4.9 Estimation of a higher-order model

The joint attribute distribution can be modeled using various methods. This section mainly focuses on the so-called higher-order approach, which was originally proposed by de la Torre and Douglas (2004) for the DINA model. It has been extended in this package for all condensation rules. Particularly, three approaches are available for the higher-order attribute structure: common slop apprach, where the probability of mastering attribute \(k\) for individual \(i\) is defined as \[ {P(\alpha_k=1|\theta_i,\lambda_{0k},\lambda_{1})=\frac{exp(\lambda_{1}\theta_i+\lambda_{0k})}{1+exp(\lambda_{1}\theta_i+\lambda_{0k})}} \] By setting \(\lambda_{1}=1\), we have a intercept only model.

For the varied slope approach, the probability of mastering attribute \(k\) for individual \(i\) is defined as \[ {P(\alpha_k=1|\theta_i,\lambda_{0k},\lambda_{1k})=\frac{exp(\lambda_{1k}\theta_i+\lambda_{0k})}{1+exp(\lambda_{1k}\theta_i+\lambda_{0k})}} \] where \({\theta_i}\) is the ability of examinee \({i}\). \({\lambda_{0k}}\) and \({\lambda_{1k}}\) are the intercept and slope parameters for attribute \({k}\), respectively. The probability of joint attributes can be written as \[ {P(\boldsymbol{\alpha}|\theta_i,\boldsymbol{\lambda})=\prod_k P(\alpha_k|\theta_i,\boldsymbol{\lambda})}. \]

To estimate higher-order G-DINA model, call GDINA function and specify the data and Q-matrix as the first two arguments. To specify a higher-order rather than the saturated joint attribute distribution, we need to set att.dist = “higher.order” and then specify model in higher.order argument: model should be a number indicating the model for higher-order joint attribute distribution. Can be “Rasch”, “1PL” or “2PL”, representing the intercept only approach, common slope approach and varied slope approach

Code

# Fit the data using higher-order G-DINA model
HO.GDINA <- GDINA(dat = data1, Q = Q1, att.dist = "higher.order", higher.order = list(model = "1PL"),
    verbose = 0)

Code

coef(HO.GDINA, what = "lambda")

##    slope intercept
## A1  0.13     0.161
## A2  0.13     0.061
## A3  0.13     0.383