# Deterministic Input, Noisy Mixed Modeling for Identifying Coexisting Condensation Rules in Cognitive Diagnostic Assessments

^{1}

^{2}

^{3}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Deterministic Input, Noisy Models with Typical Condensation Rules

_{ni}is the observed (dichotomous) response of person n (n = 1, …, N) to item i (i = 1, .., I); ${\omega}_{ni}$ is the latent response of person n to item i; g

_{i}and s

_{i}are the guessing and slip parameters of item i, respectively. Typically, a monotonicity restriction, g

_{i}< 1 − s

_{i}, can be imposed (Junker and Sijtsma 2001; Culpepper 2015). With various choices of ${\omega}_{ni}$, Equation (1) can describe many psychometric models, such as the four-parameter logistic unidimensional or multidimensional item response models (Reckase 2009) and the multicomponent latent trait model (Embretson 1984).

_{nk}is the mastery status of person n of attribute k (k = 1, …, K), α

_{nk}= 1 means mastery, and α

_{nk}= 0, otherwise. The Q-matrix (Tatsuoka 1983) is an I-by-K matrix with element q

_{ik}indicating whether attribute k is required to answer item i correctly; q

_{ik}= 1 if the attribute is required, and q

_{ik}= 0 otherwise. With the conjunctive condensation rule, Equation (1) becomes the DINA model.

#### 2.2. Mixture Item Response Models

## 3. Deterministic Input, Noisy Mixed Model

#### 3.1. Model Formulation

_{i}); ${\tau}_{im}$ is the item-level mixing proportion parameter of condensation rule m in item i, which satisfies ${\tau}_{im}\in \left[0,1\right]$ and $\sum}_{m=1}^{{M}_{i}}{\tau}_{im}=1$ for each item. The item-level mixing proportion parameters can be interpreted as the proportions/weights of different cognitive processes required to solve the problem correctly. M

_{i}is the number of preselected condensation rules for item i. For simplicity, but without loss of generality, it can be assumed that M

_{i}= M for all items throughout this study.

**τ**

_{i}= (τ

_{i}

_{1}, τ

_{i}

_{2}, τ

_{i}

_{3})’ ≡ (1, 0, 0), indicating that the DINA model is forcibly assigned to unidimensional items.2

#### 3.2. Bayesian Parameter Estimation

#### 3.3. Relationship with Existing CDMs

**q**

_{i}= (1, 1, 1), g

_{i}= s

_{i}= .1, τ

_{i}

_{1}= .8, τ

_{i}

_{2}= .1) based on them. First, compared to the DINA and DINO models, the DINR and DINMix models can better reflect the differences in various attribute profiles. Second, the values of item-level mixing proportions indicate that this item contains three condensation rules simultaneously, and the proportion of the conjunctive condensation rule is the highest. The DINMix model not only differentiates individuals in the partial mastery group but also reflects the feature of the conjunctive condensation rule (i.e., individuals in the partial mastery group have low correct response probabilities).

_{i}, g

_{i}, τ

_{i}

_{1}, and τ

_{i}

_{2}). Therefore, similar to the DINA model, all items in the DINMix model have the same number of item parameters that are easier to interpret than those (e.g., main effects, two-way interactions, three-way interactions) in the GDINA model. More importantly, the DINMix and GDINA models were developed for different purposes. The former was designed for identifying coexisting condensation rules, and the latter was created for model generalization.

#### 3.4. Parameter Identifiability

## 4. Simulation Studies

#### 4.1. Study 1

#### 4.1.1. Design and Data Generation

**τ**

_{1~I/3}= (1, 0, 0). Then, we set τ(I/3 + 1)~I = (1/3, 1/3, 1/3) for uniform mixing. In contrast, for skew mixing, we set

**τ**

_{(I/3+1)~8I/15}= (0.6, 0.2, 0.2),

**τ**

_{(8I/15+1)~11I/15}= (0.2, 0.6, 0.2), and

**τ**

_{(11I/15+1)~I}= (0.2, 0.2, 0.6). Furthermore, (e) the latent structural model of attributes (LSM) at two levels of an unstructured and a multivariate normal distribution was manipulated. When an unstructured LSM was used, the true attribute profile of each person was randomly chosen from all possible patterns with equal probability; in these cases, the tetrachoric correlations among attributes were approximately zero. In contrast, when a multivariate normal distribution was used, a latent variable matrix with continuous elements was first generated from a five-dimensional multivariate normal distribution (e.g., Chiu et al. 2009):

_{ni}~Bernoulli (p

_{ni}), where p

_{ni}was given in Equation (8) in the main text.

