R/ui.R
In ginettelafit/PowerAnalysisIL: Power analysis to select the number of participants in intensive longitudinal studies.
ui <- shinyUI(pageWithSidebar(
# Application title
headerPanel("Power analysis to select the number of participants in intensive longitudinal studies"),
# Sidebar with controls to select the outputs to compute power
sidebarPanel(
shinyjs::useShinyjs(debug = TRUE),
id = "side-panel",
withMathJax(),
# Input: Selector for choosing model ----
selectInput(inputId = "Model",
label = "Choose a model (more information in panel About the Method):",
choices = c("Model 1: Group differences in mean level"=1,
"Model 2: Effect of a level-2 continuous predictor on the mean level"=2,
"Model 3: Effect of a level-1 continuous predictor (random slope)"=3,
"Model 4: Effect of a level-1 continuous predictor (fixed slope)"=4,
"Model 5: Group differences in the effect of a level-1 continuous predictor (random slope)"=5,
"Model 6: Group differences in the effect of a level-1 continuous predictor (fixed slope)"=6,
"Model 7: Cross-level interaction effects (random slope)"=7,
"Model 8: Cross-level interaction effects (fixed slope)"=8,
"Model 9: Multilevel AR(1) model"=9,
"Model 10: Multilevel AR(1) model - Group differences in the autoregressive effects"=10,
"Model 11: Multilevel AR(1) model - Cross-level interaction effects"=11)),
conditionalPanel(
condition = "input.Model == '1'",
helpText("Model 1: Group differences in mean level"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}Z_i + \\nu_{0i} \\)"),
helpText("\\( Z_i\\) is a dummy variable equal to one if participant is in Group 1 and 0 otherwise"),
helpText("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) ")
),
conditionalPanel(
condition = "input.Model == '2'",
helpText("Model 2: Effect of a level-2 continuous predictor on the mean level"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}W_i + \\nu_{0i} \\)"),
helpText("\\( W_i\\) is the level-2 variable which is normally distributed \\(N(\\mu_{W}^2,\\sigma_{W}^2)\\)"),
helpText("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) ")
),
conditionalPanel(
condition = "input.Model == '3'",
helpText("Model 3: Effect of a level-1 continuous predictor (random slope)"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\nu_{0i} \\)"),
helpText("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\nu_{1i} \\)"),
helpText("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) ")
),
conditionalPanel(
condition = "input.Model == '4'",
helpText("Model 4: Effect of a level-1 continuous predictor (fixed slope)"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\nu_{0i} \\)"),
helpText("Level 2: \\(\\gamma_{1i} = \\beta_{10} \\)"),
helpText("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) ")
),
conditionalPanel(
condition = "input.Model == '5'",
helpText("Model 5: Group differences in the effect of a level-1 continuous predictor (random slope)"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}Z_i + \\nu_{0i} \\)"),
helpText("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}Z_i + \\nu_{1i} \\)"),
helpText("\\( Z_i\\) is a dummy variable equal to one if participant is in Group 1 and 0 otherwise"),
helpText("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) ")
),
conditionalPanel(
condition = "input.Model == '6'",
helpText("Model 6: Group differences in the effect of a level-1 continuous predictor (fixed slope)"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}Z_i + \\nu_{0i} \\)"),
helpText("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}Z_i \\)"),
helpText("\\( Z_i\\) is a dummy variable equal to one if participant is in Group 1 and 0 otherwise"),
helpText("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) ")
),
conditionalPanel(
condition = "input.Model == '7'",
helpText("Model 7: Cross-level interaction effects (random slope)"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}W_i + \\nu_{0i} \\)"),
helpText("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}W_i + \\nu_{1i} \\)"),
helpText("\\( W_i\\) is the level-2 variable which is normally distributed \\(N(\\mu_{W}^2,\\sigma_{W}^2)\\)"),
helpText("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) ")
),
conditionalPanel(
condition = "input.Model == '8'",
helpText("Model 8: Cross-level interaction effects (fixed slope)"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}W_i + \\nu_{0i} \\)"),
helpText("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}W_i \\)"),
helpText("\\( W_i\\) is the level-2 variable which is normally distributed \\(N(\\mu_{W}^2,\\sigma_{W}^2)\\)"),
helpText("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) ")
),
conditionalPanel(
condition = "input.Model == '9'",
helpText("Model 9: Multilevel AR(1) Model"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}Y_{it-1} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\nu_{0i} \\)"),
helpText("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\nu_{1i} \\)"),
helpText("Independent errors \\( \\epsilon_{it}\\) are Gausssian distributed \\(N(0,\\sigma_{\\epsilon}^2)\\) ")
),
conditionalPanel(
condition = "input.Model == '10'",
helpText("Model 10: Multilevel AR(1) model - Group differences in the autoregressive effects"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}Y_{it-1} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}Z_i + \\nu_{0i} \\)"),
helpText("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}Z_i + \\nu_{1i} \\)"),
helpText("\\( Z_i\\) is a dummy variable equal to one if participant is in Group 1 and 0 otherwise"),
helpText("Independent errors \\( \\epsilon_{it}\\) are Gausssian distributed \\(N(0,\\sigma_{\\epsilon}^2)\\) ")
),
conditionalPanel(
condition = "input.Model == '11'",
helpText("Model 11: Multilevel AR(1) model - Cross-level interaction effects"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}Y_{it-1} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}W_i + \\nu_{0i} \\)"),
