On ranking climate factors affecting the living organisms based on paired comparison model, a Bayesian approach

The method of paired comparisons (PC) endeavors to rank treatments presented in pairs to panelists (or respondents, judges, jurists, etc.) and they have to select the better one based on sensory evaluations. Sometimes the situations may occur when the panelists cannot discriminate between the treatments and declare a tie. In this study, an effort is made to extend the Weibull PC model to accommodate ties. The extended Weibull PC model is analyzed using Bayesian paradigm. Four different loss functions are used under noninformative (Uniform and Jeffreys) priors. The posterior and marginal posterior distributions are derived. The posterior estimates, posterior risks, preference probabilities, posterior probabilities and predictive probabilities are evaluated to know the ranking of ecological factor. The goodness of the proposed model is assessed. The entire analysis is carried out using a real data set based on the preference for the ecological factors.


Introduction
The paired comparison (PC) is well developed and reliable technique to order/ rank two or more treatments. Under this technique, pairs of treatments are presented to one or more judges for ranking according to some specific criterion. The PC technique is also considered as an efficient technique to access the preferences when the quantitative measurement of the treatments are either unavailable or accurate assessment is impossible. This technique is gaining more attention in different fields of life. The most frequent application of the PC technique is in sensory evaluations, personal rating, marketing, medicine etc. Trained panelists are using this technique in the industry to access customer's preferences and designing products. Annis and Craig (2005) and Cattelan (2012) used PC technique to analyze sports related problems. Kafyat et al. (2020) developed Maxwell PC model and analyzed the preference data of brands of drinking water under Bayesian paradigm. Sung and Wu (2018) established a PC technique that can be used as an alternative to the likert scale.  analyzed the Weibull PC model using noninformative priors under Bayesian technique. A real data set of cellphone brands is considered for analyzing the costumer's preferences for the cellphone brands.

Background
The literature different reflects a variety of PC models. Thurstone (1927) provided fundamental framework for the PC technique assuming the responses of the panelists to follow the normal distribution and derived his PC model based on the differences in the choice behaviors. Bradley and Terry (1952) assumed that the judges responses follow the logistic distribution and suggested a model. Abbas and Aslam (2009) considered Cauchy distribution to develop PC model.  used the Weibull distribution to developed PC model. Rao and Kupper (1967) extended Bradley and Terry 1 3 (1952) model for accommodating data with no preferences i.e. tie in the data. They introduced a threshold parameter in such a way that if the difference in the magnitude of the responses is less than the threshold parameter, it will indicate that the judges are unable to differentiate between the two treatments and hence will declare a tie. Davidson (1970) also extended the Bradley and Terry (1952) model by introducing a tie parameter in the model. Aslam (2002) performed the Bayesian analysis of the two PC models with ties. van Barren (1978) suggested six extensions of the Bradley and Terry model and introduced a tie parameter and order effect of the presentation of treatments. For more details one may see Schauberger and Tutz (2017), Dittrich et al. (2012) and Wilgenburg van et al. (2017).

Motivation
Modeling the reliability in the lifetime phenomena in engineering has great importance. The Weibull random variables can be used in such cases, so in case of paired comparison perception the preference will be given to the object having more lifetime reliability i.e. take more time to its failure, as compare to the object having less failure lifetime. Sometimes a situation occurs when it becomes very difficult for the panelists to show their preferences for the treatments or they are incompetent to recognize the difference and declare a tie between the competing treatments. In our study, we introduce the PC model to accommodate ties declared by the panelists. Our study is a new addition in the field of Bayesian paired comparison, it would be very useful for the researchers and practitioners to analyze their problem with newly developed model. To the best of our knowledge this type of work has not been carried out in the literature so far. We perform the Bayesian analysis of the model using noninformative priors under different loss functions. For illustration, we used real data set collected from the students of the earth science department Quaid-i-Azam University, Islamabad, Pakistan, showing their preferences for the ecological factors under study.
As far as the comparison of the proposed model with the other models existing in the literature is concerned, some competitive models for paired comparisons are proposed therein. But for the climatic study, we have made use of our own proposed model to compare the climate factors under study.
In this study, Sect. 4 defines material and methods including the development of the PC model, Bayesian analyses of the model using prior distributions such as Uniform and Jeffreys priors and graphical presentation of the marginal densities of the model's worth parameters. Results and discussions are presented in Sects. 5. Section 6 concludes the entire study.

