# Relating the One-Parameter Logistic Diagnostic Classification Model to the Rasch Model and One-Parameter Logistic Mixed, Partial, and Probabilistic Membership Diagnostic Classification Models

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Unidimensional Item Response Models

#### 2.1. Implementation

`mirt::mirt()`can be used to estimate user-defined IRT models. When relying on this function, user-defined IRFs ${P}_{i}$ can be specified with

`mirt::createItem()`, whereas user-defined distributions ${F}_{\mathsf{\delta}}$ can be specified with

`mirt::createGroup()`. In the sirt [28] package, the function

`sirt::xxirt()`can be used in combination with

`sirt::xxirt_createDiscItem()`(for defining IRFs) and

`sirt::xxirt_createThetaDistribution()`(for defining the distribution ${F}_{\mathsf{\delta}}$).

#### 2.2. Rasch Model

#### 2.3. Generalized Logistic Item Response Model

## 3. Unidimensional Diagnostic Classification Models

#### 3.1. Two-Parameter Diagnostic Classification Model

#### 3.2. One-Parameter Logistic Diagnostic Classification Model

#### 3.3. One-Parameter Generalized Logistic Diagnostic Classification Model

## 4. Extensions of Diagnostic Classification Models to Mixed and Partial Membership

#### 4.1. Mixed Membership Diagnostic Classification Model

#### 4.2. Partial Membership Diagnostic Classification Model

#### 4.3. Probabilistic Membership Diagnostic Classification Model

## 5. Numerical Illustration

`data.read`($N=328$, $I=12$),

`data.pisaMath`($N=565$, $I=11$),

`data.pisaRead`($N=623$, $I=12$), and

`data.trees`($N=387$, $I=15$; [77]) are included in the R [26] package sirt [28]. The datasets

`data.numeracy`($N=876$, $I=15$) and

`data.ecpe`($N=2922$, $I=28$; [11,78]) can be found in the R packages TAM [79] and CDM [15], respectively. All datasets did not contain missing values.

`sirt::xxirt()`function in the sirt [28] package. The model estimation always used 100 EM iterations initially and switched afterward to Newton–Raphson optimization. Replication material can be found at https://osf.io/kfcdb/?view_only=2073df46d35f44a5bcb5dd9cc77013e5 (accessed on 15 September 2023).

`ecpe`,

`numeracy`, and

`pisaRead`). This means that the normal distribution assumption for the latent trait $\mathsf{\theta}$ is violated. For all datasets, the LCRM2 (i.e., the 1PLDCM) was inferior to the RM with a normal distribution (NO). This finding implies that a normal distribution assumption for $\mathsf{\theta}$ was more reasonable than a two-point distribution. For all datasets except for

`pisaRead`, the latent class model with two classes based on the generalized logistic link function (GLLC2) outperformed the latent class model using the logistic link function (LCRM2). Interestingly, for four datasets, the LCRM with four or five classes fit the data better than model NO in terms of AIC differences. However, in these cases, a Rasch model with a skewed distribution (SK) had a comparable fit with the best-fitting LCRMs.

## 6. Discussion

## Funding

## Institutional Review Board Statement

## Conflicts of Interest

## Abbreviations

1PL | one-parameter logistic |

1PLDCM | one-parameter logistic diagnostic classification model |

2PL | two-parameter logistic |

AIC | Akaike information criterion |

BIC | Bayesian information criterion |

DCM | diagnostic classification model |

EM | expectation maximization |

GDINA | generalized deterministic inputs, noisy “and” gate |

IRF | item response function |

IRT | item response theory |

LCRM | latent class Rasch model |

LDCM | logistic diagnostic classification model |

MML | marginal maximum likelihood |

RM | Rasch model |

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**Table 1.**AIC differences ($\mathrm{\Delta}\mathrm{AIC}$) and BIC differences ($\mathrm{\Delta}\mathrm{BIC}$) for eleven analysis models for six different datasets.

Datasets | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

read | ecpe | numeracy | pisaMath | pisaRead | trees | ||||||||||||

$\mathbf{\Delta}\mathbf{AIC}$ | $\mathbf{\Delta}\mathbf{BIC}$ | $\mathbf{\Delta}\mathbf{AIC}$ | $\mathbf{\Delta}\mathbf{BIC}$ | $\mathbf{\Delta}\mathbf{AIC}$ | $\mathbf{\Delta}\mathbf{BIC}$ | $\mathbf{\Delta}\mathbf{AIC}$ | $\mathbf{\Delta}\mathbf{BIC}$ | $\mathbf{\Delta}\mathbf{AIC}$ | $\mathbf{\Delta}\mathbf{BIC}$ | $\mathbf{\Delta}\mathbf{AIC}$ | $\mathbf{\Delta}\mathbf{BIC}$ | ||||||

NO ${}^{\u2020}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||

SK | −2 | −6 | 55 | 49 | 31 | 26 | −1 | −6 | 14 | 10 | −2 | −6 | |||||

LCRM2 | −18 | −21 | −777 | −782 | −247 | −251 | −63 | −67 | −89 | −93 | −38 | −42 | |||||

GLLC2 | −8 | −20 | −739 | −757 | −221 | −235 | −32 | −45 | −92 | −105 | −21 | −33 | |||||

LCRM3 | −2 | −13 | −82 | −100 | −43 | −58 | 7 | −6 | −9 | −22 | −4 | −16 | |||||

LCRM4 | −3 | −22 | 47 | 17 | 36 | 12 | 3 | −18 | 6 | −16 | −5 | −25 | |||||

LCRM5 | −7 | −34 | 48 | 6 | 35 | 1 | 3 | −27 | 6 | −25 | −9 | −37 | |||||

PRLLC2 | 2 | −6 | 42 | 30 | 35 | 25 | 6 | −3 | 13 | 4 | −2 | −10 | |||||

PMLLC2 | 2 | −6 | 42 | 30 | 34 | 25 | 7 | −1 | 13 | 4 | −2 | −10 | |||||

MMLLC2 | −14 | −22 | −226 | −238 | −6 | −16 | −8 | −17 | −62 | −71 | 0 | −8 | |||||

GL | 8 | 0 | 2 | −10 | 9 | 0 | 16 | 7 | 0 | −8 | 8 | 0 |

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**MDPI and ACS Style**

Robitzsch, A.
Relating the One-Parameter Logistic Diagnostic Classification Model to the Rasch Model and One-Parameter Logistic Mixed, Partial, and Probabilistic Membership Diagnostic Classification Models. *Foundations* **2023**, *3*, 621-633.
https://doi.org/10.3390/foundations3030037

**AMA Style**

Robitzsch A.
Relating the One-Parameter Logistic Diagnostic Classification Model to the Rasch Model and One-Parameter Logistic Mixed, Partial, and Probabilistic Membership Diagnostic Classification Models. *Foundations*. 2023; 3(3):621-633.
https://doi.org/10.3390/foundations3030037

**Chicago/Turabian Style**

Robitzsch, Alexander.
2023. "Relating the One-Parameter Logistic Diagnostic Classification Model to the Rasch Model and One-Parameter Logistic Mixed, Partial, and Probabilistic Membership Diagnostic Classification Models" *Foundations* 3, no. 3: 621-633.
https://doi.org/10.3390/foundations3030037