GAMMs analyses Trombone - Tongan vs English (NZE)

This notebook provides the first half of the full analysis of the article: Heyne, M., Derrick, D., and Al-Tamimi, J. (under review). “Native language influence on brass instrument performance: An application of generalized additive mixed models (GAMMs) to midsagittal ultrasound images of the tongue”. Frontiers Research Topic: Models and Theories of Speech Production. Ed. Adamantios Gafos & Pascal van Lieshout.

# specify directory to save models and summaries
output_dir = "updated_models"
# specify whether to run models -> if set to false script will attempt to load saved models from output_dir
run_models = FALSE

1 Loading packages

load_packages = c("readr","knitr","ggplot2","mgcv","itsadug","parallel","dplyr","rlist","plotly")
# dplyr, rlist, and plotly are required by the custom plotting functions
for(pkg in load_packages){
  eval(bquote(library(.(pkg))))
  if (paste0("package:", pkg) %in% search()){
    cat(paste0("Successfully loaded the ", pkg, " package.\n"))
  }else{
    install.packages(pkg)
    eval(bquote(library(.(pkg))))
    if (paste0("package:", pkg) %in% search()){
      cat(paste0("Successfully loaded the ", pkg, " package.\n"))
    }
  }
}

Successfully loaded the readr package.
Successfully loaded the knitr package.
Successfully loaded the ggplot2 package.
Successfully loaded the mgcv package.
Successfully loaded the itsadug package.
Successfully loaded the parallel package.
Successfully loaded the dplyr package.
Successfully loaded the rlist package.
Successfully loaded the plotly package.

rm(load_packages, pkg)
# detect number of cores available for model calculations
ncores = detectCores()
cat(paste0("Number of cores available for model calculations set to ", ncores, "."))

Number of cores available for model calculations set to 8.

2 Loading custom plotting function

2.1 plotly_scatterpolar_multiplot function (Matthias Heyne, 2019)

# This function plots multiple smoothing splines in the same window
plotly_scatterpolar_multiplot <- function(df, horizontal, vertical, cols2plot, print=TRUE){
  if (length(cols2plot)>2){
    print("ERROR: You specified more than 2 columns of values to plot.")
  }else{
    dat1=df
    df_name=deparse(substitute(df))
    # layout option 1
    if (length(horizontal)==2 & length(vertical)==1){
      # Note, Intensity, Language
      hori1=nrow(unique(select(dat1, horizontal[1])))
      hori2=nrow(unique(select(dat1, horizontal[2])))
      hori=hori1*hori2
      vert=nrow(unique(select(dat1, vertical[1])))
      dat1=select(dat1, c(horizontal[1],horizontal[2],vertical[1],cols2plot[1],cols2plot[2]))
      dat1=droplevels(dat1)
      var_hori1=levels(dat1[,1])
      var_hori2=levels(dat1[,2])
      var_vert1=levels(dat1[,3])
      
      # set up line types & colors
      ltypes=list("","dash") # match length of hori1
      colors=list("blue","green","orange","red") # match length of hori2
      cat(paste0("Proceeding to assemble a ", hori, "x", vert, " multiplot.\n"))
      cat(paste0("Your plot will show the columns/variables ",horizontal[1]," & ",horizontal[2]," in the horizontal direction and ",vertical[1]," in the vertical direction.\n"))
      cat(paste0(horizontal[1], " will be plotted using the following linestyles: -> "))
      for (n in 1:length(var_hori1)){
        if (n<length(var_hori1)){
          cat(paste0(var_hori1[n], ": ", ltypes[n], " - "))
        }else{
          cat(paste0(var_hori1[n], ": ", ltypes[n], "\n"))
        }
      }
      cat(paste0(horizontal[2], " will be plotted using the following colors: -> "))
      for (n in 1:length(var_hori2)){
        if (n<length(var_hori2)){
          cat(paste0(var_hori2[n], ": ", colors[n], " - "))
        }else{
          cat(paste0(var_hori2[n], ": ", colors[n], "\n"))
        }
      }
      rm(n)
      cat(paste0(vertical[1], " will be shown in the vertical direction from ", var_vert1[1], " (bottom) to ", var_vert1[length(var_vert1)], " (top).\n"))
      # assemble layout options for all subplots
      # plot_specs set as default
      plot_specs = list(sector=c(20,160), radialaxis=list(angle=90, range=c(0,max(dat1$rho_uncut_z)), tickfont=list(size=2)), 
                        angularaxis=list(thetaunit='radians', direction="clockwise", rotation=0, tickfont=list(size=4)))
      # set layout options for required number of subplots
      for (i in 1:hori){
        for (j in 1:vert){
          specsX=list.append(plot_specs, domain=list(x=c((i-1)/hori+(1/hori*0.2), i/hori-1/hori*0.2), 
                                                     y=c((j-1)/vert+(1/vert*0.1),j/vert-1/vert*0.1)))
          assign(paste0("sub_plot",((j-1)*hori)+i), specsX)
        }
      }
      rm(i, j, specsX)
      
      # assemble smoothing splines for traces
      for (j in 1:vert){
        # subset data set by vertical
        dat2=dat1[dat1[,3]==var_vert1[j],]
        for (i1 in 1:hori1){
          # subset data set by horizontal[1]
          dat3=dat2[dat2[,1]==var_hori1[i1],]
          for (i2 in 1:hori2){
            # subset data set by horizontal[2]
            dat4=dat3[dat3[,2]==var_hori2[i2],]
            if (!nrow(dat4)==0){
              if ((((j-1)*hori)+((i1-1)*hori2)+i2)==1){
                # assemble trace & assign number
                traceX=list(theta=seq(min(dat4$theta_uncut_z)*180/pi,max(dat4$theta_uncut_z)*180/pi, length=100), 
                            r=predict(smooth.spline(dat4$theta_uncut_z, dat4$rho_uncut_z),
                                      seq(min(dat4$theta_uncut_z),max(dat4$theta_uncut_z), length=100))$y,
                            line=list(color=colors[i2], dash=ltypes[i1]))
                assign(paste0("trace",((j-1)*hori)+((i1-1)*hori2)+i2),traceX)
              }else if ((((j-1)*hori)+((i1-1)*hori2)+i2)<=hori){
                # assemble trace & assign number
                traceX=list(theta=seq(min(dat4$theta_uncut_z)*180/pi,max(dat4$theta_uncut_z)*180/pi, length=100), 
                            r=predict(smooth.spline(dat4$theta_uncut_z, dat4$rho_uncut_z),
                                      seq(min(dat4$theta_uncut_z),max(dat4$theta_uncut_z), length=100))$y,
                            line=list(color=colors[i2], dash=ltypes[i1]), subplot=paste0("polar",((j-1)*hori)+((i1-1)*hori2)+i2))
                assign(paste0("trace",((j-1)*hori)+((i1-1)*hori2)+i2),traceX)
              }else{
                # assemble trace & assign number
                traceX=list(theta=seq(min(dat4$theta_uncut_z)*180/pi,max(dat4$theta_uncut_z)*180/pi, length=100), 
                            r=predict(smooth.spline(dat4$theta_uncut_z, dat4$rho_uncut_z),
                                      seq(min(dat4$theta_uncut_z),max(dat4$theta_uncut_z), length=100))$y,
                            line=list(color=colors[i2], dash=ltypes[i1]), subplot=paste0("polar",((j-1)*hori)+((i1-1)*hori2)+i2), showlegend=FALSE)
                assign(paste0("trace",((j-1)*hori)+((i1-1)*hori2)+i2),traceX)
            }
            }
            }
        }
      }
      rm(j, i1, i2, traceX, dat2, dat3, dat4)
      # plot assembled traces with assembed layout specifications
      p = plot_ly(type='scatterpolar', mode='lines')
      dont_plot=c()
      p = add_trace(p, theta=trace1$theta, r=trace1$r, line=list(color=trace1$line$color[[1]], dash=trace1$line$dash[[1]]))
      for (k in 2:(hori*vert)){
        if (exists(paste0("trace",k))){
          p = add_trace(p, theta=get(paste0("trace",k))$theta, r=get(paste0("trace",k))$r, 
                        subplot=get(paste0("trace",k))$subplot, 
                        line=list(color=get(paste0("trace",k))$line$color[[1]], dash=get(paste0("trace",k))$line$dash[[1]]))
        }else{
          dont_plot=c(dont_plot,k)
        }
      }
      
      # set layout
      layout_comp = capture.output(
        for (l in 1:(hori*vert)){
          if (is.na(match(l, dont_plot))){
            if (l==1){
              cat(paste0("layout(p, polar=sub_plot",l,", "))
            }else if (l<=hori){
              cat(paste0("polar",l,"=sub_plot",l,", "))
            }else if (l<hori*vert){
              cat(paste0("polar",l,"=sub_plot",l,", "))
            }else{
              cat(paste0("polar",l,"=sub_plot",l,", showlegend=FALSE)"))
            }
          }
        })
      p; eval(parse(text=layout_comp))
      
    # layout option 2
    }else if (length(horizontal)==1 & length(vertical)==2){
      # Subject, Note, Intensity
      hori=nrow(unique(select(dat1, horizontal[1])))
      vert1=nrow(unique(select(dat1, vertical[1])))
      vert2=nrow(unique(select(dat1, vertical[2])))
      vert=vert1*vert2
      dat1=select(dat1, c(horizontal[1],vertical[1],vertical[2],cols2plot[1],cols2plot[2]))
      # dat1[,1]=horizontal[1]; dat1[,2]=horizontal[2]; dat1[,3]=vertical[1];
      dat1=droplevels(dat1)
      var_hori1=levels(dat1[,1])
      var_vert1=levels(dat1[,2])
      var_vert2=levels(dat1[,3])
      
      # set up line types & colors
      colors=list("blue","green","orange","red","gray") # match length of vert1
      ltypes=list("","dash","dashdot","dot") # match length of vert2
      cat(paste0("Proceeding to assemble a ", hori, "x", vert, " multiplot.\n"))
      cat(paste0("Your plot will show the columns/variables ",vertical[1]," & ",vertical[2]," in the vertical direction and ",horizontal[1]," in the horizontal direction.\n"))
      cat(paste0(vertical[1], " will be plotted using the following colors: -> "))
      for (n in 1:length(var_vert1)){
        if (n<length(var_vert1)){
          cat(paste0(var_vert1[n], ": ", colors[n], " - "))
        }else{
          cat(paste0(var_vert1[n], ": ", colors[n], "\n"))
        }
      }
      cat(paste0(vertical[2], " will be plotted using the following linestyles: -> "))
      for (n in 1:length(var_vert2)){
        if (n<length(var_vert2)){
          cat(paste0(var_vert2[n], ": ", ltypes[n], " - "))
        }else{
          cat(paste0(var_vert2[n], ": ", ltypes[n], "\n"))
        }
      }
      rm(n)
      cat(paste0(horizontal[1], " will be shown in the horizontal direction from ", var_hori1[1], " (left) to ", var_hori1[length(var_hori1)], " (right).\n"))
      # assemble layout options for all subplots
      # plot_specs set as default
      plot_specs = list(sector=c(20,160), radialaxis=list(angle=90, range=c(0,max(dat1$rho_uncut_z)), tickfont=list(size=2)), 
                        angularaxis=list(thetaunit='radians', direction="clockwise", rotation=0, tickfont=list(size=4)))
      # set layout options for required number of subplots
      for (i in 1:hori){
        for (j in 1:vert){
          specsX=list.append(plot_specs, domain=list(x=c((i-1)/hori+(1/hori*0.2), i/hori-1/hori*0.2), 
                                                     y=c((j-1)/vert+(1/vert*0.1),j/vert-1/vert*0.1)))
          assign(paste0("sub_plot",((j-1)*hori)+i), specsX)
        }
      }
      rm(i, j, specsX)
      
      # assemble smoothing splines for traces
      for (i in 1:hori){
        # subset data set by horizontal
        dat2=dat1[dat1[,1]==var_hori1[i],]
        for (j1 in 1:vert1){
          # subset data set by vertical[1]
          dat3=dat2[dat2[,2]==var_vert1[j1],]
          for (j2 in 1:vert2){
            # subset data set by vertical[2]
            dat4=dat3[dat3[,3]==var_vert2[j2],]
            if (!nrow(dat4)==0){
              if (i+((j1-1)*vert)+((j2-1)*hori)+((j1-1)*vert)==1){
                # assemble trace & assign number
                traceX=list(theta=seq(min(dat4$theta_uncut_z)*180/pi,max(dat4$theta_uncut_z)*180/pi, length=100), 
                            r=predict(smooth.spline(dat4$theta_uncut_z, dat4$rho_uncut_z),
                                      seq(min(dat4$theta_uncut_z),max(dat4$theta_uncut_z), length=100))$y,
                            line=list(color=colors[j1], dash=ltypes[j2]))
                assign(paste0("trace", i+((j1-1)*vert)+((j2-1)*hori)+((j1-1)*vert)), traceX)
              }else if (i+((j1-1)*vert)+((j2-1)*hori)+((j1-1)*vert)<=hori){
                # assemble trace & assign number
                traceX=list(theta=seq(min(dat4$theta_uncut_z)*180/pi,max(dat4$theta_uncut_z)*180/pi, length=100), 
                            r=predict(smooth.spline(dat4$theta_uncut_z, dat4$rho_uncut_z),
                                      seq(min(dat4$theta_uncut_z),max(dat4$theta_uncut_z), length=100))$y,
                            line=list(color=colors[j1], dash=ltypes[j2]), subplot=paste0("polar",i+((j1-1)*vert)+((j2-1)*hori)+((j1-1)*vert)))
                assign(paste0("trace",i+((j1-1)*vert)+((j2-1)*hori)+((j1-1)*vert)),traceX)
              }else{
                # assemble trace & assign number
                traceX=list(theta=seq(min(dat4$theta_uncut_z)*180/pi,max(dat4$theta_uncut_z)*180/pi, length=100), 
                            r=predict(smooth.spline(dat4$theta_uncut_z, dat4$rho_uncut_z),
                                      seq(min(dat4$theta_uncut_z),max(dat4$theta_uncut_z), length=100))$y,
                            line=list(color=colors[j1], dash=ltypes[j2]), subplot=paste0("polar",i+((j1-1)*vert)+((j2-1)*hori)+((j1-1)*vert)), showlegend=FALSE)
                assign(paste0("trace",i+((j1-1)*vert)+((j2-1)*hori)+((j1-1)*vert)),traceX)
              }
            }
          }
        }
      }
      rm(i, j1, j2, traceX, dat2, dat3, dat4)
      
      # plot assembled traces with assembed layout specifications
      p = plot_ly(type='scatterpolar', mode='lines')
      dont_plot=c()
      p = add_trace(p, theta=trace1$theta, r=trace1$r, line=list(color=trace1$line$color[[1]], dash=trace1$line$dash[[1]]))
      for (k in 2:(hori*vert)){
        if (exists(paste0("trace",k))){
          p = add_trace(p, theta=get(paste0("trace",k))$theta, r=get(paste0("trace",k))$r, 
                        subplot=get(paste0("trace",k))$subplot, 
                        line=list(color=get(paste0("trace",k))$line$color[[1]], dash=get(paste0("trace",k))$line$dash[[1]]))
        }else{
          dont_plot=c(dont_plot,k)
        }
      }
      # set layout
      layout_comp = capture.output(
        for (l in 1:(hori*vert)){
          if (is.na(match(l, dont_plot))){
            if (l==1){
              cat(paste0("layout(p, polar=sub_plot",l,", "))
            }else if (l<=hori){
              cat(paste0("polar",l,"=sub_plot",l,", "))
            }else if (l<hori*vert){
              cat(paste0("polar",l,"=sub_plot",l,", "))
            }else{
              cat(paste0("polar",l,"=sub_plot",l,", showlegend=FALSE)"))
            }
          }
        })
      p; eval(parse(text=layout_comp))
      
    # layout option 3
    }else if (length(horizontal)==1 & length(vertical)==1){
      # Subject, tokenPooled
      hori=nrow(unique(select(dat1, horizontal[1])))
      vert=nrow(unique(select(dat1, vertical[1])))
      dat1=select(dat1, c(horizontal[1],vertical[1],cols2plot[1],cols2plot[2]))
      dat1=droplevels(dat1)
      var_hori1=levels(dat1[,1])
      var_vert1=levels(dat1[,2])
        
      # set up line types & colors
      if (unique(df$native_lg=="Tongan") && vertical=="token"){
        # levels(dfTongan$token)
        colors=list("#D50D0B","#D50D0B","#003380","#003380","#FF7B00","#FF7B00","#009737","#009737","#C20088","#C20088","#191919","#191919","#191919","#191919","#191919")
        ltypes=list("","dash","","dash","","dash","","dash","","dash","","dash","dashdot","dot","dash")
      }else if (unique(df$native_lg=="Tongan") && vertical=="tokenPooled"){
        colors=list("#D50D0B","#003380","#FF7B00","#009737","#C20088","#191919","#191919","#191919","#191919","#191919")
        ltypes=list("","","","","","","dash","dashdot","dot","dash")
      }else if (unique(df$native_lg=="NZE")){
        # levels(dfNZE$token)
        colors=list("#D50D0B","#990000","#0075DC","#E082B4","#003380","#FF7B00","#009737","#00AFC3","#C20088","#8F48B7","#ACB500","#7B4937","#6C6C6C","#191919","#191919","#191919","#191919","#191919")
        ltypes=list("","","","","","","","","","","","","","","dash","dashdot","dot","dash")
      }
      cat(paste0("Proceeding to assemble a ", hori, "x", vert, " multiplot.\n"))
      cat(paste0("Your plot will show the columns/variables ",horizontal[1]," in the horizontal direction and ",vertical[1]," in the vertical direction.\n"))
      cat(paste0(vertical[1], " will be shown in the vertical direction from ", var_vert1[1], " (bottom) to ", var_vert1[length(var_vert1)], " (top).\n"))
      
