Title:	Finite Mixtures of Mallows Models with Spearman Distance for Full and Partial Rankings
Version:	2.0.0
Maintainer:	Cristina Mollica <cristina.mollica@uniroma1.it>
Description:	Fit and analysis of finite Mixtures of Mallows models with Spearman Distance for full and partial rankings with arbitrary missing positions. Inference is conducted within the maximum likelihood framework via Expectation-Maximization algorithms. Estimation uncertainty is tackled via diverse versions of bootstrapped and asymptotic confidence intervals. The most relevant reference of the methods is Crispino, Mollica, Astuti and Tardella (2023) <doi:10.1007/s11222-023-10266-8>.
License:	GPL (≥ 3)
Repository:	CRAN
LinkingTo:	Rcpp
Imports:	BayesMallows (≥ 2.0.1), bmixture (≥ 1.7.0), data.table (≥ 1.15.0), factoextra (≥ 1.0.7), fields (≥ 15.2.0), foreach (≥ 1.5.2), ggbump (≥ 0.1.0), ggplot2 (≥ 3.4.4), gmp (≥ 0.7.4), gridExtra (≥ 2.3.0), methods (≥ 4.3.1), magrittr (≥ 2.0.3), nnet (≥ 7.3.19), Rankcluster (≥ 0.98.0), RColorBrewer (≥ 1.1.3), Rcpp (≥ 1.0.12), reshape (≥ 0.8.9), rlang (≥ 1.1.3), scales (≥ 1.3.0), spsUtil (≥ 0.2.2), stats (≥ 4.3.1)
Depends:	R (≥ 4.3.0), dplyr (≥ 1.1.4)
Suggests:	doParallel
NeedsCompilation:	yes
Author:	Cristina Mollica [aut, cre, cph], Marta Crispino [aut, cph], Lucia Modugno [ctb], Luca Tardella [ctb]
LazyData:	true
Encoding:	UTF-8
RoxygenNote:	7.3.2
Packaged:	2025-03-25 12:31:43 UTC; cristina
Date/Publication:	2025-03-25 17:50:10 UTC

Finite Mixtures of Mallows Models with Spearman Distance for Full and Partial Rankings

Description

The MSmix package provides functions to fit and analyze finite Mixtures of Mallows models with Spearman distance (a.k.a. \theta-model) for full and partial rankings with arbitrary missing positions. Inference is conducted within the maximum likelihood (ML) framework via EM algorithms. Estimation uncertainty is tackled via diverse versions of bootstrapped and asymptotic confidence intervals.

Details

The Mallows model is one of the most popular and frequently applied parametric distributions to analyze rankings of a finite set of items. However, inference for this model is challenging due to the intractability of the normalizing constant, also referred to as partition function. The present package performs ML estimation (MLE) of the Mallows model with Spearman distance from full and partial rankings with arbitrary censoring patterns. Thanks to the novel approximation of the model normalizing constant introduced by Crispino, Mollica, Astuti and Tardella (2023), as well as the existence of a closed-form expression of the MLE of the consensus ranking, MSmix can address inference even for a large number of items. The package also allows to account for unobserved sample heterogeneity through MLE of finite mixtures of Mallows models with Spearman distance via EM algorithms, in order to perform a model-based clustering of partial rankings into groups with similar preferences.

Computational efficiency is achieved with the use of a hybrid language, combining R and C++ code, and the possibility of parallel computation.

In addition to inferential techniques, the package provides various functions for data manipulation, simulation, descriptive summary and model selection.

Specific S3 classes and methods are also supplied to enhance the usability and foster exchange with other packages.

The suite of functions available in the MSmix package is composed of:

Ranking data manipulation

data_conversion

From rankings to orderings and vice versa.

data_censoring

Censoring of full rankings.

data_completion

Deterministic completion of partial rankings with full reference rankings.

data_augmentation

Generate all full rankings compatible with partial rankings.

Ranking data simulation

rMSmix

Random samples from finite mixtures of Mallows models with Spearman distance.

Ranking data description

data_description

Descriptive summaries for partial rankings.

Model estimation

fitMSmix

MLE of mixtures of Mallows models with Spearman distance via EM algorithms.

likMSmix

Likelihood evaluation for mixtures of Mallows models with Spearman distance.

Model selection

bicMSmix

BIC value for the fitted mixture of Mallows models with Spearman distance.

aicMSmix

AIC value for the fitted mixture of Mallows models with Spearman distance.

Estimation uncertainty

bootstrapMSmix

Bootstrap confidence intervals for mixtures of Mallows models with Spearman distance.

confintMSmix

Asymptotic confidence intervals for mixtures of Mallows models with Spearman distance.

Spearman distance utilities

spear_dist

Spearman distance computation for full rankings.

spear_dist_distr

Spearman distance distribution under the uniform (null) model.

partition_fun_spear

Partition function of the Mallows model with Spearman distance.

expected_spear_dist

Expected Spearman distance under the Mallows model with Spearman distance.

var_spear_dist

Variance of the Spearman distance under the Mallows model with Spearman distance.

S3 class methods

print.bootMSmix

Print the bootstrap confidence intervals of mixtures of Mallows models with Spearman distance.

print.data_descr

Print the descriptive statistics for partial rankings.

print.emMSmix

Print the MLEs of mixtures of Mallows models with Spearman distance.

print.summary.emMSmix

Print the summary of the MLEs of mixtures of Mallows models with Spearman distance.

plot.bootMSmix

Plot the bootstrap confidence intervals of mixtures of Mallows models with Spearman distance.

plot.data_descr

Plot the descriptive statistics for partial rankings.

plot.dist

Plot the Spearman distance matrix for full rankings.

plot.emMSmix

Plot the MLEs of mixtures of Mallows models with Spearman distance.

summary.emMSmix

Summary of the MLEs of mixtures of Mallows models with Spearman distance.

Datasets

ranks_antifragility

Antifragility features of innovative startups (full rankings with covariates).

ranks_horror

Arkham Horror data (full rankings).

ranks_beers

Beers data (partial rankings with different censoring patterns and a covariate).

ranks_read_genres

Reading preference data (partial top-5 rankings with covariates).

ranks_sports

Sport preferences and habits (full rankings with covariates).

Some quantities frequently recalled in the manual are the following:

N

Sample size.

n

Number of possible items.

G

Number of mixture components.

Data must be supplied as an integer N\timesn matrix with partial rankings in each row and missing positions denoted as NA (rank = 1 indicates the most-liked item). Partial sequences with a single missing entry are automatically filled in, as they correspond to full rankings. In the present setting, ties are not allowed.

Author(s)

Cristina Mollica, Marta Crispino, Lucia Modugno and Luca Tardella

Maintainer: Cristina Mollica <cristina.mollica@uniroma1.it>

References

Crispino M, Mollica C, Astuti V and Tardella L (2023). Efficient and accurate inference for mixtures of Mallows models with Spearman distance. Statistics and Computing, 33(98), DOI: 10.1007/s11222-023-10266-8.

Crispino M, Mollica C, Modugno L, Casadio Tarabusi E, and Tardella L (2024+). MSmix: An R Package for clustering partial rankings via mixtures of Mallows models with Spearman distance. (submitted).

BIC and AIC for mixtures of Mallows models with Spearman distance

Description

bicMSmix and aicMSmix compute, respectively, the Bayesian Information Criterion (BIC) and the Akaike Information Criterion (AIC) for a mixture of Mallows models with Spearman distance fitted on partial rankings.

Usage

bicMSmix(rho, theta, weights, rankings)

aicMSmix(rho, theta, weights, rankings)

Arguments

Details

The (log-)likelihood evaluation is performed by augmenting the partial rankings with the set of all compatible full rankings (see data_augmentation), and then the marginal likelihood is computed.

When n\leq 20, the (log-)likelihood is exactly computed, otherwise it is approximated with the method introduced by Crispino et al. (2023). If n>170, the approximation is also restricted over a fixed grid of values for the Spearman distance to limit computational burden.

Value

The BIC or AIC value.

References

Crispino M, Mollica C and Modugno L (2025+). MSmix: An R Package for clustering partial rankings via mixtures of Mallows Models with Spearman distance. (submitted)

Crispino M, Mollica C, Astuti V and Tardella L (2023). Efficient and accurate inference for mixtures of Mallows models with Spearman distance. Statistics and Computing, 33(98), DOI: 10.1007/s11222-023-10266-8.

Schwarz G (1978). Estimating the dimension of a model. The Annals of Statistics, 6(2), pages 461–464, DOI: 10.1002/sim.6224.

Sakamoto Y, Ishiguro M, and Kitagawa G (1986). Akaike Information Criterion Statistics. Dordrecht, The Netherlands: D. Reidel Publishing Company.

