S3 BIC method for class "fastCUB"

Description

S3 BIC method for objects of class fastCUB.

Usage

## S3 method for class 'fastCUB'
BIC(object, ...)

Arguments

Value

BIC index for the fitted model.

See Also

logLik, fastCUB

Hadamard product of a matrix with a vector

Description

Return the Hadamard product between the given matrix and vector: this operation corresponds to multiply every row of the matrix by the corresponding element of the vector, and it is equivalent to the standard matrix multiplication to the right with the diagonal matrix whose diagonal is the given vector. It is possible only if the length of the vector equals the number of rows of the matrix, otherwise it prints an error message.

Usage

Hadprod(Amat, xvett)

Arguments

Details

It is an auxiliary function needed for computing the variance-covariance matrix of the estimated model with covariates.

Value

A matrix of the same dimensions as Amat

Best-subset variable selection for CUB models via fast EM algorithm

Description

Perform a best-subset search for CUB models on the basis of the BIC index, by combining all possible covariates' specification for feeling and for uncertainty parameters

Usage

bestcub(ordinal,m,Y,W,toler=1e-4,maxiter=200,iterc=5,alpha=0.05,mix=FALSE,
tolmix=1e+2,fmix=NULL,invgen=TRUE)

Arguments

Value

A list containing the following results:

See Also

fastCUB

Shifted Binomial probabilities of ordinal responses

Description

Compute the shifted Binomial probabilities of ordinal responses.

Usage

bitcsi(m,ordinal,csi)

Arguments

Value

A vector of the same length as ordinal, where each entry is the shifted Binomial probability of the corresponding observation.

References

Piccolo D. (2003). On the moments of a mixture of uniform and shifted binomial random variables, Quaderni di Statistica, 5, 85–104

See Also

probcub00, probcubp0, probcub0q

Examples

data(univer)
m<-7
csi<-0.7
ordinal<-univer$informat
pr<-bitcsi(m,ordinal,csi)

Shifted Binomial distribution with covariates

Description

Return the shifted Binomial probabilities of ordinal responses where the feeling component is explained by covariates via a logistic link.

Usage

bitgama(m,ordinal,W,gama)

Arguments

Value

A vector of the same length as ordinal, where each entry is the shifted Binomial probability for the corresponding observation and feeling value.

See Also

logis, probcub0q, probcubpq

Examples

n<-100
m<-7
W<-sample(c(0,1),n,replace=TRUE)
gama<-c(0.2,-0.2)
csivett<-logis(W,gama)
ordinal<-rbinom(n,m-1,csivett)+1
pr<-bitgama(m,ordinal,W,gama)

S3 Method: coef for class "fastCUB"

Description

S3 method: coef for objects of class fastCUB.

Usage

## S3 method for class 'fastCUB'
coef(object, ...)

Arguments

Details

Returns estimated values of coefficients of the fitted model

Value

ML estimates of parameters of the fitted CUB model.

See Also

fastCUB, summary

Correlation matrix for estimated model

Description

Compute parameter correlation matrix for estimated model as returned by an object of class "fastCUB".

Usage

cormat(object,digits=options()$digits)

Arguments

Value

Parameters correlation matrix for fitted fastCUB models.

See Also

fastCUB, vcov

Louis' identity for the observed information matrix of the incomplete problem for CUB models

Description

Compute the matrices involved in Louis' identity for the observed information matrix

Usage

decomp(ttau,ordinal,m,param,ai,Y,W)

Arguments

Value

A list of the matrices in Louis' identity for the observed information matrix

See Also

fastCUB

Normalized dissimilarity measure

Description

Compute the normalized dissimilarity measure between observed relative frequencies and estimated (theoretical) probabilities of a discrete distribution.

Usage

dissim(proba,probb)

Arguments

Value

Numeric value of the dissimilarity index, assessing the distance to a perfect fit.

Examples

proba<-c(0.01,0.03,0.08,0.07,0.27,0.37,0.17)
probb<-c(0.04,0.04,0.05,0.10,0.21,0.32,0.24)
dissim(proba,probb)

Auxiliary function for the log-likelihood estimation of CUB models

Description

Compute the opposite of the scalar function that is maximized when running the E-M algorithm for CUB models with covariates for the feeling parameter.

