GBTM and k-means analysis of longitudinal data

Paul Schneider
2018-04-30

Step-by-Step tutorial for cluster analysis of longitudinal data using the R-packages crimCV and traj
Simply copy-paste the code chunks or look at the code only version: https://raw.githubusercontent.com/bitowaqr/traj/master/raw_code.R

Setup packages and functions

The chunk below installs and loads the required packages and also load some slightly modified functions from both packages.

url = "https://raw.githubusercontent.com/bitowaqr/traj/master/Setup_n_functions.R"
source(url)

Prepare the data

# prepare data

# Data needs the following format:
  # rows = cases
  # columns = observations at time points
  # no time variables needed

# Enter file path or select manually
  test.data = read.csv("https://raw.githubusercontent.com/bitowaqr/traj/master/test.data.csv")
  
# Otherwise select your own data:
  # test.data = read.csv(file.choose())
  
# reducing the size of the demo data set
  # test.data = test.data[1:100,] # looking at a subset 

# Missing values need to be coded as negative values for crimCV
  test.data[is.na(test.data)] = -0.00001

# data frame to matrix
  df = as.matrix(test.data[,-1])
  rownames(df) = test.data$ID

# resulting data set
  kable(head(df),format="markdown")

t.1	t.2	t.3	t.4	t.5	t.6
707.77330	707.77330	10651.47980	247.44559	247.44559	224.3058
3336.66827	4892.62165	8621.54143	5698.54852	5846.25382	6487.2372
-0.00001	-0.00001	-0.00001	621.12652	3312.48985	9529.3954
1957.65656	763.94261	949.09004	1656.94409	1561.56394	375.1921
-0.00001	-0.00001	-0.00001	-0.00001	-0.00001	1822.2672
257.52852	257.52852	257.52852	257.52852	257.52852	257.5285

GBTM CLUSTERING: crimCV

GBTM modeling using crimCV:

Set a grid to evaluate which model specifications best fit the data
Choose a model and retrieve data
Evaluate group mean estimates and averages

set the grid of models to evaluate

# Set: 
  n.cluster = c(1,2,3) # 1.which k's to evaluate, 
  p.poly = c(1,2,3) # 2. which p's to evaluate, 
  rcv = T # 3. do you want to run cross validation? T/F

Fit models

# Run all models
  cv.eval.list = list()
  index = 0
  for(k in n.cluster)
  {
    sub.index = 0
    index = index + 1
        cv.eval.list[[index]] = list()
        names(cv.eval.list)[k] = paste("Groups",k,sep="_")
        for(p in p.poly)
          {
          print(cat("\n Running models for", n.cluster[k], "Groups, and",p.poly[p],"polynomials..."))
          cat("\n running k=",k,"poly=",p," \n")
          sub.index = sub.index + 1
          temp = crimCV(df,
                        ng = k,
                        dpolyp = p,
                        rcv = rcv,     
                        model = "ZIP"
                        )
      
          cv.eval.list[[index]][[sub.index]] = temp
      
          names(cv.eval.list[[index]])[sub.index] = paste("polynomial",p,sep="_")
          }
  }

Model evaluation

# retrieve AIC, BIC and CV error for each of the models
points.for.AIC.plot = points.for.BIC.plot = points.for.cv.plot = data.frame(x=NA,value=NA,cluster=NA)
for(c in 1:length(cv.eval.list)){
  for(p in 1:length(cv.eval.list[[c]])){
    

    tryCatch({
        cv = ifelse(!is.null(cv.eval.list[[c]][[p]]$cv),cv.eval.list[[c]][[p]]$cv,NA)
        points.for.cv.plot = rbind(points.for.cv.plot,
                                   data.frame(x=p,value=cv ,cluster=c))
    }, error =function(e){})
    
    
    tryCatch({
        aic = ifelse(!is.null(cv.eval.list[[c]][[p]]$AIC),cv.eval.list[[c]][[p]]$AIC,NA)
        points.for.AIC.plot = rbind(points.for.AIC.plot,
                                    data.frame(x=p,value=aic,cluster=c))
    }, error =function(e){})
    
        tryCatch({
        bic = ifelse(!is.null(cv.eval.list[[c]][[p]]$BIC),cv.eval.list[[c]][[p]]$BIC,NA)
        points.for.BIC.plot = rbind(points.for.BIC.plot,
                                    data.frame(x=p,value=bic,cluster=c))
        }, error =function(e){})
    
