 Methodology
 Open Access
 Published:
A primer on continuoustime modeling in educational research: an exemplary application of a continuoustime latent curve model with structured residuals (CTLCMSR) to PISA Data
Largescale Assessments in Education volume 10, Article number: 5 (2022)
Abstract
One major challenge of longitudinal data analysis is to find an appropriate statistical model that corresponds to the theory of change and the research questions at hand. In the present article, we argue that continuoustime models are well suited to study the continuously developing constructs of primary interest in the education sciences and outline key advantages of using this type of model. Furthermore, we propose the continuoustime latent curve model with structured residuals (CTLCMSR) as a suitable model for many research questions in the education sciences. The CTLCMSR combines growth and dynamic modeling and thus provides descriptions of both trends and process dynamics. We illustrate the application of the CTLCMSR with data from PISA reading literacy assessment of 2000 to 2018 and provide a tutorial and annotated code for setting up the CTLCMSR model.
Introduction
Longitudinal studies and methods are essential to many research questions in education sciences. Questions of learning, growth, and development are studied by examining change over time (Millsap, 2008; Singer, 2019). Moreover, if researchers are interested in causal relationships between several constructs, the temporal order, as realized in longitudinal studies, is one important prerequisite (among others) for drawing robust conclusions (Shadish et al., 2002). One challenge associated with longitudinal studies is the choice of the appropriate statistical model for the data analysis. The statistical models should correspond with the research questions at hand and the theoretical assumptions about the constructs and the processes underlying them (Baltes & Nesselroade, 1979; Collins, 2006; Little, 2013; McArdle, 2009). A close fit between theories of change and the models used to analyze the data is a prerequisite for valid inferences.
One class of models that is frequently applied in educational research is dynamic models, which include, for example, (vector) autoregressive models (e.g., Hsiao, 2014; Lütkepohl, 2005), crosslagged panel models (e.g., Hamaker et al., 2015; Usami et al., 2019; Zyphur, Allison, et al., 2020; Zyphur, Voelkle, et al., 2020), or latent change score models (McArdle & Hamagami, 2001). These models usually treat time as discrete, which is why they are often called discretetime dynamic models (DTMs; e.g., Hsiao, 2014; Voelkle et al., 2018). DTMs, however, are limited by the widely recognized timeinterval dependency (e.g., Gollob & Reichardt, 1987; Ryan & Hamaker, 2021). This means that dynamic parameters of DTMs depend on a respective timeinterval. Thus, they cannot represent the continuous nature of developmental processes and do not provide information regarding other time intervals, which is a considerable limitation in several other regards (e.g., Voelkle et al., 2012). For this reason, continuoustime dynamic models (CTMs) have been put forward in recent years (van Montfort et al., 2018).
In the present study, we adopt these ideas and outline potential advantages of CTMs over DTMs for longitudinal data analysis in educational research. In addition, we focus on how trends can be integrated into dynamic models. Besides the analyses of process dynamics, trends are additional characteristics of longitudinal data and are often of core interest in educational research. Several models have been proposed to capture trends and dynamics simultaneously and are discussed in ongoing debates (e.g., Asparouhov et al., 2018; Hamaker, 2005; NúñezRegueiro et al., 2021; Usami et al., 2019). However, to date, most of them employ discretetime modeling to represent the dynamic part (but see, e.g., Delsing & Oud, 2008). Dynamic models that provide information on trends and dynamics on a continuoustime scale, would overcome this limitation. In the present article, we propose a combined model that provides information on both the continuoustime dynamics and trends that we call the continuoustime latent curve model with structured residuals (CTLCMSR). The model is a continuous time version of the existing latent curve model with structured residuals (LCMSR; Curran et al., 2014; Hamaker, 2005). We illustrate the application of the CTLCMSR with data from the PISA reading literacy assessment using the R package ctsem (Driver & Voelkle, 2018a, 2021). In doing so, two research questions are examined. (1) Is there a trend in the PISA countries’ mean reading scores over the period from 2000 to 2018, and do the countries differ in the slope and direction of the trend? (2) How persistent are deviations from the trend? The latter research question tackles the stability of educational systems when deviations from the trend (socalled shocks or impulses) occur. Influences leading to temporary gains (positive deviations from the trend) could be, for example, temporary investments or shortterm interventions initiated by a country's politics. Influences which might result in negative deviations can be political unrest, a shortage of teachers in a given cohort, or a global financial crisis or pandemic. Thus, the stability of a system could be thought of as the system's resilience against disturbances. Clearly, this is oftentimes a doubleedged sword, as a high persistence of positive deviations is often desired, which would be the case in an instable or nonresilient system. Contrarily, negative disturbances are often desired to dissipate quickly, which would be the case if the system is stable or resilient.
In our illustration, we show how the CTLCMSR can be used to answer both research questions. Thereby, we compare the CTLCMSR to related models, exemplarily interpret the parameter estimates of the CTLCMSR, and present some details and ways of visualizing results to emphasize the advantages of CTMs over DTMs.
