Analyzing human perceptions from survey data withstatmath.wu.ac.at/research/talks/resources/... ·...
Transcript of Analyzing human perceptions from survey data withstatmath.wu.ac.at/research/talks/resources/... ·...
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Analyzing
human perceptions
from survey data with
Nonlinear CUB models
Wien, 13th May 2016
Marica Manisera and Paola Zuccolotto -University of Brescia, Italy
Wien -13th M
ay 2016
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Credits• Manisera M., Zuccolotto P. (2014) Modelling rating data
with Nonlinear CUB models, Computational Statistics and
Data Analysis, 78, 100–118.
• Manisera M., Zuccolotto P. (2014) Modelling “don’t know”
responses in rating scales. Pattern Recognition Letters, 45,
226-234
• Manisera M., Zuccolotto P. (2014). Nonlinear CUB models:
the R code. Statistica & Applicazioni, XII, 205-223.
• Manisera M., Zuccolotto P. (2015). Identifiability of a
model for discrete frequency distributions with a
multidimensional parameter space, Journal of Multivariate
Analysis, 140, 302-316.
• Manisera M., Zuccolotto P. (2015). Visualizing Multiple
Results from Nonlinear CUB Models with R Grid
Viewports. Electronic Journal of Applied Statistical Analysis, 8,
360-373.
• Manisera M., Zuccolotto P. (2016). Treatment of ‘don’t
know’ responses in a mixture model for rating data, Metron,
74, 99-115.
• Manisera M., Zuccolotto P. (2016). Estimation of
Nonlinear CUB models via numerical optimization and EM
algorithm, Communications in Statistics - Simulation and
Computation, forthcoming.
Credits
Paola Zuccolotto
Marica Manisera
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Agenda
• Examples of rating data (real data case studies)
• The unconscious Decision Process (DP) driving individuals’
responses on a rating scale
• CUB models (D’Elia&Piccolo 2005, Computational Statistics
and Data Analysis – Iannario&Piccolo 2011, Modern Analysis of
Customer Surveys)
• NLCUB models (Manisera&Zuccolotto 2014, Computational
Statistics and Data Analysis)
Agenda
SHAPE: “Statistical Modelling of Human Perception”, STAR project -
University of Naples Federico II - CUP: E68C13000020003
SYRTO: “SYstemicRisk TOmography: Signals, Measurements,
Transmission Channels, and Policy Interventions”, grant from the
European Union Seventh Framework Programme- Project ID: 320270
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Rating data
The analysis of human perception is often carried out by
resorting to surveys and questionnaires, where
respondents are asked to express ratings about the objects
being evaluated.
The goal of the statistical tools proposed for this kind of data
is to explicitly characterize the respondents' perceptions
about a latent trait, by taking into account, at the same time,
the ordinal categorical scale of measurement of the
involved statistical variables.
Rating data
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Rating data – example 1
(superstition)
• A survey investigating confidence about assertions
concerned with superstition in Romania
• dataset by Vlăsceanu et al. (2012), downloadable from the
IQSS (Institute of Quantitative Social Science) Dataverse
Network of the Harvard University
• Respondents (n = 1161) were asked to express a
judgment about their degree of belief in some assertions,
using a 4-point Likert scale (totally disagree, disagree,
agree, totally agree)
Rating data -
exam
ples
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Rating data – example 1
(superstition)
Rating data -
exam
ples
1. Evil has red eyes
2. Number 13 brings bad luck
3. If the palm of your left hand itches, you will receive money soon
4. Lucky at cards, unlucky in love
5. If a black cat crosses the street it is a sign of bad luck
6. Zodiacal signs influence nature and personality
7. Human civilization was created by aliens
8. There are some numbers that bring good luck to certain people
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Rating data – example 2
(fraud management)
Rating data -
exam
ples
• A survey investigating the perceveid risk of being victim
of frauds when using ICT
• dataset supplied by NetConsulting (2013)
• Respondents (n = 116 managers of small, mid-sized and
large firms) were asked to express a judgment about their
degree of perceived fraud risk when using some different
IC Technologies, using a 4-point Likert scale (very low,
low, high, very high)
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Rating data – example 2
(fraud management)
Rating data -
exam
ples
SOCNET: Web 2.0 and Social Networks
CLOUD: Cloud storage and computing
BYOD: Bring Your Own Device
LEG: Legacy technologies
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Rating data – example 3
(Standard Eurobarometer 81)
Rating data -
exam
ples
• A sample survey covering the national population of
citizens of the 27 European Union Member States
• Questions asking respondents to rate their level of
agreement with some statements using a 4-point Likert
scale (totally disagree, tend to disagree, tend to agree,
totally agree)
• “don’t know” option available
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Rating data – example 3
(Standard Eurobarometer 81)
Rating data -
exam
ples
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
The unconsciuos D
P3 !!!
