# A Nonparametric Weighted Cognitive Diagnosis Model and Its Application on Remedial Instruction in a Small-Class Situation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. NPCD Model

## 3. The Proposed Method: Nonparametric Weighted Cognitive Diagnosis

#### 3.1. NWSD Model

#### 3.2. NWBD Model

#### 3.3. A Nonparametric CDM Website

- Upload the Q-matrix saved in a CSV (Comma-Separated Values) file.
- Upload students’ responses saved in a CSV file.
- Choose the nonparametric CDMs, NPCD, NWSD, or NWBD from the method list.
- Press the “Go” button to obtain the mastery profiles of students.

## 4. Simulation Studies on Artificial and Real Datasets

#### 4.1. Case 1

#### 4.2. Case 2

## 5. Experiments on Remedial Instruction

#### 5.1. Personalized Instruction Based on NWSD

#### 5.2. Personalized Instruction Based on NWSD and NWBD

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The remedial instructions for (

**a**) Experimental Group with individual video instruction and (

**b**) Control Group with traditional group instruction.

Item | Q-Matrix (w.r.t. Skills) | M-Matrix (w.r.t. Bugs) | |||||
---|---|---|---|---|---|---|---|

S1 | S2 | S3 | S4 | B1 | B2 | B3 | |

1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |

2 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |

3 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

4 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

5 | 1 | 0 | 0 | 0 | 1 | 0 | 0 |

6 | 0 | 1 | 0 | 0 | 1 | 0 | 0 |

7 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |

8 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |

9 | 1 | 1 | 0 | 0 | 1 | 0 | 0 |

10 | 1 | 0 | 1 | 0 | 0 | 0 | 1 |

11 | 1 | 0 | 0 | 1 | 0 | 0 | 1 |

12 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |

13 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |

14 | 0 | 0 | 1 | 1 | 0 | 1 | 1 |

15 | 1 | 0 | 1 | 0 | 1 | 1 | 0 |

16 | 1 | 1 | 0 | 1 | 1 | 1 | 0 |

17 | 0 | 1 | 1 | 1 | 1 | 1 | 0 |

**Table 2.**Average AARs and PARs of DINA, G-DINA, NPCD, and NWSD models in Case 1 (corresponding standard deviations are shown in parentheses).

Model | Classification Agreement Rate | I = 20 | I = 50 | I = 100 |
---|---|---|---|---|

DINA | AAR | 0.8137 (0.06) | 0.8194 (0.04) | 0.8281 (0.03) |

PAR | 0.4588 (0.13) | 0.4673 (0.09) | 0.4812 (0.07) | |

G-DINA | AAR | 0.8248 (0.05) | 0.8195 (0.03) | 0.8225 (0.03) |

PAR | 0.4741 (0.12) | 0.4591 (0.08) | 0.4624 (0.07) | |

NPCD | AAR | 0.8297 (0.04) | 0.8337 (0.03) | 0.8328 (0.02) |

PAR | 0.4965 (0.11) | 0.5023 (0.07) | 0.4981 (0.05) | |

NWSD | AAR | 0.8437 (0.05) | 0.8496 (0.03) | 0.8509 (0.02) |

PAR | 0.5130 (0.12) | 0.5243 (0.07) | 0.5253 (0.05) |

**Table 3.**Average AARs and PARs of Bug-DINO and NWBD models in Case 1 (corresponding standard deviations are shown in parentheses).

