Latent Class Model for Two Exchangeable Raters and One Item

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 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

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 $p_{x|c}$.

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

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.

Examples

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

data(data.si05)

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

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

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