See Also

likMSmix, data_augmentation

Examples

## Example 1. Simulate rankings from a 2-component mixture of Mallows models
## with Spearman distance.
set.seed(12345)
rank_sim <- rMSmix(sample_size = 50, n_items = 12, n_clust = 2)
str(rank_sim)
rankings <- rank_sim$samples
# Fit the true model.
set.seed(12345)
fit <- fitMSmix(rankings = rankings, n_clust = 2, n_start = 10)
# Comparing the BIC at the true parameter values and at the MLE.
bicMSmix(rho = rank_sim$rho, theta = rank_sim$theta, weights = rank_sim$weights,
       rankings = rank_sim$samples)
bicMSmix(rho = fit$mod$rho, theta = fit$mod$theta, weights = fit$mod$weights,
       rankings = rank_sim$samples)
aicMSmix(rho = rank_sim$rho, theta = rank_sim$theta, weights = rank_sim$weights,
       rankings = rank_sim$samples)
aicMSmix(rho = fit$mod$rho, theta = fit$mod$theta, weights = fit$mod$weights,
       rankings = rank_sim$samples)


## Example 2. Simulate rankings from a basic Mallows model with Spearman distance.
set.seed(54321)
rank_sim <- rMSmix(sample_size = 50, n_items = 8, n_clust = 1)
str(rank_sim)
# Let us censor the observations to be top-5 rankings.
rank_sim$samples[rank_sim$samples > 5] <- NA
rankings <- rank_sim$samples
# Fit the true model with the two EM algorithms.
set.seed(54321)
fit_em <- fitMSmix(rankings = rankings, n_clust = 1, n_start = 10)
set.seed(54321)
fit_mcem <- fitMSmix(rankings = rankings, n_clust = 1, n_start = 10, mc_em = TRUE)
# Compare the BIC at the true parameter values and at the MLEs.
bicMSmix(rho = rank_sim$rho, theta = rank_sim$theta, weights = rank_sim$weights,
       rankings = rank_sim$samples)
bicMSmix(rho = fit_em$mod$rho, theta = fit_em$mod$theta, weights = fit_em$mod$weights,
       rankings = rank_sim$samples)
bicMSmix(rho = fit_mcem$mod$rho, theta = fit_mcem$mod$theta, weights = fit_mcem$mod$weights,
       rankings = rank_sim$samples)
aicMSmix(rho = rank_sim$rho, theta = rank_sim$theta, weights = rank_sim$weights,
       rankings = rank_sim$samples)
aicMSmix(rho = fit_em$mod$rho, theta = fit_em$mod$theta, weights = fit_em$mod$weights,
       rankings = rank_sim$samples)
aicMSmix(rho = fit_mcem$mod$rho, theta = fit_mcem$mod$theta, weights = fit_mcem$mod$weights,
       rankings = rank_sim$samples)

Bootstrap confidence intervals for the fitted mixture of Mallows models with Spearman distance

Description

Return the bootstrap confidence intervals for the parameters of a mixture of Mallows models with Spearman distance fitted on partial rankings.

plot method for class "bootMSmix".

Usage

bootstrapMSmix(
  object,
  n_boot = 50,
  type = (if (object$em_settings$n_clust == 1) "non-parametric" else "soft"),
  conf_level = 0.95,
  all = FALSE,
  n_start = 10,
  parallel = FALSE
)

## S3 method for class 'bootMSmix'
plot(x, ...)

Arguments

Details

When n_clust = 1, two types of bootstrap are available: 1) type = "non-parametric" (default); type = "parametric", where the latter supports full rankings only.

When n_clust > 1, two types of bootstrap are available: 1) type = "soft" (default), which is the soft-separated bootstrap (Crispino et al., 2025+) and returns confidence intervals for all the parameters of the mixture of Mallows models with Spearman distance; 2) type = "separated", which is the separated bootstrap (Taushanov and Berchtold, 2019) and returns bootstrap samples for the component-specific consensus rankings and precisions.

Value

An object of class "bootMSmix", namely a list with the following named components:

itemwise_ci_rho

Character G\timesn matrix with the bootstrap itemwise confidence intervals for the component-specific consensus rankings.

ci_boot_theta

Numeric G\times2 matrix with the bootstrap confidence intervals for the component-specific precisions.

ci_boot_weights

Numeric G\times2 matrix with the bootstrap confidence intervals for the mixture weights. Returned when n_clust > 1 and type = "soft", otherwise NULL.

boot

List containing all the n_boot bootstrap MLEs. Returned when all = TRUE, otherwise NULL.

The boot sublist contains the following named components:

rho_boot

List of length n_clust. Each element is an integer n_boot \times n_items matrix with rows containing the bootstrap MLEs of a component-specific consensus ranking.

theta_boot

Numeric n_boot\times n_clust matrix with the bootstrap MLEs of the component-specific precision parameters in each row.

weights_boot

Numeric n_boot\times n_clust matrix with the bootstrap MLEs of the mixture weights in each row. Returned when n_clust > 1 and type = "soft", otherwise NULL.

A list of 3 labelled plots, namely: i) rho_heatmap: a heatmap for the component-specific bootstrap consensus ranking estimates (when n_clust > 1, this is in turn a list of heatmaps for each consensus ranking estimate); ii) theta_density: a kernel density plot for the component-specific bootstrap precision estimates; iii) weights_density: a kernel density plot for the bootstrap mixture weight estimates is returned when n_clust > 1 and the object x was obtained from the bootstrapMSmix routine with the argument type = "soft", otherwise NULL.

References

Crispino M, Mollica C and Modugno L (2025+). MSmix: An R Package for clustering partial rankings via mixtures of Mallows Models with Spearman distance. (submitted)

Taushanov Z and Berchtold A (2019). Bootstrap validation of the estimated parameters in mixture models used for clustering. Journal de la société française de statistique, 160(1).

Efron B (1982). The Jackknife, the Bootstrap, and Other Resampling Plans. Philadelphia, Pa. :Society for Industrial and Applied Mathematics.

Examples

## Example 1. Compute the bootstrap 95% confidence intervals for the Antifragility dataset.
# Let us assume no clusters.
r_antifrag <- ranks_antifragility[, 1:7]
set.seed(12345)
fit <- fitMSmix(rankings = r_antifrag, n_clust = 1, n_start = 1)
# Apply non-parametric bootstrap procedure.
set.seed(12345)
boot_np <- bootstrapMSmix(object = fit, n_boot = 200)
print(boot_np)
# Apply parametric bootstrap procedure and set all = TRUE
# to return the bootstrap MLEs of the consensus ranking.
set.seed(12345)
boot_p <- bootstrapMSmix(object = fit, n_boot = 200,
                       type = "parametric", all = TRUE)
print(boot_p)
# Plot the bootstrap estimates.
p_boot_p <- plot(boot_p)
p_boot_p$rho_heatmap()
p_boot_p$theta_density()

## Example 2. Compute the bootstrap 95% confidence intervals for the Antifragility dataset.
# Let us assume two clusters.
r_antifrag <- ranks_antifragility[, 1:7]
set.seed(12345)
fit <- fitMSmix(rankings = r_antifrag, n_clust = 2, n_start = 20)
# Apply soft bootstrap procedure and set all = TRUE
# to return the bootstrap MLEs of the consensus ranking.
set.seed(12345)
boot_soft <- bootstrapMSmix(object = fit, n_boot = 500,
                      n_start = 20, all = TRUE)
print(boot_soft)
# Plot the bootstrap estimates.
p_boot_soft <- plot(boot_soft)
p_boot_soft$rho_heatmap[[1]]()
p_boot_soft$rho_heatmap[[2]]()
p_boot_soft$theta_density()
p_boot_soft$weights_density()
# Apply separated bootstrap and compare results.
set.seed(12345)
boot_sep <- bootstrapMSmix(object = fit, n_boot = 500,
                     n_start = 20, type = "separated", all = TRUE)
print(boot_sep)
p_boot_sep <- plot(boot_sep)
p_boot_sep$rho_heatmap[[1]]()
p_boot_sep$rho_heatmap[[2]]()
p_boot_sep$theta_density()
print(boot_soft)
print(boot_sep)

Asymptotic confidence intervals for the fitted mixture of Mallows models with Spearman distance

Description

Return the asymptotic confidence intervals of the continuous parameters (component-specific precisions and weights) of a mixture of Mallows models with Spearman distance fitted to full rankings.

print method for class "ciMSmix".

Usage

confintMSmix(object, conf_level = 0.95)

## S3 method for class 'ciMSmix'
print(x, ...)

Arguments

Details

The current implementation of the asymptotic confidence intervals assumes that the observed rankings are complete.

Value

An object of class "ciMSmix", namely a list with the following named components:

ci_theta

Numeric G\times2 matrix with the confidence intervals of the component-specific precision parameters in each row.

ci_weights

Numeric G\times2 matrix with the confidence intervals of the mixture weights in each row (when G>1), otherwise NULL.

