Validates prerequisite relations or knowledge structures
kvalidate(x, rpatterns=NULL, method=c("gamma","percent","VC","DI","DA"))Depending on the desired assessment method, a data frame with results for
each domain problem (method "percent"), or a list
(methods "gamma", "VC", "DI" and "DA") with
the following components:
The gamma-value.
Number of concordant pairs.
Number of discordant pairs.
for the "gamma" method,
The VC-value.
Number of discordant pairs.
for the "VC" method,
The DI-value.
The distance table for DI.
for the "DI" method, and
The ddat-value.
The distance table for ddat.
The dpot-value.
The distance table for dpot.
The Distance Agreement Coefficient.
for the "DA" nethod.
An R object of class kstructure.
A binary data frame or matrix where each row specifies the response pattern of one individual to the set of domain problems in x.
The desired validation method (see details).
kvalidate calculates validity coefficients for prerequisite
relations and knowledge structures.
The \(\gamma\)-Index (method "gamma") validates
the prerequisite relation underlying a knowledge structure and assumes
that not every response pattern is represented by a prerequisite relation.
For this purpose it compares the number of response patterns that are
represented by a prerequisite relation (i.e., concordant pairs) with the
number of response patterns that are not represented by a prerequisite
relation (i.e., discordant pairs). Formally, the \(\gamma\)-Index
is defined as $$\gamma = \frac{N_c - N_d}{N_c + N_d}$$
where \(N_c\) is the number of concordant pairs and \(N_d\)
the number of discordant pairs. Generally, a positive \(\gamma\)-value
supports the validity of prerequisite relations.
The validation method "percent" likewise validates prerequisite
relations and assumes that more difficult or complex domain problems are
solved less frequently than less difficult or complex domain problems.
For this purpose it calculates the relative solution frequency for each
of the domain problems in Q.
The Violational Coefficient (method "VC") also validates
prerequisite relations. For this purpose, the number of violations
(i.e., the earlier mentioned discordant pairs) against a prerequisite
relation are calculated. Formally, the VC is defined as
$$VC = \frac{1}{n(|S| - m)} \sum_{x,y} v_{xy}$$
where \(n\) denotes the number of response vectors, \(|S|\)
refers to the number of pairs in the relation, \(m\) denotes the number
of items, and \(v_{xy}\) again refers to the number of discordant
pairs. Generally, a low VC supports the validity of prerequisite relations.
In contrast to the other three indices, the Discrepancy Index (method
"DI" and the Distance Agreement
Coefficient (method "DA") validate the resulting knowledge
structure. The Discrepancy Index is the average distance between the response patterns
and the knowledge structure
$$DI = \sum_{r\in R}\min_{K\in\mathcal{K}}d(r,K) \frac{1}{n}$$
where \(d\) is the symmetric set difference. Generally, a lower DI.value indicates
a better fit between a knowledge structure and a set of response patterns.
The Distance Agreement Coefficient compares the average symmetric distance between the knowledge structure and respone patterns (referred to as ddat) to the average symmetric distance between the knowledge structure and the power set of response patterns (referred to as dpot). By calculating the ratio of ddat and dpot, the DA is determined. Generally, a lower DA-value indicates a better fit between a knowledge structure and a set of response patterns. Please note that the ddat value is equal to the DI index. The DA coefficient is insofar a further development of the DI index as it takes into account the sizes of the domain and the knowledge structure and thus makes the DA values better comparable.
Goodman, L. A. & Kruskal, W. H. (1972) Measures of association for cross classification. Journal of the American Statistical Association, 67.
Kambouri, M., Koppen, M., Villano, M., & Falmagne, J.-C. (1994). Knowledge assessment: Tapping human expertise by the QUERY routine. International Journal of Human–Computer–Studies, 40, 119–151.
Schrepp, M. (1999) An empirical test of a process model for letter series completion problems. In D. Albert & J. Lukas (Eds.), Knowledge Spaces: Theories, Emprical Research, Applications. Mahwah, NJ: Lawrence Erlbaum Associates.
Schrepp, M., Held, T., & Albert, D. (1999) Component-based construction of surmise relations for chess problems. In D. Albert & J. Lukas (Eds.), Knowledge Spaces: Theories, Empirical Research, Applications. Mahwah, NJ: Lawrence Erlbaum Associates.
kstructure
kst <- kstructure(set(set("a"), set("a","b"), set("a","c"), set("d","e"),
set("a","b","d","e"), set("a","c","d","e"), set("a","b","c","d","e")))
rp <- data.frame(a=c(1,1,0,1,1,1,1,0,0,0),b=c(0,1,0,1,0,1,0,1,0,0),
c=c(0,0,0,0,1,1,1,0,1,0),d=c(0,0,1,1,1,1,0,0,0,1), e=c(0,0,1,1,1,1,0,0,0,0))
# Gamma Index
kvalidate(kst, rpatterns=rp, method="gamma")
# Percent
kvalidate(kst, rpatterns=rp, method="percent")
# Violational Coefficient
kvalidate(kst, rpatterns=rp, method="VC")
# Discrepancy Index
kvalidate(kst, rpatterns=rp, method="DI")
# Distance Agreement Coefficient
kvalidate(kst, rpatterns=rp, method="DA")
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