helpText("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}W_i + \\nu_{1i} \\)"),
helpText("\\( W_i\\) is the level-2 variable which is normally distributed \\(N(\\mu_{W}^2,\\sigma_{W}^2)\\)"),
helpText("Independent errors \\( \\epsilon_{it}\\) are Gausssian distributed \\(N(0,\\sigma_{\\epsilon}^2)\\) ")
),
conditionalPanel(
condition = "input.Model == '1' || input.Model == '5' || input.Model == '6' || input.Model == '10'",
helpText("Number of participants: introduce an increasing sequence of positive integers (comma-separated). The length of the sequence must be the same in the two groups."),
textInput("N.0","Number of participants in Group 0 (reference group)", NULL),
textInput("N.1","Number of participants in Group 1", NULL)
),
conditionalPanel(
condition = "input.Model == '2' || input.Model == '3' || input.Model == '4' || input.Model == '7' ||
input.Model == '8' || input.Model == '9' || input.Model == '11'",
helpText("Number of participants: introduce an increasing sequence of positive integers (comma-separated)."),
textInput("N","Number of participants", NULL)
),
numericInput("T", "Number of time points", NULL),
numericInput("b00", "Fixed intercept: \\( \\beta_{00} \\)", NULL),
conditionalPanel(
condition = "input.Model == '1' || input.Model == '5' || input.Model == '6' || input.Model == '10'",
numericInput("b01.Z", "Effect of the level-2 dummy variable on the intercept: \\( \\beta_{01} \\)", NULL)
),
conditionalPanel(
condition = "input.Model == '2' || input.Model == '7' || input.Model == '8' || input.Model == '11'",
numericInput("b01.W", "Effect of the level-2 continuous variable on the intercept: \\( \\beta_{01} \\)", NULL)
),
conditionalPanel(
condition = "input.Model == '3' || input.Model == '4' || input.Model == '5' || input.Model == '6' || input.Model == '7'
|| input.Model == '8' || input.Model == '9' || input.Model == '10' || input.Model == '11'",
numericInput("b10", "Fixed slope: \\( \\beta_{10} \\)", NULL)
),
conditionalPanel(
condition = "input.Model == '5' || input.Model == '6' || input.Model == '10'",
numericInput("b11.Z", "Effect of the level-2 dummy variable on the slope: \\( \\beta_{11} \\)", NULL)
),
conditionalPanel(
condition = "input.Model == '7' || input.Model == '8' || input.Model == '11'",
numericInput("b11.W", "Effect of the level-2 continuous variable on the slope: \\( \\beta_{11} \\)", NULL)
),
numericInput("sigma", "Standard deviation of level-1 errors: \\( \\sigma_\\epsilon \\)", NULL),
conditionalPanel(
condition = "input.Model == '1' || input.Model == '2' || input.Model == '3' || input.Model == '4' || input.Model == '5'
|| input.Model == '6' || input.Model == '7' || input.Model == '8'",
numericInput("rho", "Autocorrelation of level-1 errors: \\( \\rho_\\epsilon \\)", NULL)
),
numericInput("sigma.v0", "Standard deviation of random intercept: \\( \\sigma_{\\nu_0} \\)", NULL),
conditionalPanel(
condition = "input.Model == '3' || input.Model == '5' || input.Model == '7' || input.Model == '9' || input.Model == '10'
|| input.Model == '11'",
numericInput("sigma.v1", "Standard deviation of random slope: \\( \\sigma_{\\nu_1} \\)", NULL),
numericInput("rho.v", "Correlation between the random intercept and random slope: \\( \\rho_{\\nu_{01}} \\)", NULL)
),
conditionalPanel(
condition = "input.Model == '3' || input.Model == '4' || input.Model == '7' || input.Model == '8'",
numericInput("mu.X", "Mean of time-varying variable X:", NULL),
numericInput("sigma.X", "Standard deviation of time-varying variable X:", NULL)
),
conditionalPanel(
condition = "input.Model == '5' || input.Model == '6'",
numericInput("mu.X0", "Mean of time-varying variable X in Group 0:", NULL),
numericInput("sigma.X0", "Standard deviation of time-varying variable X in Group 0:", NULL),
numericInput("mu.X1", "Mean of time-varying variable X in Group 1:", NULL),
numericInput("sigma.X1", "Standard deviation of time-varying variable X in Group 1:", NULL)
),
conditionalPanel(
condition = "input.Model == '3' || input.Model == '4' || input.Model == '5' || input.Model == '6' || input.Model == '7' || input.Model == '8'",
checkboxInput(inputId = "isX.center", label = strong("Person mean centering \\(X_{it} \\) using the individual mean"), value = TRUE),
),
conditionalPanel(
condition = "input.Model == '2' || input.Model == '7' || input.Model == '8' || input.Model == '11'",
numericInput("mu.W", "Mean of level-2 variable W:", NULL),
numericInput("sigma.W", "Standard deviation of level-2 variable W:", NULL),
checkboxInput(inputId = "isW.center", label = strong("Center the level-2 variable W"), value = TRUE)
),
conditionalPanel(
condition = "input.Model == '1' || input.Model == '2' || input.Model == '3' || input.Model == '4' || input.Model == '5'
|| input.Model == '6' || input.Model == '7' || input.Model == '8'",
checkboxInput(inputId = "is.rho.zero", label = strong("Estimate AR(1) correlated errors \\(\\epsilon_{it} \\)"), value = TRUE)
),
conditionalPanel(
condition = "input.Model == '9' || input.Model == '10' || input.Model == '11'",
checkboxInput(inputId = "Ylag.center", label = strong("Person mean centering level-1 lagged variable Y"), value = FALSE)
),
numericInput("alpha", "Type I error: \\( \\alpha \\)", 0.05),
numericInput("R", "Monte Carlo Replicates", 1000),
selectInput(inputId = "Opt.Method",
label = "Choose the method to fit linear mixed-effects model",
choices = c("Maximizing the log-likelihood"=1,
"Maximizing the restricted log-likelihood"=2)),
actionButton(inputId = "input_action_time", label = "Estimate Computational Time"),
actionButton(inputId = "input_action", label = "Compute Power"),
actionButton("reset_button", "Reset Page"),
helpText("Step 1: Estimate Computational Time"),
helpText("Step 2: Compute Power"),
helpText("Note:",
"To switch models and set new parameters click the Reset Page button."),
helpText("Contact: ginette.lafit@kuleuven.be"),
helpText("Citation: Lafit, G., Adolf, J., Dejonckheere, E.,
Myin-Germeys, I., Viechtbauer, W., & Ceulemans, E. (2020, June 1).