#### 4.1.2. Analysis

#### 4.1.3. Results

**τ**

_{3}was worse than that of

**τ**

_{1}and

**τ**

_{2}, mainly because τ

_{i}

_{3}= 1 − (τ

_{i}

_{1}+ τ

_{i}

_{2}), and thus, τ

_{i}

_{3}needs to offset both estimation errors of τ

_{i}

_{1}and τ

_{i}

_{2}.

#### 4.2. Study 2

#### 4.2.1. Design and Data Generation

_{i}and s

_{i}were both fixed at 0.1. In the ACDM, the intercept parameters were all fixed at 0.1; for two-dimensional items, two main effects were fixed at 0.5 and 0.3; for three-dimensional items, three main effects were fixed at 0.35, 0.25, and 0.2. In the GDINA model, the intercept parameters were all fixed at 0.1; for two-dimensional items, two main effects, and one two-way interaction effects were fixed at 0.35, 0.25, and 0.2, respectively; for three-dimensional items, three main effects, three two-way interaction effects, and one three-way interaction effects were fixed at 0.15, 0.1, 0.05, 0.05, 0.1, 0.15, and 0.2, respectively. The unstructured latent structural model was used (i.e., the true attribute profile of each person was randomly chosen from all possible patterns with equal probability). Finally, the observed responses of each item were generated from the corresponding model presented in Figure 1. Thirty datasets were generated in each condition.

#### 4.2.2. Analysis

#### 4.2.3. Results

_{i}

_{1}> 0.9 and τ

_{i}

_{2}> 0.9 as a judgment condition, τ

_{i}

_{1}and τ

_{i}

_{2}can accurately identify the conjunctive and disjunctive condensation rules for each item across all test situations, respectively. For τ

_{i}

_{3}, we cannot make a judgment directly by using a certain cut-point (e.g., 0.9), as for τ

_{i}

_{1}and τ

_{i}

_{2}; however, by judging whether τ

_{i}

_{3}is simultaneously larger than τ

_{i}

_{1}and τ

_{i}

_{2}, the items that follow the ratio/compensatory condensation rule can still be identified. Second, when the relationships between item responses and required attributes fuzzily followed some condensation rules (i.e., the test situation (e)), τ

_{i}

_{1}, τ

_{i}

_{2}, and τ

_{i}

_{3}seemed to show the following pattern: the proportion of τ

_{i}

_{1}and τ

_{i}

_{3}was much higher than that of τ

_{i}

_{2}. Of course, this pattern may change depending on the simulated values of different item parameters in the GDINA model. Third, for test situation (f), even in such a complex test situation, the DINMix model can identify the condensation rules followed by each item by adaptively adjusting the estimates of τ

_{i}

_{1}, τ

_{i}

_{2}, and τ

_{i}

_{3}. If the identified condensation rules for some items were inconsistent with those predefined by experts, then the experts may consider revising these items.

## 5. An Empirical Example

_{im}s indicate that the conjunctive condensation rule accounts for the largest proportion of most items. When using the judgment rules obtained in simulation Study 2 (e.g., τ

_{i}

_{1}> 0.9 or τ

_{i}

_{2}> 0.9), it was difficult to determine which specific condensation rule that items 1, 4, 5, 12, 14, 16, and 18 followed, primarily because they were judged to contain coexisting condensation rules. Similarly, the results of the Wald test also suggest that no particular condensation rule applied to items 1, 4, 12, 14, and 16. This consistency also indicates that the proposed model can further explain why the Wald test cannot find a specific condensation rule for some items. Additionally, use the item 14, $3\frac{4}{5}-3\frac{2}{5}$,5 as an example. Attributes α

_{2}(separate a whole number from a fraction) and α

_{7}(subtract numerators) were required to respond correctly according to the Q-matrix. However, respondents who mastered α

_{7}but not α

_{2}could still identify that the correct answer was 2/5. The first reason is that respondents may ignore the integer part and only focus on the difference between the fraction part. The second reason is that alternative attributes that are unspecified by the Q-matrix can probably be used to answer this item, such as convert mixed number to fraction (Mislevy 1996). Apparently, for whatever reason, the expert-defined conjunctive condensation rule does not fully apply to this item, which is also what the DINMix model indicated.

