Abstract
Background
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, distancebased model, and rankordered logit model), and the visualization of ranking data with multidimensional preference analysis.
Results
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 distancebased models with Spearman’s footrule distance.
Conclusions
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.
Keywords:
Distancebased model; Luce model; Multidimensional preference analysis; Visualization; Weighted distanceBackground
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 [1], one’s place of living [2], choice of occupations [3,4], medical treatment [5], 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 [610] and medical informatics [11].
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 [12], integration of microRNA and mRNA [13], and consumer behavior in ecommerce applications [14]. 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 [15], various models have been developed for ranking data, including the Luce model [16], distancebased model [1], ϕcomponent model [17] and weighted distancebased 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.
R [20], an opensource program for statistical analysis, is gaining in popularity because of its high flexibility. Indeed, users are free to write/use packages for specific purposes. Although many packages are highly relevant to medicine [21,22], there are only a limited number of packages for analyzing and modeling ranking data. There are some basic tools for ranking data, for example the Kendall package and the pspearman package for the computation of Kendall and Spearman rank correlation. Nonetheless, to the best of the authors’ knowledge, the only statistical model currently available in R is the RMallow package (http://cran.rproject.org/web/packages/RMallow webcite) for fitting a mixture of Mallows’ models [23]. Here, we present pmr (probability models for ranking data), an R package for analyzing and modeling ranking data with a bundle of statistical tools. A review of statistical analysis for ranking data is given, prior to demonstrating the implementation of pmr. The current version of pmr and the user manual can be found in Additional files 1 and 2 respectively. In addition, four ranking models are reviewed, namely the Luce model, distancebased model, ϕcomponent model, and weighted distancebased model. For more details, readers can refer to [15,24,25]. The use cases diagram of the pmr package is shown in Figure 1.
Additional file 1. Package source of package pmr.
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Additional file 2. Reference manual of package pmr.
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This file can be viewed with: Adobe Acrobat Reader
Figure 1. Use Case Diagram of the pmr package.
Implementation
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.
We begin with a single measure of the popularity of an item. It is natural to use the mean rank attributed to an item to represent the central tendency of the ranks. Mean rank m is defined as the kdimensional vector in which the jth entry equals
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.
Apart from the mean ranks, the pairwise frequencies, that is, the frequency with which item i is ranked higher than item j for every possible item pairs (i, j), are also often used. These pairwise frequencies can be summarized in a k×k matrix called a pair matrix (P) in which the (s,t)^{th} entry equals
where I[·] is the indicator function. Note that P_{st}/N represents the empirical probability that item s is ranked higher than item t. In addition to mean ranks and pairwise frequencies, one can look more deeply into a ranking dataset by studying the socalled “marginal” distribution of the items. A marginal matrix, specifically for this use, is the k×k matrix M in which the (s,t)^{th} entry equals
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.” ([15], page 18).
Inconsistency indices for pairwise comparisons
According to the Analytic Hierarchy Process [26], a group of judges combine the rankings from different criteria to form a final ranking. The Analytic Hierachy Process has been used to determine the weights of these criteria. First, a pairwise comparison matrix A, in which the (s,t)^{th} entry a_{st} equals the number of times criterion s is preferred over criterion t, is computed. The weights are then found as the eigenvalues of the matrix A. The reliability of these weights depends on the consistency of the ranking process, which is defined as a_{st}a_{tu} = a_{su} for s, t, u = (1,…, k). Therefore, evaluating the consistency of the ranking data using A is a crucial task in analyzing ranking data and hence a number of measures have been developed for this purpose. One popular measure is Saaty’s index, which is given by
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 [27].
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 [28] is a dimension reduction technique that aims to display ranking data in a lowdimensional (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 higherdimension 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 rankorder 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 [15]. Note that under uniformity, the expected values of mean rank, pairs, and marginals are (k + 1)/2, 0.5 N, and N/k respectively.
