cccrm: Concordance Correlation Coefficient for Repeated (and Non-Repeated) Measures

Description

Estimates the Concordance Correlation Coefficient to assess agreement. The scenarios considered are non-repeated measures, non-longitudinal repeated measures (replicates) and longitudinal repeated measures. It also includes the estimation of the one-way intraclass correlation coefficient also known as reliability index. The estimation approaches implemented are variance components and U-statistics approaches. Description of methods can be found in Fleiss (1986) doi:10.1002/9781118032923 and Carrasco et al. (2013) doi:10.1016/j.cmpb.2012.09.002.

Author(s)

Maintainer: Josep Lluis Carrasco jlcarrasco@ub.edu

Authors:

• Gonzalo Peon Pena

Wald's test on the Concordance Correlation Coefficient

Description

Wald's test is applied to assess whether the CCC (ICC) is greater than a reference value. Additionally, Wald's test is also used to compare two independent CCC (ICC).

Usage

Ztest(cccfit, cccfit2 = NULL, r0 = 0, info = TRUE)

Arguments

Details

If only one ccc is provided, the function runs a one sided test to the null hypothesis value

\rho_0

.

z=\frac{\hat{\theta}-\rho_0}{SE\left(\hat{\theta}\right)}

where

\hat{\theta}

stands for the CCC estimate and

SE\left(\hat{\theta}\right)

its standard error. If a second CCC is provided, the function runs a two-sided test to the null hypothesis of equality of CCCs.

z=\frac{\hat{\theta_1}-\hat{\theta_2}}{\sqrt{Var\left(\hat{\theta_1}\right)}+Var\left(\hat{\theta_2}\right)}

. In both cases, the p-value is computed as

P\left(X>z\right)

where X follows a standard Normal distribution. The test uses the transformation indicated when the ccc object was generated.

Value

A data frame with two columns: Z, the statistical test value; and the P-value associated.

Examples

# Testing the CCC is above 0.8
ccc_mc=ccc_vc(bpres,"DIA","ID","METODE")
ccc_mc
Ztest(ccc_mc,r0=0.8)

# Comparing two CCC

bpres_Male <- bpres |> dplyr::filter(SEXO==1)
bpres_Female <- bpres |> dplyr::filter(SEXO==2)

ccc_DIA_Male=ccc_vc(bpres_Male,"DIA","ID","METODE")
ccc_DIA_Female=ccc_vc(bpres_Female,"DIA","ID","METODE")
Ztest(ccc_DIA_Male,ccc_DIA_Female)

Blood draw data

Description

Plasma cortisol area under curve (AUC) was calculated from the trapezoidal rule over the 12-h period of the hourly blood draws. The subjects were required to repeat the process in five visits. The aim of the agreement study was to assess how well the plasma cortisol AUC from hourly measurements agreed with plasma cortisol AUC that was measured every two hours.

Usage

bdaw

Format

A data frame with the following columns:

SUBJ

Subject identifier

VNUM

Visit number

AUC

Area under the curve

MET

Device identifier

Body fat data

Description

Percentage body fat was estimated from skinfold calipers and DEXA on a cohort of 90 adolescent girls. Skinfold caliper and DEXA measurements were taken at ages 12.5, 13 and 13.5. The objective was to determine the amount of agreement between the skinfold caliper and DEXA measurements of percentage body fat.

Usage

bfat

Format

A data frame with the following columns:

SUBJECT

Subject identifier

VISITNO

Visit number

BF

Percentage body fat

MET

Device identifier

Blood pressure data

Description

Systolic and diastolic blood pressure was measured in a sample of 384 subjects using a handle mercury sphygmomanometer device and an automatic device. The blood pressure was simultaneously measured twice by each instrument, thus every subject had four measurements, two by each method.

