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

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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

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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)

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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.

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

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Models •Measurement model: Poisson –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)

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Measurement model: Poisson (1)

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Measurement model: Poisson (2)

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Parameter estimation: incomplete design (JML)

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Person parameters

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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

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Two step procedure •Estimate the task parameters σ i –JML = CML •Estimate latent distribution while fixing the task parameters at their CML -estimate

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Advantage

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Structural model: distribution of reading speed (θ)

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Marginal distribution of the sum score s

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Negative Binomial (Gamma-Poisson)

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Negative binomial

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EAP

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Reliability

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Validation (tempo test)

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Validation (TMT)

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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 –α 1 en β 1 –α 2 en β 2 • EM-algorithm

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Validation (TMT)

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Norms (TMT)

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Thank you

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Example: student v Task id vi δv :δv : 15.18

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

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Class 1Class 2Overall Mean SD Averages (1000 replications)

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Standard deviations (1000 rep.) Class 1Class 2Overall Mean SD

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