# Parameter Estimation for Cognitive Diagnosis Models (CDMs) by MMLE/EM or MMLE/BM Algorithm. A function to estimate parameters for cognitive diagnosis models by MMLE/EM (de la Torre, 2009; de la Torre, 2011) or MMLE/BM (Ma & Jiang, 2020) algorithm. The function imports various functions from the `GDINA` package, parameter estimation for Cognitive Diagnostic Models (CDMs) was performed and extended. The `CDM` function not only accomplishes parameter estimation for most commonly used models (e.g., `GDINA`, `DINA`, `DINO`, `ACDM`, `LLM`, or `rRUM`). Furthermore, it incorporates Bayes modal estimation (BM; Ma & Jiang, 2020) to obtain more reliable estimation results, especially in small sample sizes. The monotonic constraints are able to be satisfied. ```r CDM( Y, Q, model = "GDINA", method = "EM", att.str = NULL, mono.constraint = FALSE, maxitr = 2000, verbose = 1 ) ``` ## Arguments - `Y`: A required $N$ × $I$ matrix or `data.frame` consisting of the responses of `N` individuals to $N$ × $I$ items. Missing values need to be coded as `NA`. - `Q`: A required binary $I$ × $K$ matrix containing the attributes not required or required master the items. The `i`th row of the matrix is a binary indicator vector indicating which attributes are not required (coded by 0) and which attributes are required (coded by 1) to master item $i$. - `model`: Type of model to be fitted; can be `"GDINA"`, `"LCDM"`, `"DINA"`, `"DINO"`, `"ACDM"`, `"LLM"`, or `"rRUM"`. Default = `"GDINA"`. - `method`: Type of method to estimate CDMs' parameters; one out of `"EM"` and `"BM"`. Default = `"EM"`. However, `"BM"` is only available when `method = "GDINA"`. - `att.str`: Specify attribute structures. `NULL`, by default, means there is no structure. Attribute structure needs be specified as a list - which will be internally handled by `att.structure` function. See examples. It can also be a matrix giving all permissible attribute profiles. - `mono.constraint`: Logical indicating whether monotonicity constraints should be fulfilled in estimation. Default = `FALSE`. - `maxitr`: A vector for each item or nonzero category, or a scalar which will be used for all items to specify the maximum number of EM or BM cycles allowed. Default = `2000`. - `verbose`: Can be `0`, `1` or `2`, indicating to print no information, information for current iteration, or information for all iterations. Default = `1`. ## Returns An object of class `CDM` containing the following components: - **`analysis.obj`**: An `GDINA` object gained from `GDINA` package or an `list` after BM algorithm, depending on which estimation is used. - **`alpha`**: Individuals' attribute parameters calculated by EAP method - **`P.alpha.Xi`**: Individual's posterior probability - **`alpha.P`**: Individuals' marginal mastery probabilities matrix - **`P.alpha`**: Attribute prior weights for calculating marginalized likelihood in the last iteration - **`model.fit`**: Some basic model-fit indices, including `Deviance`, `npar`, `AIC`, `BIC`. @seealso `fit` - **`pattern`**: The attribute mastery pattern matrix containing all possible attribute mastery pattern. - **`arguments`**: A list containing all input arguments ## Details CDMs are statistical models that fully integrates cognitive structure variables, which define the response probability of examinees on items by assuming the mechanism between attributes. In the dichotomous test, this