Model | Classification Agreement Rate | I = 20 | I = 50 | I = 100 |
---|---|---|---|---|

Bug-DINO | AAR | 0.7158 (0.06) | 0.7233 (0.05) | 0.7215 (0.03) |

PAR | 0.3683 (0.10) | 0.3825 (0.07) | 0.3799 (0.05) | |

NWBD | AAR | 0.7818 (0.06) | 0.7966 (0.05) | 0.8010 (0.03) |

PAR | 0.4850 (0.12) | 0.5121 (0.08) | 0.5201 (0.05) |

Item | Q-Matrix (w.r.t. Skills) | M-Matrix (w.r.t. Bugs) | |||||
---|---|---|---|---|---|---|---|

S1 | S2 | S3 | S4 | B1 | B2 | B3 | |

1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 |

2 | 0 | 1 | 1 | 0 | 1 | 0 | 0 |

3 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |

4 | 1 | 1 | 0 | 1 | 1 | 1 | 0 |

5 | 1 | 1 | 1 | 1 | 1 | 1 | 0 |

6 | 1 | 1 | 0 | 1 | 1 | 1 | 0 |

7 | 0 | 1 | 1 | 1 | 0 | 1 | 0 |

**Table 5.**Average AARs and PARs of DINA, G-DINA, NPCD, and NWSD models in Case 2 (corresponding standard deviations are shown in parentheses).

Model | Classification Agreement Rate | I = 20 | I = 50 | I = 100 |
---|---|---|---|---|

DINA | AAR | 0.7716 (0.06) | 0.7797 (0.04) | 0.7774 (0.03) |

PAR | 0.4776 (0.13) | 0.4849 (0.10) | 0.4783 (0.07) | |

G-DINA | AAR | 0.5438 (0.07) | 0.5498 (0.07) | 0.5505 (0.07) |

PAR | 0.0814 (0.08) | 0.1026 (0.08) | 0.1138 (0.09) | |

NPCD | AAR | 0.7894 (0.06) | 0.7893 (0.03) | 0.7875 (0.02) |

PAR | 0.5370 (0.12) | 0.5368 (0.06) | 0.5366 (0.04) | |

NWSD | AAR | 0.8387 (0.07) | 0.8408 (0.03) | 0.8392 (0.02) |

PAR | 0.6495 (0.14) | 0.6532 (0.07) | 0.6483 (0.04) |

**Table 6.**Average AARs and PARs of Bug-DINO and NWBD models in Case 2 (corresponding standard deviations are shown in parentheses).

Model | Classification Agreement Rate | I = 20 | I = 50 | I = 100 |
---|---|---|---|---|

Bug-DINO | AAR | 0.7161 (0.06) | 0.7162 (0.03) | 0.7161 (0.02) |

PAR | 0.3972 (0.10) | 0.3923 (0.06) | 0.3932 (0.04) | |

NWBD | AAR | 0.7695 (0.06) | 0.7684 (0.03) | 0.7680 (0.02) |

PAR | 0.4422 (0.12) | 0.4311 (0.07) | 0.4292 (0.05) |

Skill | Description |
---|---|

S1 | Understanding the meaning of a series |

S2 | Understanding the meaning of an arithmetic series |

S3 | Calculate the sum of a finite arithmetic series |

S4 | Understanding and applying the formulation of the sum of a finite arithmetic series |

S5 | Applying the formulation of the sum of a finite arithmetic series to a real-world problem |

Item | S1 | S2 | S3 | S4 | S5 | Item | S1 | S2 | S3 | S4 | S5 |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | 0 | 0 | 0 | 0 | 11 | 0 | 1 | 1 | 0 | 0 |

