Reproducibility for Interval-Censored Data

Description

Calculates intraclass correlation coefficient (ICC) for assessing reproducibility of interval-censored data with two repeated measurements (Kovacic and Varnai (2014) <doi:10.1097/EDE.0000000000000139>). ICC is estimated by maximum likelihood from model with one fixed and one random effect (both intercepts). Help in model checking (normality of subjects' means and residuals) is provided.

Details

Author(s)

Jelena Kovacic

Maintainer: Jelena Kovacic <jkovacic@imi.hr>

References

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

Examples

# Data generation (grouped data)
classes <- 1:6
class.limits <- cbind(classes-0.5,classes+0.5)
r1 <- sample(classes,100,replace=TRUE) # first measurement
r2 <- sample(classes,100,replace=TRUE) # second measurement

summary(intervalICC(r1,r2,predefined.classes=TRUE,classes,class.limits)) # ICC estimation

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

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:

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 jkovacic@imi.hr

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)

Normality Check for Interval-Censored Data with Repeated Measurements - Means

Description

The function checks whether interval-censored data with two repeated measurements meet the normality assumption for subjects' means. This is a prerequisite for the random effects model used in ICC calculation.

Usage

ntest.means(r1, r2, predefined.classes=FALSE, classes, c.limits, optim.method=1, bins=10)

Arguments

Details

For ICC estimation the random effects data model

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

is used, where b_i and e_{ij} are normally distributed with means 0 and variances \sigma^2_b and \sigma^2, respectively. This function assesses the assumption that the subjects' means 0.5 (Y_{i1}+Y_{i2}) are normally distributed with mean \mu and variance \sigma^2_b + 0.5 \sigma^2, as is expected under the specified model.

To test normality, chi-square goodness-of-fit test with bins subsequent data categories is used (call to chisq.test from package stats). The categories (bins) are determined using the equidistant quantiles of expected normal distribution, with corresponding maximum likelihood parameters. Maximum likelihood estimates for parameters \mu, \sigma^2_b and \sigma^2 are obtained by calling the function intervalICC. The probability corresponding to each bin is 1/bins (expected relative frequencies; this corresponds to p = rep(1/bins,bins) in chisq.test function). Since means are interval-censored and censoring intervals overlap, the observed relative frequencies are calculated in the following way. If one of the original intervals representing subjects mean spans multiple bins, each bin receives a share of votes from the original interval. This share is calculated using the expected normal density function and it is proportional to the probability of data falling within the intersection of the original interval and bin.

Value

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

Note

This function was designed as a help in assessing goodness of model fit. However, it has not been tested in simulations nor in any other way. It is the responsibility of the user to provide appropriate number of bins; the function checks only if bins is a positive integer. Testing normality with low number of bins is unreliable. On the other hand, if the number of bins is too large, chisq.test will complain since the expected frequencies will be too low.

Author(s)

Jelena Kovacic jkovacic@imi.hr

References

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

See Also

summary.ntestMeans, intervalICC, chisq.test

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)
ntest.means(r1,r2,predefined.classes=TRUE,classes,class.limits,bins=10)

Normality Check for Interval-Censored Data with Repeated Measurements - Residuals

Description

The function checks whether interval-censored data with two repeated measurements meet the normality assumption for subjects' residuals. This is a prerequisite for the random effects model used in ICC calculation.

Usage

ntest.res(r1, r2, predefined.classes=FALSE, classes, c.limits, optim.method=1, bins=10)

Arguments

Details

For ICC estimation the random effects data model

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

is used, where b_i and e_{ij} are normally distributed with means 0 and variances \sigma^2_b and \sigma^2, respectively. This function assesses the assumption that the subjects' "residuals" Y_{i1} - 0.5 (Y_{i1}+Y_{i2}) and Y_{i2} - 0.5 (Y_{i1}+Y_{i2}) are normally distributed with mean 0 and variance 0.5 \sigma^2, as is expected under the specified model.

To test normality, chi-square goodness-of-fit test with bins subsequent data categories is used (call to chisq.test from package stats). The categories (bins) are determined using the equidistant quantiles of expected normal distribution, with corresponding maximum likelihood parameters. Maximum likelihood estimates for parameters \mu, \sigma^2_b and \sigma^2 are obtained by calling the function intervalICC. The probability corresponding to each bin is 1/bins (expected relative frequencies; this corresponds to p = rep(1/bins,bins) in chisq.test function). Since residuals are interval-censored and censoring intervals overlap, the observed relative frequencies are calculated in the following way. If one of the original intervals representing subjects residual spans multiple bins, each bin receives a share of votes from the original interval. This share is calculated using the expected normal density function and it is proportional to the probability of data falling within the intersection of the original interval and bin.

Residuals for the first time point (Y_{i1} - 0.5 (Y_{i1}+Y_{i2})) and residuals for the second (Y_{i2} - 0.5 (Y_{i1}+Y_{i2})) are tested separately; therefore two test results in the output are given.

Value

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

Note

This function was designed as a help in assessing goodness of model fit. However, it has not been tested in simulations nor in any other way. It is the responsibility of the user to provide appropriate number of bins; the function checks only if bins is a positive integer. Testing normality with low number of bins is unreliable. On the other hand, if the number of bins is too large, chisq.test will complain since the expected frequencies will be too low.

Author(s)

Jelena Kovacic jkovacic@imi.hr

References

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

See Also

summary.ntestRes, intervalICC, chisq.test

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)
ntest.res(r1,r2,predefined.classes=TRUE,classes,class.limits,bins=10)

Summary for ICCfit Objects

Description

The function summarizes the results of ICC estimation.

Usage

## S3 method for class 'ICCfit'
summary(object, ...)

Arguments

Details

For more details about ICC estimation and output values shortly described below, please refer to the documentation for intervalICC.

Value

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

Author(s)

Jelena Kovacic jkovacic@imi.hr

References

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

See Also

intervalICC

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)
icc.est <- intervalICC(r1,r2,predefined.classes=TRUE,classes,class.limits)
summary(icc.est)

Summary for ntestMeans Objects

Description

The function summarizes the results of normality check for means.

Usage

## S3 method for class 'ntestMeans'
summary(object, ...)

Arguments

Details

For more details about normality check and output values shortly described below, please refer to the documentation for ntest.means.

Value

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

Author(s)

Jelena Kovacic jkovacic@imi.hr

References

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

See Also

ntest.means

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)
nm <- ntest.means(r1,r2,predefined.classes=TRUE,classes,class.limits,bins=10)
summary(nm)

Summary for ntestRes Objects

Description

The function summarizes the results of normality check for residuals.

Usage

## S3 method for class 'ntestRes'
summary(object, ...)

Arguments

Details

For more details about normality check and output values shortly described below, please refer to the documentation for ntest.res.

Value

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

Author(s)

Jelena Kovacic jkovacic@imi.hr

References

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

See Also

ntest.res

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)
nr <- ntest.res(r1,r2,predefined.classes=TRUE,classes,class.limits,bins=10)
summary(nr)