Influential or leverage units in ADEA
Fernando Fernandez-Palacin1
Manuel Munoz-Marquez2
2023-11-08
Introduction
An influential or leverage unit is one that produces significant changes in results. In this context, it refers to a unit that has a substantial impact on the model load.
For more information about loads see the help of the package or see (Fernandez-Palacin, Lopez-Sanchez, and Munoz-Marquez 2018) and (Villanueva-Cantillo and Munoz-Marquez 2021).
Let’s load and examine the tokyo_libraries dataset using the following code:
data(tokyo_libraries)
head(tokyo_libraries)
#> Area.I1 Books.I2 Staff.I3 Populations.I4 Regist.O1 Borrow.O2
#> 1 2.249 163.523 26 49.196 5.561 105.321
#> 2 4.617 338.671 30 78.599 18.106 314.682
#> 3 3.873 281.655 51 176.381 16.498 542.349
#> 4 5.541 400.993 78 189.397 30.810 847.872
#> 5 11.381 363.116 69 192.235 57.279 758.704
#> 6 10.086 541.658 114 194.091 66.137 1438.746Searching for influential units
The adea_load_leverage function searches for units that cause substantial changes in loads. The following call demonstrates this:
input <- tokyo_libraries[, 1:4]
output <- tokyo_libraries[, 5:6]
adea_load_leverage(input, output)
#> load load.diff DMUs
#> 1 0.6028718 0.14740482 23
#> 2 0.4004102 0.05505682 6The output reveals that units 23 and 6 produce changes greater than the default value for load.diff, which is set at 0.05. The output is sorted in decreasing order of “load.diff,” which represents the change in the load model.
While the previous calls only consider changes when removing units one by one, the ndel parameter allows for testing the removal of more than one unit at a time. The following call tests all combinations of two units:
adea_load_leverage(input, output, load.diff = 0.1, ndel = 2)
#> load load.diff DMUs
#> 1 0.8333337 0.3778667 9, 23
#> 2 0.6315800 0.1761130 12, 23
#> 3 0.6315800 0.1761130 10, 23
#> 4 0.6315800 0.1761130 15, 23
#> 5 0.6315800 0.1761130 4, 23
#> 6 0.6315800 0.1761130 11, 23
#> 7 0.6315800 0.1761130 22, 23
#> 8 0.6315800 0.1761130 16, 23
#> 9 0.6315800 0.1761130 14, 23
#> 10 0.6315800 0.1761130 18, 23
#> 11 0.6315800 0.1761130 20, 23
#> 12 0.6315800 0.1761130 3, 23
#> 13 0.6225027 0.1670357 2, 23
#> 14 0.6107273 0.1552603 7, 23
#> 15 0.6028718 0.1474048 23
#> 16 0.6020337 0.1465667 13, 23
#> 17 0.6010336 0.1455666 1, 23
#> 18 0.5980232 0.1425562 8, 23
#> 19 0.5879663 0.1324993 21, 23
#> 20 0.3334068 0.1220602 6, 9
#> 21 0.3430363 0.1124307 5, 6
#> 22 0.5599886 0.1045216 17, 23This results in a long list, and to limit the number of groups in the output, you can set nmax to a specific value, as demonstrated in the following call:
adea_load_leverage(input, output, load.diff = 0.1, ndel = 2, nmax = 10)
#> load load.diff DMUs
#> 1 0.8333337 0.3778667 9, 23
#> 2 0.6315800 0.1761130 12, 23
#> 3 0.6315800 0.1761130 10, 23
#> 4 0.6315800 0.1761130 15, 23
#> 5 0.6315800 0.1761130 4, 23
#> 6 0.6315800 0.1761130 11, 23
#> 7 0.6315800 0.1761130 22, 23
#> 8 0.6315800 0.1761130 16, 23
#> 9 0.6315800 0.1761130 14, 23
#> 10 0.6315800 0.1761130 18, 23It’s important to note that the best option for removing two units is not the same as removing the two units individually in the one-by-one analysis. This discrepancy arises due to interactions between the effects of the units.
From this point forward, decision-makers or researchers must handle these units carefully to avoid biases in DEA results.
Each call to adea_load_leverage requires solving a large linear program, making it computationally demanding and potentially time-consuming. Patience is essential when working with this function.
References
Fernandez-Palacin, Fernando, Marı́a Auxiliadora Lopez-Sanchez, and Manuel Munoz-Marquez. 2018. “Stepwise selection of variables in DEA using contribution loads.” Pesquisa Operacional 38 (1): 31–52. http://dx.doi.org/10.1590/0101-7438.2018.038.01.0031.
Villanueva-Cantillo, Jeyms, and Manuel Munoz-Marquez. 2021. “Methodology for Calculating Critical Values of Relevance Measures in Variable Selection Methods in Data Envelopment Analysis.” European Journal of Operational Research 290 (2): 657–70. https://doi.org/10.1016/j.ejor.2020.08.021.
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Universidad de Cádiz, fernando.fernandez@uca.es↩︎
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Universidad de Cádiz, manuel.munoz@uca.es↩︎