intervalICC {iRepro}

Intraclass Correlation Coefficient for Interval-Censored Data

Description

The function calculates intraclass correlation coefficient (ICC) for interval-censored data with two repeated measurements. ICC is estimated by maximum likelihood from model with one fixed and one random effect (both intercepts).

Usage

intervalICC(r1, r2, predefined.classes=FALSE, classes, c.limits, optim.method=1)

Arguments

r1

data corresponding to the first measurement. If predefined.classes=TRUE (appropriate for grouped data), this is a vector of length n, where each observation is one of the labels given in classes. Otherwise, if predefined.classes=FALSE, r1 is a matrix or a data frame with n rows and 2 columns, with columns representing lower and upper bounds of censoring intervals (e.g., if i-th observation lies in the interval [a, b], then r1[i,]=c(a,b)).

r2

data corresponding to the second measurement. If predefined.classes=TRUE (appropriate for grouped data), this is a vector of length n, where each observation is one of the labels given in classes. Otherwise, if predefined.classes=FALSE, r2 is a matrix or a data frame with n rows and 2 columns, with columns representing lower and upper bounds of censoring intervals (e.g., if i-th observation lies in the interval [a, b], then r2[i,]=c(a,b)).

predefined.classes

logical, indicating whether observations belong to predefined classes (e.g. grouped data in questionnaires) or each observation has its own lower and upper limit (default; FALSE).

classes

a vector with unique labels for the k predefined classes. Required if predefined.classes=TRUE.

c.limits

a matrix or a data frame with k rows and 2 columns, corresponding to lower and upper bounds of censoring intervals for classes. Required if predefined.classes=TRUE.

optim.method

an integer (1 or 2) specifying the optimization method to be used in maximum likelihood estimation (default is 1). Details are given below.

Details

ICC is estimated by maximum likelihood from random effects model

Y_{ij} = \mu + b_i + e_{ij},

where b_i and e_{ij} are independent and normally distributed with means 0 and variances \sigma^2_b and \sigma^2, respectively. If data were uncensored, this would be analogous to

lme(ratings~1, random=~1|id, method="ML", data=observed)

in nlme package, where

observed=as.data.frame(rbind(cbind(r1,1:n), cbind(r2,1:n)))

and colnames(observed)=c("ratings","id"). To maximize log-likelihood, constrOptim from stats package is used (method=BFGS).

Two available optimization methods, specified by optim.method, correspond to two mathematically equivalent expressions for log-likelihood. The option optim.method=1 resulted in slightly more accurate estimates in simulations with grouped data, but optim.method=2 was more numerically stable. See the reference for more details.

Value

An object of class "ICCfit". The object is a list with the components:

icc

maximum likelihood estimate (MLE) of ICC

sigma2.b

MLE of between-class variance \sigma^2_b

sigma2.w

MLE of within-class variance \sigma^2

mu

MLE of mean \mu

loglikelihood

log-likelihood evaluated at MLE parameters

Note

If there are many observations with same values (i.e. with the same lower and upper bounds), it is advisable to group all observations into classes and use option predefined.classes=TRUE; this will reduce computation time.

Subjects with only one measurement are omitted from ICC calculation.

Author(s)

Jelena Kovacic [e-mail]

References

Kovacic J, Varnai VM. Intraclass correlation coefficient for grouped data. Epidemiology 2014;25(5):769–770.

See Also

summary.ICCfit

Examples

# Example with 6 predefined classes (grouped data)
classes <- 1:6
class.limits <- cbind(classes-0.5,classes+0.5)
r1 <- sample(classes,30,replace=TRUE)
r2 <- sample(classes,30,replace=TRUE)

intervalICC(r1,r2,predefined.classes=TRUE,classes,class.limits)

# The same result can be obtained with predefined.classes=FALSE option, 
# although with slower computation time
rtg1 <- matrix(nrow=30,ncol=2)
rtg2 <- matrix(nrow=30,ncol=2)
# when predefined.classes=FALSE, ratings must be given with lower and upper bounds 
# for each observation:
for(i in 1:length(classes)){
  rtg1[r1==classes[i],1] <- class.limits[i,1]
  rtg1[r1==classes[i],2] <- class.limits[i,2]
  rtg2[r2==classes[i],1] <- class.limits[i,1]
  rtg2[r2==classes[i],2] <- class.limits[i,2]
}

intervalICC(rtg1,rtg2,predefined.classes=FALSE)