Paired comparisons are simple in Facets. With paired-comparison data, the measurement model is:
measure of object A - measure of object B = log-odds(choice of A over B)
The person doing the measuring is not part of the measurement model. We want to include that person so that we can identify who has unusual characteristics, or where there is miscoding in the data. Since the person measure is irrelevant, it can be anything. Anchoring all persons at 0, as a dummy facet, is convenient. If any persons have displacements, then there is an error in the data.
Example 1: Paired comparison of objects by persons. In each pairing, one object wins.
Facet 1 is the objects to be compared. Each object is an element in the facet
Facet 2 is the persons doing the comparing. Each person is an element in the facet. This facet is a "dummy" facet. It is not used for measurement. It is used for fit analysis and interactions only.
So, here is what the Facets specification and data file look like:
Facets= 3 ; each observation has 3 elements in the data, 2 objects + 1 person
Entered= 1, 1, 2 ; the first two elements are for facet 1, the third element is for facet 2
Models= ?, -?,?,D ; the observation is "element of facet 1 - element of facet 1 + element of facet 2" produces a dichotomous 0/1 observation
Labels=
1, Objects ; the object facet
1=A
2=B
3=C
....
*
2, Persons, D ; the person facet: this is a Dummy facet. It is ignored for estimation
4=Mary
5=George
.....
*
Data=
1,2,4,1 ; Object A is compared with Object B by Mary. Object A wins
2,3,5,0 ; Object B is compared with Object C by George. Object B loses
or, better because it is more stable computationally,
Models= ?, -?,?,D, 0.5 ; weight each observation by 0.5
Data= ; every observation in data file twice
1,2,4,1 ; Object A is compared with Object B by Mary. Object A wins
2,1,4,1 ; ObjectB is compared with Object A by Mary. Object B Loses
2,3,5,0 ; Object B is compared with Object C by George. Object B loses
3,2,5,1 ; Object C is compared with Object B by George. Object C wins
Example 2: Paired comparison of objects by persons. In each pairing, one object wins and one object loses, or they are tied, draw, equal.
Score: 2=Win 1=Tie 0=Loss.
Facet 1 is the objects to be compared. Each object is an element in the facet
Facet 2 is the persons doing the comparing. Each person is an element in the facet. This facet is a "dummy" facet. It is not used for measurement. It is used for fit analysis and interactions only.
So, here is what the Facets specification and data file look like:
Facets= 3 ; each observation has 3 elements in the data, 2 objects + 1 person
Entered= 1, 1, 2 ; the first two elements are for facet 1, the third element is for facet 2
Models= ?, -?,?,R2, 0.5 ; the observation is "element of facet 1 - element of facet 1 + element of facet 2" produces a polymous 0/1/2 observation that is weighted 0.5 because each observation is twice in the data file.
Labels=
1, Objects ; the object facet
1=A
2=B
3=C
....
*
2, Persons, D ; the person facet: this is a Dummy facet. It is ignored for estimation
4=Mary
5=George
.....
*
Data=
; each observation twice (mirrored):
1,2,4,2 ; Object A is compared with Object B by Mary. Object A wins
2,1,4,0 ; Object B is compared with Object A by Mary. Object B loses
2,3,5,0 ; Object B is compared with Object C by George. Object B loses
3,2,5,2 ; Object C is compared with Object B by George. Object C wins
1,2,5,1 ; Object A is compared with Object B by George. Object A ties
2,1,5,1 ; Object B is compared with Object A by Mary. Object A ties
Example 3. Baseball.
Example 4. Flavor Strength of Gels
Bayesian imputation for unrealistically huge logit ranges or inestimable elements
A frequently-encountered problem in the analysis of paired-comparison data is an almost Guttman ordering of the pairings. This can lead to unrealistically huge logit ranges for the estimates of the elements or inestimable elements.
To solve this problem, we apply a little Bayesian logic. We know that the range of paired performances is not exceedingly wide, and we can also easily imagine a performance better than any of those being paired, and also a performance worse than any of the those being paired. Let's hypothesize that a reasonable logit distance between those two hypothetical performances is, say, 20 logits.
https://www.rasch.org/rmt/rmt151w.htm is a parallel situation for sports teams.
1.Hypothesize a "best"performance against which every other performance is worse. Anchor it at 10 logits.
2.Hypothesize a "worst" performance against which every other performance is better. Anchor it at -10 logits
3.Hypothesize a dummy judge who compares the best and worst performances against all the other performances.
4.Include these dummy observations in the analysis.
5.Analyze the actual observations + the dummy observations. The analysis should make sense, and the logit range of the performances will be about 20 logits. For reporting, we don't want the dummy material, so we write an Anchorfile= from this analysis.
6.We then use the Anchorfile as the Facets specification file, commenting out the "best" and "worst" performance elements and the dummy judge. We analyze only the actual observations. All the elements are anchored at their estimates from the actual+dummy analysis. In this anchored analysis, the "displacements" indicate the impact of the dummy data on the estimates.
7.If you perceive that the logit range of 20 logits is too big or too small, please adjust the "best" and "worst" anchored values.
Paired-Comparison Bias/Interaction Analysis:
Facets produces meaningful numbers in the Bias/interaction analysis when:
1) Use mirrored data, but set the weight = 1.0, instead of 0.5. For the main analysis, use weight 0.5.
2) Arrange the data so that the Models= is ..., -?,?,... instead of ...,?,-?,...