This article is aimed at two audiences. In the first section, we introduce CTMs in general and give arguments for why the use of CTMs can be beneficial. This section may be particularly interesting to readers who are new to continuoustime modeling. For readers who are already familiar with this type of modeling, we provide detailed information on the implementation and interpretation of the newly proposed CTLCMSR model in the methods section. Furthermore, readers get a practical introduction to the use, modeling, and interpretation of CTMs. Finally, the strengths and limitations of the illustrative example in particular and CTMs in general are discussed. The discussion also includes an outlook on more complex (multivariate) applications of the CTLCMSR.
How variables develop over time and how we measure this development
When longitudinal studies are conducted in educational research, the data are usually repeated measurements of the same constructs and target units (e.g., subjects). For instance, questionnaires or achievement tests are repeated multiple times, each after a certain period of time. Based on such snapshots in time, we gain insights about the state of our target units (Fig. 1A). Between these measurement occasions, we have no data and thus no information about the changes of the variables under study. The stability, inertia, development, change, or longitudinal association of multiple variables in multivariate analysis must be inferred from these repeated snapshotlike data. In many theories of change, however, the observation units and variables of interest are not assumed to exist only at the measurement occasions or to change in discrete steps (Fig. 1B). Rather, we assume that observation units and variables under investigation exist and develop continuously over time (Fig. 1C). This is applicable to variables at the level of individuals, such as the development of students' competencies, cognitive abilities, selfconcepts, attitudes, and motivations. Moreover, this continuoustime perspective is transferable to social systems and variables at the level of school classes, variables describing parent–childteacher interactions, or the performance of entire education systems as studied, for instance, in the international largescale assessments (LSAs) such as PISA and TIMSS. Although there are typically several years between each measurement occasion in LSAs, we do not assume that the characteristics of education systems cease to exist in the interim. Thus, from a theoretical point of view, it is desirable to adopt the idea of continuously developing variables also in the statistical models used. As shall become apparent in the following sections, this not only allows us to align models closer with theories of change, but also provides further advantages, such as greater flexibility for the procedures of data collection, improvements for crossstudy comparisons (e.g., Voelkle et al., 2012), or the exploration of the unfolding of dynamic effects (e.g., Hecht & Zitzmann, 2021a).
Discretetime dynamic models (DTMs)
Dynamic models represent the state of variables at time \({t}_{i}\) as a function of previous states (e.g., at time \({t}_{i1}\)) of the same variables (Deboeck & Boker, 2015; Hsiao, 2014; Voelkle et al., 2018). Typical questions that can be addressed with dynamic models are questions related to the stability or inertia of constructs over time, to find out, for example, how fast the states of variables change (or can be changed). Dynamic models can also be used to address questions about the variability of states of a construct or questions about the predictability of future states (based on the current state) (Deboeck & Boker, 2015; Loossens et al., 2021). Wellknown dynamic models in educational research are multivariate models, such as the crosslagged panel models (CLPM). In terms of Granger causality, such models usually aim to provide insights into the strength and predominant direction of the relationship between the variables of interest (Granger, 1969; Hamaker et al., 2015).
In educational research, many dynamic processes are modeled by firstorder models such as the firstorder autoregressive model (AR(1)) or its multivariate variant, the vector autoregressive model (VAR(1))^{Footnote 1} (Sivo & Fan, 2008). In VAR(1) models, the present state of variables is predicted by the previous state. A discretetime VAR(1) model can be represented by the following equation (cf. Oud & Delsing, 2010; Voelkle et al., 2012):
Here, \(\mathbf{x}\) is a column vector of length \(V\) (number of variables), observed at time \({t}_{i}\) (with \(i= 1,\dots , T\), where \(T\) is the number of measurement occasions) and \({\Delta t}_{i}\) the time interval between two measurement occasions. \(\mathbf{A}\) is a \(V\times V\) matrix relating the observed variables over time (autoregressions on the main diagonal, crosslagged effects on the offdiagonals), \(\mathbf{b}\) is an intercept vector of length \(V\), and \(\mathbf{w}\) a vector of prediction errors or white noise (e.g., Deboeck & Preacher, 2016; Oud & Delsing, 2010). Covariance matrix \({\mathbf{Q}}_{\Delta {t}_{i}}\) contains prediction error variances and covariances. From this representation of a VAR(1) model, it is apparent that the coefficients of interest, in particular the autoregressive and crosslagged coefficients of the matrix \(\mathbf{A}\), depend on the timeinterval \(\Delta {t}_{i}\) (\(\Delta {t}_{i}=\) \({t}_{i}{t}_{i1})\) between successive measurement occasions. For example, if this interval length is three months, the estimated coefficients are conditional on exactly this timeinterval and are not valid for any other interval. Thus, the process described by the parameters corresponds to the development of a variable or an observation unit that exists only at the time points of measurement, illustrated in Fig. 1A. However, as discussed earlier, in most cases this representation is not consistent with theory that assumes that variables and observation units exist and change continuously over time (Fig. 1C). This timeinterval dependency is also referred to in literature as the interval problem of DTMs (e.g., Ryan & Hamaker, 2021). This dependency results in several drawbacks, which have been described in detail elsewhere (e.g., Hecht et al., 2019; Oud & Delsing, 2010; Voelkle et al., 2012) and will only be briefly mentioned here: (1) The information about the dynamics of the variables under study that DTMs reveal only relates to the particular timeinterval used (e.g., Deboeck & Preacher, 2016; Kuiper & Ryan, 2018). (2) There is no information about how the dynamic effects change with increasing timeintervals (e.g., Hecht & Zitzmann, 2021a). Moreover, (3) (approximately) equal spacing between measurement occasions must be realized across all units in order to be able to properly apply DTMs^{Footnote 2} (see Driver et al., 2017). Establishing equal spacing is a major challenge, especially when using flexible survey designs (de HaanRietdijk et al., 2017) whose importance has been increasingly emphasized for educational research in recent years (Zirkel et al., 2015). The timeinterval dependency of DTMs also has disadvantages since (4) results from studies examining different timeintervals between measurement occasions are not directly comparable with each other, making it difficult to conduct metaanalyses on these results (Dormann et al., 2020). Continuoustime dynamic models (CTMs) resolve these problems.
