Calculation of Probabilities and Moments for the Generalized Logistic Item Response Model
Calculation of probabilities and moments for the generalized logistic item response model (Stukel, 1988).
pgenlogis(x, alpha1=0, alpha2=0) genlogis.moments(alpha1, alpha2)
x |
Vector |
alpha1 |
Upper tail parameter α_1 in the generalized logistic item response model. The default is 0. |
alpha2 |
Lower tail parameter α_2 parameter in the generalized logistic item response model. The default is 0. |
The class of generalized logistic link functions contain the most important link functions using the specifications (Stukel, 1988):
logistic link function L:
L(x) \approx G_{ ( α_1=0, α_2=0)}[ x ]
probit link function Φ:
Φ(x) \approx G_{ ( α_1=0.165, α_2=0.165)}[ 1.47 x ]
loglog link function H:
H(x) \approx G_{ (α_1=-0.037, α_2=0.62)}[ -0.39+1.20x-0.007x^2]
cloglog link function H:
H(x) \approx G_{ ( α_1=0.62, α_2=-0.037)}[ 0.54+1.64x+0.28x^2+0.046x^3]
Vector of probabilities or moments
Stukel, T. A. (1988). Generalized logistic models. Journal of the American Statistical Association, 83(402), 426-431. doi: 10.1080/01621459.1988.10478613
sirt::pgenlogis( x=c(-.3, 0, .25, 1 ), alpha1=0, alpha2=.6 )
## [1] 0.4185580 0.5000000 0.5621765 0.7310586
####################################################################
# compare link functions
x <- seq( -3,3, .1 )
#***
# logistic link
y <- sirt::pgenlogis( x, alpha1=0, alpha2=0 )
plot( x, stats::plogis(x), type="l", main="Logistic Link", lwd=2)
points( x, y, pch=1, col=2 )
#***
# probit link
round( sirt::genlogis.moments( alpha1=.165, alpha2=.165 ), 3 )
## M SD Var
## 0.000 1.472 2.167
# SD of generalized logistic link function is 1.472
y <- sirt::pgenlogis( x * 1.47, alpha1=.165, alpha2=.165 )
plot( x, stats::pnorm(x), type="l", main="Probit Link", lwd=2)
points( x, y, pch=1, col=2 )
#***
# loglog link
y <- sirt::pgenlogis( -.39 + 1.20*x -.007*x^2, alpha1=-.037, alpha2=.62 )
plot( x, exp( - exp( -x ) ), type="l", main="Loglog Link", lwd=2,
ylab="loglog(x)=exp(-exp(-x))" )
points( x, y, pch=17, col=2 )
#***
# cloglog link
y <- sirt::pgenlogis( .54+1.64*x +.28*x^2 + .046*x^3, alpha1=.062, alpha2=-.037 )
plot( x, 1-exp( - exp(x) ), type="l", main="Cloglog Link", lwd=2,
ylab="loglog(x)=1-exp(-exp(x))" )
points( x, y, pch=17, col=2 )Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.