Introduction
Variable selection in Data Envelopment Analysis (DEA) is a crucial
consideration that requires careful attention before the results of an
analysis can be applied in a real-world context. This is because the
outcomes can vary significantly depending on the variables included in
the model. Therefore, variable selection is a fundamental step in every
DEA application.
One of the methods proposed for this is what is known as the feature
selection method. This method constructs a linear programming problem to
maximize some objective function related to the dmu efficiencies.
fsdea function implements the feature selection method
in the article (Benítez-Peña, Bogetoft, and
Romero Morales 2020)
Let’s load and inspect 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 1438.746
Lets’t take inputs as: Area.I1, Books.I2,
Staff.I3 and Populations.I4variables and
outputs as Regist.O1 and Borrow.O2
input <- tokyo_libraries[, 1:4]
output <- tokyo_libraries[, 5:6]
First, let’s do a standard DEA analysis with:”
dea <- dea(input, output)
dea
#> [1] 0.3500108 0.7918292 0.5733000 0.7186833 1.0000000 1.0000000 0.6967419
#> [8] 0.5803315 1.0000000 0.7051438 0.5689146 0.7583527 0.7474946 0.7215430
#> [15] 0.8440736 0.5822710 1.0000000 0.7867065 1.0000000 0.8485716 0.7872304
#> [22] 0.7849437 1.0000000
Summarizing DEA, we can calculate the average efficiency:
summary(dea)
#>
#> Model name
#> Orientation input
#> Inputs Area.I1 Books.I2 Staff.I3 Populations.I4
#> Outputs Regist.O1 Borrow.O2
#> nInputs 4
#> nOutputs 2
#> nVariables 6
#> nEfficients 6
#> Eff. Mean 0.775919227646031
#> Eff. sd 0.174702408743164
#> Eff. Min. 0.350010840234134
#> Eff. 1st Qu. 0.700942885344481
#> Eff. Median 0.784943740381793
#> Eff. 3rd Qu. 0.924285790399849
#> Eff. Max. 1
Reducing the number of variables
Suposse that we want a model with only 5 variables, then the
following call does the job:
dea5v <- fsdea(input, output, nvariables = 5)
dea5v
#> [1] 0.3428216 0.7918292 0.5733000 0.7164871 1.0000000 1.0000000 0.6967419
#> [8] 0.5803315 1.0000000 0.7051438 0.5689146 0.7583527 0.7219103 0.7215430
#> [15] 0.8440736 0.5822710 1.0000000 0.7867065 1.0000000 0.8485716 0.7872304
#> [22] 0.7849437 1.0000000
#> Selected inputs : Books.I2, Staff.I3, Populations.I4
#> Selected outputs: Regist.O1, Borrow.O2
We can calculate the average efficiency for the new model by
summarizing it:
summary(dea5v)
#>
#> Model name
#> Orientation input
#> nInputs 4
#> nOutputs 2
#> nVariables 6
#> nEfficients 6
#> Eff. Mean 0.77439880335151
#> Eff. sd 0.175803112249109
#> Eff. Min. 0.34282160923055
#> Eff. 1st Qu. 0.700942885344481
#> Eff. Median 0.784943740381793
#> Eff. 3rd Qu. 0.924285790399849
#> Eff. Max. 1
#> iSelected Books.I2, Staff.I3, Populations.I4
#> oSelected Regist.O1, Borrow.O2
Observe that average efficiency has decreased from 0.7849437 to
0.7849437. This could indicate that the new model is not an improvement
over the previous one.
To delve deeper into the results, we can plot the efficiencies: 
The graph reveals that most efficiencies are very similar, a fact
that is confirmed by the correlation coefficient 0.9995329 which is very
close to 1.
Reducing the number of outputs
In the previous case, reducing the number of variables led to a
decrease in one input. We achieved the same result with the call
fsdea(input, output, ninputs = 3). However, perhaps our
goal is to reduce an output. Let’s achieve this with the following
call:
dea1o <- fsdea(input, output, noutputs = 1)
dea1o
#> [1] 0.3026132 0.6425505 0.5733000 0.7186833 0.6733832 1.0000000 0.6967419
#> [8] 0.4476942 1.0000000 0.7051438 0.5336592 0.7583527 0.5915395 0.7215430
#> [15] 0.7832606 0.5822710 0.8451129 0.7867065 1.0000000 0.8485716 0.7387768
#> [22] 0.7849437 1.0000000
#> Selected inputs : Area.I1, Books.I2, Staff.I3, Populations.I4
#> Selected outputs: Borrow.O2
Observe that, again, average efficiency has decreased from 0.7849437
to 0.7276021.
To delve deeper into the results, we can plot the new efficiencies:
In this case, the differences are more significant compared to the
previous two models. This is confirmed by a lower correlation
coefficient 0.8910575.
All these could indicate that to enhance the model, it may be
necessary to remove multiple variables simultaneously.