Selection of the Number of Participants in Intensive Longitudinal Studies:
A User-friendly Shiny App and Tutorial to Perform Power Analysis in Multilevel
Regression Models that Account for Temporal Dependencies. https://doi.org/10.31234/osf.io/dq6ky")
),
mainPanel(
tabsetPanel(
tabPanel("Power Analysis",
tags$h4("Population Models"),
htmlOutput("img"),
tags$h4("Estimated Computational Time"),
textOutput("TimeHat"),
tags$h4("Simulation Progress"),
verbatimTextOutput("text"),
tags$h4("Power Analysis"),
plotOutput("powerplot",height = "1000px")),
tabPanel("Summary Fixed Effects",
tags$h4("Summary Fixed Effects"),
tags$h6("Note:"),
tags$h6("Mean is the average of the estimated parameter over the Monte Carlo replicates"),
tags$h6("Std.error is the standard error of the estimated parameter over the Monte Carlo replicates"),
tags$h6("Bias is the average of the difference between the estimated parameter and true parameter over the Monte Carlo replicates"),
tags$h6("(1-alpha)% Coverage is the average of the (1-alpha)% confidence intervals that include the true parameter over the Monte Carlo replicates"),
tags$h6("Power is the number of times the null hypothesis is rejected over the Monte Carlo replicates"),
formattableOutput("power")
),
tabPanel("Summary Random Effects",
tags$h4("Summary Random Effects"),
tags$h6("Note:"),
tags$h6("Mean is the average of the estimated parameter over the Monte Carlo replicates"),
tags$h6("Std.error is the standard error of the estimated parameter over the Monte Carlo replicates"),
tags$h6("Bias is the average of the difference between the estimated parameter and true parameter over the Monte Carlo replicates"),
formattableOutput("covariance")
),
tabPanel("Monte Carlo Simulation",
tags$h4("Summary Monte Carlo Simulation"),
plotOutput("gmplot",height = "1000px"),
tags$h6("Note:"),
tags$h6("The distributions of the estimated parameters correspond to the case which the largest sample size"),
tags$h6("Dashed lines are the true model parameters")),
tabPanel("Y Trajectories",
tags$h4("Simulated trajectory of the outcome variable"),
plotOutput("yplot05"),plotOutput("yplot5"),plotOutput("yplot95"),
tags$h6("Note:"),
tags$h6("The trajectories of the outcome variable correspond to the case which the largest sample size"),
tags$h6("The plots show the simulated trajectory of the outcome variable for three different
participants: the participant in the 5% percentile, the participant in the 50% percentile and
the participant in the 95% percentile.")),
tabPanel("About the Method",
tags$h4("Simulation Approach to Estimate Power in Multilevel Linear Models"),
tags$p("Algorithm:"),
tags$p("1. Given a model based on the hypothesized theory, set up the population parameters. The parameter values can be decided from previous studies or a pilot study."),
tags$p("2. Set the sample size and generate a data set based on the model and its population parameters."),
tags$p("3. Test the significant of the null hypothesis (i.e. the hypothesized effect is zero) using the generated data with a Wald test."),
tags$p("4. Repeat steps 2 and 3 for R times, where R is the number of Monte Carlo replications."),
tags$p("5. Compute the power which is the number of times the hypothesized effect is significant over the Monte Carlo replications."),
tags$h4("-------------------------------------"),
tags$h4("Model 1: Group differences in mean level"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}Z_i + \\nu_{0i} \\)"),
tags$p("Variables:"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( Z_i\\) is a dummy variable equal to one if participant is in Group 1 and 0 otherwise"),
tags$p("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("\\(N_0\\) is the number of individuals in the reference group (i.e. Group 0)"),
tags$p("\\(N_1\\) is the number of individuals in Group 1"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("Model 2: Effect of a level-2 continuous predictor on the mean level"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}W_i + \\nu_{0i} \\)"),
tags$p("Variables:"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( W_i\\) is the level-2 predictor which is normally distributed \\(N(\\mu_{W}^2,\\sigma_{W}^2)\\)"),
tags$p("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("\\(N \\) is the total number of individuals"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("Model 3: Effect of a level-1 continuous predictor (random slope)"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\nu_{0i} \\)"),
tags$p("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\nu_{1i} \\)"),
tags$p("Variables:"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( X_{it}\\) is the level-1 predictor which is normally distributed \\(N(\\mu_X,\\sigma_X^2)\\)"),
tags$p("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("Random slope \\( \\gamma_{1i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_1}^2)\\)"),
tags$p("Correlation between the random intercept and random slope: \\( Cor(\\nu_{0i},\\nu_{1i}) = \\rho_{\\nu_{01}} \\)"),
tags$p("\\(N \\) is the total number of individuals"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("Model 4: Effect of a level-1 continuous predictor (fixed slope)"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\nu_{0i} \\)"),
tags$p("Level 2: \\(\\gamma_{1i} = \\beta_{10} \\)"),
tags$p("Variables:"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( X_{it}\\) is the level-1 predictor which is normally distributed \\(N(\\mu_X,\\sigma_X^2)\\)"),
tags$p("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("\\(N \\) is the total number of individuals"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("Model 5: Group differences in the effect of a level-1 continuous predictor (random slope)"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}Z_i + \\nu_{0i} \\)"),
tags$p("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}Z_i + \\nu_{1i} \\)"),
tags$p("Variables:"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( X_{0it}\\) is the level-1 predictor for individuals in the reference group (i.e. Group 0) which is normally distributed \\(N(\\mu_{X_0},\\sigma_{X_0}^2)\\)"),
tags$p("\\( X_{1it}\\) is the level-1 predictor for individuals in Group 1 which is normally distributed \\(N(\\mu_{X_1},\\sigma_{X_1}^2)\\)"),
tags$p("\\( Z_i\\) is a dummy variable equal to one if participant is in Group 1 and 0 otherwise"),
tags$p("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("Random slope \\( \\gamma_{1i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_1}^2)\\)"),
tags$p("Correlation between the random intercept and random slope: \\( Cor(\\nu_{0i},\\nu_{1i}) = \\rho_{\\nu_{01}} \\)"),
