The Reduced RUM as a Logit Model: Parameterization and Constraints

Abstract

Cognitive diagnosis models (CDMs) for educational assessment are constrained latent class models. Examinees are assigned to classes of intellectual proficiency defined in terms of cognitive skills called attributes, which an examinee may or may not have mastered. The Reduced Reparameterized Unified Model (Reduced RUM) has received considerable attention among psychometricians. Markov Chain Monte Carlo (MCMC) or Expectation Maximization (EM) are typically used for estimating the Reduced RUM. Commercial implementations of the EM algorithm are available in the latent class analysis (LCA) routines of Latent GOLD and Mplus, for example. Fitting the Reduced RUM with an LCA routine requires that it be reparameterized as a logit model, with constraints imposed on the parameters. For models involving two attributes, these have been worked out. However, for models involving more than two attributes, the parameterization and the constraints are nontrivial and currently unknown. In this article, the general parameterization of the Reduced RUM as a logit model involving any number of attributes and the associated parameter constraints are derived. As a practical illustration, the LCA routine in Mplus is used for fitting the Reduced RUM to two synthetic data sets and to a real-world data set; for comparison, the results obtained by using the MCMC implementation in OpenBUGS are also provided.

Appendices

Appendix 1: Mplus Syntax: The Synthetic Data Set Involving $K=3$ Attributes

1.1 Model Definition and Constraints

The notation follows Rupp et al. (2010) that may serve as general reference on writing Mplus code for fitting CDMs. The preamble of the Mplus command file specifies the number of $J=30$ test items. In the subsequent MODEL declaration, for each of the 8 proficiency classes separately, all $J=30$ test items must be specified (for details, consult Templin & Hoffman’s, 2013, comprehensive tutorial). The MODEL declaration must also contain the syntax for computing the means of the proficiency classes that are needed later on for modeling the higher-order structure among the attributes (more on this below):

In the MODEL CONSTRAINT section of the syntax, the constraints on the model parameters must be defined for each item individually. An item parameter is denoted by “Lj_ab” in the Mplus command language, where “j” is the item number; “a” specifies whether the parameter refers to a main effect $(\mathtt{a}=1)$ or an interaction term of order $\mathtt{a}> 1$; and “b” indicates the attribute(s) involved. As an example, consider item 1 that requires the mastery of only the first attribute: $\mathbf {q}_1 = (100)^T$. The IRF of item 1 is

$$\begin{aligned} P(Y_1 = 1 \mid \varvec{\alpha }) = \frac{ \mathrm{e}^{ \beta _{10} + \beta _{11} \alpha _1 } }{ 1 + \mathrm{e}^{ \beta _{10} + \beta _{11} \alpha _1 } } \end{aligned}$$

The intercept parameter, $\beta _{10}$, is written as L1_0; the coefficient of the main effect, $\beta _{11}$, is written as L1_11. Recall that $\beta _{10}$ is unconstrained; hence, the only parameter constraint is $\beta _{11} > 0$, which is written in the Mplus command language as $\mathtt{L1\_11>0}$. As a more complex example, consider item 7 that requires the mastery of all $K=3$ attributes; thus, $\mathbf {q}_7 = (111)^T$. The IRF of item 7 is

$$\begin{aligned} P(Y_7=1 \mid \varvec{\alpha })&= \frac{ \mathrm{e}^{ \beta _{70} + \beta _{71}\alpha _1 + \beta _{72}\alpha _2 + \beta _{73}\alpha _3 + \beta _{712}\alpha _1\alpha _2 + \beta _{713}\alpha _1\alpha _3 + \beta _{723}\alpha _2\alpha _3 + \beta _{7123}\alpha _1\alpha _2\alpha _3 } }{ 1 + \mathrm{e}^{ \beta _{70} + \beta _{71}\alpha _1 + \beta _{72}\alpha _2 + \beta _{73}\alpha _3 + \beta _{712}\alpha _1\alpha _2 + \beta _{713}\alpha _1\alpha _3 + \beta _{723}\alpha _2\alpha _3 + \beta _{7123}\alpha _1\alpha _2\alpha _3 } }\\ \text{ subject } \text{ to: }&(1)\,\, 0 < \beta _{7k} \qquad k=1,2 \\&(2)\,\, 0 < \beta _{73} < \ln \left( \big (1 + \mathrm{e}^{ \beta _{70} + \beta _{71} } \big )\big (1 + \mathrm{e}^{ \beta _{70} + \beta _{72} }\big )\right) \\&\quad - \ln \left( \big (1 + \mathrm{e}^{ \beta _{70} } \big )^2 \mathrm{e}^{ \beta _{71} + \beta _{72} } - \big (1 + \mathrm{e}^{ \beta _{70} + \beta _{71} } \big ) \big (1 + \mathrm{e}^{ \beta _{70} + \beta _{72} } \big ) \right) - \beta _{70} \end{aligned}$$

