Type: Package
Title: Estimate Correlations Between Repeatedly Measured Endpoints (E.g., Reliability) Based on Linear Mixed-Effects Models
Version: 1.1
Author: Wim Van der Elst, Geert Molenberghs, Dieter Hilgers, & Nicole Heussen
Maintainer: Wim Van der Elst <Wim.vanderelst@gmail.com>
Description: In clinical practice and research settings in medicine and the behavioral sciences, it is often of interest to quantify the correlation of a continuous endpoint that was repeatedly measured (e.g., test-retest correlations, ICC, etc.). This package allows for estimating these correlations based on mixed-effects models. Part of this software has been developed using funding provided from the European Union's 7th Framework Programme for research, technological development and demonstration under Grant Agreement no 602552.
Depends: nlme
Imports: psych
License: GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
Repository: CRAN
NeedsCompilation: no
Packaged: 2022-04-18 14:21:10 UTC; wim
Date/Publication: 2022-04-18 15:24:38 UTC

An example dataset

Description

Example.Data is a hypothetical dataset constructed to demonstrate some of the functions in the package. Data are provided for a hypothetical experiment in which a stimulus is provided under different experimental conditions. The outcome is a normally distributed variable. The entire experiment is repeated multiple times (cycle) in each patient.

Usage

data(Example.Data)

Format

A data.frame with 360 observations on 5 variables.

Id

The Subject identifier.

Cycle

The same experiment is repeated multiple times in a patient. Cycle indicates the order of these repeated experiments.

Condition

The experimental condition under which the outcome was measured.

Time

The time point at which the outcome was measured.

Outcome

A continuous outcome.

Explore within-subject correlations (reliabilities)

Description

This function allows for exploring the within-subject (test-retest) correlation (R) structure in the data, taking relevant covariates into account. Estimated correlations as a function of time lag (= absolute difference between measurement moments t_1 and t_2) are provided as well as their confidence intervals (based on a non-parametric bootstrap).

Usage

Explore.WS.Corr(OLS.Model=" ", Dataset, Id, Time, 
Alpha=0.05, Smoother.Span=.2, Number.Bootstrap=100, 
Seed=1)

Arguments

OLS.Model

OLS.Model is a formula passed to lm (to obtain the OLS residuals, i.e., to take covariates into account in the computation of R). OLS.Model should thus be a formula that specifies the outcome of interest followed by a ~ sign and the covariates to be taken into account, e.g.

OLS.Model="Outcome~1+as.factor(Time) + as.factor(Treatment)".

Dataset

A data.frame that should consist of multiple lines per subject ('long' format).

Id

The subject indicator.

Time

The time indicator. Should be coded as 1, 2, etc.

Alpha

The \alpha-level to be used in the non-parametric bootstrap-based Confidence Interval for R. Default Alpha=0.05

Smoother.Span

A smoothing (loess) technique is used to estimate R as a function of time lag. The smoother span gives the proportion of points in the plot which influence the smooth at each value. Larger values give more smoothness. For details, see https://stat.ethz.ch/R-manual/R-patched/library/stats/html/lowess.html. Defauls Smoother.Span=.2.

Number.Bootstrap

The number of non-parametric bootstrap samples to be used to estimate the Confidence Interval for R. Default Number.Bootstrap=100

Seed

The seed to be used in the bootstrap. Default Seed=1.

Value

Est.Corr

The estimated correlations R as a function of time lag. A smoothing (loess) technique is used to estimate R as a function of time lag (based on the output in All.Corrs).

All.Corrs

A matrix that contains the estimated correlations R for all individual time lags.

Bootstrapped.Corrs

A matrix that contains the estimated correlations R as a function of time lag in the bootstrapped samples.

Alpha

The \alpha level used in the estimation of the confidence interval.

CI.Upper

The upper bounds of the confidence intervals.

CI.Lower

The lower bounds of the confidence intervals.

Author(s)

Wim Van der Elst, Geert Molenberghs, Ralf-Dieter Hilgers, & Nicole Heussen

References

Van der Elst, W., Molenberghs, G., Hilgers, R., & Heussen, N. (2015). Estimating the reliability of repeatedly measured endpoints based on linear mixed-effects models. A tutorial. Submitted.

