Average root mean square error

Description

Average root mean square error (AMSE).

Usage

amse(Bpar, B0)

Arguments

Details

Let \hat{\theta}_{ij} be the estimated parameter value for the jth parameter in the ith sample (replicate), i = 1, 2, \ldots B, j = 1, 2, \ldots P, and let \theta_{j} be the corresponding true parameter value, the Average root mean square error is defined as follows:

AMSE=\frac{1}{B}\sum_{i}\sqrt{\frac{1}{P} \sum_{j} \left[ \frac{\hat{\theta}_{ij}-\theta_{j}}{\theta_{j}} \right]^2}

Value

Gives the AMSE value.

Note

If \theta_{j} = 0, the ratio \left[ \frac{\hat{\theta}_{ij}-\theta_{j}}{\theta_{j}} \right] is modified as follows: \left[ \frac{\hat{\theta}_{ij}-0}{1} \right]

Author(s)

Massimiliano Pastore & Luigi Lombardi

References

Yang-Wallentin, F., Joreskog, K. G., Luo, H. (2010). Confirmatory Factor Analysis of Ordinal Variables With Misspecified Models, Structural Equation Modeling: A Multidisciplinary Journal, 17, 392-423.

See Also

arb

Average relative bias

Description

Average relative bias (ARB).

Usage

arb(Bpar, B0)

Arguments

Details

Let \hat{\theta}_{ij} be the estimated parameter value for the jth parameter in the ith sample (replicate), i = 1, 2, \ldots B, j = 1, 2, \ldots P, and let \theta_{j} be the corresponding true parameter value, the Average relative bias is defined as follows:

ARB=\frac{100}{B}\sum_{i}\frac{1}{P} \sum_{j} \left( \frac{\hat{\theta}_{ij}-\theta_{j}}{\theta_{j}} \right)

Value

Gives the ARB value.

Note

If \theta_{j} = 0, the ratio \left( \frac{\hat{\theta}_{ij}-\theta_{j}}{\theta_{j}} \right) is modified as follows: \left( \frac{\hat{\theta}_{ij}-0}{1} \right)

Author(s)

Massimiliano Pastore & Luigi Lombardi

References

Yang-Wallentin, F., Joreskog, K. G., Luo, H. (2010). Confirmatory Factor Analysis of Ordinal Variables With Misspecified Models, Structural Equation Modeling: A Multidisciplinary Journal, 17, 392-423.

See Also

amse

Generalized Beta Distribution.

Description

The generalized beta distribution extends the classical beta distribution beyond the [0,1] range (Whitby, 1971).

Usage

dgBeta(x, a = min(x), b = max(x), gam = 1, del = 1)

Arguments

Details

The Generalized Beta Distribution is defined as follows:

G(x;a,b,\gamma,\delta) = \frac{1}{B(\gamma,\delta)(b-a)^{\gamma+\delta-1}} (x-a)^{\gamma-1}(b-x)^{\delta-1}

where B(\gamma,\delta) is the beta function. The parameters a \in R and b \in R (with a < b) are the left and right end points, respectively. The parameters \gamma > 0 and \delta > 0 are the governing shape parameters for a and b respectively. For all the values of the r.v. X that fall outside the interval [a, b], G simply takes value 0. The generalized beta distribution reduces to the beta distribution when a = 0 and b = 1.

Value

Gives the density.

Author(s)

Massimiliano Pastore & Luigi Lombardi

References

Whitby, O. (1971). Estimation of parameters in the generalized beta distribution (Technical Report NO. 29). Department of Statistics: Standford University.