Usage

bpres

Format

A data frame with the following columns:

ID

Subject identifier

SIS

Systolic blood pressure in mmHg

DIA

Diastolic blood pressure in mmHg

METODE

Device identifier

NM

Identifier of replicates

ALTURA

Height in cm

EDAD

Age in years

FRECUENC

Heart rate

INFOR_AR

Have the subject been informed about he is hypertense?

PESO

Weight in Kg

SEXO

Gender. 1 for Male. 2 for Female

TA

Was the subject's blood pressure measured the last year? 1=Yes, 2=No, 9=Unknown

TNSI_MED

Does the subject receive treatment for hypertension? 1=Yes, 2=No, 3=Doubtful, 8=not applicable, 9=Insufficient data.

Repeated Measures Concordance Correlation Coefficient estimated by U-statistics

Description

Estimation of the concordance correlation coefficient for repeated measurements using the U-statistics approach. The function is also applicable for the non-repeated measurements scenario.

Usage

cccUst(dataset, ry, rmet, rtime = NULL, Dmat = NULL, delta = 1, cl = 0.95)

Arguments

Value

A vector that includes the point estimate, confidence interval and standard error of the CCC. Additionally the Fisher's Z-transformation value and its standard error are also provided.

References

King, TS and Chinchilli, VM. (2001). A generalized concordance correlation coefficient for continuous and categorical data. Statistics in Medicine, 20, 2131:2147.

King, TS; Chinchilli, VM; Carrasco, JL. (2007). A repeated measures concordance correlation coefficient. Statistics in Medicine, 26, 3095:3113.

Carrasco, JL; Phillips, BR; Puig-Martinez, J; King, TS; Chinchilli, VM. (2013). Estimation of the concordance correlation coefficient for repeated measures using SAS and R. Computer Methods and Programs in Biomedicine, 109, 293-304.

Examples

# Non-longitudinal scenario
newdat=bpres[bpres$NM==1,]
estccc=cccUst(newdat,"DIA","METODE")
estccc

estccc=cccUst(bdaw,"AUC","MET","VNUM")
estccc

estccc=cccUst(bfat,"BF","MET","VISITNO",Dmat=diag(c(2,1,1)))
estccc

Concordance correlation Coefficient estimation from a linear mixed model

Description

It computes the Concordance Correlation Coefficient and its asymptotic confidence interval.

Usage

ccc_est(model, D = NULL, cl = 0.95, transf = "F2", sd_est = TRUE, ...)

Arguments

Value

A ccc class object.

See Also

ccc_vc

Examples

set.seed(1984)
df <- ccc_sim_data(n=50,b = c(0,1), mu = -0.25, sa = 1.5, se = 1, nrep=2)
mod <- lme_model(df,"y","id",rmet="met")
ccc_est(mod)

Concordance Correlation Coefficient estimation by variance components.

Description

Estimation of the non-longitudinal concordance correlation coefficient at each time using the variance components approach.

Usage

ccc_est_by_time(
  dataset,
  ry,
  rind,
  rmet,
  rtime,
  covar = NULL,
  int = F,
  cl = 0.95,
  control.lme = list(),
  future_seed = TRUE,
  transf = "F2",
  workers = 15,
  plotit = TRUE,
  test = FALSE,
  nboot = 500,
  adj.method = "holm",
  ...
)

Arguments

Details

The concordance correlation coefficient is estimated using the variance components approach. Confidence intervals are built using the asymptotic Normal distribution approach. Variance-covariance matrix of CCC estimates is estimated by non-parametric balanced randomized cluster bootstrap approach (Davison and Hinkley, 1997; Field and Welsh, 2007). Overall equality of CCCs is tested following the non-parametric bootstrap approach suggested in Vanbelle (2017).

Value

A ccc class object. Generic function summary show a summary of the results. The output is a list with the following components:

• ccc. CCC estimates at each level of time variable.

• plot. Plot of the CCC along with their confidence intervals.

• res_test. Test of equality of CCCs.

• ph_table. Pairwise comparison of CCCs.

References

Davison A.C., Hinkley D.V. (1997). Bootstrap Methods and Their Application. Cambridge: Cambridge University Press.

