Competency Deletion Procedure (CDP)

Description

The competency deletion procedure (CDP) reduces a set of competencies defined on a set of skills down to one of its possible reducts.

Usage

cdp(Competencies, CS, prefRel = c("NSP", "MBP", "MSP", "FSP"), 
    DesignMat = NULL, verbose = FALSE)

Arguments

Competencies	a character vector representing the set of competencies which is to be reduced.
CS	a state-by-skill indicator matrix representing the competence structure. An element equals 1 if the skill is present in the state; otherwise, it equals 0.
prefRel	the deletion rule that determines the order in which competencies are deleted. It can either be a string defining one of the preference relations defined in Anselmi et al. (2024) – except for "LNP" –, or an object of the class DeletionRule, which is a character vector containing all collections of equally preferred competencies as elements, ordered from the most to the least preferred collection.
DesignMat	an indicator matrix with the dimensions nrow(CS) \(\times\) nrow(CS). An element equals 1 if the corresponding pair of two competence states shall be discerned during the procedure; otherwise, it equals 0. Alternatively NULL can be chosen instead.
verbose	an argument deciding if the values delta, Competencies, CS and DesingMat are printed in the output object.

Details

See Anselmi et al. (2022) for details on the competency deletion procedure (CDP) for competence structures.

See Anselmi et al. (2024) for details on the preference relations for the competency deletion procedure (CDP) for competence structures.

Using a manually defined preference relation for the argument prefRel requires a character vector with its class attribute set to DeletionRule. See examples.

If NULL is used for the argument DesignMat, all possible pairs of competence states (with respect to CS) are discerned.

Value

An object of class cdp having the following components:

reduct	the vector of the final set of competencies which is a reduct.
niter	the number of deletions/iterations needed.
ordered_deletions	the competencies which were deleted, ordered by the iteration in which they were deleted.
delta	a matrix of the discerned pairs of competence states in rows and the competencies from the reduct in columns. An entry in a row means that this pair of competence states can be discerned by the competency in which column is the entry.
pref_rel	the preference relation used as DeletionRule object.
pref_rel_method	the method used to create the preference relation.
Competencies, CS, DesignMat	See Arguments.

References

Anselmi, P., Heller, J., Stefanutti, L. & Robusto, E. (2022). Constructing, improving, and shortening tests for skill assessment. Journal of Mathematical Psychology, 106, 102621. https://doi.org/10.1016/j.jmp.2021.102621

Anselmi, P., Heller, J., Stefanutti, L., & Robusto, E. (2024). Constructing tests for skill assessment with competence-based test development. British Journal of Mathematical and Statistical Psychology, 77(3), 429-458. https://doi.org/10.1111/bmsp.12335

See Also

delineate.

Examples

library("pks")

## Create competence structure
CStr <- as.binmat(c("0000", "0100", "1100", "1010", "1110", "0111", "1111"))
rownames(CStr) <- as.pattern(CStr)

## Create competency set
T_ <- c("a", "b", "c", "ab", "ac", "ad", "bc", "bd", "cd", "abc", "abd",
        "acd", "bcd", "abcd")

## Create design matrices
DesignMat <- upper.tri(matrix(0, nrow(CStr), nrow(CStr))) + 0
colnames(DesignMat) <- rownames(DesignMat) <-
  as.pattern(CStr, useNames = TRUE)

DesignMat2 <- DesignMat
DesignMat2[1, ] <- c(0, 0, 0, 1, 1, 1, 1)
DesignMat2[3, ] <- c(0, 0, 0, 1, 1, 0, 0)

## Create manually defined preference relation
ManDef_PR <- c("a, ac, ad, abc, abd, abcd",
               "b, bc, cd, acd", 
               "ab, bd, bcd",
               "c") 
class(ManDef_PR) <- "DeletionRule"