See Also

plot.Explore.WS.Corr

Examples

# Open data
data(Example.Data)

# Explore correlation structure
Expl_Corr <- Explore.WS.Corr(OLS.Model="Outcome~as.factor(Time)+ 
  as.factor(Cycle) + as.factor(Condition)", Dataset=Example.Data, 
  Id="Id", Time="Time", Alpha=.05, Number.Bootstrap=50, Seed=123)

# explore results
summary(Expl_Corr)

# plot with correlations for all time lags, and 
# add smoothed (loess) correlation function 
plot(Expl_Corr, Indiv.Corrs=TRUE)
# plot bootstrapped smoothed (loess) correlation function 
plot(Expl_Corr)

Fit fractional polynomials

Description

Fits regression models with m terms of the form X^{p}, where the exponents p are selected from a small predefined set S of both integer and non-integer values.

Usage

Fract.Poly(Covariate, Outcome, S=c(-2,-1,-0.5,0,0.5,1,2,3), Max.M=5, Dataset)

Arguments

Covariate

The covariate to be considered in the models.

Outcome

The outcome to be considered in the models.

S

The set S from which each power p^{m} is selected. Default S={-2, -1, -0.5, 0, 0.5, 1, 2, 3}.

Max.M

The maximum order M to be considered for the fractional polynomial. This value can be 5 at most. When M=5, then fractional polynomials of order 1 to 5 are considered. Default Max.M=5.

Dataset

A data.frame that should consist of multiple lines per subject ('long' format).

Value

Results.M1

The results (powers and AIC values) of the fractional polynomials of order 1.

Results.M2

The results (powers and AIC values) of the fractional polynomials of order 2.

Results.M3

The results (powers and AIC values) of the fractional polynomials of order 3.

Results.M4

The results (powers and AIC values) of the fractional polynomials of order 4.

Results.M5

The results (powers and AIC values) of the fractional polynomials of order 5.

Author(s)

Wim Van der Elst, Geert Molenberghs, Ralf-Dieter Hilgers, & Nicole Heussen

References

Van der Elst, W., Molenberghs, G., Hilgers, R., & Heussen, N. (2015). Estimating the reliability of repeatedly measured endpoints based on linear mixed-effects models. A tutorial. Submitted.

Examples

# Open data
data(Example.Data)

# Fit fractional polynomials, mox. order = 3
FP <- Fract.Poly(Covariate = Time, Outcome = Outcome, 
Dataset = Example.Data, Max.M=3)

# Explore results
summary(FP)
# best fitting model (based on AIC) for m=3, 
# powers:  p_{1}=3,  p_{2}=3, and  p_{3}=2

# Fit model and compare with observed means

    # plot of mean 
Spaghetti.Plot(Dataset = Example.Data, Outcome = Outcome,
Time = Time, Id=Id, Add.Profiles = FALSE, Lwd.Me=1, 
ylab="Mean Outcome")

    # Coding of predictors (note that when p_{1}=p_{2},
    # beta_{1}*X ** {p_{1}} + beta_{2}*X ** {p_{1}} * log(X)
    #  and when p=0, X ** {0}= log(X)    )
term1 <- Example.Data$Time**3
term2 <- (Example.Data$Time**3) * log(Example.Data$Time) 
term3 <- Example.Data$Time**2 

    # fit model
Model <- lm(Outcome~term1+term2+term3, data=Example.Data)
Model$coef # regression weights (beta's)  

    # make prediction for time 1 to 47
term1 <- (1:47)**3 
term2 <- ((1:47)**3) * log(1:47)   
term3 <- (1:47)**2 
    # compute predicted values
pred <- Model$coef[1] + (Model$coef[2] * term1) + 
  (Model$coef[3] * term2) + (Model$coef[4] * term3)
   # Add predicted values to plot
lines(x = 1:47, y=pred,  lty=2)
legend("topright", c("Observed", "Predicted"), lty=c(1, 2))

Plot a heatmap of the correlation structure

Description

This function plots a heatmap of the correlation structure (reliability) in the data. It is a wrapper function for the cor.plot function of the psych package.

Usage

Heatmap(Dataset, Id, Outcome, Time, ...)

Arguments

Dataset

A data.frame that should consist of multiple lines per subject ('long' format).

