irtProb (version 1.0)

m4pl: Multidimensional One, Two, Three and Four Person Parameters Logistic Distributions

Description

Density, distribution function, quantile function and random generation for the multidimensional one, two, three and four person parameters logistic distributions.

Usage

pm4pl(theta = 0, S = 0, C = 0, D = 0, s = 1/1.702, b = 0, c = 0, d = 1,
       lower.tail = TRUE, log.p = FALSE)
      
 dm4pl(theta = 0, S = rep(0,length(theta)), C = rep(0,length(theta)),
       D = rep(0,length(theta)), b = 0, s = 1/1.702, c = 0, d = 1, log.p = FALSE)
      
 qm4pl(p = 0.05, S = 0, C = 0, D = 0, s = 1/1.702, b = 0, c = 0, d = 1,
       lower.tail = TRUE, log.p = FALSE)
      
 rm4pl(N = 100, S = 0, C = 0, D = 0, s = 1/1.702, b = 0, c = 0, d = 1)

Arguments

theta

numeric; vector of person proficiency ($\theta$) levels scaled on a normal z score.

S

numeric; positive vector of personal fluctuation parameters ($\sigma$).

C

numeric; positive vector of personal pseudo-guessing parameters ($\chi$, a probability between 0 and 1).

D

numeric; positive vector of personal inattention parameters ($\delta$, a probability between 0 and 1).

N

numeric; number of observations.

p

numeric; vector of probability.

s

numeric; positive vector of item fluctuation parameters.

b

numeric; vector of item difficulty parameters.

c

numeric; positive vector of item pseudo-guessing parameters (a probability between 0 and 1).

d

numeric; positive vector of item inattention parameters (a probability between 0 and 1).

lower.tail

logical; if TRUE (default), probabilities are $P(X_{j} <= x_{ij})$,="" otherwise,="" $p(x_{j}=""> x_{ij})$.

log.p

logical; if TRUE, probabilities p are given as log(p).

Value

• pm4pl
  numeric; gives the distribution function (cdf).
• dm4pl
  numeric; gives the density (derivative of p4pl).
• qm4pl
  numeric; gives the quantile function (inverse of p4pl).
• rm4pl
  numeric; generates theta random deviates.

Details

The multidimensional 4 persons parameters logistic distribution (cdf) is equal to: $$P(x_{ij} = 1|\theta _j ,\sigma _j ,\chi _j ,\delta _j ,s_i ,b_i ,c_i ,d_i ) = (\chi _j + c_i ) + {{(d_i - \delta _j ) - (\chi _j + c_i )} \over {1 + e^{{{ - Da_i (\theta _j - b_i )} \over {\sqrt {\sigma _j^2 + s_i^2 } }}} }}$$ where the parameters are defined in the section arguments and i and j are respectively the items and the persons indices. The $\sigma_j$, $\chi_j$ and $\delta_j$ parameters are respectively the personnal fluctuation, pseudo-guessing and inattention parameters. The multidimensional 4 persons parameters logistic model (M4PL) was described by Raiche, Magis and Blais (2008; Raiche, Magis and Beland, 2009; Raiche, Blais and Magis, 2009).

References

Blais, J.-G., Raiche, G. and Magis, D. (2009). La detection des patrons de reponses problematiques dans le contexte des tests informatises. In Blais, J.-G. (Ed.): Evaluation des apprentissages et technologies de l'information et de la communication : enjeux, applications et modeles de mesure. Ste-Foy, Quebec: Presses de l'Universite Laval. Raiche, G., Magis, D. and Beland, S. (2009). La correction du resultat d'un etudiant en presence de tentatives de fraudes. Communication presentee a l'Universite du Quebec a Montreal. Retrieved from http://www.camri.uqam.ca/camri/camriBase/ Raiche, G., Magis, D. and Blais, J.-G. (2008). Multidimensional item response theory models integrating additional inattention, pseudo-guessing, and discrimination person parameters. Communication at the annual international Psychometric Society meeting, Durham, New Hamshire. Retrieved from http://www.camri.uqam.ca/camri/camriBase/

See Also

grm4pl, ggrm4pl, pggrm4pl

Examples

## ....................................................................
# Approximation of p4pl() by pm4pl()
 theta  <- 0
 S      <- 0; C <- 0; D <- 0
 a      <- 1.702; s <- sqrt(1/a^2); b <- 0; c <- 0; d <- 1
 p.4pl  <-  p4pl(theta,a=a,b,c,d);
 p.m4pl <- pm4pl(theta,S,C,D,s,b,c,d)
 print(c(p4pl=p.4pl, m4pl=p.m4pl))

