Probability Models for Ranking Data

Description

Probability models for ranking data.

Details

This package includes various probability models for ranking data. Current, only complete rankings are supported by this package.

Author(s)

Paul H. Lee and Philip L. H. Yu

Maintainer: Paul H. Lee <honglee@graduate.hku.hk>

References

Marden, J. I. (1995) Analyzing and Modeling Rank Data. Chapman and Hall.

Lee, P. H., and Yu, P. L. H. (2013) An R package for analyzing and modeling ranking data. BMC Medical Research Methodology, 13, 65.

The Analytic hierarchy process (AHP)

Description

Computing the weights, Saaty's (1977) and Koczkodaj's (1997) inconsistencies for analytic hierarchy process (AHP).

Usage

ahp(dset, sim_size=500)

Arguments

Value

Author(s)

Paul H. Lee and Philip L. H. Yu

References

Koczkodaj, W. W., Herman, M. W., and Orlowski, M. (1997) Using consistency-driven pairwise comaprisons in knowledge-based systems. Proceedings of the sixth international conference on information and knowledge management, ACM Press, 91-96.

Saaty, T. L. (1980) The Analytic Hierarchy Process. McGraw-Hill, New York.

Saaty, T. L. (1977) A scaling methods for priorities in hierarchical structure. Journal of Mathematical Psychology, 15, 234-281.

Examples

## create an artificial A matrix abc, example taken from Koczkodaj et al. (1997)
abc <- matrix(data = 1:16, nrow = 4, ncol = 4, byrow = TRUE)
abc[1,1] <- 1
abc[1,2] <- 2
abc[1,3] <- 5
abc[1,4] <- 4
abc[2,1] <- 1/2
abc[2,2] <- 1
abc[2,3] <- 3
abc[2,4] <- 1.9
abc[3,1] <- 1/5
abc[3,2] <- 1/3
abc[3,3] <- 1
abc[3,4] <- 0.7
abc[4,1] <- 1/4
abc[4,2] <- 1/1.9
abc[4,3] <- 1/0.7
abc[4,4] <- 1

## compute the weights, Saaty's and Koczkodaj's inconsistencies
## ahp(abc)

big4

Description

Ranking of 4 English Premier League teams from season 1992-1993 to 2012-2013.

Usage

data(big4)

Format

A data frame with 18 observations on the following 5 variables.

Arsenal

the rank of team "Arsenal"

Chelsea

the rank of team "Chelsea"

Liverpool

the rank of team "Liverpool"

Manchester.United

the rank of team "Manchester United"

n

number of observations

Details

The comparative performance of the "Big Four" since the start of the English Premier League.

Source

Wikipedia. (2013) http://en.wikipedia.org/wiki/Premier_League

References

Wikipedia. (2013) http://en.wikipedia.org/wiki/Premier_League

Examples

data(big4)
## maybe str(big4) ; plot(big4) ...

breasttissue

Description

Ranking of 4 breast cancer categories of 106 observations. The original dataset was a categorical one, and it was converted to the current ranking dataset by sorting the expected probabilities (computed by fitting a stepwise logistic regression) of the 4 categories.

Usage

data(breasttissue)

Format

A data frame with 106 observations on the following 4 variables.

adi

the rank of category "adipose"

car

the rank of category "carcinoma"

con

the rank of category "connective"

new

the rank of cateogry "fibro-adenoma, mastopathy or glandular"

Details

The rankings were generated by first fitting a stepwise logistic regression to the original dataset, and then sorted by the expected probabilities of the 4 categories.

Source

Frank, A., Asuncion, A. (2010) UCI Machine Learning Repository. http://archive.ics.uci.edu/ml

References

Jossinet, J. (1996) Variability of impedivity in normal and pathological breast tissue. Medical and Biological Engineering and Computing 34, 346-350.

da Silva, J. E., de Sa, J. P., Jossinet, J. (2000) Classification of breast tissue by electrical impedance spectroscopy. Medical and Biological Engineering and Computing 38, 26-30.

Examples

data(breasttissue)
## maybe str(breasttissue) ; rankagg(breasttissue) ...

Distance-based Models

Description

Distance-based Models for ranking data. The distance-based models assume that rankings closer to the modal ranking are more likely to be observed.

Usage

dbm(dset, dtype="tau")

Arguments

Details

Fit the distance-based models for the dataset and return a mle object. Standard methods on mle (e.g., @coef, @vcov) apply. The modal ranking and the Chi-square residual are given in the output.