Id

The subject indicator.

Outcome

The outcome indicator.

Time

The time indicator.

...

Other arguments to be passed to cor.plot.

Author(s)

Wim Van der Elst, Geert Molenberghs, Ralf-Dieter Hilgers, & Nicole Heussen

References

Van der Elst, W., Molenberghs, G., Hilgers, R., & Heussen, N. (2015). Estimating the reliability of repeatedly measured endpoints based on linear mixed-effects models. A tutorial. Submitted.

See Also

plot.Explore.WS.Corr

Examples

# Open data
data(Example.Data)

# Make heatmap
Heatmap(Dataset=Example.Data, Id = "Id", 
Outcome="Outcome", Time = "Time")

# Make heatmap in black and white
Heatmap(Dataset=Example.Data, Id = "Id", 
Outcome="Outcome", Time = "Time", colors=FALSE)

Compare the fit of linear mixed-effects models

Description

This function compares the fit of Model 1 (random intercept) and 2 (random intercept and Gausssian serial correlation), and of Model 2 (random intercept and Gausssian serial correlation) and 3 (random intercept, slope and Gausssian serial correlation)

Usage

Model.Fit(Model.1, Model.2)

Arguments

Model.1

An object of class WS.Corr.Mixed, the first fitted model.

Model.2

Another object of class WS.Corr.Mixed, the second fitted model.

Author(s)

Wim Van der Elst, Geert Molenberghs, Ralf-Dieter Hilgers, & Nicole Heussen

References

Van der Elst, W., Molenberghs, G., Hilgers, R., & Heussen, N. (2015). Estimating the reliability of repeatedly measured endpoints based on linear mixed-effects models. A tutorial. Submitted.

See Also

WS.Corr.Mixed

Examples

data(Example.Data)

# Code predictors for time
Example.Data$Time2 <- Example.Data$Time**2
Example.Data$Time3 <- Example.Data$Time**3
Example.Data$Time3_log <- (Example.Data$Time**3) * (log(Example.Data$Time))

# model 1
Model1 <- WS.Corr.Mixed(
Fixed.Part=Outcome ~ Time2 + Time3 + Time3_log + as.factor(Cycle) 
+ as.factor(Condition), Random.Part = ~ 1|Id, 
Dataset=Example.Data, Model=1, Id="Id", 
Number.Bootstrap = 0, Seed = 12345)

# model 2
Model2 <- WS.Corr.Mixed(
Fixed.Part=Outcome ~ Time2 + Time3 + Time3_log + as.factor(Cycle) 
+ as.factor(Condition), Random.Part = ~ 1|Id, 
Correlation=corGaus(form= ~ Time, nugget = TRUE),
Dataset=Example.Data, Model=2, Id="Id", 
Number.Bootstrap = 0, Seed = 12345)

# model 3
Model3 <- WS.Corr.Mixed(
  Fixed.Part=Outcome ~ Time2 + Time3 + Time3_log + as.factor(Cycle) 
  + as.factor(Condition), Random.Part = ~ 1 + Time|Id, 
  Correlation=corGaus(form= ~ Time, nugget = TRUE),
  Dataset=Example.Data, Model=3, Id="Id", 
  Number.Bootstrap = 0, Seed = 12345)

# compare models 1 and 2
Model.Fit(Model.1=Model1, Model.2=Model2)

# compare models 2 and 3
Model.Fit(Model.1=Model2, Model.2=Model3)

Make a Spaghetti plot

Description

Makes a spaghetti plot, i.e., a plot that depicts the outcome as a function of time for each individual subject.

Usage

Spaghetti.Plot(Dataset, Outcome, Time, Id, Add.Profiles=TRUE, Add.Mean=TRUE, 
Add.Median=FALSE, Col=8, Lwd.Me=3, xlim, ylim, ...)

Arguments

Dataset

A data.frame that should consist of multiple lines per subject ('long' format).

Outcome

The name of the outcome variable.

Time

The name of the time indicator.

Id

The subject indicator.

Add.Profiles

Logical. Should the individual profiles be added? Default Add.Profiles=TRUE.

Add.Mean

Logical. Should a line that depicts the mean as a function of time be added? Default Add.Mean=TRUE.