where

$$\begin{aligned} \beta _{712}= & {} \ln \left( \frac{1 + \mathrm{e}^{ \beta _{70}}}{\big (1 + \mathrm{e}^{ \beta _{70} + \beta _{71}}\big )\big (1 + \mathrm{e}^{ \beta _{70} + \beta _{72}}\big ) - \big (1 + \mathrm{e}^{ \beta _{70}}\big ) \mathrm{e}^{ \beta _{70} + \beta _{71} + \beta _{72}}} \right) \\ \beta _{713}= & {} \ln \left( \frac{1 + \mathrm{e}^{ \beta _{70}}}{\big (1 + \mathrm{e}^{ \beta _{70} + \beta _{71}}\big )\big (1 + \mathrm{e}^{ \beta _{70} + \beta _{73}}\big ) - \big (1 + \mathrm{e}^{ \beta _{70}}\big ) \mathrm{e}^{ \beta _{70} + \beta _{71} + \beta _{73}}} \right) \\ \beta _{723}= & {} \ln \left( \frac{1 + \mathrm{e}^{ \beta _{70}}}{\big (1 + \mathrm{e}^{ \beta _{70} + \beta _{72}}\big )\big (1 + \mathrm{e}^{ \beta _{70} + \beta _{73}}\big ) - \big (1 + \mathrm{e}^{ \beta _{70}}\big ) \mathrm{e}^{ \beta _{70} + \beta _{72} + \beta _{73}}} \right) \\ \beta _{7123}= & {} \ln \left( \frac{ (1 + \mathrm{e}^{ \beta _{70} } )^2 }{ \big ( 1 + \mathrm{e}^{ \beta _{70} + \beta _{71}}\big ) \big (1 + \mathrm{e}^{ \beta _{70} + \beta _{72}}\big ) \big (1 + \mathrm{e}^{ \beta _{70} + \beta _{73} }\big ) -\big (1 + \mathrm{e}^{ \beta _{70} } \big )^2 \mathrm{e}^{ \beta _{70} + \beta _{71} + \beta _{72} + \beta _{73}} } \right) \\&\quad - \big (\beta _{713} + \beta _{712} + \beta _{723}\big ) \end{aligned}$$

All model parameters in the IRF of item 7 are defined in the first line of the MODEL CONSTRAINT section, followed by the specification of the model parameters that characterize the different proficiency classes. Third, the constraints on the three main effects, $\beta _{71}$ (L7_11 in Mplus syntax), $\beta _{72}$ (L7_12), and $\beta _{73}$ (L7_13), that guarantee monotonicity are defined. (In addition, note the complex syntax of Constraint 2 that defines the upper bound on the coefficient of the last main effect, $\beta _{73}$ (L7_13).)   Finally, the equations of the interaction terms of item 7 must be implemented. (Recall that the parameters of all interaction terms are functions of the parameters of the constituting main effects.)

1.2 Converting the Parameter Estimates from the Logit to the Traditional Form of the Reduced RUM

The parameter estimates of the Reduced RUM in traditional form are retrieved from the EM-based parameter estimates obtained for the logit model. As an example of how to perform the parameter conversion, consider item 7 (see also p. 9):

$$\begin{aligned} \pi ^{*}_7= & {} \frac{ \mathrm{e}^{ \beta _{70} + \beta _{71} + \beta _{72} + \beta _{73} } (1 + \mathrm{e}^{ \beta _{70} })^2 }{ (1 + \mathrm{e}^{ \beta _{70} + \beta _{71} }) (1 + \mathrm{e}^{ \beta _{70} + \beta _{72} }) (1 + \mathrm{e}^{ \beta _{70} + \beta _{73} }) }\\ r^{*}_{71}= & {} \frac{ 1 + \mathrm{e}^{ \beta _{70} + \beta _{71} } }{ \mathrm{e}^{ \beta _{71} } (1 + \mathrm{e}^{ \beta _{70} }) }\\ r^{*}_{72}= & {} \frac{ 1 + \mathrm{e}^{ \beta _{70} + \beta _{72} } }{ \mathrm{e}^{ \beta _{72} } (1 + \mathrm{e}^{ \beta _{70} }) }\\ r^{*}_{73}= & {} \frac{ 1 + \mathrm{e}^{ \beta _{70} + \beta _{73} } }{ \mathrm{e}^{ \beta _{73} } (1 + \mathrm{e}^{ \beta _{70} }) } \end{aligned}$$