References

Crispino M, Mollica C and Modugno L (2025+). MSmix: An R Package for clustering partial rankings via mixtures of Mallows Models with Spearman distance. (submitted)

Marden JI (1995). Analyzing and modeling rank data. Monographs on Statistics and Applied Probability (64). Chapman & Hall, ISSN: 0-412-99521-2. London.

McLachlan G and Peel D (2000). Finite Mixture Models. Wiley Series in Probability and Statistics, John Wiley & Sons.

Examples

## Example 1. Simulate rankings from a 2-component mixture of Mallows models
## with Spearman distance.
set.seed(123)
d_sim <- rMSmix(sample_size = 75, n_items = 8, n_clust = 2)
rankings <- d_sim$samples
# Fit the basic Mallows model with Spearman distance.
set.seed(123)
fit1 <- fitMSmix(rankings = rankings, n_clust = 1, n_start = 10)
# Compute the asymptotic confidence intervals for the MLEs of the precision.
ci95_fit1 <- confintMSmix(object = fit1)
print(ci95_fit1)
# Fit the true model.
set.seed(123)
fit2 <- fitMSmix(rankings = rankings, n_clust = 2, n_start = 10)
# Compute the asymptotic confidence intervals for the MLEs of the weights and precisions.
ci95_fit2 <- confintMSmix(object = fit2)
print(ci95_fit2)

Data augmentation of partial rankings

Description

For a given partial ranking matrix, generate all possible full rankings which are compatible with each partially ranked sequence. Partial rankings with at most 10 missing positions and arbitrary patterns of censoring are supported.

Usage

data_augmentation(rankings, fill_single_na = TRUE)

Arguments

Details

The data augmentation of a full ranking returns the complete ranking itself arranged in a row vector. The function can be applied on partial observations expressed in ordering format as well. A message informs the user when data augmentation may be heavy.

Value

A list of N elements corresponding to the matrices of full rankings compatible with each partial sequence.

References

Crispino M, Mollica C, Astuti V and Tardella L (2023). Efficient and accurate inference for mixtures of Mallows models with Spearman distance. Statistics and Computing, 33(98), DOI: 10.1007/s11222-023-10266-8.

Examples

## Example 1. Data augmentation of a single partial top-9 ranking.
data_augmentation(c(3, 7, 5, 1, NA, 4, NA, 8, 2, 6, NA, 9))

## Example 2. Data augmentation of partial ranking matrix with different censoring patterns.
rank_data <- rbind(c(NA, 4, NA, 1, NA),
                   c(NA, NA, NA, NA, 1),
                   c(2, NA, 1, NA, 3),
                   c(4, 2, 3, 5, 1),
                   c(NA, 4, 1, 3, 2))
data_augmentation(rank_data)

Censoring of full rankings

Description

Convert full rankings into either top-k rankings or into partial rankings with missing data in arbitrary positions.

Usage

data_censoring(
  rankings,
  topk = TRUE,
  nranked = NULL,
  probs = rep(1, ncol(rankings) - 1)
)

Arguments

Details

Both forms of partial rankings can be obtained into two ways: (i) by specifying, in the nranked argument, the number of positions to be retained in each partial ranking; (ii) by setting nranked = NULL (default) and specifying, in the probs argument, the probabilities of retaining respectively 1, 2, ..., (n-1) positions in the partial rankings (recall that a partial sequence with (n-1) observed entries corresponds to a full ranking).

When topk = FALSE, the exact positions that must be retained into the partial sequences after censoring are uniformly generated, regardless of the specification of the nranked argument.

Value

A list of two named objects:

part_rankings

Integer N\timesn matrix with partial (censored) rankings in each row. Missing positions are coded as NA.

nranked

Integer vector of length N with the actual number of items ranked in each partial sequence after censoring.

Examples

## Example 1. Censoring the Antifragility dataset into partial top rankings
# Top-3 censoring (assigned number of top positions to be retained)
n <- 7
r_antifrag <- ranks_antifragility[, 1:n]
data_censoring(r_antifrag, topk = TRUE, nranked = rep(3,nrow(r_antifrag)))
# Random top-k censoring with assigned probabilities
set.seed(12345)
data_censoring(r_antifrag, topk = TRUE, probs = 1:(n-1))

## Example 2. Simulate full rankings from a basic Mallows model with Spearman distance
n <- 10
N <- 100
set.seed(12345)
rankings <- rMSmix(sample_size = N, n_items = n)$samples
# Censoring in arbitrary positions with assigned number of ranks to be retained
set.seed(12345)
nranked <- round(runif(N,0.5,1)*n)
set.seed(12345)
arbitr_ranks1 <- data_censoring(rankings, topk = FALSE, nranked = nranked)
arbitr_ranks1
identical(arbitr_ranks1$nranked, nranked)
# Censoring in arbitrary positions with random number of ranks to be retained
set.seed(12345)
probs <- runif(n-1, 0, 0.5)
set.seed(12345)
arbitr_ranks2 <- data_censoring(rankings, topk = FALSE, probs = probs)
arbitr_ranks2
prop.table(table(arbitr_ranks2$nranked))
round(prop.table(probs), 2)

Completion of partial rankings with reference full rankings

Description

Deterministic completion of partial rankings with the relative positions of the unranked items in the reference full rankings. Partial rankings with arbitrary patterns of censoring are supported.

Usage

data_completion(rankings, ref_rho)

Arguments

Details

The arguments rankings and ref_rho must be objects with the same class (matrix or data frame) and same dimensions.

The completion of a full ranking returns the complete ranking itself.

Value

Integer N\timesn matrix with the completed rankings in each row.

References

Crispino M, Mollica C and Modugno L (2025+). MSmix: An R Package for clustering partial rankings via mixtures of Mallows Models with Spearman distance. (submitted)

Examples

## Example 1. Completion of a single partial ranking.
data_completion(rankings = c(3, NA, NA, 1, NA), ref_rho = c(4, 5, 1, 3, 2))

## Example 2. Completion of partial rankings with arbitrary censoring patterns.
rankings <- rbind(c(3, NA, NA, 7, 2, NA, NA), c(NA, 6, NA, 5, NA, NA, 1),7:1)
data_completion(rankings = rankings, ref_rho = rbind(c(4, 5, 6, 1, 3, 7, 2),
                7:1, 1:7))

Switch data format from rankings to orderings and vice versa

Description

Convert the format of the input dataset from rankings to orderings and vice versa. Differently from existing analogous functions supplied by other R packages, data_conversion supports also partial rankings/orderings with arbitrary patterns of censoring.

Usage

data_conversion(data, subset = NULL)

Arguments

Value

Integer N\timesn matrix of partial sequences with inverse format with respect to the input data.

Examples

## Example 1. Switch the data format for a single complete observation.
data_conversion(c(4, 5, 1, 3, 2))

## Example 2. Switch the data format for partial sequences with arbitrary censoring patterns.
data_conversion(rbind(c(NA, 2, 5, NA, NA), c(4, NA, 2, NA, 3), c(4, 5, 1, NA, NA),
                      c(NA, NA, NA, NA, 2), c(NA, 5, 2, 1, 3), c(3, 5, 1, 2, 4)))

Descriptive summaries for partial rankings

Description

Compute various data summaries for a partial ranking dataset. Differently from existing analogous functions supplied by other R packages, data_description supports partial observations with arbitrary patterns of censoring.

print method for class "data_descr".

Usage

data_description(
  rankings,
  marg = TRUE,
  borda_ord = FALSE,
  paired_comp = TRUE,
  subset = NULL,
  item_names = NULL
)

## S3 method for class 'data_descr'
print(x, ...)

Arguments

Details

The implementation of data_description is similar to that of rank_summaries from the PLMIX package. Differently from the latter, data_description works with any kind of partial rankings (not only top rankings) and allows to summarize subsamples thanks to the additional subset argument.

The Borda ranking, obtained from the ordering of the mean rank vector, corresponds to the MLE of the consensus ranking of the Mallows model with Spearman distance. If mean_rank contains some NAs, the corresponding items occupy the bottom positions in the borda_ordering according to the order they appear in item_names.

Value

An object of class "data_descr", which is a list with the following named components:

References

Mollica C and Tardella L (2020). PLMIX: An R package for modelling and clustering partially ranked data. Journal of Statistical Computation and Simulation, 90(5), pages 925–959, ISSN: 0094-9655, DOI: 10.1080/00949655.2020.1711909.

Marden JI (1995). Analyzing and modeling rank data. Monographs on Statistics and Applied Probability (64). Chapman & Hall, ISSN: 0-412-99521-2. London.