Add.Median

Logical. Should a line that depicts the medean as a function of time be added? Default Add.Mean=FALSE.

Col

The color of the individual profiles. Default Col=8 (grey).

Lwd.Me

The line width of the lines with mean and/or median. Default Lwd.Me=3.

xlim

The (min, max) values for the x-axis.

ylim

The (min, max) values for the y-axis.

...

Other arguments to be passed to the plot() function.

Author(s)

Wim Van der Elst, Geert Molenberghs, Ralf-Dieter Hilgers, & Nicole Heussen

References

Van der Elst, W., Molenberghs, G., Hilgers, R., & Heussen, N. (2015). Estimating the reliability of repeatedly measured endpoints based on linear mixed-effects models. A tutorial. Submitted.

Examples

# Open data
data(Example.Data)

# Plot individual profiles + mean
Spaghetti.Plot(Dataset = Example.Data, Outcome = Outcome, Id=Id, Time = Time)

# Plot individual profiles + median
Spaghetti.Plot(Dataset = Example.Data, Outcome = Outcome, Id=Id, Time = Time,
Add.Mean = FALSE, Add.Median = TRUE)

Estimate within-subject correlations (reliabilities) based on a mixed-effects model.

Description

This function allows for the estimation of the within-subject correlations using a general and flexible modeling approach that allows at the same time to capture hierarchies in the data, the presence of covariates, and the derivation of correlation estimates. Non-parametric bootstrap-based confidence intervals can be requested.

Usage

WS.Corr.Mixed(Dataset, Fixed.Part=" ", Random.Part=" ", 
Correlation=" ", Id, Time=Time, Model=1, 
Number.Bootstrap=100, Alpha=.05, Seed=1)

Arguments

Dataset

A data.frame that should consist of multiple lines per subject ('long' format).

Fixed.Part

The outcome and fixed-effect part of the mixed-effects model to be fitted. The model should be specified in agreement with the lme function requirements of the nlme package. See examples below.

Random.Part

The random-effect part of the mixed-effects model to be fitted (specified in line with the lme function requirements). See examples below.

Correlation

An optional object describing the within-group correlation structure (specified in line with the lme function requirements). See examples below.

Id

The subject indicator.

Time

The time indicator. Default Time=Time.

Model

The type of model that should be fitted. Model=1: random intercept model, Model=2: random intercept and Gaussian serial correlation, Model=3: random intercept, slope, and Gaussian serial correlation, and Model=4: random intercept + slope. Default Model=1.

Number.Bootstrap

The number of bootstrap samples to be used to estimate the Confidence Intervals around R. Default Number.Bootstrap=100. As an alternative to obtain confidence intervals, the Delta method can be used (see WS.Corr.Mixed.SAS).

Alpha

The \alpha-level to be used in the bootstrap-based Confidence Interval for R. Default Alpha=0.05

Seed

The seed to be used in the bootstrap. Default Seed=1.

Details

Warning 1

To avoid problems with the lme function, do not specify powers directly in the function call. For example, rather than specifying Fixed.Part=ZSV ~ Time + Time**2 in the function call, first add Time**2 to the dataset (Dataset$TimeSq <- Dataset$Time ** 2) and then use the new variable name in the call: Fixed.Part=ZSV ~ Time + TimeSq

Warning 2 To avoid problems with the lme function, specify the Random.Part and Correlation arguments like e.g., Random.Part = ~ 1| Subject and Correlation=corGaus(form= ~ Time, nugget = TRUE)

not like e.g., Random.Part = ~ 1| Subject and Correlation=corGaus(form= ~ Time| Subject, nugget = TRUE)

(i.e., do not use Time| Subject)

Value

Model

The type of model that was fitted (model 1, 2, or 3.)

D

The D matrix of the fitted model.

Tau2

The \tau^2 component of the fitted model. This component is only obtained when serial correlation is requested (Model 2 or 3), \varepsilon_{2} \sim N(0, \tau^2 H_{i})).

Rho

The \rho component of the fitted model which determines the matrix H_{i}, \rho(|t_{ij}-t_{ik}|). This component is only obtained when serial correlation is considered (Model 2 or 3).

Sigma2

The residual variance.

AIC

The AIC value of the fitted model.