See Also

dgBetaD

Examples

curve(dgBeta(x))
curve(dgBeta(x,gam=3,del=3))
curve(dgBeta(x,gam=1.5,del=2.5))

x <- 1:7
GA <- c(1,3,1.5,8); DE <- c(1,3,4,2.5)
par(mfrow=c(2,2))
for (j in 1:4) {
  plot(x,dgBeta(x,gam=GA[j],del=DE[j]),type="h",
       panel.first=points(x,dgBeta(x,gam=GA[j],del=DE[j]),pch=19),
       main=paste("gamma=",GA[j]," delta=",DE[j],sep=""),ylim=c(0,.6),
       ylab="dgBeta(x)")  
}

Generalized Beta distribution for discrete variables

Description

Generalized Beta distribution for discrete variables.

Usage

dgBetaD(x, a = min(x), b = max(x), gam = 1, del = 1, ct = 1)

Arguments

Details

Let X be a discrete r. v. with range

R_X=\{a,a+1,a+2,\ldots, a+t-1,a+t = b \}

and where a \in \mathrm{N} \cup \{0 \} and t \in \mathrm{N}. The Generalized Discrete Beta Distribution for the r.v. X is defined as follows:

DG(x;a,b,\gamma,\delta)= \left\{ \begin{array}{cl} \frac{G^*(x;a,b,\gamma,\delta)}{\sum_{x' \in R_X} G^*(x';a,b,\gamma,\delta)} & x \in R_X\\ 0 & x \notin R_X \end{array} \right.

where G^* is a modified version of the generalized beta distribution dgBeta defined as

G^*(x;a,b,\gamma,\delta)=\frac{1}{B(\gamma,\delta)(b-a+2c)^{\gamma+\delta-1}} (x-a+c)^{\gamma-1}(b-x+c)^{\delta-1}

Value

Gives the density.

Author(s)

Massimiliano Pastore & Luigi Lombardi

References

Lombardi, L., Pastore, M. (2014). sgr: A Package for Simulating Conditional Fake Ordinal Data. The R Journal, 6, 164-177.

Pastore, M., Lombardi, L. (2014). The impact of faking on Cronbach's Alpha for dichotomous and ordered rating scores. Quality & Quantity, 48, 1191-1211.

See Also

dgBeta

Examples

x <- 1:7
GA <- c(1,3,1.5,8); DE <- c(1,3,4,2.5)
par(mfrow=c(2,2))
for (j in 1:4) {
  plot(x,dgBetaD(x,gam=GA[j],del=DE[j]),type="h",
       panel.first=points(x,dgBetaD(x,gam=GA[j],del=DE[j]),pch=19),
       main=paste("gamma=",GA[j]," delta=",DE[j],sep=""),ylim=c(0,.6),
       ylab="dgBetaD(x)")  
}

Internal function.

Description

Set different instances of the conditional replacement distribution.

Usage

model.fake.par(fake.model = c("uninformative", "average", "slight", "extreme"))

Arguments

Value

Gives a list with \gamma and \delta parameters.

Author(s)

Massimiliano Pastore

References

Lombardi, L., Pastore, M. (2014). sgr: A Package for Simulating Conditional Fake Ordinal Data. The R Journal, 6, 164-177.

Pastore, M., Lombardi, L. (2014). The impact of faking on Cronbach's Alpha for dichotomous and ordered rating scores. Quality & Quantity, 48, 1191-1211.

Examples

model.fake.par() # default
model.fake.par("average")

Internal function.

Description

This function allows to set different replacement distributions for different subsets of cells in the data matrix.

Usage

partition.replacement(Dx, PM, Q = NULL, Pparm = NULL,
        fake.model = NULL,p = NULL)

Arguments

Details

PM has size dim(Dx) and contains a numeric code for each distinct class in the partition. If a cell of the partition matrix PM contains 0, then the corresponding Dx cell value is not modified (no replacements condition class).

Pparm is a list containing three elements. Each element is a P\times 2 matrix where P is the total number of classes in the partition (see examples for further details).

p: Overall probability of replacement: p[,1] indicates the faking good probability, p[,2] indicates the faking bad probability.

gam: Gamma parameter: gam[,1] and gam[,2] indicate the faking good and the faking bad parameters for the lower bound a.

del: Delta parameter: del[,1] and del[,2] indicate the faking good and the faking bad parameters for the upper bound b.