Field, C.A., Welsh, A.H. (2007). Bootstrapping Clustered Data. Journal of the Royal Statistical Society. Series B (Statistical Methodology) 69(3):369-390.

Vanbelle S. (2017). Comparing dependent kappa coefficients obtained on multilevel data. Biometrical Journal 59(5):1016-1034.

Examples

## Not run: 
ccc_est_by_time(bdaw, "AUC", "SUBJ", "MET", "VNUM")
ccc_est_by_time(bpres, "SIS", "ID", "METODE", "NM",test=TRUE)

## End(Not run)

Data simulation using fixed and random effects

Description

The fixed effects and standard deviations of random effects can be set to specific values or, alternatively, obtained from an object of class lme.

Usage

ccc_sim_data(
  n = 30,
  nrep = 1,
  nsim = 1,
  model = NULL,
  b = NULL,
  g = NULL,
  mu = 0,
  sa = 1,
  sab = 0,
  sag = 0,
  bg = NULL,
  se = 1,
  future_seed = TRUE,
  workers = 15,
  extra.info = TRUE,
  ...
)

Arguments

Details

Random effects are simulated as normal distributions with mean 0 and the correspondign standard deviations. The simulated data is obtained as the addition of the simulated values and the fixed efffects. Parallel computation is used except if data is simulated from an object of class 'lme'. In this case. data is simulated using the simulateY function from nlmeU package.

Value

A data frame with the simulated data.

See Also

ccc_vc

Examples

# # Reliability data: 
# 50 subjects, one method, one time, 2 replicates
# Overall mean: -0.25; Subjects standard deviation: 1.5, Random error standard deviation: 1
set.seed(101)
df <- ccc_sim_data(n=50, b = NULL, g = NULL, mu = -0.25, sa = 1.5, se = 1, nrep=2)

# Method comparison data (non-longitudinal)
# 50 subjects, two methods, 2 replicates
# Overall mean: -0.25; Subjects standard deviation: 1.5, Random error standard deviation: 1
# Difference of means between methods 2 and 1: 1
# Three data sets simulated

set.seed(202)
df <- ccc_sim_data(n=50, nsim=3,b = c(0,1), mu = -0.25, sa = 1.5, se = 1, nrep=2)

# Method comparison data (longitudinal)
# 50 subjects, two methods, 3 times, 1 replicate, 
# Overall mean: -0.25; Subjects standard deviation: 1.5, Random error standard deviation: 1
# Difference of means between methods 2 and 1: 1
# Difference of means between times 3,2 and 1 respectively: 0.5 and 0.25.
# Subject-methods interaction standard deviation: 0.25
# Subject-times interaction standard deviation: 0.5
# Same difference of means at each time

set.seed(202)
df <- ccc_sim_data(n=50, b = c(0,1), g=c(0,0.25,0.5), mu = -0.25, sa = 1.5, 
sab=0.25,sag=0.5,se = 1, nrep=2)

# Simulate data using the estimates of a linear mixed model
set.seed(2024)
df3 <- ccc_sim_data(n=50, b = c(0,1), g=c(0,0.25,0.5), mu = -0.25, sa = 1.5, 
                    sab=0.25,sag=0.5,bg=c(0,0.5,0.75,0,1,1),se = 1, nrep=2)
mod3 <- lme_model(df3,"y","id","times","met",control.lme=nlme::lmeControl(opt = 'optim'))
ccc_sim_data(nsim=10,model=mod3)

Concordance Correlation Coefficient estimation by variance components.

Description

Estimation of the concordance correlation coefficient for either non-repeated, non-longitudinal, or longitudinal repeated measurements using the variance components from a linear mixed model. The appropriate intraclass correlation coefficient is used as estimator of the concordance correlation coefficient.

