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❘❛s❝❤ ▼♦❞❡❧s ❛♥❞ t❤❡ ❘ ♣❛❝❦❛❣❡ ❡❘▼

❘❡✐♥❤♦❧❞ ❍❛t③✐♥❣❡r

■♥st✐t✉t❡ ❢♦r ❙t❛t✐st✐❝s ❛♥❞ ▼❛t❤❡♠❛t✐❝s ❲❯ ❱✐❡♥♥❛

▼✉♥✐❝❤ ✷✵✶✵ ✶

■♥tr♦

❲❤❛t ✐s ❡❘♠❄

• ❡❘♠ s❤♦rt ❢♦r ❡①t❡♥❞❡❞ ❘❛s❝❤ ♠♦❞❡❧❧✐♥❣
• ✐s ❛♥ ❘ ♣❛❝❦❛❣❡
• ✐s ♦♣❡♥ s♦✉r❝❡✿ ♥♦ ❧✐❝❡♥s❡ ❢❡❡s✱ s♦✉r❝❡ ❝♦❞❡ ❛✈❛✐❧❛❜❧❡✱
• P▲✿ s❤❛r❡✱ ❝❤❛♥❣❡✱ ❛♥❞ r❡❞✐str✐❜✉t❡ ✉♥❞❡r ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥s
• ❢♦r ❘❛s❝❤ ❢❛♠✐❧② ♠♦❞❡❧s✿

✉t✐❧✐t✐❡s ❢♦r ✜tt✐♥❣✱ t❡st✐♥❣✱ ❛♥❞ ❞✐s♣❧❛②✐♥❣ r❡s✉❧ts

• ❝✉rr❡♥t❧② ✐♠♣❧❡♠❡♥t❡❞ ♠♦❞❡❧s✿

▲P❈▼✱ P❈▼✱ ▲❘❙▼✱ ❘❙▼✱ ▲▲❚▼✱ ❘▼✱ ✭▲▲❘❆✮

• ✉s❡s ❈▼▲ ❡st✐♠❛t✐♦♥

▼✉♥✐❝❤ ✷✵✶✵ ✷

■♥tr♦

❲❤❛t ✐s ■t❡♠ ❘❡s♣♦♥s❡ ❚❤❡♦r② ✭■❘❚✮❄ ■❘❚ ✐s ❜✉✐❧t ❛r♦✉♥❞ t❤❡ ❝❡♥tr❛❧ ✐❞❡❛✿ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛ s✉❜❥❡❝t✬s ❝❡rt❛✐♥ r❡❛❝t✐♦♥ t♦ ❛ st✐♠✉❧✉s ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❛s ❛ ❢✉♥❝t✐♦♥ ❝❤❛r❛❝t❡r✐s✐♥❣ t❤❡ s✉❜❥❡❝t✬s ❧♦❝❛t✐♦♥ ♦♥ ❛ ❧❛t❡♥t tr❛✐t ♣❧✉s ♦♥❡ ♦r ♠♦r❡ ♣❛r❛♠❡t❡rs ❝❤❛r❛❝t❡r✐s✐♥❣ t❤❡ st✐♠✉❧✉s

−4 −2 2 4 0.0 0.2 0.4 0.6 0.8 1.0 Latent Dimension Probability

▼✉♥✐❝❤ ✷✵✶✵ ✸

❚❤❡ ❘❛s❝❤ ▼♦❞❡❧

❚❤❡ ❘❛s❝❤ ▼♦❞❡❧ ✭❘▼✮ ✭❘❛s❝❤✱ ✶✾✻✵✮ P(Xvi = 1∣θv,βi) = exp(θv − βi) 1 + exp(θv − βi) Xvi ✳ ✳ ✳ ♣❡rs♦♥ v ❣✐✈❡s ❝♦rr❡❝t ❛♥s✇❡r t♦ ✐t❡♠ i θv ✳ ✳ ✳ ❵❛❜✐❧✐t②✬ ♦❢ ♣❡rs♦♥ v βi ✳ ✳ ✳ ❵❞✐✣❝✉❧t②✬ ♦❢ ✐t❡♠ i I1 I2 I3 I4 rv P1 ✶ ✵ ✵ ✵ ✶ P2 ✶ ✵ ✶ ✵ ✷ P3 ✶ ✶ ✵ ✵ ✷ P4 ✵ ✶ ✶ ✶ ✸ si ✸ ✷ ✷ ✶ ✕ ❘❛✇ ❙❝♦r❡s✿ ∑i xvi = rv ∑v xvi = si

▼✉♥✐❝❤ ✷✵✶✵ ✹

❚❤❡ ❘❛s❝❤ ▼♦❞❡❧

❙❡✈❡r❛❧ ■❈❈s

−4 −2 2 4 0.0 0.2 0.4 0.6 0.8 1.0 Latent Dimension Probability to Solve Item 1 Item 2 Item 3

▼✉♥✐❝❤ ✷✵✶✵ ✺

❚❤❡ ❘❛s❝❤ ▼♦❞❡❧

❘❛s❝❤ ▼♦❞❡❧ ❆ss✉♠♣t✐♦♥s ✴ Pr♦♣❡rt✐❡s ✉♥✐❞✐♠❡♥s✐♦♥❛❧✐t② P(Xvi = 1∣θv,βi,ϕ ϕ ϕ) = P(Xvi = 1∣θv,βi) r❡s♣♦♥s❡ ♣r♦❜❛❜✐❧✐t② ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ♦t❤❡r ✈❛r✐❛❜❧❡s ϕ ϕ ϕ s✉✣❝✐❡♥❝② f(xvi,...,xvk∣θv) = g(rv∣θv)h(xvi,...,xvk) r❛✇ s❝♦r❡ rv = ∑i xvi ✭s✉♠ ♦❢ r❡s♣♦♥s❡s✮ ❝♦♥t❛✐♥s ❛❧❧ ✐♥❢♦r♠❛✲ t✐♦♥ ♦♥ ❛❜✐❧✐t②✱ r❡❣❛r❞❧❡ss ✇❤✐❝❤ ✐t❡♠s ❤❛✈❡ ❜❡❡♥ s♦❧✈❡❞ ❝♦♥❞✐t✐♦♥❛❧ ✐♥❞❡♣❡♥❞❡♥❝❡ XviXvj∣θv,∀i,j ❢♦r ✜①❡❞ θ t❤❡r❡ ✐s ♥♦ ❝♦rr❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❛♥② t✇♦ ✐t❡♠s ♠♦♥♦t♦♥✐❝✐t② ❢♦r θv > θw ∶ f(xvi∣θv,βi) > f(xwi∣θw,βi),∀θv,θw r❡s♣♦♥s❡ ♣r♦❜❛❜✐❧✐t② ✐♥❝r❡❛s❡s ✇✐t❤ ❤✐❣❤❡r ✈❛❧✉❡s ♦❢ θ

▼✉♥✐❝❤ ✷✵✶✵ ✻

❚❤❡ ❘❛s❝❤ ▼♦❞❡❧

P❛r❛♠❡t❡r ❊st✐♠❛t✐♦♥ ■t❡♠ P❛r❛♠❡t❡r ❊st✐♠❛t✐♦♥

▸ ❧✐❦❡❧✐❤♦♦❞ ❜❛s❡❞ ♠❡t❤♦❞s✿

❞✐✛❡r ✐♥ t❤❡✐r tr❡❛t♠❡♥t ♦❢ ♣❡rs♦♥ ♣❛r❛♠❡t❡rs

• ❥♦✐♥t ▼▲ ❡st✐♠❛t✐♦♥ ✭❏▼▲✮
• ❝♦♥❞✐t✐♦♥❛❧ ▼▲ ❡st✐♠❛t✐♦♥ ✭❈▼▲✮
• ♠❛r❣✐♥❛❧ ▼▲ ❡st✐♠❛t✐♦♥✭▼▼▲✮