# Comparison of p4pl() and pm4pl() according to diverse values of theta
#  while person parameters vary
 theta  <- seq(-4,4,length=100); N <- length(theta)
 S      <- rep(0.9,N); C <- rep(0,N); D <- rep(0.9,N)
 a      <- 1.702; s <- 1/1.702
 p.4pl  <- p4pl(theta=theta, a=a, b=b, c=c, d=d)
 p.m4pl <- pm4pl(theta=theta, S=S, C=C, D=D, s=s, b=b, c=c, d=d)
 print(c(difference.maximale = max(p.4pl - p.m4pl)))
 round(rbind(theta=theta, p4pl=p.4pl, m4pl=p.m4pl, dif=p.4pl-p.m4pl),3)
 plot(theta,p.4pl,type="l", col="black", ylab="Probability")
 lines(theta,p.m4pl,col="red"); max(p.4pl-p.m4pl)
 
# Recovery of probability by quantile
 pm4pl(theta=3,b=0); pm4pl(theta=qm4pl(p=0.20))

# Density function dm4pl()
# Comparison 01 between d4pl() and dm4pl()
 theta  <- seq(-4,4, length=100)
 # theta <- 0 # We can also experiment with an unique value of theta
 N      <- length(theta)
 S      <- rep(0,N); C <- rep(0,N); D <- rep(0,N)
 a      <- 1.702
 d.4pl  <- d4pl(theta=theta,a=a)
 d.m4pl <- dm4pl(theta=theta,S=0,s=1/a)
 stats  <- round(cbind(theta=theta, d4pl=d.4pl, m4pl=d.m4pl, dif=d.4pl-d.m4pl),3)
 print(stats)
 print(max(d.4pl - d.m4pl))
 plot(theta,d.4pl,type="l", col="black", ylab="Density")
 lines(theta,d.m4pl,col="red")

# Comparison 02 between d4pl() and dm4pl()
 theta  <- seq(-4,4, length=10)
 N      <- length(theta)
 S      <- rep(.3,N); C <- rep(0,N); D <- rep(0,N)
 a      <- 1.702
 d.4pl  <- d4pl(theta=theta,a=a)
 d.m4pl <- dm4pl(theta=theta, S=S, C=C, D=D, s=1/a)
 stats  <- round(cbind(theta=theta, d4pl=d.4pl, m4pl=d.m4pl, dif=d.4pl-d.m4pl),3)
 print(stats)
 print(max(d.4pl - d.m4pl))
 plot(theta,d.4pl,type="l", col="black", ylab="Density")
 lines(theta,d.m4pl,col="red")

# Comparison of q4pl and qm4pl
#  followed by recovery of quantiles
 pm4pl(theta=1, ); p4pl(theta=1, a=1.702)
 qm4pl(p=0.99,); q4pl(p=0.99, a=1.702)
 qm4pl(pm4pl(theta=1)); q4pl(p4pl(theta=1, a=1.702), a=1.702)
 qm4pl(p=pm4pl(theta=3))
 qm4pl(p=seq(0.01,0.99, length=10))

# Generation of theta values by rm4pl()
# ... Exemple 01 - A 4pl() equivalent distribution must be recovered when a=1.702
 res   <- rm4pl(N=1000)
 stats <- c(mean=mean(res), sd=sd(res), skewness=skewness(res),
            kurtosis=3-kurtosis(res))
 print(stats)
 # pdf of this distribution
 theta   <- seq(-4,4,length=100); C=rep(0,N); D=rep(0,N)
 density <- dm4pl(theta,C=C,D=D)
 plot(theta, density, type="l")

# ... Exemple 02 - Distribution with D != 0
 require(moments)
 S     <- 1/1.702
 B     <- 0
 C     <- 0.00
 D     <- 0.90
 res   <- rm4pl(N=1000, S=S, C=C, D=D)
 stats <- c(mean=mean(res), sd=sd(res), skewness=skewness(res), kurtosis=3-kurtosis(res))
 print(stats)
 # pdf of this distribution
 theta   <- seq(-4,4,length=100)
 density <- dm4pl(theta, S=S, C=C, D=D)
 print(c(max=max(density)))
 plot(theta, density, type="l")
## ....................................................................