## 6. Summary and Discussion

## Supplementary Materials

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Notes

1 | Specifically, it follows the data-first philosophy (i.e., by developing more complex models to realize the analysis of data in complex situations). In contrast, for the perspective of measurement, considering the viewpoint of garbage-in, garbage-out for predetermined measurement objects (e.g., attributes and cognitive processes) to ensure the reliability and validity of measurement, factors unrelated to the measurement should be excluded (i.e., be revised or be deleted) as much as possible. |

2 | Any one element in τ_{i} can be set to 1 because all CDMs (for binary attributes) are identical to each other for unidimensional items. |

3 | According to the p-values of the Wald test, items 1, 4, 12, 14, and 16 are applicable to the GDINA model, while the remaining items are applicable to the DINA model. |

4 | The author does, however, note that this Q-matrix has been suggested to be revised by some previous studies (e.g., Chen et al. 2018). However, it is still fair to compare several models with the same Q-matrix, especially when there is no definite conclusion about the revision of this Q-matrix. |

5 |

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**Figure 1.**K-by-I Q’ matrix for simulation Study 1. Note: blank means ‘0,’ and gray means ‘1’; ‘*’ denotes items used in the I = 15 conditions.

**Figure 2.**Summary of the recovery of attributes in simulation Study 1. Note: LSM = latent structural model; IQ = item quality; N = sample size; I = test length; ACCR = attribute correct classification rate; PCCR = attribute pattern correct classification rate.

**Figure 3.**Root mean square error of attribute profile proportions in simulation Study 1. Note: LSM = latent structural model; IQ = item quality; N = sample size; I = test length; Mean = mean of the root mean square errors of 32 attribute profiles.

**Figure 4.**K-by-I Q’ matrix for simulation Study 2 and the data generation model for each item. Note: blank means ‘0’ and gray means ‘1’; DINA = deterministic input, noisy ‘and’ gate model; DINO = deterministic input, noisy ‘or’ gate model; DINR = deterministic input, noisy ratio model; ACDM = additive cognitive diagnosis model; GDINA = generalized DINA model; DINMix = deterministic input, noisy mixed model.

**Figure 5.**Estimates of the item-level mixing proportion parameter in simulation Study 2 from the DINMix model. Note: The value is the mean value of 30 replications; τ

_{1}= item-level mixing proportion parameter for the conjunctive condensation rule; τ

_{2}= item-level mixing proportion parameter for the disjunctive condensation rule; τ

_{3}= item-level mixing proportion parameter for the ratio condensation rule; DINMix = deterministic input, noisy mixed model; GDINA = generalized DINA model.

**Figure 6.**Summary of the recovery of item parameters in simulation Study 2. Note, RMSE = root mean square error; DINA = deterministic input, noisy ‘and’ gate model; DINO = deterministic input, noisy ‘or’ gate model; DINR = deterministic input, noisy ratio model; ACDM = additive cognitive diagnosis model; GDINA = generalized DINA model; DINMix = deterministic input, noisy mixed model.

**Figure 7.**Item-level –2LCPO of six models in simulation Study 2. Note: The value is the mean value of 30 replications; –2LCPO = –2 log conditional predictive ordinate; DINA = deterministic input, noisy ‘and’ gate model; DINO = deterministic input, noisy ‘or’ gate model; DINR = deterministic input, noisy ratio model; ACDM = additive cognitive diagnosis model; GDINA = generalized DINA model; DINMix = deterministic input, noisy mixed model.