2 | 0 | 1 | 0 | 0 | 0 | 12 | 0 | 1 | 0 | 1 | 0 |

3 | 0 | 1 | 0 | 0 | 0 | 13 | 0 | 1 | 0 | 0 | 1 |

4 | 0 | 0 | 1 | 0 | 0 | 14 | 0 | 0 | 1 | 1 | 0 |

5 | 0 | 0 | 0 | 1 | 0 | 15 | 0 | 0 | 1 | 0 | 1 |

6 | 0 | 0 | 0 | 0 | 1 | 16 | 1 | 1 | 1 | 0 | 0 |

7 | 1 | 1 | 0 | 0 | 0 | 17 | 1 | 1 | 0 | 1 | 0 |

8 | 1 | 0 | 1 | 0 | 0 | 18 | 1 | 1 | 0 | 0 | 1 |

9 | 1 | 0 | 0 | 1 | 0 | 19 | 1 | 0 | 1 | 0 | 1 |

10 | 1 | 0 | 0 | 0 | 1 | 20 | 1 | 1 | 1 | 1 | 1 |

Group | Mean Pretest Score | Mean Post-Test Score | t-Value |
---|---|---|---|

Experimental Group | 46.596 | 73.617 | 14.598 *** |

Control Group | 66.851 | 70.105 | 2.295 * |

Variable | Level | Mean ^{a} (SE) | F Values | Post Hoc ^{b} |
---|---|---|---|---|

Pretest | 112.031 *** | |||

DTRIP | Experimental Group | 78.691 | 54.960 *** | Experimental Group > Control Group *** |

Control Group | 65.032 |

^{a}= Covariates appearing in the model are evaluated at the following value: 56.72.

^{b}= Adjustment for multiple comparisons: least significant difference (equivalent to no adjustments).

Type | Name | Description |
---|---|---|

Skill | S1 | Understanding the definition of average acceleration |

S2 | Understanding the conversion between speed units | |

S3 | Understanding that the speed of the object will change when the object moves with acceleration | |

S4 | Understanding the change of speed when the directions of speed and acceleration change | |

S5 | Understanding the V-t diagram of constant acceleration motion is an oblique straight line | |

S6 | Judging the direction of acceleration by the V-t diagram | |

S7 | Understanding the area enclosed by the V-t diagram and the time axis represents “displacement” | |

Bugs | B1 | Calculating speed change by using large speed and small speed |

B2 | Calculating the average acceleration by using the speed on the V-t diagram divided by the time | |

B3 | If the acceleration is a positive (negative) value, then the object will increase (decrease) speed | |

B4 | The acceleration is a positive value when the V-t diagram appears in the first quadrant.The acceleration is a negative value when the V-t diagram appears in the fourth quadrant. | |

B5 | If the figure is drawn up (down), then the displacement direction is the positive (negative) direction. |

Item | Q-Matrix (w.r.t. Skills) | M-Matrix (w.r.t. Bugs) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

S1 | S2 | S3 | S4 | S5 | S6 | S7 | B1 | B2 | B3 | B4 | B5 | |

1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

2 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 |

5 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

6 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

7 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

8 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |

9 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

10 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 |

11 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |

12 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |

13 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |

14 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

15 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 |

16 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |

17 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 |

18 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 |

19 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 |

20 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |

Variable | Level | Mean (SE) | F Values | Post Hoc ^{b} |
---|---|---|---|---|

Covariate | 8.429 ** | |||

DTRIP | Experimental Group | 87.833 | 11.965 ** | Experimental Group > Control Group ** |

Control Group | 75.524 |

^{b}= Adjustment for multiple comparisons: the least significant difference (equivalent to no adjustments).

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**MDPI and ACS Style**

Li, C.-H.; Ju, Y.-J.; Hsieh, P.-J.
A Nonparametric Weighted Cognitive Diagnosis Model and Its Application on Remedial Instruction in a Small-Class Situation. *Sustainability* **2022**, *14*, 5773.
https://doi.org/10.3390/su14105773

**AMA Style**

Li C-H, Ju Y-J, Hsieh P-J.
A Nonparametric Weighted Cognitive Diagnosis Model and Its Application on Remedial Instruction in a Small-Class Situation. *Sustainability*. 2022; 14(10):5773.
https://doi.org/10.3390/su14105773

**Chicago/Turabian Style**

Li, Cheng-Hsuan, Yi-Jin Ju, and Pei-Jyun Hsieh.
2022. "A Nonparametric Weighted Cognitive Diagnosis Model and Its Application on Remedial Instruction in a Small-Class Situation" *Sustainability* 14, no. 10: 5773.
https://doi.org/10.3390/su14105773