X !!!
2 !!!
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
The unconsciuos D
P3 !!!
3 !!!
2 !!!
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
The unconsciuos D
P3 !!!
2 !!!
2 !!!
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
The unconsciuos D
P3 !!!
2 !!!
2 !!!
Feeling
approach
Uncertainty
approach
Expressed rating
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
The unconsciuos D
P3 !!!
2 !!!
2 !!!
reasoned and logical thinking, the set of
emotions, perceptions, subjective evaluations that individuals have
with regard to the latent trait being
evaluated
indecision inherently present in any human choice, not depending
on the individuals’ position on the latent
variable
Expressed rating
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
CU
B m
odels
3 !!!
2 !!!
2 !!!
Feeling
approach CUB:(shifted) Binomial
random variable (V)
Expressed rating
How do CUB models fit into this framework?
indecision inherently present in any human choice, not depending
on the individuals’ position on the latent
variable
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
CU
B m
odels
2 !!!
2 !!!
Feeling
approach CUB:(shifted) Binomial
random variable (V)
Expressed rating
How do CUB models fit into this framework?
indecision inherently present in any human choice, not depending
on the individuals’ position on the latent
variable
m = 5; x = 0.3
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
CU
B m
odels
2 !!!
2 !!!
Feeling
approach CUB:(shifted) Binomial
random variable (V)
Uncertainty
approach CUB: Uniform random
variable (U)
Expressed rating
How do CUB models fit into this framework?
m = 5; x = 0.3
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
CU
B m
odels
2 !!!
Feeling
approach CUB:(shifted) Binomial
random variable (V)
Uncertainty
approach CUB: Uniform random
variable (U)
How do CUB models fit into this framework?
Expressed rating
m = 5; x = 0.3 m = 5
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
CU
B m
odels
2 !!!
Feeling
approach CUB:(shifted) Binomial
random variable (V)
Uncertainty
approach CUB: Uniform random
variable (U)
Expressed rating CUB:
mixture of V and U (R)
How do CUB models fit into this framework?
m = 5; x = 0.3 m = 5
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
CU
B m
odels
3 !!!
2 !!!
2 !!!
Feeling
approach CUB:(shifted) Binomial
random variable (V)
Uncertainty
approach CUB: Uniform random
variable (U)
Expressed rating CUB:
mixture of V and U (R)
How do CUB models fit into this framework?
m = 5; x = 0.3; p = 0.6
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
CU
B m
odels
3 !!!
2 !!!
2 !!!
Feeling
approach CUB:(shifted) Binomial
random variable (V)
Uncertainty
approach CUB: Uniform random
variable (U)
Expressed rating CUB:
mixture of V and U (R)
How do CUB models fit into this framework?
Feeling parameter:
1 - x
Uncertainty parameter:
1 - p
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CU
B m
odels –ex
ample 1F
eeli
ng
para
mete
r:
1 -x
Uncertainty parameter:
1 - p
Example 1 (superstition)
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
NL
CU
B m
odels
3 !!!
2 !!!
2 !!!
Feeling
approach NLCUB:a random variable (A)
Expressed rating
How do NLCUB models fit into this framework?
indecision inherently present in any human choice, not depending
on the individuals’ position on the latent
variable
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
NL
CU
B m
odels
3 !!!
2 !!!
2 !!!
Feeling
approach NLCUB:a random variable (A)
Expressed rating
How do NLCUB models fit into this framework?
indecision inherently present in any human choice, not depending
on the individuals’ position on the latent
variable
T = 9; x = 0.3
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
NL
CU
B m
odels
3 !!!
2 !!!
2 !!!
Feeling
approach NLCUB:a random variable (A)
Expressed rating
How do NLCUB models fit into this framework?
indecision inherently present in any human choice, not depending
on the individuals’ position on the latent
variable
P(A
=1)
P(A
=2)
P(A
=3)
P(A
=4)
P(A
=5)
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
NL
CU
B m
odels
3 !!!
2 !!!
2 !!!
Feeling
approach NLCUB:a random variable (A)
Expressed rating
How do NLCUB models fit into this framework?
indecision inherently present in any human choice, not depending
on the individuals’ position on the latent
variable
g 1=
2
g 2=
3
g 3=
1
g 4=
3
g 5=
1
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
NL
CU
B m
odels
3 !!!