      # assemble layout options for all subplots
      # plot_specs set as default
      plot_specs = list(sector=c(20,160), radialaxis=list(angle=90, range=c(0,max(dat1$rho_uncut_z)), tickfont=list(size=2)), 
                        angularaxis=list(thetaunit='radians', direction="clockwise", rotation=0, tickfont=list(size=4)))
      # set layout options for required number of subplots
      for (i in 1:hori){
        for (j in 1:vert){
          specsX=list.append(plot_specs, domain=list(x=c((i-1)/hori+(1/hori*0.2), i/hori-1/hori*0.2), 
                                                     y=c((j-1)/vert+(1/vert*0.1),j/vert-1/vert*0.1)))
          assign(paste0("sub_plot",((j-1)*hori)+i), specsX)
        }
      }
      rm(i, j, specsX)
      
      # assemble smoothing splines for traces
      for (i in 1:hori){
        # subset data set by horizontal
        dat2=dat1[dat1[,1]==var_hori1[i],]
        for (j in 1:vert){
          # subset data set by vertical[1]
          dat3=dat2[dat2[,2]==var_vert1[j],]
          if (!nrow(dat3)==0){
            if (i+(j-1)*hori==1){
              # assemble trace & assign number
              traceX=list(theta=seq(min(dat3$theta_uncut_z)*180/pi,max(dat3$theta_uncut_z)*180/pi, length=100), 
                          r=predict(smooth.spline(dat3$theta_uncut_z, dat3$rho_uncut_z),
                                    seq(min(dat3$theta_uncut_z),max(dat3$theta_uncut_z), length=100))$y,
                          line=list(color=colors[j], dash=ltypes[j]))
              assign(paste0("trace", i+(j-1)*hori), traceX)
            }else if (i+(j-1)*hori<=hori){
              # assemble trace & assign number
              traceX=list(theta=seq(min(dat3$theta_uncut_z)*180/pi,max(dat3$theta_uncut_z)*180/pi, length=100), 
                            r=predict(smooth.spline(dat3$theta_uncut_z, dat3$rho_uncut_z),
                                      seq(min(dat3$theta_uncut_z),max(dat3$theta_uncut_z), length=100))$y,
                            line=list(color=colors[j], dash=ltypes[j]), subplot=paste0("polar",i+(j-1)*hori))
              assign(paste0("trace", i+(j-1)*hori), traceX)
            }else{
              # assemble trace & assign number
              traceX=list(theta=seq(min(dat3$theta_uncut_z)*180/pi,max(dat3$theta_uncut_z)*180/pi, length=100), 
                          r=predict(smooth.spline(dat3$theta_uncut_z, dat3$rho_uncut_z),
                                    seq(min(dat3$theta_uncut_z),max(dat3$theta_uncut_z), length=100))$y,
                          line=list(color=colors[j], dash=ltypes[j]), subplot=paste0("polar",i+(j-1)*hori), showlegend=FALSE)
              assign(paste0("trace", i+(j-1)*hori), traceX)
            }
          }
        }
      }
      rm(i, j, traceX, dat2, dat3)
      
      # plot assembled traces with assembed layout specifications
      p = plot_ly(type='scatterpolar', mode='lines')
      dont_plot=c()
      p = add_trace(p, theta=trace1$theta, r=trace1$r, line=list(color=trace1$line$color[[1]], dash=trace1$line$dash[[1]]))
      for (k in 2:(hori*vert)){
        if (exists(paste0("trace",k))){
          p = add_trace(p, theta=get(paste0("trace",k))$theta, r=get(paste0("trace",k))$r, 
                        subplot=get(paste0("trace",k))$subplot, 
                        line=list(color=get(paste0("trace",k))$line$color[[1]], dash=get(paste0("trace",k))$line$dash[[1]]))
        }else{
          dont_plot=c(dont_plot,k)
        }
      }
      
      # set layout
      layout_comp = capture.output(
        for (l in 1:(hori*vert)){
          if (is.na(match(l, dont_plot))){
            if (l==1){
              cat(paste0("layout(p, polar=sub_plot",l,", "))
            }else if (l<=hori){
              cat(paste0("polar",l,"=sub_plot",l,", "))
            }else if (l<hori*vert){
              cat(paste0("polar",l,"=sub_plot",l,", "))
            }else{
              cat(paste0("polar",l,"=sub_plot",l,", showlegend=FALSE)"))
            }
          }
        })
      p; eval(parse(text=layout_comp))
    }else{
      cat("Sorry, this layout is not yet implemented in the function. Currently the options are either 2 variables shown horizontally and 1 shown vertically or 1 horizontally and 2 vertically.\n")
      cat("Usage: plotly_scatterpolar_multiplot(df, horizontal, vertical, cols2plot, print=TRUE) ->\n where df refers to the data.frame to plot, horizontal & vertical specify the column names to use as grouping variables,\n and cols2plot refers to the 2 columns of values to plot.\n")
      cat("Use the c(x, y) notation to specify multiple colums for horizontal and/or vertical and for the cols2plot columns.\n")
  }
  }
}

3 Dataset

3.1 Manipulation

df <- read.csv("GAMM_Trombone_data.csv", sep=',', stringsAsFactors = F)
# remove empty column
df$X = NULL
df$tokenPooled <- factor(df$tokenPooled)
df$subject <- factor(df$subject)
df$native_lg <- factor(df$native_lg)
# df$playing_proficiency[df$playing_proficiency == "intermediate"] <- "amateur"
df$playing_proficiency <- factor(df$playing_proficiency, levels = c("amateur","intermediate","semi-professional","professional"))
df$block <- factor(df$block)
df$point <- as.numeric(df$point)
df$note_intensity <- factor(df$note_intensity, levels = c("piano","mezzopiano","mezzoforte","forte"))

3.2 Two new datasets

3.2.1 Notes

dfNotes <- subset(df,df$tokenPooled == "Bb2"|
                    df$tokenPooled == "Bb3"|
                    df$tokenPooled == "D4"|
                    df$tokenPooled == "F3"|
                    df$tokenPooled == "F4"
                  )
dfNotes$tokenPooled <- factor(dfNotes$tokenPooled, levels = c("Bb2","F3","Bb3","D4","F4"))
dfNotes <- subset(dfNotes,dfNotes$note_intensity == "forte"|
                    dfNotes$note_intensity == "mezzoforte"|
                    dfNotes$note_intensity == "mezzopiano"|
                    dfNotes$note_intensity == "piano"
                  )
dfNotes$note_intensity <- factor(dfNotes$note_intensity, levels = c("piano","mezzopiano","mezzoforte","forte"))
levels(dfNotes$tokenPooled)

[1] "Bb2" "F3"  "Bb3" "D4"  "F4" 

levels(dfNotes$note_intensity)

[1] "piano"      "mezzopiano" "mezzoforte" "forte"     

levels(dfNotes$playing_proficiency)

[1] "amateur"           "intermediate"      "semi-professional" "professional"     

3.2.2 Vowels

dfVowelsFull <- subset(df,is.na(df$note_intensity))
dfVowelsFull$tokenPooled <- factor(dfVowelsFull$tokenPooled)
levels(dfVowelsFull$tokenPooled)

 [1] "a"  "ɐ"  "ɐː" "ɒ"  "e"  "ə"  "ə#" "ɛ"  "ɘ"  "F3" "i"  "iː" "o"  "oː" "ɵː" "u"  "ʉː" "ʊ" 

# creating a new dataset dfVowels that contains comparable vowels
# in the two languages
dfVowels <- subset(dfVowelsFull,dfVowelsFull$tokenPooled == "ɐ"| 
                         dfVowelsFull$tokenPooled == "ɐː"| 
                         dfVowelsFull$tokenPooled == "a"| 
                         dfVowelsFull$tokenPooled == "e"| 
                         dfVowelsFull$tokenPooled == "i"| 
                         dfVowelsFull$tokenPooled == "iː"|
                         dfVowelsFull$tokenPooled == "ɛ"| 
                         dfVowelsFull$tokenPooled == "o"| 
                         dfVowelsFull$tokenPooled == "oː"| 
                         dfVowelsFull$tokenPooled == "u"| 
                         dfVowelsFull$tokenPooled == "ʉː")
dfVowels$tokenPooled <- factor(dfVowels$tokenPooled)
levels(dfVowels$tokenPooled)

 [1] "a"  "ɐ"  "ɐː" "e"  "ɛ"  "i"  "iː" "o"  "oː" "u"  "ʉː"

# creating a new variable TokenPooledNew and changing names of
# vowels in NZE to match those in Tongan.
dfVowels$tokenPooledNew <- dfVowels$tokenPooled
dfVowels$tokenPooledNew <- as.character(dfVowels$tokenPooledNew)
dfVowels$tokenPooledNew[dfVowels$tokenPooledNew == "ɐ"] <- "a"
dfVowels$tokenPooledNew[dfVowels$tokenPooledNew == "ɐː"] <- "a"
dfVowels$tokenPooledNew[dfVowels$tokenPooledNew == "ʉː"] <- "u"
dfVowels$tokenPooledNew[dfVowels$tokenPooledNew == "oː"] <- "o"
dfVowels$tokenPooledNew[dfVowels$tokenPooledNew == "iː"] <- "i"
dfVowels$tokenPooledNew[dfVowels$tokenPooledNew == "e" & dfVowels$native_lg == "NZE"] <- "i"
dfVowels$tokenPooledNew[dfVowels$tokenPooledNew == "ɛ"] <- "e"
dfVowels$tokenPooledNew <- as.factor(dfVowels$tokenPooledNew)
levels(dfVowels$tokenPooledNew)

[1] "a" "e" "i" "o" "u"

dfVowels$speech_preceding_soundNew <- dfVowels$speech_preceding_sound_IPA
dfVowels <- subset(dfVowels, dfVowels$speech_preceding_soundNew != "ɹ" &
                     dfVowels$speech_preceding_soundNew != "ʒ" &
                     dfVowels$speech_preceding_soundNew != "ð" &
                     dfVowels$speech_preceding_soundNew != "N" &
                     dfVowels$speech_preceding_soundNew != "o" &
                     dfVowels$speech_preceding_soundNew != "v" &
                     dfVowels$speech_preceding_soundNew != "w" &
                     dfVowels$speech_preceding_soundNew != "z"
                     )
dfVowels$speech_preceding_soundNew <- factor(dfVowels$speech_preceding_soundNew)
levels(dfVowels$speech_preceding_soundNew)

 [1] "a"    "b"    "d"    "e"    "f"    "g"    "h"    "i"    "k"    "l"    "m"    "n"    "NULL"
[14] "ŋ"    "p"    "s"    "ʃ"    "t"    "u"    "ʔ"   

dfVowels$speech_following_soundNew <- dfVowels$speech_following_sound
dfVowels <- subset(dfVowels, dfVowels$speech_following_soundNew != "ə" &
                     dfVowels$speech_following_soundNew != "ɫ" &
                     dfVowels$speech_following_soundNew != "ɹ" &
                     dfVowels$speech_following_soundNew != "ɾ" &
                     dfVowels$speech_following_soundNew != "ʒ" &
                     dfVowels$speech_following_soundNew != "θ" &
                     dfVowels$speech_following_soundNew != "a" &
                     dfVowels$speech_following_soundNew != "b" &
                     dfVowels$speech_following_soundNew != "ð" &
                     dfVowels$speech_following_soundNew != "g" &
                     dfVowels$speech_following_soundNew != "i" &
                     dfVowels$speech_following_soundNew != "o" &
                     dfVowels$speech_following_soundNew != "u" &
                     dfVowels$speech_following_soundNew != "e" &
                     dfVowels$speech_following_soundNew != "w"
                     )
dfVowels$speech_following_soundNew <- factor(dfVowels$speech_following_soundNew)
levels(dfVowels$speech_following_soundNew)

 [1] "4"             "5"             "approximant_r" "d"             "D"            
 [6] "f"             "final_schwa"   "glottal_stop"  "h"             "k"            
[11] "l"             "m"             "n"             "N"             "NULL"         
[16] "p"             "s"             "S"             "t"             "T"            
[21] "v"             "z"             "Z"            

3.3 Tables to check structure

kable(table(dfNotes$tokenPooled,dfNotes$note_intensity),format="html")

	piano	mezzopiano	mezzoforte	forte
Bb2	3000	3300	115300	9700
F3	7800	11800	225800	17800
Bb3	9700	5400	202800	13400
D4	5000	1900	75300	5600
F4	1200	100	25500	2400

kable(table(dfNotes$tokenPooled,dfNotes$native_lg),format="html")

	NZE	Tongan
Bb2	68700	62600
F3	130300	132900
Bb3	115000	116300
D4	43000	44800
F4	14500	14700

kable(table(dfNotes$note_intensity,dfNotes$native_lg),format="html")

	NZE	Tongan
piano	16100	10600
mezzopiano	14400	8100
mezzoforte	314300	330400
forte	26700	22200

kable(table(dfNotes$playing_proficiency,dfNotes$native_lg),format="html")

	NZE	Tongan
amateur	169200	205700
intermediate	62900	0
semi-professional	44400	124700
professional	95000	40900

kable(table(dfNotes$age_range,dfNotes$native_lg),format="html")

	NZE	Tongan
15-20	0	101700
20-25	43100	36900
25-30	61400	81700
30-35	126600	125300
35-40	15300	0
40-45	0	25700
60-65	37600	0
65-70	87500	0

kable(table(dfVowels$tokenPooledNew, dfVowels$native_lg),format="html")

	NZE	Tongan
a	100700	98400
e	56500	56400
i	91800	90400
o	31200	70300
u	39100	44300

kable(table(dfVowels$age_range, dfVowels$native_lg),format="html")

	NZE	Tongan
15-20	0	116400
20-25	37400	36800
25-30	67300	75100
30-35	106600	112900
35-40	4600	0
40-45	0	18600
60-65	33000	0
65-70	70400	0

kable(table(dfVowels$speech_preceding_soundNew, dfVowels$native_lg),format="html")

	NZE	Tongan
a	0	10400
b	30800	0
d	36900	0
e	0	8000
f	4000	5600
g	23400	0
h	10300	3300
i	0	11900
k	39900	48900
l	38100	52200
m	6600	17500
n	21700	34500
NULL	10500	38300
ŋ	0	8000
p	4400	20400
s	31800	19900
ʃ	21200	0
t	39700	47400
u	0	9100
ʔ	0	24400

kable(table(dfVowels$speech_following_soundNew, dfVowels$native_lg),format="html")

	NZE	Tongan
4	400	0
5	100	0
approximant_r	300	0
d	39600	0
D	3900	0
f	11200	30200
final_schwa	200	0
glottal_stop	600	22000
h	100	14800
k	33600	35200
l	0	1200
m	27200	27200
n	48200	40500
N	3800	22600
NULL	19900	47300
p	19800	19300
s	29500	25800
S	6100	0
t	49100	56500
T	7200	0
v	9800	17200
z	7300	0
Z	1400	0

kable(table(dfVowels$speech_preceding_soundNew, dfVowels$tokenPooledNew),format="html")

	a	e	i	o	u
a	6100	900	2400	800	200
b	6700	7100	9900	3700	3400
d	14000	2900	10200	2700	7100
e	1900	1900	2600	1000	600
f	2000	2800	1900	2500	400
g	9200	2700	6500	1500	3500
h	1700	5000	3600	1600	1700
i	2100	900	2500	4700	1700
k	30300	16000	18300	17000	7200
l	29500	14200	24700	12400	9500
m	8100	5800	6700	1800	1700
n	16000	4800	21800	7800	5800
NULL	9900	11200	12200	7600	7900
ŋ	3300	0	0	1300	3400
p	5300	5400	7900	4400	1800
s	11200	9500	17800	7600	5600
ʃ	8700	2300	5800	1900	2500
t	22500	14100	24100	15200	11200
u	4000	1700	700	1400	1300
ʔ	6600	3700	2600	4600	6900

kable(table(dfVowels$speech_following_soundNew, dfVowels$tokenPooledNew),format="html")

	a	e	i	o	u
4	0	300	0	100	0
5	0	100	0	0	0
approximant_r	100	0	100	100	0
d	9500	10800	10700	4500	4100
D	2000	0	1900	0	0
f	15400	5100	9200	9000	2700
final_schwa	0	0	100	100	0
glottal_stop	5500	5000	3400	2200	6500
h	5700	700	1800	4200	2500
k	24000	10700	20900	7200	6000
l	100	200	500	200	200
m	20600	7700	13000	4500	8600
n	20000	14500	27100	16200	10900
N	11500	4300	1700	7600	1300
NULL	21900	5500	13400	14900	11500
p	9900	12800	9300	2200	4900
s	13100	7800	22000	5000	7400
S	300	1600	2500	900	800
t	26000	19500	32200	18200	9700
T	2800	0	1800	0	2600
v	9800	4600	6800	3700	2100
z	900	1700	3200	600	900
Z	0	0	600	100	700

3.4 Visualising the data

3.4.1 Notes

Before running anything, we start by visualising the data

3.4.1.1 By Note, By Intensity and By Language

# run multiplot function
plotly_scatterpolar_multiplot(df=dfNotes, horizontal=c("native_lg","note_intensity"), vertical="tokenPooled", cols2plot=c("theta_uncut_z","rho_uncut_z"))

Proceeding to assemble a 8x5 multiplot.
Your plot will show the columns/variables native_lg & note_intensity in the horizontal direction and tokenPooled in the vertical direction.
native_lg will be plotted using the following linestyles: -> NZE:  - Tongan: dash
note_intensity will be plotted using the following colors: -> piano: blue - mezzopiano: green - mezzoforte: orange - forte: red
tokenPooled will be shown in the vertical direction from Bb2 (bottom) to F4 (top).

