Measurement models for diagnostic classification

The measurement model defines how the items relate to the attributes. The currently supported options for measurement models are: loglinear cognitive diagnostic model (LCDM); deterministic input, noisy "and", gate (DINA); deterministic input, noisy "or" gate (DINO); noisy-input,, deterministic "and" gate (NIDA); noisy-input, deterministic "or" gate (NIDO);, noncompensatory reparameterized unified model (NC-RUM); and compensatory, reparameterized unified model (C-RUM). See details for additional information on each model.

Usage

lcdm(max_interaction = Inf)

dina()

dino()

crum()

nida()

nido()

ncrum()

Arguments

max_interaction: For the LCDM, the highest item-level interaction to include in the model.

Value

A measurement model object.

Details

The LCDM (Henson et al., 2009; Henson & Templin, 2019) is a general diagnostic classification model that subsumes the other more restrictive models. The probability of a respondent providing a correct response is parameterized like a regression model. For each item, the LCDM includes an intercept, which represents the log-odds of providing a correct response when none of the attributes required by the item are present. Main effect parameters represent the increase in the log-odds for each required attribute that is present. Finally, interaction terms define the change in log-odds (after the main effects) when more than one of the required attributes are present.

Non-compensatory models

The DINA model (de la Torre & Douglas, 2004; Junker & Sijtsma, 2001) is a restrictive non-compensatory model. For each item two parameters are estimated. A guessing parameter defines the probability of a respondent providing a correct response when not all of the required attributes are present. Conversely, a slipping parameter defines the probability of providing an incorrect response when all of the required attributes are present. Thus, the DINA model takes an "all-or-nothing" approach. Either a respondent has all of the attributes required for an item, or they do not. There is no increase in the probability of providing a correct response if only a subset of the required attributes is present.

The NIDA model (Junker & Sijtsma, 2001) is a non-compensatory model that is less restrictive than the DINA model. Where the DINA model takes an "all-or-nothing" approach, the NIDA model defines the probability of responding correctly based on each attribute that has been mastered. In doing this, the NIDA model estimates parameters for each attribute and holds these parameters constant across items. Thus, respondents have increased probability of responding correctly based on the specific attributes that have been mastered. However, the parameters are held constant across items. That is, the effect of non-proficiency on an attribute is the same for all items measuring that attribute.

The reduced NC-RUM (DiBello et al., 1995; Hartz, 2002) is a non-compensatory model that is less restrictive than both the DINA and NIDA model, as the NC-RUM does not constrain parameters across items or attributes. Thus, the NC-RUM is most similar to the LCDM; however, the equivalent LCDM parameterization of the NC-RUM constrains interaction parameters to be positive, which differs from the full LCDM specification.

Compensatory models

The DINO model (Templin & Henson, 2006) is the inverse of the DINA model. Whereas the DINA model is "all-or-nothing", the DINO model can be thought of as "anything-or-nothing". In the DINO model, the guessing parameter defines the probability of a correct response when none of the required attributes are present. The slipping parameter is the probability of an incorrect response when any of the required attributes is present. Therefore, when using the DINO model, the presence of any of the required attributes results in an increased probability of a correct response, and there is no additional increase in probability for the presence of more than one of the required attributes.

The NIDO model (Templin, 2006) is a compensatory model that defines the probability of responding correctly based on each attribute that has been mastered. Like the NIDA model, the NIDO model holds these parameters constant across items. In the NIDO model, the probability of responding correctly increases with each mastered attribute without assuming a cumulative effect of mastering multiple attributes.

The C-RUM (Hartz, 2002) is similar to the LCDM, but is constrained to only include the intercept and main effect parameters. That is, no interaction terms are included for the C-RUM.

References

de la Torre, J., & Douglas, J. A. (2004). Higher-order latent trait models for cognitive diagnosis. Psychometrika, 69(3), 333-353. doi:10.1007/BF02295640

DiBello, L. V., Stout, W. F., & Roussos, L. (1995). Unified cognitive psychometric assessment likelihood-based classification techniques. In P. D. Nichols, S. F. Chipman, & R. L. Brennan (Eds.), Cognitively diagnostic assessment (pp. 361-390). Erlbaum.

Hartz, S. M. (2002). A Bayesian framework for the unified model for assessing cognitive abilities: Blending theory with practicality (Publication No. 3044108) [Doctoral dissertation, University of Illinois at Urbana-Champaign]. ProQuest Dissertations Publishing.

Henson, R. A., Templin, J. L., & Willse, J. T. (2009). Defining a family of cognitive diagnosis models using log-linear models with latent variables. Psychometrika, 74(2), 191-210. doi:10.1007/s11336-008-9089-5

Henson, R., & Templin, J. L. (2019). Loglinear cognitive diagnostic model (LCDM). In M. von Davier & Y.-S. Lee (Eds.), Handbook of Diagnostic Classification Models (pp. 171-185). Springer International Publishing. doi:10.1007/978-3-030-05584-4_8

Junker, B. W., & Sijtsma, K. (2001). Cognitive assessment models with few assumptions, and connections with nonparametric item response theory. Applied Psychological Measurement, 25(3), 258-272. doi:10.1177/01466210122032064

Templin, J. L. (2006). CDM user's guide. Unpublished manuscript.

Templin, J. L., & Henson, R. A. (2006). Measurement of psychological disorders using cognitive diagnosis models. Psychological Methods, 11(3), 287-305. doi:10.1037/1082-989X.11.3.287

Examples

lcdm()
#> <dcmstan::LCDM>
#>  @ model     : chr "lcdm"
#>  @ model_args:List of 1
#>  .. $ max_interaction: num Inf

lcdm(max_interaction = 3)
#> <dcmstan::LCDM>
#>  @ model     : chr "lcdm"
#>  @ model_args:List of 1
#>  .. $ max_interaction: num 3

dina()
#> <dcmstan::DINA>
#>  @ model     : chr "dina"
#>  @ model_args: list()

dino()
#> <dcmstan::DINO>
#>  @ model     : chr "dino"
#>  @ model_args: list()

nida()
#> <dcmstan::NIDA>
#>  @ model     : chr "nida"
#>  @ model_args: list()

nido()
#> <dcmstan::NIDO>
#>  @ model     : chr "nido"
#>  @ model_args: list()

ncrum()
#> <dcmstan::NCRUM>
#>  @ model     : chr "ncrum"
#>  @ model_args: list()

crum()
#> <dcmstan::CRUM>
#>  @ model     : chr "crum"
#>  @ model_args: list()