See Also

plot.data_descr, print.data_descr

Examples

## Example 1. Sample statistics for the Antifragility dataset.
r_antifrag <- ranks_antifragility[, 1:7]
descr <- data_description(rankings = r_antifrag)
descr

## Example 2. Sample statistics for the Sports dataset.
r_sports <- ranks_sports[, 1:8]
descr <- data_description(rankings = r_sports, borda_ord = TRUE)
descr

## Example 3. Sample statistics for the Sports dataset by gender.
r_sports <- ranks_sports[, 1:8]
desc_f <- data_description(rankings = r_sports, subset = (ranks_sports$Gender == "Female"))
desc_m <- data_description(rankings = r_sports, subset = (ranks_sports$Gender == "Male"))
desc_f
desc_m

Expectation of the Spearman distance

Description

Compute (either the exact or the approximate) (log-)expectation of the Spearman distance under the Mallows model with Spearman distance.

Usage

expected_spear_dist(theta, n_items, log = TRUE)

Arguments

Details

When n\leq 20, the expectation is exactly computed by relying on the Spearman distance distribution provided by OEIS Foundation Inc. (2023). When n>20, it is approximated with the method introduced by Crispino et al. (2023) and, if n>170, the approximation is also restricted over a fixed grid of values for the Spearman distance to limit computational burden.

The expected Spearman distance is independent of the consensus ranking of the Mallows model with Spearman distance due to the right-invariance of the metric. When \theta=0, this is equal to \frac{n^3-n}{6}, which is the expectation of the Spearman distance under the uniform (null) model.

Value

Either the exact or the approximate (log-)expected value of the Spearman distance under the Mallows model with Spearman distance.

References

Crispino M, Mollica C, Astuti V and Tardella L (2023). Efficient and accurate inference for mixtures of Mallows models with Spearman distance. Statistics and Computing, 33(98), DOI: 10.1007/s11222-023-10266-8.

OEIS Foundation Inc. (2023). The On-Line Encyclopedia of Integer Sequences, Published electronically at https://oeis.org.

Kendall MG (1970). Rank correlation methods. 4th ed. Griffin London.

See Also

spear_dist_distr, partition_fun_spear

Examples

## Example 1. Expected Spearman distance under the uniform (null) model,
## coinciding with (n^3-n)/6.
n_items <- 10
expected_spear_dist(theta = 0, n_items = n_items, log = FALSE)
(n_items^3-n_items)/6

## Example 2. Expected Spearman distance.
expected_spear_dist(theta = 0.5, n_items = 10, log = FALSE)

## Example 3. Log-expected Spearman distance as a function of theta.
expected_spear_dist_vec <- Vectorize(expected_spear_dist, vectorize.args = "theta")
curve(expected_spear_dist_vec(x, n_items = 10),
  from = 0, to = 0.1, lwd = 2, col = 2, ylim = c(3, 5.5),
  xlab = expression(theta), ylab = expression(log(E[theta](D))),
  main = "Log-expected Spearman distance")

## Example 4. Log-expected Spearman distance for varying number of items
# and values of the concentration parameter.
expected_spear_dist_vec <- Vectorize(expected_spear_dist, vectorize.args = "theta")
curve(expected_spear_dist_vec(x, n_items = 10),
  from = 0, to = 0.1, lwd = 2, col = 2, ylim = c(3, 9),
  xlab = expression(theta), ylab = expression(log(E[theta](D))),
  main = "Log-expected Spearman distance")
curve(expected_spear_dist_vec(x, n_items = 20), add = TRUE, col = 3, lwd = 2)
curve(expected_spear_dist_vec(x, n_items = 30), add = TRUE, col = 4, lwd = 2)
legend("topright", legend = c(expression(n == 10), expression(n == 20), expression(n == 30)),
  col = 2:4, lwd = 2, bty = "n")

MLE of mixtures of Mallows models with Spearman distance via EM algorithms

Description

Perform the MLE of mixtures of Mallows model with Spearman distance on full and partial rankings via EM algorithms. Partial rankings with missing data in arbitrary positions are supported.

print method for class "emMSmix".

Usage

fitMSmix(
  rankings,
  n_clust = 1,
  n_start = 1,
  n_iter = 200,
  mc_em = FALSE,
  eps = 10^(-6),
  init = list(list(rho = NULL, theta = NULL, weights = NULL))[rep(1, n_start)],
  plot_log_lik = FALSE,
  comp_log_lik_part = FALSE,
  plot_log_lik_part = FALSE,
  parallel = FALSE,
  theta_max = 3,
  theta_tol = 1e-05,
  theta_tune = 1,
  subset = NULL,
  item_names = NULL
)

## S3 method for class 'emMSmix'
print(x, ...)

Arguments

Details

The EM algorithms are launched from n_start initializations and the best solution in terms of maximized log-likelihood value (based on full or augmented rankings) is returned.

When mc_em = FALSE, the scheme introduced by Crispino et al. (2023) is performed, where partial rankings are augmented with all compatible full rankings. This type of data augmentation is supported up to 10 missing positions in the partial rankings.

When mc_em = TRUE, the - computationally more efficient - Monte Carlo EM algorithm introduced by Crispino et al. (2025+) is implemented. In the case of a large number of censored positions and sample sizes, the mc_em = TRUE must be preferred.

Regardless of the fitting method adopted for inference on partial rankings, note that setting the argument comp_log_lik_part = TRUE for the computation of the observed-data log-likelihood values (based on partial rankings) can slow down the procedure in the case of a large number of censored positions and sample sizes.

Value

An object of class "emMSmix", namely a list with the following named components:

mod

List of named objects describing the best fitted model in terms of maximized log-likelihood over the n_start initializations. See Details.

max_log_lik

Maximized log-likelihood values for each initialization.

partial_data

Logical: whether the dataset includes some partially-ranked sequences.

convergence

Binary convergence indicators of the EM algorithm for each initialization: 1 = convergence has been achieved, 0 = otherwise.

record

Best log-likelihood values sequentially achieved over the n_start initializations.

em_settings

List of settings used to fit the model.

call

The matched call.

The mod sublist contains the following named objects:

rho

Integer G\timesn matrix with the MLEs of the component-specific consensus rankings in each row.

theta

Numeric vector with the MLEs of the G component-specific precision parameters.

weights

Numeric vector with the MLEs of the G mixture weights.

z_hat

Numeric N\timesG matrix of the estimated posterior component membership probabilities. Returned when n_clust > 1, otherwise NULL.

map_classification

Integer vector of N mixture component memberships based on the MAP allocation from the z_hat matrix. Returned when n_clust > 1, otherwise NULL.

log_lik

Numeric vector of the log-likelihood values (based on full or augmented rankings) at each iteration.

best_log_lik

Maximized log-likelihood value (based on full or augmented rankings) of the fitted model.

bic

BIC value of the fitted model based on best_log_lik.

log_lik_part

Numeric vector of the observed-data log-likelihood values (based on partial rankings) at each iteration. Returned when rankings contains some partial sequences that can be completed with data_augmentation and plot_log_lik_part = TRUE, otherwise NULL. See Details.

best_log_lik_part

Maximized observed-data log-likelihood value (based on partial rankings) of the fitted model. Returned when rankings contains some partial sequences that can be completed with data_augmentation, otherwise NULL. See Details.

bic_part

BIC value of the fitted model based on best_log_lik_part. Returned when rankings contains some partial sequences that can be completed with data_augmentation, otherwise NULL. See Details.

conv

Binary convergence indicator of the best fitted model: 1 = convergence has been achieved, 0 = otherwise.

augmented_rankings

Integer N\timesn matrix with rankings completed through the Monte Carlo step in each row. Returned when rankings contains some partial sequences and mc_em = TRUE, otherwise NULL.

References

Crispino M, Mollica C and Modugno L (2025+). MSmix: An R Package for clustering partial rankings via mixtures of Mallows Models with Spearman distance. (submitted)

Crispino M, Mollica C, Astuti V and Tardella L (2023). Efficient and accurate inference for mixtures of Mallows models with Spearman distance. Statistics and Computing, 33(98), DOI: 10.1007/s11222-023-10266-8.

Sørensen Ø, Crispino M, Liu Q and Vitelli V (2020). BayesMallows: An R Package for the Bayesian Mallows Model. The R Journal, 12(1), pages 324–342, DOI: 10.32614/RJ-2020-026.

Beckett LA (1993). Maximum likelihood estimation in Mallows’s model using partially ranked data. In Probability models and statistical analyses for ranking data, pages 92–107. Springer New York.