▸ ♦t❤❡r ♠❡t❤♦❞s ❛✈❛✐❧❛❜❧❡✿

❧❡ss ♦❢t❡♥ ✉s❡❞ ♥♦t ❝♦✈❡r❡❞ ❤❡r❡ P❡rs♦♥ P❛r❛♠❡t❡r ❊st✐♠❛t✐♦♥

• ▼▲ ❛♥❞ ✇❡✐❣❤t❡❞ ▼▲ ❡st✐♠❛t✐♦♥
• ❇❛②❡s ❛♣♣r♦❛❝❤❡s

▼✉♥✐❝❤ ✷✵✶✵ ✼

❚❤❡ ❘❛s❝❤ ▼♦❞❡❧

❏♦✐♥t ▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ✭❏▼▲✮ Lu = exp(∑v θvrv)exp(−∑i βisi) ∏v ∏i(1 + exp(θv − βi)) s✉✣❝✐❡♥t st❛t✐st✐❝s ❛r❡✿ rv = ∑i xvi ❢♦r θv si = ∑v xvi ❢♦r βi ♣r♦❜❧❡♠✿ ✐t❡♠ ♣❛r❛♠❡t❡r ❡st✐♠❛t❡s ✐♥❝♦♥s✐st❡♥t ❛s n → ∞ ❜✐❛s❡❞ ✐♥ ✜♥✐t❡ s❛♠♣❧❡s ✇✐t❤ k(k − 1) ▼❛r❣✐♥❛❧ ▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ✭▼▼▲✮ ✐♥t❡❣r❛t❡ ♦✉t t❤❡ ♣❡rs♦♥ ♣❛r❛♠❡t❡r Lm = ∏

r

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ exp(−∑

i

βisi)∫ exp(θr) ∏k

i=1(1 + exp(θ − βi))

dG(θ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

nr

❞✐str✐❜✉t✐♦♥ ❢♦r θ ♠✉st ❜❡ s♣❡❝✐✜❡❞✱ ✉s✉❛❧❧② θ ∼ N(0,1) ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ✐♥ ❘ ✉s✐♥❣ t❤❡ ❧t♠ ♣❛❝❦❛❣❡ ✭❘✐③♦♣♦✉❧♦s✱✷✵✵✾✮

▼✉♥✐❝❤ ✷✵✶✵ ✽

❚❤❡ ❘❛s❝❤ ▼♦❞❡❧

❈♦♥❞✐t✐♦♥❛❧ ▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ✭❈▼▲✮ ❝♦♥❞✐t✐♦♥ ♦♥ rv Lc = exp(−∑

i

βisi)/∏

r

∑

x∣r

exp(−∑

i

xiβi)nr ✕ ♣❡rs♦♥ ♣❛r❛♠❡t❡rs ❞♦ ♥♦t ♦❝❝✉r ✐♥ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❧✐❦❡❧✐❤♦♦❞ ✕ ✐t❡♠s ❝❛♥ ❜❡ ❝♦♠♣❛r❡❞ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ♣❡rs♦♥s ✭s❡♣❛r❛t✐♦♥✮ ✕ ❧❡❛❞s t♦ s♣❡❝✐✜❝ ♦❜❥❡❝t✐✈✐t② ✕ ♣❡rs♦♥ ❢r❡❡ ✐t❡♠ ❝❛❧✐❜r❛t✐♦♥ ✕ ❵s❛♠♣❧❡✲✐♥❞❡♣❡♥❞❡♥❝❡✬✿ ❛❝t✉❛❧ s❛♠♣❧❡ ♥♦t ♦❢ r❡❧❡✈❛♥❝❡ ❢♦r ✐♥❢❡r❡♥❝❡ ♦♥ ✐t❡♠ ♣❛r❛♠❡t❡rs ❈▼▲ ❡st✐♠❛t❡s ❛r❡ ✉♥❜✐❛s❡❞ ❛♥❞ ❝♦♥s✐st❡♥t ❛s n → ∞ ❢♦r ❡st✐♠❛❜✐❧✐t② s❡t β1 = 0 ♦r ∑βi = 0 ✐t❡♠s ✇✐t❤ s❝♦r❡ si = 0 ♦r n ❛♥❞ ♣❡rs♦♥ ✇✐t❤ rv = 0 ♦r k ❛r❡ r❡♠♦✈❡❞ ♣r✐♦r t♦ ❡st✐♠❛t✐♦♥

▼✉♥✐❝❤ ✷✵✶✵ ✾

❚❤❡ ❘❛s❝❤ ▼♦❞❡❧

▼▼▲ ✈s ❈▼▲ ▼▼▲ ❆❞✈❛♥t❛❣❡s✿

• ❣✐✈❡s ❛❧s♦ ❡st✐♠❛t❡s ❢♦r ♣❡rs♦♥s ✇✐t❤ rv = 0 ♦r rv = k
• ✇❤❡♥ r❡s❡❛r❝❤ ❛✐♠s ❛t ♣❡rs♦♥ ❞✐str✐❜✉t✐♦♥
• ❛❧❧♦✇s ❡st✐♠❛t✐♦♥ ♦❢ ❛❞❞✐t✐♦♥❛❧ ♣❛r❛♠❡t❡rs

✭✷P▲✱ ✸P▲ ♠♦❞❡❧s✮

• ❢❛st❡r ✇✐t❤ ❧❛r❣❡ k

❈▼▲ ❆❞✈❛♥t❛❣❡s✿

• ✇❤❡♥ ❘▼ ✐s ✉s❡❞ ❛s ♠❡❛s✉r❡♠❡♥t ♠♦❞❡❧ ✭s❝❛❧❡ ❝♦♥str✉❝t✐♦♥✮
• ▼▼▲ ♣❛r❛♠❡t❡rs ❝❛♥ ❜❡ ❜✐❛s❡❞ ✐❢ G(θ) ✐♥❝♦rr❡❝t❧② s♣❡❝✐✜❡❞
• ❈▼▲ ❝❧♦s❡r t♦ ❝♦♥❝❡♣t ♦❢ ♣❡rs♦♥✲❢r❡❡ ❛ss❡ss♠❡♥t
• ❛❧❧♦✇s ❢♦r s♣❡❝✐✜❝ ♦❜❥❡❝t✐✈✐t②
• s❡✈❡r❛❧ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ♥♦t ❛✈❛✐❧❛❜❧❡ ✇✐t❤ ▼▼▲

❞✐str✐❜✉t✐♦♥❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❈▼▲ ❛♥❞ ▼▼▲ ❡st✐♠❛t❡s ❛r❡ t❤❡ s❛♠❡ ❛s②♠t♦t✐❝❛❧❧②

