An R package for analyzing and modeling ranking data
© Lee and Yu; licensee BioMed Central Ltd. 2013
Received: 25 September 2012
Accepted: 25 April 2013
Published: 14 May 2013
In medical informatics, psychology, market research and many other fields, researchers often need to analyze and model ranking data. However, there is no statistical software that provides tools for the comprehensive analysis of ranking data. Here, we present pmr, an R package for analyzing and modeling ranking data with a bundle of tools. The pmr package enables descriptive statistics (mean rank, pairwise frequencies, and marginal matrix), Analytic Hierarchy Process models (with Saaty’s and Koczkodaj’s inconsistencies), probability models (Luce model, distance-based model, and rank-ordered logit model), and the visualization of ranking data with multidimensional preference analysis.
Examples of the use of package pmr are given using a real ranking dataset from medical informatics, in which 566 Hong Kong physicians ranked the top five incentives (1: competitive pressures; 2: increased savings; 3: government regulation; 4: improved efficiency; 5: improved quality care; 6: patient demand; 7: financial incentives) to the computerization of clinical practice. The mean rank showed that item 4 is the most preferred item and item 3 is the least preferred item, and significance difference was found between physicians’ preferences with respect to their monthly income. A multidimensional preference analysis identified two dimensions that explain 42% of the total variance. The first can be interpreted as the overall preference of the seven items (labeled as “internal/external”), and the second dimension can be interpreted as their overall variance of (labeled as “push/pull factors”). Various statistical models were fitted, and the best were found to be weighted distance-based models with Spearman’s footrule distance.
In this paper, we presented the R package pmr, the first package for analyzing and modeling ranking data. The package provides insight to users through descriptive statistics of ranking data. Users can also visualize ranking data by applying a thought multidimensional preference analysis. Various probability models for ranking data are also included, allowing users to choose that which is most suitable to their specific situations.
KeywordsDistance-based model Luce model Multidimensional preference analysis Visualization Weighted distance
Ranking data arises when a number of items are to be ranked. By the nature of the ranking data, they can be divided into two types. The first type is characterized by a small number of items to be ranked, and they frequently represent the preference of these items among a group of judges (individuals). These items can be candidates in an election , one’s place of living , choice of occupations [3, 4], medical treatment , and so on. In analyzing these data, the focus is on the judges’ perception and preference of some specific (or all) items. In recent years, this type of ranking data have also becoming more popular in the medical literature, particularly in health economics [6–10] and medical informatics .
The second type of ranking data is characterized by a large number of items, and they frequently represent the ordering of these items in which researchers would like to determine or predict which items were ranked at the top positions. Examples of such ranking data include search engine results , integration of microRNA and mRNA , and consumer behavior in e-commerce applications . Due to the large number of items, these ranking datasets often contain missing or tie rankings, which are impossible to analyze without computers. With the decreasing cost of powerful computers, more researchers have paid attention to this type of ranking data, especially those in machine learning and knowledge discovery.
Analyzing and modeling ranking data is an efficient way to understand people’s perceptions and preferences for different items. Over the years, besides statistical tests for hypothesis testing , various models have been developed for ranking data, including the Luce model , distance-based model , ϕ-component model  and weighted distance-based model [18, 19].
The maximum likelihood estimator (MLE) of the aforementioned models does not have a closed form, yet the MLE can be obtained using iterative algorithms. However, at present, only summary statistics and a visualization of ranking data are available (partially and indirectly) in some statistical software (for example, procedure MDPREF in SAS), not to mention hypothesis testing and probability models for ranking data. The lack of software and tools for analyzing ranking data is not a problem for statisticians who are used to writing programs for their own means. However many scientists are not familiar with programming. We believe that a single package for the analysis of ranking data could offer users a more complete analysis, allowing them to use a single program instead of shifting their ranking datasets from one application to another.
In this section, we give a review of statistical analyses for ranking data. For a better description of ranking data, some notations must be defined. For a set of k items, labelled 1,…, k, a ranking π is a mapping function from 1,…, k to 1,…, k, where π(i) is the rank given to item i. For example, π(2) = 3 means that item 2 has a rank of 3. The inverse of the ranking function (sometimes referred to as ordering) π -1(i) is defined as the item that has rank i. For example, π -1(5) = 6 means that the item with rank 5 is item 6.
