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ORIGINAL RESEARCH
published: 07 April 2016
doi: 10.3389/fpsyg.2016.00486
Frontiers in Psychology | www.frontiersin.org 1April 2016 | Volume 7 | Article 486
Edited by:
Holmes Finch,
Ball State University, USA
Reviewed by:
Alessandro Giuliani,
Istituto Superiore di Sanità (Italian
NIH), Italy
Ahmed M. ElKenawy,
Mansoura University, Egypt
*Correspondence:
Tanja Krone
tanja.krone@gmail.com
Specialty section:
This article was submitted to
Quantitative Psychology and
Measurement,
a section of the journal
Frontiers in Psychology
Received: 17 December 2015
Accepted: 21 March 2016
Published: 07 April 2016
Citation:
Krone T, Albers CJ and
Timmerman ME (2016) Comparison of
Estimation Procedures for Multilevel
AR(1) Models. Front. Psychol. 7:486.
doi: 10.3389/fpsyg.2016.00486
Comparison of Estimation
Procedures for Multilevel AR(1)
Models
Tanja Krone *, Casper J. Albers and Marieke E. Timmerman
Department of Psychometrics and Statistics, Heymans Institute, University of Groningen, Groningen, Netherlands
To estimate a time series model for multiple individuals, a multilevel model may be used.
In this paper we compare two estimation methods for the autocorrelation in Multilevel
AR(1) models, namely Maximum Likelihood Estimation (MLE) and Bayesian Markov
Chain Monte Carlo. Furthermore, we examine the difference between modeling fixed
and random individual parameters. To this end, we perform a simulation study with a fully
crossed design, in which we vary the length of the time series (10 or 25), the number of
individuals per sample (10 or 25), the mean of the autocorrelation (−0.6 to 0.6 inclusive,
in steps of 0.3) and the standard deviation of the autocorrelation (0.25 or 0.40). We
found that the random estimators of the population autocorrelation show less bias and
higher power, compared to the fixed estimators. As expected, the random estimators
profit strongly from a higher number of individuals, while this effect is small for the fixed
estimators. The fixed estimators profit slightly more from a higher number of time points
than the random estimators. When possible, random estimation is preferred to fixed
estimation. The difference between MLE and Bayesian estimation is nearly negligible.
The Bayesian estimation shows a smaller bias, but MLE shows a smaller variability (i.e.,
standard deviation of the parameter estimates). Finally, better results are found for a
higher number of individuals and time points, and for a lower individual variability of the
autocorrelation. The effect of the size of the autocorrelation differs between outcome
measures.
Keywords: time series analysis, autocorrelation, Bayesian MCMC, multisubject, maximum likelihood estimation,
simulation study
1. INTRODUCTION
The electronic revolution allows for new and exciting research possibilities. One such possibility
that has become increasingly easy to use is ecological momentary assessment (c.f., Shiffman et al.,
2008; Bos et al., 2015) through electronic devices such as the mobile phone. This advancement
allows, with little hassle for the individuals, multiple measurements per individual per day at
the researcher’s discretion (Bolger and Laurenceau, 2013). The data provided through ecological
momentary assessment, often denoted as intensive longitudinal data (Hamaker et al., 2015), give
ample opportunities for studying complex processes, involving the trends and dynamics of human
behavior and experience. The latter pertains to studying how aspects of behavior and/or experience
evolve across time, and how aspects mutually influence each other. Using these kinds of data,
studies have been done pertaining to, for example, emotional complexity and age (Brose et al.,
2015), dynamics of depression (Kuppens et al., 2010; Erbas et al., 2014; Kashdan and Farmer, 2014),
and the relation between affect and stress (Scott et al., 2014).
Krone et al. Multilevel AR(1) Models
Intensive longitudinal data of several individuals fall under
the category of multilevel data. Multilevel data are collected
according to a nested sampling design, resulting in data with
a hierarchical structure (e.g., Snijders and Bosker, 1999; Hox,
2010). A two-level example is univariate longitudinal data of
multiple individuals, where the time points at level 1 are nested
within the individuals at level 2. In psychological sciences,
momentary assessment data pertain to longitudinal series of
limited length, collected among a limited number of individuals,
creating a multilevel data set. To analyze these data, one can use
multilevel models. In the analysis of longitudinal data, we can
discern two different focuses: the trend and the dynamics across
time. To study the trend across time, a multilevel regression
model for repeated measures can be used. Herewith, one could
use either dummy-variables (also known as indicator or design
variables), to indicate effects pertaining to each time point, or
time itself as a predictor (e.g., Snijders and Bosker, 1999).
To study the dynamics across time, a model is needed
for describing the relationships between scores at successive
measurements. This is typically done using an autoregressive
model (Box and Jenkins, 1976). The simplest variant is an
autoregressive model of the first order, the AR(1) model for
short. For multilevel data with multiple individuals, Suls et al.
(1998) used an autoregressive component in a multilevel model
with random coefficients to assess change in mood over time.
