group decision and negotiations,...

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May 19 2004 CORS/INFORM Joint Meeting, Banff, Canada

Estimation of parameters of multi-

attribute utility functions in the presence of

response error

Group Decision and Negotiations, 2004


Researchers– Jamshid Etezadi, Concordia University– Tak Mak, Concordia University– Gregory Kersten, Concordia & Ottawa U

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AgendaIntroduction and notationsMeasurement modelEstimation of weightsSimulationConclusionFuture plans


IntroductionAll procedures for assessment of preferences whether holistic or decomposition contain measurement error.The existing assessment methods for analysis of preferences ignore presence of error in assessment. The present DS or EN systems do not accommodate response error.

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ObjectivesModel response errorUsing the additive utility model provide a practical but mathematically rigorous estimation procedure to assess the relative importance of the attributes under consideration.Based on decomposition assessments, provide a guidelines to implement the proposed procedure in future EN systems.


Multiattribute Utility model (notations)

The linear additive utility model for n attributes may be written as:

Wj > 0 and 0 ≤ uj ( xj ) ≤ 1, Thus:

)()(1

jj

n

jj xuWxU ∑

=

=

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=∑=

n

jjW

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Assessment of weights in presence of error

Holistic procedure:Fit the model to a set of holistic judgments and estimate

weights by minimizing response error. See Etezadi & Ciampi (1983) for accommodating response error.

Decomposition procedure:There are a variety of methods. Since Σ wj = 1. The weights are dependent. Thus the

measurement error in assessing wj cannot be assumed to be independent.Instead of measuring individual weights assess the relative

importance of attributes to the most and the least desirable attribute using a method such as swing.To accommodate response error we require

repeated measures of weights.


Two ratio assessments based on decompositionIdentify the most and the least desirable attributesSuppose xn is the most desirable attribute and x1 the least desirable attribute.

Method 1. Assess the relative ratiosrj = wj/wn; j = 1, …, n-1.Method 2. Assess the relative ratiosrj’ = wj/w1; j = 2, …, n,

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A response error model for the assessed weights

Using the classical measurement model (Lord 1968), a measurement of an object may be modeled as:

yi = τ + ei (1)where τ is the true value of the object, yi is the observed measurement, and ei is the measurement error.Since we assess the ratios rj instead of the individual weights wj , the measurement error may be model multiplicative. That is, we assume the assessor either over or under estimate the true relative weight γj by a factor ej . Thus the model may be written as:

rj = γj ej (2)Where γj = Wj/Wn, ( j = 1, …, n-1) are the true ratios and ej measurement errors.


Measurement Model

Since rj, γj and ej are all positives without loss of generality, (2) may be written as:

ln (rj) = ln (γj) + fj ; j = 1, …, n-1, (3)

where fj = ln (ei), is assumed to be normally distributed, fj ~ N (0, σf

2) for all j.

This resembles the classical measurement model (1)

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Measurement Model (cont.)ln (rj) = ln (γj) + fj ; j = 1, …, n-1, (3)

Let ln (γj) = ln (Wj/Wn) = hj; j = 1, …, n-1. (4)

Then we may write (3) as:ln (rj) = hj + fj ; j = 1, …, n-1. (5)

Similarly to (3), we can write:ln (rj

’) = ln (γj’) + fj’ ; j = 2, …, n. (6)

where: rj’ = wj/w1 is the assessed relative importance

of attribute xj to x1, and γ’j = Wj/W1 the

corresponding true relative importance.


Measurement Model (cont.)γ’

j = Wj/W1We may write:γ’

j = (Wj/Wn) / (W1/Wn) = γj/γ1. (7)And then,ln (γ’

j) = ln (γj) - ln (γ1). (8)Thus form (6) we have :ln (r’

j) = hj - h1 + fj’ : j = 2, …, n. (9)From the above equations we obtainWn = 1/(exp (h1) + … + exp (hn-1) +1).

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Estimation procedureThe two measurement modelsln (rj) = hj + fj ; j = 1, …, n-1 andln (r’

j) = hj - h1 + fj’ : j = 2, …, nMay be written as:y = Dh + ε. (10)

Where y is a 2(n-1) column vector containing log of the assessed values of the two ratios rj and rj

’,y’ = [ln (r1), ln (r2), …, ln (rn-1), ln (r2

’), ln (r3’), …, ln (rn

’) ].And D a 2(n-1)×(n-1) model matrix with constants equal to 0, 1 or -1 as demonstrated below

h’ = [h1, h2, …, hn-1]. And ε is the vector of random error terms corresponding to the values of fj and dj.