tags$p("\\(N_0\\) is the number of individuals in the reference group (i.e. Group 0)"),
tags$p("\\(N_1\\) is the number of individuals in Group 1"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("Model 6: Group differences in the effect of a level-1 continuous predictor (fixed slope)"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}Z_i + \\nu_{0i} \\)"),
tags$p("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}Z_i \\)"),
tags$p("Variables:"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( X_{0it}\\) is the level-1 predictor for individuals in the reference group (i.e. Group 0) which is normally distributed \\(N(\\mu_{X_0},\\sigma_{X_0}^2)\\)"),
tags$p("\\( X_{1it}\\) is the level-1 predictor for individuals in Group 1 which is normally distributed \\(N(\\mu_{X_1},\\sigma_{X_1}^2)\\)"),
tags$p("\\( Z_i\\) is a dummy variable equal to one if participant is in Group 1 and 0 otherwise"),
tags$p("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("\\(N_0\\) is the number of individuals in the reference group (i.e. Group 0)"),
tags$p("\\(N_1\\) is the number of individuals in Group 1"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("Model 7: Cross-level interaction effects (random slope)"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}W_i + \\nu_{0i} \\)"),
tags$p("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}W_i + \\nu_{1i} \\)"),
tags$p("Variables:"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( X_{it}\\) is the level-1 predictor which is normally distributed \\(N(\\mu_X,\\sigma_X^2)\\)"),
tags$p("\\( W_i\\) is the level-2 predictor which is normally distributed \\(N(\\mu_{W}^2,\\sigma_{W}^2)\\)"),
tags$p("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("Random slope \\( \\gamma_{1i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_1}^2)\\)"),
tags$p("Correlation between the random intercept and random slope: \\( Cor(\\nu_{0i},\\nu_{1i}) = \\rho_{\\nu_{01}} \\)"),
tags$p("\\(N \\) is the total number of individuals"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("Model 8: Cross-level interaction effects (fixed slope)"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}W_i + \\nu_{0i} \\)"),
tags$p("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}W_i \\)"),
tags$p("Variables:"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( X_{it}\\) is the level-1 predictor which is normally distributed \\(N(\\mu_X,\\sigma_X^2)\\)"),
tags$p("\\( W_i\\) is the level-2 predictor which is normally distributed \\(N(\\mu_{W}^2,\\sigma_{W}^2)\\)"),
tags$p("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("\\(N \\) is the total number of individuals"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("Model 9: Multilevel AR(1) Model"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}Y_{it-1} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\nu_{0i} \\)"),
tags$p("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\nu_{1i} \\)"),
tags$p("Variables:"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( Y_{it-1}\\) is the lagged dependent variable"),
tags$p("Independent errors \\( \\epsilon_{it}\\) are Gausssian distributed \\(N(0,\\sigma_{\\epsilon}^2)\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("Random slope \\( \\gamma_{1i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_1}^2)\\)"),
tags$p("Correlation between the random intercept and random slope: \\( Cor(\\nu_{0i},\\nu_{1i}) = \\rho_{\\nu_{01}} \\)"),
tags$p("\\(N \\) is the total number of individuals"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("Model 10: Multilevel AR(1) model - Group differences in the autoregressive effects"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}Y_{it-1} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}Z_i + \\nu_{0i} \\)"),
tags$p("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}Z_i + \\nu_{1i} \\)"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( Y_{it-1}\\) is the lagged dependent variable"),
tags$p("\\( Z_i\\) is a dummy variable equal to one if participant is in Group 1 and 0 otherwise"),
tags$p("Independent errors \\( \\epsilon_{it}\\) are Gausssian distributed \\(N(0,\\sigma_{\\epsilon}^2)\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("Random slope \\( \\gamma_{1i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_1}^2)\\)"),
tags$p("Correlation between the random intercept and random slope: \\( Cor(\\nu_{0i},\\nu_{1i}) = \\rho_{\\nu_{01}} \\)"),
tags$p("\\(N_0\\) is the number of individuals in the reference group (i.e. Group 0)"),
tags$p("\\(N_1\\) is the number of individuals in Group 1"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("Model 11: Multilevel AR(1) model - Cross-level interaction effects"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}Y_{it-1} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}W_i + \\nu_{0i} \\)"),
tags$p("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}W_i + \\nu_{1i} \\)"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( Y_{it-1}\\) is the lagged dependent variable"),
tags$p("\\( W_i\\) is the level-2 variable which is normally distributed \\(N(\\mu_{W}^2,\\sigma_{W}^2)\\)"),
tags$p("Independent errors \\( \\epsilon_{it}\\) are Gausssian distributed \\(N(0,\\sigma_{\\epsilon}^2)\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("Random slope \\( \\gamma_{1i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_1}^2)\\)"),
tags$p("Correlation between the random intercept and random slope: \\( Cor(\\nu_{0i},\\nu_{1i}) = \\rho_{\\nu_{01}} \\)"),
tags$p("\\(N \\) is the total number of individuals"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("R Syntax"),
tags$p("Model 1: lme(Y ~ Z ,random = ~1 | subjno,data,correlation = corAR1())"),
tags$p("Model 2: lme(Y ~ W ,random = ~1 | subjno,data,correlation = corAR1())"),
tags$p("Model 3: lme(Y ~ X ,random = ~1 + X | subjno,data,correlation = corAR1())"),
tags$p("Model 4: lme(Y ~ X ,random = ~1 | subjno,data,correlation = corAR1())"),
tags$p("Model 5: lme(Y ~ Z + X + Z*X ,random = ~1 + X | subjno,data,correlation = corAR1())"),
tags$p("Model 6: lme(Y ~ Z + X + Z*X ,random = ~1 | subjno,data,correlation = corAR1())"),
tags$p("Model 7: lme(Y ~ W + X + W*X ,random = ~1 + X | subjno,data,correlation = corAR1())"),
tags$p("Model 8: lme(Y ~ W + X + W*X ,random = ~1 | subjno,data,correlation = corAR1())"),
tags$p("Model 9: lme(Y ~ lag(Y) ,random = ~1 + lag(Y) | subjno,data)"),
tags$p("Model 10: lme(Y ~ Z + lag(Y) + Z*lag(Y) ,random = ~1 + lag(Y) | subjno,data)"),
tags$p("Model 11: lme(Y ~ W + lag(Y) + W*lag(Y) ,random = ~1 + lag(Y) | subjno,data)"))
))
))
ginettelafit/PowerAnalysisIL documentation built on July 8, 2023, 5:57 a.m.