In this model, the autoregressive parameter was composed of a
population parameter, a parameter dependent on the predictor
neuroticism, and a subject dependent noise parameter. The same
approach was used by Kuppens et al. (2010), who included
self-esteem as a predictor for the autocorrelation. Both authors
interpreted the autoregressive parameter as reflecting the degree
of inertia, which is the tendency to retain the status-quo over
time. An often encountered problem in time series analysis is
the violation of the assumption of independent errors, due to
autocorrelated noise. To account for this effect, a multilevel
model including autocorrelated noise was proposed by Goldstein
et al. (1994). Note that although Goldstein et al. (1994) denote
this model as an autoregressive model, it is actually a moving
average model, according to the common terminology (Box and
Jenkins, 1976).
At the moment, it is unclear how efficient the estimation
methods of different multilevel model variants are for intensive
longitudinal data. Several simulation studies have been
conducted to compare the different estimators for single
case AR(1) models (Huitema and McKean, 1991; DeCarlo and
Tryon, 1993; Arnau and Bono, 2001; Solanas et al., 2010; Krone
et al., 2015). While the empirical standard error is lowest for the
classical estimation method denoted by r1(Walker, 1931), the
bias is lowest for iterative estimators (Krone et al., 2015). For all
methods, the empirical power is low for series with less than 50
time points. For a true autocorrelation below |0.40|, the power
is below 80% for all compared estimation methods (Krone et al.,
2015). This is consistent with the advice of a lower bound of 50
time points for any time series modeled with an AR(1) model, as
given by Box and Jenkins (1976).
In this paper, we focus on the AR(1) model in a
multilevel setting, for relatively short time series and numbers
of individuals. We do so because these characteristics are
typical for intensive longitudinal data, and the properties
of multilevel AR(1) model estimators have been investigated
scarcely. Furthermore, the inclusion of multiple individuals
may have a profound effect on the bias, variability and power
of the estimators. In a recent paper, Jongerling et al. (2015)
compared the MLE and the Bayesian multilevel AR(1) estimators.
Their simulation design included manipulations of the intercept
variance and of the covariance between the autocorrelation and
the error variance. However, their design lacked manipulations
of the mean and variance of the autocorrelation, central to
the current paper. Further, they only used person centering
in MLE models and only used a random effect for the error
variance in the Bayesian model, which means that their design
was not fully-crossed. Jongerling et al. (2015) concluded that
the estimation may be improved by including a random effect
for the error variance and by refraining from person-centering.
The differences in bias they found are small and inconsistent; in
certain conditions increasing sample size and time series length
also seems to increase rather than decrease the bias. As such, their
model estimates may be biased. While they raise an interesting
point with regard to individual error variances and person-
centering the data, we will first consider a more basic comparison
between estimation methods using the same model.
For multilevel models, several of the estimation methods
used in single subject designs are unavailable. Two closed
form estimators that can be used for multilevel models are
generalized least squares (GLS) and generalized estimation
equations (GEE) (Liang and Zeger, 1986). Although these
methods have the benefit of being faster than iterative methods,
i.e., MLE and Bayesian MCMC, the resulting estimates show
bias and need a large amount of data points to achieve an
acceptable standard error (Hox, 2010). Better fitting estimators
for the ML-AR(1) model are iterative estimators, specifically
the Maximum Likelihood Estimation (MLE) and the Bayesian
MCMC estimation (Hox, 2010). In an earlier study, we also
found this for single subject data, which leads us to use MLE and
Bayesian MCMC in this paper (Krone et al., 2015).
In this paper we use a simulation study to quantify
the differences between two model variants for multilevel
autocorrelated data, and between two estimation methods,
being MLE and Bayesian Markov-Chain Monte Carlo (Bayesian
MCMC) estimation. In the next part of this paper, we discuss the
multilevel model and the estimation methods. This is followed
by an explanation of the simulation study design, the results of
the simulation study, and a discussion on the implications for
designing empirical studies involving intensive longitudinal data
and properly modeling the resulting data.
2. THE MULTILEVEL AUTOREGRESSIVE
LAG 1 MODEL
The ML-AR(1) model we use is a random coefficients model (e.g.,
Snijders and Bosker, 1999; Hox, 2010). The model has two levels:
the first level holds the time points, as the second level holds the
individuals. The level 1 model is based on the AR(1) model for a
Frontiers in Psychology | www.frontiersin.org 2April 2016 | Volume 7 | Article 486
Krone et al. Multilevel AR(1) Models
single individual (Box and Jenkins, 1976):
yt,n=µn+φn(yt−1,n−µn)+et,n,et,n∼N(0, σe),(1)
where yt,nis the score of individual n(n=1,2,...,N) at time t
(t=1,2,...,T), µnthe intercept, φnthe autocorrelation, and et,n
is the error term. The error terms follow a normal distribution
with mean zero and standard deviation σeand are independent
of each other and of the observations yt,n.
In this paper we compare two ways of modeling multilevel
data: the random model and the fixed model. The difference
between these models is based on the theory behind the sampling
of individuals, and is expressed in the level 2 model. In the
random model, as used in the random coefficients approach,
the individuals are assumed to be drawn randomly from a
certain population. As such, the parameters of the individuals are
assumed to be drawn randomly from the population distribution
of the parameter concerned. It is common, but not required, to
assume a normal distribution for the individual parameters. We
will use the normality assumption in this paper.