Under uniformity, the test statistic when using mean rank, pairs, and marginals are ([15], page 58, Table 3.1)
and they follow a χ^{2} distribution with k1, , and (k1)^{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, MANOVAlike tests can be used [15].
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 rankorder 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.
Suppose n judges are asked to rank k items. Luce [29] proposed a ranking process where independent utilities V = (V_{1}, V_{2},..,V_{k}) ≥ 0 are assigned to item 1,2, …,k. The probability of observing ranking π_{n} is
and the resulting models is referred to as the Luce models [16]. The Luce models can be interpreted as a vase model [15]: 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 [30], generalized repeated insertion model [31], and random atomic measures [32].
The Luce model can be extended to incorporate covariates. We can include M covaraites of judge n, x_{mn}, m = 1, 2, …, M, into the utilities, that is,
where β_{jm}, m = 0, 1, 2, …, M are parameters specific to item j. This extension of the Luce model is known as the rankordered logit (ROL) model [3335].
Statistical models for ranking data: distancebased model
In what follows, we will introduce the distancebased model for ranking data. Before doing so, we need to have a clear definition of the “distance” between two rankings. A distance function is useful in measuring the discrepancy between two rankings. The usual properties of a distance function between two rankings π and σ are:
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.
Some popular rightinvariant distances are Spearman’s rho [36], given by
Spearman’s rho square, given by
Spearman’s footrule, given by
and Kendall’s tau, given by
where I() is the indicator function. There are other distances applicable to ranking data, and readers can refer to [24] for details.
It is reasonable to assume that there is a modal ranking π_{0}, and we expect most of the judges to have rankings close to π_{0}. According to this framework, Diaconis [1] developed a class of distancebased models,
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’ ϕmodel [37]. In distancebased 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 [17].
Statistical models for ranking data: ϕcomponent model
Fligner and Verducci [17] extended the distancebased models by decomposing the distance metric d(π_{,}σ) into k1 distance metrics,
where each d_{i}(π_{,}σ)is independent. Both Kendall’s tau and Cayley’s distance [38] can be decomposed in this form, and Fligner and Verducci [17] developed two new classes of ranking models for these, called ϕcomponent models and cyclic structure models, respectively.
Fligner and Verducci [17] showed that Kendall’s tau satisfies [1]:
where
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 k1 stages, V_{1} to V_{k1}, where V_{i} = m can be interpreted as m mistakes made in stage i.
By applying a dispersion parameter λ_{i} to stage V_{i}, the Mallows’ ϕmodel is extended to:
where Λ = {λ_{i}, i = 1,…, k = 1} and C(Λ) is the proportionality constant, which equals
These models were named k1 parameter models by Fligner and Verducci [17], but were also named ϕcomponent models in other papers [24]. Mallows’ ϕ models are special cases of ϕcomponent models when λ_{1} = … = λ_{k1}.
Statistical models for ranking data: weighted distancebased model
Lee and Yu [18,19] proposed an extension of the distancebased model by replacing the (equalweighted) distance with a new weighted distance measure, so that different weights can be assigned to different ranks.
Motivated by the weighted Kendall’s tau correlation coefficient [39], Lee and Yu [18,19] defined the weighted Kendall’s tau distance by
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).
Other distance measures can be generalized to a weighted distance in a similar manner to this generalization of Kendall’s tau distance. For example the weighted Spearman’s rho is
The weighted Spearman’s rho square is
and the weighted Spearman’s footrule is
Apart from the weighted Kendall’s tau [39] and weighted Spearman’s rho square [40], many other weighted rank correlations have been proposed [41].
Applying a weighted distance measure d_{w} to the distancebased model, the probability of observing a ranking π becomes
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 distancebased ranking models can retain the nature of distance, and at the same time maintain a greater flexibility. Readers are referred to [19] for the details of these properties.
Label ranking method using knearest 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 knearest neighbor method has been developed for label ranking [42]. If we want to predict the ranking of judge i, we can first select the knearest neighbor (by Euclidean distance) of i. Second, a statistical model (the Luce model in [42]) 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 sevenitem ranking dataset q4[11], 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)).