**Figure 8.**Item-level –2LCPO of seven models for the fraction subtraction data. Note: DINA = deterministic input, noisy ‘and’ gate model; DINO = deterministic input, noisy ‘or’ gate model; DINR = deterministic input, noisy ratio model; ACDM = additive cognitive diagnosis model; GDINA = generalized DINA model; DINMix = deterministic inputs, noisy mixed model; Wald-selected = selected mixing model via Wald test.

**Figure 9.**The estimated item parameters for the fraction subtraction data from the DINMix models. Note: g = guessing parameter; s = slip parameter; τ

_{1}= item-level mixing proportion parameter for the conjunctive condensation rule; τ

_{2}= item-level mixing proportion parameter for the disjunctive condensation rule; τ

_{3}= item-level mixing proportion parameter for the ratio condensation rule; DINMix = deterministic input, noisy mixed model.

Attribute Profile | Number of Mastered Attributes | DINA | DINO | DINR | DINMix |
---|---|---|---|---|---|

(0, 0, 0) | 0 | 0.1 | 0.1 | 0.1 | 0.1 |

(1, 0, 0) | 1 | 0.1 | 0.9 | 0.367 | 0.207 |

(0, 1, 0) | 1 | 0.1 | 0.9 | 0.367 | 0.207 |

(0, 0, 1) | 1 | 0.1 | 0.9 | 0.367 | 0.207 |

(1, 1, 0) | 2 | 0.1 | 0.9 | 0.633 | 0.233 |

(1, 0, 1) | 2 | 0.1 | 0.9 | 0.633 | 0.233 |

(0, 1, 1) | 2 | 0.1 | 0.9 | 0.633 | 0.233 |

(1, 1, 1) | 3 | 0.9 | 0.9 | 0.9 | 0.9 |

**q**

_{i}= (1, 1, 1), g

_{i}= s

_{i}= 0.1, and τ

_{i}

_{1}= 0.8, τ

_{i}

_{2}= 0.1, where

**q**

_{i}is the required attribute profile of item i, g

_{i}, and s

_{i}is the guessing and slip parameters of item i, respectively; τ

_{i}

_{1}is the item-level mixing proportion parameter for the conjunctive condensation rule; τ

_{i}

_{2}is the item-level mixing proportion parameter for the disjunctive condensation rule; DINA = deterministic input, noisy ‘and’ gate model; DINO = deterministic input, noisy ‘or’ gate model; DINR = deterministic input, noisy ratio model; DINMix = deterministic input, noisy mixed model.

LSM | IQ | N | I | TM | g | s | τ_{1} | τ_{2} | τ_{3} | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Bias | RMSE | Bias | RMSE | Bias | RMSE | Bias | RMSE | Bias | RMSE | |||||

Un | High | 500 | 15 | Unif | 0.017 | 0.048 | 0.020 | 0.054 | −0.041 | 0.100 | −0.048 | 0.107 | 0.089 | 0.150 |

Skew | 0.015 | 0.049 | 0.020 | 0.049 | 0.039 | 0.089 | 0.023 | 0.107 | −0.061 | 0.154 | ||||

30 | Unif | 0.013 | 0.037 | 0.013 | 0.035 | −0.036 | 0.083 | −0.041 | 0.090 | 0.077 | 0.121 | |||

Skew | 0.009 | 0.036 | 0.011 | 0.035 | −0.023 | 0.089 | −0.013 | 0.084 | 0.037 | 0.134 | ||||

1000 | 15 | Unif | 0.009 | 0.040 | 0.014 | 0.038 | −0.029 | 0.085 | −0.024 | 0.093 | 0.053 | 0.132 | ||

Skew | 0.008 | 0.036 | 0.007 | 0.036 | 0.013 | 0.076 | 0.013 | 0.087 | −0.026 | 0.130 | ||||

30 | Unif | 0.005 | 0.025 | 0.008 | 0.026 | −0.042 | 0.078 | −0.035 | 0.075 | 0.077 | 0.126 | |||

Skew | 0.006 | 0.024 | 0.007 | 0.024 | −0.025 | 0.073 | −0.026 | 0.072 | 0.050 | 0.119 | ||||