▼✉♥✐❝❤ ✷✵✶✵ ✶✵

❚❤❡ ❘❛s❝❤ ▼♦❞❡❧

P❡rs♦♥ P❛r❛♠❡t❡r ❊st✐♠❛t✐♦♥ ✉s✐♥❣ t❤❡ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❧✐❦❡❧✐❤♦♦❞ Lu = exp(∑v θvrv)exp(−∑i βisi) ∏v ∏i(1 + exp(θv − βi)) ❛♥❞ ❛ss✉♠✐♥❣ t❤❡ βs t♦ ❜❡ ❦♥♦✇♥ ✭❢r♦♠ ♣r✐♦r ❡st✐♠❛t✐♦♥✮ s❧✐❣❤t❧② ❜✐❛s❡❞ ✭❜✐❛s s♠❛❧❧❡r t❤❛♥ s✳❡✳✬s ♦❢ ❡st✐♠❛t❡s✮ ♥♦ ❡st✐♠❛t❡s ❢♦r rv = 0 ❛♥❞ rv = k ❝❛♥ ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ✉s✐♥❣✱ ❡✳❣✳✱ s♣❧✐♥❡ ✐♥t❡r♣♦❧❛t✐♦♥ ✇❡✐❣❤t❡❞ ▼▲ ❡st✐♠❛t✐♦♥✿ ❧✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥ ✐s s❦❡✇❡❞✱ ❛❞❞✐t✐♦♥❛❧ s♦✉r❝❡ ♦❢ ❡st✐♠❛t✐♦♥ ❜✐❛s ❲❛r♠ ✭✶✾✽✾✮ s✉❣❣❡sts ✉♥❜✐❛s✐♥❣ ❝♦rr❡❝t✐♦♥✱ ❝♦♠♣✉t❛t✐♦♥❛❧❧② ✉♥✲ ❢❡❛s✐❜❧❡

▼✉♥✐❝❤ ✷✵✶✵ ✶✶

❡❘♠

❚❤❡ ❘ ♣❛❝❦❛❣❡ ❡❘♠ ✭❡①t❡♥❞❡❞ ❘❛s❝❤ ♠♦❞❡❧❧✐♥❣✮

❃ ❧✐❜r❛r②✭❡❘♠✮

♠❛✐♥ ❢✉♥❝t✐♦♥s ❝♦♥❝❡r♥✐♥❣ ✜t ♦❢ t❤❡ ❘▼✿

• ❘▼✭❞❛t❛✮ ✜ts t❤❡ ❘▼ ❛♥❞ ❣❡♥❡r❛t❡s ♦❜❥❡❝t ♦❢ ❝❧❛ss ❞❘♠
• ♣❡rs♦♥✳♣❛r❛♠❡t❡r✭❞r♠♦❜❥✮ ❣❡♥❡r❛t❡s ♦❜❥❡❝t ♦❢ ❝❧❛ss ♣♣❛r
• ♣❧♦ts ❢r♦♠ ❞r♠ ♦❜❥❡❝t✿

✕ ♣❧♦tP■♠❛♣✭✮✱ ♣❧♦t■❈❈✭✮✱ ♣❧♦t❥♦✐♥t■❈❈✭✮

• ♣❧♦ts ❢r♦♠ ♣♣❛r ♦❜❥❡❝t✿

✕ ♣❧♦t✭✮

• ❡①tr❛❝t ✐♥❢♦r♠❛t✐♦♥ ❢r♦♠ ❞r♠ ♦❜❥❡❝t✿

✕ ❝♦❡❢✭✮✱ ✈❝♦✈✭✮✱ ❝♦♥❢✐♥t✭✮✱ ❧♦❣▲✐❦✭✮✱ ♠♦❞❡❧✳♠❛tr✐①✭✮

• ❡①tr❛❝t ✐♥❢♦r♠❛t✐♦♥ ❢r♦♠ ♣♣❛r ♦❜❥❡❝t✿

✕ ❝♦♥❢■♥t✭✮✱ ❧♦❣▲✐❦✭✮

▼✉♥✐❝❤ ✷✵✶✵ ✶✷

❡❘♠

❋✐tt✐♥❣ t❤❡ ❘▼

❃ r♠✳r❡s ❁✲ ❘▼✭❞❛t❛✮ ❃ r♠✳r❡s ❘❡s✉❧ts ♦❢ ❘▼ ❡st✐♠❛t✐♦♥✿ ❈❛❧❧✿ ❘▼✭❳ ❂ ❞❛t❛✮ ❈♦♥❞✐t✐♦♥❛❧ ❧♦❣✲❧✐❦❡❧✐❤♦♦❞✿ ✲✶✺✻✳✸✶✵✵ ◆✉♠❜❡r ♦❢ ✐t❡r❛t✐♦♥s✿ ✶✷ ◆✉♠❜❡r ♦❢ ♣❛r❛♠❡t❡rs✿ ✹ ❇❛s✐❝ P❛r❛♠❡t❡rs ❡t❛✿ ❡t❛ ✶ ❡t❛ ✷ ❡t❛ ✸ ❡t❛ ✹ ❊st✐♠❛t❡ ✵✳✹✷✾✷✻✽✺ ✲✶✳✶✼✹✸✺✹✷ ✲✵✳✶✹✾✻✼✸✷ ✵✳✵✷✻✻✼✷✻✷ ❙t❞✳❊rr ✵✳✶✾✹✺✻✶✽ ✵✳✷✷✹✸✸✵✾ ✵✳✶✾✶✽✽✷✹ ✵✳✶✾✶✶✽✸✼✾

✕ ❞❡❢❛✉❧t ✐s✿ ❘▼✭❞❛t❛♠❛tr✐①✱ s✉♠✵ ❂ ❚❘❯❊✱ ♦t❤❡r ♦♣t✐♦♥s✮ ✕ s✉♠✵ ❞❡✜♥❡s ❝♦♥str❛✐♥ts ✭❢♦r ❡st✐♠❛❜✐❧✐t②✮✿ ❚❘❯❊ ✳ ✳ ✳ s✉♠ ③❡r♦✱ ❋❆▲❙❊ ✳ ✳ ✳ ✜rst ✐t❡♠ s❡t t♦ ✵ ✕ t❤❡ ♦✉t♣✉t ❣✐✈❡s ❡❛s✐♥❡ss ✭♥♦t ❞✐✣❝✉❧t②✮ ♣❛r❛♠❡t❡rs✦