2 !!!
2 !!!
Feeling
approach NLCUB:a random variable (A)
Expressed rating
How do NLCUB models fit into this framework?
indecision inherently present in any human choice, not depending
on the individuals’ position on the latent
variable
![Page 30: Analyzing human perceptions from survey data withstatmath.wu.ac.at/research/talks/resources/... · SHAPE: “Statistical Modelling of Human Perception”, STAR project - University](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f0415357e708231d40c3c09/html5/thumbnails/30.jpg)
Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
NL
CU
B m
odels
3 !!!
2 !!!
2 !!!
Feeling
approach NLCUB:a random variable (A)
Expressed rating
How do NLCUB models fit into this framework?
indecision inherently present in any human choice, not depending
on the individuals’ position on the latent
variable
T = 9; x = 0.3g1 = 2; g2 = 3; g3 = 1; g4 = 3; g5 = 1
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
NL
CU
B m
odels
2 !!!
2 !!!
Feeling
approach NLCUB:a random variable (A)
Uncertainty
approach NLCUB: Uniform random
variable (U)
Expressed rating
How do NLCUB models fit into this framework?
T = 9; x = 0.3g1 = 2; g2 = 3; g3 = 1; g4 = 3; g5 = 1
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
NL
CU
B m
odels
2 !!!
Feeling
approach NLCUB:a random variable (A)
Uncertainty
approach NLCUB: Uniform random
variable (U)
Expressed rating
How do NLCUB models fit into this framework?
T = 9; x = 0.3g1 = 2; g2 = 3; g3 = 1; g4 = 3; g5 = 1 m = 5
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
NL
CU
B m
odels
2 !!!
Feeling
approach NLCUB:a random variable (A)
Uncertainty
approach NLCUB: Uniform random
variable (U)
Expressed rating NLCUB:
mixture of A and U (R)
How do NLCUB models fit into this framework?
T = 9; x = 0.3g1 = 2; g2 = 3; g3 = 1; g4 = 3; g5 = 1 m = 5
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
NL
CU
B m
odels
2 !!!
Feeling
approach NLCUB:a random variable (A)
Uncertainty
approach NLCUB: Uniform random
variable (U)
Expressed rating NLCUB:
mixture of A and U (R)
How do NLCUB models fit into this framework?
m = 5T = 9; x = 0.3; p = 0.6g1 = 2; g2 = 3; g3 = 1; g4 = 3; g5 = 1
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
The F
eeling approach
3 !!!
Let us focus on the Feeling approach
Feeling
approach
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
The F
eeling approach
• We assume that the Feeling approach proceeds
through T consecutive steps.
• At each step a basic judgment is formulated.
• Step-by-step, the basic judgments are accumulated
and transformed into provisional ratings.
• The rating at the end of the Feeling approach is given
by the last provisional rating.
3 !!!
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
The F
eeling approach
• We assume that the Feeling approach proceeds
through T consecutive steps.
• At each step a basic judgment is formulated.
• Step-by-step, the basic judgments are accumulated
and transformed into provisional ratings.
• The rating at the end of the Feeling approach is given
by the last provisional rating.
3 !!!
![Page 38: Analyzing human perceptions from survey data withstatmath.wu.ac.at/research/talks/resources/... · SHAPE: “Statistical Modelling of Human Perception”, STAR project - University](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f0415357e708231d40c3c09/html5/thumbnails/38.jpg)
3 !!!
Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
The F
eeling approach
• We assume that the Feeling approach proceeds
through T consecutive steps.
• At each step a basic judgment is formulated.
• Step-by-step, the basic judgments are accumulated
and transformed into provisional ratings.
• The rating at the end of the Feeling approach is given
by the last provisional rating.
An evaluation about the latent trait, but a
simpler task than the rating expression
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Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
The F
eeling approach
• We assume that the Feeling approach proceeds
through T consecutive steps.
• At each step a basic judgment is formulated.
• Step-by-step, the basic judgments are accumulated
and transformed into provisional ratings.
• The rating at the end of the Feeling approach is given
by the last provisional rating.
3 !!!
![Page 40: Analyzing human perceptions from survey data withstatmath.wu.ac.at/research/talks/resources/... · SHAPE: “Statistical Modelling of Human Perception”, STAR project - University](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f0415357e708231d40c3c09/html5/thumbnails/40.jpg)
Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
The F
eeling approach
• We assume that the Feeling approach proceeds
through T consecutive steps.
• At each step a basic judgment is formulated.