Low | 500 | 15 | Unif | 0.012 | 0.059 | 0.022 | 0.065 | −0.019 | 0.080 | −0.020 | 0.085 | 0.039 | 0.077 | |

Skew | 0.013 | 0.063 | 0.018 | 0.064 | 0.084 | 0.117 | 0.054 | 0.153 | −0.138 | 0.215 | ||||

30 | Unif | 0.011 | 0.049 | 0.017 | 0.048 | −0.030 | 0.096 | −0.015 | 0.099 | 0.045 | 0.081 | |||

Skew | 0.008 | 0.050 | 0.012 | 0.047 | −0.009 | 0.121 | 0.003 | 0.124 | 0.006 | 0.167 | ||||

1000 | 15 | Unif | 0.006 | 0.048 | 0.015 | 0.055 | −0.026 | 0.099 | −0.012 | 0.097 | 0.039 | 0.087 | ||

Skew | 0.001 | 0.052 | 0.010 | 0.049 | 0.081 | 0.131 | 0.036 | 0.144 | −0.117 | 0.200 | ||||

30 | Unif | 0.006 | 0.037 | 0.010 | 0.039 | −0.034 | 0.092 | −0.031 | 0.094 | 0.065 | 0.104 | |||

Skew | 0.001 | 0.036 | 0.008 | 0.036 | −0.013 | 0.099 | −0.009 | 0.107 | 0.022 | 0.145 | ||||

MVN | High | 500 | 15 | Unif | 0.007 | 0.031 | 0.007 | 0.031 | −0.039 | 0.089 | −0.039 | 0.090 | 0.078 | 0.130 |

Skew | 0.009 | 0.032 | 0.006 | 0.030 | 0.052 | 0.104 | 0.036 | 0.113 | −0.089 | 0.173 | ||||

30 | Unif | 0.005 | 0.026 | 0.004 | 0.025 | −0.043 | 0.086 | −0.039 | 0.086 | 0.083 | 0.125 | |||

Skew | 0.005 | 0.026 | 0.005 | 0.026 | −0.017 | 0.095 | −0.014 | 0.089 | 0.031 | 0.147 | ||||

1000 | 15 | Unif | 0.004 | 0.023 | 0.005 | 0.023 | −0.037 | 0.087 | −0.038 | 0.093 | 0.076 | 0.156 | ||

Skew | 0.002 | 0.021 | 0.006 | 0.022 | 0.019 | 0.085 | 0.027 | 0.094 | −0.046 | 0.146 | ||||

30 | Unif | 0.003 | 0.018 | 0.004 | 0.018 | −0.040 | 0.079 | −0.044 | 0.080 | 0.084 | 0.133 | |||

Skew | 0.002 | 0.018 | 0.003 | 0.018 | −0.019 | 0.077 | −0.019 | 0.078 | 0.037 | 0.132 | ||||

Low | 500 | 15 | Unif | −0.001 | 0.040 | 0.002 | 0.041 | −0.032 | 0.097 | −0.036 | 0.097 | 0.069 | 0.095 | |

Skew | −0.004 | 0.041 | −0.001 | 0.039 | 0.087 | 0.134 | 0.041 | 0.141 | −0.128 | 0.201 | ||||

30 | Unif | 0.004 | 0.034 | 0.007 | 0.034 | −0.033 | 0.096 | −0.028 | 0.097 | 0.061 | 0.090 | |||

Skew | 0.002 | 0.034 | 0.006 | 0.033 | −0.008 | 0.126 | 0.001 | 0.124 | 0.007 | 0.167 | ||||

1000 | 15 | Unif | 0.000 | 0.031 | 0.001 | 0.029 | −0.040 | 0.105 | −0.029 | 0.103 | 0.069 | 0.110 | ||

Skew | −0.003 | 0.031 | 0.004 | 0.031 | 0.057 | 0.109 | 0.044 | 0.127 | −0.102 | 0.176 | ||||

30 | Unif | 0.001 | 0.025 | 0.005 | 0.024 | −0.037 | 0.096 | −0.033 | 0.092 | 0.071 | 0.106 | |||