▼✉♥✐❝❤ ✷✵✶✵ ✶✸

❡❘♠ ❃ s✉♠♠❛r②✭r♠✳r❡s✮ ❘❡s✉❧ts ♦❢ ❘▼ ❡st✐♠❛t✐♦♥✿ ❈❛❧❧✿ ❘▼✭❳ ❂ ❞❛t❛✮ ❈♦♥❞✐t✐♦♥❛❧ ❧♦❣✲❧✐❦❡❧✐❤♦♦❞✿ ✲✶✺✻✳✸✶✵✵ ◆✉♠❜❡r ♦❢ ✐t❡r❛t✐♦♥s✿ ✶✷ ◆✉♠❜❡r ♦❢ ♣❛r❛♠❡t❡rs✿ ✹ ❇❛s✐❝ P❛r❛♠❡t❡rs ✭❡t❛✮ ✇✐t❤ ✵✳✾✺ ❈■✿ ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ❧♦✇❡r ❈■ ✉♣♣❡r ❈■ ❡t❛ ✶ ✵✳✹✷✾ ✵✳✶✾✺ ✵✳✵✹✽ ✵✳✽✶✶ ❡t❛ ✷ ✲✶✳✶✼✹ ✵✳✷✷✹ ✲✶✳✻✶✹ ✲✵✳✼✸✺ ❡t❛ ✸ ✲✵✳✶✺✵ ✵✳✶✾✷ ✲✵✳✺✷✻ ✵✳✷✷✻ ❡t❛ ✹ ✵✳✵✷✼ ✵✳✶✾✶ ✲✵✳✸✹✽ ✵✳✹✵✶ ■t❡♠ ❊❛s✐♥❡ss P❛r❛♠❡t❡rs ✭❜❡t❛✮ ✇✐t❤ ✵✳✾✺ ❈■✿ ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ❧♦✇❡r ❈■ ✉♣♣❡r ❈■ ❜❡t❛ ■✶ ✵✳✽✻✽ ✵✳✷✵✻ ✵✳✹✻✹ ✶✳✷✼✷ ❜❡t❛ ■✷ ✵✳✹✷✾ ✵✳✶✾✺ ✵✳✵✹✽ ✵✳✽✶✶ ❜❡t❛ ■✸ ✲✶✳✶✼✹ ✵✳✷✷✹ ✲✶✳✻✶✹ ✲✵✳✼✸✺ ❜❡t❛ ■✹ ✲✵✳✶✺✵ ✵✳✶✾✷ ✲✵✳✺✷✻ ✵✳✷✷✻ ❜❡t❛ ■✺ ✵✳✵✷✼ ✵✳✶✾✶ ✲✵✳✸✹✽ ✵✳✹✵✶ ▼✉♥✐❝❤ ✷✵✶✵ ✶✹

❡❘♠

❊①tr❛❝t✐♥❣ ■♥❢♦r♠❛t✐♦♥ t❤❡ ✐t❡♠ ♣❛r❛♠❡t❡r ❡st✐♠❛t❡s

❃ ❝♦❡❢✭r♠✳r❡s✮ ❡t❛ ✶ ❡t❛ ✷ ❡t❛ ✸ ❡t❛ ✹ ✵✳✹✷✾✷✻✽✺✸ ✲✶✳✶✼✹✸✺✹✷✺ ✲✵✳✶✹✾✻✼✸✶✾ ✵✳✵✷✻✻✼✷✻✷

t❤❡ ✈❛r✐❛♥❝❡✲❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ♦❢ ✐t❡♠ ♣❛r❛♠❡t❡r ❡st✐♠❛t❡s

❃ ✈❝♦✈✭r♠✳r❡s✮ ❬✱✶❪ ❬✱✷❪ ❬✱✸❪ ❬✱✹❪ ❬✶✱❪ ✵✳✵✸✼✽✺✹✸✵✻ ✲✵✳✵✶✷✺✺✹✶✼ ✲✵✳✵✵✽✵✼✸✻✷✽ ✲✵✳✵✵✼✾✺✾✹✹✹ ❬✷✱❪ ✲✵✳✵✶✷✺✺✹✶✼✺ ✵✳✵✺✵✸✷✹✸✻ ✲✵✳✵✶✶✼✶✻✵✺✼ ✲✵✳✵✶✶✼✽✵✵✽✽ ❬✸✱❪ ✲✵✳✵✵✽✵✼✸✻✷✽ ✲✵✳✵✶✶✼✶✻✵✻ ✵✳✵✸✻✽✶✽✽✻✼ ✲✵✳✵✵✼✹✽✹✹✻✹ ❬✹✱❪ ✲✵✳✵✵✼✾✺✾✹✹✹ ✲✵✳✵✶✶✼✽✵✵✾ ✲✵✳✵✵✼✹✽✹✹✻✹ ✵✳✵✸✻✺✺✶✷✹✶ ▼✉♥✐❝❤ ✷✵✶✵ ✶✺

❡❘♠

❊①tr❛❝t✐♥❣ ■♥❢♦r♠❛t✐♦♥ ✭❝♦♥t✬❞✮ ❝♦♥✜♥❞❡♥❝❡ ✐♥t❡r✈❛❧s ❢♦r t❤❡ ✐t❡♠ ♣❛r❛♠❡t❡r ❡st✐♠❛t❡s

❃ ❝♦♥❢✐♥t✭r♠✳r❡s✱ ✧❜❡t❛✧✮ ✷✳✺ ✪ ✾✼✳✺ ✪ ❜❡t❛ ■✶ ✵✳✹✻✹✹✹✷✽✹ ✶✳✷✼✶✼✷✾✼ ❜❡t❛ ■✷ ✵✳✵✹✼✾✸✹✸✺ ✵✳✽✶✵✻✵✷✼ ❜❡t❛ ■✸ ✲✶✳✻✶✹✵✸✹✼✻ ✲✵✳✼✸✹✻✼✸✼ ❜❡t❛ ■✹ ✲✵✳✺✷✺✼✺✺✽✹ ✵✳✷✷✻✹✵✾✺ ❜❡t❛ ■✺ ✲✵✳✸✹✽✵✹✵✼✷ ✵✳✹✵✶✸✽✻✵

t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❧♦❣ ❧✐❦❡❧✐❤♦♦❞

❃ ❧♦❣▲✐❦✭r♠✳r❡s✮ ✬❈♦♥❞✐t✐♦♥❛❧ ❧♦❣ ▲✐❦✳✬ ✲✶✺✻✳✸✶✵✵ ✭❞❢❂✹✮ ▼✉♥✐❝❤ ✷✵✶✵ ✶✻

❡❘♠

P❧♦t ■❈❈s

❃ ♣❧♦t❥♦✐♥t■✭r♠✳r❡s✱ ①❧✐♠ ❂ ❝✭✲✺✱ ✺✮✮

−4 −2 2 4 0.0 0.2 0.4 0.6 0.8 1.0 Latent Dimension Probability to Solve Item 1 Item 2 Item 3 Item 4 Item 5

▼✉♥✐❝❤ ✷✵✶✵ ✶✼

❡❘♠

P❧♦t s✐♥❣❧❡ ■❈❈

❃ ♣❧♦t■✭r♠✳r❡s✱ ✐ ❂ ✸✮

−4 −2 2 4 0.0 0.2 0.4 0.6 0.8 1.0 Latent Dimension Probability to Solve

▼✉♥✐❝❤ ✷✵✶✵ ✶✽

❡❘♠

P❧♦t ■❈❈s

❃ ♣❧♦t■❈❈✭r♠✳r❡s✱ ✐t❡♠✳s✉❜s❡t ❂ ✶✿✹✱ ❛s❦ ❂ ❋✱ ❡♠♣■❈❈ ❂ ❧✐st✭✧r❛✇✧✮✱ ✰ ❡♠♣❈■ ❂ ❧✐st✭❧t② ❂ ✧s♦❧✐❞✧✮✮