• Step-by-step, the basic judgments are accumulated
and transformed into provisional ratings.
• The rating at the end of the Feeling approach is given
by the last provisional rating.
3 !!!
![Page 41: Analyzing human perceptions from survey data withstatmath.wu.ac.at/research/talks/resources/... · SHAPE: “Statistical Modelling of Human Perception”, STAR project - University](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f0415357e708231d40c3c09/html5/thumbnails/41.jpg)
Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
The F
eeling approach
• We assume that the Feeling approach proceeds
through T consecutive steps.
• At each step a basic judgment is formulated.
• Step-by-step, the basic judgments are accumulated
and transformed into provisional ratings.
• The rating at the end of the Feeling approach is given
by the last provisional rating.
We can obtain several different models,
depending on the assumptions we make about:
• the distribution of the basic judgments
• the accumulation function
• the transformation function
![Page 42: Analyzing human perceptions from survey data withstatmath.wu.ac.at/research/talks/resources/... · SHAPE: “Statistical Modelling of Human Perception”, STAR project - University](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f0415357e708231d40c3c09/html5/thumbnails/42.jpg)
Are you satisfied with XYZ?Express a rating from 1 (=completely unsatisfied) to 5 (=completely satisfied)
The F
eeling approach
We can obtain several different models,
depending on the assumptions we make about:
• the distribution of the basic judgments
• the accumulation function
• the transformation function
Both CUB and NLCUB models can be derived
following this paradigm, when some specific assumptions
(... ... ...) are made about these three points
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The F
eeling approach
![Page 44: Analyzing human perceptions from survey data withstatmath.wu.ac.at/research/talks/resources/... · SHAPE: “Statistical Modelling of Human Perception”, STAR project - University](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f0415357e708231d40c3c09/html5/thumbnails/44.jpg)
Which is the advantage of fitting CUB and NLCUB
models into this paradigm?
The transition probabilities
TRANSITION PROBABILITIES
The probability of increasing one (provisional)
rating point in the next step of the decision process
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Which is the advantage of fitting CUB and NLCUB
models into this paradigm?
The transition probabilities
In CUB models:
for all t and s
TRANSITION PROBABILITIES
The probability of increasing one (provisional)
rating point in the next step of the decision process
![Page 46: Analyzing human perceptions from survey data withstatmath.wu.ac.at/research/talks/resources/... · SHAPE: “Statistical Modelling of Human Perception”, STAR project - University](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f0415357e708231d40c3c09/html5/thumbnails/46.jpg)
Which is the advantage of fitting CUB and NLCUB
models into this paradigm?
The transition probabilities
In NLCUB models:
TRANSITION PROBABILITIES
The probability of increasing one (provisional)
rating point in the next step of the decision process
![Page 47: Analyzing human perceptions from survey data withstatmath.wu.ac.at/research/talks/resources/... · SHAPE: “Statistical Modelling of Human Perception”, STAR project - University](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f0415357e708231d40c3c09/html5/thumbnails/47.jpg)
Which is the advantage of fitting CUB and NLCUB
models into this paradigm?
The transition probabilities
In NLCUB models:
Different values for
different t and s
TRANSITION PROBABILITIES
The probability of increasing one (provisional)
rating point in the next step of the decision process
![Page 48: Analyzing human perceptions from survey data withstatmath.wu.ac.at/research/talks/resources/... · SHAPE: “Statistical Modelling of Human Perception”, STAR project - University](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f0415357e708231d40c3c09/html5/thumbnails/48.jpg)
The transition probabilities
TRANSITION PROBABILITIES
“Perceived closeness” between rating s and s + 1
for example
“Perceived distance” between rating s and s + 1
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The transition plot
TRANSITION PLOT
A broken line joining points , where
i.e.: the cumulated “perceived distances”.
It gives an idea of the state of mind of
respondents toward the rating scale.
