The Poisson-Gamma model for speed tests Norman Verhelst Frans Kamphuis National Institute for Educational Measurement Arnhem, The Netherlands

The student monitoring system Measurement of individual development Common scale Estimation of distribution (norms) Twice per grade (M3, E3,…,M8) Several subjects Arithmetic Reading comprehension Technical reading

Two types of speed tests Basic observation is the time to complete a task AVI cards Basic observation is the number of completed subtasks within the time limit Tempotests (TT) Three Minute Test (TMT)

Example tempotest (E4) Op de politieschool spelen ze ook rook koor een soort toneel Het lijkt wel wat op ‘politie en boefje spelen stelpen slepen’. Net zoals op de basisschool. Wat poe doe boe je bij een gevecht? Je pistool trekken? Nee, dat mag zomen zomaar zomer niet.

Example TMT Easy version Hard version as fee oom uur zee oor … poot (=150) Hard version banden geluid tante beker kuiken koffer … brandweerwagen (=150)

Models Measurement model: Poisson Structural model: Gamma What is the relation between the (latent) ability and the test performance? Structural model: Gamma The distribution of the latent ability in one or more populations? (M3, E3, M4,…,M8)

Measurement model: Poisson (1)

Measurement model: Poisson (2)

Parameter estimation: incomplete design (JML)

Person parameters

Design TMT 3 difficulty levels (1, 2, 3) For each level: three parallell versions (a, b, c) Each student participates twice: medio and end of same grade At each administration: 3 cards of levels 1, 2 and 3 (in that sequence) M3: only cards 1 and 2

Two step procedure Estimate the task parameters σi JML = CML Estimate latent distribution while fixing the task parameters at their CML -estimate

Advantage

Structural model: distribution of reading speed (θ)

Marginal distribution of the sum score s

Negative Binomial (Gamma-Poisson)

Negative binomial

EAP

Reliability

Validation (tempo test)

Validation (tempo test)

Validation (TMT)

Latent class model Population consists of two latent classes of size π and 1 - π respectively The latent variable is gamma distributed in each class Parameters π α1 en β1 α2 en β2 EM-algorithm

Validation (TMT)

Validation (TMT)

Norms (TMT)

Thank you

Example: student v Task i dvi 1 8 0.93 - 2 1.11 8.88 3 6 0.85 4 1.05 8 0.93 - 2 1.11 8.88 3 6 0.85 4 1.05 6.30 5 1.09 δv : 15.18

Problems SE(π) large Local maxima? Thick right tail of observations >2 classes? Initial estimates Homogeneity of test material Local independence

Averages (1000 replications) Class 1 Class 2 Overall Mean 28.15 44.07 35.99 SD 2.71 3.22 0.43

Standard deviations (1000 rep.) Class 1 Class 2 Overall Mean 13.31 17.44 17.66 SD 2.21 1.68 0.47