Descriptive statistics for ranking data
Descriptive statistics give an overall picture of the ranking dataset. Not only do descriptive statistics provide a summary of the ranking dataset, but they also lead us in an appropriate direction to analyze the dataset. Therefore, it is suggested that researchers consider descriptive statistics prior to any sophisticated analysis of ranking data.
where π i , i = 1, 2, …, k! represents all possible rankings of the k items, N i is the observed frequency of ranking i, and π i (j) is the rank given to item j in ranking i.
Note that M st is the frequency of item s being ranked t th . It is called a marginal matrix because “the i th row gives the observed marginal distribution of the ranks assigned to item i, and the j th column gives the marginal distribution of objects given the rank j.” (, page 18).
Inconsistency indices for pairwise comparisons
where λ max is the largest eigenvalue of A, and RI k is the average value of for a k×k random matrix. Another popular measure is Koczkodaj’s index, which equals for each triad (a, b, c) in A}.
Other consistency indices exist besides these two .
Visualizing ranking data: multidimensional preference analysis
Because ranking data often have a high dimension, visualization is a good first step towards their analysis. Multidimensional preference analysis  is a dimension reduction technique that aims to display ranking data in a low-dimensional (preferably 2D or 3D) space. It is applicable to ranking data with five or more items where the dataset cannot be displayed in a 2D/3D plot. Let X be an N×k matrix of ranking data such that x ij represents the rank of item j assigned by judge i, centered by the overall mean rank, i.e., (k + 1)/2. Suppose the singular value decomposition of X is X = UDV’. A 2D representation of the multidimensional preference analysis denotes the items and judges by the first two columns of and , respectively. Items are usually plotted as points, whereas judges are plotted as vectors from the origin. To give a better graphical display, the length of the ranking vectors can be scaled to fit the position of the items. It is not difficult to see that the perpendicular projection of all k item points onto a judge vector will closely approximate the ranking of the k items by that judge if the 2D solution fits the data well. Otherwise, we may look for a higher-dimension solution.
Statistical inferences for ranking data
Apart from exploring ranking data using descriptive statistics and graphs to identify the structure of the data, statistical inferences can be made to test the significance of a data structure. The two most commonly used inferences are the test for uniformity in a set of ranking data and the test for common rank-order preference for two sets of ranking data.
When we say that a ranking dataset is uniform, we mean that all possible rankings have the same probability of being observed. Hence, under uniformity, the expected frequencies of every ranking should be N/k!, and the standard χ 2 test can be applied to test the uniformity. However, when k! is too large compare with N, this is not always applicable, because we may encounter rankings with fewer than five observation. In such a case, mean rank, pairs, or marginals can be used to test the uniformity instead of ranking proportions . Note that under uniformity, the expected values of mean rank, pairs, and marginals are (k + 1)/2, 0.5 N, and N/k respectively.
and they follow a χ 2 distribution with k-1, , and (k-1)2 degrees of freedom, respectively.
The χ 2 test could be used to test for any difference between two ranking datasets. Before doing so, we align the matrix (in the case of pairs and marginals) into a q × 1 vector, for both datasets. We can now use the standard χ 2 test. For comparison between three or more ranking datasets, MANOVA-like tests can be used .
Statistical models for ranking data: the Luce model
After conducting a descriptive analysis for ranking data, we may have some understanding about the empirical distribution of the rank-order preferences of different items and their popularity. To further understand the data and make inferences about its structure, an efficient method is to establish some statistical models for ranking data. Over the years, various statistical models for ranking data have been developed. In this subsection, we review a commonly used approach, the Luce model.
and the resulting models is referred to as the Luce models . The Luce models can be interpreted as a vase model : imagine there are infinitely many balls inside a vase, and each ball is labeled j., j = 1, 2, …, k. The proportion of balls labeled with j is proportional to Vj. Then, the Luce models correspond to the ranking process whereby the first ball drawn is labeled π -1 (1), the second ball drawn is labeled π -1 (2) (with all balls labeled π -1 (1) removed from the vase), and the process continues until all balls in the vase have the same label.
The loglikelihood function is globally concave, and hence a global maximum exists. The MLE of the parameters can thus be obtained using standard methods, e.g., the Newton–Raphson algorithm. Besides MLE, Bayesian method can also be used for parameter estimation, using expectation propagation , generalized repeated insertion model , and random atomic measures .