  }}

points.for.AIC.plot = points.for.AIC.plot[-1,]  
points.for.BIC.plot = points.for.BIC.plot[-1,] 
points.for.cv.plot = points.for.cv.plot[-1,]

AIC.gbtm.plot = ggplot(points.for.AIC.plot) + 
  geom_line(aes(x=x,y=value,col=as.factor(cluster))) +
  #geom_point(aes(x=x,y=value,col=as.factor(cluster))) +
  geom_text(aes(x=x,y=value,col=as.factor(cluster),label=cluster),size=5) +
  xlab("Polynomial") +
  ylab("AIC")

BIC.gbtm.plot = ggplot(points.for.BIC.plot) + 
  geom_line(aes(x=x,y=value,col=as.factor(cluster))) +
  # geom_point(aes(x=x,y=value,col=as.factor(cluster))) +
  geom_text(aes(x=x,y=value,col=as.factor(cluster),label=cluster),size=5) +
  xlab("Polynomial") +
  ylab("BIC")

cv.error.gbtm.plot = ggplot(points.for.cv.plot) + 
  geom_line(aes(x=x,y=value,col=as.factor(cluster))) +
  geom_text(aes(x=x,y=value,col=as.factor(cluster),label=cluster),size=5) +
  #geom_point(aes(x=x,y=value,col=as.factor(cluster))) +
  xlab("Polynomial")  +
  ylab("LOOCV Absolute error") 

plot.legend = get_legend(AIC.gbtm.plot + theme(legend.position = "bottom"))

model.eval.plot = 
  plot_grid(
    plot_grid(cv.error.gbtm.plot + theme(legend.position = "none"),
                            BIC.gbtm.plot + theme(legend.position = "none"),
                            AIC.gbtm.plot + theme(legend.position = "none"),
                            ncol=3),
    plot.legend,nrow = 2,rel_heights = c(10,1))

# plot model evaluation
model.eval.plot

Set the parameters for your model of choice

# Select k and p 
k.set = 3
p.set = 3

# plot details
y.axis.label = "Cost per patient month"
x.axis.label = "Months before death"
plot.title = "Cost per patient month over the last 6 months before death"

Retrive data from your model of choice

# retrieve the final model

  # select model
  ind.k = which(grepl(k.set,names(cv.eval.list)))
  ind.p = which(grepl(p.set,names(cv.eval.list[[ind.k]])))
  # retrieve participants membership
  gbtm.members = data.frame(ID =  rownames(df),
                                 cluster = apply(summary(cv.eval.list[[ind.k]][[ind.p]]),
                                                 1,
                                                 function(x)which(x ==max(x))))

  members.per.cluster = data.frame(table(gbtm.members$cluster))

Plot estimated trajectories

# estimated trajectories

modelled.list = plot(cv.eval.list[[ind.k]][[ind.p]],size=1,plot=F)
modelled.list$time = modelled.list$time 

  model.plot.modelled = 
    ggplot(modelled.list) +
    geom_line(aes(x=time,y=value,col=cluster)) +
    scale_y_continuous(name=y.axis.label) +
    scale_x_continuous(name=x.axis.label) +
    ggtitle(plot.title) +
    scale_color_manual(lab=paste("Group ",members.per.cluster$Var1," (n=",members.per.cluster$Freq,")",sep=""),
                       values=c(2,3,4),
                       name="Estimated group trajectories") +
    theme_minimal()
  
  model.plot.modelled

Retrieve group function terms

# retrieve model function terms with intercept and * for p < .05
  long.test.dat = melt(df)
  names(long.test.dat)  = c("ID","time","value")
  long.test.dat$time = as.numeric(gsub("t.","",long.test.dat$time))
  long.test.dat = merge(long.test.dat,gbtm.members,"ID")
  long.test.dat$cluster = as.factor(long.test.dat$cluster)

  polynomial.model.results = summary(lm(value ~ -1+poly(time,p.set):cluster+cluster, long.test.dat))
  model.spec = round(polynomial.model.results$coefficients[,1],2)
  sig.model.specification = ifelse(polynomial.model.results$coefficients[,4]<0.05,"*"," ")
  model.spec = paste(model.spec,sig.model.specification,sep="")
  model.spec = formatC(model.spec)
  model.spec = matrix(data=model.spec, ncol=p.set+1)
  
  colnames(model.spec) = c("Intercept",paste("Polynomial",1:p.set))
  rownames(model.spec) = c(paste("Group",1:k.set))
  
  kable(model.spec,format="markdown")