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ui <- shinyUI(pageWithSidebar(
# Application title
headerPanel("Power analysis to select the number of participants in intensive longitudinal studies"),
# Sidebar with controls to select the outputs to compute power
sidebarPanel(
shinyjs::useShinyjs(debug = TRUE),
id = "side-panel",
withMathJax(),
# Input: Selector for choosing model ----
selectInput(inputId = "Model",
label = "Choose a model (more information in panel About the Method):",
choices = c("Model 1: Group differences in mean level"=1,
"Model 2: Effect of a level-2 continuous predictor on the mean level"=2,
"Model 3: Effect of a level-1 continuous predictor (random slope)"=3,
"Model 4: Effect of a level-1 continuous predictor (fixed slope)"=4,
"Model 5: Group differences in the effect of a level-1 continuous predictor (random slope)"=5,
"Model 6: Group differences in the effect of a level-1 continuous predictor (fixed slope)"=6,
"Model 7: Cross-level interaction effects (random slope)"=7,
"Model 8: Cross-level interaction effects (fixed slope)"=8,
"Model 9: Multilevel AR(1) model"=9,
"Model 10: Multilevel AR(1) model - Group differences in the autoregressive effects"=10,
"Model 11: Multilevel AR(1) model - Cross-level interaction effects"=11)),
conditionalPanel(
condition = "input.Model == '1'",
helpText("Model 1: Group differences in mean level"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}Z_i + \\nu_{0i} \\)"),
helpText("\\( Z_i\\) is a dummy variable equal to one if participant is in Group 1 and 0 otherwise"),
helpText("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) ")
),
conditionalPanel(
condition = "input.Model == '2'",
helpText("Model 2: Effect of a level-2 continuous predictor on the mean level"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}W_i + \\nu_{0i} \\)"),
helpText("\\( W_i\\) is the level-2 variable which is normally distributed \\(N(\\mu_{W}^2,\\sigma_{W}^2)\\)"),
helpText("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) ")
),
conditionalPanel(
condition = "input.Model == '3'",
helpText("Model 3: Effect of a level-1 continuous predictor (random slope)"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\nu_{0i} \\)"),
helpText("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\nu_{1i} \\)"),
helpText("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) ")
),
conditionalPanel(
condition = "input.Model == '4'",
helpText("Model 4: Effect of a level-1 continuous predictor (fixed slope)"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\nu_{0i} \\)"),
helpText("Level 2: \\(\\gamma_{1i} = \\beta_{10} \\)"),
helpText("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) ")
),
conditionalPanel(
condition = "input.Model == '5'",
helpText("Model 5: Group differences in the effect of a level-1 continuous predictor (random slope)"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}Z_i + \\nu_{0i} \\)"),
helpText("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}Z_i + \\nu_{1i} \\)"),
helpText("\\( Z_i\\) is a dummy variable equal to one if participant is in Group 1 and 0 otherwise"),
helpText("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) ")
),
conditionalPanel(
condition = "input.Model == '6'",
helpText("Model 6: Group differences in the effect of a level-1 continuous predictor (fixed slope)"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}Z_i + \\nu_{0i} \\)"),
helpText("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}Z_i \\)"),
helpText("\\( Z_i\\) is a dummy variable equal to one if participant is in Group 1 and 0 otherwise"),
helpText("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) ")
),
conditionalPanel(
condition = "input.Model == '7'",
helpText("Model 7: Cross-level interaction effects (random slope)"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}W_i + \\nu_{0i} \\)"),
helpText("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}W_i + \\nu_{1i} \\)"),
helpText("\\( W_i\\) is the level-2 variable which is normally distributed \\(N(\\mu_{W}^2,\\sigma_{W}^2)\\)"),
helpText("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) ")
),
conditionalPanel(
condition = "input.Model == '8'",
helpText("Model 8: Cross-level interaction effects (fixed slope)"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}W_i + \\nu_{0i} \\)"),
helpText("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}W_i \\)"),
helpText("\\( W_i\\) is the level-2 variable which is normally distributed \\(N(\\mu_{W}^2,\\sigma_{W}^2)\\)"),
helpText("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) ")
),
conditionalPanel(
condition = "input.Model == '9'",
helpText("Model 9: Multilevel AR(1) Model"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}Y_{it-1} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\nu_{0i} \\)"),
helpText("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\nu_{1i} \\)"),
helpText("Independent errors \\( \\epsilon_{it}\\) are Gausssian distributed \\(N(0,\\sigma_{\\epsilon}^2)\\) ")
),
conditionalPanel(
condition = "input.Model == '10'",
helpText("Model 10: Multilevel AR(1) model - Group differences in the autoregressive effects"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}Y_{it-1} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}Z_i + \\nu_{0i} \\)"),
helpText("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}Z_i + \\nu_{1i} \\)"),
helpText("\\( Z_i\\) is a dummy variable equal to one if participant is in Group 1 and 0 otherwise"),
helpText("Independent errors \\( \\epsilon_{it}\\) are Gausssian distributed \\(N(0,\\sigma_{\\epsilon}^2)\\) ")
),
conditionalPanel(
condition = "input.Model == '11'",
helpText("Model 11: Multilevel AR(1) model - Cross-level interaction effects"),
helpText("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}Y_{it-1} + \\epsilon_{it} \\)"),
helpText("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}W_i + \\nu_{0i} \\)"),
helpText("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}W_i + \\nu_{1i} \\)"),