Statistical models for ranking data: distance-based model
For ranking data, we require that the distance, apart from having these usual properties, must be right invariant, i.e., d(π, σ) = d(π○γ, σ○γ), where π○γ(i) = π(γ(i)). This requirement ensures that the relabeling of items has no effect on the distance.
where I() is the indicator function. There are other distances applicable to ranking data, and readers can refer to  for details.
where λ > 0 is the dispersion parameter, C(λ) is the proportionality constant, and d(π , π 0) is an arbitrary right invariant distance. When we use Kendall’s tau as the distance function, the model is called Mallows’ ϕ-model . In distance-based models, rankings nearer to the modal ranking π 0 have a higher probability of occurrence and this is controlled by λ. The distribution of rankings will be more concentrated around π 0 for a smaller value of λ.
A closed form for the proportionality constant C(λ) only exists for some distances. In principle, it can be solved numerically by summing over all possible π. The computational time increases exponentially with the number of items .
Statistical models for ranking data: ϕ-component model
where each d i (π , σ)is independent. Both Kendall’s tau and Cayley’s distance  can be decomposed in this form, and Fligner and Verducci  developed two new classes of ranking models for these, called ϕ-component models and cyclic structure models, respectively.
Here, V 1 represents the number of adjacent transpositions required to place the best item in π 0 in the first position. V 2 is the number of adjacent transpositions required to place the second best item in π 0 in the second position, and so on. Therefore, the ranking can be described as k-1 stages, V 1 to V k-1, where V i = m can be interpreted as m mistakes made in stage i.
These models were named k-1 parameter models by Fligner and Verducci , but were also named ϕ-component models in other papers . Mallows’ ϕ -models are special cases of ϕ-component models when λ 1 = … = λ k-1.
Statistical models for ranking data: weighted distance-based model
Lee and Yu [18, 19] proposed an extension of the distance-based model by replacing the (equal-weighted) distance with a new weighted distance measure, so that different weights can be assigned to different ranks.
It is important to note that this weighted distance satisfies all the usual distance properties, in particular the symmetry property, i.e., T w (π , σ) = T w (σ, π).
Generally speaking, if w i is large, few people will tend to disagree that the item ranked i in π 0 should not be ranked i. This is because such disagreement will greatly increase the distance and hence the probability of observing it will become very small. If w i is close to zero, people have little or no preference on how the item ranked i in π 0 is ranked, because a change in its rank will not affect the distance at all. The extension of weighted distance-based ranking models can retain the nature of distance, and at the same time maintain a greater flexibility. Readers are referred to  for the details of these properties.
Label ranking method using k-nearest neighbor algorithm
Label ranking is defined as the problem of classifying a judge’s ranking over a set of items given the covariate of this judge and a training dataset. ROL can be used for this, as it produces utility scores that can generate rankings for the judges. However, when the number of items and covariate are large, ROL may not be feasible due to its long computation time. Recently, a local k-nearest neighbor method has been developed for label ranking . If we want to predict the ranking of judge i, we can first select the k-nearest neighbor (by Euclidean distance) of i. Second, a statistical model (the Luce model in ) is fitted to these k neighbors and the parameters will be used to predict the ranking of judge i.
Results and discussion
In this section, we will use a seven-item ranking dataset q4, in which 566 Hong Kong physicians ranked the top five incentives (1: competitive pressures; 2: increased savings; 3: government regulation; 4: improved efficiency; 5: improved quality care; 6: patient demand; 7: financial incentives) to the computerization of clinical practice. Items not ranked were imputed using the mean rank. The dataset is not available in the pmr package but is available upon request. Note that most of the functions in pmr require the input ranking data to be organized in an aggregated format, that is, a summary matrix with rankings and their corresponding frequencies. To transform the individual ranking data to an aggregated format, the rankagg function can be used (q4agg < - rankagg(q4)).
From the descriptive statistics, we can deduce that item 4, improved efficiency, is the most preferred item, and item 3, government regulation, is the least preferred item.
The χ 2 test statistic equals 524.8747 and the corresponding p-value equals 1.82345 × 10-110. Thus, the ranking was not uniformly distributed.
The χ 2 test statistic equals 66.415 and the corresponding p-value equals 0.04. Thus, we have found a significant difference between physicians’ preferences with respect to their monthly income.
Multidimensional preference analysis  can help us understand more about the physicians’ ranking process and their preferences over the seven items by decomposing the rankings into a few dimensions. This can be performed using the mdpref function (R code: mdpref(q4agg,rank.vector = T)). The output is as follows:
The coordinates of the items and rankings, and the proportion of variance explained by the first two dimensions are stored in the values $item, $ranking and $explain respectively. The final two columns of the $ranking matrix are the coordinates of the first two columns of .