Note that it is possible to define a faking model using the fake.model assignment. In such cases the user must specify also the overall probability of replacement using parameter p.

Value

Returns the fake data matrix.

Author(s)

Massimiliano Pastore

See Also

rdatarepl, replacement.matrix

Examples

require(MASS)
set.seed(20130207)
R <- matrix(c(1,.3,.3,1),2,2)
Dx <- rdatagen(n=20,R=R,Q=5)$data

## partition matrix
PM <- matrix(0,nrow(Dx),ncol(Dx))
PM[3:10,2] <- 1
PM[3:10,1] <- 1
partition.replacement(Dx,PM) # warning! zero replacements

## using fake.model
partition.replacement(Dx,PM,fake.model="uninformative",p=matrix(c(.3,.2),ncol=2))

###
p <- c(.5,0)
gam <- c(1,1)
del <- c(1,1)
Pparm <- list(p=p,gam=gam,del=del)
partition.replacement(Dx,PM,Pparm=Pparm) 

### another partition
PM[11:15,2] <- 2
(Pparm <- list(p=matrix(c(0,.5,.5,0),2,2),
      gam=matrix(c(1,4,1,4),2,2),del=matrix(c(1,2,1,2),2,2)))
partition.replacement(Dx,PM,Pparm=Pparm) 

Probability of faking.

Description

The function gives the conditional probability of replacement p(f=k|d=h,\theta_F) for discrete values in the range 1, \ldots, Q.

Usage

pfake(k, h = k, p = c(0,0), Q = 5, gam = c(1,1), del = c(1,1),
      fake.model = c("uninformative", "average", "slight", "extreme"))

Arguments

Value

Gives the conditional probability distribution based on the following equation

p(f=k|d=h,\theta_F)= \left\{ \begin{array}{cl} DG(k;h+1,Q,\gamma_{+},\delta_{+}) \pi_{+} & 1 \leq h < k \leq Q \\ DG(k;q,h-1,\gamma_{-},\delta_{-}) \pi_{-} & 1 \leq k < h \leq Q \\ 1-(\pi_{+}+\pi{-}) & 1 < h=k < Q \\ 1- \pi_{+} & k=h=1 \\ 1- \pi_{-} & k=h=Q \end{array} \right.

with \theta_F and DG being the parameter vector (\gamma_{+},\gamma_{-},\delta_{+},\delta_{-},\pi_{+},\pi_{-}) and the generalized Beta distribution for discrete variables (dgBetaD) with bounds a=h+1 (resp. a=1) and b=Q (resp b=h-1). The parameter \pi_{+} denotes the probability of faking good, \pi_{-} indicates the probability of faking bad. Note that the faking probabilities must satisfy the following condition: \pi_{+}+\pi_{-} \leq 1.

Author(s)

Massimiliano Pastore & Luigi Lombardi

References

Lombardi, L., Pastore, M. (2014). sgr: A Package for Simulating Conditional Fake Ordinal Data. The R Journal, 6, 164-177.

Pastore, M., Lombardi, L. (2014). The impact of faking on Cronbach's Alpha for dichotomous and ordered rating scores. Quality & Quantity, 48, 1191-1211.