Usage

effe01(gama, esterno01, m)

Arguments

Details

It is called as an argument for optim within CUB function for models with covariates for feeling or for both feeling and uncertainty

Auxiliary function for the log-likelihood estimation of CUB models

Description

Compute the opposite of the scalar function that is maximized when running the E-M algorithm for CUB models with covariates for the uncertainty parameter.

Usage

effe10(bet, esterno10)

Arguments

Details

It is called as an argument for optim within CUB function for models with covariates for uncertainty or for both feeling and uncertainty

Main function for fast estimation CUB models

Description

Main function to estimate and validate a CUB model for explaining uncertainty and feeling for given ratings, with or without covariates, on the basis of Louis' identity for the information matrix and the derived accelerated estimation.

Usage

fastCUB(Formula, data, ...)

Arguments

Details

This is the main function for CUB models, which calls for the corresponding functions whenever covariates are specified. It performs maximum likelihood estimation via the E-M algorithm for CUB models and extensions based on the Louis'identity for the observed information matrix.

Value

An object of the class "fastCUB": returns a list containing the following results:

See Also

probcub00, probcubp0, probcub0q, probcubpq,

fastCUB package

Description

The package implements Louis' identity for CUB models for rating data, to retrieve the observed information matrix within the EM algorithm. On this basis, an accelerated estimation procedure is derived and best-subset variable selection is implemented.

Details

Author(s)

Rosaria Simone

References

Simone R. (2021). An accelerated EM algorithm for mixture models with uncertainty for rating data, Computational Statistics, 36, 691-714
Louis T.A. (1982). Finding the Observed Information Matrix when Using the EM Algorithm, Journal of the Royal Statistical Society, Series B, 44, 226–233

Main function for CUB models without covariates

Description

Function to estimate and validate a CUB model without covariates for given ordinal responses.

Usage

fastcub00(m,ordinal,starting=NULL,maxiter,toler,iterc=3,invgen=TRUE)

Arguments

Value

An object of the class "fastCUB"

See Also

probbit, probcub00

Main function for CUB models with covariates for the feeling component

Description

Function to estimate and validate a CUB model for given ordinal responses, with covariates for explaining the feeling component.

Usage

fastcub0q(m,ordinal,W,starting=NULL,maxiter,toler,iterc=3,invgen=TRUE)

Arguments

Value

An object of the class "fastCUB"

Main function for CUB models with covariates for the uncertainty component

Description

Estimate and validate a CUB model for given ordinal responses, with covariates for explaining the feeling component via a logistic transform.

Usage

fastcubp0(m,ordinal,Y,starting=NULL,maxiter,toler,iterc=3,invgen=TRUE)

Arguments

Value

An object of the class "fastCUB"

Main function for CUB models with covariates for both the uncertainty and the feeling components

Description

Estimate and validate a CUB model for given ordinal responses, with covariates for explaining both the feeling and the uncertainty components by means of logistic transform.

Usage

fastcubpq(m,ordinal,Y,W,starting=NULL,maxiter,toler,iterc=3,invgen=TRUE)

Arguments

Value

An object of the class "fastCUB"

See Also

loglikcubpq, inibestgama

S3 method "fitted" for class "fastCUB"

Description

S3 method fitted for objects of class fastCUB.

Usage

## S3 method for class 'fastCUB'
fitted(object, ...)

Arguments

Details

Returns the fitted probability distribution for GEM models with no covariates. If only one dichotomous covariate is included in the model to explain some components, it returns the fitted probability distribution for each profile.

See Also

fastCUB

Preliminary estimators for CUB models without covariates

Description

Compute preliminary parameter estimates of a CUB model without covariates for given ordinal responses. These preliminary estimators are used within the package code to start the E-M algorithm.