The fixed model makes no assumption with regard to the
sampling of the individuals. To reflect this, the parameters of the
fixed model are estimated freely. This implicitly defines the level
2 model, as the joint distribution of the individually estimated
parameters for all individuals is hereby defined. Due to the
free parameter estimation, these model estimates would be the
same as when the time series of each individual were modeled
separately. This implies that the standard deviation of the error is
σe,n, and hence may vary across individuals.
For the random model, a level 2 model must be defined which
captures the assumed population distributions of the parameters.
The level 2 model we use is:
µn=γ0,0+U0,n,(2)
φn=γ0,1+U1,n,(3)
with:
U0,n∼N(0, σU0,n),(4)
U1,n∼N(0, σU1,n). (5)
where γ0,0is the population intercept, U0,nis the individual
specific deviation from the population intercept for individual n,
γ0,1is the population autocorrelation and U1,nis the individual
specific deviation from the population autocorrelation. Note that
the standard deviation of the error, σe, is assumed to be equal
across the population of individuals (unlike the fixed model),
and independent of both U0,nand U1,n. The composite model,
expressing both levels in one model, is:
yt,n=γ0,0+γ0,1(yt−1,n−γ0,0−U0,n)+U0,n+
U1,n(yt−1,n−γ0,0−U0,n)+et,n,et,n∼N(0, σe).(6)
2.1. Estimation Methods
2.1.1. MLE
For MLE, the distinction can be made between full maximum
likelihood (FML) and restricted maximum likelihood (RML, also
known as REML). The difference lies in how the likelihood is
estimated: FML includes both the regression coefficients and
the variance components in the likelihood, whereas RML only
includes the variance components. The regression coefficients for
RML are estimated in a secondary step (Hox, 2010). In general,
the FML is easier to calculate. Furthermore, the FML allows for an
overall chi-square test for two models that differ in the fixed part,
which the RML generally does not. However, when estimating
the variance, the FML model is biased since it does not take into
account the number of fixed parameters (Bryk and Raudenbush,
1992, p. 46), while the RML has asymptotically unbiased variance
estimates.
For the random model using MLE (henceforth denoted as
MLE-R), we will use RML (Harville, 1977) with the “Bound
Optimization BY Quadratic Approximation” algorithm (Powell,
2009). The method we use estimates the random parameters
under the assumptions of normality, in line with typical
applications in social sciences (Hox, 2010; Goldstein, 2011). The
multilevel implementation of the MLE we use is not specifically
made for autocorrelation measures, and may thus produce non-
stationary autocorrelation values, i.e., |ˆ
φn|>1. The number
of non-stationary results obtained will be touched upon in the
results section.
For the fixed model using MLE (henceforth denoted as
MLE-F), we will use the ‘Broyden-Fletcher-Goldfarb-Shanno’
algorithm (Byrd et al., 1995). The estimation method we use
is especially programmed for autocorrelation estimation and, as
such, produces stationary autocorrelation estimates. For both
MLE approaches, the algorithm may fail to reach convergence.
The number of non-convergent results will be touched upon
in the results section. Furthermore, both MLE approaches are
unable to handle missing data, other than by removing the
whole case from the analysis. To retain the data, an Expectation-
Maximization algorithm (Dempster et al., 1977), also used in
latent variable modeling, may be used. However, in this paper we
will assume that the full data is available.
2.1.2. Bayesian MCMC
Estimation through Bayesian MCMC is very versatile with
respect to the models and distributions that can be estimated.
The MCMC-method we use for both the fixed and random
(denoted as BAY-F and BAY-R, respectively) Bayesian estimators
is Hamiltonian Monte Carlo (HMC), a generalization of the
Metropolis-Hastings algorithm (Metropolis et al., 1953; Hastings,
1970) that allows for an efficient estimation of the parameters
(Gelman et al., 2013). An added advantage of the Bayesian
approach is the possibility to deal with missing data optimally,
i.e., without casewise deletion. For AR(1) models it is possible
to apply the autoregressive model on the estimated score of the
missing time point, instead of on the observed score itself. This
allows the estimation to continue past the missing data points,
adjusting the estimation as soon as the next time point is observed
again.
2.2. Procedure
In this study, we aim to examine the comparative quality of
MLE and Bayesian MCMC estimation for the autocorrelation
Frontiers in Psychology | www.frontiersin.org 3April 2016 | Volume 7 | Article 486
Krone et al. Multilevel AR(1) Models
FIGURE 1 | Flowchart of the study design of the simulation study.
parameter in random and fixed ML-AR(1) models. This results
in four estimators which will be compared: MLE-F, MLE-R,
BAY-F, and BAY-R. For the Bayesian MCMC estimations, we
use the program Rstan (Stan Development Team, 2014). For the
estimation of the MLE-R, we use the package lme4 for R (Bates
et al., 2015). All other analyses, including data generation, are
done using the functions available in the base installation of the
program R (R Core Team, 2015).
3. SIMULATION STUDY
3.1. Simulation Design
To compare the four estimators for the autocorrelation, we
set up a simulation design with data generation, data analyses,
assessment of computational issues and analyses of the results as
shown in Figure 1, with 40 conditions in total. The conditions
stem from a fully crossed experimental design, including
the following factors, with number of factor levels between
parentheses: the length of the time series T(2), the number of
individuals per dataset N(2), the standard deviation σU1,n(2),
and mean γ0,1(5) of the autocorrelation distribution, as used in
Equations (5) and (3). Both Tand Nare either 10 or 25, σU1,n
is either 0.25 or 0.40, γ0,1is set from −0.60 up to 0.60 inclusive,
taking steps of 0.30 for the values in between.