See Also

summary.emMSmix, plot.emMSmix

Examples

## Example 1. Fit the 3-component mixture of Mallows models with Spearman distance
## to the Antifragility dataset.
r_antifrag <- ranks_antifragility[, 1:7]
set.seed(123)
mms_fit <- fitMSmix(rankings = r_antifrag, n_clust = 3, n_start = 10)
mms_fit$mod$rho; mms_fit$mod$theta; mms_fit$mod$weights

## Example 2. Fit the Mallows model with Spearman distance
## to simulated partial rankings through data augmentation.
rank_data <- rbind(c(NA, 4, NA, 1, NA), c(NA, NA, NA, NA, 1), c(2, NA, 1, NA, 3),
                   c(4, 2, 3, 5, 1), c(NA, 4, 1, 3, 2))
mms_fit <- fitMSmix(rankings = rank_data, n_start = 10)
mms_fit$mod$rho; mms_fit$mod$theta

## Example 3. Fit the Mallows model with Spearman distance
## to the Reading genres dataset through Monte Carlo EM.
top5_read <- ranks_read_genres[, 1:11]
mms_fit <- fitMSmix(rankings = top5_read, n_start = 10, mc_em = TRUE)
mms_fit$mod$rho; mms_fit$mod$theta

(Log-)likelihood for mixtures of Mallows models with Spearman distance

Description

Compute the (log-)likelihood for the parameters of a mixture of Mallows models with Spearman distance on partial rankings. Partial rankings with missing data in arbitrary positions are supported.

Usage

likMSmix(
  rho,
  theta,
  weights = (if (length(theta) == 1) NULL),
  rankings,
  log = TRUE
)

Arguments

Details

The (log-)likelihood evaluation is performed by augmenting the partial rankings with the set of all compatible full rankings (see data_augmentation), and then the marginal likelihood is computed.

When n\leq 20, the (log-)likelihood is exactly computed. When n>20, the model normalizing constant is not available and is approximated with the method introduced by Crispino et al. (2023). If n>170, the approximation is also restricted over a fixed grid of values for the Spearman distance to limit computational burden.

Value

The (log)-likelihood value.

References

Crispino M, Mollica C and Modugno L (2025+). MSmix: An R Package for clustering partial rankings via mixtures of Mallows Models with Spearman distance. (submitted)

Crispino M, Mollica C, Astuti V and Tardella L (2023). Efficient and accurate inference for mixtures of Mallows models with Spearman distance. Statistics and Computing, 33(98), DOI: 10.1007/s11222-023-10266-8.

See Also

bicMSmix, aicMSmix, data_augmentation

Examples

## Example 1. Likelihood of a full ranking of n=5 items under the uniform (null) model.
likMSmix(rho = 1:5, theta = 0, weights = 1, rankings = c(3,5,2,1,4), log = FALSE)
# corresponds to...
1/factorial(5)

## Example 2. Simulate rankings from a 2-component mixture of Mallows models
## with Spearman distance.
set.seed(12345)
d_sim <- rMSmix(sample_size = 75, n_items = 8, n_clust = 2)
str(d_sim)
# Fit the true model.
rankings <- d_sim$samples
fit <- fitMSmix(rankings = rankings, n_clust = 2, n_start = 10)
# Compare log-likelihood values of the true parameter values and the MLE.
likMSmix(rho = d_sim$rho, theta = d_sim$theta, weights = d_sim$weights,
       rankings = d_sim$samples)
likMSmix(rho = fit$mod$rho, theta = fit$mod$theta, weights = fit$mod$weights,
       rankings = d_sim$samples)

## Example 3. Simulate rankings from a basic Mallows model with Spearman distance.
set.seed(12345)
d_sim <- rMSmix(sample_size = 25, n_items = 6)
str(d_sim)
# Censor data to be partial top-3 rankings.
rankings <- d_sim$samples
rankings[rankings>3] <- NA
# Fit the true model with data augmentation.
set.seed(12345)
fit <- fitMSmix(rankings = rankings, n_clust = 1, n_start = 10)
# Compare log-likelihood values of the true parameter values and the MLEs.
likMSmix(rho = d_sim$rho, theta = d_sim$theta, weights = d_sim$weights,
       rankings = d_sim$samples)
likMSmix(rho = fit$mod$rho, theta = fit$mod$theta, weights = fit$mod$weights,
       rankings = d_sim$samples)

Partition function of the Mallows model with Spearman distance

Description

Compute (either the exact or the approximate) (log-)partition function of the Mallows model with Spearman distance.

Usage

partition_fun_spear(theta, n_items, log = TRUE)

Arguments

Details

When n\leq 20, the partition function is exactly computed by relying on the Spearman distance distribution provided by OEIS Foundation Inc. (2023). When n>20, it is approximated with the method introduced by Crispino et al. (2023) and, if n>170, the approximation is also restricted over a fixed grid of values for the Spearman distance to limit computational burden.

The partition function is independent of the consensus ranking of the Mallows model with Spearman distance due to the right-invariance of the metric. When \theta=0, the partition function is equivalent to n!, which is the normalizing constant of the uniform (null) model.

Value

Either the exact or the approximate (log-)partition function of the Mallows model with Spearman distance.

References

Crispino M, Mollica C, Astuti V and Tardella L (2023). Efficient and accurate inference for mixtures of Mallows models with Spearman distance. Statistics and Computing, 33(98), DOI: 10.1007/s11222-023-10266-8.

OEIS Foundation Inc. (2023). The On-Line Encyclopedia of Integer Sequences, Published electronically at https://oeis.org.

See Also

spear_dist_distr, expected_spear_dist

Examples

## Example 1. Partition function of the uniform (null) model, coinciding with n!.
partition_fun_spear(theta = 0, n_items = 10, log = FALSE)
factorial(10)

## Example 2. Partition function of the Mallows model with Spearman distance.
partition_fun_spear(theta = 0.5, n_items = 10, log = FALSE)

## Example 3. Log-partition function of the Mallows model with Spearman distance
## as a function of theta.
partition_fun_spear_vec <- Vectorize(partition_fun_spear, vectorize.args = "theta")
curve(partition_fun_spear_vec(x, n_items = 10), from = 0, to = 0.1, lwd = 2,
  xlab = expression(theta), ylab = expression(log(Z(theta))),
  main = "Log-partition function of the Mallows model with Spearman distance",
  ylim = c(7, log(factorial(10))))

## Example 4. Log-partition function of the Mallows model with Spearman distance
## for varying number of items and values of the concentration parameter.
partition_fun_spear_vec <- Vectorize(partition_fun_spear, vectorize.args = "theta")
curve(partition_fun_spear_vec(x, n_items = 10),
  from = 0, to = 0.1, lwd = 2, col = 2,
  xlab = expression(theta), ylab = expression(log(Z(theta))),
  main = "Log-partition function of the Mallows model with Spearman distance",
  ylim = c(0, log(factorial(30))))
curve(partition_fun_spear_vec(x, n_items = 20), add = TRUE, col = 3, lwd = 2)
curve(partition_fun_spear_vec(x, n_items = 30), add = TRUE, col = 4, lwd = 2)
legend("topright", legend = c(expression(n == 10), expression(n == 20), expression(n == 30)),
  col = 2:4, lwd = 2, bty = "n")

Plot descriptive statistics for partial rankings

Description

plot method for class "data_descr".

Usage

## S3 method for class 'data_descr'
plot(
  x,
  cex_text_mean = 1,
  cex_symb_mean = 12,
  marg_by = "item",
  cex_text_pc = 3,
  cex_range_pc = c(8, 20),
  ...
)

Arguments

Details

The plots of the marginals distributions and pairwise comparisons are constructed if the object x was obtained from the data_description routine with arguments marg = TRUE and pc = TRUE; otherwise, a NULL element in the output list is returned.

Value

A list of 5 labelled plots displaying descriptive summaries of the partial ranking dataset, namely: i) n_ranked_distr: a barplot of the frequency distribution (%) of the number of items actually ranked in each partial sequence, ii) picto_mean_rank: a basic pictogram of the mean rank vector, iii) marginals: a heatmap of the marginal distributions (either by item or by rank), iv) ecdf: the ecdf of the marginal rank distributions and v) pc: a bubble plot of the pairwise comparison matrix.

References

Wickham H (2016). ggplot2: Elegant Graphics for Data Analysis. Springer-Verlag New York. ISBN 978-3-319-24277-4, https://ggplot2.tidyverse.org.