Usage

inibest(m,freq)

Arguments

Value

A vector (\pi,\xi) of the initial parameter estimates for a CUB model without covariates, given the absolute frequency distribution of ordinal responses

References

Iannario M. (2009). A comparison of preliminary estimators in a class of ordinal data models, Statistica & Applicazioni, VII, 25–44
Iannario M. (2012). Preliminary estimators for a mixture model of ordinal data, Advances in Data Analysis and Classification, 6, 163–184

See Also

inibestgama

Examples

m<-9
freq<-c(10,24,28,36,50,43,23,12,5)
estim<-inibest(m,freq) 
pai<-estim[1]
csi<-estim[2]

Preliminary parameter estimates of a CUB model with covariates for feeling

Description

Compute preliminary parameter estimates for the feeling component of a CUB model fitted to ordinal responses These estimates are set as initial values for parameters to start the E-M algorithm.

Usage

inibestgama(m,ordinal,W)

Arguments

Value

A vector of length equal to NCOL(W)+1, whose entries are the preliminary estimates of the parameters for the feeling component, including an intercept term as first entry.

References

Iannario M. (2008). Selecting feeling covariates in rating surveys, Rivista di Statistica Applicata, 20, 103–116
Iannario M. (2009). A comparison of preliminary estimators in a class of ordinal data models, Statistica & Applicazioni, VII, 25–44
Iannario M. (2012). Preliminary estimators for a mixture model of ordinal data, Advances in Data Analysis and Classification, 6, 163–184

See Also

inibest

Examples

data(univer)
m<-7; ordinal<-univer$global; cov<-univer$diploma
ini<-inibestgama(m,ordinal,W=cov)

Recursive computation of the inverse of a matrix

Description

Compute the variance-covariance matrix of the incomplete score vector involved in Louis' identity for the observed information matrix

Usage

invmatgen(G,H,listE)

Arguments

Value

The inverse of matrix G + H computed recursively thanks to matrices listed in listE

References

Miller K. (1981). On the inverse of the sum of matrices, Mathematics Magazine, 54, 67–72

See Also

fastCUB

Sequence of combinatorial coefficients

Description

Compute the sequence of binomial coefficients {m-1}\choose{r-1}, for r= 1, \dots, m, and then returns a vector of the same length as ordinal, whose i-th component is the corresponding binomial coefficient {m-1}\choose{r_i-1}

Usage

kkk(m, ordinal)

Arguments

logLik S3 Method for class "fastCUB"

Description

S3 method: logLik() for objects of class "fastCUB".

Usage

## S3 method for class 'fastCUB'
logLik(object, ...)

Arguments

Value

Log-likelihood at the final ML estimates for parameters of the fitted fastCUB model.

See Also

fastCUB

The logistic transform

Description

Create a matrix YY binding array Y with a vector of ones, placed as the first column of YY. It applies the logistic transform componentwise to the standard matrix multiplication between YY and param.

Usage

logis(Y,param)

Arguments

Value

Return a vector whose length is NROW(Y) and whose i-th component is the logistic function at the scalar product between the i-th row of YY and the vector param.

Examples

n<-50 
Y<-sample(c(1,2,3),n,replace=TRUE) 
param<-c(0.2,0.7)
logis(Y,param)

Log-likelihood function of a CUB model without covariates

Description

Compute the log-likelihood function of a CUB model without covariates for a given absolute frequency distribution.

Usage

loglikcub00(m, freq, pai, csi)

Arguments

Log-likelihood function of a CUB model with covariates for the feeling component

Description

Compute the log-likelihood function of a CUB model fitting ordinal data, with q covariates for explaining the feeling component.

Usage

loglikcub0q(m, ordinal, W, pai, gama)

Arguments

Log-likelihood function of a CUB model with covariates for the uncertainty component

Description

Compute the log-likelihood function of a CUB model fitting ordinal responses with covariates for explaining the uncertainty component.

Usage

loglikcubp0(m, ordinal, Y, bbet, ccsi)

Arguments

Log-likelihood function of a CUB model with covariates for both feeling and uncertainty

Description

Compute the log-likelihood function of a CUB model fitting ordinal data with covariates for explaining both the feeling and the uncertainty components.

Usage

loglikcubpq(m, ordinal, Y, W, bet, gama)

Arguments

Plot facilities for fastCUB objects

Description

Plot facilities for objects of class "fastCUB".

Usage

makeplot(object)

Arguments

Details

Returns a plot comparing fitted probabilities and observed relative frequencies for GEM models without covariates. If only one explanatory dichotomous variable is included in the model for one or all components, then the function returns a plot comparing the distributions of the responses conditioned to the value of the covariate.