helpText("\\( W_i\\) is the level-2 variable which is normally distributed \\(N(\\mu_{W}^2,\\sigma_{W}^2)\\)"),
helpText("Independent errors \\( \\epsilon_{it}\\) are Gausssian distributed \\(N(0,\\sigma_{\\epsilon}^2)\\) ")
),
conditionalPanel(
condition = "input.Model == '1' || input.Model == '5' || input.Model == '6' || input.Model == '10'",
helpText("Number of participants: introduce an increasing sequence of positive integers (comma-separated). The length of the sequence must be the same in the two groups."),
textInput("N.0","Number of participants in Group 0 (reference group)", NULL),
textInput("N.1","Number of participants in Group 1", NULL)
),
conditionalPanel(
condition = "input.Model == '2' || input.Model == '3' || input.Model == '4' || input.Model == '7' ||
input.Model == '8' || input.Model == '9' || input.Model == '11'",
helpText("Number of participants: introduce an increasing sequence of positive integers (comma-separated)."),
textInput("N","Number of participants", NULL)
),
numericInput("T", "Number of time points", NULL),
numericInput("b00", "Fixed intercept: \\( \\beta_{00} \\)", NULL),
conditionalPanel(
condition = "input.Model == '1' || input.Model == '5' || input.Model == '6' || input.Model == '10'",
numericInput("b01.Z", "Effect of the level-2 dummy variable on the intercept: \\( \\beta_{01} \\)", NULL)
),
conditionalPanel(
condition = "input.Model == '2' || input.Model == '7' || input.Model == '8' || input.Model == '11'",
numericInput("b01.W", "Effect of the level-2 continuous variable on the intercept: \\( \\beta_{01} \\)", NULL)
),
conditionalPanel(
condition = "input.Model == '3' || input.Model == '4' || input.Model == '5' || input.Model == '6' || input.Model == '7'
|| input.Model == '8' || input.Model == '9' || input.Model == '10' || input.Model == '11'",
numericInput("b10", "Fixed slope: \\( \\beta_{10} \\)", NULL)
),
conditionalPanel(
condition = "input.Model == '5' || input.Model == '6' || input.Model == '10'",
numericInput("b11.Z", "Effect of the level-2 dummy variable on the slope: \\( \\beta_{11} \\)", NULL)
),
conditionalPanel(
condition = "input.Model == '7' || input.Model == '8' || input.Model == '11'",
numericInput("b11.W", "Effect of the level-2 continuous variable on the slope: \\( \\beta_{11} \\)", NULL)
),
numericInput("sigma", "Standard deviation of level-1 errors: \\( \\sigma_\\epsilon \\)", NULL),
conditionalPanel(
condition = "input.Model == '1' || input.Model == '2' || input.Model == '3' || input.Model == '4' || input.Model == '5'
|| input.Model == '6' || input.Model == '7' || input.Model == '8'",
numericInput("rho", "Autocorrelation of level-1 errors: \\( \\rho_\\epsilon \\)", NULL)
),
numericInput("sigma.v0", "Standard deviation of random intercept: \\( \\sigma_{\\nu_0} \\)", NULL),
conditionalPanel(
condition = "input.Model == '3' || input.Model == '5' || input.Model == '7' || input.Model == '9' || input.Model == '10'
|| input.Model == '11'",
numericInput("sigma.v1", "Standard deviation of random slope: \\( \\sigma_{\\nu_1} \\)", NULL),
numericInput("rho.v", "Correlation between the random intercept and random slope: \\( \\rho_{\\nu_{01}} \\)", NULL)
),
conditionalPanel(
condition = "input.Model == '3' || input.Model == '4' || input.Model == '7' || input.Model == '8'",
numericInput("mu.X", "Mean of time-varying variable X:", NULL),
numericInput("sigma.X", "Standard deviation of time-varying variable X:", NULL)
),
conditionalPanel(
condition = "input.Model == '5' || input.Model == '6'",
numericInput("mu.X0", "Mean of time-varying variable X in Group 0:", NULL),
numericInput("sigma.X0", "Standard deviation of time-varying variable X in Group 0:", NULL),
numericInput("mu.X1", "Mean of time-varying variable X in Group 1:", NULL),
numericInput("sigma.X1", "Standard deviation of time-varying variable X in Group 1:", NULL)
),
conditionalPanel(
condition = "input.Model == '3' || input.Model == '4' || input.Model == '5' || input.Model == '6' || input.Model == '7' || input.Model == '8'",
checkboxInput(inputId = "isX.center", label = strong("Person mean centering \\(X_{it} \\) using the individual mean"), value = TRUE),
),
conditionalPanel(
condition = "input.Model == '2' || input.Model == '7' || input.Model == '8' || input.Model == '11'",
numericInput("mu.W", "Mean of level-2 variable W:", NULL),
numericInput("sigma.W", "Standard deviation of level-2 variable W:", NULL),
checkboxInput(inputId = "isW.center", label = strong("Center the level-2 variable W"), value = TRUE)
),
conditionalPanel(
condition = "input.Model == '1' || input.Model == '2' || input.Model == '3' || input.Model == '4' || input.Model == '5'
|| input.Model == '6' || input.Model == '7' || input.Model == '8'",
checkboxInput(inputId = "is.rho.zero", label = strong("Estimate AR(1) correlated errors \\(\\epsilon_{it} \\)"), value = TRUE)
),
conditionalPanel(
condition = "input.Model == '9' || input.Model == '10' || input.Model == '11'",
checkboxInput(inputId = "Ylag.center", label = strong("Person mean centering level-1 lagged variable Y"), value = FALSE)
),
numericInput("alpha", "Type I error: \\( \\alpha \\)", 0.05),
numericInput("R", "Monte Carlo Replicates", 1000),
selectInput(inputId = "Opt.Method",
label = "Choose the method to fit linear mixed-effects model",
choices = c("Maximizing the log-likelihood"=1,
"Maximizing the restricted log-likelihood"=2)),
actionButton(inputId = "input_action_time", label = "Estimate Computational Time"),
actionButton(inputId = "input_action", label = "Compute Power"),
actionButton("reset_button", "Reset Page"),
helpText("Step 1: Estimate Computational Time"),
helpText("Step 2: Compute Power"),
helpText("Note:",
"To switch models and set new parameters click the Reset Page button."),
helpText("Contact: ginette.lafit@kuleuven.be"),
helpText("Citation: Lafit, G., Adolf, J., Dejonckheere, E.,
Myin-Germeys, I., Viechtbauer, W., & Ceulemans, E. (2020, June 1).
