#Analyzing dyadic sequence data - research questions and implied statistical models
Please note, a full R-script of this vignette is provided under the name DySeq_script_v0.1.R in this repository!
This vignette provides a hands-on-tutorial for all analyses covered in
"Analyzing dyadic sequence data - research questions and implied statistical models" by --- blinded for reviewing --- published in --- still in review ---
Please make sure to install all required packages, including "Dyseq", which provides the example data!
Content
- Prerequisite Steps
- packages from CRAN
- DySeq-package from Github oranges
- Example Data
- Loading the Data
- Details on Data
- Graphical Analysis
- State-Distribution Plot
- Entropy Plot
- Number of Transitions
- Research question 1
- Pearson Correlation
- Research Question 2
- Aggregated Logit Models
- step 1: State-Transition Tables
- step 2: Multiple Logit-Regressions
- step 3: Aggregating
- step 4: APIM
- Multi-Level APIM
- Data praparation
- Applying MLM via lme4
- Basic Markov Model
- converting data
- obtaining the transition matrix
- Research question 3
- Hidden Markov Model
- Research question 4
- Mixture Markov Models
- Multi-Level APIM
- OM-distances
- clustering
- interpreting clusters
- Additional function
- number of expected cell problems
- find number of needed time intervals
make sure the following packages are installed:
install.packages("TraMineR") # for graphical analysis and research question 4
install.packages("RColorBrewer") # for grey-shaded graphics
install.packages("gmodels") # must be installed!
install.packages("MASS") # must be installed!
install.packages("survival") # must be installed for research question 3!
install.packages("fpc") # must be installed for research question 4!
install.packages("cluster") # must be installed for research question 4!
install.packages("devtools") # must be installed for installing packages from github
install.packages("lme4") # must be installed for the multi-level APIM
install.packages("lmerTest") # must be installed for the multi-level APIM
install.packages("seqHMM") # must be installed for Markov modelsThe Package "devtools" must be installed and loaded for installing packages from Github Make sure to delete previous Versions of DySeq before installing a new version from GitHub!
remove.packages("DySeq") # remove older Version von DySeq
library(devtools) # load devtools for enabling the install_github()-function
install_github("PeFox/DySeq") # Install DySeq from GitHubAlternatively, you can install DySeq from CRAN. However, the version on CRAN does not feature
the multi-level approach until the next release!
The data stems from a study, which was promoted as a study on close relationship and stress. Each row represents one dyad, each of them containing two sequences. Dyads are 64 heterosexuel couples. The females partners were stressed using the Trier Social Stress Test (TSST; Kirschbaum, Pirke, Hellhammer, 1993). Directly after the stress induction, both partners joint again and the couple was left alone for eight minutes. During this period (a 'fake' waiting condition) the two partners were filmed for 8 minutes divided into 48 intervals of ten seconds length. It was coded if the female partners showed stress communication (SC) within an interval (sequence 1; Colums 50:97) and if the male partner showd dyadic coping reactions (DC; sequence 2; columns 2:49). For rurther insides about dyadic coping and/or stress communication, see Bodenmann (2015).
library(DySeq) # loading the DySeq Package
help(DySeq) # get more information about the package
mydata<-CouplesCope # getting the data
help(CouplesCope) # get more information about the data set
# The majority of the following approaches will need combined states!
# The function StateExpand can combine two sequences into one!
my.expand<-StateExpand(CouplesCope, 2:49, 50:97)Following objects from the previous sections are needed:
- mydata (the example data)
- my.expand (the combined sequences)
Following Packages are needed:
- DySeq
- TraMineR
- RColorBrewer (for advanced colors)
library(TraMineR) # awesome package for sequence analysis in general!
library(RColorBrewer) # more colours!
citation("TraMineR") # please cite Packages if you use them!
citation("RColorBrewer")# Create labels for plot
couple.labels <-c("none", # no reaction
"SC only", # only stress communication
"DC only", # only dyadic coping
"SC+DC") # stress and dyadic
# Create a stslist object (TraMineR S3-Class)
couple.seq <- seqdef(my.expand, # the combined states
labels = couple.labels) # the label
# State-Distribution plot
seqdplot(couple.seq,
cex.legend=0.8) # adjust size of the legend
# Alternatively a grey version (using RColorBrewer)
# And legend aligned right
attr(couple.seq , "cpal") <- brewer.pal(4, "Greys") # see figure 2
seqdplot(couple.seq, cex.legend=0.8, withlegend="right")Entropy <- seqstatd(couple.seq)$Entropy
plot(Entropy, main= "Entropy", col="black", xlab = "Time in 10 sec. intervall", type ="l")SeqL<-seqtransn(couple.seq)
hist(SeqL, main="Number of transitions", xlab="State-transitions")Note: The TraMineR-Package features a lot of other useful functions for analyzing and plotting sequence data. Gabadinho and colleagues put great effort in providing a vast, comprehensive, yet easy-to-read vignette. Please keep in mind to cite packges!