Usage

ccc_vc(
  dataset,
  ry,
  rind,
  rmet = NULL,
  rtime = NULL,
  vecD = NULL,
  covar = NULL,
  int = F,
  rho = 0,
  cl = 0.95,
  control.lme = list(),
  transf = "F2",
  boot = FALSE,
  boot_param = FALSE,
  boot_ci = "BCa",
  nboot = 300,
  parallel = FALSE,
  future_seed = TRUE,
  workers = 15,
  sd_est = TRUE,
  apVar = TRUE,
  ...
)

Arguments

Details

The concordance correlation coefficient is estimated using the appropriate intraclass correlation coefficient (see Carrasco and Jover, 2003; Carrasco et al., 2009; Carrasco et al, 2013).

The scenarios considered are: a) reliability assessment (several measurements taken with one method); b) methods comparison data with non-repeated measurements (only one measurement by subject and method); c) Methods comparison data with non-longitudinal repeated measurements, i.e. replicates (multiple measurements by subject and method); and d) Methods comparison data with longitudinal repeated measurements (multiple longitudinal measurements by subject and method).

The variance components estimates are obtained from a linear mixed model (LMM) estimated by restricted maximum likelihood. The function lme from package nlme (Pinheiro et al., 2021) is used to estimate the LMM.

The standard error of CCC and its confidence interval can be obtained: a) asymptotically, using Taylor's series expansion of 1st order (Ver Hoef, 2012); b) using balanced randomized cluster bootstrap approach (Davison and Hinkley, 1997; Field and Welsh, 2007); c) using parametric bootstrap (Davison and Hinkley, 1997).

When estimating asymptotically the standard error, the confidence intervals are built using the point estimate of the CCC/ICC, its standard error, and the appropriate quantile of the standard Normal distribution. However, the approximation to the asymptotic Normal distribution is improved if the CCC/ICC is transformed using the Fisher's Z-transformation (Fisher, 1925), or the Konishi-Gupta transformation (Konishi and Gupta, 1989). In case the number of replicates is equal to 2, both transformations give the same result.

Value

A ccc class object. Generic function summary show a summary of the results. The output is a list with the following components:

• ccc. CCC/ICC estimate

• model. nlme object with the fitted linear mixed model.

• vc. Variance components estimates.

• sigma. Variance components asymptotic covariance matrix.

References

Carrasco, JL; Jover, L. (2003). Estimating the generalized concordance correlation coefficient through variance components. Biometrics, 59, 849:858.

Carrasco, JL; King, TS; Chinchilli, VM. (2009). The concordance correlation coefficient for repeated measures estimated by variance components. Journal of Biopharmaceutical Statistics, 19, 90:105.

Davison A.C., Hinkley D.V. (1997). Bootstrap Methods and Their Application. Cambridge: Cambridge University Press.

Field, C.A., Welsh, A.H. (2007). Bootstrapping Clustered Data. Journal of the Royal Statistical Society. Series B (Statistical Methodology). 69(3), 369-390.

Fisher, R. A. (1925) Statistical Methods for Research Workers. Edinburgh: Oliver

Konishi, S. and Gupta, A. K. (1989) Testing the equality of several intraclass correlation coefficients. J Statist. Planng Inf., 21, 93-105.

Pinheiro J, Bates D, DebRoy S, Sarkar D, R Core Team (2021). nlme: Linear and Nonlinear Mixed Effects Models. R package version 3.1-152, https://CRAN.R-project.org/package=nlme.

Ver Hoef, J.M. (2012) Who Invented the Delta Method?, The American Statistician, 66:2, 124-127.

