).This issue too appears to have been ignored in the literature.Two feasible classes are:DL− criteria : Ψ(M) =− logdet(LM+LT ),AL− criteria : Ψ(M) =−trace(LM+LT ).Here M+ denotes the Moore-Penrose inverse of M and L defines a set of (k− 1)linearly independent differences between the pi parameters.The DL-criterion would be invariant to any such choice of L.In general a locally optimal design problem is, for given p, to choose λ optimallysubject to λi j ≥ 0,∑∑i< j λi j = 1, i.e. solve Problem (P) for φ(λ ) =Ψ{M(λ )} forsome Ψ{·}.We need derivatives with respect to λi j, which we denote by di j, for optimalitychecking and numerical purposes. We have:for the DL-criterion, di j = wi jvTi jM+LT (LM+LT )−1LM+vi jfor the AL-criterion, di j = wi jvTi jM+LT LM+vi j.Of note is that these are positive, as is the case with all standard design criteria.For the multiplicative algorithm a feasible choice is f (d,δ ) = dδ , the original formof this function when the algorithm was first conceived for determining optimaldesigns. The choices of δ we opt for here correspond to choices which have beenshown to be monotonic for the standard D-criterion and A-criterion, namely δ =1,1/2 respectively.Coffee exampleWe choose to determine locally optimal designs at the current maximum likelihoodestimates; i.e. at p∗ =(0.190257, 0.122731, 0.155456, 0.106993, 0.091339, 0.149406, 0.080953, 0.102865).We use the following choices of f (d,δ ): for the DL-criterion: f (d,δ ) = d; for theAL-criterion: f (d,δ ) = d1/2.Iterations begin at λ (0)i j = 1/(J(J−1)).We take L to be the matrix defining the 7 differences p1− p j, j = 2,3, . . . ,8.We sumarise the implications if a further experiment is to be run and parametervalues are in the region of the maximum likelihood estimates: for DL-optimality nocomparisons would be made between coffee types 1 and 3 and between coffee types1 and 6; under both designs maximum weight is put on the comparisons betweencoffee types 1 and 7, which have the largest and smallest estimated Bradley Terryparameters; the AL-optimal weights of the 7 comparisons with the first coffee type8 Bernard Torsneyexceed 0.07 while the remainder are less than 0.03, which is in keeping with thefocus of the choice of L on differences with this coffee type.For comparison we note that uniform weights of 1/28 = 0.0357143.7 DiscussionThere are several extensions of this work in respect of both parameter estimation andlocal optimal designing (arguably new): for rankings; for “no preference” options;for factorially structured treatments.ReferencesBradley, R. A. and Terry, M. E. (1952). The rank analysis of incomplete blockdesigns I, The method of paired comparisons. Biometrika 39, 324-345.Bradley, R. A. and El-Helbawy, A. T. (1976). Treatment contrasts in paired com-parisons: basic procedures with applications to factorials. Biometrika 63, 255-262.Kiefer, J. (1974) General equivalence theory for optimum designs (approximatetheory) Annals of Statistics 2, 849-879.Kuk, A.C. Y. (1995). Modelling paired comparison data with large numbers ofdraws and large variability of draw percentages among players. Statistician 44,523-528.Mandal, S. and Torsney, B. (2000) Algorithms for the construction of optimisingdistributions. Communications in Statistics (Theory and Methods) 29, 1219-1231.Thurstone, L. L. (1927b). The method of paired comparisons for social values.Journal of Abnormal and Social Psychology 21, 384-400.Torsney, B. (1988) Computing Optimizing Distributions with Applications in De-sign, Estimation and Image Processing Optimal Design and Analysis of Experi-ments, Editors Y. Dodge, V.V. Fedorov, H.P. Wynn. North Holland., 361-370.Torsney, B. (2004) Fitting Bradley Terry models using a multiplicative algorithm.Proceedings in Computational Statistics (COMPSTAT2004, August 2004, Prague,Czeck Republic), Editor - Jaromir Antoch, Physica Verlag, 214-226.Torsney, B. and Alahmadi, A. M. (1992) Further developments of algorithms forconstructing optimizing distributions. Model Oriented data Analysis (V. Fedorov,W.G. Muller, I.N. Vuchkov Eds). Proceedings of 2nd IIASA- Workshop, St. Kyrik,Bulgaria, 1990. Physica Verlag , 121-129.Whittle, P. (1973). Some general points in the theory of optimal experimentaldesign J. Roy. Statist, Soc. B 35, 123-130.

Algorithms for the construction of optimising distributions.

Computing Optimizing Distributions with Applications in Design, Estimation and Image Processing Optimal Design and Analysis of Experiments, Editors

Fitting Bradley Terry models using a multiplicative algorithm.

Further developments of algorithms for constructing optimizing distributions. Model Oriented data Analysis

General equivalence theory for optimum designs (approximate theory)

Jaromir Antoch,

Modelling paired comparison data with large numbers of draws and large variability of draw percentages among players.

Some general points in the theory of optimal experimental

The method of paired comparisons for social values.

The rank analysis of incomplete block designs I, The method of paired comparisons.

Treatment contrasts in paired comparisons: basic procedures with applications to factorials.

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Estimation and optimal designing under latent variable models for paired comparisons studies via a multiplicative algorithm

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