## Paired comparisons with ties: Bradley-Terry and Rasch

The Bradley-Terry model for paired comparisons was formulated as a descriptive model, but can be written as a Rasch measurement model (Rasch RMT 9:2, 424):

loge (Pn>m/Pn<m) = Bn - Bm

where Pn>m is the probability that n is preferred to m. Bn and Bm are the desirability of n and m. This model does not allow for ties. Davidson and Beaver (D&B, 1977) propose an extension to the Bradley-Terry model that allows for ties by means of a parameter v, so that

Pn=m = v * sqrt(Pn>m * Pn<m)

D&B's model, however, does not permit separation of the parameters, and so it does not provide the interpretative power or inferential stability of a Rasch measurement model. Nevertheless, Matthews and Morris (M&M, 1995) apply D&B's model to their paired-comparison of the pain-alleviating effects of 4 local anaesthetic creams: 2 with active ingredients and 2 look- alike placebos (see Table).

```Applied      Applied Second
First      B-active B-placebo A-active A-placebo
B-active      -     4,3,0     6,0,1*   8,0,0
B-placebo  0,4,3      -       4,2,2    7,0,1*
A-active   0,0,7    1,0,7       -      5,1,1
A-placebo  1*,0,7   0,0,7     2,3,2      -
```

Numbers of patients record in the order: preferring first, no preference, preferring second. * are the three very unexpected observations.

In M&M's experiment, one cream had to be applied first. This gives that cream an advantage, analogous to that of the home team at a sports event. A Rasch model for paired comparisons with ties and an order effect (for which cream is applied first) is:

loge(Pn>m/Pn=m) = Bn + F - Bm - T
loge(Pn=m/Pn<m) = Bn + F - Bm + T

where F parameterizes the advantage of being treated first, and T parameterizes the tie, "no preference", zone. The lack of subscripts indicates that F and T are regarded as constant across all comparisons.

The measurement system resulting from a Facets analysis is informative (see Figure). Active creams are always preferred to their placebo look-alikes. B-Active cream is most preferred. A- active is not preferred even as much as B-placebo. Since the order effect (difference between First and Second use) is less than the difference between any pair of creams, the differences between the creams has a general import.

```-----------------------------------
|   Measure   |  Creams   | Order |
-----------------------------------
|      2      + B-Active  +       |
| (preferred) |           |       |
|             |           |       |
|             |           |       |
|             |           |       |
|             |           |       |
|             |           |       |
|             |           |       |
|             |           |       |
|             | B-placebo |       |
|      1      +           +       |
|             |           |       |
|             |           |       |
|             |           |       |
|             |           |       |
|             |           |       |
|             |           |       |
|             |           |       |
|             | A-active  |       |
|(not prefer.)|           | First |
|      0      * A-placebo * Second|
-----------------------------------
```

There are three very unexpected preferences. The most unexpected response is in the last data row. It is by the patient who preferred A-placebo over B-active (see Table), indicating how powerful a force the mind can be in pain-control!

The fact that, in general, each active cream was preferred to its placebo version, together with the generally consistent pattern of responses, reassures the analyst about both the quality of the data collection and the reasonableness of the analysis.

Davidson RR, Beaver RJ (1977) On extending the Bradley-Terry model to incorporate within-pair order effects. Biometrics, 33, 693-702.

Matthews JNS, Morris KP (1995) An application of Bradley-Terry- type models to the measurement of pain. Applied Statistics, 44(2) 243-255.

Paired comparisons with ties: Bradley-Terry and Rasch. Linacre JM. … Rasch Measurement Transactions, 1995, 9:2 p.425

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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