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

2. 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

3. 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)

4. 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.

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

6. 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)

7. Measurement model: Poisson (1)

8. Measurement model: Poisson (2)

9. Parameter estimation: incomplete design (JML)

10. Person parameters

11. 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

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

14. Advantage

15. Structural model: distribution of reading speed (θ)

16. Marginal distribution of the sum score s

17. Negative Binomial (Gamma-Poisson)

18. Negative binomial

19. EAP

20. Reliability

21. Validation (tempo test)

22. Validation (tempo test)

23. Validation (TMT)

24. 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

26. Validation (TMT)

27. Validation (TMT)

28. Norms (TMT)

29. Thank you

30. Example: student v Task i dvi 1 8 0.93 - 2 1.11 8.88 3 6 0.85 4 1.058
0.93 - 2
1.11
8.88 3 6
0.85 4
1.05
6.30 5
1.09
δv :
15.18

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

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

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