−4 −2 2 4 0.0 0.4 0.8

ICC plot for item I1

Latent Dimension Probability to Solve

• −4

−2 2 4 0.0 0.4 0.8

ICC plot for item I2

Latent Dimension Probability to Solve

• −4

−2 2 4 0.0 0.4 0.8

ICC plot for item I3

Latent Dimension Probability to Solve

• −4

−2 2 4 0.0 0.4 0.8

ICC plot for item I4

Latent Dimension Probability to Solve

• ▼✉♥✐❝❤ ✷✵✶✵

✶✾

❡❘♠

P❧♦t P❡rs♦♥✲■t❡♠ ▼❛♣

❃ ♣❧♦tP■♠❛♣✭r♠✳r❡s✮

I5 I4 I3 I2 I1 −1 1

Latent Dimension

• I5
• I5
• I4
• I4
• I3
• I3
• I2
• I2
• I1
• I1

Person−Item Map

Person Parameter Distribution

▼✉♥✐❝❤ ✷✵✶✵ ✷✵

❡❘♠

P❡rs♦♥ P❛r❛♠❡t❡r ❊st✐♠❛t✐♦♥

❃ ♣♣ ❁✲ ♣❡rs♦♥✳♣❛r❛♠❡t❡r✭r♠✳r❡s✮ ❃ ♣♣ P❡rs♦♥ P❛r❛♠❡t❡rs✿ ❘❛✇ ❙❝♦r❡ ❊st✐♠❛t❡ ❙t❞✳❊rr♦r ✵ ✲✷✳✻✸✶✵✾✼✾ ◆❆ ✶ ✲✶✳✺✶✽✾✾✻✼ ✶✳✶✹✾✽✺✾✾ ✷ ✲✵✳✹✻✶✺✵✾✶ ✵✳✾✺✻✺✹✷✻ ✸ ✵✳✹✸✼✹✾✸✸ ✵✳✾✻✸✻✸✵✷ ✹ ✶✳✺✷✶✼✺✽✵ ✶✳✶✻✻✾✸✾✻ ✺ ✷✳✻✻✺✾✾✶✼ ◆❆

✐❢ ◆❆s ✐♥ t❤❡ ❞❛t❛✱ ❞✐✛❡r❡♥t ♣❡rs♦♥ ♣❛r❛♠❡t❡rs ❛r❡ ❡st✐♠❛t❡❞ ❢♦r ❡✈❡r② ◆❆✲♣❛tt❡r♥ ❣r♦✉♣

▼✉♥✐❝❤ ✷✵✶✵ ✷✶

❡❘♠

▼❡t❤♦❞s ❢♦r P❡rs♦♥ P❛r❛♠❡t❡r ❊st✐♠❛t✐♦♥ ❘❡s✉❧ts

❃ ❧♦❣▲✐❦✭♣♣✮ ✬❯♥❝♦♥❞✐t✐♦♥❛❧ ✭❥♦✐♥t✮ ❧♦❣ ▲✐❦✳✬ ✲✶✵✳✽✺✸✾✽ ✭❞❢❂✹✮ ❃ ❝♦♥❢✐♥t✭♣♣✮ ✷✳✺ ✪ ✾✼✳✺ ✪ P✶ ✲✸✳✼✼✷✻✽✶ ✵✳✼✸✹✻✽✼✷ P✷ ✲✶✳✹✺✶✶✽✼ ✷✳✸✷✻✶✼✸✾ P✸ ✲✶✳✹✺✶✶✽✼ ✷✳✸✷✻✶✼✸✾ P✺ ✲✷✳✸✸✻✷✾✽ ✶✳✹✶✸✷✼✾✾ P✻ ✲✶✳✹✺✶✶✽✼ ✷✳✸✷✻✶✼✸✾ P✼ ✲✷✳✸✸✻✷✾✽ ✶✳✹✶✸✷✼✾✾ ✳✳✳

❛tt❡♥t✐♦♥✿ ❝♦♥❢✐♥t✭♣♣✮ ❣✐✈❡s ✈❛❧✉❡s ❢♦r ❛❧❧ s✉❜❥❡❝ts ✐❢ t❤❡r❡ ❛r❡ ◆❛s ✐♥ t❤❡ ❞❛t❛✱ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s ❛r❡ ♣r✐♥t❡❞ ❢♦r ❡❛❝❤ ◆❆ ❣r♦✉♣

▼✉♥✐❝❤ ✷✵✶✵ ✷✷

❡❘♠

P❧♦t ♦❢ P❡rs♦♥ P❛r❛♠❡t❡r ❊st✐♠❛t❡s

❃ ♣❧♦t✭♣♣✮

• 1

2 3 4 5 −2 −1 1 2

Plot of the Person Parameters

Person Raw Scores Person Parameters (Theta)

▼✉♥✐❝❤ ✷✵✶✵ ✷✸

❚❡st✐♥❣ t❤❡ ❘❛s❝❤ ▼♦❞❡❧

❚❡st✐♥❣ t❤❡ ❘▼ ✕ ❖✈❡r✈✐❡✇ ❘▼ ❛❧❧♦✇s t♦ ❡✈❛❧✉❛t❡ t❤❡ q✉❛❧✐t② ♦❢ ♠❡❛s✉r❡♠❡♥t ❝r✉❝✐❛❧ ❛ss✉♠♣t✐♦♥s ❡♠♣✐r✐❝❛❧❧② t❡st❛❜❧❡ ❛✐♠✿ ✜♥❞ s❡t ♦❢ ✐t❡♠s t❤❛t ❝♦♥❢♦r♠ t♦ t❤❡ ❘▼ ✭❵❞❛t❛ ✜t ♠♦❞❡❧✬✮ ✈❛r✐♦✉s t❡sts✴❞✐❛❣♥♦st✐❝s ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞ s♦♠❡ ✐♠♣❧❡♠❡♥t❡❞ ✐♥ ❡❘♠✿ ✕ ❆♥❞❡rs❡♥ ▲❘ t❡st ✕ ❲❛❧❞✲t②♣❡ t❡st ✕ ♥♦♥♣❛r❛♠❡tr✐❝ t❡sts ✕ ✐t❡♠✴♣❡rs♦♥ ✜t ✐♥❞✐❝❡s ✕ ❣r❛♣❤✐❝❛❧ ♣r♦❝❡❞✉r❡s

▼✉♥✐❝❤ ✷✵✶✵ ✷✹

❚❡st✐♥❣ t❤❡ ❘❛s❝❤ ▼♦❞❡❧

❆♥❞❡rs❡♥✬s ▲✐❦❡❧✐❤♦♦❞ ❘❛t✐♦ ❚❡st ✭❆♥❞❡rs❡♥✱ ✶✾✼✸✮ ✕ ❵❣❧♦❜❛❧✬ t❡st ✭❛❧❧ ✐t❡♠s ✐♥✈❡st✐❣❛t❡❞ s✐♠✉❧t❛♥❡♦✉s❧②✮ ✕ ♣♦✇❡r❢✉❧ ❛❣❛✐♥st ✈✐♦❧❛t✐♦♥s ♦❢ s✉✣❝✐❡♥❝② ❛♥❞ ♠♦♥♦t♦♥✐❝✐t② ✕ ❝❛♥ ❞❡t❡❝t ❉■❋ ✭❞✐✛❡r❡♥t✐❛❧ ✐t❡♠ ❢✉♥❝t✐♦♥✐♥❣ ♦r ✐t❡♠ ❜✐❛s✮✿ ❜❛s✐❝ ✐❞❡❛✿ ❝♦♥s✐st❡♥t ✐t❡♠ ♣❛r❛♠❡t❡r ❡st✐♠❛t❡s ✭❵✐♥✈❛r✐❛♥❝❡✬✮ ♦❜t❛✐♥❡❞ ❢r♦♠ ❛♥② s✉❜❣r♦✉♣ ✇❤❡r❡ t❤❡ ♠♦❞❡❧ ❤♦❧❞s ❞✐✈✐❞❡ t❤❡ s❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ s❝♦r❡ r✱ r = 1,...,J − 1 ♦❜t❛✐♥ J − 1 ❧✐❦❡❧✐❤♦♦❞s ♦❢ t❤❡ ❢♦r♠ L(r)

c

= exp(−∑

j

βjs(r)