Generic function for coefficient names

Description

Generic function for names of parameter estimates of object of class "fastCUB".

Usage

parnames(object)

Arguments

Value

Parameter names

See Also

summary

S3 method: print for class "fastCUB"

Description

S3 method print for objects of class fastCUB.

Usage

## S3 method for class 'fastCUB'
print(x, ...)

Arguments

Value

Brief summary results of the fitting procedure, including parameter estimates, their standard errors and the executed call.

Probability distribution of a shifted Binomial random variable

Description

Return the shifted Binomial probability distribution.

Usage

probbit(m,csi)

Arguments

Value

The vector of the probability distribution of a shifted Binomial model.

See Also

bitcsi, probcub00

Examples

m<-7
csi<-0.7
pr<-probbit(m,csi)
plot(1:m,pr,type="h",main="Shifted Binomial probability distribution",xlab="Categories")
points(1:m,pr,pch=19)

Probability distribution of a CUB model without covariates

Description

Compute the probability distribution of a CUB model without covariates.

Usage

probcub00(m,pai,csi)

Arguments

Value

The vector of the probability distribution of a CUB model.

References

Piccolo D. (2003). On the moments of a mixture of uniform and shifted binomial random variables. Quaderni di Statistica, 5, 85–104

See Also

bitcsi, probcub0q, probcubp0, probcubpq

Examples

m<-9
pai<-0.3
csi<-0.8
pr<-probcub00(m,pai,csi)
plot(1:m,pr,type="h",main="CUB probability distribution",xlab="Ordinal categories")
points(1:m,pr,pch=19)

Probability distribution of a CUB model with covariates for the feeling component

Description

Compute the probability distribution of a CUB model with covariates for the feeling component.

Usage

probcub0q(m,ordinal,W,pai,gama)

Arguments

Value

A vector of the same length as ordinal, whose i-th component is the probability of the i-th observation according to a CUB distribution with the corresponding values of the covariates for the feeling component and coefficients specified in gama.

References

Piccolo D. (2006). Observed Information Matrix for MUB Models, Quaderni di Statistica, 8, 33–78
Piccolo D. and D'Elia A. (2008). A new approach for modelling consumers' preferences, Food Quality and Preference, 18, 247–259
Iannario M. and Piccolo D. (2012). CUB models: Statistical methods and empirical evidence, in: Kenett R. S. and Salini S. (eds.), Modern Analysis of Customer Surveys: with applications using R, J. Wiley and Sons, Chichester, 231–258

See Also

bitgama, probcub00, probcubp0, probcubpq

Examples

data(relgoods)
m<-10
naord<-which(is.na(relgoods$Physician))
nacov<-which(is.na(relgoods$Gender))
na<-union(naord,nacov)
ordinal<-relgoods$Physician[-na]
W<-relgoods$Gender[-na]
pai<-0.44; gama<-c(-0.91,-0.7)
pr<-probcub0q(m,ordinal,W,pai,gama)

Probability distribution of a CUB model with covariates for the uncertainty component

Description

Compute the probability distribution of a CUB model with covariates for the uncertainty component.

Usage

probcubp0(m,ordinal,Y,bet,csi)

Arguments

Value

A vector of the same length as ordinal, whose i-th component is the probability of the i-th observation according to a CUB model with the corresponding values of the covariates for the uncertainty component and coefficients for the covariates specified in bet.

References

Piccolo D. (2006). Observed Information Matrix for MUB Models, Quaderni di Statistica, 8, 33–78
Piccolo D. and D'Elia A. (2008). A new approach for modelling consumers' preferences, Food Quality and Preference, 18, 247–259
Iannario M. and Piccolo D. (2012). CUB models: Statistical methods and empirical evidence, in: Kenett R. S. and Salini S. (eds.), Modern Analysis of Customer Surveys: with applications using R, J. Wiley and Sons, Chichester, 231–258

See Also

bitgama, probcub00, probcubpq, probcub0q

Examples

data(relgoods)
m<-10
naord<-which(is.na(relgoods$Physician))
nacov<-which(is.na(relgoods$Gender))
na<-union(naord,nacov)
ordinal<-relgoods$Physician[-na]
Y<-relgoods$Gender[-na]
bet<-c(-0.81,0.93); csi<-0.20
probi<-probcubp0(m,ordinal,Y,bet,csi)

Probability distribution of a CUB model with covariates for both feeling and uncertainty

Description

Compute the probability distribution of a CUB model with covariates for both the feeling and the uncertainty components.