See Also

data_description

Examples

## Example 1. Plot the mean rank vector and marginal distributions for the Antifragility dataset.
r_antifrag <- ranks_antifragility[, 1:7]
desc <- data_description(r_antifrag)
p_desc <- plot(desc)
p_desc$picto_mean_rank()
p_desc$marginals()

## Example 2. Plot the distribution of the number of ranked items and the
# pairwise comparison matrix for the Sports dataset.
r_sports <- ranks_sports[, 1:8]
desc <- data_description(rankings = r_sports, borda_ord = TRUE)
p_desc <- plot(desc, cex_text_mean = 1.2)
p_desc$n_ranked_distr()
p_desc$pc()

## Example 3. Plot the ecdf's for the marginal rank distributions for the Sports dataset by gender.
r_sports <- ranks_sports[, 1:8]
desc_f <- data_description(rankings = r_sports, subset = (ranks_sports$Gender == "Female"))
p_desc_f <- plot(desc_f, cex_text_mean = 1.2)
p_desc_f$ecdf()
desc_m <- data_description(rankings = r_sports, subset = (ranks_sports$Gender == "Male"))
p_desc_m <- plot(desc_m, cex_text_mean = 1.2)
p_desc_m$ecdf()

Plot the Spearman distance matrix

Description

plot method for class "dist". It displays a heatmap which is useful to preliminarily explore the presence of patterns (groups) of similar preferences in the ranking dataset.

Usage

## S3 method for class 'dist'
plot(
  x,
  order = TRUE,
  show_labels = TRUE,
  lab_size = 3,
  gradient = list(low = "#00AFBB", mid = "white", high = "#FC4E07"),
  ...
)

Arguments

Details

plot.dist can visualize a distance matrix of any metric, provided that its class is "dist". It can take a few seconds if the size of the distance matrix is large.

The heatmap can be also obtained by setting the arguments rho = NULL and plot_dist_mat = TRUE when applying spear_dist.

Value

A heatmap of the Spearman distance matrix between all pairs of full rankings.

References

Alboukadel K and Mundt F (2020). factoextra: Extract and Visualize the Results of Multivariate Data Analyses. R package version 1.0.7. https://CRAN.R-project.org/package=factoextra

See Also

spear_dist

Examples

## Example 1. Plot the Spearman distance matrix of the Antifragility ranking dataset.
r_antifrag <- ranks_antifragility[, 1:7]
dist_mat <- spear_dist(rankings = r_antifrag)
plot(dist_mat, show_labels = FALSE)

## Example 2. Plot the Spearman distance matrix of the Sports ranking dataset.
r_sports <- ranks_sports[, 1:8]
dist_mat <- spear_dist(rankings = r_sports)
plot(dist_mat, show_labels = FALSE)
# Plot the Spearman distance matrix for the subsample of males.
dist_m <- spear_dist(rankings = r_sports, subset = (ranks_sports$Gender == "Male"))
plot(dist_m)

Plot the MLEs for the fitted mixture of Mallows models with Spearman distance

Description

plot method for class "emMSmix".

Usage

## S3 method for class 'emMSmix'
plot(x, max_scale_w = 20, mar_lr = 0.4, mar_tb = 0.2, ...)

Arguments

Value

A list of 2 labelled plots, namely: i) bump_plot: a bump plot comparing the component-specific consensus rankings of the fitted mixture of Mallows models with Spearman distance (the size of the dots of each consensus ranking is proportional to the weight of the corresponding component); and ii) est_clust_prob: a heatmap of the estimated component membership probabilities is returned when n_clust > 1, otherwise NULL.

References

Sjoberg D (2020). ggbump: Bump Chart and Sigmoid Curves. R package version 0.1.10. https://CRAN.R-project.org/package=ggbump.

Wickham H et al. (2019). Welcome to the tidyverse. Journal of Open Source Software, 4(43), 1686, DOI: 10.21105/joss.01686.

Wickham H (2016). ggplot2: Elegant Graphics for Data Analysis. Springer-Verlag New York. ISBN 978-3-319-24277-4, https://ggplot2.tidyverse.org.

See Also

fitMSmix, summary.emMSmix

Examples

## Example 1. Fit and plot a 3-component mixture of Mallows models with Spearman distance
## to the Antifragility dataset.
r_antifrag <- ranks_antifragility[, 1:7]
set.seed(123)
mms_fit <- fitMSmix(rankings = r_antifrag, n_clust = 3, n_start = 10)
p_mms_fit <- plot(mms_fit)
p_mms_fit$bump_plot()
p_mms_fit$est_clust_prob()

Print of the bootstrap confidence intervals for the fitted mixture of Mallows models with Spearman distance

Description

print method for class "bootMSmix".

Usage

## S3 method for class 'bootMSmix'
print(x, ...)

Arguments

Random samples from a mixture of Mallows models with Spearman distance

Description

Draw random samples of full rankings from a mixture of Mallows models with Spearman distance.

Usage

rMSmix(
  sample_size = 1,
  n_items,
  n_clust = 1,
  rho = NULL,
  theta = NULL,
  weights = NULL,
  uniform = FALSE,
  mh = TRUE
)

Arguments

Details

When n_items > 10 or mh = TRUE, the random samples are obtained by using the Metropolis-Hastings algorithm, described in Vitelli et al. (2018) and implemented in the sample_mallows function of the package BayesMallows package.

When theta = NULL, the concentration parameters are randomly generated from a uniform distribution on the interval (1/n^{2},3/n^{1.5}) containing typical values for the precisions.

When uniform = FALSE, the mixing weights are sampled from a symmetric Dirichlet distribution with shape parameters all equal to 2G, to favor populated and balanced clusters, and the consensus parameters are sampled to favor well-separated clusters, i. e., at least at Spearman distance equal to \frac{2}{G}\binom{n+1}{3} from each other.

Value

A list of the following named components:

References

Vitelli V, Sørensen Ø, Crispino M, Frigessi A and Arjas E (2018). Probabilistic Preference Learning with the Mallows Rank Model. Journal of Machine Learning Research, 18(158), pages 1–49, ISSN: 1532-4435, https://jmlr.org/papers/v18/15-481.html.

Sørensen Ø, Crispino M, Liu Q and Vitelli V (2020). BayesMallows: An R Package for the Bayesian Mallows Model. The R Journal, 12(1), pages 324–342, DOI: 10.32614/RJ-2020-026.

Chenyang Zhong (2021). Mallows permutation model with L1 and L2 distances I: hit and run algorithms and mixing times. arXiv: 2112.13456.

Examples

## Example 1. Drawing from a mixture with randomly generated parameters of separated clusters.
set.seed(12345)
rMSmix(sample_size = 50, n_items = 25, n_clust = 5)


## Example 2. Drawing from a mixture with uniformly generated parameters.
set.seed(12345)
rMSmix(sample_size = 100, n_items = 9, n_clust = 3, uniform = TRUE)


## Example 3.  Drawing from a mixture with customized parameters.
r_par <- rbind(1:5, c(4, 5, 2, 1, 3))
t_par <- c(0.01, 0.02)
w_par <- c(0.4, 0.6)
set.seed(12345)
rMSmix(sample_size = 50, n_items = 5, n_clust = 2, theta = t_par, rho = r_par, weights = w_par)

Antifragility Data (complete rankings with covariates)

Description

The Antifragility dataset came up from an on-line survey conducted during spring 2021 by Sapienza University of Rome in collaboration with the Italian incubator Digital Magics, to investigate the construct of antifragility in innovative startups. Antifragility reflects the capacity of a company to adapt and improve its activity in the case of stresses, volatility and disorders triggered by critical and unexpected events, such as the COVID-19 outbreak which motivated the survey. On the basis of their experience and knowledge, a sample of N=99 startups provided their complete rankings of n=7 desirable antifragility properties in order of importance. The antifragility features are: 1 = Absorption, 2 = Redundancy, 3 = Small stressors, 4 = Non-monotonicity, 5 = Requisite variety, 6 = Emergence and 7 = Uncoupling.

Usage

data(ranks_antifragility)

Format

A data frame gathering N=99 complete rankings of the n=7 antifragility features in each row (rank 1 = most preferred item). The definition of the antifragility aspects is detailed below:

Absorption

Ability to absorb stress and shocks while remaining in the planned state.

Redundancy

Overcapacity to defend from risks and prevent faults.

Small_stressors

Ability to exert low levels of stress on the organization.

Non_monotonicity

Capacity to learn from failures and errors.

Requisite_variety

Need for regulatory agents (i.e., government agency) to monitor and control organization’s outcomes and behaviors.

Emergence

Existence of cause-effect relationships between organization’s activity at micro level and its outcomes at macro level.

Uncoupling

Existence of strong interconnection between agents inside and outside the organization.

Industry_sector

Industry sector of the startup.

Market

Market type in which the startup operates.

Innovation_type

Main innovation type of the startup.

Approach_to_crisis

Main approach implemented by the startup during Covid-19 outbreak.

Crisis_impact

Impact of Covid-19 outbreak on the startup.

Age

Age of the startup (years).

N_employees

Number of employees in the startup.

Region

Italian region of the startup.

Job_title

Job title of the startup participant in the survey.

Experience

Years of job experience of the startup participant in the survey.