3.4.1.2 By Note, By Intensity, and By subject

3.4.1.2.1 NZE

Let’s start with the NZE data. We see that speakers are variable in how they are producing the notes according to their intensities (which is normal).

dfNotesNZE <- subset(dfNotes, dfNotes$native_lg == "NZE")
dfNotesNZE$subject <- factor(dfNotesNZE$subject)
plotly_scatterpolar_multiplot(df=dfNotesNZE, horizontal="subject", vertical=c("tokenPooled","note_intensity"), cols2plot=c("theta_uncut_z","rho_uncut_z"))

Proceeding to assemble a 10x20 multiplot.
Your plot will show the columns/variables tokenPooled & note_intensity in the vertical direction and subject in the horizontal direction.
tokenPooled will be plotted using the following colors: -> Bb2: blue - F3: green - Bb3: orange - D4: red - F4: gray
note_intensity will be plotted using the following linestyles: -> piano:  - mezzopiano: dash - mezzoforte: dashdot - forte: dot
subject will be shown in the horizontal direction from S1 (left) to S5 (right).

3.4.1.2.2 Tongan

Moving on to the Tongan data, we see again that speakers are variable in how they are producing the notes according to their intensities (which is normal).

dfNotesTongan <- subset(dfNotes, dfNotes$native_lg == "Tongan")
dfNotesTongan$subject <- factor(dfNotesTongan$subject)
plotly_scatterpolar_multiplot(df=dfNotesTongan, horizontal="subject", vertical=c("tokenPooled","note_intensity"), cols2plot=c("theta_uncut_z","rho_uncut_z"))

Proceeding to assemble a 10x20 multiplot.
Your plot will show the columns/variables tokenPooled & note_intensity in the vertical direction and subject in the horizontal direction.
tokenPooled will be plotted using the following colors: -> Bb2: blue - F3: green - Bb3: orange - D4: red - F4: gray
note_intensity will be plotted using the following linestyles: -> piano:  - mezzopiano: dash - mezzoforte: dashdot - forte: dot
subject will be shown in the horizontal direction from S14 (left) to S4 (right).

3.4.1.3 New variables

The two observations on variations by note and intensity (that are within subject) are important. We will create a variable that combines subject, note and intensity. This will allow us later on to use these instead of subject only to model the within subject variation with respect to the two other predictors (note and intensity). We also create a new variable that combines language, note and intensity of the note to be used as a fixed effect and in adjustments.

dfNotes$noteIntenInt <- interaction(dfNotes$tokenPooled, dfNotes$note_intensity)
levels(dfNotes$noteIntenInt)

 [1] "Bb2.piano"      "F3.piano"       "Bb3.piano"      "D4.piano"       "F4.piano"      
 [6] "Bb2.mezzopiano" "F3.mezzopiano"  "Bb3.mezzopiano" "D4.mezzopiano"  "F4.mezzopiano" 
[11] "Bb2.mezzoforte" "F3.mezzoforte"  "Bb3.mezzoforte" "D4.mezzoforte"  "F4.mezzoforte" 
[16] "Bb2.forte"      "F3.forte"       "Bb3.forte"      "D4.forte"       "F4.forte"      

str(dfNotes$noteIntenInt)

 Factor w/ 20 levels "Bb2.piano","F3.piano",..: 11 11 11 11 11 11 11 11 11 11 ...

dfNotes$langNoteInt <- interaction(dfNotes$native_lg, 
dfNotes$tokenPooled, dfNotes$note_intensity)
levels(dfNotes$langNoteInt)

 [1] "NZE.Bb2.piano"         "Tongan.Bb2.piano"      "NZE.F3.piano"         
 [4] "Tongan.F3.piano"       "NZE.Bb3.piano"         "Tongan.Bb3.piano"     
 [7] "NZE.D4.piano"          "Tongan.D4.piano"       "NZE.F4.piano"         
[10] "Tongan.F4.piano"       "NZE.Bb2.mezzopiano"    "Tongan.Bb2.mezzopiano"
[13] "NZE.F3.mezzopiano"     "Tongan.F3.mezzopiano"  "NZE.Bb3.mezzopiano"   
[16] "Tongan.Bb3.mezzopiano" "NZE.D4.mezzopiano"     "Tongan.D4.mezzopiano" 
[19] "NZE.F4.mezzopiano"     "Tongan.F4.mezzopiano"  "NZE.Bb2.mezzoforte"   
[22] "Tongan.Bb2.mezzoforte" "NZE.F3.mezzoforte"     "Tongan.F3.mezzoforte" 
[25] "NZE.Bb3.mezzoforte"    "Tongan.Bb3.mezzoforte" "NZE.D4.mezzoforte"    
[28] "Tongan.D4.mezzoforte"  "NZE.F4.mezzoforte"     "Tongan.F4.mezzoforte" 
[31] "NZE.Bb2.forte"         "Tongan.Bb2.forte"      "NZE.F3.forte"         
[34] "Tongan.F3.forte"       "NZE.Bb3.forte"         "Tongan.Bb3.forte"     
[37] "NZE.D4.forte"          "Tongan.D4.forte"       "NZE.F4.forte"         
[40] "Tongan.F4.forte"      

str(dfNotes$langNoteInt)

 Factor w/ 40 levels "NZE.Bb2.piano",..: 22 22 22 22 22 22 22 22 22 22 ...

4 GAMM

4.1 Ordering predictors

We are intersted in the effect of Native Language on the tongue position of musical notes in relation to their intensity. We create three new ordered predictors

dfNotes$native_lg.ord <- as.ordered(dfNotes$native_lg)
contrasts(dfNotes$native_lg.ord) <- "contr.treatment"
dfNotes$tokenPooled.ord <- as.ordered(dfNotes$tokenPooled)
contrasts(dfNotes$tokenPooled.ord) <- "contr.treatment"
dfNotes$note_intensity.ord <- as.ordered(dfNotes$note_intensity)
contrasts(dfNotes$note_intensity.ord) <- "contr.treatment"
dfNotes$langNoteInt.ord <- as.ordered(dfNotes$langNoteInt)
contrasts(dfNotes$langNoteInt.ord) <- "contr.treatment"
dfVowels$native_lg.ord <- as.ordered(dfVowels$native_lg)
contrasts(dfVowels$native_lg.ord) <- "contr.treatment"
dfVowels$tokenPooledNew.ord <- as.ordered(dfVowels$tokenPooledNew)
contrasts(dfVowels$tokenPooledNew.ord) <- "contr.treatment"

We create a new variable (start) when Point of tongue == 1. Our dataset is already ordered by speaker, by note, by intensity and by points of measurements.

dfNotes$start <- dfNotes$points==1

4.2 Running model with no random effects

4.2.1 Model specification

We start by running a model with no random effects. Just to evaluate structure.

if (run_models == TRUE){
  mdl.sys.time1 <- system.time(Notes.gam.noAR.noRandom <- bam(rho_uncut_z ~ langNoteInt.ord +
                          ## 1d smooths
                          s(theta_uncut_z, bs="cr", k=10) +
                          ## 1d smooths * factors
                          s(theta_uncut_z, k=10, bs="cr", by=langNoteInt.ord),
                          data=dfNotes,discrete=TRUE,nthreads=ncores))
mdl.sys.time1
 
# # save model so that it can be reloaded later
 saveRDS(Notes.gam.noAR.noRandom, paste0(output_dir,"/Notes.gam.noAR.noRandom.rds"))
 capture.output(summary(Notes.gam.noAR.noRandom), file =
                  paste0(output_dir,"/summary_Notes.gam.noAR.noRandom.txt"))
}else{
# reload model
Notes.gam.noAR.noRandom = readRDS(paste0(output_dir,"/Notes.gam.noAR.noRandom.rds"))
}

4.2.2 Summary

summary(Notes.gam.noAR.noRandom)

Family: gaussian 
Link function: identity 

Formula:
rho_uncut_z ~ langNoteInt.ord + s(theta_uncut_z, bs = "cr", k = 10) + 
    s(theta_uncut_z, k = 10, bs = "cr", by = langNoteInt.ord)

Parametric coefficients:
                                     Estimate Std. Error t value Pr(>|t|)    
(Intercept)                          213.0172     0.5668 375.837  < 2e-16 ***
langNoteInt.ordTongan.Bb2.piano        7.4726     1.7031   4.388 1.15e-05 ***
langNoteInt.ordNZE.F3.piano            2.4840     0.9487   2.618 0.008838 ** 
langNoteInt.ordTongan.F3.piano         8.8295     0.9296   9.498  < 2e-16 ***
langNoteInt.ordNZE.Bb3.piano           6.0375     0.8775   6.880 5.99e-12 ***
langNoteInt.ordTongan.Bb3.piano        8.9175     0.9513   9.374  < 2e-16 ***
langNoteInt.ordNZE.D4.piano            7.1371     2.1611   3.302 0.000958 ***
langNoteInt.ordTongan.D4.piano        18.4862     2.1446   8.620  < 2e-16 ***
langNoteInt.ordNZE.F4.piano           14.2002     2.3560   6.027 1.67e-09 ***
langNoteInt.ordTongan.F4.piano        10.4258     1.1362   9.176  < 2e-16 ***
langNoteInt.ordNZE.Bb2.mezzopiano      6.4927     0.5078  12.787  < 2e-16 ***
langNoteInt.ordTongan.Bb2.mezzopiano   4.0760     3.8063   1.071 0.284245    
langNoteInt.ordNZE.F3.mezzopiano       2.6786     1.0106   2.650 0.008041 ** 
langNoteInt.ordTongan.F3.mezzopiano   11.2574     1.6972   6.633 3.29e-11 ***
langNoteInt.ordNZE.Bb3.mezzopiano      6.0516     0.4798  12.612  < 2e-16 ***
langNoteInt.ordTongan.Bb3.mezzopiano  23.5412     3.2381   7.270 3.60e-13 ***
langNoteInt.ordNZE.D4.mezzopiano      -0.1294     2.8748  -0.045 0.964084    
langNoteInt.ordTongan.D4.mezzopiano    8.0803     3.6105   2.238 0.025224 *  
langNoteInt.ordTongan.F4.mezzopiano   -5.1305     5.1158  -1.003 0.315916    
langNoteInt.ordNZE.Bb2.mezzoforte      0.1050     0.6632   0.158 0.874196    
langNoteInt.ordTongan.Bb2.mezzoforte   6.1475     1.3390   4.591 4.41e-06 ***
langNoteInt.ordNZE.F3.mezzoforte       6.0076     0.5984  10.039  < 2e-16 ***
langNoteInt.ordTongan.F3.mezzoforte   10.3537     0.6393  16.195  < 2e-16 ***
langNoteInt.ordNZE.Bb3.mezzoforte      5.3990     0.6021   8.966  < 2e-16 ***
langNoteInt.ordTongan.Bb3.mezzoforte   5.0364     0.7853   6.414 1.42e-10 ***
langNoteInt.ordNZE.D4.mezzoforte       6.2444     0.6859   9.104  < 2e-16 ***
langNoteInt.ordTongan.D4.mezzoforte    7.0215     1.1541   6.084 1.17e-09 ***
langNoteInt.ordNZE.F4.mezzoforte       5.0801     0.8427   6.029 1.65e-09 ***
langNoteInt.ordTongan.F4.mezzoforte    7.7414     1.4961   5.174 2.29e-07 ***
langNoteInt.ordNZE.Bb2.forte           4.4002     0.6970   6.313 2.74e-10 ***
langNoteInt.ordTongan.Bb2.forte        3.8021     1.5454   2.460 0.013882 *  
langNoteInt.ordNZE.F3.forte            5.5036     0.6859   8.024 1.02e-15 ***
langNoteInt.ordTongan.F3.forte         5.0509     1.2315   4.101 4.11e-05 ***
langNoteInt.ordNZE.Bb3.forte           5.9310     0.7772   7.631 2.32e-14 ***
langNoteInt.ordTongan.Bb3.forte        6.1954     1.4324   4.325 1.52e-05 ***
langNoteInt.ordNZE.D4.forte            8.6245     2.6400   3.267 0.001087 ** 
langNoteInt.ordTongan.D4.forte        14.6699     2.0012   7.331 2.29e-13 ***
langNoteInt.ordNZE.F4.forte           10.1427     1.6105   6.298 3.02e-10 ***
langNoteInt.ordTongan.F4.forte        16.6990     3.1356   5.326 1.01e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Approximate significance of smooth terms:
                                                        edf Ref.df       F  p-value    
s(theta_uncut_z)                                      8.578  8.769 442.830  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.piano      4.407  5.330  27.024  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.F3.piano          6.530  7.323  27.087  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.F3.piano       4.721  5.668  61.471  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.piano         7.001  7.831  34.162  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.piano      5.352  6.263  93.381  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.D4.piano          6.736  7.424  16.204  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.D4.piano       6.258  6.968 122.810  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.F4.piano          4.220  5.098  13.499 3.05e-13 ***
s(theta_uncut_z):langNoteInt.ordTongan.F4.piano       1.972  2.483  91.361  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.Bb2.mezzopiano    1.124  1.234   2.465   0.0847 .  
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.mezzopiano 2.517  3.099  21.589 1.03e-13 ***
s(theta_uncut_z):langNoteInt.ordNZE.F3.mezzopiano     6.158  7.009   9.795 2.75e-12 ***
s(theta_uncut_z):langNoteInt.ordTongan.F3.mezzopiano  7.492  8.022 113.815  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.mezzopiano    1.013  1.025   0.113   0.7413    
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.mezzopiano 6.128  6.790  40.603  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.D4.mezzopiano     3.881  4.560   5.628 5.83e-05 ***
s(theta_uncut_z):langNoteInt.ordTongan.D4.mezzopiano  4.965  5.708 111.063  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.F4.mezzopiano  2.999  3.667  21.821 1.07e-15 ***
s(theta_uncut_z):langNoteInt.ordNZE.Bb2.mezzoforte    7.709  8.229  22.253  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.mezzoforte 8.482  8.766 192.150  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.F3.mezzoforte     8.676  8.877  15.192  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.F3.mezzoforte  8.678  8.895 213.719  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.mezzoforte    8.500  8.796  36.025  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.mezzoforte 8.830  8.967 196.997  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.D4.mezzoforte     8.778  8.951  63.919  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.D4.mezzoforte  7.837  8.223 188.242  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.F4.mezzoforte     8.471  8.837  44.451  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.F4.mezzoforte  7.355  7.805 132.911  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.Bb2.forte         5.957  6.918   9.256 2.85e-11 ***
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.forte      7.249  7.840 170.201  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.F3.forte          5.907  6.832  13.291  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.F3.forte       7.807  8.330 215.267  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.forte         6.427  7.262  20.115  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.forte      6.997  7.595 167.566  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.D4.forte          6.388  7.108  30.162  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.D4.forte       6.403  7.072 160.842  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.F4.forte          3.414  4.182   9.874 3.71e-08 ***
s(theta_uncut_z):langNoteInt.ordTongan.F4.forte       5.480  6.183  54.626  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

R-sq.(adj) =  0.493   Deviance explained = 49.3%
fREML = 3.1342e+06  Scale est. = 270.46    n = 742800

4.3 Models with random effects

Our second model includes random effects for subject. Given the observed variations between the notes produced by speakers and according to intensity in both languages, we examine two models:

1. A model with interaction between language, note and intensity as fixed effects, and by speaker random smooths
2. A model with interaction between language, note and intensity as fixed effects, and by speaker random smooths with adjustments to note and intensity.