Figure 2 shows the multidimensional preference graph. The 2D plot explains around 42 % of the total variance. The first dimension can be interpreted as the overall preference of the seven items (labeled as “internal/external”). The leftmost item (item 4) and rightmost item (item 3) are the most and the least preferred items, respectively. The second dimension can be interpreted as the overall variance of the seven items (labeled as “push/pull factors”). The bottommost item (item 7) has the largest variance and the topmost item (item 2) has the second largest variance among the seven items.
Descriptive statistics and plots provide an insight to the data, but modeling will be more useful if we wish to have a deeper understanding. The Luce model (pl), distance-based model (dbm), ϕ-component model (phicom) and weighted distance-based model (wdbm) can be fitted using the pmr, which requires the stats4 package. We will demonstrate the model fitting procedure. Spearman’s footrule distance usually gives the best fit [18, 19] and hence it will be used in our demonstration of distance-based models.
The warning messages are a result of some of the predicted probabilities being close to zero. The parameter estimates of the distance-based model can be obtained using the R code q4.dbm < - dbm(q4agg); q4.dbm@coef, and the distance type can be specified using the argument dtype (default: Kendall’s tau; rho: Spearman’s rho; rho2: Spearman’s rho square; foot: Spearman’s footrule).
From the model parameters, we can conclude that item 4 is ranked 1 st , but the judges preference for this position is not particularly strong. Note that the modal ranking in the weighted distance-based model is different from that using the mean rank.
where is the Pearson residual, and O i , E i are the observed and expected frequencies of ranking i, respectively. The sum of square Pearson residual will automatically be given in the output, together with the corresponding degrees of freedom.
According to the results of the ROL model, female physicians preferred items 1 and 4, and private physicians did not prefer items 1, 2, and 7.
Assume that we want to predict the preference of a list of physicians with known covariates q4covtest. One possible method is to assign the utility ranks of the seven items for these physicians using the parameters obtained from the ROL model. Another method is to use the local k-nearest neighbor algorithm with the R code local.knn(q4,q4covtest,q4cov,knn.k = k). The value of k must be pre-specified. The pmr package provides the cross-validation version of the local k-nearest neighbor local.knn.cv(q4,q4covtest,q4cov). By default this uses 10-fold cross validation and tests the cross-validation prediction error of k (defined as the total Kendall’s distance) from 1 to 20.
In this paper, we presented the pmr R package, the first package for analyzing and modeling ranking data. The package provides insight to users through descriptive statistics of ranking data. Users can also visualize ranking data by applying a thought multidimensional preference analysis. Various probability models for ranking data are also included, allowing users to choose that which is most suitable to their specific situations. Besides the models introduced in this paper, there are other functions included in the pmr package that have not been presented here due to scope limitations, including the Analytic Hierarchy Process model (ahp) [26, 43], multidimensional preference analysis (mdpref), and rank plots (rankplot) . Details of these functions can be found at http://cran.r-project.org/web/packages/pmr/pmr.pdf. Future works on developing the package will include the incorporation of latent class models.
In the pmr package, we aimed at including traditional ranking models like the Luce model and distance-based model, and many recently-developed models for ranking data were not included (examples included decision tree models for ranking data [18, 45, 46] and multistage models [47, 48]). Nevertheless, since many of these models belong to extensions of traditional ranking models, we believe that the development of new ranking models can rely on the programming code provided by package pmr.
Availability and requirements
Project name: Probability Models for Ranking Data
Project home page: http://cran.r-project.org/web/packages/pmr/index.html
Operating system(s): Platform independent
Programming language: R
Other requirements: R 2.15.0 or above
Any restrictions to use by non-academics: none
The research of Philip L. H. Yu was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7473/05H).
