# Bootstrap confidence intervals for the fitted mixture of Mallows models with Spearman distance 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"`. ```r 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 - `object`: An object of class `"emMSmix"` returned by `fitMSmix`. - `n_boot`: Number of desired bootstrap samples. Defaults to 50. - `type`: Character indicating which bootstrap method must be used. Available options are: `"non-parametric"` or `"parametric"` for the $G=1$ case, and `"soft"` or `"separated"` for the $G>1$ case. Defaults to `"non-parametric"` when `n_clust = 1` and to `"soft"` when `n_clust > 1`. See Details. - `conf_level`: Numeric: value in the interval (0,1] indicating the desired confidence level of the interval estimates. Defaults to 0.95. - `all`: Logical: whether the bootstrap samples of the MLEs for all the parameters must be returned. Defaults to `FALSE`. - `n_start`: Number of starting points for the MLE on each bootstrap sample. Defaults to 10. - `parallel`: Logical: whether parallelization over multiple initializations of the EM algorithm must be used. Used when `rankings` contains some partial rankings. Defaults to `FALSE`. - `x`: An object of class `"bootMSmix"` returned by `bootstrapMSmix`. - `...`: Further arguments passed to or from other methods (not used). ## Returns An object of class `"bootMSmix"`, namely a list with the following named components: - **`itemwise_ci_rho`**: Character $G$$x$$n$ matrix with the bootstrap itemwise confidence intervals for the component-specific consensus rankings. - **`ci_boot_theta`**: Numeric $G$$x$$2$ matrix with the bootstrap confidence intervals for the component-specific precisions. - **`ci_boot_weights`**: Numeric $G$$x$$2$ 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` $x$ `n_items` matrix with rows containing the bootstrap MLEs of a component-specific consensus ranking. - **`theta_boot`**: Numeric `n_boot`$x$ `n_clust` matrix with the bootstrap MLEs of the component-specific precision parameters in each row. - **`weights_boot`**: Numeric `n_boot`$x$ `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`. ## 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. ## Examples ```r ## 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) ``` ## 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**.

bootstrapMSmix function