	Intercept	Polynomial 1	Polynomial 2	Polynomial 3
Group 1	2760.21*	133725.18*	13492.78*	-5078.66
Group 2	4251.83*	96114.93*	-47743.14*	2492.84
Group 3	685.78*	36158.03*	-10915.68	-1298.57

Plot average group trajectories

# Retrieve observed group trajectories
  long.test.dat$value[long.test.dat$value<0] = NA
  long.test.dat.means = aggregate(value ~ cluster + time, long.test.dat, function(x) mean( x , na.rm = T))
  
  model.plot.from.data = ggplot(long.test.dat.means) +
    geom_line(aes(x=time,y=value,col=cluster)) +
    scale_y_continuous(name=y.axis.label) +
    scale_x_continuous(name=x.axis.label) +
    ggtitle(plot.title) +
    scale_color_manual(lab=paste("Group ",members.per.cluster$Var1," (n=",members.per.cluster$Freq,")",sep=""),
                       values=c(2,3,4),
                       name="Observed group trajectories") +
    theme_minimal()
  
  model.plot.from.data

Setup for combined plot

# give names to clusters? 
cluster.names = paste(
  c("First cluster", 
    "Second",
    "Third"),
    " (n=",format(as.numeric(members.per.cluster$Freq),digits=1),")",sep="")


# set a y limits to have all sub plots on the same scale?
set.y.limit = c(0,15000)

# Group average overview and individual trajectories

pop.average.traj = aggregate(value ~time, long.test.dat, function(x) mean(x,na.rm=T))
model.plot.modelled.plus.pop.average = 
    ggplot() +
  geom_line(data=modelled.list,aes(x=time,y=value,col=cluster,linetype="Estimated")) +
  geom_line(data=pop.average.traj,aes(x=time,y=value,col="Total",linetype="Average")) +  
    scale_y_continuous(name=y.axis.label) +
    scale_x_continuous(name=x.axis.label) +
    ggtitle(plot.title) +
    scale_color_manual(lab=c(paste("Group ",members.per.cluster$Var1," (n=",members.per.cluster$Freq,")",sep=""),
                             paste("Total", " (n=",sum(members.per.cluster$Freq),")",sep="")),
                       values=c(2:(k.set+1),1),
                       name="Estimated group trajectories") +
  scale_linetype_manual(lab=c("Average","Estimated"),values = c(2,1), name="")+
  guides(color = guide_legend(order = 1),
      linetype = guide_legend(order=0)) +
    theme_minimal()


  individual.plot.list = list()
  times.ex = unique(long.test.dat$time)
  for(i in 1:length(unique(long.test.dat$cluster))){
    individual.plot.list[[i]] = 
      ggplot() +
      theme_minimal() +
          geom_line(data=long.test.dat[long.test.dat$cluster==i,],
                    aes(x=time,y=value,group=ID,linetype="Average"),col=i+1,alpha=0.3,size=0.4) +
      geom_line(data=long.test.dat.means[long.test.dat.means$cluster==i,],
                    aes(x=time,y=value,linetype="Average"),col=i+1,size=1) +
       geom_line(data=modelled.list[modelled.list$cluster==i,],
                    aes(x=time,y=value,col=cluster,linetype="Estimate"),col=i+1,size=1) +
        scale_linetype_manual(lab=c("Average","Estimated"),values = c(2,1), name="") +
      theme(legend.position = "none") +
      ylab("") +
            # ggtitle(plot.titles.for.mega[i]) +
            coord_cartesian(ylim=set.y.limit) 
  }
  # individual.plot.list[[length(individual.plot.list)+1]] = get.legend
  
  gbtm.mega.plot = 
    plot_grid(model.plot.modelled.plus.pop.average+
                coord_cartesian(ylim=set.y.limit),
              plot_grid(plotlist=individual.plot.list,ncol=round(k.set/2,0)),ncol=1,rel_heights = c(2,3))
  
  gbtm.mega.plot

k-means clustering

k-means of what?