- Diaconis P: Group representations in probability and statistics. 1988, Hayward: Institute of Methematical StatisticsGoogle Scholar
- Duncan OD, Brody C: Analyzing rankings of three items. Social structure and behavior. Edited by: Hauser RM, Mechanic D, Haller AO, Hauser TS. 1982, New York: Academic, 269-310.View ArticleGoogle Scholar
- Goldberg AI: The relevance of cosmopolitan/local orientations to professional values and behavior. Sociol Work Occup. 1975, 3: 331-356.View ArticleGoogle Scholar
- Yu PLH, Chan LKY: Bayesian analysis of wandering vector models for displaying ranking data. Stat Sin. 2001, 11: 445-461.Google Scholar
- Plumb AAO, Grieve FM, Khan SH: Survey of hospital clinicians’ preferences regarding the format of radiology reports. Clin Radiol. 2009, 64: 386-394. 10.1016/j.crad.2008.11.009.View ArticlePubMedGoogle Scholar
- Salomon JA: Reconsidering the use of rankings in the valuation of health states: a model for estimating cardinal values from ordinal data. Popul Health Metrics. 2003, 1: 1-12. 10.1186/1478-7954-1-1.View ArticleGoogle Scholar
- Krabbe PFM, Salomon JA, Murray CJL: Quantificaition of health states with rank-based nonmetric multidimensional scaling. Med Decis Making. 2007, 27: 395-405. 10.1177/0272989X07302131.View ArticlePubMedGoogle Scholar
- McCabe C, Brazier J, Gilks P, Tsuchiya A, Roberts J, O’Hagan A, Stevens K: Use rank data to estimate health state utility models. J Health Econ. 2006, 25: 418-431. 10.1016/j.jhealeco.2005.07.008.View ArticlePubMedGoogle Scholar
- Craig BM, Busschbach JJV, Salomon JA: Modeling ranking, time trade-off, and visual analog scale values for EQ-5d health states: a review and comparison of methods. Med Care. 2009, 47: 634-641. 10.1097/MLR.0b013e31819432ba.View ArticlePubMedPubMed CentralGoogle Scholar
- Ratcliffe J, Brazaier J, Tsuchiya A, Symonds T, Brown M: Using DCE and ranking data to estimate cardinal values for health states for deruving a preference-based single index from the sexual quality of life questionnaire. Health Econ. 2009, 18: 1261-1276. 10.1002/hec.1426.View ArticlePubMedGoogle Scholar
- Leung GM, Yu PLH, Wong IOL, Johnston JM, Tin KYK: Incentives and barriers that influence clinical computerization in Hong Kong: a population-based physician survey. J Am Med Inform Assoc. 2003, 10: 201-212. 10.1197/jamia.M1202.View ArticlePubMedPubMed CentralGoogle Scholar
- Park ST, Pennock DM: Applying collaborative filtering techniques to movie search for better ranking and browsing. Proc KDD 2007. 2007Google Scholar
- Lin S, Ding J: Integration of ranked lists via Cross Entropy Monte Carlo with applications to mRNA and microRNA studies. Biometrics. 2009, 65: 9-18. 10.1111/j.1541-0420.2008.01044.x.View ArticlePubMedGoogle Scholar
- Ganesan K, Zhai C: Opinion-based entity ranking. Inf Retr. 2012, 15: 116-150. 10.1007/s10791-011-9174-8.View ArticleGoogle Scholar
- Marden JI: Analyzing and modeling rank data. 1995, London: Chapman and HallGoogle Scholar
- Luce RD: Individual choice behavior. 1959, New York: John Wiley and SonsGoogle Scholar
- Fligner MA, Verducci JS: Distance based ranking models. J R Stat Soc B. 1986, 48 (3): 359-369.Google Scholar
- Lee PH, Yu PLH: Distance-based tree models for ranking data. Comput Stat Data Anal. 2010, 54 (6): 1672-1682. 10.1016/j.csda.2010.01.027.View ArticleGoogle Scholar
- Lee PH, Yu PLH: Mixtures of weighted distance-based models for ranking data with applications in political studies. Comput Stat Data Anal. 2012, 56 (8): 2486-2500. 10.1016/j.csda.2012.02.002.View ArticleGoogle Scholar
- R: a language and environment for statistical computating. [http://www.R-project.org]
- Holleczek B, Gondos A, Brenner H: Period R - an R package to calculate long-term cancer survival estimates using period analysis. Methods Inf Med. 2009, 48 (2): 123-128.PubMedGoogle Scholar
- Kreuz M, Rosolowski M, Berger H, Schwaenen C, Wessendorf S, Loeffler M, Hasenclever D: Development and implementation of an analysis tool for array-based comparative genomic hybridization. Methods Inf Med. 2007, 46 (5): 608-613.PubMedGoogle Scholar