References

Ghasemi A and Alizadeh M (2017). Evaluating organizational antifragility via fuzzy logic. The case of an Iranian company producing banknotes and security paper. Operations research and decisions, 27(2), pages 21–43, DOI: 10.5277/ord170202.

Examples

str(ranks_antifragility)
head(ranks_antifragility)

Beers Data (partial rankings with covariate)

Description

The Beers dataset was collected through an on-line survey on beer preferences administrated to the participants of the 2018 Pint of Science festival held in Grenoble. A sample of N=105 respondents provided their partial rankings of n=20 beers according to their personal preferences. The dataset also includes a covariate concerning respondents' residence.

Usage

data(ranks_beers)

Format

A data frame gathering N=105 partial rankings of the beers in the first n=20 columns (rank 1 = most preferred item) and an individual covariate in the last column. The partial rankings are characterized by different censoring patterns (that is, not exclusively top sequences), with missing positions coded as NA. The variables are detailed below:

Stella

Rank assigned to Stella Artois.

Kwak

Rank assigned to Kwak Brasserie.

KronKron

Rank assigned to Kronenbourg (Kronenbourg).

Faro

Rank assigned to Faro Timmermans.

Kron1664

Rank assigned to 1664 (Kronenbourg).

Chimay

Rank assigned to Chimay Triple.

Pelforth

Rank assigned to Pelforth Brune.

KronCarls

Rank assigned to Carlsberg (Kronenbourg).

KronKanter

Rank assigned to Kanterbraeu (Kronenbourg).

Hoegaarden

Rank assigned to Hoegaarden Blanche.

Grimbergen

Rank assigned to Grimbergen Blonde.

Pietra

Rank assigned to Pietra Brasserie.

Affligem

Rank assigned to Affligem Brasserie.

Goudale

Rank assigned to La Goudale.

Leffe

Rank assigned to Leffe Blonde.

Heineken

Rank assigned to Heineken.

Duvel

Rank assigned to Duvel Brasserie.

Choulette

Rank assigned to La Choulette.

Orval

Rank assigned to Orval.

Karmeliet

Rank assigned to Karmeliet Triple.

Residence

Residence.

References

Crispino M (2018). On-line questionnaire of the 2018 Pint of Science festival in Grenoble available at https://docs.google.com/forms/d/1TiOIyc-jFXZF2Hb9echxZL0ZOcmr95LIBIieQuI-UJc/viewform?ts=5ac3a382&edit_requested=true.

Examples

str(ranks_beers)
head(ranks_beers)

Arkham Horror Data (complete rankings)

Description

The Arkham Horror dataset came up from an on-line survey conducted by Curtis Miller to investigate popularity of different player roles of Arkham Horror: The Card Game. A sample of N=241 respondents provided their complete rankings of n=5 player modes in order of preference. The player roles are: 1 = Survivor, 2 = Rogue, 3 = Guardian, 4 = Seeker, 5 = Mystic. Statistical analyses of these data can be found at the links provided in References.

Usage

data(ranks_horror)

Format

A data frame gathering N=421 complete rankings of the n=5 player roles in each row (rank 1 = most preferred item). The variables are detailed below:

Survivor

Rank assigned to Survivor role.

Rogue

Rank assigned to Rogue role.

Guardian

Rank assigned to Guardian role.

Seeker

Rank assigned to Seeker role.

Mystic

Rank assigned to Mystic role.

References

Curtis Miller's personal website: https://ntguardian.wordpress.com/2019/02/18/.

Examples

str(ranks_horror)
head(ranks_horror)

Reading Genres Data (partial rankings with covariates)

Description

The Reading Genres dataset was collected through an on-line survey conducted in Italy to investigate reading preferences in the context of the 2019 project Patto per la lettura – Conta chi legge. The questionnaire was administrated by the municipality of Latina (Latium, Italy), in collaboration with Sapienza University of Rome and the School of Government of the University of Tor Vergata. A sample of N=507 respondents provided their partial top-5 rankings of n=11 reading genres according to their personal preferences. The reading genres are: 1 = Classic, 2 = Novel, 3 = Thriller, 4 = Fantasy, 5 = Biography, 6 = Teenage, 7 = Horror, 8 = Comics, 9 = Poetry, 10 = Essay and 11 = Humor. The dataset also includes several covariates concerning respondents' socio-demographics characteristics and other free time activities.

Usage

data(ranks_read_genres)

Format

A data frame gathering N=507 partial top-5 rankings of the reading genres in the first n=11 columns (rank 1 = most preferred item) and individual covariates in the remaining columns. Missing positions are coded as NA. The variables are detailed below:

Classic

Rank assigned to Classic.

Novel

Rank assigned to Novel.

Thriller

Rank assigned to Thriller.

Fantasy

Rank assigned to Fantasy.

Biography

Rank assigned to Biography.

Teenage

Rank assigned to Teenage.

Horror

Rank assigned to Horror.

Comics

Rank assigned to Comics.

Poetry

Rank assigned to Poetry.

Essay

Rank assigned to Essay.

Humor

Rank assigned to Humor.

Gender

Gender.

Region

Italian region of residence.

Age

Age (years).

N_children

Number of children.

Education

Education level.

Final_mark

Final grade of the education degree, scaled in the interval [6,10].

N_books

Number of books read in the last 12 months.

References

Crispino M, Mollica C, Astuti V and Tardella L (2023). Efficient and accurate inference for mixtures of Mallows models with Spearman distance. Statistics and Computing, 33(98), DOI: 10.1007/s11222-023-10266-8.

Mollica (2019). On-line questionnaire of the Italian 2019 project Patto per la lettura – Conta chi legge available at https://form.jotformeu.com/90275118835359.

Examples

str(ranks_read_genres)
head(ranks_read_genres)

Sports Data (complete rankings with covariates)

Description

The Sports dataset was collected through an on-line questionnaire on sport preferences and habits administered within the 2016 Big Data Analystics in Sports (BDsports) project, developed by the Big and Open Data Innovation Laboratory (BODaI-Lab) of the University of Brescia. A sample of N=647 respondents provided their complete rankings of n=8 popular sports according to their personal preferences. The sports are: 1 = Soccer, 2 = Swimming, 3 = Volleyball, 4 = Cycling, 5 = Basket, 6 = Boxe and martial arts, 7 = Tennis and 8 = Jogging. The dataset also includes several covariates concerning respondents' socio-demographics characteristics and other sport-related information.

Usage

data(ranks_sports)

Format

A data frame gathering N=647 complete rankings of the sports in the first n=8 columns (rank 1 = most preferred item) and individual covariates in the remaining columns. The variables are detailed below:

Soccer

Rank assigned to Soccer.

Swimming

Rank assigned to Swimming.

Volleyball

Rank assigned to Volleyball.

Cycling

Rank assigned to Cycling.

Basket

Rank assigned to Basket.

Boxe_and_martial_arts

Rank assigned to Boxe and Martial Arts.

Tennis

Rank assigned to Tennis.

Jogging

Rank assigned to Jogging.

Gender

Gender.

Birth_month

Month of birth.

Birth_year

Year of birth.

Education

Education level.

Residence

Geographical area of residence.

Work

Type of work.

Smoking

Smoking status.

Sport_frequency

Number of times per week that the respondent plays sports.

Sport_hours

Number of hours per week that the respondent watches sports.

Sport_played

Sport played by the respondent.

Personality

Main aspect of respondent's personality.

Sport_motivation

Main reason why the respondent plays sport.

Sport_type

Favorite sport type.

Sport_relationships

Do you think that sport, especially in team games, allows you to make new friends?

Water

Quantity of water consumed per day.

Alcohol

Frequency of alcohol consumption.

Fastfood

Frequency of fast food consumption.

Food_supplements

Opinion about the use of food supplements in sports.

Massmedia

Prevalent mass media used to inquire about sport.

TV_space

Do you think that sport currently occupies the space it deserves on TV? Rating from 1=definitely not to 7=definitely yes with 4=indifferent.

Magazine_space

Do you think that sport currently occupies the space it deserves on the magazines? Rating from 1=definitely not to 7=definitely yes with 4=indifferent.

Radio_space

Do you think that sport currently occupies the space it deserves on the radio? Rating from 1=definitely not to 7=definitely yes with 4=indifferent.

Internet_space

Do you think that sport currently occupies the space it deserves on internet? Rating from 1=definitely not to 7=definitely yes with 4=indifferent.

Paid_channels

Do you think it is right that some sports are only accessible on paid channels? Rating from 1=definitely not to 7=definitely yes with 4=indifferent.

Subscriptions

Any past or current subscription to a sport magazine/channel.

Psycol_well_being

Do you think that practicing sports affects psychological well-being? Rating from 1=definitely not to 7=definitely yes with 4=indifferent.

Physical_well_being

Do you think that practicing sports affects physical well-being? Rating from 1=definitely not to 7=definitely yes with 4=indifferent.