?step1measures # info shows the 24 measurements on which k-means is performend

The 24 measures are:

Range
Mean-over-time*
Standard deviation (SD)
Coefficient of variation (CV)
Change
Mean change per unit time
Change relative to the first score
Change relative to the mean over time
Slope of the linear model*
R^2: Proportion of variance explained by the linear model
Maximum of the first differences
SD of the first differences
SD of the first differences per time unit
Mean of the absolute first differences*
Maximum of the absolute first differences
Ratio of the maximum absolute difference to the mean-over-time
Ratio of the maximum absolute first difference to the slope
Ratio of the SD of the first differences to the slope
Mean of the second differences
Mean of the absolute second differences
Maximum of the absolute second differences
Ration of the maximum absolute second difference to the mean-over-time
Ratio of the maximum absolute second difference to mean absolute first difference
Ratio of the mean absolute second difference to the mean absolute first difference

Automated function for plotting results

from step1measures --> step2factors --> step3clusters

# Takes a 'data frame' and creates traj cluster analysis with mean and individual plots
    traj.k.mean = function( fill.data.matrix = test.data[,-1],
                            ID = test.data$ID ,
                            nclusters = NULL # set number of clusters, or NULL to let R decide
                       )
      {
      times = dim(fill.data.matrix)[2]
      colnames(fill.data.matrix) = paste("x",1:dim(fill.data.matrix)[2],sep="")
      time.mat =  matrix(data=rep(1:dim(fill.data.matrix)[2],each=dim(fill.data.matrix)[1]),
                         nrow=dim(fill.data.matrix)[1],
                         ncol=dim(fill.data.matrix)[2])
      colnames(time.mat) = paste("t",1:dim(fill.data.matrix)[2],sep="")
      cluster.data = cbind(ID = ID, fill.data.matrix,time.mat)
      
      fill.data.matrix = cbind(ID,fill.data.matrix)
      time.mat = cbind(ID,time.mat)
      
      cat("\n Clustering needas at least 5 observation per case... Cases with fewer observations are being removed...! \n")
      s1 = step1measures(Data = fill.data.matrix, 
                         Time = time.mat,
                         ID = T )
      s2all = step2factors(s1)
      s3all = step3clusters(s2all, nclusters = nclusters)  # 5 clusters
      # clust.build.plot(s3all,y.max.lim =y.max.lim)
      k.membership.table = data.frame(table(s3all$clusters$cluster))
      k.membership = s3all$clusters
      k.k = unique(k.membership$cluster)
      
      
      cluster.plot.df = data.frame(cluster=NA,value=NA,time=NA)
      for(i in k.k){
        mean.per.cluster = colMeans(fill.data.matrix[fill.data.matrix$ID %in% k.membership$ID[k.membership$cluster==i],-1],na.rm=T)
        cluster.name = paste(i," (",length(fill.data.matrix[fill.data.matrix$ID %in% k.membership$ID[k.membership$cluster==i],1]),")",sep="")
        cluster.plot.df=rbind(cluster.plot.df,
                              data.frame(cluster=cluster.name,
                                         value=mean.per.cluster,
                                         time = 1:length(mean.per.cluster)))
      }
      cluster.plot.df = cluster.plot.df[-1,]
      cluster.plot.df$time = as.numeric(cluster.plot.df$time)
      n.per.cluster = as.numeric(table(cluster.plot.df$cluster))
      cluster.plot.df$cluster = as.factor(cluster.plot.df$cluster)
      
      mean.cluster.plot = 
        ggplot(cluster.plot.df) + 
        geom_line(aes(x=time,y=value,col=cluster)) +
        guides(color=guide_legend("Cluster (n)"))  # add guide properties by aesthetic
      
      # Individual plot
      indiv.data = merge(fill.data.matrix,k.membership,"ID")
      indiv.data = melt(indiv.data,id.vars=c("ID","cluster"))
      indiv.data$variable = as.character(indiv.data$variable)
      indiv.data$variable = gsub("x","",indiv.data$variable)
      indiv.data$variable = as.numeric(indiv.data$variable)
      indiv.data$cluster = as.factor(indiv.data$cluster)
      
      lev.clust = levels(indiv.data$cluster)
      n.clust = as.numeric(by(indiv.data$ID,indiv.data$cluster,function(x)length(unique(x))))
      names.clust = data.frame(cluster = lev.clust,n = paste(lev.clust," (",n.clust,")",sep=""))
      indiv.data = merge(indiv.data,names.clust,by="cluster")
      indiv.data$ID = as.factor(indiv.data$ID)

