sirt (version 1.5-0)

lc.2raters: Latent Class Model for Two Exchangeable Raters and One Item

Description

This function computes a latent class model for ratings on an item based on exchangeable raters (Uebersax & Grove, 1990). Additionally, several measures of rater agreement are computed (see e.g. Gwet, 2010).

Usage

lc.2raters(data, conv = 0.001, maxiter = 1000, progress = TRUE)

## S3 method for class 'lc.2raters':
summary(object,...)

Arguments

data

Data frame with item responses (must be ordered from 0 to $K$) and two columns which correspond to ratings of two (exchangeable) raters.

conv

Convergence criterion

maxiter

Maximum number of iterations

progress

An optional logical indicating whether iteration progress should be displayed.

object

Object of class lc.2raters

...

Further arguments to be passed

Value

• A list with following entries
• classprob.1rater.like
  Classification probability $P(c|x)$ of latent category $c$ given a manifest rating $x$ (estimated by maximum likelihood)
• classprob.1rater.post
  Classification probability $P(c|x)$ of latent category $c$ given a manifest rating $x$ (estimated by the posterior distribution)
• classprob.2rater.like
  Classification probability $P(c|(x,y))$ of latent category $c$ given two manifest ratings $x$ and $y$ (estimated by maximum likelihood)
• classprob.2rater.post
  Classification probability $P(c|(x,y))$ of latent category $c$ given two manifest ratings $x$ and $y$ (estimated by posterior distribution)
• f.yi.qk
  Likelihood of each pair of ratings
• f.qk.yi
  Posterior of each pair of ratings
• probs
  Item response probabilities $p_{x|c}$
• pi.k
  Estimated class proportions $\pi_c$
• pi.k.obs
  Observed manifest class proportions
• freq.long
  Frequency table of ratings in long format
• freq.table
  Symmetrized frequency table of ratings
• agree.stats
  Measures of rater agreement. These measures include percentage agreement (agree0, agree1), Cohen's kappa and weighted Cohen's kappa (kappa, wtd.kappa.linear), Gwet's AC1 agreement measures (AC1; Gwet, 2008, 2010) and Aickin's alpha (alpha.aickin; Aickin, 1990).
• data
  Used dataset
• N.categ
  Number of categories

Details

For two exchangeable raters which provide ratings on an item, a latent class model with $K+1$ classes (if there are $K+1$ item categories $0,...,K$) is defined. Where $P(X = x, Y=y | c)$ denotes the probability that the first rating is $x$ and the second rating is $y$ given the true but unknown item category (class) $c$. Ratings are assumed to be locally independent, i.e. $$P(X=x , Y=y | c ) = P( X =x | c) \cdot P(Y=y | c ) = p_{x|c} \cdot p_{y|c}$$ Note that $P(X=x|c)=P(Y=x|c)=p_{x|c}$ holds due to the exchangeability of raters. The latent class model estimates true class proportions $\pi_c$ and conditional item probabilities $p_{x|c}$.

References

Aickin, M. (1990). Maximum likelihood estimation of agreement in the constant predictive probability model, and its relation to Cohen's kappa. Biometrics, 46, 293-302. Gwet, K. L. (2008). Computing inter-rater reliability and its variance in the presence of high agreement. British Journal of Mathematical and Statistical Psychology, 61, 29-48. Gwet, K. L. (2010). Handbook of Inter-Rater Reliability. Advanced Analytics, Gaithersburg. http://www.agreestat.com/ Uebersax, J. S., & Grove, W. M. (1990). Latent class analysis of diagnostic agreement. Statistics in Medicine, 9, 559-572.

See Also

See also rm.facets and rm.sdt for specifying rater models. See also the irr package for measures of rater agreement.

Examples

#############################################################################
# EXAMPLE 1: Latent class models for rating datasets data.si05
#############################################################################

data(data.si05)

#*** Model 1: one item with two categories
mod1 <- lc.2raters( data.si05$Ex1)
summary(mod1)

#*** Model 2: one item with five categories
mod2 <- lc.2raters( data.si05$Ex2)
summary(mod2)

#*** Model 3: one item with eight categories
mod3 <- lc.2raters( data.si05$Ex3)
summary(mod3)