Examples

## Not run: 
# Scenario 1. Reliability 
newdat <- bpres |> dplyr::filter(METODE==1)
icc_rel<-ccc_vc(newdat,"DIA","ID")
icc_rel
summary(icc_rel)


# Confidence interval using non-parametric bootstrap

icc_rel_bt<-ccc_vc(newdat,"DIA","ID",boot=TRUE,sd_est=FALSE,
nboot=500,parallel=TRUE)
icc_rel_bt
summary(icc_rel_bt)


#' # Scenario 2. Non-longitudinal methods comparison.
# Only 1 measure by subject and method. 
# No subjects-method interaction included in the model.

newdat <- bpres |> dplyr::filter(NM==1)
ccc_mc<-ccc_vc(newdat,"DIA","ID","METODE")
ccc_mc
summary(ccc_mc)


# Confidence interval using parametric bootstrap

ccc_mc_bt<-ccc_vc(newdat,"DIA","ID",boot=TRUE,boot_param=TRUE,
sd_est=FALSE,nboot=500,parallel=TRUE)
ccc_mc_bt
summary(ccc_mc_bt)


# Scenario 3. Non-longitudinal methods comparison. 
# Two measures by subject and method. 
# No subject-method interaction included in the model.

ccc_mc=ccc_vc(bpres,"DIA","ID","METODE")
ccc_mc
summary(ccc_mc)



# Scenario 4. Methods comparison in longitudinal repeated measures setting.
ccc_mc_lon<-ccc_vc(bdaw,"AUC","SUBJ","MET","VNUM")
ccc_mc_lon
summary(ccc_mc_lon)

# Scenario 5. Methods comparison in longitudinal repeated measures setting.
# More weight given to readings from first time.

ccc_mc_lonw<-ccc_vc(bfat,"BF","SUBJECT","MET","VISITNO",vecD=c(2,1,1))
ccc_mc_lonw
summary(ccc_mc_lonw)

## End(Not run)

Concordance Correlation Coefficient estimated by Variance Components

Description

Concordance Correlation Coefficient for longitudinal repeated measures estimated by variance components

Usage

ccclon(
  dataset,
  ry,
  rind,
  rtime,
  rmet,
  covar = NULL,
  rho = 0,
  cl = 0.95,
  control.lme = list()
)

Arguments

Details

This function has been deprecated. See ccc_vc.

Concordance Correlation Coefficient estimated by Variance Components

Description

Concordance Correlation Coefficient for longitudinal repeated measures estimated by variance components

Usage

ccclonw(
  dataset,
  ry,
  rind,
  rtime,
  rmet,
  vecD,
  covar = NULL,
  rho = 0,
  cl = 0.95,
  control.lme = list()
)

Arguments

Details

This function has been deprecated. See ccc_vc.

Concordance Correlation Coefficient estimated by Variance Components

Description

Estimation of the concordance correlation coefficient for non-repeated measurements and non-longitudinal repeated measurements (replicates) using the variance components from a linear mixed model. The appropriate intraclass correlation coefficient is used as estimator of the concordance correlation coefficient.

Usage

cccvc(
  dataset,
  ry,
  rind,
  rmet,
  covar = NULL,
  int = FALSE,
  cl = 0.95,
  control.lme = list()
)

Arguments

Details

This function has been deprecated. See ccc_vc.

Fits a Linear Mixed Effects Model

Description

Fits a Linear Mixed Effects Model

Usage

lme_model(
  dataset,
  ry,
  rind,
  rtime = NULL,
  rmet = NULL,
  vecD = NULL,
  covar = NULL,
  rho = 0,
  int = FALSE,
  cl = 0.95,
  control.lme = list(),
  apVar = TRUE,
  ...
)

Arguments

Value

an object of class lme.

Examples

# Reliability ICC
set.seed(2024)
df <- ccc_sim_data(b = NULL, g = NULL, mu = -0.25, sa = 1.5, se = 1)
mod1 <- lme_model(df,"y","id")
mod1

#Non-longitudinal Methods comparison data
set.seed(2024)
df2 <- ccc_sim_data(n=50,b = c(0,1), mu = -0.25, sa = 1.5, se = 1, nrep=2)
mod2 <- lme_model(df2,"y","id",rmet="met")
mod2