Usage

probcubpq(m,ordinal,Y,W,bet,gama)

Arguments

Value

A vector of the same length as ordinal, whose i-th component is the probability of the i-th rating according to a CUB distribution with given covariates for both uncertainty and feeling, and specified coefficients vectors bet and gama, respectively.

References

Piccolo D. (2006). Observed Information Matrix for MUB Models, Quaderni di Statistica, 8, 33–78
Piccolo D. and D'Elia A. (2008). A new approach for modelling consumers' preferences, Food Quality and Preference, 18, 247–259
Iannario M. and Piccolo D. (2012). CUB models: Statistical methods and empirical evidence, in: Kenett R. S. and Salini S. (eds.), Modern Analysis of Customer Surveys: with applications using R, J. Wiley and Sons, Chichester, 231–258

See Also

bitgama, probcub00, probcubp0, probcub0q

Examples

data(relgoods)
m<-10
naord<-which(is.na(relgoods$Physician))
nacov<-which(is.na(relgoods$Gender))
na<-union(naord,nacov)
ordinal<-relgoods$Physician[-na]
W<-Y<-relgoods$Gender[-na]
gama<-c(-0.91,-0.7); bet<-c(-0.81,0.93)
probi<-probcubpq(m,ordinal,Y,W,bet,gama)

Relational goods and Leisure time dataset

Description

Dataset consists of the results of a survey aimed at measuring the evaluation of people living in the metropolitan area of Naples, Italy, with respect to of relational goods and leisure time collected in December 2014. Every participant was asked to assess on a 10 point ordinal scale his/her personal score for several relational goods (for instance, time dedicated to friends and family) and to leisure time. In addition, the survey asked respondents to self-evaluate their level of happiness by marking a sign along a horizontal line of 110 millimeters according to their feeling, with the left-most extremity standing for "extremely unhappy", and the right-most extremity corresponding to the status "extremely happy".

Usage

data(relgoods)

Format

The description of subjects' covariates is the following:

ID

An identification number

Gender

A factor with levels: 0 = man, 1 = woman

BirthMonth

A variable indicating the month of birth of the respondent

BirthYear

A variable indicating the year of birth of the respondent

Family

A factor variable indicating the number of members of the family

Year.12

A factor with levels: 1 = if there is any child aged less than 12 in the family, 0 = otherwise

EducationDegree

A factor with levels: 1 = compulsory school, 2 = high school diploma, 3 = Graduated-Bachelor degree, 4 = Graduated-Master degree, 5 = Post graduated

MaritalStatus

A factor with levels: 1 = Unmarried, 2 = Married/Cohabitee, 3 = Separated/Divorced, 4 = Widower

Residence

A factor with levels: 1 = City of Naples, 2 = District of Naples, 3 = Others Campania, 4 = Others Italia, 5 = Foreign countries

Glasses

A factor with levels: 1 = wearing glasses or contact lenses, 0 = otherwise

RightHand

A factor with levels: 1 = right-handed, 0 = left-handed

Smoking

A factor with levels: 1 = smoker, 0 = not smoker

WalkAlone

A factor with levels: 1 = usually walking alone, 0 = usually walking in company

job

A factor with levels: 1 = Not working, 2 = Retired, 3 = occasionally, 4 = fixed-term job, 5 = permanent job

PlaySport

A factor with levels: 1 = Not playing any sport, 2 = Yes, individual sport, 3 = Yes, team sport

Pets

A factor with levels: 1 = owning a pet, 0 = not owning any pet

1. Respondents were asked to evaluate the following items on a 10 point Likert scale, ranging from 1 = "never, at all" to 10 = "always, a lot":

   WalkOut

   How often the respondent goes out for a walk

   Parents

   How often respondent talks at least to one of his/her parents

   MeetRelatives

   How often respondent meets his/her relatives

   Association

   Frequency of involvement in volunteering or different kinds of associations/parties, etc

