Least Squares Distance Method of Cognitive Validation

This function estimates the least squares distance method of cognitive validation (Dimitrov, 2007; Dimitrov & Atanasov, 2012) which assumes a multiplicative relationship of attribute response probabilities to explain item response probabilities. The argument distance allows the estimation of a squared loss function (distance="L2") and an absolute value loss function (distance="L1").

The function also estimates the classical linear logistic test model (LLTM; Fischer, 1973) which assumes a linear relationship for item difficulties in the Rasch model.

Usage

lsdm(data, Qmatrix, theta=seq(-3,3,by=.5), wgt_theta=rep(1, length(theta)), distance="L2",
   quant.list=c(0.5,0.65,0.8), b=NULL, a=rep(1,nrow(Qmatrix)), c=rep(0,nrow(Qmatrix)) )

# S3 method for lsdm
summary(object, file=NULL, digits=3, ...)

# S3 method for lsdm
plot(x, ...)

Arguments

data: An $I \times L$ matrix of dichotomous item responses. The data consists of $I$ item response functions (parametrically or nonparametrically estimated) which are evaluated at a discrete grid of $L$ theta values (person parameters) and are specified in the argument theta.
Qmatrix: An $I \times K$ matrix where the allocation of items to attributes is coded. Values of zero and one and all values between zero and one are permitted. There must not be any items with only zero Q-matrix entries in a row.
theta: The discrete grid points $\theta$ where item response functions are evaluated for doing the LSDM method.
wgt_theta: Optional vector for weights of discrete $\theta$ points
quant.list: A vector of quantiles where attribute response functions are evaluated.
distance: Type of distance function for minimizing the discrepancy between observed and expected item response functions. Options are "L2" which is the squared distance (proposed in the original LSDM formulation in Dimitrov, 2007) and the absolute value distance "L1" (see Details).
b: An optional vector of item difficulties. If it is specified, then no data input is necessary.
a: An optional vector of item discriminations.
c: An optional vector of guessing parameters.
object: Object of class lsdm
file: Optional file name for summary output
digits: Number of digits aftert decimal in summary
...: Further arguments to be passed
x: Object of class lsdm

Details

The least squares distance method (LSDM; Dimitrov 2007) is based on the assumption that estimated item response functions $P(X_i=1 | \theta)$ can be decomposed in a multiplicative way (in the implemented conjunctive model): $$ P( X_i=1 | \theta ) \approx \prod_{k=1}^K [ P( A_k=1 | \theta ) ]^{q_{ik}} $$ where $P( A_k=1 | \theta )$ are attribute response functions and $q_{ik}$ are entries of the Q-matrix. Note that the multiplicative form can be rewritten by taking the logarithm $$ \log P( X_i=1 | \theta ) \approx \sum_{k=1}^K q_{ik} \log [ P( A_k=1 | \theta ) ] $$ The item and attribute response functions are evaluated on a grid of $\theta$ values. Using the definitions of matrices $\bold{L}=\{ \log P( X_i=1 ) | \theta ) \} $, $\bold{Q}=\{ q_{ik} \} $ and $\bold{X}=\{ \log P( A_k=1 | \theta ) \} $, the estimation problem can be formulated as $ \bold{L} \approx \bold{Q} \bold{X}$. Two different loss functions for minimizing the discrepancy between $ \bold{L}$ and $\bold{Q} \bold{X}$ are implemented. First, the squared loss function computes the weighted difference $|| \bold{L} - \bold{Q} \bold{X}||_2=\sum_i ( l_i - \sum_t q_{it} x_{it})^2$ (distance="L2") and has been originally proposed by Dimitrov (2007). Second, the absolute value loss function $|| \bold{L} - \bold{Q} \bold{X}||_1=\sum_i | l_i - \sum_t q_{it} x_{it} |$ (distance="L1") is more robust to outliers (i.e., items which show misfit to the assumed multiplicative LSDM formulation).

After fitting the attribute response functions, empirical item-attribute discriminations $w_{ik}$ are calculated as the approximation of the following equation $$ \log P( X_i=1 | \theta )= \sum_{k=1}^K w_{ik} q_{ik} \log [ P( A_k=1 | \theta ) ] $$

Value

A list with following entries

mean.mad.lsdm0: Mean of $MAD$ statistics for LSDM
mean.mad.lltm: Mean of $MAD$ statistics for LLTM
attr.curves: Estimated attribute response curves evaluated at theta
attr.pars: Estimated attribute parameters for LSDM and LLTM
data.fitted: LSDM-fitted item response functions evaluated at theta
theta: Grid of ability distributions at which functions are evaluated
item: Item statistics (p value, $MAD$, ...)
data: Estimated or fixed item response functions evaluated at theta
Qmatrix: Used Q-matrix
lltm: Model output of LLTM (lm values)
W: Matrix with empirical item-attribute discriminations

References

Al-Shamrani, A., & Dimitrov, D. M. (2016). Cognitive diagnostic analysis of reading comprehension items: The case of English proficiency assessment in Saudi Arabia. International Journal of School and Cognitive Psychology, 4(3). 1000196. http://dx.doi.org/10.4172/2469-9837.1000196

DiBello, L. V., Roussos, L. A., & Stout, W. F. (2007). Review of cognitively diagnostic assessment and a summary of psychometric models. In C. R. Rao and S. Sinharay (Eds.), Handbook of Statistics, Vol. 26 (pp. 979-1030). Amsterdam: Elsevier.

Dimitrov, D. M. (2007). Least squares distance method of cognitive validation and analysis for binary items using their item response theory parameters. Applied Psychological Measurement, 31, 367-387. http://dx.doi.org/10.1177/0146621606295199

Dimitrov, D. M., & Atanasov, D. V. (2012). Conjunctive and disjunctive extensions of the least squares distance model of cognitive diagnosis. Educational and Psychological Measurement, 72, 120-138. http://dx.doi.org/10.1177/0013164411402324

Dimitrov, D. M., Gerganov, E. N., Greenberg, M., & Atanasov, D. V. (2008). Analysis of cognitive attributes for mathematics items in the framework of Rasch measurement. AERA 2008, New York.

Fischer, G. H. (1973). The linear logistic test model as an instrument in educational research. Acta Psychologica, 37, 359-374. http://dx.doi.org/10.1016/0001-6918(73)90003-6

Sonnleitner, P. (2008). Using the LLTM to evaluate an item-generating system for reading comprehension. Psychology Science, 50, 345-362.