vignette('TraMineR-state-sequence') # get the TraMineR Vignette
citation("TraMineR") # get information on how to cite TraMineRD. Research Question 1: Is there an association between a particular behavior by one and the reaction by the other partner?
Following objects from the previous sections are needed:
- mydata (the example data)
Following Packages are needed:
- DySeq
# NumbOccur counts how often a certain behavior is shown within each sequence
SC.sumscores<-NumbOccur(x=mydata[,2:49], # col: 2:49 represent stress communication (SC)
y=1, # 0=no SC; 1=SC was shown
prop=FALSE) # absolute (TRUE) or relative (FALSE) frequency?
DC.sumscores<-NumbOccur(mydata[,50:97], 1, prop=FALSE) # Same for dyadic coping (DC)
# Correlation of both frequencies
cor.test(SC.sumscores, DC.sumscores)
# plotting the correlation
plot(SC.sumscores, DC.sumscores, ylab="Number of DC", xlab="Number of SC")
abline(lm(SC.sumscores~DC.sumscores))
Following objects from the previous sections are needed:
- mydata (the example data)
- my.expand (the combined sequences)
- couple.seq (a a stslist object from TraMineR)
Following Packages are needed:
- DySeq
- lme4 (for MLM approach)
- lmerTest (for MLM approach)
- TraMineR (for the basic Markov model)
- seqHMM (For the basic Markov model)
First step is transforming the sequences into transition tables. That can be done by DySeq's function StateTrans, which stores the transition tables in a list of the class 'state.trans'.
my.trans<-StateTrans(my.expand, # the combined sequences
first=FALSE) The argument 'first' specifies which sequence should be used as a dependend variable. Dependend variable in this context means, which behavior is shown at time t (columns), and dependend on the previous behavior at t-1 (rows). When my.expand was created via the StateExpand-function (see B.), SC was defined as the first sequence and DC as the second sequence. Therefore, DC is now the dependend variable of the transition tables.
# Just print a state.trans object to get the relative frequencies accross all dyads!
my.trans
# A single case can be plottet by using the [[]] operator
my.trans[[1]] # inspects the first table!
# If the original data.frame containes a ID-variable, the following can be used:
ID<-mydata$code # stores the ID-variable in the object ID
my.trans[[which(ID==129)]] # inspects the dyad with ID=129
# If relative frequencies are preferred for single case analysis, just divide the transition table by its sum.
# The following shows an example for the 41th transition table, which belongs to the couple with ID 129.
# round(x,3) rounds the frequencies to three digits
round(my.trans[[41]]/sum(my.trans[[41]],3))Second step is computing a logit model for each dyad. That is a very cumbersome procedure, yet the Package DySeq provides the function LogSeq for doing so with only one function!
my.logseq<-LogSeq(my.trans, # a list containing transition plots (from step 1)
delta=0.5, # adds a frequency of .5 on every cell. Needed if zero cells exist!
single.case=TRUE) # if TRUE, stores single case results (and the function becomes slower)If a researcher is interested in single case analysis, the function single.LogSeq() can be used to obtain these in a ready-to-interpret table! p-value test whether betas are different from zero or not. Note that the p-value for the intercept is not implemented yet!
# Single case analsis for transition plot 41 (aka couple 129)
single.LogSeq(my.logseq, 41)Next step is aggregating the results. This is automatically done by printing the results of the last step! No further functions are needed! Again, p-values test whether the betas are not equal zero.
my.logseq # prints the aggrregated results!Plotting the results will produce an interaction-plot. Mapping the probabilities of showing the dependend variable against the combinations of previous behavior:
plot(my.logseq)
Re-running the procedure a second time but switching the dependend variable will result in the same analysis but with stress communication as dependent variable.
my.trans.SC<-StateTrans(my.expand, first=TRUE) # This time, the first sequence is the DV
my.logseq.SC<-LogSeq(my.trans.SC, delta=0.5)
my.logseq.SC # Contains actor and partner effects for the femaleOur previous results from step 3 contained the effects for men, while our second results (my.logseq.SC) contain the effects for women. Thus, combining both results in a APIM as shown in the article.