Continuoustime dynamic models (CTMs)
The assumption of variables that exist and unfold continuously is consistent with CTMs (e.g., Kuiper & Ryan, 2018; Oud & Delsing, 2010). By using CTMs, we obtain not only the same information provided by DTMs if all underlying assumptions are met, but also parameters that are independent of the length of the timeintervals of a given study. These continuoustime parameters allow for gaining insights into the underlying continuoustime process and for deriving discretetime parameters for various timeintervals (Hecht et al., 2019; Voelkle et al., 2012). In addition, CTMs overcome the practical drawbacks and limitations of DTMs outlined above (e.g., Oud & Delsing, 2010; Voelkle & Oud, 2013). However, the implementation of CTMs is mathematically more challenging than that of DTMs because CTMs require differential calculus (Deboeck & Boulton, 2016; Voelkle et al., 2018). Fortunately, the application of CTMs has been greatly facilitated in recent years by software packages like the R package ctsem (Driver et al., 2017; Driver & Voelkle, 2017, 2018a, 2021). As a result, the application of a wide range of CTMs is now possible in a relatively straightforward manner.
The basic stochastic differential equation (SDE) used in ctsem can be represented as follows (Oud & Delsing, 2010; Voelkle et al., 2012; for an overview of different notation styles of the same equation see, e.g., Hecht & Zitzmann, 2021a):
The derivative \(\frac{\mathrm{d}\mathbf{x}\left(t\right)}{\mathrm{d}t}\) provides information about the rate of change of the variables in the column vector \(\mathbf{x}\) of length \(V\) at the time \(t\) (Deboeck & Boulton, 2016; Ryan et al., 2018). The term can be thought of as the change \(\mathrm{d}x\) in the variable \(x\) over the timeinterval \(\mathrm{d}t\), which, in case of the derivative, is infinitesimally small (\(\Delta {t}_{i}\) → 0). The expression \(\frac{\mathrm{d}\mathbf{x}\left(t\right)}{\mathrm{d}t}\) is also referred to as velocity (Deboeck et al., 2015). Given the information regarding the rate of change at time \(t\), we know how the process under study is changing at that time (Ryan et al., 2018). The rates of change of the variables \(\mathbf{x}\), shown on the left side of Eq. 2, is explained by a deterministic and a stochastic part on the right side. The deterministic part is composed of the socalled drift matrix \(\mathbf{A}\), which is a \(V\times V\) matrix and includes the continuoustime autoeffects (equivalent to autoregressive effects in DTMs but independent from the length of timeintervals) on the main diagonal and the continuoustime crosseffects on the offdiagonals, the state of the variables \(\mathbf{x}\) at time \(t\), and a \(V\times 1\) continuoustime intercept vector \(\mathbf{b}\). \(\mathbf{G}\frac{\mathrm{d}\mathbf{W}(t)}{\mathrm{d}t}\) represents the stochastic part of the equation and can be described as a continuoustime error process (Oud & Delsing, 2010; Ryan et al., 2018). \(\mathbf{W}(t)\) is the socalled Wiener process, which is a random walk in continuoustime and the Cholesky factor \(\mathbf{G}\) indicates the effect of \(\mathbf{W}\) on the change in \(\mathbf{x}\left(t\right)\) (for details see Driver & Voelkle, 2018a; Oud & Jansen, 2000). Because the variance of the Wiener process depends on the length of the timeinterval over which it is integrated, it can represent the accumulation of errors over longer time periods, resulting in larger error variances (Deboeck & Preacher, 2016; Oravecz et al., 2011). The associated \(V\times V\) variance–covariance matrix \(\mathbf{Q}=\mathbf{G}\mathbf{G}\mathbf{^{\prime}}\) is often called diffusion matrix and contains the process error variances on the main diagonal and the process error covariances on the offdiagonals.
