Conjoint Measurement

Transcription

1 Conjoint Measurement A major success of Representational Measurement, with clear qualitative (nonmetric) tests for interval (metric) representation in terms of an additive model Provides non-metric analogue to N-way ANOVA Kruskal calls it MONANOVA Realisable algorithm for best-fitting noninteractive solution Provides realistic way to assess importance of component characteristics in a complex stimulus.

2 Conjoint Measurement PROBLEM: What are the conditions under which a monotonic/ordinal re-scaling of a 2 (or N)- way Table will yield (2, N) interval scales which make them additive? A representation is sought that is additive in the component factors, f and g : m(i,j) = f (i) + g (j) for all (i,j) Subject to the rescaling function k being (weakly) monotonic with the data: k(m(i,j) ) k(m(i,j )) iff m(i,j) m(i,j )

3 Conjoint Measurement QUALITATIVE conditions for additive representation: CANCELLATION and SOLVABILITY: CANCELLABILITY: m(a,x) <= m(b,y) and m(b,z) <= m(c,x) implies m(a,z) <= m(c,y) A sort of transitivity of differences ; can be re-written as: (AY >= BX) & (BX>=CZ) IMPLIES (AY >= CZ) Where XY signifies the interval x-y This is readily testable empirically

4 Conjoint Measurement SOLVABILITY: For all a,b, in A and for all x,y in X, there exists a c in A and a z in X, such that: m(a,x) ~ and m(c,y) and m(a,x) ~ m(b,z) ~ is an indifference relation This axiom implies that functions f and g are unbounded If these two axioms are satisfied, then and ADDITIVE REPRESENTATION exists, and the scales f and g are unique up to a linear transformation (metric, interval level).

5 Conjoint Measurement A Good example: Occupation-City Judgments (Mean attractiveness ratings): Occup / City A B C D Lawyer/Doctor teacher Accountant Mean attractiveness ratings (1-9), Sidowski & Andersen 1967

6 Conjoint Measurement: Good example --Note: Clear evidence of Interaction: Difference between Lawyer-doctor and Accountant is approximately constant across cities BUT: rating for Teacher is not parallel, It approaches Lawyer-doctor in City A, and Accountant in City D non-additivity/interaction ANOVA indicates highly significant interaction (p< 0.001)

7 Conjoint Measurement: Good example o o BUT These data satisfy all the Cancellation axioms (Cancellation & Solvability) for Conjoint Additive Measurement, and Therefore an order-preserving [monotone] additive representation is possible

8 Conjoint Measurement: Good example One such ordinal re-scaling (produced by CONJOINT in NewMDSX) is: ===================================== A B C D Occup: Lawyer-Doctor Teacher Accountant (Effect values in italics) ===================================================

9 Conjoint Measurement: Good example Which shows (in this case) that: -- the interaction is an artefect of the assumption that the rating scale is interval-level! -- Or, Interaction can be removed by a monotonic rescaling From Sidowsi & Anderson 1971 (Judgments of city-occupation combinations). ref TUM pp169, 262 And See Krantz et al 1971, pp

10 Conjoint Measurement: Not-so-Good example A monotonic re-scaling, seeking low badness-of-fit may well exploit the WEAK MONOTONICITY criterion: If δ(i,j) < δ(k,l) then d (i,j) <= d(k,l) Allowing (model) tie-ing in the face of data inequality: When δ(i,j) < δ(k,l) then d (i,j) may equal d(k,l) Repeatedly use of this tieing of untied data can signify desperate attempts to overcome interaction i.e. backhanded admission of genuine interaction

11 Conjoint Measurement: Not-so-Good example Take the following example of the average number of children (dep var) in terms of the occupational achievement of the father: First / Current Prof. Semi-SkUnsk Professional Skilled Unskilled The axioms are far from being fulfilled, so the program seeks an ordinal transformation yielding best additive fit and succeeds only by tie-ing In this context, use STRONG monotonicity and SECONDARY approach to ties If δ(i,j) < δ(k,l) then d (i,j) < d(k,l)

12 Conjoint Measurement: Not-so-Good example Take the following example of the average number of children (dep var) in terms of the occupational achievement of the father: First / Current Prof. Semi-SkUnsk Professional Skilled Unskilled The axioms are far from being fulfilled, so the program seeks an ordinal transformation yielding best additive fit and succeeds only by tie-ing In this context, use STRONG monotonicity and SECONDARY approach to ties If δ(i,j) < δ(k,l) then d (i,j) < d(k,l)

13 Conjoint Measurement: Conclusion CM can be extended: to higher-way data tables To other compositions (subtractive, multiplicative) Though it is mostly used for the additive model Program implementations: Kruskal s MONANOVA Guttman-Lingoes CM series Roskam s UNICON NewMDSX CONJOINT implements all the above.

14 Conjoint Measurement: Conclusion CM is one of the success stories of Representational Measurement Stating the testable qualitative [ordinal] conditions that a N-way Table of values has to satisfy in order to guarantee Additivity (non-interaction) in the N (unidimensional) Effect scales, which are Metric (up to a linear transform). Even in the case of fallible data, a non-metric algorithm exists to get a least worst-fitting solution thereby illustrating Coombs dictum that more conservative measurement assumptions can nonetheless achieve a betterfounded, justifiable and higher-level solution than pseudoquantification can.