A Latent Class Model for Agreement of Two Raters

Estimates a latent class model for agreement of two raters (Schuster & Smith, 2006). See Details for the description of the model.

Usage

lc2_agreement(y, w=rep(1, nrow(y)), type="homo", method="BFGS", ...)

# S3 method for lc2_agreement
summary(object, digits=3,...)

# S3 method for lc2_agreement
logLik(object, ...)

# S3 method for lc2_agreement
anova(object, ...)

Arguments

y: A data frame containing the values of two raters in columns
w: Optional vector of weights
type: Type of model specification. Can be "unif", "equal", "homo" or "hete". See Details.
method: Optimization method used in stats::optim
...: Further arguments passed to stats::optim
object: Object of class l2_agreement
digits: Number of digits for rounding

Details

The latent class model for two raters decomposes a portion of ratings which conform to true agreement and another portion of ratings which conform to a random rating of a category. Let $X_r$ denote the rating of rater $r$, then for $ i \neq j$, it is assumed that $$P(X_1=i, X_2=j)=\phi_{1i} \phi_{2j} ( 1 - \gamma )$$ For $i=j$ it is assumed that $$P(X_1=i, X_2=i)=\tau_i \gamma + \phi_{1i} \phi_{2i} ( 1 - \gamma )$$ where $\gamma$ denotes the proportion of true ratings.

All $\tau_i$ and $\phi_{ri}$ parameters are estimated using type="hete". If the $\phi$ parameters are assumed as invariant across the two raters (i.e. $\phi_{1i}=\phi_{2i}=\phi_{i}$), then type="homo" must be specified. The constraint $\tau_i=\phi_i$ is imposed by type="equal". All $\phi_i$ parameters are set equal to each other using type="unif".

Value

model_output: Output of the fitted model
saturated_output: Output of the saturated model
LRT_output: Output of the likelihood ratio test of model fit
partable: Parameter table
parmsummary: Parameter summary
agree_true: True agreement index shich is the $\gamma$ parameter
agree_chance: Agreement by chance
rel_agree: Conditional reliability of agreement
optim_output: Output of optim from the fitted model
nobs: Number of observations
type: Model type
ic: Information criteria
loglike: Log-likelihood
npars: Number of parameters
y: Used dataset
w: Used weights

References

Schuster, C., & Smith, D. A. (2006). Estimating with a latent class model the reliability of nominal judgments upon which two raters agree. Educational and Psychological Measurement, 66(5), 739-747.

Examples

#############################################################################
# EXAMPLE 1: Dataset in Schuster and Smith (2006)
#############################################################################

data(data.immer08)
dat <- data.immer08

# select ratings and frequency weights
y <- dat[,1:2]
w <- dat[,3]

#*** Model 1: Uniform distribution phi parameters
mod1 <- immer::lc2_agreement( y=y, w=w, type="unif")
summary(mod1)

#*** Model 2: Equal phi and tau parameters
mod2 <- immer::lc2_agreement( y=y, w=w, type="equal")
summary(mod2)

if (FALSE) {
#*** Model 3: Homogeneous rater model
mod3 <- immer::lc2_agreement( y=y, w=w, type="homo")
summary(mod3)

#*** Model 4: Heterogeneous rater model
mod4 <- immer::lc2_agreement( y=y, w=w, type="hete")
summary(mod4)

#--- some model comparisons
anova(mod3,mod4)
IRT.compareModels(mod1,mod2,mod3,mod4)
}