Examples

x <- 1:7
GA <- c(1,3,1.5,8); DE <- c(1,3,4,2.5)

### fake good
par(mfrow=c(2,2))
for (j in 1:4) {
  y <- NULL
  for (i in x) y <- c(y,pfake(x[i],h=4,Q=7,
                gam=c(GA[j],GA[j]),del=c(DE[j],DE[j]),p=c(.4,0)))
  plot(x,y,type="h",panel.first=points(x,y,pch=19),
       main=paste("gamma=",GA[j]," delta=",DE[j],sep=""),ylim=c(0,.7),
       ylab="Replacement probability") 
}

### fake bad
for (j in 1:4) {
  y <- NULL
  for (i in x) y <- c(y,pfake(x[i],h=4,Q=7,
                gam=c(GA[j],GA[j]),del=c(DE[j],DE[j]),p=c(0,.4)))
  plot(x,y,type="h",panel.first=points(x,y,pch=19),
       main=paste("gamma=",GA[j]," delta=",DE[j],sep=""),ylim=c(0,.7),
       ylab="Replacement probability") 
}

### fake good and fake bad
P = c(.4,.4)
par(mfrow=c(2,4))
for (j in x) {
  y <- NULL
  for (i in x) {
    y <- c(y,pfake(x[i],h=x[j],Q=max(x),gam=c(GA[1],GA[1]),del=c(DE[1],DE[1]),p=P))
  }
  plot(x,y,type="h",panel.first=points(x,y,pch=19),
       main=paste("h=",x[j],sep=""),ylim=c(0,1),
       ylab="Replacement probability") 
  print(sum(y,na.rm=TRUE)) 
}

### using the fake.model argument
x <- 1:5 
models <- c("uninformative","average","slight","extreme")
par(mfrow=c(2,2))
for (j in 1:4) {
  y <- NULL
  for (i in x) y <- c(y,pfake(x[i],h=2,Q=max(x),
            fake.model=models[j],p=c(.45,0)))       # fake good
  plot(x,y,type="h",panel.first=points(x,y,pch=19),
       main=paste(models[j],"model"),ylim=c(0,1),
       ylab="Replacement probability") 
}

par(mfrow=c(2,2))
for (j in 1:4) {
  y <- NULL
  for (i in x) y <- c(y,pfake(x[i],h=4,Q=max(x),
            fake.model=models[j],p=c(0,.45)))       # fake bad
  plot(x,y,type="h",panel.first=points(x,y,pch=19),
       main=paste(models[j],"model"),ylim=c(0,1),
       ylab="Replacement probability") 
}

Probability of faking bad.

Description

The function gives the conditional probability of replacement p(f=k|d=h,\theta_F) for discrete values in the range 1, \ldots, Q.

Usage

pfakebad(k, h = k, p = 0, Q = 5, gam = 1, del = 1)

Arguments

Value

Gives the conditional probability based on the following equation

p(f=k|d=h,\theta_F)= \left\{ \begin{array}{cl} 1 & h=k=1 \\ GD(k;1,h-1,\gamma,\delta) \pi & 1 \leq k < h \leq Q \\ 1-\pi & 1 < h=k \leq Q \\ 0 & 1 \leq h < k \leq Q \end{array} \right.

with \theta_F and GD being the parameter vector (\gamma,\delta,\pi) and the generalized Beta distribution for discrete variables (dgBetaD) with bounds a=h+1 and b=Q. The parameter \pi denotes the probability of faking bad.

Author(s)

Massimiliano Pastore & Luigi Lombardi

References

Pastore, M., Lombardi, L. (2014). The impact of faking on Cronbach's Alpha for dichotomous and ordered rating scores. Quality & Quantity, 48, 1191-1211.

Examples

x <- 1:7
GA <- c(1,3,1.5,8); DE <- c(1,3,4,2.5)
par(mfrow=c(2,2))
for (j in 1:4) {
  y <- NULL
  for (i in x) y <- c(y,pfakebad(x[i],h=5,Q=7,gam=GA[j],del=DE[j],p=.4))
  plot(x,y,type="h",panel.first=points(x,y,pch=19),
       main=paste("gamma=",GA[j]," delta=",DE[j],sep=""),ylim=c(0,.7),
       ylab="Replacement probability")  
}

Probability of faking good.

Description

The function gives the conditional probability of replacement p(f=k|d=h,\theta_F) for discrete values in the range 1, \ldots, Q.