4.3.1 Optimal models

4.3.1.1 Model specification

if (run_models == TRUE){
  mdl.sys.time5 <- system.time(Notes.gam.noAR.Mod1 <- bam(rho_uncut_z ~ langNoteInt.ord +
                          ## 1d smooths
                          s(theta_uncut_z, bs="cr", k=10) +
                          ## 1d smooths * factors
                          s(theta_uncut_z, k=10, bs="cr", by=langNoteInt.ord) +
                          s(theta_uncut_z, subject, bs="fs", k=10, m=1),
                          data=dfNotes, discrete=TRUE, nthreads=ncores))
 mdl.sys.time5
 
# save model so that it can be reloaded later
 saveRDS(Notes.gam.noAR.Mod1, paste0(output_dir,"/Notes.gam.noAR.Mod1.rds"))
 capture.output(summary(Notes.gam.noAR.Mod1), file =
                  paste0(output_dir,"/summary_Notes.gam.noAR.Mod1.txt"))
}else{
# reload model
Notes.gam.noAR.Mod1 = readRDS(paste0(output_dir,"/Notes.gam.noAR.Mod1.rds"))
}
## Model 2
if (run_models == TRUE){
 mdl.sys.time6 <- system.time(Notes.gam.noAR.Mod2 <- bam(rho_uncut_z ~ langNoteInt.ord +
                          ## 1d smooths
                          s(theta_uncut_z, bs="cr", k=10) +
                          ## 1d smooths * factors
                          s(theta_uncut_z, k=10, bs="cr", by=langNoteInt.ord) +
                         ## random smooths by subject, note and intensity
                         s(theta_uncut_z, subject, bs="fs", k=10, m=1, by=noteIntenInt),                
                         data=dfNotes, discrete=TRUE, nthreads=ncores))
 mdl.sys.time6
 
  # save model so that it can be reloaded later
 saveRDS(Notes.gam.noAR.Mod2, paste0(output_dir,"/Notes.gam.noAR.Mod2.rds"))
 capture.output(summary(Notes.gam.noAR.Mod2), file = 
                  paste0(output_dir,"/summary_Notes.gam.noAR.Mod2.txt"))
}else{
# reload model
Notes.gam.noAR.Mod2 = readRDS(paste0(output_dir,"/Notes.gam.noAR.Mod2.rds"))
}

4.3.1.2 Checking k

4.3.1.2.1 Model 1

gam.check(Notes.gam.noAR.Mod1)

Method: fREML   Optimizer: perf chol
$grad
 [1]  4.597180e-10  6.341532e-10  1.194125e-10 -7.896097e-10 -4.394556e-10  3.959424e-10
 [7] -5.696865e-11  1.404099e-10 -1.336464e-11  3.476179e-10  2.997602e-11 -2.401662e-03
[13] -8.296164e-11 -2.472526e-09  1.191092e-10  3.971712e-10 -7.720669e-11  1.769371e-10
[19]  2.947056e-10  6.129501e-10  9.078249e-11 -2.482232e-10  2.998922e-08  2.000697e-10
[25] -1.664499e-09 -7.270717e-11  2.790864e-10 -7.344969e-11 -4.506138e-10  6.367484e-11
[31]  9.202106e-11 -1.727858e-10 -8.074346e-10  7.524115e-11  1.592025e-09 -1.030509e-11
[37]  9.588619e-10 -2.684120e-11  3.654255e-10 -8.963625e-09  1.274270e-10 -2.002344e-08

$hess
           [,1]          [,2]          [,3]          [,4]          [,5]          [,6]
   2.605823e+00  5.012861e-04  5.318245e-03  9.643342e-04  9.327650e-02  3.584340e-03
   5.012861e-04  2.192992e+00 -1.341285e-03  6.568124e-03  1.162043e-03 -5.020418e-03
   5.318245e-03 -1.341285e-03  2.153807e+00  1.002775e-04 -4.731157e-02 -1.266306e-03
   9.643342e-04  6.568124e-03  1.002775e-04  2.372189e+00 -6.452043e-04 -5.359462e-02
   9.327650e-02  1.162043e-03 -4.731157e-02 -6.452043e-04  1.453578e+00 -6.429710e-04
   3.584340e-03 -5.020418e-03 -1.266306e-03 -5.359462e-02 -6.429710e-04  1.151205e+00
   1.124551e-02  6.496255e-04 -1.293983e-02 -9.475523e-04 -1.987892e-02  1.703578e-04
  -2.160924e-03 -3.007292e-03 -8.607714e-05 -4.508422e-03 -6.295946e-04 -2.750255e-02
  -3.561384e-03  2.115890e-05 -1.940893e-02 -1.019878e-04 -8.350815e-03 -2.247671e-04
   1.953489e-03 -5.016340e-03 -1.951715e-03 -9.942705e-03  7.161469e-04 -1.854246e-02
  -3.530021e-03  1.910112e-04  1.151904e-01  1.004015e-03  3.859976e-02  1.143434e-03
   3.276668e-05  1.372479e-05 -9.359491e-07 -2.121244e-05  4.304682e-06 -7.519772e-05
   1.750921e-02  8.374202e-04 -4.415803e-03 -1.288875e-03 -4.123862e-02  8.992806e-04
  -5.139993e-03  2.155965e-02  3.087907e-03 -3.015937e-02 -5.036422e-04 -3.788018e-04
   1.427977e-02  3.398550e-04  1.211289e-02  4.094691e-04  6.696846e-03  1.641448e-03
   3.442201e-03 -5.178372e-03 -1.953005e-03  1.158395e-02  1.932358e-04  7.245857e-03
  -8.163562e-03  1.795375e-04  1.131962e-02 -5.704507e-05 -2.803600e-03  2.814824e-04
  -2.312779e-03  1.808694e-03  4.268026e-04 -6.332382e-03 -2.964937e-04 -1.506917e-02
  -7.177385e-04 -4.180115e-04  5.457187e-05 -2.115630e-04 -1.596727e-04  1.555707e-03
  -5.113316e-02 -4.885984e-03  7.638214e-02  6.826474e-03  9.881642e-02  2.242192e-03
   2.410296e-02 -2.140135e-02 -6.129948e-03 -8.890876e-03 -3.916885e-03 -1.695663e-02
  -2.914458e-01 -3.316918e-03  1.251081e-01  1.124229e-03  2.880843e-01  2.346491e-03
   1.105395e-01  3.968861e-02  1.856194e-02 -2.629679e-02 -4.225838e-02  1.563352e-02
           [,7]          [,8]          [,9]         [,10]         [,11]         [,12]
   1.124551e-02 -2.160924e-03 -3.561384e-03  1.953489e-03 -3.530021e-03  3.276668e-05
   6.496255e-04 -3.007292e-03  2.115890e-05 -5.016340e-03  1.910112e-04  1.372479e-05
  -1.293983e-02 -8.607714e-05 -1.940893e-02 -1.951715e-03  1.151904e-01 -9.359491e-07
  -9.475523e-04 -4.508422e-03 -1.019878e-04 -9.942705e-03  1.004015e-03 -2.121244e-05
  -1.987892e-02 -6.295946e-04 -8.350815e-03  7.161469e-04  3.859976e-02  4.304682e-06
   1.703578e-04 -2.750255e-02 -2.247671e-04 -1.854246e-02  1.143434e-03 -7.519772e-05
   2.001559e+00  2.478908e-04 -6.692286e-03  2.730831e-04  1.000928e-02  1.731293e-06
   2.478908e-04  2.355253e+00 -1.909814e-04  4.697002e-03 -1.187054e-04 -2.043225e-05
  -6.692286e-03 -1.909814e-04  1.240432e+00 -1.585636e-04 -2.832318e-02  1.248037e-06
   2.730831e-04  4.697002e-03 -1.585636e-04  1.860632e+00  7.202741e-04  6.280270e-06
   1.000928e-02 -1.187054e-04 -2.832318e-02  7.202741e-04  1.958161e+00  2.059233e-06
   1.731293e-06 -2.043225e-05  1.248037e-06  6.280270e-06  2.059233e-06  2.386540e-03
  -5.826161e-02  9.198693e-04  1.525312e-03  4.431826e-04  2.033510e-02  2.118459e-06
  -6.210102e-04  7.451712e-03  4.362318e-04  7.517835e-03  4.743696e-05 -4.449718e-05
  -2.460104e-02  3.152826e-04 -5.821206e-02  2.817653e-04 -2.002687e-01  5.916435e-06
   2.826211e-04  1.918360e-04 -2.686261e-04 -1.232852e-03  2.351721e-04  6.897814e-07
  -2.049026e-03  4.153310e-05 -2.077156e-02  1.784596e-04 -4.280206e-02  6.722659e-07
   8.381883e-05 -1.089470e-02 -1.928575e-05  6.913276e-03 -2.762632e-04 -2.753721e-05
  -1.672066e-04  1.973802e-03 -3.020979e-05  8.422714e-04 -6.741772e-05  3.560547e-05
   5.189056e-02  5.451869e-04  4.982341e-03 -2.334708e-03 -1.993118e-01 -5.813571e-06
  -3.526735e-03  2.697949e-02 -1.876196e-03 -5.024370e-02  1.498921e-03 -4.530682e-05
   3.670964e-02  2.526583e-03 -2.039448e-02 -2.221923e-03 -1.406984e-01 -1.177906e-05
  -3.501503e-03  1.207420e-02 -2.070318e-03  8.388635e-03 -1.386570e-02 -7.340560e-05
          [,13]         [,14]         [,15]         [,16]         [,17]         [,18]
   1.750921e-02 -5.139993e-03  1.427977e-02  3.442201e-03 -8.163562e-03 -2.312779e-03
   8.374202e-04  2.155965e-02  3.398550e-04 -5.178372e-03  1.795375e-04  1.808694e-03
  -4.415803e-03  3.087907e-03  1.211289e-02 -1.953005e-03  1.131962e-02  4.268026e-04
  -1.288875e-03 -3.015937e-02  4.094691e-04  1.158395e-02 -5.704507e-05 -6.332382e-03
  -4.123862e-02 -5.036422e-04  6.696846e-03  1.932358e-04 -2.803600e-03 -2.964937e-04
   8.992806e-04 -3.788018e-04  1.641448e-03  7.245857e-03  2.814824e-04 -1.506917e-02
  -5.826161e-02 -6.210102e-04 -2.460104e-02  2.826211e-04 -2.049026e-03  8.381883e-05
   9.198693e-04  7.451712e-03  3.152826e-04  1.918360e-04  4.153310e-05 -1.089470e-02
   1.525312e-03  4.362318e-04 -5.821206e-02 -2.686261e-04 -2.077156e-02 -1.928575e-05
   4.431826e-04  7.517835e-03  2.817653e-04 -1.232852e-03  1.784596e-04  6.913276e-03
   2.033510e-02  4.743696e-05 -2.002687e-01  2.351721e-04 -4.280206e-02 -2.762632e-04
   2.118459e-06 -4.449718e-05  5.916435e-06  6.897814e-07  6.722659e-07 -2.753721e-05
   1.851815e+00 -9.090741e-04 -3.430274e-02  4.064845e-04  8.457589e-03  1.366917e-04
  -9.090741e-04  1.454382e+00  1.256797e-03  1.911629e-02 -1.780102e-04 -1.880340e-03
  -3.430274e-02  1.256797e-03  1.873324e+00 -4.896715e-04 -6.062616e-02  5.770607e-05
   4.064845e-04  1.911629e-02 -4.896715e-04  1.890224e+00  1.432312e-04  1.082619e-03
   8.457589e-03 -1.780102e-04 -6.062616e-02  1.432312e-04  1.921398e+00 -1.762012e-05
   1.366917e-04 -1.880340e-03  5.770607e-05  1.082619e-03 -1.762012e-05  1.781030e+00
  -4.161393e-04  3.278134e-03 -3.566567e-04 -1.990091e-04 -3.602391e-05 -2.225433e-03
   3.316679e-02  7.126852e-03 -1.207792e-01 -3.470945e-03  5.371893e-03 -1.498070e-04
  -5.008181e-03  4.649701e-02 -7.216575e-03 -1.771996e-02 -4.947747e-04  5.352350e-02
   6.907925e-02  1.350858e-03 -8.372983e-02 -5.998502e-04 -3.719846e-02  1.081107e-03
  -2.110858e-03 -1.111350e-01 -1.428449e-02  3.861536e-02 -1.244212e-03  9.540621e-03
          [,19]         [,20]         [,21]         [,22]         [,23]         [,24]
  -7.177385e-04 -5.113316e-02  2.410296e-02 -2.914458e-01  0.1105394775  1.863044e-01
  -4.180115e-04 -4.885984e-03 -2.140135e-02 -3.316918e-03  0.0396886058  4.750001e-03
   5.457187e-05  7.638214e-02 -6.129948e-03  1.251081e-01  0.0185619352 -2.553086e-03
  -2.115630e-04  6.826474e-03 -8.890876e-03  1.124229e-03 -0.0262967891 -4.144312e-03
  -1.596727e-04  9.881642e-02 -3.916885e-03  2.880843e-01 -0.0422583833 -2.578566e-01
   1.555707e-03  2.242192e-03 -1.695663e-02  2.346491e-03  0.0156335216 -1.089849e-03
  -1.672066e-04  5.189056e-02 -3.526735e-03  3.670964e-02 -0.0035015031 -5.941672e-02
   1.973802e-03  5.451869e-04  2.697949e-02  2.526583e-03  0.0120742033 -9.792391e-04
  -3.020979e-05  4.982341e-03 -1.876196e-03 -2.039448e-02 -0.0020703182 -2.866675e-02
   8.422714e-04 -2.334708e-03 -5.024370e-02 -2.221923e-03  0.0083886348  3.133895e-03
  -6.741772e-05 -1.993118e-01  1.498921e-03 -1.406984e-01 -0.0138656964  7.430220e-02
   3.560547e-05 -5.813571e-06 -4.530682e-05 -1.177906e-05 -0.0000734056  1.123796e-05
  -4.161393e-04  3.316679e-02 -5.008181e-03  6.907925e-02 -0.0021108582 -1.120088e-01
   3.278134e-03  7.126852e-03  4.649701e-02  1.350858e-03 -0.1111349849 -5.112802e-03
  -3.566567e-04 -1.207792e-01 -7.216575e-03 -8.372983e-02 -0.0142844890  4.546169e-02
  -1.990091e-04 -3.470945e-03 -1.771996e-02 -5.998502e-04  0.0386153594  2.574431e-03
  -3.602391e-05  5.371893e-03 -4.947747e-04 -3.719846e-02 -0.0012442123 -1.943718e-02
  -2.225433e-03 -1.498070e-04  5.352350e-02  1.081107e-03  0.0095406208 -6.256886e-04
   1.425163e+00  1.307782e-04  2.476317e-02  2.297252e-04  0.0121864611 -5.196789e-04
   1.307782e-04  2.118300e+00  1.098788e-02 -3.037775e-01  0.0293737281  3.713034e-01
   2.476317e-02  1.098788e-02  1.970729e+00  8.219763e-03  0.0090400875 -1.070062e-02
   2.297252e-04 -3.037775e-01  8.219763e-03  1.437187e+00  0.1346754463  4.824060e-01
   1.218646e-02  2.937373e-02  9.040088e-03  1.346754e-01  2.0540735595 -9.799389e-02
          [,25]         [,26]         [,27]         [,28]         [,29]         [,30]
  -0.0109200321  9.424059e-02 -1.443619e-02  5.691162e-02 -2.463889e-04 -1.640267e-02
   0.0124615679  1.866912e-03  2.160027e-02  1.043388e-03  7.155162e-03  9.803022e-04
   0.0021120257 -5.463668e-03  8.540198e-05 -9.989298e-03  2.827651e-03  5.520732e-02
   0.0632387105 -1.202336e-03  3.864184e-02 -6.472053e-04 -3.077838e-02 -1.726245e-03
   0.0017202681 -1.239712e-01 -4.951594e-04 -9.230165e-02 -1.924702e-03  3.567010e-02
   0.0766916309 -4.289512e-04  3.987772e-02 -5.750278e-04 -2.862443e-02  1.385764e-03
   0.0007769706 -2.100504e-02 -7.513207e-05 -8.069038e-03 -9.689961e-04 -3.441119e-02
  -0.0441389939 -5.801406e-04 -8.299130e-04 -5.378681e-04 -3.550867e-03  1.102922e-03
   0.0003516007 -4.750658e-03 -5.744607e-04 -2.745752e-03  1.452595e-04  3.109046e-04
   0.0621633229  1.378810e-03  5.313239e-02  7.894934e-04 -1.088709e-03  3.431238e-04
  -0.0009002990  2.833726e-02 -2.756187e-03  1.655355e-02 -1.375262e-04 -4.514281e-02
  -0.0001276604  4.844118e-06 -4.239892e-04  2.413802e-06 -3.658916e-06  3.570340e-06
   0.0019441917 -4.153693e-02 -1.373462e-03 -1.510895e-02 -1.123951e-03 -1.022002e-01
  -0.0305868910 -1.852941e-03 -4.021388e-02 -1.111598e-03 -2.140134e-02 -7.759210e-04
   0.0012840525  1.525776e-02 -4.461264e-03  9.917581e-03  3.231305e-04 -6.865081e-02
  -0.0009931109  9.634109e-04 -1.151789e-02  5.594816e-04  1.231708e-02  4.106165e-04
  -0.0001904536 -8.620991e-04 -5.088863e-04  1.421052e-04 -2.291423e-04  9.837426e-04
  -0.0133393819 -3.839397e-04 -6.689243e-02 -2.918784e-04 -4.566778e-03  4.306393e-04
   0.0081582262 -1.975529e-04 -1.612569e-02 -5.654399e-05  5.302072e-04 -4.180651e-04
   0.0001193108  1.350013e-01 -8.460544e-03  7.272871e-02  9.057195e-03 -3.973988e-02
   0.1931569170 -3.274669e-03  2.152602e-01 -8.714242e-04 -6.158915e-03 -8.613381e-03
  -0.0040593269  2.753084e-01  8.748913e-04  2.039639e-01  5.650859e-03 -1.152506e-01
  -0.0537739092 -4.901030e-02  1.879801e-02 -2.949261e-02 -4.532730e-02  1.121624e-02
          [,31]         [,32]         [,33]         [,34]         [,35]         [,36]
   1.059369e-02 -1.333259e-02  5.897999e-03 -1.093860e-02  1.853938e-02  1.915607e-03
  -2.690077e-04  1.347504e-03 -5.250485e-03 -1.137380e-03 -9.923597e-03 -6.723875e-04
   1.419763e-03  1.166352e-01  3.965550e-03  4.837698e-02 -1.861190e-04 -9.781191e-02
  -3.174394e-02 -1.537879e-03 -7.034565e-03  1.765196e-03 -1.096471e-04 -1.423215e-04
  -2.994202e-03  1.110578e-02  5.786229e-04  1.021890e-02  2.511200e-03 -3.543226e-02
  -3.605867e-02  3.870685e-04  5.545969e-03 -6.409174e-04  9.184798e-03 -7.738774e-04
  -1.949715e-03 -1.880933e-02 -5.601822e-04  4.259786e-02 -8.558434e-05 -1.312259e-02
  -2.446119e-04  3.873156e-04  3.794879e-03 -6.083196e-04  4.229298e-02  4.049169e-05
  -1.735069e-04 -1.548547e-02  1.090919e-03  9.936337e-03  9.613387e-04 -1.379674e-02
  -7.356313e-03  1.158998e-03 -7.006485e-03 -3.564043e-04 -2.408564e-02 -1.034261e-03
   6.685078e-04 -3.849861e-02  9.702982e-04 -1.692960e-02  1.886471e-03  7.423005e-02
   2.504571e-05  5.893864e-07  9.473749e-05 -3.763200e-06 -2.127863e-05 -9.301557e-07
  -2.790053e-03 -6.854218e-02 -7.098751e-04  7.644030e-02 -1.319401e-03 -3.138178e-02
  -1.588186e-03 -2.716598e-03  2.769055e-03  8.335987e-04  2.564745e-02  1.452863e-03
  -7.741032e-04  3.594905e-02  3.237069e-03  8.863383e-02  2.584819e-03  4.581512e-02
   8.458165e-03  1.450234e-03  9.869051e-03 -3.970545e-04  3.343685e-04 -9.859736e-04
  -2.250131e-04 -1.458794e-02  3.192754e-05  7.107722e-03  3.657848e-04  8.528148e-03
   2.723050e-04 -1.567021e-04  3.220751e-02 -3.301834e-04  2.527442e-02  1.894823e-04
  -4.007053e-03 -1.922369e-04  2.770360e-03  3.110580e-04 -8.917428e-03 -1.299010e-05
   1.242231e-02 -3.656499e-03  9.530481e-03 -1.440387e-01 -5.685523e-04  2.835918e-02
  -2.023581e-02  5.503400e-04 -7.417148e-02  8.334404e-03 -4.894286e-03 -2.996339e-03
   7.701753e-03 -1.245110e-01 -2.059872e-03 -3.908017e-02 -8.834257e-03  7.787371e-02
   7.077793e-03 -4.403908e-03 -1.931575e-02  9.252135e-04  4.375022e-02  1.078930e-02
          [,37]         [,38]         [,39]         [,40]         [,41]         [,42]
  -1.978689e-03 -2.097860e-04 -1.031141e-02 -2.985702e-01  2.134990e-04 -3.328111e+00
  -2.528599e-03 -8.089858e-04 -5.971001e-03 -6.887900e-03 -1.677578e-03 -3.144996e+00
   3.233495e-04 -8.118942e-02 -9.598916e-05  9.727858e-02  8.799713e-03 -2.685404e+00
   7.281175e-03  1.738891e-04  5.526787e-03  2.789436e-02  1.153067e-03 -3.018486e+00
  -7.660370e-05 -2.473419e-02 -1.467088e-03  4.470283e-02 -6.537484e-03 -3.539809e+00
  -6.211941e-03 -5.168141e-05  2.055156e-02 -4.287527e-03 -2.157283e-03 -2.379551e+00
   1.508139e-04 -9.644774e-03 -5.428751e-04  3.613257e-04 -1.516333e-03 -3.292778e+00
  -1.421211e-02  3.815129e-04  7.534758e-03  2.356229e-03  2.549408e-04 -2.788538e+00
  -9.370279e-06 -1.059826e-02 -4.812508e-04  7.773987e-03  2.780245e-05 -1.853104e+00
   1.436896e-02 -1.036918e-03 -7.175085e-03 -9.955920e-03 -1.517894e-03 -2.422667e+00
  -5.565677e-04  4.261505e-02 -5.997492e-04 -5.602651e-02 -5.953180e-03 -2.276551e+00
  -1.418142e-05 -2.264579e-07  2.599112e-04  8.088634e-05 -2.684443e-06 -1.623705e-03
  -2.046961e-05 -2.411149e-02 -5.636199e-04  1.327395e-02 -5.138883e-03 -3.115953e+00
   1.981786e-02  1.819897e-03  1.269015e-02 -8.407391e-04 -1.017179e-02 -2.769772e+00
  -4.160925e-04 -1.114804e-02 -2.073318e-03 -3.954180e-03 -5.762824e-03 -2.400317e+00
  -4.288054e-03 -1.117677e-03 -6.778227e-04  1.381986e-02  7.402344e-05 -3.132712e+00
  -2.982804e-05 -5.836321e-04 -1.987138e-04 -7.342678e-03 -9.380168e-05 -2.033496e+00
  -2.133482e-02  2.723796e-04  1.468471e-02  3.985723e-03  1.118045e-04 -2.069745e+00
  -7.149188e-03 -1.123335e-04 -9.363734e-03  1.181004e-04  2.738036e-04 -1.662068e+00
  -2.048167e-03 -4.382315e-03  1.663099e-03  1.119849e-01  4.693282e-03 -3.031820e+00
   8.758692e-02 -3.884546e-03 -4.406886e-02 -4.738562e-02  3.884010e-03 -2.603059e+00
   2.600781e-04  2.285161e-02  4.033810e-03 -2.819951e-03  1.026599e-02 -3.289092e+00
   5.852510e-02  1.269887e-02  1.292290e-02  1.141696e-01  2.354994e-02 -3.385242e+00
 [ reached getOption("max.print") -- omitted 19 rows ]