LogLik

The log likelihood value of the fitted model.

R

The estimated reliabilities.

CI.Upper

The upper bounds of the bootstrapped confidence intervals.

CI.Lower

The lower bounds of the bootstrapped confidence intervals.

Alpha

The \alpha level used in the estimation of the confidence interval.

Coef.Fixed

The estimated fixed-effect parameters.

Std.Error.Fixed

The standard errors of the fixed-effect parameters.

Time

The time values in the dataset.

Fitted.Model

A fitted model of class lme.

Author(s)

Wim Van der Elst, Geert Molenberghs, Ralf-Dieter Hilgers, & Nicole Heussen

References

Van der Elst, W., Molenberghs, G., Hilgers, R., & Heussen, N. (2015). Estimating the reliability of repeatedly measured endpoints based on linear mixed-effects models. A tutorial. Submitted.

See Also

Explore.WS.Corr, WS.Corr.Mixed.SAS

Examples

# open data
data(Example.Data)

# Make covariates used in mixed model
Example.Data$Time2 <- Example.Data$Time**2
Example.Data$Time3 <- Example.Data$Time**3
Example.Data$Time3_log <- (Example.Data$Time**3) * (log(Example.Data$Time))

# model 1: random intercept model
Model1 <- WS.Corr.Mixed(
Fixed.Part=Outcome ~ Time2 + Time3 + Time3_log + as.factor(Cycle) 
+ as.factor(Condition), Random.Part = ~ 1|Id, 
Dataset=Example.Data, Model=1, Id="Id", Number.Bootstrap = 50, 
Seed = 12345)

  # summary of the results
summary(Model1)
  # plot the results
plot(Model1)

## Not run: time-consuming code parts
# model 2: random intercept + Gaussian serial corr
Model2 <- WS.Corr.Mixed(
Fixed.Part=Outcome ~ Time2 + Time3 + Time3_log + as.factor(Cycle) 
+ as.factor(Condition), Random.Part = ~ 1|Id, 
Correlation=corGaus(form= ~ Time, nugget = TRUE),
Dataset=Example.Data, Model=2, Id="Id", Seed = 12345)

  # summary of the results
summary(Model2)

  # plot the results
    # estimated corrs as a function of time lag (default plot)
plot(Model2)
    # estimated corrs for all pairs of time points
plot(Model2, All.Individual = T)

# model 3
Model3 <- WS.Corr.Mixed(
  Fixed.Part=Outcome ~ Time2 + Time3 + Time3_log + as.factor(Cycle) 
  + as.factor(Condition), Random.Part = ~ 1 + Time|Id, 
  Correlation=corGaus(form= ~ Time, nugget = TRUE),
  Dataset=Example.Data, Model=3, Id="Id", Seed = 12345)

  # summary of the results
summary(Model3)

  # plot the results
    # estimated corrs for all pairs of time points
plot(Model3)
    # estimated corrs as a function of time lag

## End(Not run)

Estimate within-subject (test-retest) correlations based on a mixed-effects model using the SAS proc MIXED output.

Description

This function allows for the estimation of the within-subject correlations using a general and flexible modeling approach that allows at the same time to capture hierarchies in the data, the presence of covariates, and the derivation of correlation estimates. The output of proc MIXED (SAS) is used as the input for this function. Confidence intervals for the correlations based on the Delta method are provided.

Usage

WS.Corr.Mixed.SAS(Model, D, Sigma2, Asycov, Rho, Tau2, Alpha=0.05, Time)

Arguments

Model

The type of model that should be fitted. Model=1: random intercept model, Model=2: random intercept and serial correlation, and Model=3: random intercept, slope, and serial correlation. Default Model=1.

D

The D matrix of the fitted model.

Sigma2

The residual variance.

Asycov

The asymptotic correlation matrix of covariance parameter estimates.

Rho

The \rho component of the fitted model which determines the matrix H_{i}, \rho(|t_{ij}-t_{ik}|). This component is only needed when serial correlation is involved, i.e., when Model 2 or 3 used.

Tau2

The \tau^2 component of the fitted model. This component is only needed when serial correlation is involved (i.e., when Model 2 or 3 used), \varepsilon_{2} \sim N(0, \tau^2 H_{i})).