## Classic examples from Anselmi et al. (2024)

## Basic version using preference relation "NSP"
cdp(Competencies = T_, CS = CStr, prefRel = "NSP")

Competency Deletion Procedure (CDP) Results

Achieved Reduct:
a b cd abc

Number of Iterations: 10
Deleted Competencies in chronological order:
c bd abd abcd bc bcd ac ad ab acd

Chosen method for creating the preference relation: NSP
Preference Relation:
NSP does not use a Preference Relation

## Larger output option and preference relation "FSP"
cdp(Competencies = T_, CS = CStr, prefRel = "FSP", verbose = TRUE)

Competency Deletion Procedure (CDP) Results

Achieved Reduct:
a b c ad

Number of Iterations: 10
Deleted Competencies in chronological order:
abcd abd acd bcd abc cd bd ab ac bc

Chosen method for creating the preference relation:
                      FSP
Preference Relation:
a, b, c
ab, ac, ad, bc, bd, cd
abc, abd, acd, bcd
abcd


Bag Matrix (Delta)
         a b c ad
{}/ab    1 1 0  0
{}/abc   1 1 1  0
{}/abcd  1 1 1  1
{}/ac    1 0 1  0
{}/b     0 1 0  0
{}/bcd   0 1 1  0
ab/abc   0 0 1  0
ab/abcd  0 0 1  1
ab/ac    0 1 1  0
ab/bcd   1 0 1  0
abc/abcd 0 0 0  1
abc/bcd  1 0 0  0
ac/abc   0 1 0  0
ac/abcd  0 1 0  1
ac/bcd   1 1 0  0
b/ab     1 0 0  0
b/abc    1 0 1  0
b/abcd   1 0 1  1
b/ac     1 1 1  0
b/bcd    0 0 1  0
bcd/abcd 1 0 0  1


Used Competencies (T):
a b c ab ac ad bc bd cd abc abd acd bcd abcd

Used Competence Structure (CS):
     a b c d
0000 0 0 0 0
0100 0 1 0 0
1100 1 1 0 0
1010 1 0 1 0
1110 1 1 1 0
0111 0 1 1 1
1111 1 1 1 1

Used Design Matrix:
     {} b ab ac abc bcd abcd
{}    0 1  1  1   1   1    1
b     0 0  1  1   1   1    1
ab    0 0  0  1   1   1    1
ac    0 0  0  0   1   1    1
abc   0 0  0  0   0   1    1
bcd   0 0  0  0   0   0    1
abcd  0 0  0  0   0   0    0

## Classic complete design matrix and preference relation "MBP"
cdp(Competencies = T_, CS = CStr, prefRel = "MBP", DesignMat = DesignMat)

Competency Deletion Procedure (CDP) Results

Achieved Reduct:
b c ab cd

Number of Iterations: 10
Deleted Competencies in chronological order:
abd acd ad abcd bcd abc bd bc ac a

Chosen method for creating the preference relation:
                      MBP
Preference Relation:
a, c, ab, ac, bc
b, bd, cd, abc, bcd
ad, abd, acd, abcd

## Reduced design matrix and preference relation "MSP" 
cdp(Competencies = T_, CS = CStr, prefRel = "MSP", DesignMat = DesignMat2)

Competency Deletion Procedure (CDP) Results

Achieved Reduct:
ab ac bcd

Number of Iterations: 11
Deleted Competencies in chronological order:
a c b bd bc cd ad abd acd abc abcd

Chosen method for creating the preference relation:
                      MSP
Preference Relation:
abcd
abc, abd, acd, bcd
ab, ac, ad, bc, bd, cd
a, b, c

## Manually defined preference relation 
cdp(Competencies = T_, CS = CStr, prefRel = ManDef_PR)

Competency Deletion Procedure (CDP) Results

Achieved Reduct:
a b ad bc

Number of Iterations: 10
Deleted Competencies in chronological order:
c bd bcd ab acd cd abd abc abcd ac

Chosen method for creating the preference relation: 
        manually defined
Preference Relation:
a, ac, ad, abc, abd, abcd
b, bc, cd, acd
ab, bd, bcd
c