skillspace.hierarchy function

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)

Arguments

• B: A matrix or a string containing restrictions of the hierarchy. If B is a $K \times K$ matrix containing where $K$

  denotes the number of skills, then B[ii,jj]=1 means that if an examinee mastered skill jj, then he or she should also master skill ii.

  Alternatively, a string can be also conveniently used for defining a hierarchy (see Examples).

• skill.names: Vector of names in skills

Details

The reduced skill space output can be used as an argument in din

or gdina to directly test for a hierarchy in attributes.

Returns

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 skillspace.complete which were removed for defining skillspace.reduced

References

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 Also

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.

Examples

#############################################################################
# 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
## End(Not run)