The proposed model
The details for the development of the Weibull PC model is elaborated in . According to the Weibull PC model, the probability of preferring the treatment T j over T k , denoted by j.jk , may be defined as: and similarly the probability of preferring the treatment T k over T j , denoted by k.jk , is defined as: Davidson (1970) used Luce (1959 choice axiom in Bradley and Terry model (1952) for accommodating ties. The Weibull PC model is extended by introducing a new tie parameter ( ) based on the Davidson (1970) criteria. We considered the ratio scale of the preference probabilities j.jk and j.jk as following: Thus the probability of no preference is denoted by o.jk , and is defined as proportional to the geometric mean of the two preference probabilities j.jk and k.jk , i.e.
where is the constant of proportionality and also known as tie parameter. As the sum of probability is equals one, so by using (3) and (4) we obtain the following probabilities; Here j , j = 1,2,…,m denotes the treatment parameters and is the tie parameter. The expressions (5), (6) and (7) represent the extended Weibull PC model with ties. Here.
j.jkr will take the value 1 if the treatment T j is preferred to T k otherwise 0 in rth repetition of a comparison.
k.jkr will take the value 1 when treatment T k is preferred to T j otherwise 0 in rth repetition of a comparison.
o.jkr stands for the treatment T j is tied to the treatment T k in rth repetition of a comparison.
jkr is the number of times the treatment T j is tied to the treatment T k . n jk is the total number of times the treatment T j is compared with the treatment T k and the T j and T k are tied. Such that: n jk = n j.jk + n k.jk + n o.jk

Prior distributions for the proposed model
For the Bayesian analysis of the proposed model: two noninformative Uniform and Jeffreys priors are assumed (Laplace 1812; Bayes 1763). The uniform prior allocates identical probability to each and every unit. Symbolically we can write it as: while the Jeffreys prior can be obtained as: where I( ) denotes the Fisher information matrix. For m = 2 we have Fisher's Information matrix as: As the Jeffreys prior have long and complicated algebraic expression which is not easy for applying for m = 5, so we drive the Jeffreys prior numerically for m = 5, designed in SAS package.

Bayesian analysis of the model
The likelihood function of the observed outcomes of the trial "a" and the parameters is: where S j = 3.n j.jk + 3 2 n o.jk . We impose a constraint on the parameters of the model as ∑ m j=1 j = 1 , which shows that the parameters are well defined. The joint posterior distribution for the parameters given data under the noninformative priors are defined as: Here W = 1 denotes the uniform prior, W = 2 stands for the Jeffreys prior and R is the normalizing constant and defined as: The marginal posterior distribution for the parameter of 1 using uniform prior is Similarly we can obtain the other marginal posterior distributions.
The data set given in Table 1 used for the analysis is about the preference of ecological factors {Topographic Factors (TO), Edaphic Factors (ED), Pyric Factors (PY), Limiting Factors (LI) and Biotic factors (BI)} which is collected from the students of earth science department of Quaid-i-Azam University, Islamabad, Pakistan.

Graphical representation of the marginal distributions
The graphs of the marginal posterior distributions for the worth parameters are given in Figs. 1 and 2 respectively. Figures 3 and 4 represent the marginal posterior distributions for the parameters using the Jeffreys priors.
Glancing through the above figures, we observe that all graphs are showing approximately symmetrical behavior under the both of the priors.

The preference probabilities
The probability that defines the chance of preferring the ecological factor T j over T k in a single comparison. We denote preference probability by p j.jk . The posterior estimates are used to calculate preference probabilities. Since the posterior estimates obtained under SELF have minimum posterior risks, so these estimates are used to find the preference probabilities for the worth parameters based on the noninformative priors and are given in Table 4. The value p 1.12 = 0.4104 for the   (TO, ED) indicates that TO has 41.04% probability of being preferred against ED and p 2.12 = 0.5070 indicating 50.70% preference in the favor of ED against TO and there are 8.26% probability that none of the ecological factor will be preferred. In the similar ways, we can interpret the remaining preference probabilities.
When we observe the ranking order, we found that there exists complete synchronization between the posterior estimates and the preference probabilities.