All analyses of ranking data start from descriptive statistics. Using the R code destat(q4agg), the destat function produces the mean rank vector, the pairs matrix, and the marginal:
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.
Statistical inferences about ranking data can be performed using the destat function. For instance, if we want to test whether the ranking over seven items is uniform using mean rank, the following R code can be input:
and would give the output:
The χ^{2} test statistic equals 524.8747 and the corresponding pvalue equals 1.82345 × 10^{110}. Thus, the ranking was not uniformly distributed.
This example illustrates how to test the uniformity of a ranking dataset using the destat function, and we will now explain how to compare two ranking datasets using the same function. For example, we may wish to test the hypothesis that physicians with monthly incomes above and below HK$100,000 (rankings stored in q4agg.highincome and q4agg.lowincome respectively) have different preferences towards computerization incentives. According to the marginal matrix using the χ^{2} test, the following codes:
give the output:
The χ^{2} test statistic equals 66.415 and the corresponding pvalue equals 0.04. Thus, we have found a significant difference between physicians’ preferences with respect to their monthly income.
Multidimensional preference analysis [28] 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:
and the 2D plot is given in Figure 2.
Figure 2. Multidimensional preference of the q4 dataset (1: competitive pressures; 2: increased savings; 3: government regulation; 4: improved efficiency; 5: improved quality care; 6: patient demand; 7: financial incentives).
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), distancebased model (dbm), ϕcomponent model (phicom) and weighted distancebased 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 distancebased models.
The parameter estimates of the Luce model can be obtained using the R code q4.pr < pl(q4agg); q4.pr@coef, and the output is as follows:
The warning messages are a result of some of the predicted probabilities being close to zero. The parameter estimates of the distancebased 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).
The loglikelihood is a suitable criterion for determining which model should be used. The model with the largest loglikelihood is selected. We can compute the loglikelihood of all models using the minimum value (@min) of the negative loglikelihood function, which is builtin for maximum likelihood models:
and the output is as follows:
The best model (with the smallest negative loglikelihood) is the weighted footrule model. The parameters are given by the R code q4.wdbm@coef as follows:
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 distancebased model is different from that using the mean rank.
As the “best” model does not imply that it gives an adequate fit to the data, we need to assess the goodnessoffit. The sum of squares Pearson residuals (χ^{2}) [18,19] can be used for this purpose, and is provided in pmr. It is given by
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.
We can also examine the effect of physicians’ gender and type (private/public) on their preferences (gender and type stored in q4cov) using the ROL model. This can be fitted using the rol function in the pmr package with the R code q4.rol < rol(q4,q4cov); q4.rol@coef where covariate stores the gender and type of every physicians. The output is as follows:
These parameters are difficult to interpret without their corresponding significance levels. To obtain the pvalues, the following R code can be used:
which gives the output:
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 knearest neighbor algorithm with the R code local.knn(q4,q4covtest,q4cov,knn.k = k). The value of k must be prespecified. The pmr package provides the crossvalidation version of the local knearest neighbor local.knn.cv(q4,q4covtest,q4cov). By default this uses 10fold cross validation and tests the crossvalidation prediction error of k (defined as the total Kendall’s distance) from 1 to 20.
Conclusions
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) [44]. Details of these functions can be found at http://cran.rproject.org/web/packages/pmr/pmr.pdf webcite. 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 distancebased model, and many recentlydeveloped 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.rproject.org/web/packages/pmr/index.html webcite
Operating system(s): Platform independent
Programming language: R
Other requirements: R 2.15.0 or above
License: GPL2
Any restrictions to use by nonacademics: none
Competing interests
The authors declare that they have no competing of interests.
Authors’ contributions
PHL wrote the package pmr and drafted the manuscript. PLHY helped in the development of the package pmr and significantly revised the manuscript. All authors read and approved the final manuscript.
Acknowledgements
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).
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