Sport_nutrition

Do you think nutrition affects sport? Rating from 1=definitely not to 7=definitely yes with 4=indifferent.

Overall_health

Do you think that practicing sports affects health? Rating from 1=definitely not to 7=definitely yes with 4=indifferent.

Stress

Self-reported stress level on a scale between 0 and 100.

Economic_status

Level of satisfaction for one's own economic status: 0=not at all, 1=a little bit, 2=enough, 3=satisfied, 4=a lot. It is the only covariate with some NA's.

References

Simone, R., Cappelli, C. and Di Iorio, F., (2019). Modelling marginal ranking distributions: the uncertainty tree. Pattern Recognition Letters, 125, pages 278–288, DOI: 10.1016/j.patrec.2019.04.026.

Simone, R. and Iannario, M., (2018). Analysing sport data with clusters of opposite preferences. Statistical Modelling, 18(5-6), pages 505–524, DOI: 10.1177/1471082X18798455.

Examples

str(ranks_sports)
head(ranks_sports)

Arrange the output of MLE of mixtures of Mallows models with Spearman distance via EM algorithms

Description

Arrange the output of the object of class "emMSmix" according to a given relabelling of the mixture component labels.

Usage

rearrange_output_mix(output, ord)

Arguments

Spearman distance

Description

Compute either the Spearman distance between each row of a full ranking matrix and a reference complete ranking, or the Spearman distance matrix between all pairs of full rankings.

Usage

spear_dist(
  rankings,
  rho = NULL,
  subset = NULL,
  diag = FALSE,
  upper = FALSE,
  plot_dist_mat = FALSE
)

Arguments

Details

When rho = NULL, spear_dist recalls the dist function from the base package to compute the Spearman metric as squared Euclidian distance between all pairs of rows in rankings; otherwise, it recalls the compute_rank_distance routine of the BayesMallows package for the computation of the Spearman distance between each row in rankings and the full ranking rho.

Value

When rho = NULL, an object of class "dist" corresponding to the Spearman distance matrix; otherwise, a vector with the Spearman distances between each row in rankings and the full ranking rho.

References

Sørensen Ø, Crispino M, Liu Q and Vitelli V (2020). BayesMallows: An R Package for the Bayesian Mallows Model. The R Journal, 12(1), pages 324–342, DOI: 10.32614/RJ-2020-026.

See Also

plot.dist

Examples

## Example 1. Spearman distance between two full rankings.
spear_dist(rankings = c(4, 8, 6, 9, 2, 11, 3, 5, 1, 12, 7, 10), rho = 1:12)

## Example 2. Spearman distance between the Antifragility ranking dataset and the Borda ranking.
r_antifrag <- ranks_antifragility[, 1:7]
borda <- rank(data_description(rankings = r_antifrag)$mean_rank)
spear_dist(rankings = r_antifrag, rho = borda)

## Example 3. Spearman distance matrix of the Sports ranking dataset.
r_sports <- ranks_sports[, 1:8]
dist_mat <- spear_dist(rankings = r_sports)
dist_mat
# Spearman distance matrix for the subsample of females.
dist_f <- spear_dist(rankings = r_sports, subset = (ranks_sports$Gender == "Female"))
dist_f

Spearman distance distribution under the uniform ranking model

Description

Provide (either the exact or the approximate) frequency distribution of the Spearman distance under the uniform (null) ranking model.

Usage

spear_dist_distr(n_items, log = TRUE)

Arguments

Details

When n\leq 20, the exact distribution provided by OEIS Foundation Inc. (2023) is returned by relying on a call to the get_cardinalities routine of the BayesMallows package. When n>20, the approximate distribution introduced by Crispino et al. (2023) is returned. If n>170, the approximation is also restricted over a fixed grid of values for the Spearman distance to limit computational burden.

Value

A list of two named objects:

References

OEIS Foundation Inc. (2023). The On-Line Encyclopedia of Integer Sequences, Published electronically at https://oeis.org.

Crispino M, Mollica C, Astuti V and Tardella L (2023). Efficient and accurate inference for mixtures of Mallows models with Spearman distance. Statistics and Computing, 33(98), DOI: 10.1007/s11222-023-10266-8.

Sørensen Ø, Crispino M, Liu Q and Vitelli V (2020). BayesMallows: An R Package for the Bayesian Mallows Model. The R Journal, 12(1), pages 324–342, DOI: 10.32614/RJ-2020-026.

See Also

spear_dist, expected_spear_dist, partition_fun_spear

Examples

## Example 1. Exact Spearman distance distribution for n=20 items.
distr <- spear_dist_distr(n_items = 20, log = FALSE)
plot(distr$distances,distr$logcard,type="l",ylab="Frequency",xlab="d",
main='Distribution of the Spearman distance\nunder the null model')


## Example 2. Approximate Spearman distance distribution for n=50 items with log-frequencies.
distr <- spear_dist_distr(n_items = 50)
plot(distr$distances,distr$logcard,type="l",ylab="Log-frequency",xlab="d",
    main='Log-distribution of the Spearman distance\nunder the null model')

Summary of the fitted mixture of Mallows models with Spearman distance

Description

summary method for class "emMSmix".

print method for class "summary.emMSmix".

Usage

## S3 method for class 'emMSmix'
summary(object, digits = 3, ...)

## S3 method for class 'summary.emMSmix'
print(x, ...)

Arguments

Value

A list with the following named components:

See Also

fitMSmix, plot.emMSmix

Examples

## Example 1. Fit and summary of a 3-component mixture of Mallows models with Spearman distance
## for the Antifragility dataset.
r_antifrag <- ranks_antifragility[, 1:7]
set.seed(123)
mms_fit <- fitMSmix(rankings = r_antifrag, n_clust = 3, n_start = 10)
summary(mms_fit)

Variance of the Spearman distance

Description

Compute (either the exact or the approximate) (log-)variance of the Spearman distance under the Mallows model with Spearman distance.

Usage

var_spear_dist(theta, n_items, log = TRUE)

Arguments

Details

When n\leq 20, the variance is exactly computed by relying on the Spearman distance distribution provided by OEIS Foundation Inc. (2023). When n>20, it is approximated with the method introduced by Crispino et al. (2023) and, if n>170, the approximation is also restricted over a fixed grid of values for the Spearman distance to limit computational burden.

The variance of the Spearman distance is independent of the consensus ranking of the Mallows model with Spearman distance due to the right-invariance of the metric. When \theta=0, this is equal to \frac{n^2(n-1)(n+1)^2}{36}, which is the variance of the Spearman distance under the uniform (null) model.

Value

Either the exact or the approximate (log-)variance of the Spearman distance under the Mallows model with Spearman distance.

References

Crispino M., Mollica C., Astuti V. and Tardella L. (2023). Efficient and accurate inference for mixtures of Mallows models with Spearman distance. Statistics and Computing, 33(98), DOI: 10.1007/s11222-023-10266-8.

OEIS Foundation Inc. (2023). The On-Line Encyclopedia of Integer Sequences, Published electronically at https://oeis.org

Kendall, M. G. (1970). Rank correlation methods. 4th ed. Griffin London.

Examples

## Example 1. Variance of the Spearman distance under the uniform (null) model,
## coinciding with n^2(n-1)(n+1)^2/36.
n_items <- 10
var_spear_dist(theta = 0, n_items= n_items, log = FALSE)
n_items^2*(n_items-1)*(n_items+1)^2/36

## Example 2. Variance of the Spearman distance.
var_spear_dist(theta = 0.5, n_items = 10, log = FALSE)

## Example 3. Log-variance of the Spearman distance as a function of theta.
var_spear_dist_vec <- Vectorize(var_spear_dist, vectorize.args = "theta")
curve(var_spear_dist_vec(x, n_items = 10),
  from = 0, to = 0.1, lwd = 2, col = 2,
  xlab = expression(theta), ylab = expression(log(V[theta](D))),
  main = "Log-variance of the Spearman distance")

## Example 4. Log-variance of the Spearman distance for varying number of items
# and values of the concentration parameter.
var_spear_dist_vec <- Vectorize(var_spear_dist, vectorize.args = "theta")
curve(var_spear_dist_vec(x, n_items = 10),
  from = 0, to = 0.1, lwd = 2, col = 2, ylim = c(5, 14),
  xlab = expression(theta), ylab = expression(log(V[theta](D))),
  main = "Log-variance of the Spearman distance")
curve(var_spear_dist_vec(x, n_items = 20), add = TRUE, col = 3, lwd = 2)
curve(var_spear_dist_vec(x, n_items = 30), add = TRUE, col = 4, lwd = 2)
legend("topright", legend = c(expression(n == 10), expression(n == 20), expression(n == 30)),
  col = 2:4, lwd = 2, bty = "n")