A Cognitive Diagnosis Model for Cognitively-Based Multiple-Choice Options

1. A Cognitive Diagnosis Model forCognitively-Based Multiple-Choice Options Jimmy de la Torre Department of Educational Psychology Rutgers, The State University of New Jersey

2. All wrong answers are wrong; But some wrong answers are more wrong than others.

3. Introduction • Assessments should educate and improve student performance, not merely audit it • In other words, assessments should not only ascertain the status of learning, but also further learning • Due to emphasis on accountability, more and more resources are allocated towards assessments that only audit learning • Tests used to support school and system accountability do not provide diagnostic information about individual students

4. Tests based on unidimensional IRT models report single-valued scores that submerge any distinct skills • These scores are useful in establishing relative order but not evaluation of students' specific strengths and weaknesses • Cluster scores have been used, but these scores are unreliable and provide superficial information about the underlying processes • Needed are assessments that can provide interpretative, diagnostic, highly informative, and potentially prescriptive information

5. Some psychometric models allow the merger of advances in cognitive and psychometric theories to provide inferences more relevant to learning • These models are called cognitive diagnosis models (CDMs) • CDMs are discrete latent variable models • They are developed specifically for diagnosing the presence or absence of multiple fine-grained skills, processes or problem-solving strategies involved in an assessment

6. Fundamental difference between IRT and CDM: A fraction subtraction example • IRT: performance is based on a unidimensional continuous latent trait • Students with higher latent traits have higher probability of answering the question correctly

8. Fundamental difference between IRT and CDM: A fraction subtraction example • IRT: performance is based on a unidimensional continuous latent trait • Students with higher latent traits have higher probability of answering the question correctly • CDM: performance is based on binary attribute vector • Successful performance on the task requires a series of successful implementations of the attributes specified for the task

9. Required attributes: (1) Borrowing from whole (2) Basic fraction subtraction (3) Reducing • Other attributes: (4) Separating whole from fraction (5) Converting whole to fraction

10. 1 0.75 0.5 0.25 0

11. Background • Denote the response and attribute vectors of examinee i by and • Each attribute pattern is a unique latent class; thus, K attributes define latent classes • Attribute specification for the items can be found in the Q-matrix, a J x K binary matrix • DINA (Deterministic Input Noisy “And” gate) is a CDM model that can be used in modeling the distribution of given

12. In the DINA model • where is the latent group classification of examinee i with respect to item j • P(H|g) is the probability that examinees in group g will respond with h to item j • In more conventional notation of the DINA = guessing, = slip

13. Of the various test formats, multiple-choice (MC) has been widely used for its ability to sample and accommodate diverse contents • Typical CDM analyses of MC tests involve dichotomized scores (i.e., correct/incorrect) • The approach ignores the diagnostic insights about student difficulties and alternative conceptions in the distractors • Wrong answers can reveal both what students know and what they do not know

14. Purpose of the paper is to propose a two-component framework for maximizing the diagnostic value of MC assessments • Component 1: Prescribes how MC options can be designed to contain more diagnostic information • Component 2: Describes a CDM model that can exploit such information • Viability (i.e., estimability, efficiency) of the proposed framework is evaluated using a simulation study

15. Component 1: Cognitively-Based MC Options • For the MC format, , where each number represents a different option • An option is coded or cognitively-based if it is constructed to correspond to some of the latent classes • Each coded option has an attribute specification • Attribute specifications for non-coded options are implicitly represented by the zero-vector

16. A Fraction Subtraction Example A) B) C) D)

17. Attributes Required for Each Option of

18. The option with the largest number of required attributes is the key

19. Attributes Required for Each Option of

20. The option with the largest number of required attributes is the key • Distractors are created to reflect the type of responses students who lack one or more of the required attributes for the key are likely to give

21. Attributes Required for Each Option of

22. The option with the largest number of required attributes is the key • Distractors are created to reflect the type of responses students who lack one or more of the required attributes for the key are likely to give • Knowledge states represented by the distractors should be in the subset of the knowledge state that corresponds to the key • Number of latent classes under the proposed framework is equal to , the number of coded options plus 1

23. 000 100 010 001 110 101 011 111 “0”

24. “0” “1” 000 100 010 001 110 101 011 111

25. “1” “2” “3” 000 100 010 001 110 101 011 111

26. “1” “2” “4” “3” “0” 000 100 010 001 110 101 011 111

27. Component 2: The MC-DINA Model • Let be the Q-vector for option h of item j, and • With respect to item j, examinee i is in group • Probability of examinee i choosing option h of item j is

28. This is the DINA model extended to coded MC options, hence, MC-DINA model • Each item has parameters • Expected response for a group, say h, is its coded option h: “correct” response for group h • MC-DINA model can still be used even if only the key is coded as long as the distractors are distinguished from each other • The MC-DINA model is equivalent to the DINA model if no distinctions are made between the distractors

36. 0 1 2 3 4

38. DINA Model for Nominal Response N-DINA Model

39. Group A C D B 0 1

41. P(1|0) – guessing parameter P(0|1) – slip parameter Plain DINA Model

42. Estimation • Like in IRT, JMLE of the MC-DINA model parameters can lead to inconsistent estimates • Using MMLE, we maximize prior probability of , the marginalized likelihood of examinee i

43. Estimation • Like in IRT, JMLE of the MC-DINA model parameters can lead to inconsistent estimates • Using MMLE, we maximize • The estimator based on an EM algorithm is where is the expected number of examinees in group g choosing option h of item j

44. A Simulation Study • Purpose: To investigate how • well the item parameters and SE can be estimated • accurately the attributes can be classified • MC-DINA compares with the traditional DINA • 1000 examinees, 30 items, 5 attributes • Parameters: • Number of replicates: 100

45. Required attribute per item: 1, 2 or 3 (10 each) • Exhaustive hierarchically linear specification: • One-attribute item • Two-attribute item • Three-attribute item 0 0 0

46. Results Bias, Mean and Empirical SE Across 30 Items

47. Bias, Mean and Empirical SE by Item Classification (True Probability: 0.25)

48. Bias, Mean and Empirical SE by Item Classification (True Probability: 0.82)

49. Bias, Mean and Empirical SE by Item Classification (True Probability: 0.06)

50. Review of Parameter Estimation Results • Algorithm provides accurate estimates of the model parameters and SEs • SE of does not depend on item type • When , • What factor affects the precision of ? , expected number of examinees in group g of item j