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Influential or leverage units in ADEA

Fernando Fernandez-Palacin¹

Manuel Munoz-Marquez²

2022-02-09

Introduction

An influential or leverage units is a one that produce a big changes in results. In this case is a unit that produce a big change in model load.

For more information about loads help of the package about adea or see (Fernandez-Palacin, Lopez-Sanchez, and Munoz-Marquez 2018) and (Villanueva-Cantillo and Munoz-Marquez 2021).

Let’s load and have a look at the tokyo_libraries dataset with

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  1438.746

Searching for influential units

adea_load_leverage function looks for units that produce higher change in loads. The following call

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    6

shows that units 23 and 6 produce changes greater than the default value for load.diff which is 0.05.

The output is sorted in decreasing order of load.diff which is the change in load model.

Those call only considers changes taking units one by one, but using ndel parameter remove of more than one unit at the same time can be tested. The following call tests all groups 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 20, 23
#> 3  0.6315800 0.1761130 15, 23
#> 4  0.6315800 0.1761130  4, 23
#> 5  0.6315800 0.1761130 12, 23
#> 6  0.6315800 0.1761130 14, 23
#> 7  0.6315800 0.1761130 16, 23
#> 8  0.6315800 0.1761130 18, 23
#> 9  0.6315800 0.1761130 22, 23
#> 10 0.6315800 0.1761130 11, 23
#> 11 0.6315800 0.1761130 10, 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, 23

This results in a very long list, so the number or groups in output can be limited, for example, to 10, as 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 20, 23
#> 3  0.6315800 0.1761130 15, 23
#> 4  0.6315800 0.1761130  4, 23
#> 5  0.6315800 0.1761130 12, 23
#> 6  0.6315800 0.1761130 14, 23
#> 7  0.6315800 0.1761130 16, 23
#> 8  0.6315800 0.1761130 18, 23
#> 9  0.6315800 0.1761130 22, 23
#> 10 0.6315800 0.1761130 11, 23

This shows that the best option to remove two units is not the same as remove the two firsts in the one by one analysis. This is because that there are interactions between the units effects.

From this point, decision maker or researcher have to handle this units properly, to avoid biases in DEA results.

Each call to adea_load_leverage requires to solve a big set of a large linear program, so is a very demanding of computation resource, and can require a very large time, so be patient.

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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