Model rank =  590 / 590 

Basis dimension (k) checking results. Low p-value (k-index<1) may
indicate that k is too low, especially if edf is close to k'.

                                                          k'    edf k-index p-value
s(theta_uncut_z)                                        9.00   7.66    1.02    0.93
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.piano        9.00   7.29    1.02    0.93
s(theta_uncut_z):langNoteInt.ordNZE.F3.piano            9.00   6.37    1.02    0.95
s(theta_uncut_z):langNoteInt.ordTongan.F3.piano         9.00   7.04    1.02    0.92
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.piano           9.00   8.08    1.02    0.94
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.piano        9.00   5.76    1.02    0.91
s(theta_uncut_z):langNoteInt.ordNZE.D4.piano            9.00   7.59    1.02    0.92
s(theta_uncut_z):langNoteInt.ordTongan.D4.piano         9.00   6.58    1.02    0.94
s(theta_uncut_z):langNoteInt.ordNZE.F4.piano            9.00   4.71    1.02    0.89
s(theta_uncut_z):langNoteInt.ordTongan.F4.piano         9.00   5.85    1.02    0.93
s(theta_uncut_z):langNoteInt.ordNZE.Bb2.mezzopiano      9.00   5.55    1.02    0.92
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.mezzopiano   9.00   1.01    1.02    0.92
s(theta_uncut_z):langNoteInt.ordNZE.F3.mezzopiano       9.00   7.23    1.02    0.96
s(theta_uncut_z):langNoteInt.ordTongan.F3.mezzopiano    9.00   6.54    1.02    0.90
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.mezzopiano      9.00   5.80    1.02    0.91
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.mezzopiano   9.00   7.27    1.02    0.91
s(theta_uncut_z):langNoteInt.ordNZE.D4.mezzopiano       9.00   5.07    1.02    0.96
s(theta_uncut_z):langNoteInt.ordTongan.D4.mezzopiano    9.00   5.14    1.02    0.93
s(theta_uncut_z):langNoteInt.ordTongan.F4.mezzopiano    9.00   4.32    1.02    0.95
s(theta_uncut_z):langNoteInt.ordNZE.Bb2.mezzoforte      9.00   7.06    1.02    0.92
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.mezzoforte   9.00   6.21    1.02    0.92
s(theta_uncut_z):langNoteInt.ordNZE.F3.mezzoforte       9.00   7.58    1.02    0.93
s(theta_uncut_z):langNoteInt.ordTongan.F3.mezzoforte    9.00   7.77    1.02    0.91
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.mezzoforte      9.00   8.21    1.02    0.95
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.mezzoforte   9.00   6.82    1.02    0.91
s(theta_uncut_z):langNoteInt.ordNZE.D4.mezzoforte       9.00   8.83    1.02    0.90
s(theta_uncut_z):langNoteInt.ordTongan.D4.mezzoforte    9.00   2.92    1.02    0.93
s(theta_uncut_z):langNoteInt.ordNZE.F4.mezzoforte       9.00   8.81    1.02    0.93
s(theta_uncut_z):langNoteInt.ordTongan.F4.mezzoforte    9.00   7.79    1.02    0.92
s(theta_uncut_z):langNoteInt.ordNZE.Bb2.forte           9.00   8.04    1.02    0.92
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.forte        9.00   7.24    1.02    0.92
s(theta_uncut_z):langNoteInt.ordNZE.F3.forte            9.00   7.01    1.02    0.96
s(theta_uncut_z):langNoteInt.ordTongan.F3.forte         9.00   5.58    1.02    0.95
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.forte           9.00   7.38    1.02    0.94
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.forte        9.00   4.95    1.02    0.93
s(theta_uncut_z):langNoteInt.ordNZE.D4.forte            9.00   6.38    1.02    0.93
s(theta_uncut_z):langNoteInt.ordTongan.D4.forte         9.00   6.05    1.02    0.94
s(theta_uncut_z):langNoteInt.ordNZE.F4.forte            9.00   5.47    1.02    0.94
s(theta_uncut_z):langNoteInt.ordTongan.F4.forte         9.00   2.53    1.02    0.89
s(theta_uncut_z,subject)                              200.00 183.48    1.02    0.92

4.3.1.2.2 Model 2

gam.check(Notes.gam.noAR.Mod2)

Method: fREML   Optimizer: perf chol
$grad
 [1] -5.222272e-10  8.350765e-12 -1.231396e-10 -9.292633e-11  1.811995e-11 -5.204726e-11
 [7] -2.396273e-03 -1.588668e-03  1.092700e-10 -2.696555e-03 -4.656371e-03 -3.161904e-11
[13] -7.747108e-04 -1.405542e-12 -1.321252e-03 -2.320477e-11 -3.048507e-03 -2.795972e-04
[19]  3.805001e-11 -5.218573e-03 -4.260883e-03 -2.468364e-04  2.313252e-10  5.363745e-10
[25] -1.447102e-10  1.140523e-10  2.561795e-11 -6.216494e-11 -2.303289e-03 -1.954304e-03
[31] -4.756062e-11 -3.355731e-04  6.134826e-11  8.658185e-12  1.446976e-11  7.443729e-11
[37] -8.418866e-11 -2.753393e-03 -3.874406e-11 -3.331856e-09 -2.475824e-03 -1.514877e-11
[43] -2.442762e-05 -1.058709e-11 -6.909762e-11 -7.964474e-11  3.163019e-05  5.464074e-11
[49] -1.688094e-12  6.773426e-11  3.899170e-11  7.136691e-11 -1.316147e-11  8.218137e-11
[55]  7.227330e-11 -1.309122e-10 -2.919032e-03 -6.818727e-04 -6.394885e-14 -3.251444e-10
[61] -6.520207e-11 -1.413596e-09 -1.934790e-10 -3.347367e-10  2.191847e-10 -1.819132e-10
[67] -1.343290e-10 -3.437748e-10 -1.449276e-10 -1.353300e-10  1.212830e-10 -5.768186e-11
[73]  2.577392e-03 -2.322054e-10 -4.619860e-11  9.106316e-11  2.079813e-03  2.257821e-10
[79] -1.168684e-03  2.734014e-07