Alpha

The \alpha-level to be used in the computation of the Confidence Intervals around the within-subject correlation. The Confidence Intervals are based on the Delta method. Default Alpha=0.05.

Time

The time points available in the dataset on which the analysis was conducted.

Value

Model

The type of model that was fitted.

R

The estimated within-subject correlations.

Alpha

The \alpha-level used to computed the Confidence Intervals around R.

CI.Upper

The upper bounds of the confidence intervals (Delta-method based).

CI.Lower

The lower bounds of the confidence intervals (Delta-method based).

Time

The time values in the dataset.

Author(s)

Wim Van der Elst, Geert Molenberghs, Ralf-Dieter Hilgers, & Nicole Heussen

References

Van der Elst, W., Molenberghs, G., Hilgers, R., & Heussen, N. (2015). Estimating the reliability of repeatedly measured endpoints based on linear mixed-effects models. A tutorial. Submitted.

See Also

WS.Corr.Mixed

Examples

# Open data 
data(Example.Data)

# Estimate R and Delta method-based CI 
# based on SAS output of fitted Model 2

# First specify asycov matrix
Asy_mat <- matrix(c(129170, -10248, -12.0814, -74.8605,
                    -10248, 25894, 21.0976, -50.1059,
                    -12.0814, 21.0976, 0.07791, 1.2120,
                    -74.8605, -50.1059, 1.212, 370.65), nrow = 4)
Model2_SAS <-  WS.Corr.Mixed.SAS(Model="Model 2", 
D=500.98, Tau2=892.97, Rho=3.6302, Sigma2=190.09, 
Asycov = Asy_mat, Time=c(1:45))                               
summary(Model2_SAS)
plot(Model2_SAS)

Plot of exploratory within-subject correlations (reliabilities)

Description

Provides an exploratory plot that allows for examining the within-subject correlations R (reliabilities) as a function if time lag.

Usage

## S3 method for class 'Explore.WS.Corr'
plot(x, Est.Corrs=TRUE, Indiv.Corrs=FALSE, 
Add.CI=FALSE, Add.CI.Smoothed=TRUE, Smoother.Span=0.2, 
Add.Boot.Corrs=FALSE, Add.CI.Polygon=FALSE, 
ylim=c(-1, 1), xlab="Time Lag", ylab="Reliability", ...)

Arguments

x

A fitted object of class Explore.WS.Corr.

Est.Corrs

Logical. Should the smoothed (loess) correlation function as a function of time lag be added? Default TRUE.

Indiv.Corrs

Logical. Should the estimated correlations for all individual time lags be added? Default FALSE.

Add.CI

Logical. Should a bootstrapped 100(1-\alpha)% Confidence Interval be added around the smoothed correlation function? Default FALSE.

Add.CI.Smoothed

Logical. Should a smoothed bootstrapped 100(1-\alpha)% Confidence Interval be added around the smoothed correlation function? Default FALSE.

Smoother.Span

The smoother span to be used. The smoother span gives the proportion of points in the plot which influence the smooth at each value. Larger values give more smoothness. For details, see https://stat.ethz.ch/R-manual/R-patched/library/stats/html/lowess.html. Defauls Smoother.Span=.2.

Add.Boot.Corrs

Logical. Should the inidividual bootstrapped smoothed (loess) correlation functions be added? Default FALSE.

Add.CI.Polygon

Logical. Similar to Add.CI but adds a grey polygon to mark the a bootstrapped 100(1-\alpha)% Confidence Interval (instead of dashed lines). Default FALSE.

ylim

The minimum and maximum values of the Y-axis. Default ylim=c(-1,1).

xlab

The label of the X-axis. Default xlab="Time Lag".

ylab

The label of the Y-axis. Default ylab="Reliability".

...

Other arguments to be passed to the plot function.

Author(s)

Wim Van der Elst, Geert Molenberghs, Ralf-Dieter Hilgers, & Nicole Heussen

References

Van der Elst, W., Molenberghs, G., Hilgers, R., & Heussen, N. (2015). Estimating the reliability of repeatedly measured endpoints based on linear mixed-effects models. A tutorial. Submitted.

