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Hierarchical Bayesian Continuous Time Dynamic Modeling

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Continuous time dynamic models are similar to popular discrete time models such as autoregressive cross-lagged models, but through use of stochastic differential equations can accurately account for differences in time intervals between measurements, and more parsimoniously specify complex dynamics. As such they offer powerful and flexible approaches to understand ongoing psychological processes and interventions, and allow for measurements to be taken a variable number of times, and at irregular intervals. However, limited developments have taken place regarding the use of continuous time models in a fully hierarchical context, in which all model parameters are allowed to vary over individuals. This has meant that questions regarding individual differences in parameters have had to rely on single-subject time series approaches, which require far more measurement occasions per individual. We present a hierarchical Bayesian approach to estimating continuous time dynamic models, allowing for individual variation in all model parameters. We also describe an extension to the ctsem package for R, which interfaces to the Stan software and allows simple specification and fitting of such models. To demonstrate the approach, we use a subsample from the German socioeconomic panel and relate overall life satisfaction and satisfaction with health.
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Hierarchical Bayesian Continuous Time Dynamic Modeling
Charles C Driver
Max Planck Institute for Human Development
Manuel C Voelkle
Humboldt University Berlin
Max Planck Institute for Human Development
Continuous time dynamic models are similar to popular discrete time models such as autore-
gressive cross-lagged models, but through use of stochastic dierential equations can accur-
ately account for dierences in time intervals between measurements, and more parsimoni-
ously specify complex dynamics. As such they oer powerful and flexible approaches to un-
derstand ongoing psychological processes and interventions, and allow for measurements to be
taken a variable number of times, and at irregular intervals. However, limited developments
have taken place regarding the use of continuous time models in a fully hierarchical context, in
which all model parameters are allowed to vary over individuals. This has meant that questions
regarding individual dierences in parameters have had to rely on single-subject time series
approaches, which require far more measurement occasions per individual. We present a hier-
archical Bayesian approach to estimating continuous time dynamic models, allowing for indi-
vidual variation in all model parameters. We also describe an extension to the ctsem package
for R, which interfaces to the Stan software and allows simple specification and fitting of such
models. To demonstrate the approach, we use a subsample from the German socio-economic
panel and relate overall life satisfaction and satisfaction with health.
Keywords: Continuous time, dynamic model, state space, non-linear mixed-eects, stochastic
dierential equation, hierarchical time series
In this work we bring continuous time dynamic model-
ing together with hierarchical Bayesian modeling, and de-
scribe the resultant model and software for estimation. In
this approach, the estimation of subject specific continuous
time dynamic model parameters is enhanced by using data
from other subjects to inform a prior distribution, with no re-
strictions on the length of time series or number of subjects.
This allows for questions regarding individual dierences in
dynamics, without the requirement of very large numbers
of repeated observations that single-subject time series ap-
proaches can have, nor the convergence problems with low
numbers of subjects that some random-eects approaches
have (Eager & Roy, 2017).
Both hierarchical Bayes and continuous time dynamic
modeling oer an improved use of data over common al-
ternatives. The hierarchical approach uses information from
Charles C Driver, Centre for Lifespan Psychology, Max Planck
Institute for Human Development. Manuel C Voelkle, Psycholo-
gical Research Methods, Humboldt University, Berlin, and Centre
for Lifespan Psychology, Max Planck Institute for Human Devel-
opment. This work also constitutes part of the first authors doc-
toral thesis, and has been distributed in pre-print form. Corres-
pondence concerning this article should be addressed to Charles C
Driver, Max Planck Institute for Human Development, Lentzeallee
94, 14195 Berlin, Germany. E-mail: driver@mpib-berlin.mpg.de
other subjects to better estimate parameters of each subjects‘
specific model, while the continuous time approach uses in-
formation about the time interval between measurements of
each subject. The nature of this improvement is such that
it also allows for new questions on individual dierences
in the continuous time model parameters. That is, in data-
sets that do not contain sucient information for single-
subject time series approaches – as for instance with the
German socioeconomic panel (GSOEP) – we can now es-
timate subject-specific continuous time dynamic models, by
incorporating information from other subjects. As an intro-
duction we first review some of the background and motiva-
tion for dynamic models in psychology, consider separately
hierarchical modeling and continuous time models, and then
consider the combination of both approaches. We then de-
scribe the full model more formally, discuss the software im-
plementation we have developed, demonstrate performance
of the approach and software with a simulation study, and
finish with an example application of a dynamic model of
wellbeing based on the GSOEP data. Although the article
aims at creating a general understanding of the model, read-
ers who are primarily interested in its application, may easily
skip over some of the more detailed description of the subject
likelihood and population distribution in the model section.
2DRIVER
Dynamic models
Models containing dependencies on earlier states and
stochastic inputs, such as the autoregressive and cross lagged
model, oer a flexible and powerful approach to examine
how psychological constructs function over time within a
subject. For the sake of expediency we will refer to such
models as dynamic models, but note that such a term could
in general encompass a broader range of models than dealt
with here. Similar to trajectory oriented approaches such
as latent growth curves, dynamic models generally involve
some smooth long term trajectory. Instead of treating devi-
ations from a smooth overall trajectory as mere uninformat-
ive noise, however, such deviations can often be informative
as to future states. So, while someone’s life satisfaction last
year doesn’t completely determine their current life satisfac-
tion, it does have some predictive value, over and above their
typical level for the last 30 years. If we were to develop
a formal model out of such a statement, we would have a
simple dynamic model involving some dependence on past
states, and a stable baseline state. Since such a model cannot
perfectly predict the way satisfaction may change in time, we
will also need a variance term to quantify the degree of un-
certainty regarding predictions – if satisfaction last year was
substantially below baseline, we may expect that satisfaction
is still at a level below baseline, but with some uncertainty.
The more time that passes before we check satisfaction again,
the greater the uncertainty, as more external events and in-
ternal processes that we cannot fully predict will occur. As
such, the more time that passes, the less informative our pre-
vious observation will be, until at some point, the previous
observation no longer helps our prediction.
The model described so far assumes we have a perfect
measure of life satisfaction, but this is highly unlikely –
regardless of exactly how we measure, the same internal
satisfaction state could result in a range of measured out-
comes due to spurious factors that are unrelated to satisfac-
tion. When such measurement errors are left in the dynamic
model, any parameter estimates regarding the dynamics of
the true process we are interested in are likely to be biased
(Schuurman, Houtveen & Hamaker, 2015). Consequently,
state-space models (Hamilton, 1986) have been developed,
which partition observed deviations from a prediction into
innovation variance, representing meaningful but unpredict-
able fluctuations in the process itself, and measurement error,
representing uncorrelated random noise in the observations.
The innovation variance may be considered meaningful, in
that although it represents unpredictable changes in the pro-
cess, once such changes in the processes are observed, they
are useful for future predictions of the states of the processes.
This is in contrast to measurement error, which will not oer
any predictive value for the future. By fitting such a dynamic
model to the data, we can ask questions like ‘how long does a
change in this persons’ life satisfaction typically persist?’. If
we include a second set of observations, this time regarding
their health, we can then ask ‘to what extent do changes in
life satisfaction and health covary?’, and also consider tem-
poral dynamics between the two, with ’to what extent does
a change in health predict a later change in life satisfaction,
after controlling for shared sources of change?’. The ma-
jor distinction between such an approach and a trajectory
only model such as a latent growth curve, is that the rela-
tion between fluctuations is addressed, rather than only the
shape of overall change. So while it may be that over 30
years, a person exhibits some general decline in health and
apparently stable life satisfaction, when the relation between
fluctuations is taken into account, one might instead see that
life satisfaction tends to fluctuate downwards when health
satisfaction does, then slowly recover due to other factors.
Common examples of dynamic models include the autore-
gressive and cross-lagged panel model, (Finkel, 1995;
Hertzog & Nesselroade, 2003) autoregressive moving aver-
age models (Box, Jenkins, Reinsel & Ljung, 2015), or the
latent change score formulations sometimes used (John J.
McArdle, 2009), when they include innovation variance for
each time point. They can be applied with observational
or experimental data, or a mix of the two, and can be used
to tell us to what extent changes in a psychological process
are predictive of later changes of either the same process, or
some other process of interest. Some examples of the usage
of dynamic models within psychology include the analysis
of symptom networks in psychopathology (Bringmann et al.,
2013), dierences in individuals‘ aective dynamics (Koval,
Sütterlin & Kuppens, 2016), and the influence of partners on
one another (Hamaker, Zhang & van der Maas, 2009).
Hierarchical Dynamic models
While in some circumstances, a well developed dynamic
model for a single subject may be important, in many situ-
ations scientists are instead interested in generalising from
observed subjects to some population, as well as understand-
ing dierences between subjects. For such purposes, one
typically needs repeated measures data from multiple sub-
jects. How then, should data from dierent individuals be
combined? Were all subjects exactly alike, combining the in-
formation would be very simple and eective. Were the sub-
jects entirely distinct, with no similarities, combining the in-
formation would tell us nothing – an estimate of the average
weight of a type of leaf, is not improved by including a rock
in the sample. The estimation approaches herein endeavour
to tread the middle ground, in that while dierences between
subjects are acknowledged, similarities are leveraged to im-
prove estimation. Such an approach can broadly be called
hierarchical dynamic modelling, a term encompassing both
frequentist and Bayesian approaches. To understand how
hierarchical dynamic models function, and the benefits they
oer, it is helpful to first consider two extremes of possible
3
approaches to dynamic models with multiple subjects. At
one end of the continuum are panel models with a single set
of fixed-eects parameters governing the dynamics of every
subject – ‘all leaves weigh exactly the same’ – and at the
other lies person-specific time series – ‘one leaf is to another,
as to a rock’.
Panel models containing autoregressive and cross-lagged
parameters are regularly estimated with a single set of fixed-
eects parameters governing the behavior of the dynamic
system for many subjects – one assumes that the system char-
acterizing the psychological processes of one subject is ex-
actly the same as for other subjects1. The benefits of this
assumption are that the number of parameters in the model
is relatively low, and the data from every individual is rel-
evant for estimating every parameter. This assumption usu-
ally makes fitting the model to data much simpler, and can
increase the precision and accuracy of parameter estimates
when the assumption is valid. However, it is a very strong
assumption and can result in highly biased parameter estim-
ates. This has long been recognized (i.e., Balestra & Nerlove,
1966) to be the case in regards to the intercept parameter of
the dynamic model, in that when subject specific dierences
in the average level of the processes are not accounted for, the
temporal dynamics parameters (auto and cross eects) can
be terribly biased – instead of representing some average of
the temporal dynamics for all subjects, they instead become
a mixture of the within-subject temporal dynamics and the
between-subjects dierences in average levels. While many
poor implementations and claims of large cross eects and
causality can still be found, this issue is at least widely re-
cognized and readily resolved. See Halaby (2004) and Ha-
maker, Kuiper and Grasman (2015) for more details, and Cat-
tell (1963), Molenaar (2004), Voelkle, Brose, Schmiedek and
Lindenberger (2014), Wang and Maxwell (2015) for more on
between and within subject issues in general.
At the other end of the spectrum is idiographic, or indi-
vidual specific, time series approaches, where one fits the
model separately for each subject (see for example Steele,
Ferrer & Nesselroade, 2014). Such approaches ensure that
the estimated parameters are directly applicable to a specific
individual. With ‘suciently large’ amounts of data for each
subject, this is likely to be the simplest and best approach.
However, in applications of dynamic models there may be
many correlated parameters, coupled with noisy measure-
ments that are correlated in time, ensuring that a ‘suciently
large’ amount of data per subject may in fact be ‘unattainably
large’, particularly when one wishes to distinguish measure-
ment error and relate multiple processes. Models estimated
with less than ideal amounts of data tend to suer from fi-
nite sample biases, as for example in the autoregressive para-
meter when the number of time points is low (Marriott &
Pope, 1954, 3/4), and also from higher uncertainty and more
inaccurate point estimates. Were parameters independent of
one another then inaccurate estimates of a parameter of little
interest may be tolerable, but in dynamic models there are
typically strong dependencies between parameters, such that
inaccurate estimation of any single parameter can also re-
duce accuracy for all other parameters. An example of this
dependency is that when, as typically occurs, the parameter
governing the subjects‘ base level is overfit and explains
more observed variance than it should, the related auto-eect
parameter is typically underestimated, in turn aecting many
other parameters in the model. While some may be inclined
to view the complete independence between models for each
subject as a strength of the single-subject time series ap-
proach, there is little value to such independence if it also
comes at the cost of making the best model for the sub-
jects empirically unidentifiable, or the estimates substantially
biased. Recent work by Liu (2017) demonstrates some of
these issues. Liu compared a multilevel and person-specific
approach to modelling multiple-subjects autoregressive time
series, without measurement error. They examined the ef-
fect of time series length, number of subjects, distribution of
coecients, and model heterogeneity (diering order mod-
els). So long as the model converges and is not under spe-
cified for any subjects (i.e., contains lags of a high enough
order), they find that the multilevel approach provides better
estimates, even when distributional assumptions regarding
the parameters are not met. Of course, not every possibility
has been tested, and it is likely possible to find specific cases
where this result does not hold.
Hierarchical approaches to dynamic models, such as those
from Lodewyckx, Tuerlinckx, Kuppens, Allen and Sheeber
(2011), are essentially a generalization which encompasses
the two extremes already discussed – a single model for all
individuals, or a distinct model for each. Such a hierarchical
formulation could take shape either in a frequentist random-
eects setting, or a hierarchical Bayesian setting.