$hess
           [,1]          [,2]          [,3]          [,4]          [,5]          [,6]
   3.739219e+00  2.150094e-02  1.233462e-02  2.895253e-02 -1.273032e-02  2.400762e-02
   2.150094e-02  2.639166e-01 -8.111032e-03 -1.233008e-02 -3.178739e-03 -1.116114e-02
   1.233462e-02 -8.111032e-03  2.951862e-01 -1.433907e-02 -3.212881e-04 -9.926526e-03
   2.895253e-02 -1.233008e-02 -1.433907e-02  9.323236e-01  3.690898e-03 -1.701150e-02
  -1.273032e-02 -3.178739e-03 -3.212881e-04  3.690898e-03  1.716922e-01  4.167569e-03
   2.400762e-02 -1.116114e-02 -9.926526e-03 -1.701150e-02  4.167569e-03  7.920162e-01
  -4.565461e-05  1.584773e-05  2.004673e-05  3.093920e-05 -1.012116e-05  2.672589e-05
   3.515328e-04 -8.877418e-05 -1.705543e-04 -2.317190e-04  8.670706e-05 -1.830336e-04
  -1.092927e-02  3.354776e-03  7.709000e-03  1.351940e-02 -9.841460e-03  1.242516e-02
   4.234221e-05 -1.072672e-05 -2.471389e-05 -3.432032e-05  1.498256e-05 -2.788293e-05
  -2.599818e-04  1.426588e-04  1.749421e-04  2.048318e-04  3.980359e-06  1.528377e-04
   8.629746e-03 -3.450466e-03 -3.935068e-03 -7.164342e-03  3.019521e-03 -6.788055e-03
           [,7]          [,8]          [,9]         [,10]         [,11]         [,12]
  -4.565461e-05  3.515328e-04 -1.092927e-02  4.234221e-05 -2.599818e-04  8.629746e-03
   1.584773e-05 -8.877418e-05  3.354776e-03 -1.072672e-05  1.426588e-04 -3.450466e-03
   2.004673e-05 -1.705543e-04  7.709000e-03 -2.471389e-05  1.749421e-04 -3.935068e-03
   3.093920e-05 -2.317190e-04  1.351940e-02 -3.432032e-05  2.048318e-04 -7.164342e-03
  -1.012116e-05  8.670706e-05 -9.841460e-03  1.498256e-05  3.980359e-06  3.019521e-03
   2.672589e-05 -1.830336e-04  1.242516e-02 -2.788293e-05  1.528377e-04 -6.788055e-03
   2.390279e-03  3.583597e-07 -2.570318e-05  4.082966e-08 -3.590105e-07  1.198904e-05
   3.583597e-07  1.629841e-03  1.884914e-04 -4.980448e-07  2.174345e-06 -8.669557e-05
  -2.570318e-05  1.884914e-04  5.074544e-01  2.966495e-05 -1.106578e-04  6.514558e-03
   4.082966e-08 -4.980448e-07  2.966495e-05  2.689582e-03  2.272947e-07 -1.414858e-05
  -3.590105e-07  2.174345e-06 -1.106578e-04  2.272947e-07  4.634628e-03  5.769053e-05
   1.198904e-05 -8.669557e-05  6.514558e-03 -1.414858e-05  5.769053e-05  1.451856e-01
          [,13]         [,14]         [,15]         [,16]         [,17]         [,18]
  -3.094108e-04 -2.592338e-03 -5.450309e-05  2.237123e-02 -2.253035e-05  9.127725e-05
   6.073118e-05 -3.549889e-03  2.923942e-05 -9.369437e-03  7.490036e-06  1.040034e-06
   2.091152e-04  5.121692e-03  6.080866e-05 -7.050686e-03  8.125874e-06 -4.791532e-05
   2.749158e-04 -4.659774e-03  7.740101e-05 -1.125168e-02  1.097623e-05 -6.471130e-05
  -1.608471e-04  1.177553e-02 -2.677789e-05  1.250315e-03 -1.611147e-06  4.646551e-05
   2.100436e-04 -1.009772e-02  6.134533e-05 -9.714236e-03  8.607872e-06 -4.919592e-05
  -5.241099e-07  1.085912e-05 -1.399502e-07  1.585450e-05 -1.837240e-08  1.118429e-07
   3.925781e-06 -4.355060e-05  9.958679e-07 -1.105098e-04  1.196100e-07 -1.065258e-06
  -2.811214e-04  1.279642e-02 -6.920117e-05  5.465505e-03 -6.282922e-06  7.214447e-05
   5.434709e-07 -1.081008e-05  1.395288e-07 -1.421217e-05  1.313144e-08 -1.784474e-07
  -2.994892e-06 -4.081463e-05 -8.854351e-07  1.134384e-04 -1.341655e-07  5.211065e-07
   1.058399e-04 -5.324920e-03  2.913120e-05 -3.477986e-03  3.243601e-06 -2.838238e-05
          [,19]         [,20]         [,21]         [,22]         [,23]         [,24]
  -3.274723e-03  1.866758e-04  7.331886e-05 -4.956315e-05  3.453877e-02  1.067170e-01
  -1.744747e-03  4.614511e-05 -1.885602e-05  4.337098e-06 -6.139318e-03 -4.498531e-04
   3.249535e-03  1.055323e-04 -2.122462e-05  7.599894e-06  2.689770e-03  9.750410e-03
   6.895522e-03  1.018102e-04 -7.467281e-05  1.071937e-05 -5.142913e-03  6.459458e-03
  -5.004158e-03 -2.982802e-05  5.879582e-05 -8.256877e-06  5.860890e-03  5.551644e-03
   6.912837e-03  6.081338e-05 -8.370276e-05  9.405360e-06 -9.307623e-03  5.824547e-03
  -1.573624e-05 -1.930258e-07  9.791086e-08 -2.242368e-08  1.279898e-05 -9.309202e-06
   1.119332e-04  1.272449e-06 -1.018538e-06  1.501242e-07 -2.496786e-05  7.179764e-05
  -9.749873e-03 -9.127035e-05  8.959306e-05 -1.318590e-05  9.252928e-03 -6.982865e-03
   2.104726e-05  1.542482e-07 -2.305786e-07  2.072170e-08 -1.106031e-05  9.652775e-06
  -3.546355e-05 -1.675260e-06 -4.758647e-09 -1.479145e-07  1.179655e-05 -7.907256e-05
   4.581427e-03  2.558805e-05 -4.497993e-05  4.676267e-06 -4.621483e-03  3.420841e-03
          [,25]         [,26]         [,27]         [,28]         [,29]         [,30]
   1.226104e-02 -6.054072e-03  1.643114e-02 -2.336169e-02  2.900606e-04  3.654734e-05
  -1.682313e-03  3.712714e-03 -1.464032e-03 -2.567522e-03 -6.991736e-05 -8.048525e-06
  -3.298002e-03  1.081638e-02  3.254692e-03 -4.507587e-03 -2.146523e-04 -2.835420e-05
  -4.412362e-03  1.621142e-02 -7.211024e-03  4.369636e-03 -3.292289e-04 -3.471046e-05
  -1.277315e-03 -1.160769e-02  1.311938e-02 -1.344791e-02  2.213250e-04  1.737693e-05
  -5.606369e-03  1.393800e-02 -1.111116e-02  7.325124e-03 -2.897145e-04 -2.550622e-05
   9.632156e-06 -2.980039e-05  1.486656e-05 -1.323412e-05  4.965884e-07  4.105754e-08
  -3.379778e-05  2.314273e-04 -1.010155e-04  8.768583e-05 -4.820771e-06 -5.304050e-07
   5.612630e-03 -1.918872e-02  1.350425e-02 -1.159121e-02  3.712965e-04  3.135690e-05
  -1.181121e-05  3.578607e-05 -1.960720e-05  1.647650e-05 -8.965100e-07 -1.000342e-07
   4.532836e-05 -1.484116e-04 -3.353955e-05  4.663180e-05  2.213937e-06  2.420304e-07
  -3.199759e-03  7.253036e-03 -6.152937e-03  4.805620e-03 -1.535832e-04 -1.290980e-05
          [,31]         [,32]         [,33]         [,34]         [,35]         [,36]
   1.526930e-02 -4.219910e-05  3.282684e-02 -3.853697e-02  3.472859e-03 -1.741884e-02
  -1.163835e-02  1.950299e-05 -9.134113e-03  1.444622e-02 -6.039480e-03  5.337883e-03
  -4.511609e-03  3.489681e-05  2.769865e-03  1.221816e-02  2.444394e-03  8.534030e-03
  -1.205271e-02  4.142425e-05 -8.787064e-03  1.924898e-02 -4.714296e-03  1.237418e-02
   5.957920e-03 -1.045021e-05  1.175864e-02 -2.029823e-03  8.086089e-03 -5.363043e-03
  -1.331080e-02  3.062368e-05 -1.241362e-02  1.710445e-02 -7.323821e-03  1.023444e-02
   1.757851e-05 -7.299742e-08  1.382844e-05 -3.117616e-05  7.245068e-06 -2.143790e-05
  -9.925022e-05  5.175822e-07 -7.095231e-05  1.989527e-04 -2.995267e-05  1.585958e-04
   1.043426e-02 -3.186293e-05  1.027197e-02 -1.149384e-02  6.981138e-03 -1.088182e-02
  -1.459439e-05  6.643862e-08 -1.056781e-05  2.593994e-05 -4.512623e-06  2.266273e-05
   9.173466e-05 -5.282637e-07  3.386642e-06 -2.070257e-04 -4.669255e-06 -1.255515e-04
  -5.058320e-03  1.339188e-05 -4.925844e-03  6.864367e-03 -2.798981e-03  4.731649e-03
          [,37]         [,38]         [,39]         [,40]         [,41]         [,42]
   3.300158e-03 -2.440536e-04  1.515509e-02  3.249768e-02  7.883717e-05  5.197862e-02
   2.447627e-03  4.014184e-05 -5.076922e-03  5.956120e-01 -1.721228e-04 -1.898407e-02
  -2.131251e-03  1.323539e-04 -6.210848e-03 -3.462442e-02 -3.113297e-05  2.221233e-01
  -4.455107e-03  2.111527e-04 -8.843505e-03 -3.452407e-02 -3.304677e-05  1.057075e-01
   9.160424e-03 -1.489599e-04  2.395000e-03  1.895220e-02  3.495076e-06 -1.086118e-02
  -3.856131e-03  1.811441e-04 -7.195901e-03 -2.011649e-02 -2.112066e-05 -1.271708e-02
   8.125858e-06 -4.129716e-07  1.319244e-05  5.051300e-05  4.415289e-08  2.123373e-05
  -9.589611e-05  3.017278e-06 -1.064404e-04 -5.598689e-04 -4.016588e-07 -1.751272e-04
   9.106574e-03 -2.395732e-04  6.053812e-03  3.235601e-02  1.547460e-05 -2.679755e-03
  -1.680303e-05  4.398085e-07 -1.576195e-05 -9.106365e-05 -5.228602e-08 -1.840287e-05
   1.049036e-05 -1.887959e-06  8.759181e-05  3.785424e-04  4.103956e-07  2.652843e-04
  -2.593381e-03  9.470925e-05 -3.036827e-03 -1.045297e-02 -7.138049e-06 -2.545782e-03
          [,43]         [,44]         [,45]         [,46]         [,47]         [,48]
   2.635989e-05 -1.481265e-02 -1.837017e-02 -1.114633e-02  8.260741e-06 -3.126815e-02
  -4.521647e-06 -8.644738e-03 -1.017492e-03  9.211840e-03 -1.877284e-06  2.173448e-03
   1.966612e-04 -6.185919e-03 -3.253473e-03 -2.433562e-03 -3.058417e-06 -1.681805e-04
  -8.170723e-05 -3.512804e-03 -1.269106e-03 -2.028655e-03 -2.435522e-06  7.144570e-03
  -3.364442e-06  2.576325e-01  1.399548e-01  8.109595e-03 -1.811051e-06 -1.097688e-02
  -3.082946e-06  2.809239e-01 -2.758261e-02 -4.746043e-04 -1.178611e-06  9.668171e-03
   6.216872e-09  2.834935e-07  1.079971e-06 -1.442796e-03 -4.318859e-07 -1.782087e-05
  -4.989199e-08  1.724867e-05 -4.829399e-06  1.548122e-03 -1.746650e-06  8.565269e-05
  -1.001166e-06 -7.478541e-03 -9.639794e-04  8.992834e-03 -3.506411e-07  3.781570e-01
  -4.961488e-09  5.137608e-06 -2.785202e-07 -3.043866e-05 -3.041573e-09 -2.124152e-03
   7.138113e-08  8.793660e-05  3.457937e-05 -3.682892e-05  3.822189e-08 -3.102375e-05
  -7.281557e-07  5.770568e-04 -5.645545e-05 -2.673023e-03 -2.322994e-07  4.609268e-03
          [,49]         [,50]         [,51]         [,52]         [,53]         [,54]
   1.590416e-03 -6.037523e-02  3.407396e-03 -6.528217e-03  4.872454e-03  1.920957e-03
  -5.190489e-04  2.364491e-02 -1.061469e-03 -2.346540e-03 -9.508350e-04  2.269802e-04
  -8.308846e-04  1.725201e-02  3.694330e-04 -1.568377e-03 -2.006578e-03  7.160268e-03
  -1.026293e-03  2.359367e-02  1.106531e-03  5.286455e-03 -1.476019e-03  1.004500e-02
  -9.283287e-05  7.410302e-03 -3.606302e-03 -1.142598e-02 -8.763141e-04 -3.299802e-03
  -7.837396e-04  1.972909e-02  1.044938e-03  9.191425e-03 -5.715688e-04  9.406625e-03
   1.682437e-06 -3.583193e-05 -2.447212e-06 -2.026220e-05  1.778281e-06 -2.525148e-05
  -1.289709e-05  1.854512e-04  2.593271e-05  6.491624e-05 -1.839957e-05  1.537252e-04
  -2.413037e-03 -3.516757e-03 -3.372920e-03 -1.530203e-02  1.106421e-05 -1.358519e-02
   1.454577e-05  9.286968e-06  4.641634e-06  1.062393e-05 -2.229721e-06  2.532938e-05
   1.163699e-05 -4.011102e-04 -2.134259e-04 -1.071105e-05  2.558746e-05 -1.147918e-04
  -3.689114e-04  2.059533e-01 -3.182110e-02  5.671549e-03 -1.174260e-04  6.179868e-03
          [,55]         [,56]         [,57]         [,58]         [,59]         [,60]
   1.010091e-02 -1.091533e-02  8.952351e-05  5.837364e-06  5.681035e-27 -4.799675e-02
   1.579218e-03  1.303786e-02 -2.750875e-05 -3.477619e-06 -3.309502e-28  1.442764e-03
   3.813152e-03 -2.633467e-03 -3.376736e-05  1.851428e-07  4.129027e-27  2.586508e-03
   1.778430e-03 -1.499573e-03 -4.256115e-05  4.000026e-06  5.240981e-27 -3.043317e-02
   3.197103e-03  1.011468e-02  4.450944e-06 -4.996000e-07 -5.157100e-27  4.287587e-02
   4.079453e-04  4.500366e-05 -3.162064e-05  5.689790e-06  4.002079e-27 -4.339336e-02
  -2.001546e-06  6.555842e-06  6.182609e-08 -1.305771e-08 -9.491574e-30  5.719921e-05
   1.461834e-05 -1.167955e-04 -4.887573e-07  5.995590e-08  8.595399e-29 -4.185080e-04
   2.994104e-04  1.123870e-02  2.177431e-05 -6.505493e-06 -7.465953e-27  5.634868e-02
   1.408804e-06 -2.760286e-05 -6.502173e-08  1.650455e-08  1.425973e-29 -9.434677e-05
  -4.847511e-05 -2.955410e-05  4.852547e-07  5.749122e-09 -4.538535e-29 -1.127599e-04
  -4.304391e-05 -2.776498e-03 -1.186896e-05  4.498253e-06  2.318982e-27 -2.535919e-02
          [,61]         [,62]         [,63]         [,64]         [,65]         [,66]
  -4.424785e-03 -1.811129e-01  1.299360e-03 -1.212064e-01 -7.598785e-03 -7.596663e-02
   1.909691e-03  1.064722e-02  1.335177e-03 -1.345293e-03  6.707868e-04 -5.383923e-03
  -2.698422e-04  1.213013e-02 -3.768402e-04 -3.084354e-03 -1.704748e-03 -2.553358e-03
  -4.323269e-04  2.623169e-02 -4.880348e-04 -1.112692e-03  3.399878e-04 -2.498967e-03
   3.773048e-03 -2.996179e-02  2.581496e-03 -6.572909e-03 -1.681865e-03 -6.982900e-03
  -1.073934e-04  2.069269e-02  1.379867e-04 -3.669756e-03  1.383797e-03 -4.596559e-03
   1.176689e-06 -4.808208e-05  8.658448e-07  3.587242e-06 -1.119748e-06  2.047761e-06
  -2.474975e-05  3.918896e-04 -2.194312e-05  2.963869e-05  4.326119e-08  2.418376e-05
   3.083527e-03 -2.584579e-02  1.523899e-03  5.949873e-03 -9.692417e-04  1.829603e-03
  -4.526544e-06  5.028637e-05 -3.249005e-06 -2.163515e-06  3.618646e-07  1.021648e-06
  -6.088773e-06 -2.456567e-04 -1.713093e-06  2.700462e-05  1.501671e-05  3.058999e-05
  -3.944749e-04  8.808158e-03 -9.715874e-05 -2.655210e-03  6.662207e-04 -1.982237e-03
          [,67]         [,68]         [,69]         [,70]         [,71]         [,72]
  -4.140214e-02 -5.408217e-02 -3.471658e-02  4.463672e-02  1.425549e-02 -9.823463e-02
   1.147488e-03 -6.849389e-03  1.034123e-03  5.427021e-03  6.423612e-04  3.141403e-02
  -4.779388e-03 -7.901484e-03 -5.139484e-03 -6.790491e-03  5.745022e-03  2.210017e-02
   6.232373e-04 -2.035537e-02  3.520483e-04 -1.393202e-02  3.489781e-03  3.978696e-02
  -5.624905e-03  1.269645e-02 -6.673477e-03  1.991099e-02  1.154685e-03 -7.279067e-03
   2.677773e-03 -2.539507e-02  2.276156e-03 -1.109447e-02  1.260406e-03  3.649793e-02
  -2.560926e-06  3.429463e-05 -1.982027e-06  1.769843e-05 -5.203371e-06 -7.184623e-05
   9.414633e-06 -2.281397e-04  7.263905e-06 -2.627127e-04  5.014505e-05  4.212495e-04
  -1.756778e-03  2.974359e-02 -2.044903e-03  1.731120e-02 -2.506116e-03 -2.639375e-02
   1.714324e-06 -5.231244e-05  1.189123e-06 -4.826423e-05  7.232527e-06  5.063540e-05
   4.069185e-05  7.682282e-05  4.509627e-05  1.050162e-06 -6.672325e-05 -4.470010e-04
   1.273785e-03 -1.378734e-02  1.025419e-03 -6.402921e-03  6.152634e-04  1.518484e-02
          [,73]         [,74]         [,75]         [,76]         [,77]         [,78]
   3.583007e-05 -1.166339e-01 -1.247084e-02 -2.492670e-03  1.741338e-05  1.918727e-02
  -1.586354e-06  1.173269e-02  1.607155e-04  1.113173e-02  4.068661e-08 -9.383989e-03
   1.862070e-06  2.940639e-03 -2.438003e-03  1.252470e-02  5.656715e-06  2.744075e-03
   8.309517e-08  1.459837e-02 -1.855806e-04  1.532413e-02  1.435221e-06  1.349165e-02
   1.433563e-06 -6.431046e-03 -2.802059e-03  2.500803e-04 -1.785988e-06 -2.382155e-02
  -2.330525e-07  1.462702e-02  7.623114e-04  1.296754e-02 -1.930749e-06  1.729869e-02
  -2.521855e-10 -2.317073e-05 -4.353767e-07 -2.765917e-05 -1.003544e-09 -4.042186e-05
   3.121772e-10  1.515550e-04 -1.005874e-06  1.587208e-04  3.411741e-08  2.478111e-04
  -1.100606e-06 -5.428721e-03 -8.499488e-04 -1.194704e-02 -2.648436e-06 -3.438186e-02
   7.673594e-10  1.762437e-05  1.484798e-07  2.288596e-05  4.307353e-09  4.807871e-05
  -9.342239e-09 -8.918991e-05  2.315934e-05 -2.069970e-04 -6.800161e-08 -3.114856e-05
   1.676069e-07  5.418989e-03  3.917517e-04  5.661248e-03 -1.209708e-06  1.244319e-02
          [,79]         [,80]
   1.356053e-05 -3.835982e+00
   3.912791e-06 -8.400919e-01
  -3.421644e-06 -5.901800e-01
  -6.489829e-06 -1.083965e+00
   7.424382e-06 -7.473124e-01
  -5.507864e-06 -1.151671e+00
   1.300291e-08 -4.895556e-04
  -1.335029e-07 -5.232824e-03
   9.404771e-06 -7.930655e-01
  -2.364314e-08 -2.109711e-04
   1.982545e-08 -3.152078e-03
  -3.989904e-06 -4.183449e-01
 [ reached getOption("max.print") -- omitted 68 rows ]

Model rank =  4390 / 4390 

Basis dimension (k) checking results. Low p-value (k-index<1) may
indicate that k is too low, especially if edf is close to k'.