probability is the probability of answering correctly. According to the specificity or generality of CDM assumptions, it can be divided into reduced CDM and saturated CDM. Reduced CDMs possess specific assumptions about the mechanisms of attribute interactions, leading to clear interactions between attributes. Representative reduced models include the Deterministic Input, Noisy and Gate (DINA) model (Haertel, 1989; Junker & Sijtsma, 2001; de la Torre & Douglas, 2004), the Deterministic Input, Noisy or Gate (DINO) model (Templin & Henson, 2006), and the Additive Cognitive Diagnosis Model (A-CDM; de la Torre, 2011), the reduced Reparametrized Unified Model (rRUM; Hartz, 2002), among others. Compared to reduced models, saturated models, such as the Log-Linear Cognitive Diagnosis Model (LCDM; Henson et al., 2009) and the general Deterministic Input, Noisy and Gate model (G-DINA; de la Torre, 2011), do not have strict assumptions about the mechanisms of attribute interactions. When appropriate constraints are applied, saturated models can be transformed into various reduced models (Henson et al., 2008; de la Torre, 2011). The LCDM is a saturated CDM fully proposed within the framework of cognitive diagnosis. Unlike reduced models that only discuss the main effects of attributes, it also considers the interaction between attributes, thus having more generalized assumptions about attributes. Its definition of the probability of correct response is as follows: $$ P(X_{pi}=1|\boldsymbol{\alpha}_{l}) =\frac{\exp \left[\lambda_{i0} + \boldsymbol{\lambda}_{i}^{T} \boldsymbol{h} (\boldsymbol{q}_{i}, \boldsymbol{\alpha}_{l}) \right]}{1 + \exp \left[\lambda_{i0} + \boldsymbol{\lambda}_{i}^{T} \boldsymbol{h} (\boldsymbol{q}_{i}, \boldsymbol{\alpha}_{l}) \right]} $$ $$ \boldsymbol{\lambda}_{i}^{T} \boldsymbol{h}(\boldsymbol{q}_{i}, \boldsymbol{\alpha}_{l}) =\sum_{k=1}^{K^\ast}\lambda_{ik}\alpha_{lk} +\sum_{k=1}^{K^\ast-1}\sum_{k'=k+1}^{K^\ast}\lambda_{ikk'}\alpha_{lk}\alpha_{lk'} +\cdots + \lambda_{12 \cdots K^\ast}\prod_{k=1}^{K^\ast}\alpha_{lk} $$ Where, $P(X_{pi}=1|\boldsymbol{\alpha}_{l})$ represents the probability of an examinee with attribute mastery pattern $\boldsymbol{\alpha}_{l}$ ($l=1,2,\cdots,L$ and $L=2^{K^\ast}$) correctly answering item i. Here, $K^\ast = \sum_{k=1}^{K} q_{ik}$ denotes the number of attributes in the collapsed q-vector, $\lambda_{i0}$ is the intercept parameter, and c("$\\boldsymbol{\\lambda}_{i}=(\\lambda_{i1}, \\lambda_{i2}, \\cdots, \\lambda_{i12},\n$", "$\\cdots, \\lambda_{i12{\\cdots}K^\\ast})$") represents the effect vector of the attributes. Specifically, $\lambda_{ik}$ is the main effect of attribute $k$, $\lambda_{ikk'}$ is the interaction effect between attributes $k$ and $k'$, and $\lambda_{j12{\cdots}K^\ast}$ represents the interaction effect of all required attributes. The G-DINA, proposed by de la Torre (2011), is another saturated model that offers three types of link functions: identity link, log link, and logit link, which are defined as follows: $$ P(X_{pi}=1|\boldsymbol{\alpha}_{l}) =\delta_{i0} + \sum_{k=1}^{K^\ast}\delta_{ik}\alpha_{lk} +\sum_{k=1}^{K^\ast-1}\sum_{k'=k+1}^{K^\ast}\delta_{ikk'}\alpha_{lk}\alpha_{lk'} +\cdots + \delta_{12{\cdots}K^\ast}\prod_{k=1}^{K^\ast}\alpha_{lk} $$ $$ log \left[P(X_{pi}=1|\boldsymbol{\alpha}_{l}) \right] =v_{i0} + \sum_{k=1}^{K^\ast}v_{ik}\alpha_{lk} +\sum_{k=1}^{K^\ast-1}\sum_{k'=k+1}^{K^\ast}v_{ikk'}\alpha_{lk}\alpha_{lk'} +\cdots + v_{12{\cdots}K^\ast}\prod_{k=1}^{K^\ast}\alpha_{lk} $$ $$ logit \left[P(X_{pi}=1|\boldsymbol{\alpha}_{l}) \right] =\lambda_{i0} + \sum_{k=1}^{K^\ast}\lambda_{ik}\alpha_{lk} +\sum_{k=1}^{K^\ast-1}\sum_{k'=k+1}^{K^\ast}\lambda_{ikk'}\alpha_{lk}\alpha_{lk'} +\cdots + \lambda_{12{\cdots}K^\ast}\prod_{k=1}^{K^\ast}\alpha_{lk} $$ Where $\delta_{i0}$, $v_{i0}$, and $\lambda_{i0}$ are the intercept parameters for the three link functions, respectively; $\delta_{ik}$, $v_{ik}$, and $\lambda_{ik}$ are the main effect parameters of $\alpha_{lk}$ for the three link functions, respectively; $\delta_{ikk'}$, $v_{ikk'}$, and $\lambda_{ikk'}$ are the interaction effect parameters between $\alpha_{lk}$ and $\alpha_{lk'}$ for the three link functions, respectively; and $\delta_{i12{\cdots }K^\ast}$, $v_{i12{\cdots}K^\ast}$, and $\lambda_{i12{\cdots}K^\ast}$ are the interaction effect parameters of $\alpha_{l1}{\cdots}\alpha_{lK^\ast}$ for the three link functions, respectively. It can be observed that when the logit link is adopted, the G-DINA model is equivalent to the LCDM model. Specifically, the A-CDM can be formulated as: $$ P(X_{pi}=1|\boldsymbol{\alpha}_{l}) =\delta_{i0} + \sum_{k=1}^{K^\ast}\delta_{ik}\alpha_{lk} $$ The rRUM, can be written as: $$ log \left[P(X_{pi}=1|\boldsymbol{\alpha}_{l}) \right] =\lambda_{i0} + \sum_{k=1}^{K^\ast}\lambda_{ik}\alpha_{lk} $$ The item response function for the linear logistic model (LLM) can be given by: $$ logit\left[P(X_{pi}=1|\boldsymbol{\alpha}_{l}) \right] =\lambda_{i0} + \sum_{k=1}^{K^\ast}\lambda_{ik}\alpha_{lk} $$ In the DINA model, every item is characterized by two key parameters: guessing (g) and slip (s). Within the traditional framework of DINA model parameterization, a latent variable $\eta$, specific to examinee $p$ who has the attribute mastery pattern $\boldsymbol{\alpha}_{l}$ and responses to $i$, is defined as follows: $$ \eta_{li}=\prod_{k=1}^{K}\alpha_{lk}^{q_{ik}} $$ If examinee $p$ whose attribute mastery pattern is $\boldsymbol{\alpha}_{l}$ has acquired every attribute required by item i, $\eta_{pi}$ is given a value of 1. If not, $\eta_{pi}$ is set to 0. The DINA model's item response function can be concisely formulated as such: $$ P(X_{pi}=1|\boldsymbol{\alpha}_{l}) =(1-s_j)^{\eta_{li}}g_j^{(1-\eta_{li})} =\delta_{i0}+\delta_{i12{\cdots}K}\prod_{k=1}^{K^\ast}\alpha_{lk} $$ $(1-s_j)^{\eta_{li}}g_j^{(1-\eta_{li})}$ is the original expression of the DINA model, while $\delta_{i0}+\delta_{i12{\cdots}K}\prod_{k=1}^{K^\ast}\alpha_{lk}$ is an equivalent form of the DINA model after adding constraints in the G-DINA model. Here, $g_j = \delta_{i0}$ and $1-s_j = \delta_{i0}+\delta_{i12{\cdots}K}\prod_{k=1}^{K^\ast}\alpha_{lk}$. In contrast to the DINA model, the DINO model suggests that an examinee can correctly respond to an item if he/she have mastered at least one of the item's measured attributes. Additionally, like the DINA model, the DINO model also accounts for parameters related to guessing and slipping. Therefore, the main difference between DINO and DINA lies in their respective $\eta_{li}$ formulations. The DINO model can be given by: $$ \eta_{li} = 1-\prod_{k=1}^{K}(1 - \alpha_{lk})^{q_{lk}} $$ ## Examples ```r ################################################################ # Example 1 # # fit using MMLE/EM to fit the GDINA models # ################################################################ set.seed(123) library(Qval) ## generate Q-matrix and data to fit K <- 3 I <- 30 Q <- sim.Q(K, I) IQ <- list( P0 = runif(I, 0.0, 0.2), P1 = runif(I, 0.8, 1.0) ) data.obj <- sim.data(Q = Q, N = 500, IQ = IQ, model = "GDINA", distribute = "mvnorm") ## using MMLE/EM to fit GDINA model CDM.obj <- CDM(data.obj$dat, Q, model = "GDINA", method = "EM", maxitr = 2000, verbose = 1) ################################################################ # Example 2 # # fit using MMLE/BM to fit the DINA # ################################################################ set.seed(123) library(Qval) ## generate Q-matrix and data to fit K <- 5 I <- 30 Q <- sim.Q(K, I) IQ <- list( P0 = runif(I, 0.0, 0.2), P1 = runif(I, 0.8, 1.0) ) data.obj <- sim.data(Q = Q, N = 500, IQ = IQ, model = "DINA", distribute = "horder") ## using MMLE/BM to fit GDINA model CDM.obj <- CDM(data.obj$dat, Q, model = "GDINA", method = "BM", maxitr = 1000, verbose = 2) ################################################################ # Example 3 # # fit using MMLE/EM to fit the GDINA with attribute structures # ################################################################ set.seed(123) library(Qval) ## generate Q-matrix and data to fit K <- 3 I <- 30 att.str <- list(c(1,2), c(2,3)) Q <- sim.Q(K, I, att.str) IQ <- list( P0 = runif(I, 0.0, 0.2), P1 = runif(I, 0.8, 1.0) ) data.obj <- sim.data(Q = Q, N = 500, IQ = IQ, att.str=att.str, model = "GDINA") ## using MMLE/EM to fit GDINA model with attribute structures CDM.obj <- CDM(data.obj$dat, Q, model = "GDINA", att.str=att.str, method = "EM", maxitr = 2000, verbose = 1) summary(CDM.obj) ``` ## References de la Torre, J. (2009). DINA Model and Parameter Estimation: A Didactic. Journal of Educational and Behavioral Statistics, 34(1), 115-130. DOI: 10.3102/1076998607309474. 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. de la Torre, J. (2011). The Generalized DINA Model Framework. Psychometrika, 76(2), 179-199. DOI: 10.1007/s11336-011-9207-7. Haertel, E. H. (1989). Using restricted latent class models to map the skill structure of achievement items. Journal of Educational Measurement, 26(4), 301-323. DOI: 10.1111/j.1745-3984.1989.tb00336.x. Hartz, S. M. (2002). A Bayesian framework for the unified model for assessing cognitive abilities: Blending theory with practicality (Unpublished doctoral dissertation). University of Illinois at Urbana-Champaign. Henson, R. A., Templin, J. L., & Willse, J. T. (2008). 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. Huebner, A., & Wang, C. (2011). A note on comparing examinee classification methods for cognitive diagnosis models. Educational and Psychological Measurement, 71, 407-419. DOI: 10.1177/0013164410388832. 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. Ma, W., & Jiang, Z. (2020). Estimating Cognitive Diagnosis Models in Small Samples: Bayes Modal Estimation and Monotonic Constraints. Applied Psychological Measurement, 45(2), 95-111. DOI: 10.1177/0146621620977681. 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. Tu, D., Chiu, J., Ma, W., Wang, D., Cai, Y., & Ouyang, X. (2022). A multiple logistic regression-based (MLR-B) Q-matrix validation method for cognitive diagnosis models: A confirmatory approach. Behavior Research Methods. DOI: 10.3758/s13428-022-01880-x. ## See Also `validation`. ## Author(s) Haijiang Qin

CDM function