ε’ = [f1, f2, …, fn-1, f2’, …, fn’].


Estimation procedure

⎡ 1 0 0 ... 0 0 ⎤ ⎢ 0 1 0 ... 0 0 ⎥ ⎢ . . . ... . . ⎥ ⎢ 0 0 0 ... 0 1 ⎥D = ⎢ -1 1 0 ... 0 0 ⎥ ⎢ -1 0 1 ... 0 0 ⎥ ⎢ . . . ... . . ⎥ ⎢ -1 0 0 ... 0 1 ⎥ ⎣ -1 0 0 ... 0 0 ⎦

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Generalization

The two measurement model (5) and (9) may be generalized to accommodate bias in measurementln (rj) = c1 + hj + fj ; j = 1, …, n-1, (11)

ln (r’j) = c2 + hj - h1 + dj; j = 2, …, n. (12)


Simulation Study

0.60

0.3389-1.0820-0.2065-0.87550.41670.25

0.2652-1.32730.4645-1.79180.16670.10

0.0775-2.5572-0.0723-2.48490.08330.05

rj= simulated ratios

γj+ fjfj= normal error

γj=Ln(Wj/Wn)

True ratiosWj/W1

True values Wj

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Simulation Study (cont.)

16.80842.82190.33702.48512.000.60

4.28831.4559-0.15361.6095.000.25

2.10630.74490.05180.6932.000.10

0.05

rj’= simulated ratios

γ’j+ fj’fj’=

normal error

γ’j=Ln

(Wj/W1)True ratiosWj/W1

True values Wj


SPSS output from analysis of the 1st data set

.011-5.724-.345191-1.094h3

.004-8.226-.496.191-1.572h2

.000-17.732-1.142.144-2.561h1

Sig.tStandardized

Coefficients

Std. Error

Unstandardized

Coefficients

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Estimates of weights for Data sets 1

0.61740.6000

0.20670.2500

0.12820.1000

0.04770.0500

Estimated weightsTrue weights W


Introduce biasData set 2 (c1 = -.7 and c2 = .7) Data set 3 (c1 = .7 and c2 = .5)

27.71233.848127.0708.63553.4734.2422.6830.1680.42.5340.1320.17.1560.0380.08

3rd data set 2nd data setTrue ratios

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SPSS output from analysis of the 2nd data set

.131-4.776-.344.311-1.483h3

.100-6.308-.455.311-1.959h2

.083-7.591-1.013.407-3.087h1

Sig.tStandardized Coefficients

Beta

Std. Error

Unstandardized

Coefficients


Estimates of weights for Data sets 2 & 3

0.70710.70740.6000

0.22720.14060.2500

0.14070.08740.1000

0.04630.03230.0500

3rd data set2st data setW

weightsEstimatedTrue weights

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Reanalysis of the 3rd data set SPSS output set c1=c2 =c

.075-3.431-.314.338-1.159h3

.040-4.846-.444.338-1.637h2

.006-12.954-1.040.187-2.428h1

.0863.190.230.732C

Sig.tStandardized Coefficients

Beta

Std. Error

UnstandardizedCoefficients


Estimates of weights for Data sets 1 & 3 reanalyzed

0.62630.61740.6000

0.19650.20670.2500

0.12190.12820.1000

0.05530.04770.0500

3rd data setc1=.7, c2=.5

1st data setNo bias

W

weightsEstimatedTrue weights

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Estimates of the overall utilities and confidence intervals

)()(1

jj

n

jj xuWxU ∑

=

=

')ˆ,...,ˆ(})(

{))ˆˆ

(ln( AhhACovU1U

WU1

UVar 1n12n

−−≈

−

))(),...,(( 1111 UuUuA nn −−= −−γγ

)ln(U1

UQ−

=


Estimates of utility

0.4015± .0037

0.48380.35840.40U2

0.8485± .0398

0.76620.89160.85U1

0.6263.2510.70590.42140.60w4

0.1965.5.750.18010.28780.25w3

0.1219.75.50.08850.22500.10w2

0.05531.250.02550.06570.05w1

Proposed Estimates

utilities A2

utilities A1

Based on second Assess

Based on first Assess.

True values

Parameters

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conclusionsThe proposed method is simple practical and requires only 2(n-1) judgmentsCapable of differentiating signal from noiseCan accommodate systematic over or under estimation of relative importance in assessment – biasGives confidence interval for the estimates and provides an opportunity to test a number of hypothesis on weights or utilities


Future ResearchDevelop a parallel method for assessment of utilitiesGeneralize the estimation procedure to cover multiplicative utility modelsImprove the distribution assumption for measurement errors

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Thank you!

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