3. The Rao-Kupper Model

The Rao-Kupper Model

声明：本文为本人毕业研究报告《The Exploration of Pairwise Comparison in Football Application》中的部分内容摘录与整理，仅用于学习与交流。

Introduction

Rao and Kupper [1] extended the Bradley-Terry model [2] for paired comparison experiments by incorporating a threshold parameter $\eta$, thereby introducing the possibility of handling ties (draws, unsure). This development resolves the inherent limitation of the Bradley-Terry model [2], which requires a clear preference between any two treatments.

In the Rao-Kupper model, a new parameter $\theta =\exp (\eta )$ is set, adjusting the model to maintain the form of a ratio while acknowledging indistinguishable outcomes in paired comparisons, thus enhancing the model’s applicability to real-world situations. Additionally, the range of preference strengths ${\pi }_i$ is specifically defined.

Model Formulation 3

Consider a set of stimuli $\{O_1,O_2,...,O_n\}$, where each stimulus $O_i$ is associated an inherent preference strength, ${\pi }_i$, where ${\pi }_i\ge 0$ for all $i$ and $\sum _{i=1}^n{\pi }_i=1$.

To address indistinguishable sensory differences, involve a threshold $\eta$. Then redefine $p_{ij}$ based on Bradley-Terry model’s equation (see Eq. 9), now accounting for the scenario where the difference between the stimuli perceived strengths is less than the threshold, and thus, a tie is declared:

$$p_{ij}=\frac{1}{4}{\int }_{-(S_i-S_j)+\eta }^{\mathrm{\infty }}{\text{sech}}^2\left(\frac{y}{2}\right)dy,$$

where $S_i={\log }_e({\pi }_i)$ and $S_j={\log }_e({\pi }_j)$ as previously defined, with $\eta$ representing the minimum perceptual difference required for a preference decision.

Introducing the probability of a tie $p_0$, which is based on the range of sensory differences that fall within the $\eta$ threshold:

$$p_0=\frac{1}{4}{\int }_{-(S_i-S_j)-\eta }^{-(S_i-S_j)+\eta }{\text{sech}}^2\left(\frac{y}{2}\right)dy,$$

By letting $\theta =\exp (\eta )$ and redefining ${\pi }_i=\exp (S_i)$, the preference probabilities can be succinctly expressed in a form that emphasises the threshold parameter’s role:

$$\begin{array}{rl}{p}_{ij}& =\frac{{\pi }_i}{{\pi }_i+\theta {\pi }_j},\\ p_{ji}& =\frac{{\pi }_j}{\theta {\pi }_i+{\pi }_j},\\ p_0& =\frac{({\theta }^2-1){\pi }_i{\pi }_j}{(\theta {\pi }_i+{\pi }_j)(\theta {\pi }_j+{\pi }_i)},\end{array}$$

where ${\pi }_i$ is the true treatment rating and $\theta$ is the threshold parameter.

$p_{ij}$ is the probability that $i$ is preferred over $j$, adjusted by $\theta$ to influence the comparison. Similarly, $p_{ji}$ shows the probability that $j$ is preferred over $i$. $p_0$ represents the probability of a tie.

Example 3

Let’s consider a scenario where we want to evaluate the probability of different outcomes in matches based on the perceived strength of the teams. Let’s say we are assessing the likelihood of Team $A$ winning over Team $B$, Team $B$ winning over Team $A$, or the match ending in a draw.

Let preference strengths to each team are ${\pi }_A=0.6$ and ${\pi }_B=0.4$, where ${\pi }_A+{\pi }_B=1$, and assume $\theta =\exp (\eta )=1.1$, implying a $10\mathrm{\%}$ difference is needed to decisively say one team is likely to beat another. The probabilities of each case:

• Team A Winning: $p_{AB}=\frac{{\pi }_A}{{\pi }_A+\theta {\pi }_B}\approx 57.69\mathrm{\%}$
• Team B Winning: $p_{BA}=\frac{{\pi }_B}{\theta {\pi }_A+{\pi }_B}\approx 37.74\mathrm{\%}$
• Draw: $p_0=\frac{({\theta }^2-1){\pi }_A{\pi }_B}{(\theta {\pi }_A+{\pi }_B)(\theta {\pi }_B+{\pi }_A)}\approx 4.57\mathrm{\%}$

Model Derivation 3

Based on the previous Model Derivation 2 (Method 1), we can easily obtain the above integrals by setting $\eta =\log (\theta )$.

Now, by adding the new parameter $\eta$ to the previous logistic CDF:

$$p_{ij}=F(\mathrm{\Delta }S+\eta )=\frac{1}{1+e^{-({\log }_e({\pi }_i)-{\log }_e({\pi }_j))+\eta }}$$

Substitute ${\log }_e({\pi }_i)-{\log }_e({\pi }_j)={\log }_e\left(\frac{{\pi }_i}{{\pi }_j}\right)$,

$$\begin{array}{rl}{p}_{ij}& =\frac{1}{1+e^{-{\log }_e\left(\frac{{\pi }_i}{{\pi }_j}\right)+\eta }}\\ & =\frac{1}{1+e^{\eta }\cdot e^{-{\log }_e\left(\frac{{\pi }_i}{{\pi }_j}\right)}}\\ & =\frac{1}{1+e^{\eta }\cdot \frac{{\pi }_j}{{\pi }_i}}\end{array}$$

Replace $e^{\eta }$ with $\theta$,

$$p_{ij}=\frac{1}{1+\theta \cdot \frac{{\pi }_j}{{\pi }_i}}=\frac{{\pi }_i}{{\pi }_i+\theta {\pi }_j}.$$

Similarly, for $p_{ji}$, the probability that $j$ is preferred over $i$:

Given that the total probability sum to 1

$$p_{ij}+p_{ji}+p_0=1,$$

then express $p_0$ the probability of a tie as:

$$\begin{array}{rl}{p}_0& =1-\frac{{\pi }_i}{{\pi }_i+\theta {\pi }_j}-\frac{{\pi }_j}{{\pi }_j+\theta {\pi }_i}\\ & =\frac{({\theta }^2-1){\pi }_i{\pi }_j}{(\theta {\pi }_i+{\pi }_j)(\theta {\pi }_j+{\pi }_i)}.\end{array}$$

Conclusion

To sum up, in real-world scenarios, judges or decision-makers might not discern a difference between two treatments when their comparative strengths are very close. $\eta$ models this threshold, allowing for a more nuanced representation of preferences, including the possibility of ties.

However, although the Rao-Kupper model [1] can maintain the representation of $p_{ij}$ or $p_{ji}$ using a simple proportional model, $p_0$ is still expressed in a complex model. Furthermore, the denominators of these three models are uniform. The Davidson model [3] addresses this issue by using a more succinct model to represent the case of a tie.

References

[1] P. V. Rao and L. L. Kupper. Ties in paired-comparison experiments: A generalization of the Bradley-Terry model. Journal of the American Statistical Association, 62(317):194–204, Mar 1967. Available
[2] Ralph Allan Bradley. Some statistical methods in taste testing and quality evaluation. Biometrics, 9(1):22–38, 1953.
[3] Roger R. Davidson. On extending the Bradley-Terry model to accommodate ties in paired comparison experiments. Journal of the American Statistical Association, 65(329):317–328, 1970.