Author(s)

Paul H. Lee and Philip L. H. Yu

See Also

wdbm

Examples

library(pmr)
## create an artificial dataset
X1 <- c(1,1,2,2,3,3)
X2 <- c(2,3,1,3,1,2)
X3 <- c(3,2,3,1,2,1)
n <- c(6,5,4,3,2,1)
test <- data.frame(X1,X2,X3,n)

## fit the distance-based model with Spearman's rho distance
## dbm(test,dtype="rho")

Descriptive statistics of a ranking dataset

Description

Computing the descriptive statistics (mean rank, pairs matrix, and marginals matrix) of a ranking dataset.

Usage

destat(dset)

Arguments

Value

Author(s)

Paul H. Lee and Philip L. H. Yu

Examples

## create an artificial dataset
X1 <- c(1,1,2,2,3,3)
X2 <- c(2,3,1,3,1,2)
X3 <- c(3,2,3,1,2,1)
n <- c(6,5,4,3,2,1)
test <- data.frame(X1,X2,X3,n)

## compute the descriptive statistics of the artificial dataset
## destat(test)

idea

Description

Ranking of 5 items according to the similarity with the word "idea".

Usage

data(idea)

Format

A data frame with 98 observations on the following 6 variables.

X1

the rank of word "thought"

X2

the rank of word "play"

X3

the rank of word "theory"

X4

the rank of word "dream"

X5

the rank of word "attention"

n

number of observations

Details

98 college students were asked to rank five words according to the similarity with the word "idea". The five words were (1) thought, (2) play, (3) theory, (4) dream, and (5) attention.

Source

Fligner, M. A., and Verducci, J. S. (1986) Distance based ranking models. Journal of the Royal Statistical Society Series B, 48(3), 359-369.

References

Fligner, M. A., and Verducci, J. S. (1986) Distance based ranking models. Journal of the Royal Statistical Society Series B, 48(3), 359-369.

Examples

data(idea)
## maybe str(idea) ; plot(idea) ...

leisure.black

Description

Ranking of 3 items according to the preference of the group in an activity.

Usage

data(leisure.black)

Format

A data frame with 13 observations on the following 4 variables.

X1

Male

X2

Female

X3

Both

n

number of observations

Details

13 black women, aged 70-79, were asked whom they would prefer to spend their leisure time.

Source

Hollander, M. and Sethuraman, J. (1978) Testing for agreement between groups of judges. Biometrika, 65, 403-411.

References

Marden, J. I. (1995) Analyzing and Modeling Rank Data. Chapman and Hall.

Examples

data(leisure.black)
## maybe str(leisure.black) ; plot(leisure.black) ...

leisure.white

Description

Ranking of 3 items according to the preference of the group in an activity.

Usage

data(leisure.white)

Format

A data frame with 14 observations on the following 4 variables.

X1

Male

X2

Female

X3

Both

n

number of observations

Details

14 white women, aged 70-79, were asked whom they would prefer to spend their leisure time.

Source

Hollander, M. and Sethuraman, J. (1978) Testing for agreement between groups of judges. Biometrika, 65, 403-411.

References

Marden, J. I. (1995) Analyzing and Modeling Rank Data. Chapman and Hall.

Examples

data(leisure.white)
## maybe str(leisure.white) ; plot(leisure.white) ...

Local k-nearest neighbor method for label ranking.

Description

Predict the ranking of a group of judges based on a training dataset with rankings and covariates. First, for each judge, the k-nearest neighbors (by Euclidean distance) are selected. Second, the prediction of rankings are done based on the rankings of these neighbors. Users can chooce two methods of prediction: by mean rank or by Luce model.

Usage

local.knn(dset,covariate.test,covariate,knn.k=1,method="mean")

Arguments

Author(s)

Paul H. Lee and Philip L. H. Yu

References

Cheng, W., Dembczynski, K., Hullermeier, E. (2010). Label ranking methods based on the Plackett-Luce model. Proceedings of ICML 2010.

See Also

local.knn.cv

Examples

## create an artificial dataset
X1 <- c(1,1,2,2,3,3)
X2 <- c(2,3,1,3,1,2)
X3 <- c(3,2,3,1,2,1)
co <- c(6,5,4,3,2,1)
co.test <- 1.2
train <- data.frame(X1,X2,X3)

## local k-nearest neighbor method of the artificial dataset
## local.knn(train,co.test,co)

Local k-nearest neighbor method for label ranking.

Description

Local k-nearest neighbor method with the parameter k determined using cross-validation error (defined as the sum of Kendall's distance).

Usage

local.knn.cv(dset,covariate.test,covariate,cv=10,k.max=20,method.cv="mean")

Arguments

Author(s)

Paul H. Lee and Philip L. H. Yu

References

Cheng, W., Dembczynski, K., Hullermeier, E. (2010). Label ranking methods based on the Plackett-Luce model. Proceedings of ICML 2010.