j

)/γ(r;β1,...,βJ)nr t❤❡ t♦t❛❧ ❧✐❦❡❧✐❤♦♦❞ ✐s Lc = ∏

r

L(r)

c ▼✉♥✐❝❤ ✷✵✶✵ ✷✺

❚❡st✐♥❣ t❤❡ ❘❛s❝❤ ▼♦❞❡❧

❆♥❞❡rs❡♥✬s ▲✐❦❡❧✐❤♦♦❞ ❘❛t✐♦ ❚❡st ✭❝♦♥t✬❞✮ t❤❡♥

Λ =

Lc ∏r L(r)

c

= 1, ♦♥❧② ✐❢ t❤❡ ❘▼ ❤♦❧❞s Z = −2lnΛ ✐s ❛s②♠♣t♦t✐❝❛❧❧② χ2✲❞✐str✐❜✉t❡❞ ✇✐t❤ d f = (J − 2)(J − 1) t❡st ❝❛♥ ❜❡ ✉s❡❞ ❢♦r ❛♥② ♣❛rt✐t✐♦♥ ♦❢ t❤❡ s❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ ❡①tr❛♥❡♦✉s ✈❛r✐❛❜❧❡s ✭❡✳❣✳✱ ❣❡♥❞❡r✱ ❛❣❡✱ ✳✳✳✮ ❲❛❧❞ ❚❡st ❛❧❧♦✇s ❢♦r t❡st✐♥❣ s✐♥❣❧❡ ✐t❡♠s ✐❞❡❛ ✐s ❛❣❛✐♥✿ s❛♠♣❧❡ ✐♥t♦ s✉❜✲ ❣r♦✉♣s ✭✉s✉❛❧❧② ✷✮ ✉s✐♥❣ s❡♣❛r❛t❡ ❡st✐♠❛t❡s ˆ β(1)

j

❛♥❞ ˆ β(2)

j

✭❛♥❞ ˆ σ(1)

βj ✱ ˆ

σ(2)

βj ✮✱

Sj = (ˆ β(1)

j

− ˆ β(2)

j

)/ √ ˆ σ(1)

β

+ ˆ σ(2)

β

≈ N(0,1)

▼✉♥✐❝❤ ✷✵✶✵ ✷✻

❚❡st✐♥❣ t❤❡ ❘❛s❝❤ ▼♦❞❡❧

◆♦♥♣❛r❛♠❡tr✐❝ ✭❵❡①❛❝t✬✮ ❚❡sts ■❞❡❛✿

• P❛r❛♠❡t❡r ❡st✐♠❛t❡s ❞❡♣❡♥❞ ♦♥❧② ♦♥ ♠❛r❣✐♥❛❧s r ❛♥❞ s
• ❢♦r ❛♥② st❛t✐st✐❝ ♦❢ t❤❡ ❞❛t❛ ♠❛tr✐①✱ ♦♥❡ ❝❛♥ ❛♣♣r♦①✐♠❛t❡ t❤❡

♥✉❧❧ ❞✐str✐❜✉t✐♦♥

• t❛❦❡ r❛♥❞♦♠ s❛♠♣❧❡ ❢r♦♠ t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❡q✉❛❧❧② ❧✐❦❡❧② ❞❛t❛

♠❛tr✐❝❡s✱ ❝♦♠♣✉t❡ ♥✉❧❧ ❞✐str✐❜✉t✐♦♥ ♦❢ st❛t✐st✐❝

• ✈❛❧✐❞ ❛♥❞ ♣♦✇❡r❢✉❧✱ ❡✈❡♥ ✐♥ s♠❛❧❧ s❛♠♣❧❡s

P❡rs♦♥✴■t❡♠ ❋✐t ♦❜❥❡❝t✐✈❡ ✐s t♦ ❞❡t❡❝t ♥♦t✐❝❡❛❜❧❡ ♣❛tt❡r♥s ❊①♣❡❝t❡❞ r❡s♣♦♥s❡✿ πvi = exp(θv − βi)/(1 + exp(θv − βi)) ❘❡s✐❞✉❛❧s✿ evi = xvi − πvi ❊①❛♠♣❧❡✿ ❖✉t✜t ▼❙◗ ❢♦r ✐t❡♠s✿ ui = 1 n ∑

v

e2

vi

πvi(1 − πvi) t❡st st❛t✐st✐❝s✱ ❡✳❣✳✱ nu2

i ✱ ❛r❡ χ2 ✇✐t❤ ❝♦rr❡s♣♦♥❞✐♥❣ d

f

▼✉♥✐❝❤ ✷✵✶✵ ✷✼

❚❡st✐♥❣ t❤❡ ❘❛s❝❤ ▼♦❞❡❧

• r❛♣❤✐❝❛❧ Pr♦❝❡❞✉r❡

✉♥❞❡r❧②✐♥❣ ✐❞❡❛ ❛❣❛✐♥ s✉❜❣r♦✉♣ ❤♦♠♦❣❡♥❡✐t②✱ ♣❧♦t ˆ β(1) ✈s ˆ β(2)

• −3

−2 −1 1 2 3 −3 −2 −1 1 2 3

Graphical Model Check

Beta for Group: Raw Scores <= Median Beta for Group: Raw Scores > Median I1 I2 I3 I4 I5

▼✉♥✐❝❤ ✷✵✶✵ ✷✽

❡❘♠

▲✐❦❡❧✐❤♦♦❞r❛t✐♦✲ ❛♥❞ ❲❛❧❞ ❚❡sts ▲❘ ❚❡st✿

❃ ❧rt ❁✲ ▲❘t❡st✭r♠✳r❡s✱ s❡ ❂ ❚❘❯❊✮ ❃ ❧rt ❆♥❞❡rs❡♥ ▲❘✲t❡st✿ ▲❘✲✈❛❧✉❡✿ ✷✳✹✵✼ ❈❤✐✲sq✉❛r❡ ❞❢✿ ✹ ♣✲✈❛❧✉❡✿ ✵✳✻✻✶

❲❛❧❞ ❚❡st✿

❃ ❲❛❧❞t❡st✭r♠✳r❡s✮ ❲❛❧❞ t❡st ♦♥ ✐t❡♠ ❧❡✈❡❧ ✭③✲✈❛❧✉❡s✮✿ ③✲st❛t✐st✐❝ ♣✲✈❛❧✉❡ ❜❡t❛ ■✶ ✲✵✳✽✸✷ ✵✳✹✵✺ ❜❡t❛ ■✷ ✲✵✳✸✺✷ ✵✳✼✷✺ ❜❡t❛ ■✸ ✵✳✹✷✽ ✵✳✻✻✽ ❜❡t❛ ■✹ ✶✳✸✵✵ ✵✳✶✾✹ ❜❡t❛ ■✺ ✲✵✳✹✶✶ ✵✳✻✽✶ ▼✉♥✐❝❤ ✷✵✶✵ ✷✾

❡❘♠

■t❡♠ ❋✐t ❙t❛t✐st✐❝s

❃ ✐t❡♠❢✐t✭♣♣✮ ■t❡♠❢✐t ❙t❛t✐st✐❝s✿ ❈❤✐sq ❞❢ ♣✲✈❛❧✉❡ ❖✉t❢✐t ▼❙◗ ■♥❢✐t ▼❙◗ ■✶ ✽✵✳✾✸✽ ✽✹ ✵✳✺✼✹ ✵✳✾✺✷ ✵✳✾✻✻ ■✷ ✼✽✳✹✾✶ ✽✹ ✵✳✻✹✾ ✵✳✾✷✸ ✵✳✾✸✹ ■✸ ✽✷✳✹✽✵ ✽✹ ✵✳✺✷✻ ✵✳✾✼✵ ✵✳✾✻✶ ■✹ ✽✺✳✶✹✹ ✽✹ ✵✳✹✹✺ ✶✳✵✵✷ ✶✳✵✷✹ ■✺ ✼✹✳✷✼✺ ✽✹ ✵✳✼✻✼ ✵✳✽✼✹ ✵✳✾✵✽

