CDM (version 2.7-7)

skillspace.hierarchy: Creation of a Hierarchical Skill Space

Description

Defines a reduced skill space for hierarchies in skills (see e.g. Leighton, Gierl, & Hunka, 2004).

Usage

skillspace.hierarchy(B, 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

skill.names

Vector of names in skills

Value

• 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

Details

The reduced skill space output can be used as an argument in din or gdina to directly test for a hierarchy in attributes.

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.

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 <- 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 <- 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 <- 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 <- 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 <- skillspace.hierarchy( B=B , skill.names=skill.names )
sp1b$skillspace.reduced

#*** Model 2:  A1 > A2 and A1 > A3
B <- "A1 > A2
      A1 > A3"      
sp2 <- skillspace.hierarchy( B=B , skill.names=skill.names )
sp2$skillspace.reduced

#*** Model 3: A1 > A3
B <- "A1 > A3"
sp3 <- skillspace.hierarchy( B=B , skill.names=skill.names )
sp3$skillspace.reduced

#############################################################################
# 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 <- 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 <- skillspace.hierarchy( B=B , skill.names=skill.names )
sp2$skillspace.reduced

#*** Model 3: Divergent hierarchy (C)
B <- "A1 > A2 > A3
      A1 > A4 > A5
      A3 > A5 > A6
      A5 > A6"
sp3 <- 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 <- skillspace.hierarchy( B=B , skill.names=skill.names )
sp4$skillspace.reduced