Using a hierarchical formulation, instead of estimating
either a single set of parameters or multiple sets of inde-
pendent parameters, one can estimate population distribu-
tions for model parameters. It is common in hierarchical
frequentist approaches to only estimate population distri-
butions, while with Bayesian approaches it is more typical
to simultaneously sample subject level parameters from the
population distribution, with the population distribution es-
sentially serving as a prior distribution for the subject level
parameters. The latter approach can help to provide the intu-
ition for the relation among person-specific, hierarchical, and
single fixed-eect parameter set approaches. Using a hier-
archical model, if one were to fix the variance of the popula-
tion distribution to dierent values, one could see that as the
1Note that this is distinct from what has become known in
econometrics as the ‘fixed-eects panel model’, which estimates
unique intercept terms for every subject, with all other parameters
constant across subjects (Halaby, 2004).
4DRIVER
variance approaches zero, the subject level parameter estim-
ates get closer and closer to the fixed-eects model with the
same set of parameter values for every subject. Conversely,
as one fixed the population variance further and further to-
wards infinity, subject level estimates would get closer and
closer to those of the person-specific approach, in which the
parameter estimates for one subject are not influenced at all
by estimates of the others. The benefit of a hierarchical ap-
proach however, is that rather than fixing the parameters of
the population distribution in advance, the population distri-
bution mean and variance can be estimated at the same time
as subject level parameters – the extent of similarity across
subjects is estimated, rather than fixed in advance to ‘not at
all similar’ or ’completely the same’. An intuitive interpret-
ation of this is that information from all other subjects serves
as prior information to help parameter estimation for each
specific subject. As is standard with Bayesian methods, one
still specifies some prior distributions in advance, but in hier-
archical Bayes models these are priors regarding our expect-
ations for the population distribution, otherwise known as
hyperpriors. So, some initial, possibly very vague, hyperpri-
ors are specified regarding possible population distributions.
These hyperpriors, coupled with data and a Bayesian infer-
ence procedure, result in an estimated population distribution
for the parameters of each subjects‘ dynamic model. This
estimated population distribution, coupled with the likeli-
hood of the dynamic model parameters for a specific subject
(calculated just as we would in the single-subject approach),
gives the posterior distribution for the subjects‘ specific para-
meters.
In the frequentist setting, while it is relatively straight-
forward to include random-eects for intercept parameters,
random-eects for regression and variance parameters of lat-
ent factors have typically been more problematic, due to
the complex integration required (see for instance Delattre,
Genon-Catalot & Samson, 2013; Leander, Almquist, Ahl-
ström, Gabrielsson & Jirstrand, 2015). This mixed-eects
approach is helpful in that stable dierences in level between
subjects – likely the largest source of bias due to unob-
served heterogeneity – are accounted for, while maintain-
ing a model that can be fit reasonably quickly using the well
and commonly understood frequentist inference architecture.
Although we show later that when individual dierences in
latent regression and variance parameters are (incorrectly)
not modeled, the magnitude of spurious cross eects is low,
many spurious results do nevertheless occur. Further, if not
modelled it is of course impossible to ask questions regarding
such individual dierences, which are a key interest in many
cases. Bayesian approaches oer good scope for random-
eects over all parameters, and indeed hierarchical Bayesian
discrete time dynamic models are implemented and in use,
see for instance Schuurman (2016), Schuurman, Ferrer, de
Boer-Sonnenschein and Hamaker (2016).
Once a hierarchical dynamic model is estimated, one may
be interested in questions regarding the population distribu-
tion, one or multiple specific subjects, or one or multiple
specific time points of a subject. Questions regarding the
population distribution could relate to: means and variances
of parameter distributions, such as “what is the average cross
eect of health on life satisfaction”, and “how much variance
is there in the eect across people?” Also possible are ques-
tions regarding correlations between population distributions
of dierent parameters, or between parameters and covari-
ates, such as, “do those who typically have worse health
show a stronger cross-eect?” Instead of whole population
questions, one could instead ask about, for example, the cross
eect of health on life satisfaction of a specific subject. Or
yet more specific, we might for instance want to predict the
health of a subject at some particular time point in the future,
given a particular life event and set of circumstances.
Continuous time dynamic models
Within psychology and many fields, classical approaches
to dynamic modeling have tended to rely on discrete time
models, which generally rely on an assumption that the time
intervals between measurements are all the same. A dis-
crete time model directly estimates the relation (regression
strength) between one measurement occasion and another,
without incorporating the time interval information. If all
measurement occasions have the same time interval between
them, this can be fine. In many cases though the time inter-
vals between occasions may dier, and if the same regression
parameter is used to account for dierent time intervals, it
is likely that the parameter is incorrect for some or all oc-
casions – the relation between someone’s earlier mood and
later mood is likely to be quite dierent if we compare a ten
minute interval to a three day interval, for instance. Obvi-
ously, parameter estimates and, thus, scientific conclusions,
are biased when observation intervals vary and this is not ad-
equately accounted for. In simple cases, so called phantom
variables (Rindskopf, 1984), with missing values for all indi-
viduals could be added in order to artificially create equally
spaced time intervals. For complex patterns of individually
varying time intervals, however, this approach can become
untenable (Voelkle & Oud, 2013).
Continuous time models overcome these problems, oer-
ing researchers the possibility to estimate parameters free
from bias due to unequal intervals, easily compare between
studies and datasets with dierent observation schedules,
gather data with variable time intervals between observa-
tions, understand changes in observed eects over time, and
parsimoniously specify complex dynamics.
Rather than estimate the regression between two meas-
urement occasions directly, the continuous time approach in-
stead estimates a set of continuous time parameters that de-
scribe how the process changes at any particular moment,
5
based on the current state of the process. A mathematical
function can then be used, in combination with the time in-
terval, to determine the appropriate discrete time parameters,
such as regression strength, governing the relation between
two specific occasions separated by some time interval. This
is a non-linear function, that operates in a way such that for
first order (dependent only on the previous occasion, and not
earlier occasions) discrete-time models with constant time
intervals, the discrete and continuous time approaches are
equivalent. The same of course holds true for other dynamic
model parameters, such as the intercepts and innovation co-
variance matrix. The ability to naturally and exactly account
for dierent time intervals means continuous time models are
particularly suited to the analysis of data from studies with
dierent measurement schemes.
While accounting for dierent time intervals naturally is
one obvious benefit, another reason for interest in continu-
ous time models is that the parameters directly represent
elements of dierential equations. This is beneficial both
in terms of allowing more meaningful interpretation of the
parameters, as well as for more readily specifying higher
order dynamics such as damped oscillations. More mean-
ingful interpretation of continuous time dynamic models is
possible because they are not just models for the specific
time points observed as with discrete time models, but rather
they describe expected behavior of the processes at any time
point, whether observed or not. Dierential equation mod-
els describing more complex (e.g., higher than first order)
dynamics have proven of some interest in psychology, with
analysis of constructs such as working memory (Gasimova
et al., 2014), emotion, (Chow, Ram, Boker, Fujita & Clore,
2005), and addiction (Grasman, Grasman & van der Maas,
2016). For some background on how dierential equation
models can be applied to psychological constructs, see De-
boeck, Nicholson, Bergeman and Preacher (2013) and Boker
(2001). Important to note here is that continuous time dy-
namic models are not the only form of dynamic model us-
ing dierential equations, as approaches such as generalized
local linear approximation (Boker, Deboeck, Edler & Keel,
2010) are also available. Rather, the terminology ’continuous
time model’ has typically been used to refer to dynamic mod-
els that explicitly incorporate the time interval in the equa-
tions, leading to an ‘exact’ solution, that does not require
the specification of an embedding dimension in advance. For
discussion on the distinction between ‘approximate’ and ‘ex-
act’ approaches to estimation of dierential equations, see
Oud and Folmer (2011) and Steele and Ferrer (2011).
Even for models that explicitly incorporate the time in-
terval, exact, analytic solutions to the stochastic dierential
equations are usually only available for linear stochastic dif-
ferential equations involving a Gaussian noise term, which
are the form of model we are concerned with here. For more
complex forms of continuous time model, various approx-
imations are typically necessary, as for instance in Särkkä,
Hartikainen, Mbalawata and Haario (2013). For further dis-
cussion of the continuous time dynamic model in psycho-
logy, Oud (2002) and Voelkle, Oud, Davidov and Schmidt
(2012) describe the basic model and applications to panel
data, and Voelkle and Oud (2013) detail higher order mod-
elling applications such as the damped linear oscillator. A
classic reference for stochastic dierential equations more
generally is the text from Gardiner (1985).
Hierarchical continuous time dynamic models
While Oud and Jansen (2000) describe a mixed-eects,
structural equation modelling approach to continuous time
dynamic models, and Driver, Oud and Voelkle (2017) the
related software, relatively little work has been done on the
generalization to fully random-eects models, in which all
parameters may vary over subjects. The continuous time
dynamic model has been specified in Bayesian form by
Boulton (2014), but this did not extend to a hierarchical ap-
proach. The most substantial foray into hierarchical continu-
ous time approaches in psychology has been with the work
by Oravecz, Tuerlinckx and Vandekerckhove (2009, 2016)
on the hierarchical Ornstein-Uhlenbeck model, which also
led to the creation of the BHOUM software for estimation.
While the software and modelling approach as it stands is
useful, there is a limitation for some applications in that the
matrix containing dynamic eects is constrained to be sym-
metric and positive definite. This means that model struc-
tures that require non-symmetric cross eects, such as the
autoregressive and cross lagged panel model or damped lin-
ear oscillator, cannot be estimated.
Chow, Lu, Sherwood and Zhu (2014) have also looked at
hierarchical dynamics in continuous time, fitting nonlinear
ordinary dierential equation models with random eects on
the parameters to ambulatory cardiovascular data from mul-
tiple subjects. While the nonlinear aspect oers increased
flexibility (at cost of additional complexity), the ordinary
dierential aspect, rather than stochastic dierential, means
that the estimated processes are deterministic, and random-
ness is assumed to occur only at the measurement level. Un-
less the process is very well understood, with all contribut-
ing factors measured, the lack of innovation variance may be
quite limiting. Lu, Chow, Sherwood and Zhu (2015) used
similar estimation algorithms and data with stochastic dier-
ential equations which can account for innovation variance,
but avoided a hierarchical approach, and assumed independ-
ence between subjects parameters.
This work describes the model and software for a hier-
archical Bayesian continuous time dynamic model, without
any restriction of symmetric positive-definite dynamics. It
builds upon the model and software, ctsem, described in
Driver et al. (2017), which oered a mixed-eects modeling
framework for the same linear stochastic dierential equation
6DRIVER
model, in which intercept and mean related parameters could
be estimated as random eects across subjects, but subject
level variance and regression parameters were restricted to
fixed eects estimation. With the change to a hierarchical
Bayesian framework, all subject level parameters may now
vary across subjects, the covariance between subject level
parameters is available, and the estimation of time-invariant
relations between covariates and all subject level parameters
is now possible (as opposed to relations with only the mean
levels of subjects’ processes).
The model
There are three main elements to our hierarchical con-
tinuous time dynamic model. There is a subject level latent
dynamic model, a subject level measurement model, and a
population level model that describes the distribution of sub-
ject level parameters across subjects. Note that while various
elements in the model depend on time, the fundamental para-
meters of the model are time-invariant. Note also that we ig-
nore subject specific subscripts when discussing the subject
level model.
Subject level latent dynamic model
The subject level dynamics are described by the stochastic
dierential equation:
dη(t)=Aη(t)+b+Mχ(t)dt+GdW(t) (1)
Vector η(t)Rvrepresents the state of the vlatent pro-
cesses at time t, so dη(t) simply means the direction of
change, or gradient, of the latent processes at time t. The
matrix ARv×vis often referred to as the drift matrix,
with auto eects on the diagonal and cross eects on the
o-diagonals characterizing the temporal dynamics of the
processes. Negative values on the auto eects are typical,
and imply that as the latent state becomes more positive, a
stronger negative influence on the expected change in the
process occurs – the process tends to revert to a baseline
(at least in the absence of other influences). Positive auto-
eects imply an explosive process, in which deviations from
a baseline do not dissipate, but rather accelerate. Cross-
eects between processes may be positive or negative, with
a positive cross-eect in the first row and second column im-
plying that as the second process becomes more positive, the
direction of change in the first process also becomes more
positive, while for a negative cross eect the first process is
negatively influenced.
The expected auto and cross regression matrix for a given
interval of time (i.e., a discrete time eect) can be calcu-
lated as per Equation 2. Figure 1 shows this calculation
over a range of time intervals, for an example bivariate pro-
cess of sociability and energy level. The first column of
the drift matrix, with values 0.4,0.1, represents the eect
of the current state of sociability, on change in sociability
and change in energy level respectively. So, when sociabil-
ity is above baseline – someone is enjoying socialising with
friends – both sociability and energy level are likely to go
down. The second column of the drift matrix, with values
0.3,0.2, represents the eect of the current state of energy
level, on change in sociability and change in energy level re-
spectively. So in this case, if energy levels are above baseline
then the expected change in sociability will be positively in-
fluenced, while the expected change in energy level will be
downwards. The converse of course holds true when energy
levels are below baseline – sociability will reduce, and en-
ergy level will rise.
A
tu=eA(tutu1)(2)
0 5 10 15
0.0 0.2 0.4 0.6 0.8 1.0
Time interval
Value
discrete DRIFT[1,1] (-0.40)
discrete DRIFT[2,1] (-0.10)
discrete DRIFT[1,2] (0.30)
discrete DRIFT[2,2] (-0.20)
Figure 1. Values of the discrete DRIFT, or auto and cross re-
gression matrix, change depending on a function of the con-
tinuous time DRIFT matrix and the time interval between
observations. The continuous time DRIFT values are shown
in brackets in the legend.
The notation is used in this work to indicate a term that
is the discrete time equivalent of the original continuous time
parameter, for the time interval tu. The time interval tuis
simply the time between measurement occasions uand u1,
where Uis the set of all measurement occasions u, from 1 to
the total number of measurement occasions. A
tuthen con-
tains the appropriate auto and cross regressions for the eect
of latent processes ηat measurement occasion u1 on ηat
measurement occasion u. So, for a simple univariate process
with A=.4, a time scale of weeks, and a time interval
of two weeks between measurement occasions, the discrete
time autoregression would be e.4(2) =0.45. Note that here
the exponential of the Amatrix is equivalent to the regular
univariate exponential, but once the Amatrix has non-zero
o-diagonals, this is no longer the case and a matrix expo-
nential function is necessary.
In Equation 1 the continuous time intercept vector bRv,
provides a constant fixed input to the latent processes η. In
combination with temporal dynamics matrix A, this vector
determines the long-term level at which the processes fluctu-
7
ate around. Without the continuous time intercept the pro-
cesses, if mean reverting, would simply fluctuate around
zero. The discrete time intercept may be calculated as per
Equation 3. For the sociability and energy level example, this
discrete time intercept would be a vector of two values, the
first representing the value to be added to the sociability state,
and the second added to the energy level state. One way to
think about this is that when the baseline of the processes is
not zero, the autoregressive nature of the system means some
portion of the process value is lost for each time interval – the
discrete time intercept simply adds a sucient amount back
in to maintain a non-zero baseline (note that this intuition is
only strictly valid for stationary systems). For interpretation
purposes we are inclined to think the asymptotic level of the
processes, bt, is more useful. The asymptotic process level
is the level the processes tend towards over time, and also the
level they start at if stationary.
b
tu=A1(A
tuI)b(3)
bt=A1b(4)
In Equation 1 the time dependent predictors χ(t) represent
exogenous inputs to the system, such as an intervention, that
may vary over time and are independent of fluctuations in
the system. Equation 1 shows a generalized form for time
dependent predictors, that could be treated a variety of ways
depending on the predictors assumed shape, or time course
(i.e., what values should χtake on between measurement oc-
casions?). We use a simple impulse form shown in Equation
5, in which the predictors are treated as impacting the pro-
cesses only at the instant of an observation occasion u, and
the eects then transmit through the system in accordance
with Aas usual. Such a form has the virtue that many al-
ternative shapes are made possible via augmentation of the
system state matrices – discussion and examples of this are
available in Driver and Voelkle (2017).
χ(t)=X
uU
xu(ttu) (5)
Here, time dependent predictors xuRlare observed at
measurement occasions uU. The Dirac delta function
(ttu) is a generalized function that is at 0 and 0 else-
where, yet has an integral of 1 (when 0 is in the range of
integration). It is useful to model an impulse to a system,
and here is scaled by the vector of time dependent predictors
xu. The eect of these impulses on processes η(t) is then
MRv×l. Put simply, the equation means that at the meas-
urement occasion ua time dependent predictor (e.g., inter-
vention) is observed, the system processes spike upwards or
downwards by Mxu. For a typical intervention that probably
only occurs once during the observation window, xuwould
then be zero for every observation uexcept when the inter-
vention occurred, where it could take on a dummy coding
value such as one, or could reflect the strength of the inter-
vention. Because Mis conceptualized as the eect of instant-
aneous impulses x, which only occur at occasions Uand are
not continuously present as for the processes η, the discrete
and continuous time forms are equivalent, at the times when
observations are made. This means that the eect of some
intervention at measurement occasion u=3 is simply Mxu=3
at the instant of the intervention, and at later measurement
occasion u=4, the remaining eect is A
tu=4Mxu=3. If the
time interval between occasions 3 and 4 is t=2.30, using
Equation 2 this translates to eA(2.30)Mxu=3.
In Equation 1, W(t)Rvrepresents independent Wiener
processes, with a Wiener process being a random-walk
in continuous time. dW(t) is meaningful in the con-
text of stochastic dierential equations, and represents the
stochastic noise term, an infinitesimally small increment of
the Wiener process. Lower triangular matrix GRv×vrep-
resents the eect of this noise on the change in η(t). Q, where
Q=GG>, represents the variance-covariance matrix of this
diusion process in continuous time. Intuitively, one may
think of dW(t) as random fluctuations, and Gas the eect of
these fluctuations on the processes. The discrete time innov-
ation covariance matrix, which represents the increase in un-
certainty about the process states over a certain time interval,
may be calculated as shown in Equations 6 and 7. Figure 2
plots this calculation over a range of time intervals for the ex-
ample sociability and energy level processes, with diusion
variances of 2 and 3 respectively, and covariance of -1.
Q
tu=QtA
tuQt(A
tu)>(6)
Qt=irow(AI+IA)1row(Q)(7)
0 5 10 15
0 2 4 6
Time interval
Value
discrete DIFFUSION[1,1] (2.00)
discrete DIFFUSION[2,1] (-1.00)
discrete DIFFUSION[2,2] (3.00)
Figure 2. Values of the innovation covariance matrix change
depending on a function of the continuous time DRIFT and
DIFFUSION matrices and the time interval between obser-
vations. The continuous time DIFFUSION values are shown
in brackets in the legend (with DRIFT the same as for Figure
1). The values at higher time intervals are no longer chan-
ging substantially, thus representing the total within-person
covariance matrix (or very close to it).
8DRIVER
Equation 6 calculates the discrete time innovation covari-
ance matrix for a given time interval. Intuitively, it shows
that the discrete time innovation covariance matrix, which
may be thought of as representing the amount and correla-
tion of random noise added to the processes over a specified
time interval, equals the asymptotic, or total, innovation co-
variance matrix Qt, minus the amount ‘remaining’ from
the earlier measurement occasion. This remaining amount
is determined by the temporal dynamics of A.Qtrepres-
ents the innovation covariance matrix as tapproaches in-
finity, which for a stationary process also represents the total
within-subject variance covariance at any point in time. This
asymptotic within-person covariance could also be thought
of as the uncertainty about process states when no meas-
urements of the processes exist. In the plotted example, we
can see that uncertainty regarding the sociability and energy
level processes is roughly at this asymptote after a time in-
terval of approximately 10. This asymptotic covariance mat-
rix (as used in Tómasson, 2013) provides a computationally
ecient basis for calculating the additional covariance Q
tu
added to the system over the time interval tu, as the asymp-
totic matrix only needs to be computed once. denotes the
Kronecker-product, row is an operation that takes elements
of a matrix row wise and puts them in a column vector, and
irow is the inverse of the row operation.
As the various discrete time calculations formula shown
in this section rely on the matrix exponential seen in Equa-
tion 2, they can be dicult to intuitively understand. Plotting
them with specific parameter values and increasing time in-
terval t, can be a valuable tool, as this demonstrates how
the implied discrete-time coecients change as a function
of the continuous time parameters and the time interval. An
R script to perform such plots for a bivariate latent process
model is available in the supplementary material.
Latent dynamic model — discrete time solution
To derive expectations for discretely sampled data, the
stochastic dierential Equation 1 may be solved and trans-
lated to a discrete time representation, for any observation
uU. Most components for this solution were already
shown in Equations 2, 3, 5, 6, and 7, here we simply bring
them together.
ηu=A
tuηu1+b
tu+Mxu+ζuζuN(0v,Q
tu) (8)
To reiterate, the notation is used in this work to indicate
a term that is the discrete time equivalent of the original con-
tinuous time parameter, for the time interval tu.ζuis the
zero mean random error term for the processes at occasion u,
which is distributed according to a multivariate normal with
covariance Q
tu. The recursive nature of this solution means
that at the first measurement occasion u=1, the system must
be initialized in some way, with A
tuηu1replaced by ηt0, and
Q
tureplaced by Q
t0. These initial states and covariances are
later referred to as T0MEANS and T0VAR respectively.
Measurement model
While in principle, non-Gaussian generalizations are pos-
sible, for the purposes of this work the latent process vector
η(t) has the linear measurement model:
y(t)=Λη(t)+τ+(t) where (t)N(0c,Θ) (9)
y(t)Rcis the clength vector of manifest variables,
ΛRc×vrepresents the factor loadings, and τRcthe mani-
fest intercepts. The manifest residual vector Rchas cov-
ariance matrix ΘRc×c. When such a measurement model
is used to adequately account for non-zero equilibrium levels
in the data, the continuous time intercept bmay be unneces-
sary – accounting for such via the measurement model has
the virtue that the measurement parameters are not dependent
on the temporal dynamics, making optimization or sampling
easier.
Subject level likelihood
The subject level likelihood, conditional on time depend-
ent predictors xand subject level parameters Φ, is as follows:
p(y|Φ,x)=Y
uU
p(yu|yu1,u2,...,1),xu,Φ) (10)
To avoid the large increase in parameters that comes with
sampling or optimizing latent states, we use a continuous-
discrete, or hybrid, Kalman filter (Kalman & Bucy, 1961)
to analytically compute subject level likelihoods, conditional
on subject parameters. For more on filtering see Jazwinski
(2007) and Särkkä (2013). The filter operates with a pre-
diction step, in which the expectation ˆ
ηu|u1and covariance
ˆ
Pu|u1of the latent states are predicted by:
ˆ
ηu|u1=A
tuˆ
ηu1|u1+b
tu+Mxu(11)
ˆ
Pu|u1=A
tu
ˆ
Pu1|u1(A
tu)>+Q
tu(12)
For the first measurement occasion u=1, the values ˆ
ηu|u1
and ˆ
Pu|u1must be provided to the filter. These parameters
may in some cases be freely estimated, but in other cases
need to be fixed or constrained, either to specific values or
by enforcing a dependency to other parameters in the model,
such as an assumption of stationarity.
Prediction steps are followed by an update step, wherein
rows and columns of matrices are filtered as necessary de-
pending on missingness of the measurements y. The update
step involves combining the observed data with the expecta-
tion and variance, using the Kalman gain matrix KRv×c,
9
which represents the ratio between the process inovation co-
variance and measurement error.
ˆ
yu|u1=Λˆ
ηu|u1+τ(13)
ˆ
Vu=Λˆ
Pu|u1Λ>+Θ(14)
ˆ
Ku=ˆ
Pu|u1Λ>ˆ
V1
u(15)
ˆ
ηu|u=ˆ
ηu|u1+ˆ
Ku(yuˆ
yu|u1) (16)
ˆ
Pu|u=(Iˆ
KuΛ)ˆ
Pu|u1(17)
The log likelihood (ll) for each subject, conditional on
subject level parameters, is typically2then (Genz & Bretz,
2009):
ll =
U
X1/2(nln(2π)+ln Vu+
(ˆ
y(u|u1) yu)V1
u(ˆ
yu|u1yu)>)(19)
Where nis the number of non-missing observations at
measurement occasion u.
Population distribution
Rather than assume complete independence or depend-
ence across subjects, we assume subject level parameters are
drawn from a population distribution, for which we also es-
timate parameters, conditional on specified hyperpriors. This
results in a joint-posterior distribution of:
p(Φ,µ,R,β|Y,z)p(Y|Φ)p(Φ|µ,R,β,z)p(µ,R,β) (20)
Where subject specific parameters Φiare determined in
the following manner:
Φi=tformµ+Rhi+βzi(21)
hiN(0,1) (22)
µN(0,1) (23)
βN(0,1) (24)
ΦiRsis the slength vector of parameters for the dy-
namic and measurement models of subject i.µRspara-
meterizes the means of the raw population distributions of
subject level parameters. RRs×sis the matrix square root
of the raw population distribution covariance matrix, para-
meterizing the eect of subject specific deviations hiRs
on Φi. The matrix square root Ris itself a transformation
of parameters sampled and transformed in various ways, as
discussed in the following section. βRs×wis the raw eect
of time independent predictors ziRwon Φi, where wis the
number of time independent predictors. Yicontains all the
data for subject iused in the subject level model – y(process
related measurements) and x(time dependent predictors). zi
contains time independent predictors data for subject i. tform
is an operator that applies a transform to each value of the
vector it is applied to. The specific transform depends on
which subject level parameter matrix the value belongs to,
and the position in that matrix — these transforms and ra-
tionale are described below, but are in general necessary be-
cause many parameters require some bounded distribution,
making a purely linear Gaussian approach untenable.
The basic structure of Equation 21 is such that everything
inside the brackets – population distribution means, subject
specific random deviations, and covariate eects – is on the
unconstrained, real number scale. This bracketed portion,
which we will later refer to as the raw subject level para-
meters, then undergoes some transformation function (tform)
that varies depending on the sort of parameter to be estimated
(e.g., standard deviations need to be positive, correlations
must be between -1 and 1). This transformation of the raw
subject level parameters then leaves us with the subject level
parameters we are actually interested in, for example the drift
matrix of temporal dynamics. For this reason, we will also
refer to the population means and covariate eects inside the
brackets as raw population means and raw covariate eects –
they are a necessity but we are not interested in them directly.
We take this approach to ensure that subject specific paramet-
ers do not violate boundary conditions, and that deviations
from a mean that is close to a boundary are more likely to be
smaller in the direction of the boundary than away from it.
For example, for a standard deviation parameter with a pop-
ulation distribution mean of 0.30, a subject specific deviation
of 0.40 is not possible because it results in a negative value,
while +0.40 would be perfectly reasonable. Figure 3 gives a
visual sense to this approach with transformations.
The approach wherein we first sample raw subject spe-
2For computational reasons we use an alternate but equivalent
form of the log likelihood. We scale the prediction errors across all
variables to a standard normal distribution, drop constant terms, cal-
culate the log likelihood of the transformed prediction error vector,
and appropriately update the log likelihood for the change in scale,
as follows:
ll =
U
Xlntr(V1/2
u)X1/2V1/2
u(ˆ
y(u|u1) yu)(18)
Where tr indicates the trace of a matrix, and V1/2is the inverse
of the Cholesky decomposition of V. The Stan software manual
discusses such a change of variables (Stan Development Team,
2016b).
10 DRIVER
cific deviations hifrom a standard normal distribution, be-
fore multiplying them by the matrix square root of the popu-
lation distribution Rand add the population mean µ, may be
unfamiliar to some. This is a non-centered parameterization,
which we implemented to improve sampling eciency. See
Bernardo et al. (2003) and Betancourt and Girolami (2013)
for discussion of non-centered parameterizations.
-4 -2 0 2 4 6
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Value
Density
Raw pop. distribution
Raw subject level param
Pop. distribution
Subject level param
Parameter
Figure 3. Subject level parameters are obtained by first
sampling from a multivariate normal raw population distri-
bution, and then transforming by some function to satisfy
boundary and other criteria. This example shows a possib-
ility for standard deviation parameters, using an exponen-
tial transformation. The height of the individual parameters
simply represents the match between raw and transformed.
Population distribution covariance
The matrix R, a square root3of the population distribu-
tion covariance matrix (RR>), accounts for parameter cor-
relations such as would be found when, for example, sub-
jects that typically score highly on measurements of one pro-
cess are also likely to exhibit stronger auto-eects on another.
Rather than simply using the conjugate inverse-Wishart prior
for the covariance matrix, we opt for an approach that first
separates scale and correlation parameters (Barnard, McCul-
loch & Meng, 2000), wherein:
Z=RR>(25)
R=SX (26)
Zis a positive semi-definite covariance matrix, Ris a mat-
rix square root of covariance Z,Xis a matrix square root of
a correlation matrix, and Sis a diagonal matrix of standard
deviations. These matrices are all of dimension s×s. This
separation approach is taken to ensure that the scale of pop-
ulation distributions does not influence the probability of the
correlation between them (Tokuda, Goodrich, Van Mechelen,
Gelman & Tuerlinckx, 2011), and similarly to ensure that
the prior distribution does not become highly informative as
variances approach zero (Gelman, 2006). Sampling Xis still
somewhat complicated however, as a) XX>must result in
a correlation matrix, and b) the resulting marginal distribu-
tion over XX>should not vary across the o-diagonal ele-
ments. One approach to handling these concerns is to sample
a Cholesky factor correlation matrix using an LKJ prior, as
implemented in the Stan software (Stan Development Team,
2016a), and is based on the work of Lewandowski, Kur-
owicka and Joe (2009). Specifics of how the prior is obtained
are complex, but the result is that of a uniform distribution
over the space of correlation matrices, and prior probabil-
ities of correlations are the same across all elements of the
matrix. A drawback of this approach however is that there
is no obvious way to maintain these characteristics for the
hierarchical case – sampling a population mean correlation
matrix, and then subject level correlation matrices based on
this mean matrix, again leaves one with the problem of vary-
ing marginal correlation probabilities across elements of the
matrix. While to obtain Rwe only need a single correlation
matrix, for subject level covariance matrices we will need
the hierarchical form, and for consistency we maintain the
same approach in both cases. To deal with these concerns,
our approach is as follows: To obtain X, a matrix square root
of a correlation matrix, we first sample lower o-diagonal
elements from a normal distribution, scale these eects to
between -1 and 1 using an inverse logit function (with ap-
propriate scale adjustment), copy these to the upper triangle
to form a symmetric matrix, and set the diagonal to 1. This
matrix is then scaled by the diagonal matrix containing the
inverse of the square roots of the sum of squares of each row
to give us X– similar to scaling a covariance matrix to a
correlation, but as we are scaling to a matrix square root we
only multiply by the scale matrix once. This ensures that
XX>always results in a correlation matrix, and the marginal
distributions are equal across elements even in the hierarch-
ical case. This algorithm along with a script to plot vari-
ous outcomes, for any dimension and number of subjects,
is included in the supplementary material. As with the LKJ
approach, estimates are somewhat regularised towards zero.
To obtain the raw population distribution matrix square
root R, the correlation matrix square root Xis pre-multiplied
by a diagonal matrix Scontaining standard deviations. These
standard deviations may be sampled a number of ways de-
pending on how much information one has about their ex-
pected scale. By default, we sample from a standard normal
distribution and exponentiate, just as for subject level stand-
ard deviations. When there are concerns about the informat-
iveness of such a prior at very small values, one may instead
sample from a standard normal prior distribution, truncated
3When speaking of matrix square roots in this context, we mean
a matrix Rthat when multiplied by its own transpose as in RR>,
yields the matrix Z. This is not a more regular matrix square root
where Zwould be given by RR, but neither is Ra Cholesky or
other well known factor, as it is symmetric and not lower or upper
triangular.
11
below at zero, without any transformation. In both cases,
the expected scale of subject level deviations for each para-
meter is set by the tform function, described in the follow-
ing section. When necessary, the prior for the scale of raw
population distributions can be altered, by multiplying the
vector of standard deviations by a scaling vector that is fixed
in advance.
tform operator - parameter transforms and priors
The tform operator achieves two things. The first, is to
convert the raw subject level parameters of each subjects dy-
namic and measurement models, from the standard normal
space we use for sampling, to a range of dierently shaped
distributions. The second, is to set the prior distribution over
our parameter space. Because the raw parameters are on a
standard normal scale, applying a simple linear transform
multiplying by two would give a prior with a standard de-
viation of two. In general, the transformations and resulting
priors we discuss here are proposed as reasonable starting
points for a range of typical situations, not as perfectly robust
catch-all solutions. The ctsem software we discuss later al-
lows for them to be easily altered. The transforms we choose,
and resulting shape of the prior distributions, depends on the
requirements of the specific parameter types. For instance
as we have already discussed, standard deviation parameters
cannot be negative, so a simple approach for those is for the
tform operator to perform an exponential operation. Such
parameter boundaries also imply that the subject level para-
meters are unlikely to be normally, or symmetrically, distrib-
uted, particularly as means of the population distributions ap-
proach the boundaries – a change in standard deviation from
1.00 to 0.01 is more dramatic than a change from 1.00 to
1.99. Along with the general shape and boundaries of the
prior distribution, the scale is of course also important. As
a general approach we have aimed for scales that support
inference using standardised and centered data, with para-
meters at relatively normal magnitudes – neither extremely
large nor extremely small, as such parameters will tend to
generate numeric problems anyway. Further, when subject
level parameters are estimated at an upper or lower bound
in such a model, it can indicate the need for model respe-
cification (such as including higher order terms) or rescaling
the time variable. Another factor to take into account with
regard to transformation, is that there is also a need to be
able to fix parameters to specific, understandable values, as
for instance with elements of the diusion matrix Q, which
for higher order models will generally require a number of
elements fixed to 0. This possibility can be lost under cer-
tain multivariate transformations. A final important factor for
deciding on a transformation is that of sampling eciency.
Sampling eciency is typically reduced when parameters are
correlated (with respect to the sampling procedure), because
a random change in one parameter requires a corresponding
non-random change in another to compensate, complicating
ecient exploration of the parameter space. While the use of
modern sampling approaches like Hamiltonian Monte Carlo
(Betancourt & Girolami, 2013) and the no U-turn sampler
(Homan & Gelman, 2014) mitigate these issues to some ex-
tent, minimizing correlations between parameters through
transformations still substantially improves performance. A
further eciency consideration is the inclusion of sucient
prior information to guide the sampler away from regions
where the likelihood of the data approaches zero and the
gradient of the likelihood is relatively flat.
Subject level covariance matrices are created in a similar
way to that described for the population distribution cov-
ariance matrix, in that we pre-multiply a correlation matrix
square root by a diagonal matrix containing the standard de-
viations to obtain a covariance matrix square root, and then
post-multiply this matrix by its transpose to obtain a full co-
variance matrix. The correlation matrix square root is ob-
tained as per the algorithm already discussed, and standard
deviation parameters within the subject level models are ob-
tained by exponentiating a multiple (we have chosen a value
of four) of the raw parameter, which results in a prior similar
to an independence Jereys, or reference scale prior (Bern-
ardo, 1979), but is regularized away from the low or high
extremes to ensure a proper posterior.
Φi j =e4x(27)
where xis the raw parameter and jdenotes the location of
any partial correlation parameters in the subject level para-
meter vector Φi. This approach, wherein mass reduces to 0
at parameter boundaries, is used for all subject level paramet-
ers subject to boundaries, because typically at such boundar-
ies other parameters of the model become empirically non-
identified and optimization or sampling procedures can run
into trouble.
Because intercept and regression type parameters need not
be bounded, for these we simply scale the standard normal to
the desired range (i.e., level of informativeness) by multiplic-
ation. We could of course also add some value if we wanted
a non-zero mean.
Diagonals of the drift matrix A– the temporal auto eects
– are transformed to be negative, with probability mass rel-
atively uniformly distributed for discrete time autoregressive
eects between 0 and 1, given a time interval of 1, but de-
clining to 0 at the extremes.
Φi j =log(e1.5x+1) (28)
where xis the raw parameter and jdenotes the location of
any partial correlation parameters in the subject level para-
meter vector Φi. We have opted to use a bounded distribu-
tion on the drift auto eects for pragmatic reasons – values
greater than 0 represent explosive, non-stationary processes
12 DRIVER
that are in most cases not theoretically plausible. While al-
lowing for such values may point to misspecification more
readily, the constrained form results in what we believe is
a computationally simpler and sensible prior distribution for
genuine eects – but the model and software allows for this
to be easily changed, and we have also successfully tested a
simple normal distribution.
We have found that odiagonals of the drift matrix A
the temporal cross eects – function best when specified in a
problem dependent manner. For basic first order processes,
they can simply be left as multiplications of the standard nor-
mal distribution. For higher order processes, it may help to
parameterize the cross eects between a lower and higher or-
der component, which determine for instance the wavelength
of an oscillation, similarly to the auto eects, ensuring neg-
ative values.
Figure 4 plots the resulting prior densities when using the
described transformations. Note that of course the density
for a variance is directly related to the standard deviation,
and the density plot for an autoregression assumes that the
time interval is 1 with no cross eects involved. For the
sake of completeness we include a prior density for all other
parameters, such as the drift cross eects, intercepts, and
regression type parameters, although these just use a simple
multiplication of the standard normal.
Software implementation
The hierarchical continuous time dynamic model has been
implemented as an extension to the ctsem software (Driver
et al., 2017) for R (R Core Team, 2014). Originally, ctsem
was designed to perform maximum likelihood estimation of
continuous time structural equation models as they are de-
scribed in Voelkle et al., 2012, in which the structural equa-
tion matrices are set up in the RAM (reticular action model)
format (J. Jack McArdle & McDonald, 1984). Individual
specific time intervals are accounted for by definition vari-
ables, and these are coupled with matrix algebra functions to
determine the expected means and covariance matrices for
each individual. The need for complex functions like the
matrix exponential made the OpenMx software (Neale et al.,
2016) an obvious choice for fitting the models. In this ori-
ginal form of ctsem however, random-eects are only pos-
sible to estimate for intercept parameters. This is a primary
motivation for this extension to a hierarchical Bayesian for-
mulation, where all parameters may vary across individuals
according to a simultaneously estimated distribution. To fit
this new hierarchical form of the model, we use a recursive
state-space formulation in which expectations for each time
point are modeled conditional on the prior time point, and
rely on the Stan software (Carpenter et al., 2017) for model
estimation and inference.
Stan is a probabilistic programming language with some
0 1 2 3 4 5
0.0 0.2 0.4 0.6 0.8 1.0
0 5 10 15 20 25
0.00 0.02 0.04 0.06 0.08 0.10
Variance
Value
Density
-5 -4 -3 -2 -1 0
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.5 1.0 1.5
Autoregression |t=1
Value
Density
-1.0 -0.5 0.0 0.5 1.0
0.0 0.2 0.4 0.6 0.8 1.0
-4 -2 0 2 4
0.0 0.1 0.2 0.3 0.4 0.5
Other parameters
Value
Density
Figure 4. Priors for population distribution means. Correla-
tion plot shown is a marginal distribution for a 6 ×6 matrix.
similarities to the BUGS language (Bayesian inference us-
ing Gibbs sampling) (Spiegelhalter, Thomas, Best, Gilks &
Lunn, 1996) language, but greater flexibility. While the
switch to a hierarchical Bayesian approach oers a range of
benefits, it comes at the price of additional computation time,
and the necessity of specifying priors. In cases where compu-
tation time or the use of priors is problematic, or one wishes
to develop a specific model structure not available with a re-
cursive state-space formulation, the classic form of ctsem for
frequentist inference may be used via some dierent function
arguments.
Software usage
The ctsem (Driver et al., 2017) software is avail-
able via the R software repository CRAN, using the R
code install.packages('ctsem'). While full details
on usage of the software are provided in the help files
of the mentioned functions and the supplementary ctsem
package vignette, ‘Introduction to Hierarchical Continuous
Time Dynamic Models With ctsem’, http://cran.r-project.
org/package=ctsem/vignettes/hierarchical.pdf, we describe
the fundamentals here. The main functions of this exten-
sion to ctsem are the ctModel and ctStanFit functions. The
ctModel function allows the user to specify the continuous
time matrices as any combination of fixed values or freely
13
estimated parameters. The ctStanFit function translates the
specification from ctModel into a model in the Stan lan-
guage, combines this model with specified data, and estim-
ates the model. Summary and plot functions are available for
the output object, and additional details are available by dir-
ectly applying rstan (Stan Development Team, 2016a) func-
tions to the rstan fit output, available within the ctStanFit out-
put as myfit$stanfit.
Simulation study
To confirm and demonstrate the performance of our spe-
cification of the hierarchical Bayesian continuous time dy-
namic model, we have conducted a small simulation study,
using version 2.5.0 of the ctsem software. For the study,
we used a model similar to that in the empirical study of
wellbeing dynamics we show in the next section, with model
structure and true parameter values of the simulation similar
to those estimated by the empirical work. The generating
model we used specified individual variability over (nearly)
all parameters, with the individual parameters distributed ac-
cording to the default priors we have already discussed –
while we do not think this is crucial for performance of the
model and software, we will not directly address questions
of this form of potential misspecification. The initial latent
variance parameters (T0VAR) were not specified as individu-
ally varying, as they are problematic to estimate when free
across subjects. One of the drift matrix auto eects was also
set to have zero variance, to test the performance of the spe-
cification when a random eect is specified in cases of no
variation (because a standard deviation of exactly zero is not
possible to obtain with this approach, for computing simula-
tion quantities such as coverage rates, we specified 0.01 as
an arbitrary, approximately zero quantity). Using the ctsem
software, we fit data from this generating model using the
true model structure, with default priors and start values. We
generated data for either 10 or 30 observation time points,
50 or 200 subjects, a cross eect of 0.00 or 0.20, and with
or without variation in time intervals between observations.
The conditions were fully crossed, and each condition was
repeated 200 times. The exact model structure and parameter
values used for data generation can be seen in Table 1, and
an R script is available in the supplementary material. Three
chains with 500 warmup and 500 sampling iterations were
used (Hamiltonian Monte Carlo typically generates far more
eective samples per iteration than other common sampling
approaches). To ensure that only usable simulation runs were
included, we rejected runs if any split Rhat scale reduction
factor (Gelman & Rubin, 1992) from the samples was greater
than 1.10, or if the average eective samples was lower than
100. The split Rhat value refers to the ratio of between
chain to within chain variance, while the number of eective
samples estimates the number of independent samples out
of the total drawn, after accounting for autocorrelation in the
chains. In the worst case conditions with only 10 time points,
this approach resulted in roughly 12% of runs being dropped.
The median number of eective samples across all paramet-
ers and conditions was 335, and the median split Rhat was
1.01.
Table 1 shows true values, mean point estimates, RMSE
(root mean squared error), 95% credible interval widths, and
coverage rates, for all estimated parameters. If the parameter
is a subject level parameter, the relevant symbol represent-
ing the parameter is also shown. For this case 200 subjects
were measured at 30 times, but for other examples (N=200 &
T=10, N=50 & T=10, N=50 & T=30) see Appendix C. Due
to limitations with the number of simulation runs possible,
the simulation measures reported will be subject to some
sampling variability, and thus should not be treated as per-
fectly precise – nevertheless they oer insight into the per-
formance of the model.
From Table 1 we can see that with 200 subjects and
30 time points, inferential properties are for the most part
very good – empirical coverage rates of the 95% inter-
vals are approximately 95%, there is minimal to no bias
in parameters, and error from point estimates is compar-
able across approaches. The only deviation from this pic-
ture is for correlations between the random eects. While
there are far too many (hundreds) of such correlations to
table individually, for the most part the generating model
had these set to zero. These zero correlations are estim-
ated well, with conservative coverage rates of 1.00 for most
parameters, with a few scattered in the 0.90 to 1.00 re-
gion. The only poor performer here was a spurious cor-
relation between the correlation parameter in the diusion
matrix and the initial latent mean for the first process – it
is not obvious to us why this occurs but it may be due to
fixing the initial latent variance across subjects. Stronger
correlations, as between the tabled manifestmeans paramet-
ers (corr_manifestmeans_Y1_manifestmeans_Y2, which sets
the correlation between baseline levels of each manifest vari-
able), exhibit a mild bias towards zero.
Table C1 shows that with only 50 subjects and 10 time
points, inferential properties are still reasonable, though not
optimal. Some biases in the population means are now ap-
parent, similar to those one would expect when fitting time
series models to single subject data with too few time points.
This pattern could also be predicted by the generally too high
estimates of population standard deviations – the population
model is providing too little regularization for the subject
level parameters. Depending on ones priorities, in cases with
less data available such as this one it may be worthwhile
to scale the prior for population standard deviations down-
wards. Tables C2 and C3 show that combinations of either
50 subjects and 30 time points (1500 measurements), or 200
subjects and 10 time points (2000 measurements), are eect-
ive for our test model and perform quite similarly. Of course,
14 DRIVER
Table 1
Simulation results for the full random eects model, with 200 subjects and 30 time points.
Parameter Symbol True value Mean point est. RMSE CI width Coverage
T0mean_eta1 η1[1] 1.00 0.99 0.10 0.40 0.95
T0mean_eta2 η1[2] 1.00 1.00 0.10 0.44 0.97
drift_eta1_eta1 A[1,1] -0.40 -0.40 0.05 0.19 0.96
drift_eta2_eta1 A[1,2] 0.00 0.00 0.02 0.08 0.98
drift_eta1_eta2 A[2,1] 0.10 0.11 0.03 0.15 0.97
drift_eta2_eta2 A[2,2] -0.20 -0.20 0.03 0.11 0.96
manifestvar_Y1_Y1 Θ[1,1] 1.00 1.00 0.06 0.25 0.95
manifestvar_Y2_Y2 Θ[2,2] 1.00 1.00 0.06 0.23 0.96
diusion_eta1_eta1 Q[1,1] 1.00 1.00 0.09 0.34 0.94
diusion_eta2_eta1 Q[2,1] 0.50 0.50 0.05 0.22 0.97
diusion_eta2_eta2 Q[2,2] 1.00 0.99 0.07 0.27 0.96
T0var_eta1_eta1 Q
1[1,1] 0.50 0.42 0.14 0.57 0.97
T0var_eta2_eta1 Q
1[2,1] 0.10 0.21 0.38 1.73 0.99
T0var_eta2_eta2 Q
1[2,2] 0.51 0.44 0.14 0.60 0.97
manifestmeans_Y1 τ[1] 0.50 0.50 0.09 0.38 0.96
manifestmeans_Y2 τ[2] 0.00 -0.01 0.11 0.42 0.95
hsd_manifestmeans_Y1 1.00 1.01 0.07 0.29 0.96
corr_manifestmeans_Y1_manifestmeans_Y2 0.50 0.46 0.09 0.26 0.86
hsd_manifestmeans_Y2 1.00 1.01 0.08 0.32 0.95
hsd_drift_eta1_eta1 0.15 0.14 0.06 0.22 0.92
hsd_drift_eta1_eta2 0.15 0.15 0.03 0.11 0.95
hsd_drift_eta2_eta1 0.01 0.03 0.02 0.06 0.98
hsd_drift_eta2_eta2 0.08 0.06 0.04 0.14 0.93
hsd_diusion_eta1_eta1 0.64 0.65 0.07 0.30 0.95
hsd_diusion_eta2_eta1 0.30 0.24 0.11 0.35 0.89
hsd_diusion_eta2_eta2 0.64 0.65 0.06 0.25 0.95
hsd_manifestvar_Y1_Y1 0.64 0.65 0.06 0.23 0.95
hsd_manifestvar_Y2_Y2 0.64 0.65 0.05 0.22 0.95
both give less precise estimates than the 200 subjects 30 time
points example already discussed.
Table 2 shows an extended set of simulation measures,
including empirical power with a 5% alpha level. These
measures are shown with respect to the cross eect parameter
drift_eta1_eta2 only, for conditions with and without a true
cross eect, over dierent combinations of Nand T. In this
case, we also included results from a more restrictive mixed-
eects style model, in which only intercept parameters were
allowed to vary across subjects. While here this more re-
strictive model is obviously misspecified, we suspect there
are many such cases as the mixed-eects form is much more
commonly used, because it is simpler to specify. For cases
where a cross eect of 0.2 exists, the results show that even
in the worst case condition of N=50 & T=10 power is tol-
erable, and with N either increased to 200, or T increased
to 30, power is very good. In general, it seems that under
this model specification, an increase in the number of data
points, regardless of whether via more Nor more T, seems to
improve results similarly. In cases where no cross eect ex-
ists, the correctly specified full model is only returning false
positives at rates equal to or lower than the 5% alpha, but
problems become apparent with the misspecified, mixed ef-
fects only model – while we should only wrongly conclude
an eect exists 5% of times, as Nand Tincrease, so too
does the likelihood of making a spurious inference. However,
some comfort for those relying on mixed-eects models can
be taken in the mean estimates and RMSE values, indicating
that while inference focusing on significance testing is likely
to be problematic, actual parameter estimates for the cross
eects are unlikely to be too far wrong.
To check performance when misspecification occurs in the
opposite direction, with random eects specified in the fitted
model while none exist in the generating model, we ran the
same simulations with a mixed-eects generating model and
a full random eects model fit to the data. As it is not a
focus of our investigation we do not provide all the tables,
but will mention only that the general trend is that popu-
lation mean parameters are estimated similarly to when a
random-eects generating model is used, though biases re-
lated to over-estimation of the population standard deviation
parameters are somewhat stronger. Empirical coverage rates
for population mean parameters were in general still around
95%.
Dynamics of overall life satisfaction and health
satisfaction
To highlight usage of the ctsem software and possibilit-
ies of the model, we assessed the dynamics of overall life
satisfaction and satisfaction with health, for a selection of
subjects from the long running German socioeconomic panel
(GSOEP) study, using version 29 of the GSOEP data. Ques-
tions regarding the fundamental structure of, and causal rela-
tions between, subjective wellbeing constructs are still very
much open (Busseri & Sadava, 2011; Schimmack, 2008).
Dynamic models have been posed as one way of understand-
ing these constructs better (Headey & Muels, 2014). Given
the long time-span over which such constructs are expected
to exhibit substantial change — in the order of months, years,
or even decades — gathering sucient data to reasonably
15
Table 2
Extended simulation results regarding only the cross-eect parameter. Columns refer to conditions with and without a true
cross eect (CE), with N =50 or 200 subjects, and T =10 or 30 time points, collapsed over varying intervals conditions. Both
full random eects and more restricted mixed eects models were fit. The mixed eects model represents a common approach,
in which only intercept parameters vary over subjects.
CE =0, N =50 CE =0, N =200 CE =0.2, N =50 CE =0.2, N =200
Measure T =10 T =30 T =10 T =30 T =10 T =30 T =10 T =30
Coverage - full 0.95 0.98 0.95 0.98 0.95 0.96 0.96 0.97
Coverage - mixed 0.89 0.73 0.87 0.60 0.89 0.62 0.74 0.28
Mean Est. - full 0.08 0.02 0.08 0.01 0.37 0.23 0.23 0.21
Mean Est. - mixed 0.04 -0.01 -0.01 -0.02 0.30 0.15 0.16 0.13
RMSE - full 0.15 0.05 0.15 0.03 0.25 0.08 0.09 0.04
RMSE - mixed 0.15 0.06 0.05 0.03 0.23 0.10 0.10 0.08
CI width - full 0.71 0.25 0.71 0.12 0.93 0.33 0.37 0.17
CI width - mixed 0.52 0.14 0.16 0.06 0.79 0.21 0.25 0.09
Power - full 0.05 0.02 0.05 0.02 0.55 0.92 0.91 1.00
Power - mixed 0.11 0.28 0.13 0.40 0.50 0.81 0.77 0.99
fit single-subject models is dicult. Further, although the
GSOEP is administered yearly, variability in timing of the
questionnaire each year results in some variability of time
intervals, which if ignored, may add noise and bias. Thus,
a hierarchical continuous time approach, in which we lever-
age variation in the time intervals between questionnaires as
additional information, and inform our estimates of specific
subjects dynamics based on many other subjects, seems par-
ticularly applicable to such data.
Core questions
While many questions might be asked using this approach,
the questions we will address here are the very general ones:
What are the temporal dynamics of overall and health satis-
faction? How much variation in such dynamics exists? Are
there relations between cross-sectional age and dynamics, or
between certain aspects of dynamics and other aspects?
Sample details
For this example we randomly sampled 200 subjects from
the GSOEP that had been observed at all 29 occasions in
our data. Such a sub-sample of course no longer benefits
from the population representative nature of the GSOEP. This
sample resulted in subject ages (calculated at the midpoint of
their participation in the study) from 30 to 77 years (mean
=49.23 and sd=10.69). In our subsample, time intervals
between measurements ranged from 0.25 to 1.75 years, with
a mean of 1 and standard deviation of 0.11.
Constructs
We are interested in the constructs of satisfaction with
health, and overall satisfaction with life. These were meas-
ured on an 11 point scale. Translations of the questions from
German are as follows: “How satisfied are you today with
the following areas of your life?” followed by a range of
items including “your health”. These scales ranged from 0,
totally unhappy, to 10, totally happy. Overall satisfaction was
assessed separately, as “How satisfied are you with your life,
all things considered?” and ranged from 0, completely dis-
satisfied, to 10, completely satisfied.
Model
The individual level dynamic model was specified as a
first order bivariate model. All parameters of the bivariate
latent process and measurement models were left free, ex-
cept for the process intercept and loading matrices. The pro-
cess intercepts were set to 0, as the measurement model here
accounts for non-zero equilibria, and the factor loading mat-
rix to an identity matrix, for model identification purposes.
All free dynamic and measurement model parameters (ex-
cept initial latent variance) were also free to vary across sub-
jects. Variation in subject level parameters was predicted by
age and age squared, with residual variation arising from a
multivariate population distribution of parameters. R code to
generate this model, plot the population distribution priors,
and view the resulting Stan code, is provided in Appendix A.
The matrix forms for the subject level model are shown in
Figure 5, with underbraced notations indicating the relevant
matrix as described in the model section of this paper, and
when appropriate, also the name of the matrix in the ctsem
software model specification.
Means of population distributions
Shown in Table 3 are the posterior density intervals, point
estimates, and diagnostic statistics of the means of the popu-
lation distributions4, attained after sampling with four chains
of 2000 iterations each, giving potential scale reduction
factors (Gelman & Rubin, 1992) below 1.01 and a minimum
of 194 eective samples per parameter. Note that the median,
4Note that any variance /covariance related parameters are re-
ported as standard deviations and unconstrained correlation square
roots – regular covariance matrices are reported in 4.
16 DRIVER
d"η1
η2#t
| {z }
dη(t)
=
"drift_overallSat_overallSat drift_overallSat_healthSat
drift_healthSat_overallSat drift_healthSat_healthSat #
| {z }
A
|{z}
DRIFT
"η1
η2#t
| {z }
η(t)
+"0
0#
|{z}
b
|{z}
CINT
dt+
"diusion_overallSat_overallSat 0
diusion_healthSat_overallSat diusion_healthSat_healthSat#
| {z }
G
|{z}
DIFFUSION
d"W1
W2#(t)
| {z }
dW(t)
"Y1
Y2#(t)
| {z }
Y(t)
="1 0
0 1#
| {z }
Λ
|{z}
LAMBDA
"η1
η2#(t)
| {z }
η(t)
+"manifestmeans_overallSat
manifestmeans_healthSat #
| {z }
τ
|{z}
MANIFESTMEANS
+"1
2#(t)
| {z }
(t)
"1
2#(t)
| {z }
(t)
N
"0
0#,"manifestvar_overallSat_overallSat 0
0 manifestvar_healthSat_healthSat#
| {z }
Θ
|{z}
MANIFESTVAR
Figure 5. Matrix specification of the subject level model of overall life satisfaction and satisfaction with health. Underbraced
notations denoting the symbol used to represent the matrix in earlier formulas, and where appropriate also the matrix name in
the ctsem specificationa.
aStrictly speaking, the diusion matrix is actually the covariance matrix GG>, but Gis the way it is specified in ctsem.
50%, is reported as the point estimate, as this is typically
closest to the true value in simulations reported in Table 1.
Table 3
Posterior intervals and point estimates for means of estim-
ated population distributions
Symbol 2.5% 50% 97.5%
T0mean_overallSat η1[1] 0.53 0.80 1.09
T0mean_healthSat η1[2] 1.10 1.40 1.72
drift_overallSat_overallSat A[1,1] -0.49 -0.33 -0.22
drift_overallSat_healthSat A[1,2] 0.02 0.08 0.19
drift_healthSat_overallSat A[2,1] -0.07 -0.01 0.04
drift_healthSat_healthSat A[2,2] -0.25 -0.17 -0.10
diusion_overallSat_overallSat Q[1,1] 0.48 0.58 0.72
diusion_healthSat_overallSat Q[2,1] 1.00 1.47 2.30
diusion_healthSat_healthSat Q[2,2] 0.51 0.60 0.70
manifestvar_overallSat_overallSat Θ[1,1] 0.78 0.85 0.91
manifestvar_healthSat_healthSat Θ[2,2] 0.98 1.05 1.11
manifestmeans_overallSat τ[1] 6.67 6.87 7.04
manifestmeans_healthSat τ[2] 6.04 6.30 6.53
T0var_overallSat_overallSat Q
1[1,1] 1.28 1.50 1.74
T0var_healthSat_overallSat Q
1[2,1] 0.35 0.57 0.91
T0var_healthSat_healthSat Q
1[2,2] 1.12 1.48 1.78
Going down the list of parameters shown in Table 3, the
T0mean parameters are positive for both overall and health
satisfaction. Because the T0means represent initial state es-
Table 4
Posterior intervals and point estimates for means of estim-
ated population distributions of covariance matrices
Matrix Symbol 2.5% 50% 97.5%
T0VAR Q
1[1,1] 1.63 2.24 3.02
T0VAR Q
1[2,1] 0.64 1.13 1.73
T0VAR Q
1[2,2] 1.25 2.17 3.15
DIFFUSION Q[1,1] 0.23 0.34 0.52
DIFFUSION Q[2,1] 0.23 0.31 0.41
DIFFUSION Q[2,2] 0.26 0.36 0.49
dtDIFFUSION Qt=1,[1,1] 0.19 0.27 0.38
dtDIFFUSION Qt=1,[2,1] 0.19 0.25 0.33
dtDIFFUSION Qt=1[2,2] 0.22 0.30 0.40
asymDIFFUSION Q
[1,1] 0.53 0.72 0.96
asymDIFFUSION Q
[2,1] 0.59 0.77 0.99
asymDIFFUSION Q
[2,2] 0.77 0.99 1.29
timates for the latent processes, and because we have spe-
cified the model such that the latent processes have long
run means of 0 (by including non-zero manifest means and
leaving the continuous time intercepts fixed to 0), the pop-
17
0 2 4 6 8 10
0.0 0.2 0.4 0.6 0.8 1.0
Time interval in years
Value
Overall satisfaction AR
Eect of overall sat. on health sat.
Eect of health sat. on overall sat.
Health satisfaction AR
Figure 6. Median and 95% quantiles of auto and cross re-
gressions over time.
ulation means for the T0mean parameters show to what ex-
tent subjects‘ initial states tend to be higher or lower than
their later states. Combining this structure with the posit-
ive values observed, suggests that as time goes by, satis-
faction scores decline somewhat, with a larger decline in
the health domain. Turning to the auto eect paramet-
ers of the drift matrix, drift_healthSat_healthSat is higher
(closer to zero) than drift_overallSat_overallSat, suggesting
that changes in health satisfaction typically persist longer
than changes in overall satisfaction. Sometimes these negat-
ive coecients are confusing for those used to discrete-time
results, but they provide a relatively intuitive interpretation
– the higher above baseline a process is, the stronger the
downwards pressure due to the auto-eect, and the further
below baseline, the stronger the upwards pressure. The cross
eect parameter drift_healthSat_overallSat is very close to
zero, suggesting that changes in overall satisfaction do not
predict later changes in health satisfaction. Conversely how-
ever, drift_overallSat_healthSat is substantially positive, so
changes in health satisfaction predict later changes in overall
satisfaction, in the same direction. To understand these tem-
poral dynamics due to the drift matrix more intuitively, the
expected auto and cross regressions over time are shown in
Figure 6, where we see for instance that the expected eect
of health satisfaction on overall satisfaction peaks at time in-
tervals of around 2-4 years. Note that this does not imply
that the relation between the two satisfaction processes is
changing with time – yet because the processes are some-
what stable, and a change at one time (in the plot, a change
of 1.00 at time zero) persists, this allows for consequences of
the initial change to continue building for some time. The
diusion parameters are dicult to interpret on their own
because a) they do not reflect total variance in the system,
but rather only the rate of incoming variance (unexplained
exogenous inputs), and b) also because of the unusual trans-
formation necessary in the o-diagonal parameters. Instead
we can consider the diusion covariance matrix, as well as
the discrete time and asymptotic forms, as output by ctsem
and shown in Table 4. The discrete time diusion (dtDIF-
FUSION) matrix tells us how much change is likely to oc-
cur in the latent processes over that interval, and the ex-
tent of covariation. The asymptotic diusion (asymDIFFU-
SION) covariance matrix gives the total latent process vari-
ance and covariance. From Table 4, the dtDIFFUSION vari-
ance parameters show that over a time span of 1 year, over-
all and health satisfaction processe have similar levels of
variance. The asymDIFFUSION variance parameters show
that in the longer term, there is somewhat more variation in
health satisfaction. The o-diagonal, covariance parameters
show substantial positive covariation, so when overall satis-
faction rises due to unmodeled factors, so too is health satis-
faction likely to rise. This is unsurprising, as we might ex-
pect that overall and health satisfaction certainly share some
common causes. Turning back to Table 3,The two mani-
fest indicators show similar standard deviations for meas-
urement error (manifestvar_overallSat_overallSat and mani-
festvar_healthSat_healthSat) for each process, which implies
that measurement limitations and short term situational influ-
ences (e.g., a sunny day) contribute similar levels of variance
to each indicator. Further, it seems that such influences con-
tribute a similar amount of variance to the observed scores
as the latent process variability – given that we fixed factor
loadings to 1.00, they are directly comparable. This is not
however suggestive that the measures are unreliable per se,
as the measurement error and total latent process variance
only reflect within-person variability – to consider reliability
one would need to also consider the between-person vari-
ance in the manifest means, which in this model account for
baseline levels of the processes. The manifest means para-
meters reflect the intercepts of the manifest indicators, and
here both are at roughly similar levels, between 6 and 7. The
absolute value here is probably not so interesting, it is rather
the individual dierences, and relations between individual
dierences and other parameters or covariates, that are of
most interest, and these are discussed later. The T0var para-
meters reflect the initial variance and covariance of the latent
processes, and again, the covariance matrix parameters from
Table 4 are likely to be more interpretable.
Covariate eects
Cross-sectional age and age squared were included as time
independent predictors for all parameters. Appendix B con-
tains a table of full results, but because we are looking at a
combined linear and quadratic eect, examining the multi-
dimensional credible region is much simpler with the plots
of Figure 7 (generated using the ctStanTIpredeects function
from ctsem). This figure shows the 50% credible intervals for
the eect of age on the model parameters, as well as the im-
plications of the model parameters such as the discrete time
eects, the asymptotic diusion variance. Discrete time (dt)
matrices were computed for a time interval of two years, and
18 DRIVER
when appropriate, correlations (cor) are shown alongside co-
variances. 50% was chosen for the sake of interpretability of
plots in this example, and we do not mean for this interval to
be taken as strong support for any interpretations – although
the estimated eects are regularised by the standard normal
prior, to reduce the chance of large spurious eects.
The top left plot of Figure 7 focuses on initial and
baseline levels of the processes, with the strongest eect be-
ing that the baseline level of health satisfaction (manifest-
means_healthSat) declines with age. Conversely, baseline
overall satisfaction seems to rise marginally. Although the
t0mean parameters, which reflect the dierence between ini-
tial and baseline latent values, appear to show some change
with age, it is quite modest and we would not make too much
of it at this point. For a more complete modelling of within-
person trends an additional latent process with zero diusion
could be included (see Driver & Voelkle, 2017, for an ex-
ample specified using ctsem).
The plots in the centre and top right display the temporal
dynamics from the drift matrix, with the centre plot show-
ing the continuous time parameters and the right a discrete
time eect matrix. While no eects stand out as highly sub-
stantial, there is a rise in the persistence of changes in over-
all satisfaction (drift_overallSat_overallSat) with older ages.
There is also something of a decrease in the eect of health
satisfaction on overall satisfaction in older ages, and a corres-
ponding increase in the alternate eect – that of overall sat-
isfaction on health. Turning to the lower left and centre plots
containing the continuous and discrete time diusion para-
meters, younger and older subjects appear to show higher
within-subject variability in overall satisfaction, while when
it comes to health satisfaction it is only the older subjects
showing increased variability. With increasing age, random
(in the sense that the model does not predict them) changes to
health and overall satisfaction appear to become highly cor-
related, with the correlation approaching 1.00 – given this it
may be plausible to model such subjects using a single latent
process. Finally, the lower right plot suggests that measure-
ments of health satisfaction, and to a lesser extent measure-
ments of overall satisfaction, become more error prone with
age.
Variance of population distributions
Shown in Table 5 are the posterior density intervals of
standard deviation parameters of the population distribu-
tions, showing to what extent individual subjects parameter
values tended to dier from the population mean values,
in ways that could not be predicted by our age covari-
ates – the unexplained between-subjects variance in a para-
meter. So while every subject has their own particular set
of parameters, the estimated mean of the parameter distri-
bution over all subjects is shown in Table 3, and the stand-
ard deviation is shown in Table 5. Individual dierences in
the T0means is unsurprising, as they simply reflect dier-
ences in the initial level of the latent process states. Look-
ing at the temporal dynamics parameters, both auto eects
(drift_overallSat_overallSat and drift_healthSat_healthSat)
show some variability, reflecting individual dierences in the
persistence of changes in overall and health satisfaction pro-
cesses. Regarding cross eects, the eect of health on overall
satisfaction drift_overallSat_healthSat) shows more variab-
ility than the reverse direction drift_healthSat_overallSat),
which seems consistent with the strength of the eects at
the population level. That is, the non-existent or very weak
average eect of overall satisfaction on health shows little
variability, while the stronger eect of health satisfaction on
later overall satisfaction varies more across subjects – the
eect is in general more important, but more so for some
people than others. The between-subjects variability in both
the diusion diagonals and manifestvar parameters suggests
that some subjects exhibit more latent variability than others,
and that some subjects measurements are noisier than others.
Dierences in latent variability may be due to genuine dif-
ferences, in that some people just experience more change in
satisfaction, but could also be due to dierent scale usage –
some people may interpret a change of 1.00 as less meaning-
ful than others, and consequently score themselves with more
variability year to year. Dierences in the manifestvar para-
meters may reflect that some people respond to the survey
questions with more randomness, or more influence of tran-
sient conditions. Between-subjects variation in the manifest
means parameters reflects individual dierences in baseline
levels of the two satisfaction processes, and we see slightly
more variation for health satisfaction here.
Table 5
Posterior intervals and point estimates for standard devi-
ations of estimated population distributions
2.5% 50% 97.5%
T0mean_overallSat 0.03 0.24 0.74
T0mean_healthSat 0.02 0.28 1.06
drift_overallSat_overallSat 0.13 0.25 0.44
drift_overallSat_healthSat 0.01 0.09 0.19
drift_healthSat_overallSat 0.00 0.02 0.06
drift_healthSat_healthSat 0.07 0.16 0.27
diusion_overallSat_overallSat 0.24 0.34 0.47
diusion_healthSat_overallSat 0.57 1.02 1.78
diusion_healthSat_healthSat 0.20 0.30 0.42
manifestvar_overallSat_overallSat 0.25 0.30 0.35
manifestvar_healthSat_healthSat 0.32 0.37 0.43
manifestmeans_overallSat 0.81 0.94 1.09
manifestmeans_healthSat 1.07 1.25 1.46
Correlations in individual dierences
Correlations can help to better understand the individual
dierences in dynamics, and those whose 95% interval do
19
30 40 50 60 70
0 2 4 6 8
Age
Effect
MANIFESTMEANS[1,1]
MANIFESTMEANS[2,1]
T0MEANS[1,1]
T0MEANS[2,1]
30 40 50 60 70
−0.4 −0.2 0.0 0.2 0.4
Age
Effect
DRIFT[1,1]
DRIFT[1,2]
DRIFT[2,1]
DRIFT[2,2]
30 40 50 60 70
0.0 0.2 0.4 0.6 0.8 1.0
Age
Effect
dtDRIFT[1,1]
dtDRIFT[1,2]
dtDRIFT[2,1]
dtDRIFT[2,2]
30 40 50 60 70
0.2 0.4 0.6 0.8 1.0 1.2
Age
Effect
DIFFUSION[1,1]
DIFFUSION[2,1]
DIFFUSION[2,2]
DIFFUSIONcor[2,1]
30 40 50 60 70
0.2 0.4 0.6 0.8 1.0 1.2
Age
Effect
dtDIFFUSION[1,1]
dtDIFFUSION[2,1]
dtDIFFUSION[2,2]
dtDIFFUSIONcor[2,1]
30 40 50 60 70
0.5 1.0 1.5 2.0
Age
Effect
MANIFESTVAR[1,1]
MANIFESTVAR[2,2]
Figure 7. Estimated eect of age on model parameters. Index 1 refers to overall satisfaction, and index 2 to health satisfaction.
Diusion parameters shown are variance and covariance, unless specified as cor (correlation). Discrete time matrices computed
with time interval of 2 years.
not contain zero are shown in Table 6, with health and overall
satisfaction abbreviated to H and O, respectively. While we
note the highly exploratory nature of this approach, we will
also remind readers of the regularising prior on such correla-
tions, reducing the likelihood of false positives.
From Table 6, we see that baseline levels of health and
overall satisfaction are highly correlated, which is not too
surprising. However, what may be somewhat surprising is
that the table is largely comprised of correlations involving
a baseline level parameter, and the pattern of results is
very similar for both health and overall satisfaction paramet-
ers. This general pattern is such that increases in baseline
levels (manifestmeans) predict reductions in both measure-
ment error (manifestvar) and latent process variance (diu-
sion diagonals). Specific to health satisfaction, the manifest-
means_H__drift_H_H parameter shows that higher baseline
levels predicts reduced persistence of changes (drift diagon-
als). There are also some correlations that do not involve
the baseline level parameters, but either regard the cross
eect of health on overall satisfaction, or variance terms.
The drift_O_H__drift_O_O parameter shows a negative re-
lation between persistence of changes in overall satisfac-
tion (the drift auto eect), and the eect of health satis-
faction on overall satisfaction (the cross eect). So, sub-
jects for whom changes in overall satisfaction do not per-
sist as long, show a stronger eect of health on overall
satisfaction. This is a compensatory correlation, in that
it serves to maintain the predictive value of changes in
health on later overall satisfaction, even though the pre-
dictive value of overall satisfaction for itself at later times
is weaker. Regarding correlations in variance parameters,
manifestvar_H_H__manifestvar_O_O shows that someone
who exhibits high measurement error for health satisfaction
is likely to also show high measurement error on overall
satisfaction, while manifestvar_H_H__diusion_H_H and
manifestvar_O_O__diusion_H_H show that subjects who
have more measurement error, also have more variance in
health satisfaction.
20 DRIVER
Table 6
Posterior intervals and point estimates for correlations in
random-eects, where 95% interval does not include zero.
2.5% 50% 97.5%
manifestmeans_H__diusion_O_O -0.53 -0.29 -0.02
manifestmeans_O__diusion_H_H -0.55 -0.29 -0.03
manifestmeans_H__diusion_H_H -0.60 -0.31 -0.04
manifestmeans_H__manifestvar_O_O -0.50 -0.32 -0.13
manifestvar_O_O__diusion_H_H 0.11 0.38 0.63
manifestmeans_O__manifestvar_H_H -0.52 -0.39 -0.25
manifestmeans_O__diusion_O_O -0.63 -0.40 -0.13
diusion_H_H__diusion_O_O 0.01 0.43 0.72
manifestmeans_H__drift_H_H -0.69 -0.43 -0.04
manifestvar_H_H__diusion_H_H 0.18 0.49 0.74
manifestmeans_H__manifestvar_H_H -0.61 -0.49 -0.36
manifestmeans_O__manifestvar_O_O -0.69 -0.55 -0.38
drift_O_H__drift_O_O -0.82 -0.56 -0.04
manifestvar_H_H__manifestvar_O_O 0.41 0.56 0.70
manifestmeans_H__manifestmeans_O 0.67 0.76 0.82
0 5 10 15 20 25 30 35
3 4 5 6 7 8 9
Years
Value
0 5 10 15 20 25 30 35
3 4 5 6 7 8 9
Years
Value
OverallSat
HealthSat
Figure 8. Observed scores for a single subject and Kalman
smoothed score estimates from the subjects dynamic model.
Individual level analyses
For individual level analysis, we may compute predicted
(based on all prior observations), updated (based on all prior
and current observations), and smoothed (based on all ob-
servations) expectations and covariances from the Kalman
filter, based on specific subjects models. All of this is readily
achieved with the ctKalman function from ctsem. This ap-
proach allows for: predictions regarding individuals‘ states
at any point in time, given any values on the time dependent
predictors (external inputs such as interventions or events);
residual analysis to check for unmodeled dependencies in
the data; or simply as a means of visualization, for com-
prehension and model sanity checking purposes. An ex-
ample of such is depicted in Figure 8, where we see observed
and smoothed scores with uncertainty (95% intervals), for a
randomly selected subject from our sample. Note that we
show the uncertainty regarding the latent process – the not
displayed measurement uncertainty results in many observa-
tions outside of the 95% interval.
Discussion
We have described a hierarchical continuous time dy-
namic model, for the analysis of repeated measures data
of multiple subjects. In addition, we have introduced a
Bayesian extension to the free and open-source software ct-
sem (Driver et al., 2017) that allows fitting this model. The
subject level model is flexible enough to allow for many
popular longitudinal model structures in psychological re-
search to be specified, including forms of latent trajectories
(Delsing & Oud, 2008), autoregressive cross-lagged models
(and thus the latent change score formulation, see Voelkle &
Oud, 2015), higher order oscillating models, or some mix-
ture of approaches. We can use the model to learn about the
measurement structure, variance and covariance of the latent
processes stochastic aspects, and temporal dynamics of the
processes. Because it is a continuous time model, there are
no restrictions on the timing of data collection. Time inter-
vals between observations can vary both within and across
individuals, and indeed such variability is likely to improve
the estimation of the model (Voelkle & Oud, 2013). The hier-
archical aspect ensures that while each individual can have
their own distinct model parameters, data collected from
other subjects is still informative, leading generally to im-
proved individual and population estimates. The inclusion of
subject specific covariates (time independent predictors) may
be used to improve estimation and inform about relationships
to parameters, but is also not necessary as any sort of control,
as heterogeneity at the individual level is accounted for by
allowing random subject specific parameters.
Results from a limited simulation study suggest that the
Bayesian form oers reliable inferential properties when the
correct model is specified, with only marginal reductions
when an overly complex population model is specified. In
the more limited data conditions, certain point estimates were
biased, though coverage rates in general remained good. Un-
der conditions similar or worse than our 50 subject 10 time
point example, additional thought regarding model complex-
ity and prior specification may be helpful. Of course, while
the generating model was largely based on the empirical res-
ults from the GSOEP data, if the generating model had para-
meter values that diered substantially from what we have
posed as a rough norm (see Figure 4), performance of the
approach may be dierent. Thus, care should of course be
taken that variables are scaled and centred, and the time
scale is such that neither extremely high or extremely low
auto eects are expected – often a time scale that gives time
intervals around 1.00 serves to achieve this. The only es-
timates showing noticeable bias in the simulation conditions
with more data (200 subjects with 30 time points) were those
of strong correlations in random eects, which are slightly
pushed towards zero. While alternate approaches to correla-
tion matrices that avoid this are in theory possible, some mild
regularisation of the many (hundreds in our example) random
21
eects correlations has in our experience helped to minimise
computational diculties and improve performance. While
more comprehensive power and simulation studies will help
to improve the understanding of this and similar models un-
der a wider range of conditions, our results serve to provide
some confidence in the procedure and software.
Now, while the model as specified is relatively flexible,
there are of course some limitations: Although parameter dif-
ferences between individuals are fully accounted for, for the
most part we do not account for parameter variability within
individuals. Unpredictable changes in the process means can
be accounted for through augmentation of the state matrices
(Delsing & Oud, 2008; Oud & Jansen, 2000), and known
changes can be accounted for using time dependent predict-
ors (Driver & Voelkle, 2017), but changes in dynamic rela-
tionships, or randomly fluctuating parameters, are at present
not accounted for. Such eects generally imply non-linear
stochastic dierential equations, and alternative, more com-
plex approaches to dynamic modelling are necessary, as for
example in Lu et al. (2015).
In the form described, the model and software is also lim-
ited to a linear measurement model with normally distributed
errors. However, an option to avoid the Kalman filter and ex-
plicitly sample latent states is included in the ctsem software,
so although most alternate measurement models are not ex-
plicitly specifiable via the ctsem functions at present, non-
linearities in the measurement model and link are possible
with some modifications to the Stan model that is output by
the software.
Additional limitations are those typically found when
dealing with either complex hierarchical models, dynamic
models, or specifically continuous time dynamic models.
These include computation time, parameter dependencies,
and parameter interpretation.
Computation time is generally high for hierarchical time
series models estimated with sampling approaches typical
to Bayesian modeling. In this case the continuous time as-
pect adds some additional computations for every measure-
ment occasion where the time interval changes. Based on
our experience so far, using ctsem and Stan in their present
forms on a modern desktop PC requires anywhere from a few
minutes for simple univariate process models with limited
between-subject random eects, to days for complex multi-
process multi-measurement models with many subjects and
covariates. The satisfaction example we discussed took ap-
proximately 6 hours to complete.
Parameter dependencies in dynamic models pose di-
culties both during estimation, and during interpretation.
While on the one hand it would be great to specify the model
using parameters that were entirely independent of each
other, this is not always possible for every parameter, and
even when it is, may limit the initial specification and or com-
pound diculties with interpretation. For instance, rather
than parameterizing the innovation covariance matrix using
the standard deviations and correlation of the continuous-
time diusion matrix, one alternative we have explored is to
estimate directly the asymptotic innovation covariance mat-
rix, Q
t=. This is beneficial in that the covariance matrix
parameters are made independent of the temporal dynam-
ics parameters, and in this case we think also assists in in-
terpretation. Unfortunately however, it limits the possibility
to specify more complex dynamics where many elements of
the continuous time diusion matrix may need to be fixed
to zero, but the asymptotic latent variance matrix cannot be
determined in advance. While the specific parameterizations
we propose may need to be adapted, in this paper we have
aimed for a middle ground approach, trying to balance com-
peting factors for typical use cases.
Interpretation of the continuous time dynamic parameters
is typically less intuitive for the applied researcher than the
related discrete time parameters. While this may be true, we
would argue that the two forms can yield dierent interpreta-
tions, and that in general it is helpful to consider both the un-
derlying continuous time parameters as well as the discrete
time implications. We hope our explanations of the various
parameters encourage people to explore these aspects. We
also hope to encourage better intuition for dynamic models
in general, by plotting expected eects (as per Figure 6) over
a range of time intervals.
While neither hierarchical random-eects models nor
continuous time dynamic models are themselves new, there
have been few approaches put forward combining the two.
We believe it is important to describe such a model and
provide the means to estimate it, because accurate estimates
for single subject dynamic models may require very large
numbers of time points, and because inaccuracies in the es-
timation of one parameter propagate through the majority of
others due to dependencies in the models. We have high-
lighted a potential application, by showing that we can es-
timate individual specific models for subjects in long term
panel studies such as the GSOEP. Such studies may con-
tain information over a very long amount of time, which is
great in terms of investigating developmental questions, but
the data are usually not dense enough to estimate individual
specific models without the help of information from other
individuals. Amongst other things, from this analysis we
found some evidence that changes in health satisfaction pre-
dict later changes in overall satisfaction, and also that people
functioning at dierent baselines on the satisfaction scales
tend to show dierent dynamics and measurement properties.
We hope that this work provides a means for more under-
standing of processes occurring within individuals, and the
factors that relate to dierences in such processes between
individuals.
22 DRIVER
Acknowledgments
The data used in this work were made available to us by
the German Socio-Economic Panel Study (SOEP) at the Ger-
man Institute for Economic Research (DIW), Berlin. Soft-
ware used for the development, analyses and documenta-
tion includes knitr (Xie, 2014), texStudio, R (R Core Team,
2014), RStudio (RStudio Team, 2016), nyx (von Oertzen,
Brandmaier & Tsang, 2015), OpenMx (Neale et al., 2016),
Stan (Carpenter et al., 2017), and RStan (Stan Development
Team, 2016a). The formal methods team at the Max Planck
Institute for Human Development, and JHL Oud, provided
valuable input. Thanks to three reviewers for their diligent
and thoughtful reviews.
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Appendix A
Dynamic model of wellbeing — R script
This script installs and loads ctsem, and specifies the dynamic model of wellbeing used in this paper (although we cannot
distribute the GSOEP data). The script also plots the priors used for the population level model, as well as examples of
possible priors for the subject level parameters – because the subject level priors depend on the estimated population level
parameters.
install.packages("ctsem")
require(ctsem)
model<-ctModel(type='stanct', LAMBDA=diag(2),
n.manifest=2, manifestNames=c('overallSat','healthSat'),
n.latent=2, latentNames=c('overallSat','healthSat'),
n.TIpred=2, TIpredNames=c("meanAge", "meanAgeSq"))
plot(model, wait=TRUE)
fakedata=matrix(1,nrow=5,ncol=6)
colnames(fakedata)=c('id','time','overallSat','healthSat','meanAge','meanAgeSq')
fakedata[,'time']=1:5
fit<-ctStanFit(fakedata,model,fit=FALSE)
cat(fit$stanmodeltext)
27
Appendix B
Time independent predictor eects
Table B1
Posterior distributions for time independent predictor eects
2.5% mean 97.5% z
Age_manifestmeans_healthSat -0.64 -0.41 -0.17 -3.49
AgeSq_drift_healthSat_overallSat -0.00 0.03 0.07 1.50
AgeSq_drift_overallSat_overallSat -0.22 -0.09 0.02 -1.50
Age_diusion_healthSat_overallSat -0.07 0.31 0.78 1.45
AgeSq_drift_overallSat_healthSat -0.10 -0.04 0.02 -1.34
Age_diusion_healthSat_healthSat -0.05 0.07 0.18 1.20
AgeSq_manifestmeans_healthSat -0.10 0.14 0.37 1.11
Age_manifestvar_healthSat_healthSat -0.03 0.03 0.10 1.03
AgeSq_diusion_overallSat_overallSat -0.05 0.05 0.16 1.02
Age_drift_healthSat_healthSat -0.15 -0.05 0.05 -0.96
Age_drift_overallSat_overallSat -0.19 -0.06 0.08 -0.87
AgeSq_drift_healthSat_healthSat -0.05 0.04 0.13 0.87
AgeSq_T0mean_overallSat -0.16 0.12 0.38 0.84
Age_diusion_overallSat_overallSat -0.17 -0.05 0.08 -0.77
Age_T0mean_healthSat -0.18 0.12 0.42 0.76
Age_drift_healthSat_overallSat -0.06 -0.02 0.02 -0.76
AgeSq_diusion_healthSat_healthSat -0.14 -0.04 0.07 -0.67
Age_manifestmeans_overallSat -0.12 0.05 0.23 0.57
AgeSq_diusion_healthSat_overallSat -0.24 0.11 0.54 0.56
AgeSq_T0mean_healthSat -0.38 -0.08 0.22 -0.49
Age_T0mean_overallSat -0.34 -0.06 0.22 -0.42
Age_manifestvar_overallSat_overallSat -0.05 0.01 0.07 0.33
Age_drift_overallSat_healthSat -0.07 -0.01 0.06 -0.24
AgeSq_manifestmeans_overallSat -0.18 0.01 0.19 0.13
AgeSq_manifestvar_overallSat_overallSat -0.06 -0.00 0.05 -0.11
AgeSq_manifestvar_healthSat_healthSat -0.06 -0.00 0.06 -0.06
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Appendix C
Additional simulation results
Table C1
Simulation results for 50 subjects and 10 time points, with all parameters varying over subjects.
Parameter Symbol True value Mean point est. RMSE CI width Coverage
T0mean_eta1 η1[1] 1.00 1.03 0.32 1.40 0.95
T0mean_eta2 η1[2] 1.00 1.00 0.35 1.62 0.96
drift_eta1_eta1 A[1,1] -0.40 -0.69 0.39 1.38 0.92
drift_eta2_eta1 A[1,2] 0.00 0.12 0.19 0.77 0.95
drift_eta1_eta2 A[2,1] 0.10 0.22 0.20 0.82 0.95
drift_eta2_eta2 A[2,2] -0.20 -0.46 0.31 1.10 0.91
manifestvar_Y1_Y1 Θ[1,1] 1.00 0.97 0.18 0.77 0.97
manifestvar_Y2_Y2 Θ[2,2] 1.00 0.94 0.18 0.74 0.95
diusion_eta1_eta1 Q[1,1] 1.00 1.08 0.35 1.42 0.96
diusion_eta2_eta1 Q[2,1] 0.50 0.50 0.19 1.05 0.99
diusion_eta2_eta2 Q[2,2] 1.00 1.15 0.32 1.27 0.95
T0var_eta1_eta1 Q
1[1,1] 0.50 0.56 0.15 1.05 1.00
T0var_eta2_eta1 Q
1[2,1] 0.10 0.27 0.35 1.87 1.00
T0var_eta2_eta2 Q
1[2,2] 0.51 0.60 0.17 1.10 1.00
manifestmeans_Y1 τ[1] 0.50 0.48 0.30 1.28 0.94
manifestmeans_Y2 τ[2] 0.00 0.01 0.34 1.50 0.96
hsd_manifestmeans_Y1 1.00 0.96 0.21 0.95 0.96
corr_manifestmeans_Y1_manifestmeans_Y2 0.50 0.27 0.25 0.94 0.94
hsd_manifestmeans_Y2 1.00 0.94 0.24 1.08 0.97
hsd_drift_eta1_eta1 0.15 0.48 0.33 2.25 0.98
hsd_drift_eta1_eta2 0.15 0.22 0.14 0.65 0.99
hsd_drift_eta2_eta1 0.01 0.16 0.14 0.54 0.87
hsd_drift_eta2_eta2 0.08 0.53 0.42 2.76 0.97
hsd_diusion_eta1_eta1 0.64 0.65 0.28 1.15 0.94
hsd_diusion_eta2_eta1 0.30 1.26 0.32 8.81 1.00
hsd_diusion_eta2_eta2 0.64 0.69 0.22 1.06 0.96
hsd_manifestvar_Y1_Y1 0.64 0.65 0.16 0.67 0.96
hsd_manifestvar_Y2_Y2 0.64 0.63 0.16 0.66 0.93
Table C2
Simulation results for 50 subjects and 30 time points, with all parameters varying over subjects.
Parameter Symbol True value Mean point est. RMSE CI width Coverage
T0mean_eta1 η1[1] 1.00 1.02 0.20 0.84 0.98
T0mean_eta2 η1[2] 1.00 1.01 0.22 0.92 0.97
drift_eta1_eta1 A[1,1] -0.40 -0.45 0.11 0.42 0.94
drift_eta2_eta1 A[1,2] 0.00 0.02 0.04 0.19 0.98
drift_eta1_eta2 A[2,1] 0.10 0.13 0.07 0.29 0.97
drift_eta2_eta2 A[2,2] -0.20 -0.24 0.07 0.26 0.94
manifestvar_Y1_Y1 Θ[1,1] 1.00 0.97 0.13 0.52 0.95
manifestvar_Y2_Y2 Θ[2,2] 1.00 0.98 0.11 0.48 0.96
diusion_eta1_eta1 Q[1,1] 1.00 1.05 0.18 0.73 0.96
diusion_eta2_eta1 Q[2,1] 0.50 0.48 0.11 0.42 0.95
diusion_eta2_eta2 Q[2,2] 1.00 1.04 0.15 0.60 0.95
T0var_eta1_eta1 Q
1[1,1] 0.50 0.49 0.14 0.87 1.00
T0var_eta2_eta1 Q
1[2,1] 0.10 0.18 0.34 1.86 1.00
T0var_eta2_eta2 Q
1[2,2] 0.51 0.51 0.14 0.91 1.00
manifestmeans_Y1 τ[1] 0.50 0.50 0.19 0.76 0.95
manifestmeans_Y2 τ[2] 0.00 -0.00 0.21 0.84 0.94
hsd_manifestmeans_Y1 1.00 1.01 0.16 0.61 0.94
corr_manifestmeans_Y1_manifestmeans_Y2 0.50 0.40 0.17 0.56 0.90
hsd_manifestmeans_Y2 1.00 1.00 0.17 0.70 0.95
hsd_drift_eta1_eta1 0.15 0.18 0.11 0.44 0.96
hsd_drift_eta1_eta2 0.15 0.17 0.06 0.24 0.95
hsd_drift_eta2_eta1 0.01 0.05 0.04 0.14 0.95
hsd_drift_eta2_eta2 0.08 0.11 0.07 0.31 0.98
hsd_diusion_eta1_eta1 0.64 0.70 0.17 0.69 0.94
hsd_diusion_eta2_eta1 0.30 0.25 0.15 0.57 0.97
hsd_diusion_eta2_eta2 0.64 0.70 0.15 0.60 0.93
hsd_manifestvar_Y1_Y1 0.64 0.68 0.12 0.52 0.94
hsd_manifestvar_Y2_Y2 0.64 0.68 0.12 0.51 0.95
29
Table C3
Simulation results for 200 subjects and 10 time points, with all parameters varying over subjects.
Parameter Symbol True value Mean point est. RMSE CI width Coverage
T0mean_eta1 η1[1] 1.00 1.02 0.16 0.67 0.96
T0mean_eta2 η1[2] 1.00 1.01 0.18 0.82 0.97
drift_eta1_eta1 A[1,1] -0.40 -0.44 0.11 0.47 0.96
drift_eta2_eta1 A[1,2] 0.00 0.02 0.05 0.24 0.97
drift_eta1_eta2 A[2,1] 0.10 0.12 0.07 0.32 0.97
drift_eta2_eta2 A[2,2] -0.20 -0.24 0.09 0.33 0.95
manifestvar_Y1_Y1 Θ[1,1] 1.00 0.99 0.09 0.37 0.96
manifestvar_Y2_Y2 Θ[2,2] 1.00 0.98 0.08 0.34 0.95
diusion_eta1_eta1 Q[1,1] 1.00 1.02 0.15 0.65 0.96
diusion_eta2_eta1 Q[2,1] 0.50 0.51 0.11 0.46 0.97
diusion_eta2_eta2 Q[2,2] 1.00 1.03 0.14 0.54 0.95
T0var_eta1_eta1 Q
1[1,1] 0.50 0.46 0.13 0.72 1.00
T0var_eta2_eta1 Q
1[2,1] 0.10 0.28 0.38 1.81 0.99
T0var_eta2_eta2 Q
1[2,2] 0.51 0.49 0.14 0.77 1.00
manifestmeans_Y1 τ[1] 0.50 0.48 0.15 0.63 0.95
manifestmeans_Y2 τ[2] 0.00 -0.01 0.18 0.78 0.96
hsd_manifestmeans_Y1 1.00 0.99 0.10 0.46 0.96
corr_manifestmeans_Y1_manifestmeans_Y2 0.50 0.39 0.14 0.53 0.92
hsd_manifestmeans_Y2 1.00 0.98 0.13 0.59 0.96
hsd_drift_eta1_eta1 0.15 0.17 0.10 0.46 0.98
hsd_drift_eta1_eta2 0.15 0.15 0.06 0.25 0.96
hsd_drift_eta2_eta1 0.01 0.07 0.06 0.17 0.94
hsd_drift_eta2_eta2 0.08 0.13 0.09 0.40 0.98
hsd_diusion_eta1_eta1 0.64 0.65 0.12 0.47 0.95
hsd_diusion_eta2_eta1 0.30 0.27 0.17 0.74 0.99
hsd_diusion_eta2_eta2 0.64 0.66 0.09 0.40 0.96
hsd_manifestvar_Y1_Y1 0.64 0.65 0.07 0.28 0.93
hsd_manifestvar_Y2_Y2 0.64 0.65 0.07 0.27 0.94
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... temporal regressions and covariances) for single individuals. To overcome these limits, the present study used Hierarchical Bayesian Continuous Time Dynamic Modelling 68 to model smartphone use and well-being as continuously changing and interacting processes, with system characteristics that vary across individuals 67,68 . To understand better these features, we can consider two extremes of a continuum in dynamic models. ...
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... To ensure that the system parameters representing dynamics actually reflect dynamics and not stable individual differences, starting points and intercept terms are allowed to (co)vary across subjects using a random-effects approach. For simplicity of exposition, other system parameters are assumed constant over subjects, though this is not necessary (For the hierarchical Bayes formulation see Driver & Voelkle, 2018a, though note that use of non-linear filtering in modern versions of ctsem also allows for regular maximum likelihood estimation with complex heterogeneity). The R script used for fitting is provided in the supplementary material, as is randomly generated data from the fitted model, since it is not possible to provide the GSOEP data. ...
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