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Quantities
Qualities
Measurementcan and should be exclusively studied with
Classification?
Ordination?
may be studied
Description?
Non-Measurement?
if heterogeneous order via
if classificatory via
Measurement defines and can be used for studying Attributes
x ′ = f(x)f(x)
x ′ = f(x)f(x)
x ′ = ax+ b
x ′ = ax
Measurement rule based assignment of numerals to
Objects orEvents
Ratio Scales
Ordinal Scales
Nominal Scales
Interval Scalesto form
X =
⟨X,!, S1, . . . , Sn⟩ !, S1, . . . , Sn
R = ⟨R,≥, R1, . . . , Rn⟩ R
Ri
X R X
R
Measurement representsQualitativeStructures
e.g. Interval, Ratio and Absolute Scalese.g. Ordinal and Hyper Ordinal Scales
e.g. Nominal Scales
viaNumericalStructures
to form
Measurement modelsObservableVariables
Latent Classes
Latent Ordered Classes
Latent Continua
as a function of
Class 1
Class 4
Class 3
Class 2
Class 1
Class 4
Class 3
Class 2
Class 1
Class 4
Class 3
Class 2
Class 1
Class 1
Class 2 Class 3
Class 4
(a) Continuous Quantitative
Variable
(b) Located Heterogeneous
Classes
(c) Located Homogeneous
Classes
(d) Ordered Classes
(e) Qualitative Classes
Quantitative Structure
OrdinalStructure
QualitativeStructure
Classification
Measurement experimentallyobtaining
QuantityValues
Ordinal Quantity
Quantity
Nominal Propertiesexperimentally studies
reasonably attributable to
Measurement
MMet MAMT
MCTM
MRMT
MOper
MTST
MLVM
Classification
Ordination
Quantification
Quantification
Ordination
Classification
Assessment
B A∗ A
B A∗ A
X
X
X
m1c1(t1 − tf) = m2c2(tf − t2)
∆Q = mc∆t
Class 1
Class 4
Class 3
Class 2
Class 1
Class 4
Class 3
Class 2
Class 1
Class 4
Class 3
Class 2
Class 1
Class 1
Class 2 Class 3
Class 4
(a) Continuous Quantitative
Variable
(b) Located Heterogeneous
Classes
(c) Located Homogeneous
Classes
(d) Ordered Classes
(e) Qualitative Classes
Quantitative Structure
OrdinalStructure
QualitativeStructure
(J+1)/2 J
T ≥ J+12
T
P
p I i
p i xip xip = 1
xip = 0
C c
πi|c c i
(xic = 1|c) = πi|c
πc c
p
( p) =C∑
c=1
πc
I∏
i=1
πi|c
πi|c
πc
∑c πc = 1
πi|c
[Pr(xic = 1|c)] = [πi|c] = βic
πi|c [0, 1]
βic (−∞,∞)
βic
θ
(X = x|θ) =I∏
i=1
(Xi = xi|θ)
θa ≤ θb(Xi = 1|θa) ≤ (Xi = 1|θb)
i(θ)
i θ
1(θ) ≤ 2(θ) ≤ ... < k(θ) θ.
C
c
πc
i βic
βic
[ (xic = 1|c)] = [πi|c] = βic,
βic ≤ βic ′ c < c ′ i
[ (xic = 1|c)] = [πi|c] = βic,
βic ≤ βi ′c i < i ′ c
[ (xic = 1|c)] = [πi|c] = βic,
βic ≤ βic ′ c < c ′ i,
βic ≤ βi ′c i < i ′ c
θ
θ
A = B+ C A = B× C
θ
δ
θ δ
δi θp
βci
[ (xip = 1|θp, δi)] = θp − δi
A B A > B
B > A
A B A B
A B
C
θ
p
t c
[ (xic = 1|θc, δi)] = θc − δi
C πc
Class 4
Class 3
Class 2
Class 1
dmClass 4
dmClass 3
dmClass 2
dmClass 1
Class 1
Class 2 Class 3
Class 4
Rasch ModelLatent Class
Rasch ModelOrdered Latent Class Analysis Unconstrained Latent
Class Analysis
Quantitative Structure
OrdinalStructure