◆♦♥♣❛r❛♠❡tr✐❝ ❚❡sts

❃ t✶✶ ❁✲ ◆Pt❡st✭❞❛t❛✱ ♠❡t❤♦❞ ❂ ✧❚✶✶✧✮ ❃ t✶✶ ◆♦♥♣❛r❛♠❡tr✐❝ ❘▼ ♠♦❞❡❧ t❡st✿ ❚✶✶ ✭❣❧♦❜❛❧ t❡st ✲ ❧♦❝❛❧ ❞❡♣❡♥❞❡♥❝❡✮ ✭s✉♠ ♦❢ ❞❡✈✐❛t✐♦♥s ❜❡t✇❡❡♥ ♦❜s❡r✈❡❞ ❛♥❞ ❡①♣❡❝t❡❞ ✐♥t❡r✲✐t❡♠ ❝♦rr❡❧❛t✐♦♥s✮ ◆✉♠❜❡r ♦❢ s❛♠♣❧❡❞ ♠❛tr✐❝❡s✿ ✺✵✵ ♦♥❡✲s✐❞❡❞ ♣✲✈❛❧✉❡✿ ✵✳✾✸✹ ▼✉♥✐❝❤ ✷✵✶✵ ✸✵

❡❘♠

• r❛♣❤✐❝❛❧ Pr♦❝❡❞✉r❡

❃ ♣❧♦t●❖❋✭❧rt✱ ❝♦♥❢ ❂ ❧✐st✭✮✮

• −3

−2 −1 1 2 3 −3 −2 −1 1 2 3

Graphical Model Check

Beta for Group: Raw Scores <= Median Beta for Group: Raw Scores > Median I1 I2 I3 I4 I5

• ▼✉♥✐❝❤ ✷✵✶✵

✸✶

❡❘♠

P♦❧②t♦♠♦✉s ▼♦❞❡❧s P❛rt✐❛❧ ❈r❡❞✐t ▼♦❞❡❧❧ ✭P❈▼✮ P(Xvi = h) = exp[h(θv + βi) + ωhi] ∑mi

l=0 exp[l(θv + βi) + ωli]

h ✳ ✳ ✳ r❡s♣♦♥s❡ ❝❛t❡❣♦r✐❡s ✭h = 0,...,mi✮ mi ✳ ✳ ✳ ♥✉♠❜❡r ♦❢ r❡s♣♦♥s❡ ❝❛t❡❣♦r✐❡s ♠❛② ❞✐✛❡r ❛❝r♦ss ✐t❡♠s ωhi ✳ ✳ ✳ ❝❛t❡❣♦r② ♣❛r❛♠❡t❡r ❘❛t✐♥❣ ❙❝❛❧❡ ▼♦❞❡❧❧ ✭❘❙▼✮ s✐♠♣❧✐✜❝❛t✐♦♥✿ mi = m ✳ ✳ ✳ ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ ❝❛t❡❣♦r✐❡s ❛r❡ ❡q✉❛❧ ❛❝r♦ss ❛❧❧ ✐t❡♠ ωhi = ωh ✳ ✳ ✳ ❵❡q✉✐st❛♥t s❝♦r✐♥❣✬

▼✉♥✐❝❤ ✷✵✶✵ ✸✷

❡❘♠

■❈❈s ❢♦r t❤❡ P❈▼

−4 −2 2 4 0.0 0.2 0.4 0.6 0.8 1.0

ICC plot for item I2

Latent Dimension probability for responding in category Category 0 Category 1 Category 2 Category 3

▼✉♥✐❝❤ ✷✵✶✵ ✸✸

❡❘♠

❈♦♠♣❛r✐s♦♥ ❘❙▼ ✈s P❈▼

QA20_6 QA20_5 QA20_4 QA20_3 QA20_2 QA20_1 −5 −4 −3 −2 −1 1 2 3 4 5

Latent Dimension

• 1

2 3 4

• 1

2 3 4

• 1

2 3 4

• 1

2 3 4

• 1

2 3 4

• 1

2 3 4

• RSM

Person Parameter Distribution QA20_6 QA20_5 QA20_4 QA20_3 QA20_2 QA20_1 −5 −4 −3 −2 −1 1 2 3 4 5

Latent Dimension

• 1

2 3 4

• ●●

1 2 3 4

• 1

2 3 4

• 1

2 3 4

• 1

2 3 4

• ●
• 1

2 3 4

• PCM

Person Parameter Distribution

▼✉♥✐❝❤ ✷✵✶✵ ✸✹

❡❘♠

❘ ❝♦♠♠❛♥❞s ♠❛✐♥ ❢✉♥❝t✐♦♥s ❝♦♥❝❡r♥✐♥❣ ✜t ♦❢ ♣♦❧②t♦♠♦✉s ♠♦❞❡❧s✿

• P❈▼✭❞❛t❛✮ ✜ts t❤❡ P❈▼ ❛♥❞ ❣❡♥❡r❛t❡s ♦❜❥❡❝t ♦❢ ❝❧❛ss ❘♠
• ❘❙▼✭❞❛t❛✮ ✜ts t❤❡ ❘❙▼ ❛♥❞ ❣❡♥❡r❛t❡s ♦❜❥❡❝t ♦❢ ❝❧❛ss ❘♠
• t❤r❡s❤♦❧❞s✭r♠♦❜❥✮ ❞✐s♣❧❛②s t❤❡ ✐t❡♠♣❛r❛♠❡t❡r ❡st✐♠❛t❡s ❛s

t❤r❡s❤♦❧❞s

• ❛❧❧ ♦t❤❡r ❢✉♥❝t✐♦♥s ❛r❡ t❤❡ s❛♠❡ ❛s ♣r❡✈✐♦✉s❧② ♣r❡s❡♥t❡❞

✭❡①❝❡♣t ❢♦r ♣❧♦t❥♦✐♥t■❈❈✭✮✮

▼✉♥✐❝❤ ✷✵✶✵ ✸✺

❡❘♠ ❙✉♠♠❛r②

❡❘♠ ❙✉♠♠❛r② ❝♦r❡ ♦❢ ❡❘♠ ✐s t❤❡ ▲✐♥❡❛r P❛rt✐❛❧ ❈r❡❞✐t ▼♦❞❡❧ ✭▲P❈▼✮✿ P(Xvi = h) = exp(hθv + βih) ∑mi

l=0 exp(lθv + βih)

✇❤❡r❡ t❤❡ βih✬s ❛r❡ ❧✐♥❡❛r❧② r❡♣❛r❛♠❡t❡r✐s❡❞ βih =

mi

∑

p

wihpηp ❛❧❧♦✇s ❢♦r ❛ ❣❡♥❡r❛❧ ❛❧❣♦r✐t❤♠✿ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ s♣❡❝✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❞❡s✐❣♥ ♠❛tr✐① W = ((wih,p)) ✈❛r✐♦✉s ♠♦❞❡❧s ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ❝✉rr❡♥t❧② ❢✉♥❝t✐♦♥s ❢♦r✿ ▲P❈▼✱ P❈▼✱ ▲❘❙▼✱ ❘❙▼✱ ▲▲❚▼✱ ❘▼