The first step is data preparation. Each transition must be recoded to be a single observation within a nested data structure. Transitions are level-1 observations, which are nested within dyads. That can be achieved by one single function from the DySeq-Package!
# Make sure all needed packages are loaded!
# DySeq is needed for this section too, but was already loaded before
library("lme4") # lme4 is a package for fitting multi-level models
library("lmerTest") # lmerTest adds p-values to several statistics in lme4
# ML_Trans transforms dyadic sequences into multi-level data!
ML_data<-ML_Trans(data=CouplesCope, # The data, which should be used!
first=2:49, # The sequence, which should be used as the DV
second=50:97) # The sequence, which serves as the IV!If the data should be used to apply a multi-level APIM, transitions must be recoded first into lagged actor (LA) and lagged partner (LP) effects. The function MLAP_Trans does so.
MLAP_data<-MLAP_Trans(ML_data) # ML_data must be the output of ML_Trans!Labels should be added or else the procedure can become confusing later on.
names(MLAP_data)[1]<-"sex"
MLAP_data$sex<-as.factor(MLAP_data$sex)
levels(MLAP_data$sex)<-c("female", "male")MLAP_Trans uses dummy-coding per default. However, for the purposes of an APIM effect-coding is better, because it is easier to interpret especially if effects should be compared between members of dyads. Furthermore, effect coding was also used in the article (-----). So for the sake of comparable results, we will stick to it for this case.
MLAP_data$Partner[MLAP_data$Partner==0]<-(-1)
MLAP_data$Actor[MLAP_data$Actor==0]<-(-1)The second step is applying and testing MLM-models. There is a vast and increasing number of packages in R, wich can run multi-level models. However, lme4 became one of the best known packages for multi-level analysis, and an increasing number of tutorials are spreading through the net. Thus, we will stick to lme4, too. Do not forget to cite lme4 and lmerTest if you use this approach! (Again, use the citation()-function to get additional information on how to cite a particular package!
The following shows the most complex modell, which is possible to estimate. There will be some estimation problems with this model, such as that estimates will be very close to thei boundaries. However, it will serve as an example to explain the function's arguments.
Warning: the actual estimation of these MLMs can be very time consuming. The more complex, the longer the estimation procedure (estimation of the most complex model can take up to ten minutes).
set.seed(1234) # setting SEED for replication purposes!
fit<-glmer(DV~1+sex+Actor+Partner+Actor*Partner+ # intercept, Actor, Partner and interaction effect for the referrence group
sex*Actor+sex*Partner+sex*Actor*Partner+ # difference for the non-referrence group (here males/DC)
(1+sex+Actor+Partner+Actor*Partner+ # Random effects for intercept, Actor, Partner and interaction effect
sex*Actor+sex*Partner+sex*Actor*Partner|ID), # Random effects for the differences between the DVs (SC vs. DV)
data=MLAP_data, # the actual data
family=binomial) # provides Link-function, so logistig regression is applied!
summary(fit) # Provides summary of results
AIC(fit) # Akaike information criterion (AIC; a comparative fit index)
BIC(fit) # Bayesian information criterion (BIC; a more conservative fit index)Models can be compared using AIC or BIC (comparative fit-indices). Smaller values indicate better model fit. In our case, a model containting random effects for the intercept, actor, partner effects were the best fitting one.
# Random Actor und Partner Effekte
set.seed(1234)
fit<-glmer(DV~1+sex+Actor+Partner+Actor*Partner+
sex*Actor+sex*Partner+sex*Actor*Partner+
(1+Actor+Partner|ID),
data=MLAP_data,
family=binomial)
summary(fit)
AIC(fit)
BIC(fit)For a closer interpretation of this model, inspect the article (----).
The TraMineR-package provides a function to fit a basic Markov model. The couple.seq object is needed, which was created in the 'graphic analysis' section!
round(seqtrate(couple.seq),2) # the round command is optional, and rounds the transition matrix to two digitsThe second way, using the seqHMM-package is a little bit more complicated at the first glance. However, it is worth to try this package too, because hidden Markov and mixture Markov model follow the exact same logic. Please cite seqHMM if you are using it for your analysis!