   RelFriends

   Quality of respondent's relationships with friends

   RelNeighbours

   Quality of the relationships with neighbors

   NeedHelp

   Easiness in asking help whenever in need

   Environment

   Level of comfort with the surrounding environment

   Safety

   Level of safety in the streets

   EndofMonth

   Family making ends meet

   MeetFriend

   Number of times the respondent met his/her friends during the month preceding the interview

   Physician

   Importance of the kindness/simpathy in the selection of respondent's physician

   Happiness

   Each respondent was asked to mark a sign on a 110mm horizontal line according to his/her feeling of happiness (left endpoint corresponding to completely unhappy, right-most endpoint corresponding to extremely happy

2. The same respondents were asked to score the activities for leisure time listed below, according to their involvement/degree of amusement, on a 10 point Likert scale ranging from 1 = "At all, nothing, never" to 10 = "Totally, extremely important, always":

   Videogames

   Reading

   Cinema

   Drawing

   Shopping

   Writing

   Bicycle

   Tv

   StayWFriend

   Spending time with friends

   Groups

   Taking part to associations, meetings, etc.

   Walking

   HandWork

   Hobby, gardening, sewing, etc.

   Internet

   Sport

   SocialNetwork

   Gym

   Quiz

   Crosswords, sudoku, etc.

   MusicInstr

   Playing a musical instrument

   GoAroundCar

   Hanging out by car

   Dog

   Walking out the dog

   GoOutEat

   Go to restaurants/pubs

Details

Period of data collection: December 2014
Mode of collection: questionnaire
Number of observations: 2459
Number of subjects' covariates: 16
Number of analyzed items: 34
Warning: with a limited number of missing values

S3 method: summary for class "fastCUB"

Description

S3 method summary for objects of class fastCUB.

Usage

## S3 method for class 'fastCUB'
summary(object, correlation = FALSE, ...)

Arguments

Value

Extended summary results of the fitting procedure, including parameter estimates, their standard errors and Wald statistics, maximized log-likelihood compared with that of the saturated model and of a Uniform sample. AIC, BIC and ICOMP indeces are also displayed for model selection. Execution time and number of exectued iterations for the fitting procedure are aslo returned.

Evaluation of the Orientation Services 2002

Description

A sample survey on students evaluation of the Orientation services was conducted across the 13 Faculties of University of Naples Federico II in five waves: participants were asked to express their ratings on a 7 point scale (1 = "very unsatisfied", 7 = "extremely satisfied"). Here dataset collected during 2002 is loaded.

Usage

data(univer)

Format

The description of subjects' covariates is:

Faculty

A factor variable, with levels ranging from 1 to 13 indicating the coding for the different university faculties

Freqserv

A factor with levels: 0 = for not regular users, 1 = for regular users

Age

Variable indicating the age of the respondent in years

Gender

A factor with levels: 0 = man, 1 = woman

Diploma

A factor with levels: 1 = classic studies, 2 = scientific studies, 3 = linguistic, 4 = Professional, 5 = Technical/Accountancy, 6 = others

Residence

A factor with levels: 1 = city NA, 2 = district NA, 3 = others

ChangeFa

A factor with levels: 1 = changed faculty, 2 = not changed faculty

Analyzed ordinal variables (Likert ordinal scale): 1 = "extremely unsatisfied", 2 = "very unsatisfied", 3 = "unsatisfied", 4 = "indifferent", 5 = "satisfied", 6 = "very satisfied", 7 = "extremely satisfied"

Informat

Level of satisfaction about the collected information

Willingn

Level of satisfaction about the willingness of the staff

Officeho

Judgment about the Office hours

Competen

Judgement about the competence of the staff

Global

Global satisfaction

Details

Period of data collection: 2002
Mode of collection: questionnaire
Number of observations: 2179
Number of subjects' covariates: 7
Number of analyzed items: 5

S3 method vcov() for class "fastCUB"

Description

S3 method: vcov for objects of class fastCUB.

Usage

## S3 method for class 'fastCUB'
vcov(object, ...)

Arguments

Value

Variance-covariance matrix of the final ML estimates for parameters of the fitted CUB model. It is computed on the basis of Louis' identity within the EM algorithm.

See Also

fastCUB