4pl: One, Two, Three and Four Parameters Logistic Distributions

In irtProb: Utilities and Probability Distributions Related to Multidimensional Person Item Response Models

Description

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

Usage

p4pl(theta = 0, a = 1, b = 0, c = 0, d = 1, lower.tail = TRUE,
log.p = FALSE)

d4pl(theta = 0, a = 1, b = 0, c = 0, d = 1,
log.p = FALSE)

q4pl(p  = 0.05, a = 1, b = 0, c = 0, d = 1, lower.tail = TRUE,
log.p = FALSE)

r4pl(N  = 100,  a = 1, b = 0, c = 0, d = 1)

Arguments

N	numeric; number of observations.
p	numeric; vector of probability.
theta	numeric; vector of person proficiency levels scaled on a normal z score.
a	numeric; positive vector of item discrimination 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).

Details

The 4 parameters logistic distribution (cdf) is equal to:

P(x_{ij} = 1|θ _j ,a_i ,b_i ,c_i ,d_i ) = c_i + \frac{{d_i - c_i }}{{1 + e^{ - Da_i (θ _j - b_i )} }},

where the parameters are defined in the section arguments and i and j are respectively the items and the persons indices. A normal version of the 4PL model was described by McDonald (1967, p. 67), Barton and Lord (1981), like Hambleton and Swaminathan (1985, p. 48-50).

Value

p4pl	numeric; gives the distribution function (cdf).
d4pl	numeric; gives the density (derivative of p4pl).
q4pl	numeric; gives the quantile function (inverse of p4pl).
r4pl	numeric; generates theta random deviates.

Note

Code inspired by the pnorm function structure from the R base package.

Author(s)

Gilles Raiche, Universite du Quebec a Montreal (UQAM),

Departement d'education et pedagogie

[e-mail], http://www.er.uqam.ca/nobel/r17165/

References

Barton, M. A. and Lord, F. M. (1981). An upper asymptote for the tree-parameter logistic item-response model. Research Bulletin 81-20. Princeton, NJ: Educational Testing Service.

Hambleton, R. K. and Swaminathan, H. (1985). Item response theory - Principles and applications. Boston, Massachuset: Kluwer.

Lord, F. M. and Novick, M. R. (1968). Statistical theories of mental test scores, 2nd edition. Reading, Massacusett: Addison-Wesley.

McDonald, R. P. (1967). Non-linear factor analysis. Psychometric Monographs, 15.

See Also

gr4pl, ggr4pl, ctt2irt, irt2ctt

Examples

## ....................................................................
# probability of a correct response
 p4pl(theta = 3, b = 0)
 
# Verification of the approximation of N(0,1) by a logistic (D=1.702)
 a <- 1; b <- 0; c <- 0; d <- 1; theta <- seq(-4, 4, length = 100)
 
# D constant 1.702 gives an approximation of a N(0,1) by a logistic
 prob.irt  <- p4pl(theta, a*1.702, b, c, d)
 prob.norm <- pnorm(theta, 0, 1)
 plot(theta, prob.irt)
 lines(theta, prob.norm, col = "red")
 
# Maximal difference between the two functions: less than 0.01
 max(prob.irt - prob.norm)
 
# Recovery of the value of the probability of a correct response p4pl()
# from the quantile value q4pl()
 p4pl(theta = q4pl(p = 0.20))

# Recovery of the quantile value from the probability of a correct
# response
 q4pl(p=p4pl(theta=3))

# Density Functions [derivative of p4pl()]
 d4pl(theta = 3, a = 1.702)
 theta   <- seq(-4, 4, length = 100)
 a       <- 3.702; b <- 0; c <- 0; d <- 1
 density <- d4pl(theta = theta, a = a, b = b, c = c, d = d)
 label   <- expression("Density - First Derivative")
 plot(theta, density, ylab = label, col = 1, type = "l")
 lines(theta, dnorm(x = theta, sd = 1.702/a), col = "red", type = "l")

## Generation of proficiency levels from r4pl() according to a N(0,1)
 data <- r4pl(N = 10000, a = 1.702, b = 0, c = 0, d = 1)
 c(mean = mean(data), sd = sd(data))
## ....................................................................

Example output

Loading required package: lattice
Loading required package: moments
[1] 0.9525741
[1] 0.009446789
[1] 0.2
[1] 3
    theta 
0.0101907 
      mean         sd 
0.02240066 1.06486553 

irtProb documentation built on May 2, 2019, 1:30 p.m.