- Murphy TB, Martin D: Mixtures of distance-based models for ranking data. Comput Stat Data Anal. 2003, 41: 645-655. 10.1016/S0167-9473(02)00165-2.View ArticleGoogle Scholar
- Critchlow DE, Fligner MA, Verducci JS: Probability models on rankings. J Math Psychol. 1991, 35: 294-318. 10.1016/0022-2496(91)90050-4.View ArticleGoogle Scholar
- Yu PLH: Statistical modelling of ranking data. Computational mathematics and modelling. edn. Edited by: Lenbury Y, Sanh NV, Wu YH, Wiwatanapataphee B. 2003, 319-326.Google Scholar
- Saaty TL: A scaling methods for priorities in hierarchical structure. J Math Psychol. 1977, 15: 234-281. 10.1016/0022-2496(77)90033-5.View ArticleGoogle Scholar
- Bozoki S, Rapcsak T: On Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrices. J Global Optim. 2008, 42 (2): 157-175. 10.1007/s10898-007-9236-z.View ArticleGoogle Scholar
- Carroll JD: Individual differences and multidimensional scaling. Multidimensional scaling: theory and applications in the behavioral sciences. Volume 1, edn. Edited by: Shepard RN, Romney AK, Nerlove SB. 1972, New York: Seminar PressGoogle Scholar
- Thurstone LL: A law of comparative judgement. Psychol Rev. 1927, 34: 273-286.View ArticleGoogle Scholar
- Guiver J, Snelson E: Bayesian inference for Plackett-Luce ranking models. Proc ICML 2009. 2009Google Scholar
- Lu T, Boutilier C: Learning mallows models with pairwise preferences. Proc ICML 2011. 2011Google Scholar
- Caron F, Teh YW: Bayesian nonparametric models for ranked data. Proc NIPS 2012. 2012Google Scholar
- Chapman RG, Staelin R: Exploiting rank ordered choice set data within the stochastic utility model. J Market Res. 1982, 19: 288-301. 10.2307/3151563.View ArticleGoogle Scholar
- Beggs S, Cardell S, Hausman JA: Assessing the potential demand for electric cars. J Econ. 1981, 16: 1-19.View ArticleGoogle Scholar
- Hausman JA, Ruud PA: Specifying and testing econometric models for ranked-ordered data. J Econ. 1987, 34 (1-2): 82-104.View ArticleGoogle Scholar
- Spearman C: The proof and measurement of association between two things. Am J Psychol. 1904, 15: 72-101. 10.2307/1412159.View ArticleGoogle Scholar
- Mallows CL: Non-null ranking models. I. Biometrika. 1957, 44: 114-130.View ArticleGoogle Scholar
- Cayley A: A note on the theory of permutations. Phil Mag. 1849, 34: 527-529.Google Scholar
- Shieh GS: A weighted Kendall’s tau statistic. Stat Prob Lett. 1998, 39: 17-24. 10.1016/S0167-7152(98)00006-6.View ArticleGoogle Scholar
- Shieh GS, Bai Z, Tsai WY: Rank tests for independence - with a weighted contamination alternative. Stat Sin. 2000, 10: 577-593.Google Scholar
- Tarsitano A: Comparing the effectiveness of rank correlation statistics. Working papers, universita della calabria, dipartimento di economia e statistica, 200906. 2009Google Scholar
- Cheng W, Dembczynski K, Hullermeier E: Label ranking methods based on the Plackett-Luce model. Proc ICML 2010. 2010Google Scholar
- Koczkodaj WW, Herman MW, Orlowski M: Using consistency-driven pairwise comaprisons in knowledge-based systems. Proc CIKM 1997. 1997Google Scholar
- Thompson GL: Graphical techniques for ranked data. Probability models and statistical analyses for ranking data. edn. Edited by: Fligner MA, Verducci JS. 1993, New York: Springer, 294-298.View ArticleGoogle Scholar
- Cheng W, Hullermeier E: A new instance-based label ranking approach using the Mallows model. Proc ISNN 2009. 2009Google Scholar
- Yu PLH, Wan WM, Lee PH: Decision tree modelling for ranking data. Preference learning. edn. Edited by: Furnkranz J, Hullermeier E. 2010, Berlin: Springer-Verlag, 83-106.View ArticleGoogle Scholar
- Fligner MA, Verducci JS: Multi-stage ranking models. J Am Stat Assoc. 1988, 83: 892-901. 10.1080/01621459.1988.10478679.View ArticleGoogle Scholar
- Xu L: A multistage ranking model. Psychometrika. 2000, 65 (2): 217-231. 10.1007/BF02294375.View ArticleGoogle Scholar
- The pre-publication history for this paper can be accessed here:http://www.biomedcentral.com/1471-2288/13/65/prepub
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