# First of all, starting values for the transition matrix must be specified
# each row must add up to one! In this case we assume that each transition is equally likely!
# One of the advantages of seqHMM is, that restrictions can be added. For example, setting a value to zero
# will restrict it to zero, same goes for setting a vaue to 1. All other values will be used as starting values for
# the estimation process.
mytrans<-matrix(.25, 4,4) # a 4*4 Matrix with all starting transition probabilties equals .25
# other starting values can be used, but all rows must add up to 1!
# Same for the initial probabilities
myinit<-c(.25,.25,.25,.25)
# Builds the acutal model
mybuild<-build_mm(couple.seq, # the data
mytrans, # starting values for transitions
myinit) # starting values for the initial probabilities (probabilities at t=0)
# Fits the model on the data
fit<-fit_model(mybuild)
# print results
fit
# get comparative fit indices!
AIC(fit$model) # Akaike information criterion (AIC; a comparative fit index)
BIC(fit$model) # Bayesian information criterion (BIC; a more conservative fit index)Note: Helske & Helske did a great job on providing a very user-friendly package. Again, please cite packages you use. Reading the vignette is strongly recommended, which can be assessed by typing vignette('seqHMM').
Research Question 3: Is there an underlying dyadic process, which might account for the observed behavior?
Needs following objects: couple.seq (created in the graphics section, using TraMineR-package)
Following packages needed: seqHMM
This question can be answered by a hidden Markov model using seqHMM. Please cite seqHMM if you are using it for your analysis! First we have to specify starting values for the hidden chain by doing so, we also decide on the number of latent states!
myhtrans<-matrix(c(.50, 0, 0.5, 1), 2,2)
# In this example we created a 2*2 matrix, the second
# state is an absorbing one, because we resticted the transition
# probabilities in a way that one cannot leave the second state!
my_emission<-matrix(c(.25),2,4)
# the emission matrix has one row per latent state
# and four columns for eacht observed state
myinit2<-c(.50, .50)
# Because the hidden chain is now a 2*2 matirx, we only need two initial probabilities
my_hmodel<-build_hmm(couple.seq, # the data
myhtrans, # starting values for transition probabilities for the hidden chain
my_emission, # starting probabilties for the emissions
myinit2) # starting probabilities for the initial probabilities for the hidden chain
fit2<-fit_model(my_hmodel)
fit2 # print the resultsResearch question 4: Are there latent groups of dyads, which might account for observing different reaction patterns?
This question can be answered by a mixture Markov model or by the OM-procedure. We start with
Needs following objects: couple.seq (created in the graphics section, using TraMineR-package)
Following packages needed: seqHMM
First of all we have to specify starting values for the each chain so we need as many chains as there a latent groups. For the sake of examplification we will assume two groups:
# These are transition matrices for the observed states,
# therefore they have the as many rows and columns as observed states:
mytrans1<-matrix(c(.25), 4,4)
mytrans2<-matrix(c(.25), 4,4)
# We have to define two matrices, because we assume two latent groups. Both of which have to their own chain.
# The transition matrices must be placed into a list, before building the model
mymixtrans<-list(mytrans1, mytrans2) # placing the matrices into a list
# There are no emissions because there are no hidden states!
# Still, we need to set the starting values for the initial probabilities. One for each chain!
myinit1<-c(.25, .25, .25, .25)
myinit2<-c(.25, .25, .25, .25)
# again both must be placed inside a list:
mymixinit<-list(myinit1, myinit2)
my_mmodel<-build_mmm(couple.seq, # the data
mymixtrans, # starting values for the two chain`s transition probabilities
mymixinit) # starting values for the two chain`s initial probabilities
fit3<-fit_model(my_mmodel,
global_step=TRUE, # additional arguments for the optimizier
local_step=TRUE, # to avoid local maximum
control_em=list(restart = list(times=10))) # rerunning the optimizer several times for avoiding
# a local maxima. That is simplified, the optimizer
# finds the wrong best fitting solution.
# times=10 specifies that the procedure runs 10 times
# and returns the best fitting one.
fit3Objects from previous sections needed:
- mydata
- my.expand
- couple.seq
- my.trans
- my.trans.SC
Packages needes:
- DySeq
- TraMineR
Substitution-cost-matrix is derived by Gabadinho's TRATE-Formula
submat <- seqsubm(couple.seq, method = "TRATE")The substitution-cost-matrix is then used to calculate the distance-matrix
dist.oml <- seqdist(couple.seq, method = "OM", sm = submat)First of all, the optimal number of clusters needs to be determined:
plot(pam(dist.oml, pamk(dist.oml)$nc))First Graph: [note: not shown in the article]
The distances describes a number of sequences minus 1 dimensional space, and therefore it is possible to plot them. However, the first plot projects the dissimilarities in a two-dimensional space. By this, information is lost and the distances shown in the plot won't fit the distances in distance-matrix perfectely. Yet this plot provides us with a first limpse into the cluster structure: here we see two clusters. However they are not completely distinct, because we caan see that they overlap a little bit! Furthermore, the smaller cluster (left) seem to be more homogenous (points are very close to each other) and the bigger one (right) seem to be more heterogenous (more dispersion).