Usage

pfakegood(k, h = k, p = 0, Q = 5, gam = 1, del = 1)

Arguments

Value

Gives the conditional probability based on the following equation

p(f=k|d=h,\theta_F)= \left\{ \begin{array}{cl} 1 & h=k=Q \\ GD(k;h+1,Q,\gamma,\delta) \pi & 1 \leq h < k \leq Q \\ 1-\pi & 1 \leq k=h < Q \\ 0 & 1 \leq k < h \leq Q \end{array} \right.

with \theta_F and GD being the parameter vector (\gamma,\delta,\pi) and the generalized Beta distribution for discrete variables (dgBetaD) with bounds a=h+1 and b=Q. The parameter \pi denotes the probability of faking good.

Author(s)

Massimiliano Pastore & Luigi Lombardi

References

Pastore, M., Lombardi, L. (2014). The impact of faking on Cronbach's Alpha for dichotomous and ordered rating scores. Quality & Quantity, 48, 1191-1211.

Examples

x <- 1:7
GA <- c(1,3,1.5,8); DE <- c(1,3,4,2.5)
par(mfrow=c(2,2))
for (j in 1:4) {
  y <- NULL
  for (i in x) y <- c(y,pfakegood(x[i],h=3,Q=7,gam=GA[j],del=DE[j],p=.4))
  plot(x,y,type="h",panel.first=points(x,y,pch=19),
       main=paste("gamma=",GA[j]," delta=",DE[j],sep=""),ylim=c(0,.7),
       ylab="Replacement probability")  
}

Data set

Description

The psydata data frame has 744 rows (observations) and 22 columns (variables).

Usage

data(psydata)

Format

This data frame contains the following variables:

• nsogg: int, subject number.

• vers: Factor, questionnaire version: V1 fake-motivating version, V3 honest-motivating version e V4 neutral version.

• sex: Factor, gender.

• eta: int, age.

• resid: Factor, residence.

• dipl: Factor, education.

• voto: int, high school's final score.

• votomax: int, maximum value for voto.

• cdl: Factor, a character string indicating the type of undergraduate program.

• aep..: int, 12 items of the AEP/A scale.

• tot: int, total score.

Author(s)

Andrea Bobbio, Massimo Nucci, Massimiliano Pastore

Simulate discrete data.

Description

Simulate discrete data from either a correlation matrix or thresholds or probabilities.

Usage

rdatagen(n = 100, R = diag(1,2), Q = NULL, th = NULL, probs = NULL)

Arguments

Value

Returns a list with four elements:

Note

Defaults work like in the mvrnorm function of the MASS package.

Author(s)

Massimiliano Pastore, Luigi Lombardi & Marco Bressan

References

Lombardi, L., Pastore, M. (2014). sgr: A Package for Simulating Conditional Fake Ordinal Data. The R Journal, 6, 164-177.

Pastore, M., Lombardi, L. (2014). The impact of faking on Cronbach's Alpha for dichotomous and ordered rating scores. Quality & Quantity, 48, 1191-1211.

Examples

require(MASS)
## only default
rdatagen()

## set correlations only
R <- matrix(c(1,.4,.4,1),2,2)
Dx <- rdatagen(n=10000,R=R)$data

par(mfrow=c(1,2))
for (j in 1:ncol(Dx)) hist(Dx[,j])

## set correlations and Q
Dx <- rdatagen(n=10000,R=R,Q=2)$data

par(mfrow=c(1,2))
for (j in 1:ncol(Dx)) barplot(table(Dx[,j])/nrow(Dx))

## set correlations and thresholds
th <- list(c(-Inf,qchisq(pbinom(0:3,4,.5),1),Inf),
    c(-Inf,qnorm(pbinom(0:2,3,.5)),Inf))
Dx <- rdatagen(n=10000,R=R,th=th)$data

par(mfrow=c(1,2))
for (j in 1:ncol(Dx)) barplot(table(Dx[,j])/nrow(Dx))