Selection of the Number of Participants in Intensive Longitudinal Studies:
A User-friendly Shiny App and Tutorial to Perform Power Analysis in Multilevel
Regression Models that Account for Temporal Dependencies. https://doi.org/10.31234/osf.io/dq6ky")
),
mainPanel(
tabsetPanel(
tabPanel("Power Analysis",
tags$h4("Population Models"),
htmlOutput("img"),
tags$h4("Estimated Computational Time"),
textOutput("TimeHat"),
tags$h4("Simulation Progress"),
verbatimTextOutput("text"),
tags$h4("Power Analysis"),
plotOutput("powerplot",height = "1000px")),
tabPanel("Summary Fixed Effects",
tags$h4("Summary Fixed Effects"),
tags$h6("Note:"),
tags$h6("Mean is the average of the estimated parameter over the Monte Carlo replicates"),
tags$h6("Std.error is the standard error of the estimated parameter over the Monte Carlo replicates"),
tags$h6("Bias is the average of the difference between the estimated parameter and true parameter over the Monte Carlo replicates"),
tags$h6("(1-alpha)% Coverage is the average of the (1-alpha)% confidence intervals that include the true parameter over the Monte Carlo replicates"),
tags$h6("Power is the number of times the null hypothesis is rejected over the Monte Carlo replicates"),
formattableOutput("power")
),
tabPanel("Summary Random Effects",
tags$h4("Summary Random Effects"),
tags$h6("Note:"),
tags$h6("Mean is the average of the estimated parameter over the Monte Carlo replicates"),
tags$h6("Std.error is the standard error of the estimated parameter over the Monte Carlo replicates"),
tags$h6("Bias is the average of the difference between the estimated parameter and true parameter over the Monte Carlo replicates"),
formattableOutput("covariance")
),
tabPanel("Monte Carlo Simulation",
tags$h4("Summary Monte Carlo Simulation"),
plotOutput("gmplot",height = "1000px"),
tags$h6("Note:"),
tags$h6("The distributions of the estimated parameters correspond to the case which the largest sample size"),
tags$h6("Dashed lines are the true model parameters")),
tabPanel("Y Trajectories",
tags$h4("Simulated trajectory of the outcome variable"),
plotOutput("yplot05"),plotOutput("yplot5"),plotOutput("yplot95"),
tags$h6("Note:"),
tags$h6("The trajectories of the outcome variable correspond to the case which the largest sample size"),
tags$h6("The plots show the simulated trajectory of the outcome variable for three different
participants: the participant in the 5% percentile, the participant in the 50% percentile and
the participant in the 95% percentile.")),
tabPanel("About the Method",
tags$h4("Simulation Approach to Estimate Power in Multilevel Linear Models"),
tags$p("Algorithm:"),
tags$p("1. Given a model based on the hypothesized theory, set up the population parameters. The parameter values can be decided from previous studies or a pilot study."),
tags$p("2. Set the sample size and generate a data set based on the model and its population parameters."),
tags$p("3. Test the significant of the null hypothesis (i.e. the hypothesized effect is zero) using the generated data with a Wald test."),
tags$p("4. Repeat steps 2 and 3 for R times, where R is the number of Monte Carlo replications."),
tags$p("5. Compute the power which is the number of times the hypothesized effect is significant over the Monte Carlo replications."),
tags$h4("-------------------------------------"),
tags$h4("Model 1: Group differences in mean level"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}Z_i + \\nu_{0i} \\)"),
tags$p("Variables:"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( Z_i\\) is a dummy variable equal to one if participant is in Group 1 and 0 otherwise"),
tags$p("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("\\(N_0\\) is the number of individuals in the reference group (i.e. Group 0)"),
tags$p("\\(N_1\\) is the number of individuals in Group 1"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("Model 2: Effect of a level-2 continuous predictor on the mean level"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}W_i + \\nu_{0i} \\)"),
tags$p("Variables:"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( W_i\\) is the level-2 predictor which is normally distributed \\(N(\\mu_{W}^2,\\sigma_{W}^2)\\)"),
tags$p("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("\\(N \\) is the total number of individuals"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("Model 3: Effect of a level-1 continuous predictor (random slope)"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\nu_{0i} \\)"),
tags$p("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\nu_{1i} \\)"),
tags$p("Variables:"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( X_{it}\\) is the level-1 predictor which is normally distributed \\(N(\\mu_X,\\sigma_X^2)\\)"),
tags$p("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("Random slope \\( \\gamma_{1i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_1}^2)\\)"),
tags$p("Correlation between the random intercept and random slope: \\( Cor(\\nu_{0i},\\nu_{1i}) = \\rho_{\\nu_{01}} \\)"),
tags$p("\\(N \\) is the total number of individuals"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("Model 4: Effect of a level-1 continuous predictor (fixed slope)"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\nu_{0i} \\)"),
tags$p("Level 2: \\(\\gamma_{1i} = \\beta_{10} \\)"),
tags$p("Variables:"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( X_{it}\\) is the level-1 predictor which is normally distributed \\(N(\\mu_X,\\sigma_X^2)\\)"),
tags$p("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("\\(N \\) is the total number of individuals"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("Model 5: Group differences in the effect of a level-1 continuous predictor (random slope)"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}Z_i + \\nu_{0i} \\)"),
tags$p("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}Z_i + \\nu_{1i} \\)"),
tags$p("Variables:"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( X_{0it}\\) is the level-1 predictor for individuals in the reference group (i.e. Group 0) which is normally distributed \\(N(\\mu_{X_0},\\sigma_{X_0}^2)\\)"),
tags$p("\\( X_{1it}\\) is the level-1 predictor for individuals in Group 1 which is normally distributed \\(N(\\mu_{X_1},\\sigma_{X_1}^2)\\)"),
tags$p("\\( Z_i\\) is a dummy variable equal to one if participant is in Group 1 and 0 otherwise"),
tags$p("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("Random slope \\( \\gamma_{1i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_1}^2)\\)"),
tags$p("Correlation between the random intercept and random slope: \\( Cor(\\nu_{0i},\\nu_{1i}) = \\rho_{\\nu_{01}} \\)"),
tags$p("\\(N_0\\) is the number of individuals in the reference group (i.e. Group 0)"),
