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Calculation of Probabilities and Moments for the Generalized Logistic Item Response Model

pgenlogis.Rd

Calculation of probabilities and moments for the generalized logistic item response model (Stukel, 1988).

Usage

pgenlogis(x, alpha1=0, alpha2=0)

genlogis.moments(alpha1, alpha2)

Arguments

x

Vector

alpha1

Upper tail parameter \(\alpha_1\) in the generalized logistic item response model. The default is 0.

alpha2

Lower tail parameter \(\alpha_2\) parameter in the generalized logistic item response model. The default is 0.

Details

The class of generalized logistic link functions contain the most important link functions using the specifications (Stukel, 1988):

  • logistic link function \(L\): $$ L(x) \approx G_{ ( \alpha_1=0, \alpha_2=0)}[ x ] $$

  • probit link function \(\Phi\): $$ \Phi(x) \approx G_{ ( \alpha_1=0.165, \alpha_2=0.165)}[ 1.47 x ] $$

  • loglog link function \(H\): $$ H(x) \approx G_{ (\alpha_1=-0.037, \alpha_2=0.62)}[ -0.39+1.20x-0.007x^2] $$

  • cloglog link function \(H\): $$ H(x) \approx G_{ ( \alpha_1=0.62, \alpha_2=-0.037)}[ 0.54+1.64x+0.28x^2+0.046x^3] $$

Value

Vector of probabilities or moments

References

Stukel, T. A. (1988). Generalized logistic models. Journal of the American Statistical Association, 83(402), 426-431. doi:10.1080/01621459.1988.10478613

Examples

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 )