Based on the continuoustime (CT) parameters of Eq. 2 it is possible to calculate the DT parameters for any given timeinterval (Hecht et al., 2019; Oud & Delsing, 2010). To better distinguish between CT parameters and its corresponding DT parameters, in the following, DT parameters that are constrained to underlying CT parameters are denoted with an asterisk (*) (cf. Hecht et al., 2019). For an overview of corresponding DT and CT parameter labels, see also Hecht and Voelkle (2021). The CT drift matrix \(\mathbf{A}\) is related to the DT autoregressive matrix \({\mathbf{A}}_{\Delta {t}_{i}}^{\boldsymbol{*}}\) via the following equation:
The CT intercept vector \(\mathbf{b}\) is related to the DT intercept vector \({{\varvec{b}}}_{\Delta {t}_{i}}^{\boldsymbol{*}}\) via the following equation:
Furthermore, the DT error covariance matrix \({\mathbf{Q}}_{\Delta {t}_{i}}^{\boldsymbol{*}}\) for interval ∆t can be calculated from the CT diffusion matrix \(\mathbf{Q}\) (Delsing & Oud, 2008):
where \(\mathbf{I}\) is an identity matrix, \(\otimes\) the Kronecker product, which results in a large matrix containing all possible products of the elements of the two initial matrices, row and irow are operators, the former puts the elements rowwise in a column vector and the latter puts them from a column vector into a matrix. To solve the system of Eqs.2, 3, 4, 5, the matrix exponential has to be used (e.g., Oud et al., 2018; Ryan et al., 2018). Details and examples of the link between the parameters and their interpretation will be described in the methodical and results section of the article. Before doing so, however, we address another challenge that often arises when analyzing longitudinal data in education sciences: systematic trends.
Trends
Analyzing longitudinal data in educational research is often also about identifying trends. For example, when studying students’ achievement repeatedly over several months and years, their achievement typically continues to increase. Such systematic trends can also occur when examining other variables, such as motivational (e.g., Wigfield et al., 2006, 2015) or personalityrelated characteristics (e.g., Specht, 2017). Furthermore, besides the theoretical interest in trends, it is also often important for dynamic models to account for such trends. Unaccounted trends can affect the autocovariance structure and thus lead to biased estimates of the parameters (e.g., Asparouhov et al., 2018; NúñezRegueiro et al., 2021; Walls & Schäfer, 2006). Therefore, there are several approaches to deal with trends in dynamic models, such as predetrending the data (e.g., Box et al., 2015; Walls & Schäfer, 2006). Another possibility is to model trends and dynamics simultaneously. Growthcurve models (GCM) are a wellknown approach to model trends (Fig. 1D) (e.g., Bollen & Curran, 2006). There are approaches that combine dynamic DTMs with GCMs, such as the autoregressive latent trajectory (ALT) models (e.g., Bianconcini & Bollen, 2018; Bollen & Curran, 2006; Curran et al., 2014; Hamaker, 2005). In addition to all the other disadvantages of DTMs already described, approaches combining elements from DTMs and GCMs come with a modelinherent unequal timeinterval dependency of parameters: While GCMparameters are usually treated and interpreted as independent from timeintervals (Fig. 1D), the DTM parameters depend on the timeinterval used in the study (Delsing & Oud, 2008). Thus, in combined models such as ALT models, the trend parameters describe a continuous process (Fig. 1D) and the dynamic parameters describe a discretetime process (Fig. 1B). To remove this inconsistency from combined models, a promising solution is to use CTMs to make the dynamic parameters independent of timeintervals as well. One of the very first explicit formulations of a combined model in continuoustime is the continuoustime autoregressive latent trajectory (CALT) model proposed by Delsing and Oud (2008; see also Oud, 2010).
However, ALT and CALT models are associated with other considerable limitations. Trends and dynamics are not clearly separated in ALT and CALT models and can influence and “compete” with each other. An associated problem is that dynamic models are recursive, which means that dynamic processes depend on their previous values. This is not the case for GCMs. These problems lead to drawbacks, such as the need for special solutions to “start up” ALT and CALT models and uncertainty of how to interpret GCM parameters in ALT and CALT models (e.g., Bianconcini & Bollen, 2018; Hamaker, 2005; Jongerling & Hamaker, 2011; Little, 2013; Oud, 2010; Usami et al., 2019).
To clearly isolate trends from the dynamic part of the model, the discretetime (DT) latentcurve model with structured residuals (LCMSR) has been suggested as one suitable alternative (Curran et al., 2014; Hamaker, 2005). In the LCMSR, the development of a variable under study is described by two different process components, trends and dynamics, which are—contrarily to the ALT or CALT models—clearly separated (Fig. 2). Because the trend component (which is similar to what is modeled in the GCM) accounts for the trends in the data, the dynamic component represents a trendadjusted (detrended) and centered process in the LCMSR (Usami et al., 2019). For this reason, systematic (linear) trends are disentangled from the autoregressive and crosslagged parameters. An additional advantage of this separation is that the parameters of both processes can be interpreted as is typically done in CTMs and GCMs. Moreover, trend parameters and dynamic parameters do not compete in a joint process in the LCMSR as in the ALT and CALT models (see Curran et al., 2014; Hamaker, 2005; Jongerling & Hamaker, 2011).^{Footnote 3}
To our knowledge, an explicit formulation of a continuoustime version of the LCMSR has not been proposed in the literature, although a continuoustime latent curve model with structured residuals (CTLCMSR; Fig. 2) comes with several advantages: (1) Using the CTLCMSR, information is obtained describing trends in the data as well as information describing the process’ dynamics on a continuoustime scale. (2) Trends and dynamic processes are clearly separated in CTLCMSR, meaning that trends and dynamics can no longer influence each other. (3) The interpretation of the parameters follows the interpretation of CTMs and GCMs. (4) All parameters of the CTLCMSR describe the process under study independently of the length of timeintervals in a given study. This is not the case for the DTLCMSR in which the dynamic parameters always depend on the length of the timeinterval (cf. Delsing & Oud, 2008; Oud, 2010). Finally, (5) both the standard linear GCM and the CTAR(1) model (which is a CTM without trend components) are nested within the CTLCMSR which enables chisquare difference testing to compare models. In addition, with the CT version of the LCMSR, we can take advantage of all the other benefits associated with using CTMs instead of DTMs.
Methods
The CTLCMSR
The basic idea of the CTLCMSR is to decompose a process into the two process components, trends and dynamics. This is technically done by introducing two continuoustime processes with special parameter values and constraints, so that one continuoustime process purely captures the trend and the other one the dynamics (see technical details below, especially Eq. 8 and 9). In the following, we therefore also refer to the trend component as the “trend (or growth) process” and to the dynamic component as the dynamic process with the need to keep in mind that these are just technical terms for the two components of one single process variable.
The univariate CTLCMSR is thus composed of two process components, one to describe the linear trend and one to model the dynamics of one process variable. When including more variables, each variable would need two processes to represent its two process components, for instance, a bivariate CTLCMSR would then include four processes, a trivariate CTLCMSR six, and so forth. The ordering of the two processes is arbitrary. We use the first process as the linear growth process that accounts for initial values and trends (Fig. 3A). The second process is then the dynamic process that provides information about the process’ dynamics on a continuoustime scale (Fig. 3B).^{Footnote 4} Since the growth process accounts for nonzero initial values (intercept parameter) and for linear trends (growth parameter), the dynamic process can be considered as centered and linearly detrended (cf. Usami et al., 2019). By combining the two process components in one model, the values can be thought of as fluctuating around the linear trend (Fig. 3C).
Just as in growthcurve modeling, CTLCMSR can account for differences between observation units in intercept and growth parameters via random effects (e.g., Bollen & Curran, 2006; Little, 2013). Thus, the dynamic process is withinunit centered. If potential differences in trends and initial values between observation units are not taken into account, within and between level effects can be confounded (see also the discussion of the use of random effects in dynamic models by, e.g., Hamaker et al., 2015; Lüdtke & Robitzsch, 2021; NúñezRegueiro et al., 2021; Usami et al., 2019).
Using the SDE (Eq. 2), growthcurve models can be specified as special cases (Driver, 2020). To illustrate the application of a CTLCMSR with the SDE, we first consider a single growthcurve process.
To obtain a GCM with the SDE, the CT autoeffect is set to a negative value very close to zero (e.g., − 0.0001). This means, that the corresponding DT autoregressive effect (\({a}_{\Delta {t}_{i}}^{*}={e}^{a\Delta t}\)) approximates 1. The freely estimated initial value \({x}_{0}\) corresponds to the growthcurve intercept parameter, while the continuoustime intercept (\({b}_{\Delta {t}_{i}}^{\boldsymbol{*}}={a}^{1}\left({e}^{a\Delta t}1\right)b\)) serves as the linear growth parameter. To illustrate the functionality, we consider the deterministic part of Eq. 2 and insert the DT parameter constraints of Eqs. 3 and 4 (Oud & Delsing, 2010):
The right part of the equation is composed of the two constraints relating the CT autoeffect to the DT autoregressive effect and the continuoustime intercept to the DT intercept. \(E[{x}_{0}]\) is the expected value of the freely estimated initial mean. To obtain the expected values for other measurement occasions, the value for \(\Delta t\) is changed. To illustrate this, let us assume \(\Delta t={t}_{1}{t}_{0}=1\) for the second measurement occasion \({t}_{1}\):
Compared to the initial measurement occasion, the expected value for the second measurement occasion increases by the additive component \(1\times b\). To obtain the expected value for the third measurement occasion \({t}_{2}\), we set \(\Delta t=2\):
The expected value for \({t}_{2}\) increases by the additive component \(2\times b\) compared to the initial measurement occasion. This can be continued for any measurement occasion. Because the CT autoeffect is fixed to a negative value close to zero, which corresponds to a DT autoregressive effect approximating 1, the continuoustime intercept part linearly increases with longer timeintervals.
To obtain a univariate CTLCMSR (Fig. 2), we extend Eq. 6 by a second (dynamic) process with a freely estimated autoeffect \(a\) and a process mean of zero (\(b=0)\):
The lambda matrix \({\varvec{\lambda}}\) = \(\left[{\lambda }_{1} {\lambda }_{2}\right]\) in Eq. 9 relates the observed values of variable \(x\) at time \(t\) to the two processes \({{x}_{lin}}_{t}\) and \({{x}_{dyn}}_{t}\), which represent the growth component and the dynamic component, respectively. The states of the two processes \({x}_{lin}\) and \({x}_{dyn}\) at time \(t\) are predicted by the state of the two processes at a previous time point \(t\Delta t\) (Eq. 8). The autoregressive matrix is the first element on the right side of the equation. On the main diagonal, the autoregressive matrix has the restricted parameter \({e}^{.0001\Delta t}\) for the linear trend component and the freely estimated autoregression \({e}^{a\Delta t}\) for the dynamic component. The offdiagonals are set to zero since the two process components are not assumed to directly affect each other. The intercept vector (the second element on the righthand side of Eq. 8) contains the known restrictions for the growth process component at the top, and is set to zero at the bottom, because the dynamic process component is centered and detrended (fluctuating around the processmean of zero).
The model can be represented in the SDE (Eq. 2) notation as follows:
Here, it becomes obvious that the trend component is deterministic because the random error term is zero. Therefore, the residuals are completely absorbed into the dynamic part and are accounted for by CT autoeffect \(a\) and CT random error variance \(\mathbf{Q}\). In addition, the first measurement occasion (T0) is estimated as
where \({int}_{MEAN}\) represent the mean intercept of the trend component, \({int}_{SD}\) the standard deviation (SD) of the random effects of the intercept, and \({T0dyn}_{SD}\) the residual SD of the first measurement occasion entering the dynamic process (the labels of the estimated parameters in Eq. 10 and 11 correspond to the labels used in the illustrated example later on and Fig. 2). In the random effects version of the CTLCMSR, there is an additional variance component for the slope of the trend component \(b\boldsymbol{ }\sim N\left[{b}_{Mean}, {b}_{SD}\right]\) and a correlation of the random effects (\({Cor}_{int,b}\)).
To turn this CTLCMSR into a model without trend component (i.e., a standard CTAR(1) model), we can set the freely estimated parameter \(b\) in Eq. 10 to zero.
An illustrative example of the CTLCMSR model using PISA data
In the following, we provide an illustrative application of the CTLCMSR to data from the PISA reading literacy assessment, including a stepbystep tutorial on how to set up the CTLCMSR in ctsem. To make it as understandable as possible, a simple example was chosen, which can, however, easily be extended to more complex models and research questions (as outlined in the outlook of this article).
Data
The data stem from the seven currently available PISA waves from 2000 to 2018. For each single wave, countryspecific mean scores were calculated across all participating students in each country (see OECD, 2009). For this step of data preparation, the Rpackage intsvy (Caro & Biecek, 2017) was used and the resulting values were additionally checked with the SPSS macros provided on the OECD website. This was primarily done to ensure that the weighting procedures were accurate (IBM SPSS, 2020; OECD, 2021). Furthermore, because data from the achievement tests were processed with the Rasch model, the use of plausible values (PVs) is recommended (OECD, 2009; Wu, 2005). Therefore, all analyses were first performed with each PV separately and the results were pooled afterwards (OECD, 2009; Rubin, 2004). In PISA, the achievement data are represented by five PVs.^{Footnote 5} Thus, five data sets were created, one for each PV (see OECD, 2009).
The data sets had a hierarchical data structure, with repeated measurements nested within countries. For analyses with the ctsem package (Driver & Voelkle, 2017), the data sets should preferable be in the long format (see also the ctsem tutorial by Hecht et al., 2019). Therefore, the data were structured, with all observations of a variable being contained in the same column (in contrast to the wide format, where there is a separate column for each measurement occasion). The corresponding time of the measurement was put in a separate column. Information about which observations were related to which country was contained in another column (see Fig. 4).
Sample
The final sample examined in this study consisted of N = 56 PISA countries. It is the same sample selected by the OECD (2019a) for the presentation of countries developmental trajectories in mean reading literacy performance. To be included, a country had to have participated in at least five PISA surveys between 2000 and 2018.
Measures
PISA reading literacy
The primary variable of interest are the countries’ mean estimates in PISA reading literacy assessment. Because the reading scales were linked across all available PISA waves from 2000 to 2018, they are basically comparable and thus suitable to be analyzed longitudinally (OECD, 2019b).
Coding time
In the data sets, the time point of measurement was represented as the years starting from 2000 as the first measurement occasion coded as \({t}_{0}\)= 0 and continuing in threeyear intervals (2003 was coded as \({t}_{1}\) = 3, 2006 as \({t}_{2}\) = 6, and so on; see Fig. 4).
Data analysis procedure
To tackle the guiding research questions (1) whether there was a linear trend in the PISA countries’ mean reading scores, whether countries differed in the slope and direction of the trend, and (2) how persistent the effect of a potential deviation from the trend (a shock) would be, we first computed a series of models. For this purpose, we used the ctsem package (Version 3.4.3; Driver et al., 2017; Driver & Voelkle, 2018a, 2021) in R (R Core Team, 2021). All models were compared with respect to their fit. Because all models were nested in the CTLCMSR, multiple chisquare difference tests were performed. To address the question of whether there is a trend in PISA countries’ mean reading literacy scores, we compared the CTLCMSR to a CTAR(1) model, which is a CTM without trend component. To answer the question of whether there is also a dynamic process in the PISA data, we compared the CTLCMSR to a standard linear GCM (specified within the continuoustime framework), which is a linear trend model without a dynamic process. The dynamic process describes the effects of shortterm influences (shocks) on countries mean reading literacy development. In addition, to answer whether there are significant differences in starting values and trends across PISAcountries, we compared fixed and random effects variants of all models (i.e., variances of the intercept and linear slope were freely estimated vs. fixed to zero).
We present an exemplary interpretation of the estimated parameters of the most relevant model, the CTLCMSR, focusing on the CT dynamic parameters. We also present and visualize DT parameters for different timeintervals derived from CT parameters and study the impact of temporary deviations from the trends on the development of countries' mean PISA reading scores to answer the second research question. Moreover, we visualize trends and dynamics of certain PISA countries as examples and show how predictions for future states can be made for various timeintervals.
Models
In addition to the CTLCMSR, we also run a linear GCM and a standard CTAR(1) model. We included fixed and random effects variants of all models, considering random effects for intercept and growth parameters but not for dynamic parameters (model specifications of all models can be found in the Additional file 1). These additional models are nested in the CTLCMSR model. To obtain a ctsem version of a standard GCM from the CTLCMSR, we set the dynamic process component to zero, assuming a homoscedastic measurement error variance.^{Footnote 6} To obtain a standard CTAR(1) model from the CTLCMSR, we fixed the growth parameter from the trend component to zero. To obtain the random effects variants of the models, we set the respective model parameters to be variant across countries (see the tutorial in the last part of this section). Pooling of estimates and variance components across the five data sets was carried out using the mice package (van Buuren et al., 2015).
Fit Indices and ChiSquare tests
To be able to evaluate the relative model fits, we used Akaike Information Criterion (AIC; Akaike, 1973) and Bayesian Information Criterion (BIC; Schwartz, 1978) as well as the Deviance (twice the negative logarithm of the likelihood). For all three indices, lower values indicate better model fits. Furthermore, we run chisquare difference tests for nested models (e.g., Bollen, & Long, 1993). To pool the chisquare test statistics, the miceadds package (Robitzsch et al., 2017) was used.
Missing data
Countries with less than five PISA participations between 2000 and 2018 were excluded from the analysis to ensure a sufficient data base for modeling trajectories. This procedure is in line with the analysis and presentation by the OECD (2019a). The remaining proportion of missing values in countries reading literacy scores was 9.4% and thus small.
Dealing with missing values is basically modelinherent in CTMs because the length of the timeinterval between two consecutive measurement occasions is taken into account (for a detailed discussion of missing data treatment in CTMs, see Oud & Voelkle, 2014). When missing values are left in the data set, they can be addressed using, for example, maximum likelihood approaches (under the MAR assumption; Driver et al., 2017; Enders, 2010), as it was done in the present application.
ctsem syntax for estimating the CTLCMSR
In the following, we give a stepbystep tutorial of how to specify and run a CTLCMSR with the data from PISA using the ctsem package^{Footnote 7} (Driver & Voelkle, 2017; Driver et al., 2017) in R (R Core Team, 2021).
For continuoustime modeling with ctsem, two functions are crucial. The ctModel function is needed to setup the required model, and the ctStanFit function is needed to fit the model to the data. Thus, in order to perform a CTLCMSR with the prepared PISA data, the first step was to specify the model. For setting up CTMs properly, the arguments of the ctModel function have to be understood. The following arguments are of key importance within the ctModel function (Driver & Voelkle, 2017; Hecht et al., 2019):
With the type argument, we chose the type of model we want to setup. In ctsem the user has the option of type = “omx” specifying a CTM interfacing to OpenMx (Neale et al., 2016), which is the original procedure of ctsem using frequentist estimation. Meanwhile, however, this part is mainly outsourced to an extra package ctsemOMX (Driver & Voelkle, 2021). Model type “stanct” interfaces to RStan^{Footnote 8} (Stan Development Team, 2020) and allows to choose between maximum likelihood, maximum a posteriori, or fully Bayesian estimation (Driver & Voelkle, 2017, 2021). In the present application we chose type = “stanct” and maximum likelihood estimation for all models.