Skew | 0.001 | 0.023 | 0.005 | 0.023 | −0.011 | 0.099 | −0.007 | 0.098 | 0.018 | 0.149 |

_{1}= item-level mixing proportion parameter for the conjunctive condensation rule; τ

_{2}= item-level mixing proportion parameter for the disjunctive condensation rule; τ

_{3}= item-level mixing proportion parameter for the ratio condensation rule; LSM = latent structural model; IQ = item quality; N = sample size; I = test length; TM = type of item-level mixing proportion; Un = unstructured LSM; MVN = multivariate normal distribution; Unif = uniform; RMSE = root mean square error.

Test Situation | Analysis Model | DIC | Test_–2LCPO | RMSE_g | RMSE_s | PCCR | RMSE_α |
---|---|---|---|---|---|---|---|

Conjunctive | DINA | 20,748.69 | 21,201.02 | 0.012 | 0.021 | 0.805 | 0.006 |

DINMix | 20,852.34 | 21,256.41 | 0.024 | 0.022 | 0.804 | 0.006 | |

Disjunctive | DINO | 20,746.62 | 21,200.64 | 0.022 | 0.013 | 0.806 | 0.006 |

DINMix | 20,827.94 | 21,251.88 | 0.022 | 0.024 | 0.803 | 0.006 | |

Ratio | DINR | 29,963.85 | 30,625.57 | 0.020 | 0.020 | 0.853 | 0.004 |

DINMix | 30,020.02 | 30,676.92 | 0.021 | 0.021 | 0.852 | 0.004 | |

Compensatory | ACDM | 29,613.86 | 30,390.21 | 0.020 | 0.020 | 0.853 | 0.003 |

DINMix | 30,084.68 | 30,713.51 | 0.022 | 0.021 | 0.844 | 0.004 | |

Fuzzily | GDINA | 28,541.86 | 29,260.53 | 0.020 | 0.022 | 0.845 | 0.004 |

DINMix | 28,743.04 | 29,372.46 | 0.023 | 0.024 | 0.840 | 0.004 | |

Separately | DINA | 29,348.12 | 29,971.11 | 0.195 | 0.035 | 0.746 | 0.007 |

DINO | 30,078.30 | 30,616.86 | 0.039 | 0.227 | 0.726 | 0.008 | |

DINR | 28,386.28 | 28,972.58 | 0.079 | 0.093 | 0.827 | 0.004 | |

ACDM | 28,266.83 | 28,911.75 | 0.079 | 0.092 | 0.828 | 0.004 | |

GDINA | 25,595.61 | 26,341.86 | 0.022 | 0.022 | 0.900 | 0.003 | |

DINMix | 25,744.26 | 26,397.83 | 0.021 | 0.022 | 0.899 | 0.003 |

Analysis Model | DIC | Test-Level –2LCPO |
---|---|---|

DINA | 8347.341 | 7882.304 |

DINO | 9038.505 | 8665.394 |

DINR | 8911.684 | 8293.461 |

ACDM | 10,528.705 | 7826.567 |

GDINA | 11,096.800 | 7690.248 |

DINMix | 8330.826 | 7803.378 |

Wald-selected | 8785.672 | 7783.750 |

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## Share and Cite

**MDPI and ACS Style**

Zhan, P.
Deterministic Input, Noisy Mixed Modeling for Identifying Coexisting Condensation Rules in Cognitive Diagnostic Assessments. *J. Intell.* **2023**, *11*, 55.
https://doi.org/10.3390/jintelligence11030055

**AMA Style**

Zhan P.
Deterministic Input, Noisy Mixed Modeling for Identifying Coexisting Condensation Rules in Cognitive Diagnostic Assessments. *Journal of Intelligence*. 2023; 11(3):55.
https://doi.org/10.3390/jintelligence11030055

**Chicago/Turabian Style**

Zhan, Peida.
2023. "Deterministic Input, Noisy Mixed Modeling for Identifying Coexisting Condensation Rules in Cognitive Diagnostic Assessments" *Journal of Intelligence* 11, no. 3: 55.
https://doi.org/10.3390/jintelligence11030055