See Also

local.knn

Examples

## create an artificial dataset
X1 <- c(1,1,2,2,3,3)
X2 <- c(2,3,1,3,1,2)
X3 <- c(3,2,3,1,2,1)
co <- c(6,5,4,3,2,1)
co.test <- 1.2
train <- data.frame(X1,X2,X3)

## local k-nearest neighbor method of the artificial dataset
## local.knn.cv(train,co.test,co)

Multidimensional preference analysis.

Description

Display a 2D plot of the position of both judges and items. The items are labeled with consecutive numbers 1, 2, ..., k while the judges are presented as vectors pointing from the origin to their most preferred items.

Usage

mdpref(dset,rank.vector=FALSE,ndim=2)

Arguments

Details

Multidimenional preference analysis is a dimension reduction technique which aims to project the high-dimensional ranking data into 2D or 3D plot. Dimension reduction is done using singular value decomposition. Note that the perpendicular projection of the item points onto a judge vector represents the ranking of these items by this judge.

Value

Author(s)

Paul H. Lee and Philip L. H. Yu

References

Carroll, J. D. (1972) Individual differences and multidimensional scaling. In Shepard, R. N., Ronney, A. K., and Nerlove, S. B. (eds.)

Examples

## create an artificial dataset
X1 <- c(1,1,2,2,3,3)
X2 <- c(2,3,1,3,1,2)
X3 <- c(3,2,3,1,2,1)
n <- c(6,5,4,3,2,1)
test <- data.frame(X1,X2,X3,n)

## multidimensional preference analysis of the artificial dataset
## mdpref(test,rank.vector=TRUE)

phi-component Models

Description

phi-component Models for ranking data. The distance-based models assume that rankings closer to the modal ranking are more likely to be observed. Phi-component models are extensions of distance-based models with Kendall's distance by allowing weights at different stages.

Usage

phicom(dset)

Arguments

Details

Fit the phi-component models for the dataset and return a mle object. Standard methods on mle (e.g., @coef, @vcov) apply. The modal ranking and the Chi-square residual are given in the output.

Author(s)

Paul H. Lee and Philip L. H. Yu

References

Fligner, M. A., and Verducci, J. S. (1986) Distance based ranking models. Journal of the Royal Statistical Society Series B, 48(3), 359-369.

See Also

dbm

Examples

## create an artificial dataset
X1 <- c(1,1,2,2,3,3)
X2 <- c(2,3,1,3,1,2)
X3 <- c(3,2,3,1,2,1)
n <- c(6,5,4,3,2,1)
test <- data.frame(X1,X2,X3,n)

## fit the phi-component model
## phicom(test)

The Luce Models

Description

The Luce Models for ranking data. The Luce models assumed that the rankings of the items are proportional to the item parameters.

Usage

pl(dset)

Arguments

Details

Fit the Luce models for the dataset and return a mle object. Standard methods on mle (e.g., @coef, @vcov) apply. The Chi-square residual are given in the output.

Author(s)

Paul H. Lee and Philip L. H. Yu

References

Luce, R. D. (1959) Individual Choice Behavior. New York: Wiley.

See Also

rinv

Examples

## create an artificial dataset
X1 <- c(1,1,2,2,3,3)
X2 <- c(2,3,1,3,1,2)
X3 <- c(3,2,3,1,2,1)
n <- c(6,5,4,3,2,1)
test <- data.frame(X1,X2,X3,n)

## fit the Luce model
## pl(test)

Summary of a ranking dataset.

Description

Turn individual rankings into a summary matrix.

Usage

rankagg(dset)

Arguments

Value

Return the dataset which summarize the original ranking data.

Author(s)

Paul H. Lee and Philip L. H. Yu

Examples

## create an artificial dataset
X1 <- c(1,1,2,2,3,3)
X2 <- c(2,3,1,3,1,2)
X3 <- c(3,2,3,1,2,1)
test <- data.frame(X1,X2,X3)

## aggregate the ranking of all the observations and create a summary matrix
## test2 <- rankagg(test)

Visualizing rankings of 3 or 4 items by plot.

Description

Display rankings of 3 items by hexagon and 4 items by truncated octahedron. Each line linking two points represents a Kendall distance of one. The size of the point is proportional to the frequency of that particular ranking.

Usage

rankplot(dset,trans=FALSE,label.type="item", circle.col="black", circle.bg=FALSE)

Arguments

Details

As a ranking dataset of k items can be represented in k-1 dimensions, ranking data with 3 or 4 items can be represented in graphs without losing any information. For visualization of ranking data with more items, multidimensional preference analysis can be used.