tags$p("\\(N_1\\) is the number of individuals in Group 1"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("Model 6: Group differences in the effect of a level-1 continuous predictor (fixed slope)"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}Z_i + \\nu_{0i} \\)"),
tags$p("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}Z_i \\)"),
tags$p("Variables:"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( X_{0it}\\) is the level-1 predictor for individuals in the reference group (i.e. Group 0) which is normally distributed \\(N(\\mu_{X_0},\\sigma_{X_0}^2)\\)"),
tags$p("\\( X_{1it}\\) is the level-1 predictor for individuals in Group 1 which is normally distributed \\(N(\\mu_{X_1},\\sigma_{X_1}^2)\\)"),
tags$p("\\( Z_i\\) is a dummy variable equal to one if participant is in Group 1 and 0 otherwise"),
tags$p("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("\\(N_0\\) is the number of individuals in the reference group (i.e. Group 0)"),
tags$p("\\(N_1\\) is the number of individuals in Group 1"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("Model 7: Cross-level interaction effects (random slope)"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}W_i + \\nu_{0i} \\)"),
tags$p("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}W_i + \\nu_{1i} \\)"),
tags$p("Variables:"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( X_{it}\\) is the level-1 predictor which is normally distributed \\(N(\\mu_X,\\sigma_X^2)\\)"),
tags$p("\\( W_i\\) is the level-2 predictor which is normally distributed \\(N(\\mu_{W}^2,\\sigma_{W}^2)\\)"),
tags$p("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("Random slope \\( \\gamma_{1i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_1}^2)\\)"),
tags$p("Correlation between the random intercept and random slope: \\( Cor(\\nu_{0i},\\nu_{1i}) = \\rho_{\\nu_{01}} \\)"),
tags$p("\\(N \\) is the total number of individuals"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("Model 8: Cross-level interaction effects (fixed slope)"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}X_{it} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}W_i + \\nu_{0i} \\)"),
tags$p("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}W_i \\)"),
tags$p("Variables:"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( X_{it}\\) is the level-1 predictor which is normally distributed \\(N(\\mu_X,\\sigma_X^2)\\)"),
tags$p("\\( W_i\\) is the level-2 predictor which is normally distributed \\(N(\\mu_{W}^2,\\sigma_{W}^2)\\)"),
tags$p("AR(1) errors \\( \\epsilon_{it}\\) with autocorrelation \\( \\rho_{\\epsilon}\\) and variance \\( \\sigma_{\\epsilon}^2\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("\\(N \\) is the total number of individuals"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("Model 9: Multilevel AR(1) Model"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}Y_{it-1} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\nu_{0i} \\)"),
tags$p("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\nu_{1i} \\)"),
tags$p("Variables:"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( Y_{it-1}\\) is the lagged dependent variable"),
tags$p("Independent errors \\( \\epsilon_{it}\\) are Gausssian distributed \\(N(0,\\sigma_{\\epsilon}^2)\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("Random slope \\( \\gamma_{1i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_1}^2)\\)"),
tags$p("Correlation between the random intercept and random slope: \\( Cor(\\nu_{0i},\\nu_{1i}) = \\rho_{\\nu_{01}} \\)"),
tags$p("\\(N \\) is the total number of individuals"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("Model 10: Multilevel AR(1) model - Group differences in the autoregressive effects"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}Y_{it-1} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}Z_i + \\nu_{0i} \\)"),
tags$p("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}Z_i + \\nu_{1i} \\)"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( Y_{it-1}\\) is the lagged dependent variable"),
tags$p("\\( Z_i\\) is a dummy variable equal to one if participant is in Group 1 and 0 otherwise"),
tags$p("Independent errors \\( \\epsilon_{it}\\) are Gausssian distributed \\(N(0,\\sigma_{\\epsilon}^2)\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("Random slope \\( \\gamma_{1i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_1}^2)\\)"),
tags$p("Correlation between the random intercept and random slope: \\( Cor(\\nu_{0i},\\nu_{1i}) = \\rho_{\\nu_{01}} \\)"),
tags$p("\\(N_0\\) is the number of individuals in the reference group (i.e. Group 0)"),
tags$p("\\(N_1\\) is the number of individuals in Group 1"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("Model 11: Multilevel AR(1) model - Cross-level interaction effects"),
tags$p("Level 1: \\(Y_{it} = \\gamma_{0i} + \\gamma_{1i}Y_{it-1} + \\epsilon_{it} \\)"),
tags$p("Level 2: \\(\\gamma_{0i} = \\beta_{00} + \\beta_{01}W_i + \\nu_{0i} \\)"),
tags$p("Level 2: \\(\\gamma_{1i} = \\beta_{10} + \\beta_{11}W_i + \\nu_{1i} \\)"),
tags$p("\\( Y_{it}\\) is the dependent variable"),
tags$p("\\( Y_{it-1}\\) is the lagged dependent variable"),
tags$p("\\( W_i\\) is the level-2 variable which is normally distributed \\(N(\\mu_{W}^2,\\sigma_{W}^2)\\)"),
tags$p("Independent errors \\( \\epsilon_{it}\\) are Gausssian distributed \\(N(0,\\sigma_{\\epsilon}^2)\\) "),
tags$p("Random intercept \\( \\gamma_{0i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_0}^2)\\)"),
tags$p("Random slope \\( \\gamma_{1i}\\) is Gausssian distributed \\(N(0,\\sigma_{\\nu_1}^2)\\)"),
tags$p("Correlation between the random intercept and random slope: \\( Cor(\\nu_{0i},\\nu_{1i}) = \\rho_{\\nu_{01}} \\)"),
tags$p("\\(N \\) is the total number of individuals"),
tags$p("\\(T \\) is the total number of repeated measurements for each individual"),
tags$h4("-------------------------------------"),
tags$h4("R Syntax"),
tags$p("Model 1: lme(Y ~ Z ,random = ~1 | subjno,data,correlation = corAR1())"),
tags$p("Model 2: lme(Y ~ W ,random = ~1 | subjno,data,correlation = corAR1())"),
tags$p("Model 3: lme(Y ~ X ,random = ~1 + X | subjno,data,correlation = corAR1())"),
tags$p("Model 4: lme(Y ~ X ,random = ~1 | subjno,data,correlation = corAR1())"),
tags$p("Model 5: lme(Y ~ Z + X + Z*X ,random = ~1 + X | subjno,data,correlation = corAR1())"),
tags$p("Model 6: lme(Y ~ Z + X + Z*X ,random = ~1 | subjno,data,correlation = corAR1())"),
tags$p("Model 7: lme(Y ~ W + X + W*X ,random = ~1 + X | subjno,data,correlation = corAR1())"),
tags$p("Model 8: lme(Y ~ W + X + W*X ,random = ~1 | subjno,data,correlation = corAR1())"),
tags$p("Model 9: lme(Y ~ lag(Y) ,random = ~1 + lag(Y) | subjno,data)"),
tags$p("Model 10: lme(Y ~ Z + lag(Y) + Z*lag(Y) ,random = ~1 + lag(Y) | subjno,data)"),
tags$p("Model 11: lme(Y ~ W + lag(Y) + W*lag(Y) ,random = ~1 + lag(Y) | subjno,data)"))
))
))
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