## set correlations and probabilities [1]
probs <- list(c(.125,.375,.375,.125),c(.125,.375,.375,.125))
Dx <- rdatagen(n=10000,R=R,probs=probs)$data

par(mfrow=c(1,2))
for (j in 1:ncol(Dx)) barplot(table(Dx[,j])/nrow(Dx))

## set correlations and probabilities [2]
probs <- c(.125,.375,.375,.125)
Dx <- rdatagen(n=10000,R=R,probs=probs)$data

par(mfrow=c(1,2))
for (j in 1:ncol(Dx)) barplot(table(Dx[,j])/nrow(Dx))

## set different values for Q
Dx <- rdatagen(n=1000,Q=c(2,4),R=R)$data

par(mfrow=c(1,2))
for (j in 1:ncol(Dx)) barplot(table(Dx[,j])/nrow(Dx))

Random replacements of data.

Description

Replaces data in the original data matrix using a specified replacement matrix.

Usage

rdatarepl(Dx, RM, printfp = TRUE)

Arguments

Details

Replacement matrices can be obtained from the replacement.matrix function.

Value

Returns a list with two elements:

Author(s)

Massimiliano Pastore

See Also

replacement.matrix

Examples

require(MASS)
set.seed(20130207)
Dx <- rdatagen(R=matrix(c(1,.3,.3,1),2,2),Q=5)$data
RM <- replacement.matrix(p=c(.6,0))
Fx <- rdatarepl(Dx,RM)$Fx

par(mfrow=c(2,2))
for (j in 1:ncol(Dx)) barplot(table(Dx[,j]),main="original data")
for (j in 1:ncol(Fx)) barplot(table(Fx[,j]),main="replaced data")

Replacement matrix.

Description

Builds the replacement matrix.

Usage

replacement.matrix(Q = 5, p = c(0,0), gam = c(1,1), del = c(1,1),
    fake.model = c("uninformative", "average", "slight", "extreme"))

Arguments

Value

Gives a Q \times Q matrix with replacement probabilities. Each row r (1 \leq r \leq Q) in the matrix indicates the conditional probability distribution

p(k=r|h=c,\pi), \qquad h=1,\ldots,Q

\pi (p) denotes the overall replacement probability.

Author(s)

Massimiliano Pastore

See Also

dgBetaD, pfake, pfakegood, pfakebad

Examples

## no replacements
replacement.matrix(Q=7) 

## faking good
replacement.matrix(Q=7,p=c(.5,0))
replacement.matrix(Q=7,p=c(.5,0),gam=8,del=2.5)

## faking bad
replacement.matrix(Q=7,p=c(0,.5))
replacement.matrix(Q=7,p=c(0,.5),gam=8,del=2.5)

## faking good and faking bad
replacement.matrix(Q=7,p=c(.3,.5),gam=c(8,8),del=c(2.5,2.5))

## using the fake.model argument
replacement.matrix(Q=7,p=c(0,.4),fake.model="extreme")
replacement.matrix(Q=7,p=c(.4,0),fake.model="extreme")
replacement.matrix(Q=7,p=c(.4,.4),fake.model="slight")

Data set

Description

Data about smoking and drug consumption among young people.

Usage

data(smokers)

Format

This data frame contains the following columns:

• age: int, 1 = adults, 2 = minors.

• smoking: int, 1 = yes, 2 = no.

• drug: int, drug addiction, 1 = yes, 2 = no.

• druguse: int, drug consumption, 1 = never, 2 = once, 3 = some times, 4 = often.

Source

Pastore, M., Lombardi, L., Mereu, F. (2007). Effects of malingering in self-report measures: A scenario analysis approach; in C. H. Skiadas (Ed.), Recent Advances in Stochastic Modeling and Data Analysis, pp. 483-491, World Scientific Publishing.