# kvalidate: Validate Prerequisite Relations or Knowledge Structures In kst: Knowledge Space Theory

## Description

Validates prerequisite relations or knowledge structures

## Usage

 1  kvalidate(x, rpatterns=NULL, method=c("gamma","percent","VC","DI","DA")) 

## Arguments

 x An R object of class kstructure. rpatterns A binary data frame or matrix where each row specifies the response pattern of one individual to the set of domain problems in x. method The desired validation method (see details).

## 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 = (Nc-Nd)/(Nc+Nd)

where Nc is the number of concordant pairs and Nd 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 = 1/(n*(|S|-m))*sum(vxy)

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 vxy 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 = ∑_{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.

## Value

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:

 gamma The gamma-value. nc Number of concordant pairs. nd Number of discordant pairs.

for the "gamma" method,

 vc The VC-value. nd Number of discordant pairs.

for the "VC" method,

 di The DI-value. di_dist The distance table for DI.

for the "DI" method, and

 ddat The ddat-value. ddat_dist The distance table for ddat. dpot The dpot-value. dpot_dist The distance table for dpot. DA The Distance Agreement Coefficient.

for the "DA" nethod.

## References

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

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 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") 

kst documentation built on Sept. 21, 2018, 6:23 p.m.