# Longitudinal Methods comparison data
set.seed(2024)
df3 <- ccc_sim_data(n=50, b = c(0,1), g=c(0,0.25,0.5), mu = -0.25, sa = 1.5, 
                    sab=0.25,sag=0.5,bg=c(0,0.5,0.75,0,1,1),se = 1, nrep=2)

mod3 <- lme_model(df3,"y","id","times","met",control.lme=nlme::lmeControl(opt = 'optim'))
mod3

Non-parametric cluster bootstrap to make inference on the concordance correlation coefficient by time variable

Description

Non-parametric cluster bootstrap to make inference on the concordance correlation coefficient by time variable

Usage

np_boot_ccc_by_time(
  dataset,
  ry,
  rind,
  rmet,
  rtime,
  covar = NULL,
  int = FALSE,
  cl = 0.95,
  control.lme = list(),
  nboot = 500,
  future_seed = 123
)

Arguments

Value

Vector of CCC bootstrap estimates.

Non-parametric cluster bootstrap to make inference on the concordance correlation coefficient

Description

Non-parametric cluster bootstrap to make inference on the concordance correlation coefficient

Usage

np_boot_ci(
  dataset,
  ry,
  rind,
  rtime = NULL,
  rmet = NULL,
  vecD = NULL,
  covar = NULL,
  rho = 0,
  int = FALSE,
  cl = 0.95,
  control.lme = list(),
  nboot = 500,
  parallel = TRUE,
  future_seed = 123
)

Arguments

Value

Vector of CCC bootstrap estimates.

Parametric bootstrap to make inference on the concordance correlation coefficient

Description

Parametric bootstrap to make inference on the concordance correlation coefficient

Usage

para_boot_ci(
  dataset,
  model,
  ry,
  rind,
  rtime = NULL,
  rmet = NULL,
  vecD = NULL,
  covar = NULL,
  rho = 0,
  int = F,
  cl = 0.95,
  control.lme = list(),
  nboot = 500,
  parallel = TRUE,
  future_seed = 121
)

Arguments

Value

Vector of CCC bootstrap estimates.

Power and confidence interval range

Description

Power and confidence interval range obtained by simulation

Usage

sim_power_ccc(
  n = 30,
  nrep = 2,
  nsim = 300,
  r0 = 0,
  alpha = 0.05,
  model = NULL,
  b = NULL,
  g = NULL,
  mu = 0,
  sa = 1,
  sab = 0,
  sag = 0,
  bg = NULL,
  se = 1,
  extra.info = TRUE,
  vecD = NULL,
  covar = NULL,
  int = FALSE,
  rho = 0,
  cl = 0.95,
  control.lme = list(),
  transf = "F2",
  future_seed = TRUE,
  workers = 15
)

Arguments

Details

The power and the range of the confidence interval are computed using the approach suggested in Choudhary and Nagaraja (2018). Data sets are simulated by setting the fixed effects values and the standard deviation of the random effects. The CCC and its standard error are estimated in each data set, along with its 95% confidence interval and the Wald test Ztest.

Value

A data frame with the following components:

• n Number of subjects

• reps Number of replicates

• CCC. Median of the CCC estimates.

• Power. Empirical power computed as proportion of times the null hypothesis is rejected using a type-I error rate of alpha.

• SEICC. Average of CCC standard errors.

• SEZ. Average of transformed CCC standard errors.

• Range IC95. Average of CCC confidence interval widths.

References

Choudhary, P.K. and Nagaraja, H.N. (2018). Measuring Agreement-Models, Methods, and Applications. John Wiley & Sons

Examples

# Power to test the CCC is above 0.8 with 35 subjects and 4 replicates.
# Two methods, three times. Simulated CCC=0.87.
sim_pw<-sim_power_ccc(n = 35, nrep=4, nsim=500, r0=0.8, b = c(-0.5,0.5), 
g=c(-0.25,0,0.25), mu = -0.25, sa = 4,sab=0.5,sag=1,
bg=c(-0.5,-0.25,0.25,-0.5,0.25,0.75),se = 1)