                                                            k'      edf k-index p-value
s(theta_uncut_z)                                      9.00e+00 8.67e+00       1    0.35
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.piano      9.00e+00 2.68e+00       1    0.38
s(theta_uncut_z):langNoteInt.ordNZE.F3.piano          9.00e+00 2.18e+00       1    0.36
s(theta_uncut_z):langNoteInt.ordTongan.F3.piano       9.00e+00 3.17e+00       1    0.34
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.piano         9.00e+00 2.49e+00       1    0.35
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.piano      9.00e+00 3.30e+00       1    0.37
s(theta_uncut_z):langNoteInt.ordNZE.D4.piano          9.00e+00 1.01e+00       1    0.39
s(theta_uncut_z):langNoteInt.ordTongan.D4.piano       9.00e+00 1.01e+00       1    0.35
s(theta_uncut_z):langNoteInt.ordNZE.F4.piano          9.00e+00 2.59e+00       1    0.39
s(theta_uncut_z):langNoteInt.ordTongan.F4.piano       9.00e+00 1.01e+00       1    0.39
s(theta_uncut_z):langNoteInt.ordNZE.Bb2.mezzopiano    9.00e+00 1.02e+00       1    0.36
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.mezzopiano 9.00e+00 1.84e+00       1    0.36
s(theta_uncut_z):langNoteInt.ordNZE.F3.mezzopiano     9.00e+00 1.01e+00       1    0.34
s(theta_uncut_z):langNoteInt.ordTongan.F3.mezzopiano  9.00e+00 5.87e+00       1    0.39
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.mezzopiano    9.00e+00 1.01e+00       1    0.41
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.mezzopiano 9.00e+00 4.21e+00       1    0.35
s(theta_uncut_z):langNoteInt.ordNZE.D4.mezzopiano     9.00e+00 1.01e+00       1    0.34
s(theta_uncut_z):langNoteInt.ordTongan.D4.mezzopiano  9.00e+00 1.01e+00       1    0.30
s(theta_uncut_z):langNoteInt.ordTongan.F4.mezzopiano  9.00e+00 4.18e+00       1    0.39
s(theta_uncut_z):langNoteInt.ordNZE.Bb2.mezzoforte    9.00e+00 1.02e+00       1    0.40
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.mezzoforte 9.00e+00 1.02e+00       1    0.36
s(theta_uncut_z):langNoteInt.ordNZE.F3.mezzoforte     9.00e+00 1.00e+00       1    0.34
s(theta_uncut_z):langNoteInt.ordTongan.F3.mezzoforte  9.00e+00 6.82e+00       1    0.41
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.mezzoforte    9.00e+00 5.19e+00       1    0.35
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.mezzoforte 9.00e+00 4.98e+00       1    0.39
s(theta_uncut_z):langNoteInt.ordNZE.D4.mezzoforte     9.00e+00 3.38e+00       1    0.41
s(theta_uncut_z):langNoteInt.ordTongan.D4.mezzoforte  9.00e+00 3.98e+00       1    0.38
s(theta_uncut_z):langNoteInt.ordNZE.F4.mezzoforte     9.00e+00 3.76e+00       1    0.38
s(theta_uncut_z):langNoteInt.ordTongan.F4.mezzoforte  9.00e+00 1.03e+00       1    0.41
s(theta_uncut_z):langNoteInt.ordNZE.Bb2.forte         9.00e+00 1.00e+00       1    0.35
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.forte      9.00e+00 4.61e+00       1    0.33
s(theta_uncut_z):langNoteInt.ordNZE.F3.forte          9.00e+00 1.00e+00       1    0.38
s(theta_uncut_z):langNoteInt.ordTongan.F3.forte       9.00e+00 5.72e+00       1    0.31
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.forte         9.00e+00 3.34e+00       1    0.41
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.forte      9.00e+00 6.21e+00       1    0.34
s(theta_uncut_z):langNoteInt.ordNZE.D4.forte          9.00e+00 1.94e+00       1    0.38
s(theta_uncut_z):langNoteInt.ordTongan.D4.forte       9.00e+00 3.09e+00       1    0.41
s(theta_uncut_z):langNoteInt.ordNZE.F4.forte          9.00e+00 1.02e+00       1    0.38
s(theta_uncut_z):langNoteInt.ordTongan.F4.forte       9.00e+00 1.68e+00       1    0.40
s(theta_uncut_z,subject):noteIntenIntBb2.piano        2.00e+02 8.66e+01       1    0.35
s(theta_uncut_z,subject):noteIntenIntF3.piano         2.00e+02 1.18e+02       1    0.40
s(theta_uncut_z,subject):noteIntenIntBb3.piano        2.00e+02 1.25e+02       1    0.28
s(theta_uncut_z,subject):noteIntenIntD4.piano         2.00e+02 1.21e+02       1    0.40
s(theta_uncut_z,subject):noteIntenIntF4.piano         2.00e+02 5.48e+01       1    0.38
s(theta_uncut_z,subject):noteIntenIntBb2.mezzopiano   2.00e+02 3.32e+01       1    0.36
s(theta_uncut_z,subject):noteIntenIntF3.mezzopiano    2.00e+02 1.24e+02       1    0.39
s(theta_uncut_z,subject):noteIntenIntBb3.mezzopiano   2.00e+02 1.11e+02       1    0.35
s(theta_uncut_z,subject):noteIntenIntD4.mezzopiano    2.00e+02 2.48e+01       1    0.35
s(theta_uncut_z,subject):noteIntenIntF4.mezzopiano    2.00e+02 5.81e-03       1    0.42
s(theta_uncut_z,subject):noteIntenIntBb2.mezzoforte   2.00e+02 1.71e+02       1    0.34
s(theta_uncut_z,subject):noteIntenIntF3.mezzoforte    2.00e+02 1.74e+02       1    0.39
s(theta_uncut_z,subject):noteIntenIntBb3.mezzoforte   2.00e+02 1.71e+02       1    0.34
s(theta_uncut_z,subject):noteIntenIntD4.mezzoforte    2.00e+02 1.59e+02       1    0.35
s(theta_uncut_z,subject):noteIntenIntF4.mezzoforte    2.00e+02 1.45e+02       1    0.36
s(theta_uncut_z,subject):noteIntenIntBb2.forte        2.00e+02 1.28e+02       1    0.38
s(theta_uncut_z,subject):noteIntenIntF3.forte         2.00e+02 1.38e+02       1    0.34
s(theta_uncut_z,subject):noteIntenIntBb3.forte        2.00e+02 1.33e+02       1    0.39
s(theta_uncut_z,subject):noteIntenIntD4.forte         2.00e+02 1.26e+02       1    0.40
s(theta_uncut_z,subject):noteIntenIntF4.forte         2.00e+02 9.38e+01       1    0.34

4.3.1.3 Summary

4.3.1.3.1 Model 1

summary(Notes.gam.noAR.Mod1)

Family: gaussian 
Link function: identity 

Formula:
rho_uncut_z ~ langNoteInt.ord + s(theta_uncut_z, bs = "cr", k = 10) + 
    s(theta_uncut_z, k = 10, bs = "cr", by = langNoteInt.ord) + 
    s(theta_uncut_z, subject, bs = "fs", k = 10, m = 1)

Parametric coefficients:
                                     Estimate Std. Error t value Pr(>|t|)    
(Intercept)                          220.0141    11.5001  19.131  < 2e-16 ***
langNoteInt.ordTongan.Bb2.piano       -9.9031    15.8093  -0.626  0.53105    
langNoteInt.ordNZE.F3.piano            2.0675     0.7197   2.873  0.00407 ** 
langNoteInt.ordTongan.F3.piano        -9.3024    15.6670  -0.594  0.55267    
langNoteInt.ordNZE.Bb3.piano           7.2213     0.7308   9.881  < 2e-16 ***
langNoteInt.ordTongan.Bb3.piano      -10.3189    15.6615  -0.659  0.50998    
langNoteInt.ordNZE.D4.piano           10.3209     1.8874   5.468 4.55e-08 ***
langNoteInt.ordTongan.D4.piano        -4.9010    15.7111  -0.312  0.75508    
langNoteInt.ordNZE.F4.piano           11.2321     1.6297   6.892 5.51e-12 ***
langNoteInt.ordTongan.F4.piano        -9.0475    15.7260  -0.575  0.56507    
langNoteInt.ordNZE.Bb2.mezzopiano      3.6730     0.6970   5.270 1.37e-07 ***
langNoteInt.ordTongan.Bb2.mezzopiano -13.5918    15.7118  -0.865  0.38700    
langNoteInt.ordNZE.F3.mezzopiano       2.3798     0.8886   2.678  0.00740 ** 
langNoteInt.ordTongan.F3.mezzopiano  -10.3496    15.6643  -0.661  0.50880    
langNoteInt.ordNZE.Bb3.mezzopiano      5.3348     0.6835   7.805 5.93e-15 ***
langNoteInt.ordTongan.Bb3.mezzopiano  -0.8983    15.9532  -0.056  0.95509    
langNoteInt.ordNZE.D4.mezzopiano      -0.6941     3.2263  -0.215  0.82966    
langNoteInt.ordTongan.D4.mezzopiano  -12.6210    15.8136  -0.798  0.42481    
langNoteInt.ordTongan.F4.mezzopiano  -23.0377    16.5419  -1.393  0.16371    
langNoteInt.ordNZE.Bb2.mezzoforte      1.4546     0.6074   2.395  0.01662 *  
langNoteInt.ordTongan.Bb2.mezzoforte  -9.8525    15.6534  -0.629  0.52908    
langNoteInt.ordNZE.F3.mezzoforte       3.6283     0.5923   6.125 9.05e-10 ***
langNoteInt.ordTongan.F3.mezzoforte   -9.3077    15.6503  -0.595  0.55203    
langNoteInt.ordNZE.Bb3.mezzoforte      5.0614     0.5947   8.511  < 2e-16 ***
langNoteInt.ordTongan.Bb3.mezzoforte -10.2151    15.6512  -0.653  0.51397    
langNoteInt.ordNZE.D4.mezzoforte       6.4908     0.6300  10.303  < 2e-16 ***
langNoteInt.ordTongan.D4.mezzoforte  -10.3672    15.6563  -0.662  0.50786    
langNoteInt.ordNZE.F4.mezzoforte       5.9797     0.7026   8.511  < 2e-16 ***
langNoteInt.ordTongan.F4.mezzoforte  -12.1222    15.6971  -0.772  0.43996    
langNoteInt.ordNZE.Bb2.forte           3.1020     0.6642   4.670 3.01e-06 ***
langNoteInt.ordTongan.Bb2.forte      -13.0803    15.6747  -0.834  0.40401    
langNoteInt.ordNZE.F3.forte            4.7461     0.6481   7.323 2.42e-13 ***
langNoteInt.ordTongan.F3.forte       -10.8924    15.6571  -0.696  0.48663    
langNoteInt.ordNZE.Bb3.forte           5.8687     0.6985   8.402  < 2e-16 ***
langNoteInt.ordTongan.Bb3.forte      -11.7036    15.6622  -0.747  0.45491    
langNoteInt.ordNZE.D4.forte            8.7341     1.6006   5.457 4.85e-08 ***
langNoteInt.ordTongan.D4.forte        -7.8214    15.6859  -0.499  0.61804    
langNoteInt.ordNZE.F4.forte            8.1765     2.0798   3.931 8.44e-05 ***
langNoteInt.ordTongan.F4.forte        -7.5539    15.6778  -0.482  0.62993    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Approximate significance of smooth terms:
                                                          edf  Ref.df        F  p-value    
s(theta_uncut_z)                                        7.656   8.023    6.950 4.27e-09 ***
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.piano        7.290   7.827   35.268  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.F3.piano            6.371   7.164   24.797  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.F3.piano         7.037   7.749   10.500 2.22e-13 ***
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.piano           8.080   8.662   14.929  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.piano        5.759   6.615    3.364  0.00103 ** 
s(theta_uncut_z):langNoteInt.ordNZE.D4.piano            7.586   7.941   21.841  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.D4.piano         6.577   7.241   20.466  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.F4.piano            4.706   5.592   16.916  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.F4.piano         5.845   6.653   10.761 1.20e-11 ***
s(theta_uncut_z):langNoteInt.ordNZE.Bb2.mezzopiano      5.553   6.454   17.323  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.mezzopiano   1.008   1.015    5.890  0.01487 *  
s(theta_uncut_z):langNoteInt.ordNZE.F3.mezzopiano       7.232   7.780   15.365  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.F3.mezzopiano    6.540   7.178    8.459 9.59e-10 ***
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.mezzopiano      5.801   6.727    8.880 1.92e-10 ***
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.mezzopiano   7.265   7.756    9.457 1.50e-12 ***
s(theta_uncut_z):langNoteInt.ordNZE.D4.mezzopiano       5.067   5.623   30.802  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.D4.mezzopiano    5.139   5.846   15.784  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.F4.mezzopiano    4.324   5.084   18.183  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.Bb2.mezzoforte      7.064   7.670   16.778  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.mezzoforte   6.206   6.869    6.801 7.04e-08 ***
s(theta_uncut_z):langNoteInt.ordNZE.F3.mezzoforte       7.578   8.117   16.954  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.F3.mezzoforte    7.770   8.285   14.297  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.mezzoforte      8.206   8.584   28.224  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.mezzoforte   6.816   7.394    8.443 1.31e-10 ***
s(theta_uncut_z):langNoteInt.ordNZE.D4.mezzoforte       8.833   8.955   47.709  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.D4.mezzoforte    2.918   3.387    2.833  0.02789 *  
s(theta_uncut_z):langNoteInt.ordNZE.F4.mezzoforte       8.813   8.966   35.696  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.F4.mezzoforte    7.788   8.040   41.574  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.Bb2.forte           8.042   8.609   21.858  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.forte        7.245   7.831   27.993  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.F3.forte            7.006   7.728   36.225  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.F3.forte         5.579   6.418    7.946 8.84e-09 ***
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.forte           7.378   8.047   28.265  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.forte        4.945   5.795    5.432 1.25e-05 ***
s(theta_uncut_z):langNoteInt.ordNZE.D4.forte            6.382   7.100   40.090  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.D4.forte         6.046   6.764   17.447  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.F4.forte            5.465   6.143   12.386 2.50e-14 ***
s(theta_uncut_z):langNoteInt.ordTongan.F4.forte         2.532   3.114    3.041  0.02435 *  
s(theta_uncut_z,subject)                              183.480 198.000 7575.846  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

R-sq.(adj) =  0.832   Deviance explained = 83.2%
fREML = 2.7247e+06  Scale est. = 89.598    n = 742800

4.3.1.3.2 Model 2

summary(Notes.gam.noAR.Mod2)

Family: gaussian 
Link function: identity 

Formula:
rho_uncut_z ~ langNoteInt.ord + s(theta_uncut_z, bs = "cr", k = 10) + 
    s(theta_uncut_z, k = 10, bs = "cr", by = langNoteInt.ord) + 
    s(theta_uncut_z, subject, bs = "fs", k = 10, m = 1, by = noteIntenInt)

Parametric coefficients:
                                      Estimate Std. Error t value Pr(>|t|)    
(Intercept)                          208.01898    3.84870  54.049  < 2e-16 ***
langNoteInt.ordTongan.Bb2.piano        7.54672    5.74105   1.315  0.18867    
langNoteInt.ordNZE.F3.piano            4.02764    5.42078   0.743  0.45748    
langNoteInt.ordTongan.F3.piano         6.93585    5.38919   1.287  0.19810    
langNoteInt.ordNZE.Bb3.piano          14.05159    6.91265   2.033  0.04208 *  
langNoteInt.ordTongan.Bb3.piano        4.30408    6.88054   0.626  0.53162    
langNoteInt.ordNZE.D4.piano            9.84484    6.17110   1.595  0.11064    
langNoteInt.ordTongan.D4.piano        11.01255    6.19898   1.777  0.07565 .  
langNoteInt.ordNZE.F4.piano           16.26831    5.72894   2.840  0.00452 ** 
langNoteInt.ordTongan.F4.piano        14.44407    5.66381   2.550  0.01077 *  
langNoteInt.ordNZE.Bb2.mezzopiano      7.98477    7.06301   1.131  0.25826    
langNoteInt.ordTongan.Bb2.mezzopiano   6.69424   10.45428   0.640  0.52196    
langNoteInt.ordNZE.F3.mezzopiano      11.10212    5.50585   2.016  0.04376 *  
langNoteInt.ordTongan.F3.mezzopiano    9.69187    5.48745   1.766  0.07736 .  
langNoteInt.ordNZE.Bb3.mezzopiano     11.99878    6.85456   1.750  0.08004 .  
langNoteInt.ordTongan.Bb3.mezzopiano  -0.04365    7.19182  -0.006  0.99516    
langNoteInt.ordNZE.D4.mezzopiano       7.20483   12.88237   0.559  0.57597    
langNoteInt.ordTongan.D4.mezzopiano   20.82488    8.40065   2.479  0.01318 *  
langNoteInt.ordTongan.F4.mezzopiano   -0.81442   13.56072  -0.060  0.95211    
langNoteInt.ordNZE.Bb2.mezzoforte      6.10145    7.60843   0.802  0.42259    
langNoteInt.ordTongan.Bb2.mezzoforte  10.12249    7.59045   1.334  0.18234    
langNoteInt.ordNZE.F3.mezzoforte      12.86554    8.79113   1.463  0.14334    
langNoteInt.ordTongan.F3.mezzoforte    9.71283    9.29607   1.045  0.29610    
langNoteInt.ordNZE.Bb3.mezzoforte     12.23200    9.04619   1.352  0.17632    
langNoteInt.ordTongan.Bb3.mezzoforte   5.96677    9.14190   0.653  0.51396    
langNoteInt.ordNZE.D4.mezzoforte      15.61654    6.94413   2.249  0.02452 *  
langNoteInt.ordTongan.D4.mezzoforte    7.74809    7.02910   1.102  0.27034    
langNoteInt.ordNZE.F4.mezzoforte       9.54996    6.58711   1.450  0.14712    
langNoteInt.ordTongan.F4.mezzoforte    5.85110    6.58902   0.888  0.37454    
langNoteInt.ordNZE.Bb2.forte           3.97016    5.61254   0.707  0.47933    
langNoteInt.ordTongan.Bb2.forte        3.48826    6.06286   0.575  0.56506    
langNoteInt.ordNZE.F3.forte           11.80956    4.97293   2.375  0.01756 *  
langNoteInt.ordTongan.F3.forte         8.09523    5.26203   1.538  0.12395    
langNoteInt.ordNZE.Bb3.forte          14.04910    5.93719   2.366  0.01797 *  
langNoteInt.ordTongan.Bb3.forte        6.83202    6.20956   1.100  0.27123    
langNoteInt.ordNZE.D4.forte           14.55603    5.86757   2.481  0.01311 *  
langNoteInt.ordTongan.D4.forte         8.32015    6.00647   1.385  0.16599    
langNoteInt.ordNZE.F4.forte           13.72840    5.82409   2.357  0.01841 *  
langNoteInt.ordTongan.F4.forte         5.31625    6.28208   0.846  0.39741    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Approximate significance of smooth terms:
                                                            edf  Ref.df        F  p-value    
s(theta_uncut_z)                                      8.672e+00   8.795  124.695  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.piano      2.680e+00   2.842    2.526 0.074545 .  
s(theta_uncut_z):langNoteInt.ordNZE.F3.piano          2.180e+00   2.303    1.336 0.286047    
s(theta_uncut_z):langNoteInt.ordTongan.F3.piano       3.168e+00   3.370    3.205 0.016257 *  
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.piano         2.495e+00   2.652    0.446 0.609886    
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.piano      3.303e+00   3.520    3.313 0.013339 *  
s(theta_uncut_z):langNoteInt.ordNZE.D4.piano          1.006e+00   1.007    2.006 0.156395    
s(theta_uncut_z):langNoteInt.ordTongan.D4.piano       1.014e+00   1.015    6.000 0.014200 *  
s(theta_uncut_z):langNoteInt.ordNZE.F4.piano          2.586e+00   2.769    2.021 0.088555 .  
s(theta_uncut_z):langNoteInt.ordTongan.F4.piano       1.006e+00   1.007    2.702 0.099877 .  
s(theta_uncut_z):langNoteInt.ordNZE.Bb2.mezzopiano    1.016e+00   1.017    4.971 0.025596 *  
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.mezzopiano 1.837e+00   1.964    3.222 0.036920 *  
s(theta_uncut_z):langNoteInt.ordNZE.F3.mezzopiano     1.014e+00   1.016    1.492 0.224093    
s(theta_uncut_z):langNoteInt.ordTongan.F3.mezzopiano  5.875e+00   6.220    5.182 2.54e-05 ***
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.mezzopiano    1.007e+00   1.008    1.116 0.291994    
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.mezzopiano 4.209e+00   4.467    5.476 0.000378 ***
s(theta_uncut_z):langNoteInt.ordNZE.D4.mezzopiano     1.007e+00   1.007    0.190 0.665001    
s(theta_uncut_z):langNoteInt.ordTongan.D4.mezzopiano  1.007e+00   1.008    3.961 0.046260 *  
s(theta_uncut_z):langNoteInt.ordTongan.F4.mezzopiano  4.181e+00   4.936   23.183  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.Bb2.mezzoforte    1.016e+00   1.021    0.675 0.414308    
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.mezzoforte 1.022e+00   1.027    3.039 0.080044 .  
s(theta_uncut_z):langNoteInt.ordNZE.F3.mezzoforte     1.002e+00   1.002    0.223 0.637699    
s(theta_uncut_z):langNoteInt.ordTongan.F3.mezzoforte  6.817e+00   7.337    3.664 0.000605 ***
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.mezzoforte    5.186e+00   5.894    1.039 0.392773    
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.mezzoforte 4.984e+00   5.547    1.800 0.074241 .  
s(theta_uncut_z):langNoteInt.ordNZE.D4.mezzoforte     3.381e+00   3.810    2.094 0.084827 .  
s(theta_uncut_z):langNoteInt.ordTongan.D4.mezzoforte  3.979e+00   4.352    2.714 0.036509 *  
s(theta_uncut_z):langNoteInt.ordNZE.F4.mezzoforte     3.755e+00   4.078    1.127 0.374094    
s(theta_uncut_z):langNoteInt.ordTongan.F4.mezzoforte  1.032e+00   1.036    7.035 0.007601 ** 
s(theta_uncut_z):langNoteInt.ordNZE.Bb2.forte         1.005e+00   1.005    0.804 0.369561    
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.forte      4.610e+00   4.922    5.088 0.000243 ***
s(theta_uncut_z):langNoteInt.ordNZE.F3.forte          1.002e+00   1.002    1.365 0.243097    
s(theta_uncut_z):langNoteInt.ordTongan.F3.forte       5.717e+00   6.084    3.729 0.001246 ** 
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.forte         3.341e+00   3.571    2.456 0.039868 *  
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.forte      6.215e+00   6.542    4.419 0.000280 ***
s(theta_uncut_z):langNoteInt.ordNZE.D4.forte          1.938e+00   2.025    2.395 0.090963 .  
s(theta_uncut_z):langNoteInt.ordTongan.D4.forte       3.089e+00   3.271    2.845 0.033453 *  
s(theta_uncut_z):langNoteInt.ordNZE.F4.forte          1.016e+00   1.018    1.594 0.207858    
s(theta_uncut_z):langNoteInt.ordTongan.F4.forte       1.676e+00   1.748    4.899 0.019808 *  
s(theta_uncut_z,subject):noteIntenIntBb2.piano        8.660e+01 147.000   49.325  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntF3.piano         1.182e+02 187.000   92.516  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntBb3.piano        1.246e+02 186.000  115.287  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntD4.piano         1.208e+02 188.000   75.218  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntF4.piano         5.479e+01 118.000   22.931  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntBb2.mezzopiano   3.319e+01  58.000   81.579  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntF3.mezzopiano    1.243e+02 188.000  167.108  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntBb3.mezzopiano   1.110e+02 187.000   70.930  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntD4.mezzopiano    2.482e+01  47.000   89.478  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntF4.mezzopiano    5.814e-03   9.000    0.000 0.180312    
s(theta_uncut_z,subject):noteIntenIntBb2.mezzoforte   1.711e+02 200.000 1389.565  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntF3.mezzoforte    1.742e+02 198.000 2877.405  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntBb3.mezzoforte   1.715e+02 200.000 2729.417  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntD4.mezzoforte    1.594e+02 199.000 1191.110  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntF4.mezzoforte    1.452e+02 199.000  451.923  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntBb2.forte        1.280e+02 188.000  108.302  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntF3.forte         1.385e+02 197.000  248.953  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntBb3.forte        1.332e+02 198.000  208.773  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntD4.forte         1.262e+02 198.000   88.992  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntF4.forte         9.380e+01 168.000   43.950  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