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The transition plot
TRANSITION PLOT – CUB model
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The transition plot
TRANSITION PLOT – NLCUB model
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NL
CU
B m
odels
NonLinear CUB models
• Derive from a different assumed mechanism in the
Feeling approach (the Uncertainty approach is
unchanged)
• Allow us to gain insight about the state of mind
toward the rating scale
• Include traditional CUB models as a special case
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NL
CU
B m
odels
NonLinear CUB models
• Derive from a different assumed mechanism in the
Feeling approach (the Uncertainty approach is
unchanged)
• Allow us to gain insight about the state of mind
toward the rating scale
• Include traditional CUB models as a special case
![Page 54: Analyzing human perceptions from survey data withstatmath.wu.ac.at/research/talks/resources/... · SHAPE: “Statistical Modelling of Human Perception”, STAR project - University](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f0415357e708231d40c3c09/html5/thumbnails/54.jpg)
NL
CU
B m
odels
NonLinear CUB models
• Derive from a different assumed mechanism in the
Feeling approach (the Uncertainty approach is
unchanged)
• Allow us to model nonlinear DPs, gaining insight
about the state of mind toward the rating scale
• Include traditional CUB models as a special case
![Page 55: Analyzing human perceptions from survey data withstatmath.wu.ac.at/research/talks/resources/... · SHAPE: “Statistical Modelling of Human Perception”, STAR project - University](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f0415357e708231d40c3c09/html5/thumbnails/55.jpg)
NL
CU
B m
odels -ex
ample 1
Example 1 (superstition)
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NL
CU
B m
odels -ex
ample 1
Example 1 (superstition)
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NL
CU
B m
odels -ex
ample 2
Example 2 (fraud management)
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NL
CU
B m
odels -ex
ample 3
Example 3
(Standard Eurobarometer 81)
• Manisera & Zuccolotto (Pattern Recognition Letters,
2014) have proposed a procedure to take into
account the presence of “don’t know” responses
(DK)
• The idea is that DKs inform about the uncertainty
of the respondents, so they can be introduced in
the CUB framework
• DKs determine an adjustment of the uncertainty
parameter
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NL
CU
B m
odels -ex
ample 3
Example 3
(Standard Eurobarometer 81)
The arrows show the shift in
uncertainty due to the presence
of DK responses.
The arrows are coloured in a
gray-level scale. The darker the
colour, the higher the degree of
nonlinearity of the transition
plot, according to a nonlinearity
index l proposed by
Manisera&Zuccolotto (QdS -
Journal of Methodological and
Applied Statistics, 2013)
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NL
CU
B –
estimation issues
Parameter estimation
(two-steps procedure)
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NL
CU
B –
estimation issues
Likelihood function for fixed g
(step 1)
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NL
CU
B –
estimation issues
Model selection
(step 2)
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NL
CU
B –
estimation issues
Model selection
(step 2)
Maximum Likelihood
Information criteria
Out-of-sample predictive measures
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NL
CU
B –
R functions
NLCUB: R functions available!
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NL
CU
B –
R functions
NLCUB: R functions available!
![Page 66: Analyzing human perceptions from survey data withstatmath.wu.ac.at/research/talks/resources/... · SHAPE: “Statistical Modelling of Human Perception”, STAR project - University](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f0415357e708231d40c3c09/html5/thumbnails/66.jpg)
NL
CU
BSummarizing...
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Basic references
Basic References
• D’Elia A., Piccolo D. (2005) A mixture model for preference data
analysis, Computational Statistics and Data Analysis, 49, 917-934.
• Iannario M., Piccolo D. (2011) CUB Models: Statistical Methods
and Empirical Evidence, in: Kenett, R. S. and Salini, S. (eds.),
Modern Analysis of Customer Surveys, Wiley, NY, 231–254.
• Manisera M., Zuccolotto P. (2013) Nonlinear CUB models: some
stylized facts. QdS - Journal of Methodological and Applied
Statistics, 15, 1-20
• Manisera M., Zuccolotto P. (2014) Modelling rating data with
Nonlinear CUB models, Computational Statistics and Data Analysis,
78, 100–118.
• Manisera M., Zuccolotto P. (2014) Modelling “don’t know”
responses in rating scales. Pattern Recognition Letters, 45, 226-234
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Basic references
Basic References
• Manisera M., Zuccolotto P. (2014). Nonlinear CUB models: the R
code. Statistica & Applicazioni, XII, 205-223.
• Manisera M., Zuccolotto P. (2015). Identifiability of a model for
discrete frequency distributions with a multidimensional parameter
space, Journal of Multivariate Analysis, 140, 302-316.
• Manisera M., Zuccolotto P. (2015). Visualizing Multiple Results
from Nonlinear CUB Models with R Grid Viewports. Electronic
Journal of Applied Statistical Analysis, 8, 360-373.
• Manisera M., Zuccolotto P. (2016). Treatment of ‘don’t know’
responses in a mixture model for rating data, Metron, 74, 99-115.
• Manisera M., Zuccolotto P. (2016). Estimation of Nonlinear CUB
models via numerical optimization and EM algorithm,
Communications in Statistics - Simulation and Computation, forthcoming.
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Thank
you for your attention
Thank you