The time series were generated according to Equation (6).
The mean and standard deviation of the error of each series in
each replication is set to zero and one, respectively. The values
of φnwere then drawn from a truncated and rescaled normal
distribution with range −1 to 1, to ensure the resulting time series
were stationary:
φn∝N(γ0,1, σU1,n)τ(−1,1). (7)
3.1.1. Parameter Priors
We performed a small simulation study to examine the sensitivity
for the choice of the hyperparameters of the priors of our
Bayesian model. We considered 2000 replications of a single
simulation condition, using 5000 iterations, taking γ0,0=0.00,
σU1,n=0.40, T=10 and N=10 (see Equation 6). This
condition is one where the prior is expected to have the most
influence, due to the high variability across individuals and the
small amount of data. The prior we use for ˆ
γ0,1for BAY-R and
ˆ
φnfor BAY-F is Berger’s symmetrized reference prior (Berger and
Yang, 1994), which has shown to better perform than the flat prior
for single case AR(1) models (Krone et al., 2015). This prior does
not need hyperparameters.
We tested several hyperparameters for the prior distributions
of µnand σefor the fixed model, and γ0,0,σU0,n,σeand σU1,n
of the random model, as shown in Table 1. Our parameter of
TABLE 1 | Different combinations of priors tested to see their influence on
the posterior results.
Test Fixed model Random model
µnσeµ σµσeσφ
1N∼(0,2) Ŵ∼(2,2) N∼(0,2) Ŵ∼(2,2) Ŵ∼(2,2) Ŵ∼(2,2)
2N∼(0,5) Ŵ∼(2,2) N∼(0,5) Ŵ∼(2,2) Ŵ∼(2,2) Ŵ∼(2,2)
3N∼(1,2) Ŵ∼(2,2) N∼(1,2) Ŵ∼(2,2) Ŵ∼(2,2) Ŵ∼(2,2)
4N∼(1,5) Ŵ∼(2,2) N∼(0,2) Ŵ∼(1,1) Ŵ∼(2,2) Ŵ∼(2,2)
5N∼(0,2) Ŵ∼(1,1) N∼(0,2) Ŵ∼(1,2) Ŵ∼(2,2) Ŵ∼(2,2)
6N∼(0,2) Ŵ∼(1,2) N∼(0,2) Ŵ∼(2,2) Ŵ∼(1,1) Ŵ∼(2,2)
7N∼(0,2) Ŵ∼(2,1) N∼(0,2) Ŵ∼(2,2) Ŵ∼(1,2) Ŵ∼(2,2)
8N∼(0,2) Ŵ∼(2,2) Ŵ∼(2,2) Ŵ∼(1,1)
9N∼(0,2) Ŵ∼(2,2) Ŵ∼(2,2) Ŵ∼(1,2)
primary interest, ˆ
γ0,1, showed small differences across the various
tests. For the random model, the estimates ranged from 0.017
(test 5) to 0.026 (test 9). For the fixed model, the estimates ranged
from 0.033 (test 6) to 0.109 (test 4).
For the random estimator, the estimated γ0,0showed small
differences across the various priors, resulting in estimates
ranging from 0.000 (test 9) to 0.004 (test 3). For the fixed
estimator, the estimated ¯µnranged from 0.000 (test 7) to 0.192
(test 2). The effect of the different priors is most notable for the
posterior of the parameter for which the prior was changed. For
the simulation study, we use the priors of test 1 of Table 1, as these
gave the best results.
3.1.2. Number of Iterations
A preliminary study was performed to decide on the number of
iterations needed for the Bayesian MCMC. Because of the more
complicated model of BAY-R compared to BAY-F, we only tested
the number of iterations for BAY-R. Ten datasets per condition
were used to find the convergence rate as expressed through the
potential scale reduction factor ˆ
R, as can be seen in Table 2. The
potential scale reduction factor shows the ratio of how much the
estimation may change when the number of iterations is doubled,
with a value of 1 indicating that no change is expected (Gelman
and Rubin, 1992; Stan Development Team, 2014). We deemed
the improvements brought by a higher number of iterations
negligible, thus we continued using 3000 total iterations, of which
1500 were burn-in.
3.1.3. Number of Replications
A preliminary study using N=10, T=10, σU1,n=0.40,
and γ0,1= −0.30, with the priors and number of iterations as
specified, showed that the outcome measures (to be introduced in
the next section) started stabilizing after around 1500 replications
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Krone et al. Multilevel AR(1) Models
TABLE 2 | Values of ˆ
Rfor different amounts of iterations for tests with 10
replications per condition using the BAY-R method and for the final
analyses.
Iterations Mean Percentage of ˆ
Rabove:
Total Burn-in ˆ
R1.05 1.1 1.5 1.7
3000 1500 1.01 2.53 0.89 0.02 0.00
4000 2000 1.01 1.97 0.68 0.02 0.00
10,000 5000 1.00 1.54 0.75 0.01 0.00
10,000 8000 1.01 2.13 0.70 0.01 0.00
Final analyses: 3000 iterations with 1500 burn-in
BAY-R 1.00 0.94 0.35 0.01 0.00
BAY-F 1.00 0.02 0.00 0.00 0.00
for all used methods, being stable for all at 2000 replications.
For example, the standard deviations of the estimated mean
γ0,1or φn, depending on estimation method, over replications
was lower than 0.01 at 2000 replications for all used estimators.
Therefore, the number of replications per condition is set to
R=2000. Given that we have 40 conditions, this amounts to
40 ×2000 =80,000 datasets generated.
3.1.4. Summary
Using this simulation design, we can define our study using
the classification for intensive longitudinal data designs as
discussed by Hamaker et al. (2015). We analyze multi-subject
data (where the single-subject case can be seen as a special case).
Since we use the classic AR(1) model, we model a univariate,
stationary, linear process in discrete time. Our variable has
a continuous distribution and is based in the time-domain.
Finally, we model the process and are primarily interested in the
parameters characterizing the process, rather than the descriptive
statistics.
In our simulation study, we consider two measures of
computational problems (i.e., non-convergence and non-
stationary estimates), and six different outcome measures for the
autocorrelation: the bias of ˆ
γ0,1, the bias of ˆσU1,n, the empirical
standard deviation of ˆ
γ0,1, the bias of the standard error of
ˆ
γ0,1, the empirical rejection rate (EPr) of ˆ
γ0,1and the point
and interval estimates of ˆ
γ0,1. For each outcome measure, we
offer a short explanation of the measurement and the obtained
results.
3.2. Results
We start with discussing the rates of non-convergence (MLE-
F) and non-stationarity (MLE-R), followed by the outcome
measures for the autocorrelation. We will only discuss the
conditions where an effect was found; thus if the random
estimator is named but not the fixed estimator, the condition
discussed does not influence the result of the fixed estimator and
vice versa. The graphs presented in this section show the outcome
measures as a function of N,T,σU1,nand γ0,1. The model
parameters will be discussed in the notation used in Equation (6),
the statistics obtained with the random and fixed estimators in
their respective notations as in Equations (6) and (1).
3.2.1. Computational Problems: Non-Convergence
and Non-Stationary Estimates
The MLE-F is occasionally unable to reach convergence in the
estimation of the model, which is connected to the inability to
estimate values outside the range of −1 to 1. Of the 40 conditions,
28 converged for all analyses performed. In total, 0.002% of the
estimates did not reach convergence. The highest percentage
of non-convergence for individual time series is 0.01% for the
condition with N=10,T=25, σU1,n=0.25,and
γ0,1=0.6. Apart from the condition with the highest number of
non-convergence, higher numbers of non-convergence are found
for conditions with larger values of |φ|and conditions with the
highest value of σU1,n.
Out of the 40 conditions, only three had purely stationary
estimates. In total 0.33% of the estimates were non-stationary.
The highest percentage of non-stationary values for the MLE-R
was 1.23%, for the condition with N=10,T=10, σU1,n=
0.40, and γ0,1= −0.60. As expected, higher numbers of non-
stationary estimates were found for higher values of |γ0,1|and
for the highest value of σU1,n.
Thus, although we found non-convergence and non-
stationarity in some cases, their low occurrence indicate that the
problems caused by these issues are minor.
3.2.2. Bias of ˆ
γ0,1
The bias of the ˆ
γ0,1indicates whether a systematic under- or
overestimation of γ0,1is found. The bias is computed as:
bias = 1
R
R
X
r=1
ˆ
γ0,1r!−γ0,1,(8)
where r(r=1,2,...,R) refers to the replication number.
The random estimators estimate ˆ
γ0,1directly. For the fixed
estimators, ˆ
γ0,1is estimated as 1
NPN
n=1ˆ
φn.
The bias decreases marginally for N=25 compared to N=
10, with the largest difference being −0.05 for MLE-R, in the
conditions with T=25, σU1,n=0.25 and γ0,1=0.6. This
prompted us to only show the results for N=10, see Figure 2.
The bias decreases for T=25 compared to T=10 for the
fixed methods. For σU1,n=0.25 compared to σU1,n=0.40,
the bias decreases for all methods. A trend is present, where the
value of the bias of ˆ
γ0,1decreases as γ0,1increases. The bias is,
in general, positive for negative values of γ0,1, and negative for
positive values of γ0,1.
As can be seen in Figure 2, the random estimators, BAY-R and
MLE-R, show a smaller bias than the fixed estimators, MLE-F
and BAY-F. This difference is larger when T=10 compared to
T=25. The difference between MLE-R and BAY-R is very small
and inconsistent over conditions. For γ0,1above 0.00, the bias of
MLE-F is larger than the bias of BAY-F; for γ0,1below 0.00, this
is the other way around.
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Krone et al. Multilevel AR(1) Models
3.2.3. Bias of ˆ
σU1,n
The bias of ˆσU1,nindicates whether ˆσU1,nis systematically under-
or overestimated, and is calculated as:
bias = 1
R
R
X
r=1
ˆσU1,n!−σU1,n. (9)
The random estimators estimate ˆσU1,n. For the fixed estimators,
ˆσU1,nis calculated per replication ras SD(ˆ
φn).
The bias of ˆσU1,nis smaller for σU1,n=0.40 than for σU1,n=
0.25 for all estimators. As the pattern over the other conditions
stays the same, we only show the results for σU1,n=0.25, as
depicted in Figure 3. For the random estimators, the bias for
N=25 is smaller than the bias for N=10. The bias is smaller
FIGURE 2 | The bias of ˆ
γ0,1for N=10 for the different estimators, different time series length Tand different values of σγ0,1as a function of γ0,1.
FIGURE 3 | The bias of ˆ
σU1,nfor σφn=0.25 for the different estimators and different group sizes Nfor timeseries of different length Tas a function of
γ0,1.
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Krone et al. Multilevel AR(1) Models
for T=25 than for T=10, with a more pronounced effect for
the fixed estimators. The effect of γ0,1is small and inconsistent
between conditions and estimators.
BAY-R shows the lowest bias, followed by MLE-R, except for
the combination of σU1,n=0.40, N=10 and T=25, where
MLE-F shows a smaller bias than both MLE-R and BAY-R. For
all conditions, the bias of ˆσU1,nis largest for BAY-F.
3.2.4. Empirical SD(ˆ
γ0,1)
The empirical, or observed, standard deviation (SD(ˆ
γ0,1))
indicates the variability of ˆ
γ0,1. The empirical SD is computed
as the standard deviation of ˆ
γ0,1over the Rreplications for the
random estimators, and as the standard deviation of 1
NPN
n=1ˆ
φn
over replications for the fixed estimators.
The empirical SD(ˆ
γ0,1) is larger for σU1,n=0.40 than for
σU1,n=0.25, on average by a factor of 1.2. The effect of all other
parameters is equal for both values of σU1,n, prompting us to only
display the SD(ˆ
γ0,1) for σU1,n=0.40, as can be seen in Figure 4.
The SD(ˆ
γ0,1) is smaller for N=25 compared to N=10, and for
T=25 compared to T=10. Extreme values of γ0,1give a lower
SD(ˆ
γ0,1), but only marginally.
The random estimators show a larger SD(ˆ
γ0,1) than the fixed
estimators. The smallest empirical SD is shown by the MLE-F,
followed by the BAY-F. The difference between the MLE-R and
BAY-R is small and practically negligible.
3.2.5. Bias of SE(ˆ
γ0,1)
The bias of the standard error indicates how well the methods
estimate the standard deviation of ˆ
γ0,1. The bias of SE(ˆ
γ0,1) is
calculated as:
bias of SE (ˆ
γ0,1)= 1
R
R
X
r=1
SE(ˆ
γ0,1r)!−SD(ˆ
γ0,1),(10)
where SE(ˆ
γ0,1r) is the standard error of ˆ
γ0,1in replication r.
For the random estimators, the SE(ˆ
γ0,1r) is the standard error
as calculated by the estimator. For the fixed estimators, the SE
is taken as 1
NPN
n=1SE(ˆ
φn).
The bias of SE(ˆ
γ0,1) is smaller when σU1,n=0.40 than when
σU1,n=0.25. However, the effect of all other parameters on the
bias of SE(ˆ
γ0,1) is equal for both values of σU1,n, prompting us
to display the results for σU1,n=0.25 only, as can be seen in
Figure 5. For the random estimators, N=25 gives a smaller
bias than N=10, for the fixed estimators this is the other way
around. The effect of Tis only present for the fixed estimators,
which show a smaller bias of SE(ˆ
γ0,1) for T=25 than for T=10.
For the fixed estimators, this effect is stronger than the effect of
N. The different values of γ0,1only influence the estimations of
the fixed estimators, which show a slightly smaller bias for higher
values of |γ0,1|.
The MLE-R shows the smallest bias of SE(ˆ
γ0,1) for all
conditions, and is the only estimator which shows a constant
negative bias. For higher values of N, the difference between
MLE-R and BAY-R disappears. For all conditions, the bias of
SE(ˆ
γ0,1) is larger for the fixed estimators than for the random
estimators.
3.2.6. Empirical Rejection Rate and Power
For each estimator and condition, we compute the empirical
probability (EPr) for rejecting H0:γ0,1=0 in favor of Hα:γ0,16=
0.00, with α=0.05. Using frequentist terminology, the EPr
equals the actual αin the condition with γ0,1=0.00; and the
power in all other conditions.
For frequentist methods, testing H0:γ0,1=0 vs. a two-
sided alternative at significance level α, is equivalent to checking
whether the (1 −α) confidence interval (CI) includes zero or
not. The CI per replication per condition and per estimator is
calculated as follows:
ˆ
γ0,1±t∗
(1−α);df =N−2SE(ˆ
γ0,1),(11)
where SE(ˆ
γ0,1) is obtained as explained in Section 3.2.5.
The proportion of replications per condition for which the
corresponding confidence interval does not contain zero is
the EPr.
For the Bayesian estimators, the EPr is the proportion of
replications per condition for which the credible interval (CrI)
as obtained through MCMC does not hold zero. For the BAY-R,
we consider the CrI of ˆ
φ, for BAY-F we use the average scores of
the CrI’s of ˆ
φnwithin each replication.
The power is higher for N=25 than for N=10 and for
T=25 compared to T=10, as can be seen in Figures 6,7. The
actual αshows no such effect. The EPr shows lower values for
σU1,n=0.40 compared to σU1,n=0.25, except for the actual
αof MLE-R. When |γ0,1|is higher, the EPr becomes higher. For
the fixed estimators, this effect is strongly dependent on T: for
T=10, the EPr only increases for γ0,1<−0.30.
The highest power is found using BAY-R when σU1,n=0.25,
and using MLE-R when σU1,n=0.40. For the fixed estimators,
the BAY-F shows a higher power than the MLE-F. The BAY-R has
an actual αconsistently at or around 0.05, while the MLE-R has
an actual αthat is too high for σU1,n=0.40, namely at 0.10. The
fixed estimators have an actual αat or even below 0.01, rather
than the desired 0.05.
3.2.7. Point and Interval Estimates of γ0,1
To illustrate the joint effects of bias and variability we consider
BAY-R and MLE-R, using the point and interval estimates of γ0,1.
As point estimate we use the mean of ˆ
γ0,1per condition. For the
interval estimation we present the 2.5 and 97.5 percentiles of the
ˆ
γ0,1across all Rreplications per condition as the lower and upper
bounds.
The point estimates and interval estimates can be seen in
Figure 8 for σU1,n=0.40. The interval is larger for N=10 and
for T=10 than for N=25 and T=25. The effect of Nis slightly
larger. σU1,n=0.25 effectuates a smaller estimation interval than
σU1,n=0.40, the latter being 1.2 to 1.3 times the former. The
influence of γ0,1on the estimation interval is negligible, as are
the differences between BAY-R and MLE-R.
3.3. Combined Conclusions of the Different
Measures
We found that the use of random estimators as opposed to fixed
estimators improves all measurements considerably, except for
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Krone et al. Multilevel AR(1) Models
FIGURE 4 | The empirical SD(ˆ
γ0,1)for σφn=040 for the different estimators and different group sizes Nfor timeseries of different length Tas a
function of γ0,1.
FIGURE 5 | The bias of SE(ˆ
γ0,1)for σφn=0.25 for the different estimators and different group sizes Nfor timeseries of different length Tas a function
of γ0,1.
the empirical SD, which is larger for the random estimators. The
BAY-R shows a slight advantage over the MLE-R with respect to
the bias of ˆσU1,nand the bias of SEˆ
γ0,1. As expected, higher values
of Nand Timprove the estimation. Further, as expected, a lower
value of σU1,nlowers the bias of ˆ
γ0,1, lowers the SD(ˆ
γ0,1) and
increases the power, but also increases the bias of ˆσU1,nand the
bias of SE(ˆ
γ0,1).
4. DISCUSSION AND CONCLUSIONS
In this paper we studied the performance of four models
for multilevel time-series data. We compared two estimation
methods, namely maximum likelihood estimation and Bayesian
MCMC, as previous work indicates that these methods perform
best for single case designs (Krone et al., 2015). We combined
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Krone et al. Multilevel AR(1) Models
FIGURE 6 | The EPR for σU1,n
=0.25 for the different estimators, different group sizes N, and for different timeseries length Tas a function of γ0,1.
FIGURE 7 | The EPR for σU1,n
=0.40 for the different estimators, different group sizes N, and for different timeseries length Tas a function of γ0,1.
this with two model variants, a random model and a fixed model,
to obtain four estimators: MLE-F, MLE-R, BAY-F, and BAY-
R. We compared their estimates in different conditions, where
we varied the time series lengths, number of subjects and the
mean and standard deviation of the autocorrelation distribution.
As outcome measures, we considered the bias, the bias of the
standard deviation, the empirical standard deviation, the bias of
the standard error, the empirical rejection rate, and the point and
interval estimates of the autocorrelation.
We found substantial differences between the fixed and the
random estimators. When compared to the fixed estimators, the
random estimators show better results for the bias, the bias of the
standard deviation, the bias of the standard error and the power.
Furthermore, the actual αas obtained with the fixed estimators,
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Krone et al. Multilevel AR(1) Models
FIGURE 8 | The point and interval estimates for σφn=0.40 for the different estimators and different group sizes N(N=10 top pane, N=25 bottom
pane) for time series length T=10 and T=25 as a function of γ0,1.
appears to be between 0.00 and 0.01, in stead of 0.05. The fixed
estimators show a better empirical standard deviation than the
random estimators. In general, the random estimators are clearly
preferred over the fixed estimators.
Smaller differences were found between the estimation
methods. In general, the Bayesian MCMC shows a smaller bias
than the MLE. The bias of the standard deviation is smaller for
BAY-R than for MLE-R, but smaller for MLE-F than for BAY-
F. The empirical standard deviation is smaller for the MLE-F
than for the BAY-F, but the difference between BAY-R and MLE-
R is negligible. The bias of the standard error is smaller for the
MLE. The power is higher for the MLE estimators, but the actual
αis better for the Bayesian MCMC. In general, the bias of the
estimated autocorrelation is smaller for the Bayesian MCMC, but
the variability is smaller for the MLE estimators.
The effect of the different conditions depends on the
model variant. A higher sample size Nimproves all
outcome measurements for the random estimators. For
the fixed estimators, a higher Nmarginally improved the
bias, the empirical standard deviation and the power of the
autocorrelation. However, although the increase in Ndecreased
the empirical standard deviation, it did not influence the
estimation of the standard error, thus increasing the bias of the
standard error for N=25.