R-sq.(adj) =  0.861   Deviance explained = 86.1%
fREML = 2.6589e+06  Scale est. = 74.134    n = 742800

4.4 Model with random effects and AR1 model

So far, our second model Notes.gam.noAR.Mod2 that takes into account the random effect structure of by speaker, by note and by intensity accounted for 86.1% of the variance in the data. It showed some differences between the two languages in terms of how tongue contours are different depending on the note and its intensity. We next need to check the autocorrelation in the residuals and acocunt for these.

4.4.1 Checking ACF

4.4.1.1 ACF full

As we see below, the autocorrelation in the residuals is massive. We need to check whether this is on all predictors or on specific ones.

acf_resid(Notes.gam.noAR.Mod2,main = "Average ACF No.AR",cex.lab=1.5,cex.axis=1.5)

4.5 Running our final model

We finally run our final model that takes into account the autocorrelations in the risiduals

4.5.1 Estimating Rho

We use the following to get an estimate of the rho to be included later on in our model

rho_est <- start_value_rho(Notes.gam.noAR.Mod2)
rho_est

[1] 0.9770577

4.5.2 Model specification

if (run_models == TRUE){
mdl.sys.time6 <-  system.time(Notes.gam.AR.Mod2 <- bam(rho_uncut_z ~ langNoteInt.ord +
                                            ## 1d smooths
                                            s(theta_uncut_z, bs="cr", k=10) +
                                            ## 1d smooths * factors
                                            s(theta_uncut_z, k=10, bs="cr", by=langNoteInt.ord) +
                                            ## random smooths by subject, note and intensity
                                            s(theta_uncut_z, subject, bs="fs", k=10, m=1,
                                              by=noteIntenInt), 
                                      data=dfNotes,AR.start=dfNotes$start, 
                                      rho=rho_est, discrete=TRUE, nthreads=ncores))
  mdl.sys.time6
 
 # save model so that it can be reloaded later
 saveRDS(Notes.gam.AR.Mod2, paste0(output_dir,"/Notes.gam.AR.Mod2.rds"))
 capture.output(summary(Notes.gam.AR.Mod2), file =
                  paste0(output_dir,"/summary_Notes.gam.AR.Mod2.txt"))
}else{
# reload model
Notes.gam.AR.Mod2 = readRDS(paste0(output_dir,"/Notes.gam.AR.Mod2.rds"))
}

4.5.3 Checking ACF

4.5.3.1 ACF full

acf_resid(Notes.gam.AR.Mod2,main = "Average ACF AR", cex.lab=1.5, cex.axis=1.5)

4.5.4 Summary

summary(Notes.gam.AR.Mod2)

Family: gaussian 
Link function: identity 

Formula:
rho_uncut_z ~ langNoteInt.ord + s(theta_uncut_z, bs = "cr", k = 10) + 
    s(theta_uncut_z, k = 10, bs = "cr", by = langNoteInt.ord) + 
    s(theta_uncut_z, subject, bs = "fs", k = 10, m = 1, by = noteIntenInt)

Parametric coefficients:
                                     Estimate Std. Error t value Pr(>|t|)    
(Intercept)                          208.5297     5.1929  40.157  < 2e-16 ***
langNoteInt.ordTongan.Bb2.piano        5.7461     7.5982   0.756  0.44951    
langNoteInt.ordNZE.F3.piano            2.7598     6.3210   0.437  0.66240    
langNoteInt.ordTongan.F3.piano         4.9391     6.4963   0.760  0.44707    
langNoteInt.ordNZE.Bb3.piano          10.1666     7.4838   1.358  0.17431    
langNoteInt.ordTongan.Bb3.piano        1.3756     7.7532   0.177  0.85918    
langNoteInt.ordNZE.D4.piano            5.9943     7.7973   0.769  0.44203    
langNoteInt.ordTongan.D4.piano         3.2696     7.7748   0.421  0.67409    
langNoteInt.ordNZE.F4.piano           11.3212     7.4149   1.527  0.12680    
langNoteInt.ordTongan.F4.piano         8.9149     7.3554   1.212  0.22550    
langNoteInt.ordNZE.Bb2.mezzopiano      2.2355     8.3557   0.268  0.78906    
langNoteInt.ordTongan.Bb2.mezzopiano  -1.2011    12.5009  -0.096  0.92346    
langNoteInt.ordNZE.F3.mezzopiano       5.5038     8.2090   0.670  0.50257    
langNoteInt.ordTongan.F3.mezzopiano    1.5723     8.0063   0.196  0.84431    
langNoteInt.ordNZE.Bb3.mezzopiano      8.4158     7.9947   1.053  0.29249    
langNoteInt.ordTongan.Bb3.mezzopiano   1.5906     8.0432   0.198  0.84324    
langNoteInt.ordNZE.D4.mezzopiano       1.2078    11.7024   0.103  0.91779    
langNoteInt.ordTongan.D4.mezzopiano   15.2717     8.5842   1.779  0.07523 .  
langNoteInt.ordTongan.F4.mezzopiano   -2.8141    10.4186  -0.270  0.78708    
langNoteInt.ordNZE.Bb2.mezzoforte      2.1800     8.1746   0.267  0.78971    
langNoteInt.ordTongan.Bb2.mezzoforte   4.3604     8.1593   0.534  0.59306    
langNoteInt.ordNZE.F3.mezzoforte      12.9677    17.3100   0.749  0.45377    
langNoteInt.ordTongan.F3.mezzoforte   -9.0879    17.4460  -0.521  0.60242    
langNoteInt.ordNZE.Bb3.mezzoforte     21.1206     8.0329   2.629  0.00856 ** 
langNoteInt.ordTongan.Bb3.mezzoforte   3.9376     8.1256   0.485  0.62797    
langNoteInt.ordNZE.D4.mezzoforte      20.5862     8.2622   2.492  0.01272 *  
langNoteInt.ordTongan.D4.mezzoforte    7.0268     8.3811   0.838  0.40181    
langNoteInt.ordNZE.F4.mezzoforte      11.7620     6.9821   1.685  0.09207 .  
langNoteInt.ordTongan.F4.mezzoforte    1.3537     7.1839   0.188  0.85053    
langNoteInt.ordNZE.Bb2.forte           2.2826     7.2333   0.316  0.75233    
langNoteInt.ordTongan.Bb2.forte        1.2167     7.5663   0.161  0.87225    
langNoteInt.ordNZE.F3.forte            9.2558     7.1718   1.291  0.19685    
langNoteInt.ordTongan.F3.forte         5.6065     7.2610   0.772  0.44003    
langNoteInt.ordNZE.Bb3.forte          12.2867     6.8755   1.787  0.07393 .  
langNoteInt.ordTongan.Bb3.forte        7.1173     6.9315   1.027  0.30451    
langNoteInt.ordNZE.D4.forte            9.7201     7.6617   1.269  0.20456    
langNoteInt.ordTongan.D4.forte         8.0273     7.6695   1.047  0.29527    
langNoteInt.ordNZE.F4.forte           10.7750     6.8925   1.563  0.11798    
langNoteInt.ordTongan.F4.forte         0.9717     7.3024   0.133  0.89414    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Approximate significance of smooth terms:
                                                            edf  Ref.df        F  p-value    
s(theta_uncut_z)                                        8.89208   8.934  179.641  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.piano        1.05657   1.068    7.273 0.006511 ** 
s(theta_uncut_z):langNoteInt.ordNZE.F3.piano            1.02064   1.026    0.874 0.343937    
s(theta_uncut_z):langNoteInt.ordTongan.F3.piano         4.91246   5.412    3.585 0.002145 ** 
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.piano           1.01948   1.027    0.109 0.751566    
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.piano        4.29449   4.829    4.182 0.001167 ** 
s(theta_uncut_z):langNoteInt.ordNZE.D4.piano            1.01221   1.014    1.919 0.165399    
s(theta_uncut_z):langNoteInt.ordTongan.D4.piano         2.65733   2.873    5.688 0.001260 ** 
s(theta_uncut_z):langNoteInt.ordNZE.F4.piano            3.77367   4.142    1.107 0.395074    
s(theta_uncut_z):langNoteInt.ordTongan.F4.piano         1.01182   1.014    3.976 0.046151 *  
s(theta_uncut_z):langNoteInt.ordNZE.Bb2.mezzopiano      1.01773   1.022    1.840 0.173802    
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.mezzopiano   4.60731   4.906    3.739 0.003906 ** 
s(theta_uncut_z):langNoteInt.ordNZE.F3.mezzopiano       3.63215   4.121    0.291 0.889135    
s(theta_uncut_z):langNoteInt.ordTongan.F3.mezzopiano    5.90150   6.423    5.518 6.74e-06 ***
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.mezzopiano      1.00711   1.009    0.464 0.498633    
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.mezzopiano   4.90630   5.283    4.675 0.000196 ***
s(theta_uncut_z):langNoteInt.ordNZE.D4.mezzopiano       1.01001   1.013    0.909 0.340842    
s(theta_uncut_z):langNoteInt.ordTongan.D4.mezzopiano    1.40989   1.467    5.604 0.014736 *  
s(theta_uncut_z):langNoteInt.ordTongan.F4.mezzopiano    5.24063   6.028   23.670  < 2e-16 ***
s(theta_uncut_z):langNoteInt.ordNZE.Bb2.mezzoforte      1.00423   1.007    0.582 0.444998    
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.mezzoforte   1.00124   1.002   13.781 0.000204 ***
s(theta_uncut_z):langNoteInt.ordNZE.F3.mezzoforte       6.88004   7.545    4.291 9.49e-05 ***
s(theta_uncut_z):langNoteInt.ordTongan.F3.mezzoforte    6.67813   7.430    3.892 0.010532 *  
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.mezzoforte      1.00395   1.007    0.344 0.561286    
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.mezzoforte   1.00096   1.002   10.457 0.001216 ** 
s(theta_uncut_z):langNoteInt.ordNZE.D4.mezzoforte       1.00063   1.001    0.795 0.372832    
s(theta_uncut_z):langNoteInt.ordTongan.D4.mezzoforte    1.00326   1.005    6.495 0.010750 *  
s(theta_uncut_z):langNoteInt.ordNZE.F4.mezzoforte       1.00931   1.013    2.046 0.153106    
s(theta_uncut_z):langNoteInt.ordTongan.F4.mezzoforte    3.50924   3.975    4.303 0.001666 ** 
s(theta_uncut_z):langNoteInt.ordNZE.Bb2.forte           2.69411   3.181    1.630 0.163113    
s(theta_uncut_z):langNoteInt.ordTongan.Bb2.forte        5.32949   5.851    5.439 2.22e-05 ***
s(theta_uncut_z):langNoteInt.ordNZE.F3.forte            5.38738   6.098    0.866 0.514672    
s(theta_uncut_z):langNoteInt.ordTongan.F3.forte         7.15532   7.634    5.717 2.92e-07 ***
s(theta_uncut_z):langNoteInt.ordNZE.Bb3.forte           6.85022   7.404    3.356 0.003461 ** 
s(theta_uncut_z):langNoteInt.ordTongan.Bb3.forte        5.77196   6.246    4.625 8.16e-05 ***
s(theta_uncut_z):langNoteInt.ordNZE.D4.forte            4.02487   4.433    1.489 0.229501    
s(theta_uncut_z):langNoteInt.ordTongan.D4.forte         5.33783   5.756    4.104 0.001461 ** 
s(theta_uncut_z):langNoteInt.ordNZE.F4.forte            1.01874   1.021    1.214 0.271640    
s(theta_uncut_z):langNoteInt.ordTongan.F4.forte         1.90475   2.014    5.554 0.004046 ** 
s(theta_uncut_z,subject):noteIntenIntBb2.piano        109.29624 149.000   24.077  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntF3.piano         139.58881 188.000   44.333  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntBb3.piano        144.71619 190.000   60.600  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntD4.piano         138.04856 190.000   40.983  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntF4.piano          70.55049 118.000   10.032  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntBb2.mezzopiano    38.69254  59.000   28.475  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntF3.mezzopiano    145.11477 190.000   74.337  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntBb3.mezzopiano   128.75921 189.000   34.134  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntD4.mezzopiano     28.90230  47.000   28.839  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntF4.mezzopiano      0.01025   9.000    0.001 0.344630    
s(theta_uncut_z,subject):noteIntenIntBb2.mezzoforte   183.98586 198.000  790.892  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntF3.mezzoforte    188.26352 199.000 1524.348  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntBb3.mezzoforte   186.25424 198.000 1474.416  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntD4.mezzoforte    181.07227 198.000  609.675  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntF4.mezzoforte    163.33833 200.000  199.277  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntBb2.forte        147.48222 190.000   57.413  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntF3.forte         157.57852 200.000  108.622  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntBb3.forte        153.15114 200.000   79.316  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntD4.forte         143.04055 200.000   39.678  < 2e-16 ***
s(theta_uncut_z,subject):noteIntenIntF4.forte         113.17115 168.000   20.944  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

R-sq.(adj) =  0.855   Deviance explained = 85.4%
fREML = 6.7861e+05  Scale est. = 7.6401    n = 742800