▼✉♥✐❝❤ ✷✵✶✵ ✸✻

❡❘♠ ❙✉♠♠❛r②

❚❤❡ ♠♦❞❡❧ ❤✐❡r❛r❝❤② ✐♥ ❡❘♠ t❤❡ ▲P❈▼ ✐s t❤❡ ♠♦st ❣❡♥❡r❛❧ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ♠♦❞❡❧ ✐♥ t❤✐s ❢❛♠✐❧② ❛❧❧ ♦t❤❡r ♠♦❞❡❧s ❛r❡ s✉❜♠♦❞❡❧s t❤❡② ❛r❡ ♦❜t❛✐♥❡❞ ❜② ❛♣♣r♦♣r✐❛t❡❧② ❞❡✜♥✐♥❣ t❤❡ ❞❡s✐❣♥ ♠❛tr✐① W

LPCM PCM LRSM RSM LLTM RM ▼✉♥✐❝❤ ✷✵✶✵ ✸✼

❡❘♠ ❙✉♠♠❛r②

❡❘♠ ❋❡❛t✉r❡s ❙❝♦♣❡✿

• ❙❝❛❧❡ ❆♥❛❧②s✐s ✭♠❡❛s✉r❡♠❡♥t ♠♦❞❡❧s✮
• ▼♦❞❡❧❧✐♥❣ ❧❛t❡♥t ❝❤❛♥❣❡ ✭st❛t✐st✐❝❛❧ ♠♦❞❡❧s✮

✉♥✐✲ ❛♥❞ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ✭▲▲❘❆✮ ▼♦❞❡❧s✿

• ❘▼✱ ❘❙▼✱ P❈▼✱ ▲▲❚▼✱ ▲❘❙▼✱ ▲P❈▼✱ ✭▲▲❘❆✮
• ❚r❡❛t♠❡♥t ♦❢ ♠✐ss✐♥❣ ✈❛❧✉❡s ✭▼❈❆❘✮
• ❉✐✛❡r❡♥t ❝♦♥str❛✐♥ts ❢♦r ♣❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥
• ❉❡s✐❣♥ ♠❛tr✐① ✭❞❡❢❛✉❧t ✴ ✉s❡r ❞❡✜♥❡❞✮

❊st✐♠❛t✐♦♥✿

• ■t❡♠♣❛r❛♠❡t❡rs✱ ❵❜❛s✐❝✬✲ ❛♥❞ ❡✛❡❝t ♣❛r❛♠❡t❡rs✱

t❤r❡s❤♦❧❞ ♣❛r❛♠❡t❡rs ✭❛❧❧ ✉s✐♥❣ ❈▼▲✮

• P❡rs♦♥♣❛r❛♠❡t❡rs ✭❏▼▲✮
• ❈♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s ✭❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s✮
• ❙✉♣♣♦rt ❢♦r st❡♣✇✐s❡ ✐t❡♠ s❡❧❡❝t✐♦♥

▼✉♥✐❝❤ ✷✵✶✵ ✸✽

❡❘♠ ❙✉♠♠❛r②

❡❘♠ ❋❡❛t✉r❡s ✭❝♦♥t✬❞✮ ❉✐❛❣♥♦st✐❝s✱ ▼♦❞❡❧ ❚❡sts✱ ❛♥❞ ❋✐t ❙t❛t✐st✐❝s✿

• ❆♥❞❡rs❡♥ ▲❘✲t❡st✱ ❲❛❧❞ ❚❡st ❢♦r s✐♥❣❧❡ ✐t❡♠s
• ●❧♦❜❛❧ ❛♥❞ ✐t❡♠ ❧❡✈❡❧ ♥♦♥♣❛r❛♠❡tr✐❝ t❡sts ✭❢♦r ❘▼✮
• ■t❡♠✜t✱ P❡rs♦♥✜t ✭✉s✐♥❣ P❡❛rs♦♥ r❡s✐❞✉❛❧s✮
• ■♥❢♦r♠❛t✐♦♥ ❝r✐t❡r✐❛ ✭❆■❈✱ ❇■❈✱ ❝❆■❈✮
• ❈❤❡❝❦ ❢♦r ❡①✐st❡♥❝❡ ♦❢ ▼▲ ❡st✐♠❛t❡s ✕

❵✇❡❧❧✲❝♦♥❞✐t✐♦♥❡❞ ❞❛t❛♠❛tr✐①✬ ✭❢♦r ❘▼✮

• s♦♠❡ ✭♥♦♥♣s②❝❤♦♠❡tr✐❝✮ ❧♦❣✐st✐❝ r❡❣r❡ss✐♦♥ ❞✐❛❣♥♦st✐❝s

P❧♦ts✿

• ●♦♦❞♥❡ss✲♦❢✲❋✐t P❧♦ts
• ■❈❈✲P❧♦ts ❢♦r s✐♥❣❧❡ ✐t❡♠s ✭✇✐t❤ ♦♣t✐♦♥❛❧ ❡♠♣✐r✐❝❛❧ ■❈❈s✮
• ❏♦✐♥t ■❈❈✲P❧♦t ✭❢♦r ❘▼✮
• P❡rs♦♥✲■t❡♠ ▼❛♣

▼✐s❝❡❧❧❛♥❡♦✉s ✿

• ❙✐♠✉❧❛t✐♦♥ ♦❢ ❞❛t❛ ♠❛tr✐❝❡s ❛❝❝♦r❞✐♥❣ t♦ ❘▼ ✈✐♦❧❛t✐♦♥s
• ✳ ✳ ✳

▼✉♥✐❝❤ ✷✵✶✵ ✸✾

❡❘♠ ❙✉♠♠❛r②

❋✉rt❤❡r ■♥❢♦s✿ ❘ ❋♦r❣❡✿ ❤tt♣✿✴✴r✲❢♦r❣❡✳r✲♣r♦❥❡❝t✳♦r❣✴ ❉❡✈❡❧♦♣♠❡♥t ♣❧❛t❢♦r♠ ❧❛t❡st r❡❧❡❛s❡s ❞♦✇♥❧♦❛❞s ❉✐s❝✉sss✐♦♥ ❛♥❞ ❤❡❧♣ ❢♦r✉♠ Pr♦❥❡❝t ❤♦♠❡♣❛❣❡ ❤tt♣✿✴✴❡r♠✳r✲❢♦r❣❡✳r✲♣r♦❥❡❝t✳♦r❣✴ P✉❜❧✐❝❛t✐♦♥s✿ ▼❛✐r ✫ ❍❛t③✐♥❣❡r ✭✷✵✵✼✮✳ ❏♦✉r♥❛❧ ❙t❛t✐st✐❝❛❧ ❙♦❢t✇❛r❡ ▼❛✐r ✫ ❛♥❞ ❍❛t③✐♥❣❡r ✭✷✵✵✼✮✳ Ps②❝❤♦❧♦❣② ❙❝✐❡♥❝❡ ❍❛t③✐♥❣❡r ✫ ❘✉s❝❤ ✭✷✵✵✾✮✳ Ps②❝❤♦❧♦❣② ❙❝✐❡♥❝❡ ◗✉❛rt❡r❧❡②

▼✉♥✐❝❤ ✷✵✶✵ ✹✵