Second Graph: silhouette plot [note: not shown in the article]
The silhuette (s) is the distance of one object (sequence) to its clusters centroid minus the distance to the nearest clusters centroid devided by the maximal possible distance. Therefore, the higher the mean s-values of a cluster, the stronger the found cluster structure.
[see: Peter J. Rousseeuw, Silhouettes:
A graphical aid to the interpretation and validation of cluster analysis,
Journal of Computational and Applied Mathematics, Volume 20, 1987, Pages 53-65]
The plot contains the following information:
-
2 clusters are identified
-
First one includes 38 observations with s=.43
-
Second one includes 26 observations with s=.59
-
The overall mean s is .5
-
typical rules of thumb are: 1 >= s >.75 strong structure
0.75 >= s > 0.50 medium structure
0.50 >= s > 0.25 week structure
0.25 >= s > 0 no real structure
The mean silhuette value indactes an overall weak cluster structure. Therefore, additional methods should be used to determine the optimal number of clusters. Here, for example, a screeplot and adendrogramm:
Screeplot: (Indicating 1 or 2 clusters)
wss <- (nrow(dist.oml)-1)*sum(apply(mydata,2,var))
for (i in 2:15) wss[i] <- sum(kmeans(mydata, centers=i)$withinss)
plot(1:15, wss, type="b", xlab="Number of Clusters", ylab="Within groups sum of squares")Dendrogramm: (Indicating 2, maybe 3, Clustersolution)
plot(agnes (dist.oml, diss=TRUE, method = "ward"), which.plots=2)After determing the number of clusters, the ward-algorithm can be used for clustering.
clusterward1 <- agnes (dist.oml, diss=TRUE, method = "ward") # the algorithm
cluster2 <- cutree (clusterward1, k=2) # saving the two-cluster solution
# adding labels
cluster2fac <- factor (cluster2, labels = c("cluster 1; fast coper", "cluster 2; slow coper"))Please note that the number of observational units of both clusters doesn't match exactly the n from the silhuette test! This is because the latter depends on another algorithm. Typically they provide very similar cluster solutions. However, in this case one sequence was assigned differently. The first plot from line 478 shows that exactly one sequence is located in the overlapping area between both clusters. Because of that, some algorithms assign it to the first, other to the second cluster. However, changing its cluster - or deleting that observation - would not lead to other results.
State-distribution plot for both clusters:
seqdplot (couple.seq, group = cluster2fac)Correlation between the cluster membership and men’s self-assessed dyadic coping ability
clust2.dummy<-as.numeric(cluster2fac)
clust2.dummy[clust2.dummy==1]<-0
clust2.dummy[clust2.dummy==2]<-1
cor.test(mydata$EDCm,clust2.dummy) LogSeq from the DySeq-Package provides an optional argument "subgroups" that allows to compare transition tables for two groups.
LogSeq(my.trans, delta=0.5, subgroups=cluster2) # Comparim aggregated logit models between clusters with DC as DV
LogSeq(my.trans.SC, delta=0.5, subgroups=cluster2) # Comparim aggregated logit models between clusters with SC as DV - number of expected cell problems
- find number of needed time intervals
- define a matrix containing the expected transition rates
- run EstFreq
- print the results
# First step:
my.trans.table<-matrix(c(0.57, 0.13, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05),4,2)
# Second step:
my.cellproblems<-EstFreq(my.trans.table, t=100, min.cell=5, k=20000)
# Third step
my.cellproblemsmy.EstTime.plot<-EstTime(my.trans.table, # contains expected transition probabilities (from the last section)
t=50:100, # limits the range of time intervals, which are testet.
k=5000) # precission of the simulation
# Warning: The function is very time consuming!
# printing will result in a plot of time point vs. expected number of low and zero frequencies
my.EstTime.plot
# if legend position should be change:
attr(my.EstTime.plot, "pos")<-"topright" # alternatives are: bottomleft (default)
my.EstTime.plot