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

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

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

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

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

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

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

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

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 Background

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

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

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

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

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

Attributes Required for Each Option of Option (1) Borrowing from whole (2) Basic fraction subtraction (3) Reducing (4) Separating whole from fraction (5) Converting whole to fraction A) B) C) D)

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

Attributes Required for Each Option of Option (1) Borrowing from whole (2) Basic fraction subtraction (3) Reducing (4) Separating whole from fraction (5) Converting whole to fraction A) B) C) D)

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

Attributes Required for Each Option of Option (1) Borrowing from whole (2) Basic fraction subtraction (3) Reducing (4) Separating whole from fraction (5) Converting whole to fraction A) B) C) D)

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

“0”

“0” “1”

“2” “3” “1”

“2”“4” “3” “1” “0”

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

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

Option GroupABCD

Option GroupABCD 0P(A|0)

Option GroupABCD 0P(A|0)P(B|0)

Option GroupABCD 0P(A|0)P(B|0)P(C|0)

Option GroupABCD 0P(A|0)P(B|0)P(C|0)P(D|0)

Option GroupABCD 0P(A|0)P(B|0)P(C|0)P(D|0) 1P(A|1)P(B|1)P(C|1)P(D|1) 2 3 4

Option GroupABCD 0P(A|0)P(B|0)P(C|0)P(D|0) 1P(A|1)P(B|1)P(C|1)P(D|1) 2P(A|2)P(B|2)P(C|2)P(D|2) 3P(A|3)P(B|3)P(C|3)P(D|3) 4P(A|4)P(B|4)P(C|4)P(D|4)

Option GroupABCD

Option GroupABCD 0 1

Option GroupABCD 0P(A|0)P(B|0)P(C|0)P(D|0) 1P(A|1)P(B|1)P(C|1)P(D|1) DINA Model for Nominal Response N-DINA Model

Option 0 1 A B CDGroup

Option Group01 0 1

Option Group01 0P(0|0)P(1|0) 1P(0|1)P(1|1) Plain DINA Model P(1|0) – guessing parameter P(0|1) – slip parameter

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

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 Estimation

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

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

Results Bias, Mean and Empirical SE Across 30 Items

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

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

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

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

Illustration of the impact of Consider the following three items

Implications The differences in sample sizes in the latent groups account for the observed differences in the SEs of the parameter estimates This underscores the importance, not only of the overall sample size I, but also the expected numbers of examinees in the latent groups in determining the precision of the estimates

Attribute Classification Accuracy Percent of Attribute Correctly Classified

Summary and Conclusion There is an urgent need for assessments that provide interpretative, diagnostic, highly informative, and potentially prescriptive scores This type of scores can inform classroom instruction and learning With appropriate construction, MC items can be designed to be more diagnostically informative Diagnostic information in MC distractors can be harnessed using the MC-DINA

Parameters of the MC-DINA model can be accurately estimated MC-DINA attribute classification accuracy is dramatically better than the traditional DINA Caveat: This framework is only the psychometric aspect of cognitive diagnosis Development of cognitively diagnostic assessment is a multi-disciplinary endeavor requiring collaboration between experts from learning science, cognitive science, subject domains, didactics, psychometrics,...

More general version of the model (e.g., attribute specifications need not be linear, exhaustive nor hierarchical) Applications to traditional MC assessments Issues related to sample size – – Sample size needed for different numbers of items and attributes, and types of attribute specifications – – Trade-off between the number of coded options and sample size necessary for stable estimates – – Feasibility of some simplifying assumptions such as equiprobability in choosing non-expected responses Further considerations

That’s all folks!