The conversion of the parameter estimates can be carried out directly in Mplus:

1.3 Modeling the Higher-Order Structure Among Attributes

As discussed in Rupp et al. (2010), several models can be used for the higher-order structure among attributes. In the simulation studies here, the saturated log-linear model is used. Following the notation in Rupp et al. (2010), the parameters are denoted as G_ab, where “a” indicates if the parameter refers to a main effect $(\mathtt{a}=1)$ or an interaction term of order $\mathtt{a}> 1$; “b” specifies the attribute(s) involved.

Appendix 2: Mplus Syntax: The Synthetic Data Set Involving $K=4$ Attributes

The most complex test item 15 requiring the mastery of $K=4$ attributes is used as an example here for the implementation of the parameters of the logistic function and their associated constraints. First, the constraints on the main effect parameters, $\beta _{151}$, $\beta _{152}$, $\beta _{153}$, and $\beta _{154}$, that guarantee monotonicity must be defined. The least upper bound on the parameter associated with the last main effect, $\beta _{154}$ (L15_14 in Mplus syntax), is derived from Equation 14 as

$$\begin{aligned} \beta _{154}< & {} \sum ^{4-1}_{k=1}\ln (1 + \mathrm{e}^{ \beta _{150} + \beta _{15k}}) - \ln \left( \prod ^{4-1}_{k=1} (\mathrm{e}^{\beta _{15k}} + \mathrm{e}^{ \beta _{150} + \beta _{15k}}) -\prod ^{4-1}_{k=1} \big (1 + \mathrm{e}^{\beta _{150} + \beta _{jk}}\big )\right) -\beta _{150} \\= & {} \ln \big (1 + \mathrm{e}^{ \beta _{150} + \beta _{151}}\big )+\ln \big (1 + \mathrm{e}^{ \beta _{150} + \beta _{152}}\big )+\ln \big (1 + \mathrm{e}^{ \beta _{150} + \beta _{153}}\big )\\&-\ln \left( \big (1\!+\!\beta _{150}\big )^3\mathrm{e}^{\beta _{151}+\beta _{152}+\beta _{153}}-\big (1 \!+\! \mathrm{e}^{\beta _{150} \!+\! \beta _{151}}\big )\big (1 \!+\! \mathrm{e}^{\beta _{150} \!+\! \beta _{152}}\big )\big (1 \!+\! \mathrm{e}^{\beta _{150} + \beta _{153}}\big )\right) -\beta _{150} \end{aligned}$$

As an example of a rather complex model parameter, the coefficient of the four-way interaction term of item 15, $\beta _{151234}$, is given (see Equation 11):

$$\begin{aligned} \beta _{151234}= & {} \ln \left( \frac{ (1 + \mathrm{e}^{\beta _{150}})^{4-1}}{\prod ^4_{k=1} (1 + \mathrm{e}^{\beta _{150} + \beta _{15k}}) - (1 + \mathrm{e}^{ \beta _{150}} )^{4-1} \mathrm{e}^{ \beta _{150} + \sum ^4_{k=1} \beta _{15k} } } \right) \\&-\left( \sum ^4_{k^{\prime }=k+1} \sum ^3_{k=1} \beta _{15kk^{\prime }} + \sum ^4_{k_3=k_2+1}\sum ^3_{k_2=k_1+1} \sum ^2_{k_1=1} \beta _{15 k_1k_2k_3}\right) \\= & {} \ln \left( \frac{(1 + \mathrm{e}^{\beta _{150}})^3}{(1 + \mathrm{e}^{\beta _{150} + \beta _{151}} )\cdots (1 + \mathrm{e}^{\beta _{150} + \beta _{154}} ) - (1 + \mathrm{e}^{ \beta _{150}})^3 \mathrm{e}^{ \beta _{150} + \beta _{151} + \beta _{152} + \beta _{153} + \beta _{154} } } \right) \\&- (\beta _{1512}\!+\! \beta _{1513} \!+\!\beta _{1514} \!+\!\beta _{1523} +\beta _{1524} \!+\!\beta _{1534}\!+\! \beta _{15123}+ \beta _{15124}+ \beta _{15134}+ \beta _{15234}) \end{aligned}$$

Here is the complete Mplus syntax for item 15: