A procedure for dimensionality analyses of response data from

various test designs

Jinming Zhang

William Stout

Introduction

Dimension Dimensional structure of the test (e.g.,

algebra and geometry) statistical dimensional structure of response

data Incorporate both Judgments about test

content and evidence from statistical analyses

Missing data: CAT & multistage testing Missing item pair measurement

DETECT index

Modified DETECT index

Example 3 Stage 1 booklet: 2 dim & Stage 2 booklet 1: 2

dim, Stage 2 booklet 2: 1 dim (additional dim), Stage 2 booklet 3: 1 dim (additional dim).

Bridge item (in Stage 1) E(1,2) and E(2,3), so E(1,3)

First-stage booklet should measure all of the constructs/contents the whole

test aim to measure, though it is unknown. classify examinees into different proficiency levels

Item 1 and 2 are measuring the same dimension

1. If E(1,2), E(2,3), and E(1,3) D(P*) is maximized in P* partition if item 1 and

2 are in the same cluster, other than in other P

2. If only E(1,2) and E(2,3), and if item 2 is a bridge item

D(P*) is maximized in P* partition if item 1 and 2 are in the same cluster when item 2 is in the same cluster or not, other than in other P

Whether a test has an approximate simple structure or not

Discordance Resulted from

Not a approximate simple structure Inaccuracy of conditional covariance

estimation Given a partition of items,

What made Dd(P*) and PropD(P*) large Unidimensionality Violation of approximate simple

structure Inaccuracy of conditional covariance

estimation

polyDETECT

Obtain the composite theta score: unidimensinoal approximation & simple structure approach.

Use percentiles of composite theta scores as cut-off points in forming AHGs. Between 25 to 100 in each group

Cross-validation:

polyDETECT

Evidence of multidimensionality

Condition 3 is hold or not? Each sizable cluster contains at least one

stage-1 item. Exist at least one sizable cluster that does not

contain any Stage-1 items, and all items in such cluster belong to the same booklet.

Exist at least one sizable cluster that does not contain any Stage-1 items, and all items in such cluster belong to at least two booklet.

Exist at least two sizable cluster that does not contain any Stage-1 items, items in each such cluster come from the same booklet but different clusters belong to different booklets.

Dealing with CAT data

Estimates of item parameters were obtained before conducting CAT. Why do dimensionality assessment on CAT data?

Sparse data set of CAT 100,000 responses are required at

least Item selection, item exposure

control, content balance to satisfy the condition 3

Simulation study M2PL Each booklet has 30 items Dimension: 1, 2, 3 Number of examinees: 750, 1500, 3000, 4000 Theta: MVN(0,sigma), correlation = 0.8 Cut-off points for low-, moderate-, and high-

scoring group: <10, 11~18, >18 About 37.82% unestimable item pairs Cross-validation Replication: 100 Composite theta: use unidimensional IRT

model

Results

Real data analyses

Missing values are large (55%~71%)

Real data analyses

Weak dimensionality (M value) High PropD(P*) indicates a large amount of

spurious information to form partition P* Confirmatory analyses: D3 >D2

But DETECT tends to underestimate, so two-cluster partition solution may be preferred.

Real data analyses

High correlation indicates a weak degree of multidimensionality

Concluding remarks

Moderate violation of approximate simple structure is still hold