ci_rbod_mdir: Robust multi-directional Benefit of the Doubt approach...

In Compind: Composite Indicators Functions

Robust multi-directional Benefit of the Doubt approach (MDRBoD)

Description

Robust Multi-directional Benefit of the Doubt (MDRBoD) allows to introduce the non-compensability among simple indicators in a standard Robust BOD in an objective manner: the preference structure, i.e., the direction, is determined directly from the data and is specific for each unit and these estimated values are calculated as the reference sample varies in order to smooth out the effect of outliers or out-of-range data.

Usage

ci_rbod_mdir(x,indic_col,M, B, interval)

Arguments

x	A data.frame containing simple indicators.
indic_col	A numeric list indicating the positions of the simple indicators.
M	The number of elements in each of the bootstrapped samples (M lower than the number of units).
B	The number of bootstrap replicates.
interval	Significance level of the confidence intervals of the estimated scores; default = 0.05, i.e. 95% confidence intervals.

Details

At each replicate b = 1, \dots, B a subsample of M units is drawn with replacement and every unit is evaluated against the order-m frontier spanned by the subsample, as in Vidoli et al. (2024), eq. 3-6: the potential improvements of each simple indicator define the unit-specific direction and \beta \in [0,1] measures the proportion of the improvement needed to reach the frontier (the evaluated unit is included in its own reference set, so that a unit dominating the subsample lies on the frontier, \beta = 0). Directions and \beta are averaged over the B replicates and the composite indicator is computed with the averaged quantities (eq. 7-10); scores are therefore bounded by 1 and converge to ci_bod_mdir as M grows.

Two confidence intervals are returned. conf follows eq. 11 (Student-t, \pm t \cdot s/\sqrt{B}, where s is the standard deviation of the replicate scores), centred on the estimated score: it measures the Monte Carlo precision of the estimate and narrows as B increases. conf_perc contains the percentile interval of the replicate scores, which describes the variability of the score with respect to the reference subsample and does not shrink with B.

Value

An object of class "CI". This is a list containing the following elements:

ci_rbod_mdir_est	Composite indicator estimated values (eq. 10).
conf	lower_ci and upper_ci; Student-t confidence interval centred on the estimated values (eq. 11).
conf_perc	lower_ci and upper_ci; percentile interval of the replicate scores.
ci_rbod_mdir_spec	Simple indicators specific scores (eq. 9).
ci_rbod_mdir_dir	Directions for each simple indicator and unit, averaged over the replicates (eq. 7).
ci_rbod_mdir_beta	Proportion of the improvement needed to reach the frontier, averaged over the replicates (eq. 8).
ci_rbod_mdir_boot	Matrix (units x replicates) of the replicate scores.
ci_method	Method used; for this function ci_method="rbod_mdir".

Author(s)

Vidoli F.

References

F. Vidoli, E. Fusco, G. Pignataro, C. Guccio (2024) "Multi-directional Robust Benefit of the Doubt model: An application to the measurement of the quality of acute care services in OECD countries", Socio-Economic Planning Sciences. https://doi.org/10.1016/j.seps.2024.101877

See Also

ci_rbod, ci_bod_mdir

Examples

data(BLI_2017)
CI <- ci_rbod_mdir(BLI_2017, c(2:5), M = 10, B = 10, interval = 0.05)
CI$ci_rbod_mdir_est
CI$conf_perc

CI <- ci_rbod_mdir(BLI_2017, c(2:12), M = 6, B = 20, interval = 0.05)

Compind documentation built on Oct. 6, 2026, 5:07 p.m.