The time series length Tinfluences the estimations for
both model variants. A higher value of Tshowed small but
positive effects on the outcome measures for the random
estimators. However, the improvement was smaller than for an
equal increase in N. For the fixed estimators, the results were
more profound, showing stronger improvements in all outcome
measures than obtained for an equal increase in N.
The standard deviation of φninfluenced the results for all
estimators and conditions. A higher σU1,ngave less favorable
results for the autocorrelation with regard to bias, empirical
standard deviation and power, but more favorable results for the
bias of the standard deviation and the bias of the standard error.
The effect of the mean of φn,γ0,1, differs per estimator and per
condition, not showing a clear pattern between estimators and
conditions. Earlier studies showed a negative relation between
the bias and γ0,1: a negative γ0,1gave a positive bias, and the
other way around (e.g., Huitema and McKean, 1991; DeCarlo and
Tryon, 1993; Solanas et al., 2010). This result was replicated.
An important question in time series analysis is how many
individuals and time points are needed to obtain acceptable
estimates for a given model. In choosing between a random or
a fixed approach to modeling, the random modeling is clearly
favored when the assumptions associated with the model do hold.
In this case, more individuals can be used to make up for a smaller
number of time points, and the other way around. When σU1,n
is up to 0.25, the random model may produce results with an
acceptable size of bias when Tor Nis at least 25, and the other
one of the two is at least 10. When σU1,nis up to 0.40, both are
required to be higher than 25. The number of individuals only
has a small effect on the results for the fixed model. Here, the
number of time points is the strongest criteria. In this study, we
still found a sizable bias for 25 time points, which is stronger for
σU1,n=0.40. This is confirmed in single subject studies, where a
Tof 50 is advised (Box and Jenkins, 1976; Krone et al., 2015).
The aim of this paper was to compare the four estimators
MLE-F, MLE-R, BAY-F, and BAY-R using a multilevel AR(1)
model. For the single subject AR(1) model, several issues and
important factors are discussed in the literature. These may
Frontiers in Psychology | www.frontiersin.org 10 April 2016 | Volume 7 | Article 486
Krone et al. Multilevel AR(1) Models
be just as relevant for a multisubject model, such as our
multilevel model. The AR(1) model, though very often used, is
not sophisticated enough for various empirical applications. This
is because the error term (et,nin Equation 6) is also affected
by the auto-correlation. Schuurman et al. (2015) demonstrates
that including so-called white noise (i.e., error not carried
over to the next time point) in the model, leads to improved
empirical model fit. Lacking this term leads to underestimation
of the absolute autocorrelation. Studying how various estimators
perform under such an extension to the (multilevel) AR(1) model
is an interesting step in future research.
The literature on the single subject AR(1) model discusses
several other factors that influence the estimation of the
autocorrelation. In our models we kept the error variance equal
for all datasets, but this does influence the estimation of the AR(1)
model (Schuurman et al., 2015), as does the error distribution
(Solanas et al., 2010). This may also influence the performance
of the different estimators as used in this paper. Another issue
is misspecification, where the model used may not be equal to
the one underlying the data. Earlier studies showed that this
influences the estimation of the autocorrelation (Tanaka and
Maekawa, 1984; Kunitomo and Yamamoto, 1985; Krone et al.,
2015). For the multilevel model, the inclusion of a random error
covariance may improve estimation, while person-centering may
have a negative effect on the estimation of the parameters
(Jongerling et al., 2015). The effect of these factors on the different
estimators in a multilevel model is also an interesting topic for
further studies.
We chose a well-known multilevel framework for our
estimators, which is often used in longitudinal analyses. An
alternative framework to model an AR-model is a State Space
Model (SSM) (Durbin and Koopman, 2012). The versatility
of the SSM means that it can be used for a vast range of
models and any distribution for which a link-function with the
normal distribution exists. Furthermore, the implementation of
measurement error parameters is straightforward in a SSM. SSM
can be modeled to allow for a multilevel AR(1) structure for
different kinds of distributions; implementations have been made
for normally distributed data (Lodewyckx et al., 2011) and data
following a Poisson distribution (Terui et al., 2010). However, the
theoretical framework to estimate a SSM with any distribution
in the exponential family is available (Petris et al., 2009; Durbin
and Koopman, 2012). A Bayesian interpretation of the state
space model is found in the Bayesian dynamic model (West and
Harrison, 1997).
We compared several estimators, but many other possibilities
remain. Future studies may look into the effect of data properties,
such as the error variance or misspecification, and different ways
of modeling the data, using for example a SSM framework.
Finally, we did not assess how the estimators handle missing data,
and what the effect of missing data is on the outcome measures.
As missing data occurs often in the social sciences, this is an
interesting and important topic for further studies.
AUTHOR CONTRIBUTIONS
TK: Main contributing author, CA and MT: co-authors of the
study-design, substantial contributions to the manuscript and
revisions thereof.
FUNDING
This work is funded by the Netherlands Organization for
Scientific Research (NWO), grant number 406-11-018.
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Conflict of Interest Statement: The authors declare that the research was
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