QualitativeStructure
iioClass 4
iioClass 3
iioClass 2
iioClass 1
monClass 4
monClass 3
monClass 2
monClass 1
iiodm mon
θ
(a) Unconstrained Latent Class (b) Class Monotonicity
(c) Invariant Item Ordering (d) Double Monotonicity
(e) Latent Class Rasch (f) Rasch Model
Item01
Item02
Item03
Item04
Item05
Item06
Item07
Item08
Item09
Item10
logi
ts
-4
-2
0
2
4
Item01
Item02
Item03
Item04
Item05
Item06
Item07
Item08
Item09
Item10
logi
ts
-4
-2
0
2
4
Item01
Item02
Item03
Item04
Item05
Item06
Item07
Item08
Item09
Item10
logi
ts
-4
-2
0
2
4
Item01
Item02
Item03
Item04
Item05
Item06
Item07
Item08
Item09
Item10
logi
ts
-4
-2
0
2
4
Item01
Item02
Item03
Item04
Item05
Item06
Item07
Item08
Item09
Item10
logi
ts
-4
-2
0
2
4
Item01
Item02
Item03
Item04
Item05
Item06
Item07
Item08
Item09
Item10
logi
ts
-4
-2
0
2
4
Structureof the relevant
attribute
Structureof the model
Structureof the collected data
Structureof the relevant
attribute
Structureof the model
Structureof the instrument
Structureof the collected data
A
A∗
A
A B
Logits
Items
Logits
Items
Latent Classes:Unconstrained (un)
OrderedLatent Classes:
Class Monotonicity (mon)
OrderedLatent Classes:
Invariant Item Ordering (iio)
OrderedLatent Classes:
Double Monotonicity (dm)
LocatedLatent Classes:
Latent Class Rasch (lcr)
ContinuousVariable:
Rasch Model (rm)
Differences of Quality
Differencesof Order
Differencesof Quantity
Non-Monotonic
SingleMonotonicity
DoubleMonotonicity
Class 4
Class 3
Class 2
Class 1
Class 4
Class 3
Class 2
Class 1
Class 4
Class 3
Class 2
Class 1
(a) Discretized Continuous
Variable
(b) Located Heterogeneous
Classes
(c) Located Homogeneous
Classes
(d) Ordered Classes
Quantitative Structure
OrdinalStructure
Class 4
Class 3
Class 2
Class 1
α
β α
β
N
( p) =C∑
c=1
πc
I∏
i=1
(πxipi|c × (1− πi|c)
1−xip)
p P i I
c C
πc c∑C
c=1 πc = 1
πi|c c
xi i c
βic c i
[πi|c] = βic
πi|c
βic
c
[Pr(xic = 1|c)] = [πi|c] = βic,
βic ≤ βic ′ c < c ′ i.
Division
Multiplication
Subtraction
Addition
Claudia
Paul
Sam
[ (xpi = 1|θp)] = θp − δi
θpiid∼ N(0,ψ)
xpi p i
θp p
δi i
ψ
qi δi
K
> 1 × ÷
8+ 3
2+ 2
5− 1
8− 3
7× 2
9÷ 3
(9− 2)× 3
(3× 8)÷ 4
(5× 3) + (8÷ 2)
[ (xpi = 1|θp)] = θp −
(η0 +
K∑
1
ηkqik
)
θpiid∼ N(0,ψ)
k
ηk
θp
0 0 0 0 1 1 1 10 0 1 1 0 0 1 10 1 0 1 0 1 0 1
F1F2F3
0 0 0 0 1 1 1 10 0 1 1 0 0 1 10 1 0 1 0 1 0 1
F1F2F3
θp
πc c
πi|c p
i c
πi|c
c
[πi|c] = θpc −K∑
k=1
ηkcqik
0 0 0 0 1 1 1 10 0 1 1 0 0 1 10 1 0 1 0 1 0 1
F1F2F3
l
l
l
l
l
l
l
l
l
l
l
l
l
l
( p|θp) =C∑
c=1
πc
I∏
i=1
[−1
(θpc −
K∑
k=1
ηkcqik
)xip
×
(1− −1
(θpc −
K∑
k=1
ηkcqik
))1−xip⎤
⎦
θpciid∼ N(µc,ψc)
ηkc
C
C
C
θpc c
θ
θ
I
I K
0 0 0 0 1 1 1 10 0 1 1 0 0 1 10 1 0 1 0 1 0 1
F1F2F3
l
l
l
l
l
l
l
l
l
l
l
l
l
l
[ (xic = 1|c)] = [πi|c] = η0c +K∑
k=1
ηkcqik = βic,
βic ≤ βic ′ c < c ′ i.
θpc
η η
η
η
η
[Pr(xic = 1|c)] = [πi|c] = η0c +K∑
k=1
ηkcqik,
ηkc ≤ ηkc ′ c < c ′ k.