The predictive probabilities
The predictive probability describes the preference of ecological factor T j over T k in a single future comparison of the two factors (T j , T k ). The predictive probability of these two factors (T 1 , T 2 ) is denoted by P 12 and can be calculated as: where 1.12 is the model preference probability given in (5), similarly the predictive probabilities of P 21 and P 012 can be calculated as following; where 2.12 and 0.12 are defined in (6) and (7) respectively. The predictive probabilities under both noninformative priors are obtained and given in Table 5.
The predictive probability P 12 predicts the future preference in the contest of ecological factors in their single future comparisons. The predictive probability for preferring the ecological factor TO on ED is 0.4109, which indicates that there are 41.09% chances that the ecological factor TO will be preferred to ED in the single future comparison. The remaining predictive probabilities can be also interpreted on the same lines. From the results obtained, it is evident that the predictive probabilities do agree with the ranking order given by the posterior means under both of the priors. Similar results are observed on the basis of the preference probabilities.

Bayesian hypothesis testing
Bayesian hypothesis testing is a simple and straightforward procedure. The posterior probabilities are calculated, and decision between the hypotheses is directly made. For the comparison of the worth of any two ecological factors T j and Tk, we consider the following hypotheses: We represent posterior probability for H jk by p jk = P( j > k ) and for H kj , we use q kj = 1 − p jk , so the posterior probability for p 12 is H 12 which is defined as: The hypothesis with higher probability will be accepted. The posterior probabilities of the hypotheses H jk and H kj (j < k= 1, 2, … , 5) are computed for the noninformative priors and given in Table 6. The posterior probability p 12 = 0.0735 for the ecological factors pair (TO, ED) indicates that the probability for the ecological factor TO is very small, so we shall reject the null hypothesis and accept the alternative hypothesis indicating a higher preference for the factor ED.
From the results, we see that the hypotheses H 21 , H 13 , H 41 , H 51 H 23 , H 24, H 25, H 34 , H 35 and H 54 are accepted while all the remaining are rejected. On the same lines we can interpret the remaining probabilities. Our results also indicate that ecological factor ED is preferred the most and ecological factor PY is preferred least, same ranking is observed through the posterior estimates which shows complete coordination among the results.  (7), respectively. We define the following hypotheses. The χ 2 test is applied to test the goodness of fit of the model.

Plausibility of the model
The obtained value of the Chi Square statistic for the Extended Weibull PC model is 7.227, with p value 0.94946. So according to decision rule, we conclude that the model under study is plausible and fit for the PC data.

Conclusion
In this study, we extended the Weibull PC model to accommodate ties. The model is analyzed under the Bayesian paradigm using noninformative (Uniform and Jeffreys) priors. Four different loss functions i.e., SELF, QLF, DLF and PLF are used for the analysis. A real data set of the preference for the ecological factors is collected from the students of earth science department Quaid-i-Azam University, Islamabad, Pakistan. The Bayesian analysis of the model constitutes the estimation of BEs along with their PRs, the preference probabilities, the predictive probability, hypothesis testing and test for the plausibility of the model. Graphs of the model parameters showed approximately a symmetrical behavior around their BEs for both of the priors. Our results indicate that BEs obtained under SELF have minimum PRs, so on the basis of BEs the ecological factors may be ranked as: ED → TO → BI → LI → PY. Our model gives encouraging results and presents the ranking which is generally prevails in our society. It is also worth mentioning that results obtained from the both priors agree a lot. The estimate of tie parameter is small and approximately identical under both of the noninformative priors. The p-value found for the model also justifies the right direction for extending the Weibull model for paired comparisons.
Author contribution KU performed the data analysis and preparation of original draft. MA did supervision and methodology validation. NA worked on investigation, review and validation. SIS did the conceptual and graphical analysis. All the authors contributed in the interpretation discussion and refinement of the paper.