See Also

Explore.WS.Corr, Heatmap

Examples

# Open data
data(Example.Data)

# Explore correlation structure
Expl_Corr <- Explore.WS.Corr(OLS.Model="Outcome~as.factor(Time)+ 
  as.factor(Cycle) + as.factor(Condition)", Dataset=Example.Data, 
  Id="Id", Time="Time", Alpha=.05, Number.Bootstrap=50, Seed=123)

# explore results
summary(Expl_Corr)

# plot with correlations for all time lags, and 
# add smoothed (loess) correlation function 
plot(Expl_Corr, Indiv.Corrs=TRUE, Add.CI=FALSE, Add.Boot.Corrs=FALSE)
# plot bootstrapped smoothed (loess) correlation function 
plot(Expl_Corr, Add.Boot.Corrs=TRUE)

Plot the within-subject correlations (reliabilities) obtained by using the mixed-effects modeling approch

Description

Plots the within-subject correlations (reliabilities) and 100(1-\alpha)% Confidence Intervals based on the fitted mixed-effect models.

Usage

## S3 method for class 'WS.Corr.Mixed'
plot(x, xlab, ylab, ylim, main, All.Individual=FALSE, ...)

Arguments

x

A fitted object of class WS.Corr.Mixed

xlab

The label of the X-axis.

ylab

The label of the Y-axis.

ylim

The min, max values of the Y-axis.

main

The main title of the plot.

All.Individual

Logical. Should correlation functions be provided that show the correlations between all indidual measurement moments R(t_{i},t_{k})? Argument is only used if Model 2 was fitted. Default All.Individual=FALSE.

...

Other arguments to be passed to the plot function.

Author(s)

Wim Van der Elst, Geert Molenberghs, Ralf-Dieter Hilgers, & Nicole Heussen

References

Van der Elst, W., Molenberghs, G., Hilgers, R., & Heussen, N. (2015). Estimating the reliability of repeatedly measured endpoints based on linear mixed-effects models. A tutorial. Submitted.

See Also

WS.Corr.Mixed, plot WS.Corr.Mixed

Examples

# open data
data(Example.Data)

# Make covariates used in mixed model
Example.Data$Time2 <- Example.Data$Time**2
Example.Data$Time3 <- Example.Data$Time**3
Example.Data$Time3_log <- (Example.Data$Time**3) * (log(Example.Data$Time))

# model 1: random intercept model
Model1 <- WS.Corr.Mixed(
Fixed.Part=Outcome ~ Time2 + Time3 + Time3_log + as.factor(Cycle) 
+ as.factor(Condition), Random.Part = ~ 1|Id, 
Dataset=Example.Data, Model=1, Id="Id", Number.Bootstrap = 50, 
Seed = 12345)

  # plot the results
plot(Model1)

## Not run: time-consuming code parts
# model 2: random intercept + Gaussian serial corr
Model2 <- WS.Corr.Mixed(
Fixed.Part=Outcome ~ Time2 + Time3 + Time3_log + as.factor(Cycle) 
+ as.factor(Condition), Random.Part = ~ 1|Id, 
Correlation=corGaus(form= ~ Time, nugget = TRUE),
Dataset=Example.Data, Model=2, Id="Id", Seed = 12345)

  # plot the results
    # estimated corrs as a function of time lag (default plot)
plot(Model2)
    # estimated corrs for all pairs of time points
plot(Model2, All.Individual = T)

# model 3
Model3 <- WS.Corr.Mixed(
  Fixed.Part=Outcome ~ Time2 + Time3 + Time3_log + as.factor(Cycle) 
  + as.factor(Condition), Random.Part = ~ 1 + Time|Id, 
  Correlation=corGaus(form= ~ Time, nugget = TRUE),
  Dataset=Example.Data, Model=3, Id="Id", Seed = 12345)

  # plot the results
    # estimated corrs for all pairs of time points
plot(Model3)
    # estimated corrs as a function of time lag

## End(Not run)

Summary

Description

summary

Usage

## S3 method for class 'Explore.WS.Corr'
summary(object, ..., Object)
## S3 method for class 'Fract.Poly'
summary(object, ..., Object)
## S3 method for class 'WS.Corr.Mixed'
summary(object, ..., Object)
## S3 method for class 'WS.Corr.Mixed.SAS'
summary(object, ..., Object)