η
●
●
●
●
●
●
●
●
Prob
abilit
ies
●
●
●
●
●
●
●
●
●●
●●
● ●
● ●
Patt. 1
Patt. 2
Patt. 3
Patt. 4
Patt. 5
Patt. 6
Patt. 7
Patt. 8
0.0
0.2
0.4
0.6
0.8
1.0
●
●●
●
●
● ●
●
●
● ●
●
●
● ●●
Prob
abilit
ies
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
● ●
●
●
● ●
●
●
● ●
●
Patt. 1
Patt. 2
Patt. 3
Patt. 4
Patt. 5
Patt. 6
Patt. 7
Patt. 8
Patt. 9
Patt. 10
Patt. 11
Patt. 12
Patt. 13
Patt. 14
Patt. 15
Patt. 16
0.0
0.2
0.4
0.6
0.8
1.0
(Cp = c| ) =
π̂c
I∏i=1
(π̂xipi|c × (1− π̂i|c)1−xip
)
C∑c=1
π̂c
I∏i=1
(π̂xipi|c × (1− π̂i|c)1−xip
)
πc
πc
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
60 Tasks 20 Tasks60 Tasks 60 Tasks 60 Tasks20 Tasks 20 Tasks 20 Tasks
4 Features 3 Features 3 Features4 Features
1000 Persons 500 Persons
Bias - Mean of the !c estimates
95% Confidence intervalfor bias
60 Tasks 20 Tasks60 Tasks 60 Tasks 60 Tasks20 Tasks 20 Tasks 20 Tasks
4 Features 3 Features 3 Features4 Features
1000 Persons 500 Persons
Mean ProportionCorrectly Recovered
Std. Dev. ProportionCorrectly Recovered
1.0
0.0
> 0.99
< 0.01
> 0.99
< 0.01
> 0.99
< 0.01
0.99
< 0.01
0.82
0.27
0.94
< 0.01
0.94
0.01
Parameters
Logi
ts
Mean distance from generating parameter
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
Cl.1 C
Cl.2 C
Cl.3 C
Cl.1 F1
Cl.2 F1
Cl.3 F1
Cl.1 F2
Cl.2 F2
Cl.3 F2
Cl.1 F3
Cl.2 F3
Cl.3 F3
Parameters
Logi
ts
Mean distance from generating parameter
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
Cl.1 C
Cl.2 C
Cl.3 C
Cl.1 F1
Cl.2 F1
Cl.3 F1
Cl.1 F2
Cl.2 F2
Cl.3 F2
Cl.1 F3
Cl.2 F3
Cl.3 F3
Cl.1 F4
Cl.2 F4
Cl.3 F4
Parameters
Logi
ts
Mean distance from generating parameter
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
Cl.1 C
Cl.2 C
Cl.3 C
Cl.1 F1
Cl.2 F1
Cl.3 F1
Cl.1 F2
Cl.2 F2
Cl.3 F2
Cl.1 F3
Cl.2 F3
Cl.3 F3
Parameters
Logi
ts
Mean distance from generating parameter
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
Cl.1 C
Cl.2 C
Cl.3 C
Cl.1 F1
Cl.2 F1
Cl.3 F1
Cl.1 F2
Cl.2 F2
Cl.3 F2
Cl.1 F3
Cl.2 F3
Cl.3 F3
Cl.1 F4
Cl.2 F4
Cl.3 F4
Parameters
Logi
ts
Mean distance from generating parameter
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
Cl.1 C
Cl.2 C
Cl.3 C
Cl.1 F1
Cl.2 F1
Cl.3 F1
Cl.1 F2
Cl.2 F2
Cl.3 F2
Cl.1 F3
Cl.2 F3
Cl.3 F3
Parameters
Logi
ts
Mean distance from generating parameter
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
Cl.1 C
Cl.2 C
Cl.3 C
Cl.1 F1
Cl.2 F1
Cl.3 F1
Cl.1 F2
Cl.2 F2
Cl.3 F2
Cl.1 F3
Cl.2 F3
Cl.3 F3
Cl.1 F4
Cl.2 F4
Cl.3 F4
Parameters
Logi
ts
Mean distance from generating parameter
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
Cl.1 C
Cl.2 C
Cl.3 C
Cl.1 F1
Cl.2 F1
Cl.3 F1
Cl.1 F2
Cl.2 F2
Cl.3 F2
Cl.1 F3
Cl.2 F3
Cl.3 F3
Parameters
Logi
ts
Mean distance from generating parameter
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
Cl.1 C
Cl.2 C
Cl.3 C
Cl.1 F1
Cl.2 F1
Cl.3 F1
Cl.1 F2
Cl.2 F2
Cl.3 F2
Cl.1 F3
Cl.2 F3
Cl.3 F3
Cl.1 F4
Cl.2 F4
Cl.3 F4
0 2 4 6 8
0.0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 8
95% Interval Generating Value Mean Recovered Value
0 5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15
95% Interval Generating Value Mean Recovered Value
0 2 4 6 8
0.0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 8
95% Interval Generating Value Mean Recovered Value
0 5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15
95% Interval Generating Value Mean Recovered Value
β
η
η