Author(s)

Paul H. Lee and Philip L. H. Yu

References

Thompson, G. L. (1993). Graphical techniques for ranked data. In Fligner, M. A., and Verducci, J. S. (eds.) Probability Models and Statistical Analyses for Ranking Data, pp. 294-298.

Examples

## create an artificial dataset
X1 <- c(1,1,2,2,3,3)
X2 <- c(2,3,1,3,1,2)
X3 <- c(3,2,3,1,2,1)
n <- c(6,5,4,3,2,1)
test <- data.frame(X1,X2,X3,n)

## plotting the artificial dataset in a hexagon
## rankplot(test)

Inverse of a ranking dataset.

Description

Computing the inverse of a summarized (or aggregated) ranking dataset. The function rankagg can be used to created a summarized ranking dataset from individual rankings.

Usage

rinv(dset)

Arguments

Value

Return the dataset which is the orderings of the original ranking data.

Author(s)

Paul H. Lee and Philip L. H. Yu

Examples

## create an artificial dataset
X1 <- c(1,1,2,2,3,3)
X2 <- c(2,3,1,3,1,2)
X3 <- c(3,2,3,1,2,1)
n <- c(6,5,4,3,2,1)
test <- data.frame(X1,X2,X3,n)

## compute the inverse of the artificial dataset
## rinv(test)

The Rank-ordered Logit Models

Description

The Rank-ordered Logit (ROL) Models for ranking data. ROL models are extensions of the Luce models by incorporating covariates.

Usage

rol(dset, covariate)

Arguments

Details

Fit the rank-ordered logit models for the dataset and return a mle object. Standard methods on mle (e.g., @coef, @vcov) apply. By default, the intercept term is included.

Author(s)

Paul H. Lee and Philip L. H. Yu

References

Beggs, S., Cardell, S., and Hausman, J. (1981) Assessing the potential demand for electric cars. Journal of Econometrics, 16: 1-19.

Chapman, R. G., and Staelin, R. (1982) Exploiting rank ordered choice set data within the stochastic utility model. Journal of Market Research, 19:288-301.

Hausman, J., and Ruud, P. A. (1987) Specifying and testing econometric models for rank-ordered data. Journal of Econometrics, 34:83-104.

See Also

pl

Examples

## create an artificial dataset
X1 <- c(1,1,2,2,3,3)
X2 <- c(2,3,1,3,1,2)
X3 <- c(3,2,3,1,2,1)
X4 <- c(6,5,4,3,2,1)
test <- data.frame(X1,X2,X3)

## fit the Luce model
## rol(test,X4)

song

Description

Ranking of 5 items according to the similarity with the word "song".

Usage

data(song)

Format

A data frame with 83 observations on the following 6 variables.

X1

the rank of word "score"

X2

the rank of word "instrument"

X3

the rank of word "solo"

X4

the rank of word "benediction"

X5

the rank of word "suit"

n

number of observations

Details

83 college students were asked to rank five words according to the similarity with the word "song". The five words were (1) score, (2) instrument, (3) solo, (4) benediction, and (5) suit.

Source

Critchlow, D. E., Fligner, M. A., and Verducci, J. S. (1991) Probability models on rakning. Journal of Mathematical Psychology, 35, 294-318.

References

Critchlow, D. E., Fligner, M. A., and Verducci, J. S. (1991) Probability models on rakning. Journal of Mathematical Psychology, 35, 294-318.

Examples

data(song)
## maybe str(song) ; plot(song) ...

Weighted Distance-based Models

Description

Weighted Distance-based Models for ranking data. The distance-based models assume that rankings closer to the modal ranking are more likely to be observed. Weighted distance-based models are extensions of distance-based models with by allowing weights for different items.

Usage

wdbm(dset, dtype="tau")

Arguments

Details

Fit the weighted distance-based models for the dataset and return a mle object. Standard methods on mle (e.g., @coef, @vcov) apply. The modal ranking and the Chi-square residual are given in the output.

Author(s)

Paul H. Lee and Philip L. H. Yu

References

Lee, P. H., and Yu, P. L. H. (2010) Distance-based tree models for ranking data. Computational Statistics and Data Analysis, 54(6), 1672-1682.

Lee, P. H., and Yu, P. L. H. (2012) Mixtures of weighted distance-based models for ranking data with applications in political studies. Computational Statistics and Data Analysis, 56(8), 2486-2500.

See Also

dbm

Examples

## create an artificial dataset
X1 <- c(1,1,2,2,3,3)
X2 <- c(2,3,1,3,1,2)
X3 <- c(3,2,3,1,2,1)
n <- c(6,5,4,3,2,1)
test <- data.frame(X1,X2,X3,n)

## fit the weighted distance-based model with Spearman's foot distance
## wdbm(test,dtype="foot")