      individual.cluster.plot = 
        ggplot(indiv.data,aes(x=variable,y=value,col= cluster)) + 
            geom_line(aes(group=ID),alpha=0.3,size=0.6) +
            stat_summary(aes(group = cluster), fun.y = mean, geom = 'line', size=1, alpha=1) +
            guides(color=guide_legend("Cluster")) 
      
      
      cluster.analysis = list(
        s1 = s1,
        s2all = s2all,
        s3all = s3all,
        membership = k.membership,
        mean.cluster.plot=mean.cluster.plot,
        individual.cluster.plot = individual.cluster.plot)
      
      return(cluster.analysis)
    }

k-24-means cluster analysis

cluster.analysis.full = traj.k.mean( fill.data.matrix = test.data[,-1], ID = test.data$ID ,nclusters = NULL)

## 
##  Clustering needas at least 5 observation per case... Cases with fewer observations are being removed...! 
## [1] "Correlation of m5 and m6 : 1"
## [1] "Correlation of m12 and m13 : 1"
## [1] "Correlation of m17 and m18 : 1"
## [1] "m6 is removed because it is perfectly correlated with m5"  
## [2] "m13 is removed because it is perfectly correlated with m12"
## [1] "Computing reduced correlation e-values..."

# results
cluster.analysis.full$s3

## Number of observations:  558 
## 
## Cluster distribution:
## 
##   1   2 
## 531  27 
## 
## Measures with max.loading in factors:  m5 m10 m14 m16
## 
## If you report these results, please cite:
## Sylvestre MP, et al. (2006). Classification of patterns of delirium severity scores over time in an elderly population. 
## International Psychogeriatrics,18(4), 667-680. doi:10.1017/S1041610206003334.

cluster.analysis.full$mean.cluster.plot

cluster.analysis.full$individual.cluster.plot

k-means clusters within GBTM clusters

# k-means clusters within GBTM clusters
unique.GBTM.clusters = unique(long.test.dat$cluster)

clusters.within.clusters.plots = list()
set.y.limit = set.y.limit 

for(k in unique.GBTM.clusters){
  temp.pat = unique(long.test.dat$ID[long.test.dat$cluster==k])
  temp.fill.data = test.data[test.data$ID %in% temp.pat,-1]
  temp.k.means = traj.k.mean( fill.data.matrix = temp.fill.data, ID = temp.pat ,nclusters = NULL)
  clusters.within.clusters.plots[[k]] = temp.k.means$individual.cluster.plot + coord_cartesian(ylim=set.y.limit) 
}

## 
##  Clustering needas at least 5 observation per case... Cases with fewer observations are being removed...! 
## [1] "Correlation of m5 and m6 : 1"
## [1] "Correlation of m12 and m13 : 1"
## [1] "Correlation of m17 and m18 : 1"
## [1] "m6 is removed because it is perfectly correlated with m5"  
## [2] "m13 is removed because it is perfectly correlated with m12"
## [1] "Computing reduced correlation e-values..."

## 
##  Clustering needas at least 5 observation per case... Cases with fewer observations are being removed...! 
## [1] "Correlation of m5 and m6 : 1"
## [1] "Correlation of m12 and m13 : 1"
## [1] "m6 is removed because it is perfectly correlated with m5"  
## [2] "m13 is removed because it is perfectly correlated with m12"
## [1] "Computing reduced correlation e-values..."

## 
##  Clustering needas at least 5 observation per case... Cases with fewer observations are being removed...! 
## [1] "Correlation of m5 and m6 : 1"
## [1] "Correlation of m12 and m13 : 1"
## [1] "Correlation of m17 and m18 : 0.999"
## [1] "m6 is removed because it is perfectly correlated with m5"  
## [2] "m13 is removed because it is perfectly correlated with m12"
## [1] "Computing reduced correlation e-values..."

Overiew plot

plot_grid(plotlist = clusters.within.clusters.plots)

References

Jason D. Nielsen (2013). crimCV: Group-Based Modelling of Longitudinal Data. R package version 0.9.3. https://CRAN.R-project.org/package=crimCV

Sylvestre MP, et al. (2006). Classification of patterns of delirium severity scores over time in an elderly population. International Psychogeriatrics, 18(4), 667-680. doi:10.1017/S1041610206003334.

Leffondree, K. et al. (2004). Statistical measures were proposed for identifying longitudinal patterns of change in quantitative health indicators. Journal of Clinical Epidemiology, 57, 1049-1062. doi : 10.1016/j.jclinepi.2004.02.012.

xcmimic/traj