Creation of a Hierarchical Skill Space
The function skillspace.hierarchy defines a reduced skill space
for hierarchies in skills (see e.g. Leighton, Gierl, & Hunka, 2004).
The function skillspace.full defines a full skill space
for dichotomous skills.
skillspace.hierarchy(B, skill.names) skillspace.full(skill.names)
B |
A matrix or a string containing restrictions of the hierarchy.
If Alternatively, a string can be also conveniently used for defining a hierarchy (see Examples). |
skill.names |
Vector of names in skills |
A list with following entries
R |
Reachability matrix |
skillspace.reduced |
Reduced skill space fulfilling the specified hierarchy |
skillspace.complete |
Complete skill space |
zeroprob.skillclasses |
Indices of skill patterns in
|
Leighton, J. P., Gierl, M. J., & Hunka, S. M. (2004). The attribute hierarchy method for cognitive assessment: A variation on Tatsuoka's rule space approach. Journal of Educational Measurement, 41, 205-237.
See din (Example 6) for an application of
skillspace.hierarchy for model comparisons.
See the GDINA::att.structure function in the
GDINA package for similar functionality.
#############################################################################
# EXAMPLE 1: Toy example with 3 skills
#############################################################################
K <- 3 # number of skills
skill.names <- paste0("A", 1:K ) # names of skills
# create a zero matrix for hierarchy definition
B0 <- 0*diag(K)
rownames(B0) <- colnames(B0) <- skill.names
#*** Model 1: A1 > A2 > A3
B <- B0
B[1,2] <- 1 # A1 > A2
B[2,3] <- 1 # A2 > A3
sp1 <- CDM::skillspace.hierarchy( B=B, skill.names=skill.names )
sp1$skillspace.reduced
## A1 A2 A3
## 1 0 0 0
## 2 1 0 0
## 4 1 1 0
## 8 1 1 1
#*** Model 2: A1 > A2 and A1 > A3
B <- B0
B[1,2] <- 1 # A1 > A2
B[1,3] <- 1 # A1 > A3
sp2 <- CDM::skillspace.hierarchy( B=B, skill.names=skill.names )
sp2$skillspace.reduced
## A1 A2 A3
## 1 0 0 0
## 2 1 0 0
## 4 1 1 0
## 6 1 0 1
## 8 1 1 1
#*** Model 3: A1 > A3, A2 is not included in a hierarchical way
B <- B0
B[1,3] <- 1 # A1 > A3
sp3 <- CDM::skillspace.hierarchy( B=B, skill.names=skill.names )
sp3$skillspace.reduced
## A1 A2 A3
## 1 0 0 0
## 2 1 0 0
## 3 0 1 0
## 4 1 1 0
## 6 1 0 1
## 8 1 1 1
#~~~ Hierarchy specification using strings
#*** Model 1: A1 > A2 > A3
B <- "A1 > A2
A2 > A3"
sp1 <- CDM::skillspace.hierarchy( B=B, skill.names=skill.names )
sp1$skillspace.reduced
# Model 1 can be also written in one line for B
B <- "A1 > A2 > A3"
sp1b <- CDM::skillspace.hierarchy( B=B, skill.names=skill.names )
sp1b$skillspace.reduced
#*** Model 2: A1 > A2 and A1 > A3
B <- "A1 > A2
A1 > A3"
sp2 <- CDM::skillspace.hierarchy( B=B, skill.names=skill.names )
sp2$skillspace.reduced
#*** Model 3: A1 > A3
B <- "A1 > A3"
sp3 <- CDM::skillspace.hierarchy( B=B, skill.names=skill.names )
sp3$skillspace.reduced
## Not run:
#############################################################################
# EXAMPLE 2: Examples from Leighton et al. (2004): Fig. 1 (p. 210)
#############################################################################
skill.names <- paste0("A",1:6) # 6 skills
#*** Model 1: Linear hierarchy (A)
B <- "A1 > A2 > A3 > A4 > A5 > A6"
sp1 <- CDM::skillspace.hierarchy( B=B, skill.names=skill.names )
sp1$skillspace.reduced
#*** Model 2: Convergent hierarchy (B)
B <- "A1 > A2 > A3
A2 > A4
A3 > A5 > A6
A4 > A5 > A6"
sp2 <- CDM::skillspace.hierarchy( B=B, skill.names=skill.names )
sp2$skillspace.reduced
#*** Model 3: Divergent hierarchy (C)
B <- "A1 > A2 > A3
A1 > A4 > A5
A1 > A4 > A6"
sp3 <- CDM::skillspace.hierarchy( B=B, skill.names=skill.names )
sp3$skillspace.reduced
#*** Model 4: Unstructured hierarchy (D)
B <- "A1 > A2 \n A1 > A3 \n A1 > A4 \n A1 > A5 \n A1 > A6"
# This specification of B is equivalent to writing separate lines:
# B <- "A1 > A2
# A1 > A3
# A1 > A4
# A1 > A5
# A1 > A6"
sp4 <- CDM::skillspace.hierarchy( B=B, skill.names=skill.names )
sp4$skillspace.reduced
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