The argument n.manifest defines the number of variables to be analyzed in a given model. Because we wanted to analyze one single variable (countries' mean reading literacy scores), we set n.manifest = 1. Furthermore, with manifestNames, the names of the variables to be analyzed as labelled in the respective data sets are defined. Reading scores were labelled “READ” in our data sets, so we set manifestNames = “READ” (cf. Fig. 4). The argument n.latent determines the number of process components we need to analyze our variables under study. Because we needed two components for the CTLCMSR (one linear trend and one dynamic component) we set n.latent = 2. With the latentNames argument, labels for the process components can be chosen. The number of names has to correspond with the n.latent argument. We chose n.latent = c(“lin”, “dyn”). The first process component should account for the trends in the data (linear trend) and the second should account for the dynamic process (fluctuating around the linear trends; Fig. 3A–C).
Next, the model matrices DRIFT, CINT, and DIFFUSION corresponding to Eq. 2 must be specified. The size of these three matrices corresponds to the number of process components (n.latent). Thus, in our application, the drift matrix \(\mathbf{A}\) (n.latent \(\times\) n.latent) was a \(2\times 2\) matrix. To model the linear trend component, the autoeffect of the first process had to be set to a negative value close to 0. In ctsem this specification is made automatically when a parameter on the main diagonal of the drift matrix is set to zero. In addition, the crosseffects between the two process components were also set to zero because trends and dynamics are independent in CTLCMSR. It was only the autoeffect of the dynamic process that was freely estimated by setting DRIFT = matrix(c(0, 0, 0, “a”)). The diffusion matrix \(\mathbf{Q}\) looks very similar because in the CTLCMSR, we also only have a single continuoustime error process: DIFFUSION = matrix(c(0, 0, 0, “q”)). The continuoustime intercept vector \(\mathbf{b}\) (n.latent \(\times\) 1) has a special role in the CTLCMSR specification in ctsem as it serves as a slope parameter in the trend process and therefore has to be freely estimated. As explained above, the continuoustime intercept was set to 0 via CINT = c(“b”, 0).
Because the first measurement occasion cannot be regressed on a previous occasion, dynamic models must be initiated somehow. By default, ctsem uses a “predetermined” model with freely estimated parameters at the first time point (Driver et al., 2017). To model the linear trends of the CTLCMSR in ctsem (using Eq. 2), T0MEANS plays a special role and serves as the (freely estimated) intercept. Since the second process is (withinunit) centered in the CTLCMSR, it was restricted to be zero. So we set T0MEANS = c(“int”, 0). Because the covariances of the initial timepoints of both processes should be uncorrelated in the CTLCMSR, we set the \(2\times 2\) matrix to T0VAR = matrix(c(“intSD”, 0, 0, “T0dynSD”)).
The two further arguments MANIFESTMEANS and MANIFESTVAR can be used to specify manifest components such as residuals or measurement errors (for more details, see Driver et al., 2017). In our application, we did not account for measurement error and set both to zero.
Finally, the LAMBDA (cf. Eq. 9) matrix relates the observed scores to the process components of the model (components are to be represented in the columns and the manifest variables in the rows): LAMBDA = matrix(c(1, 1)).^{Footnote 9} The complete specification is stored in a model object and reads as follows:
After model specification, we can print and check the model object with the command head(CT_LCM_SR$pars, 20). In addition, some further modifications can be made, such as fixing individual parameters to a certain value or specifying random effects (Driver & Voelkle, 2018b). In the current application, we tested if countries differed in terms of their initial level at the first PISA wave in 2000 (random intercept) as well as in terms of direction and slope of the trends (random slope) and specified random effects for t0m (random intercept) and b_lin (random slope): CT_LCM_SR$pars[c(1, 7),]$indvarying < TRUE. Any parameter that should vary across units (in this case parameter one, which is the intercept, and parameter seven, which is the slope) must be set to indvarying = TRUE in the model object.
Once all parameters were properly specified, the model could be fitted to the PISA data. For this purpose, the ctStanFit function was used. For maximum likelihood estimation we set the following additional arguments (see Driver & Voelkle, 2021): CT_LCM_SR_fit < ctStanFit(datalong = PISA_PV1, ctstanmodel = CT_LCM_SR, optimize = TRUE, nopriors = TRUE).
Descriptive statistics
For the sample of N = 56 countries (level2 units) included in the analysis the descriptive statistics of the mean reading literacy scores for each of the seven waves are reported in Table 1. In the present sample, the overall mean across the 56 countries and all seven waves was 473 PISA points (SD = 45.4), pooled over the five data sets.
Model comparisons and fit statistics
Model comparisons and fit statistics provided first insights regarding the research questions. In Table 2, the deviance, AIC and BIC of the six models are presented. The random effects models outperformed the fixed effects variants in all cases. This implied considerable differences in initial levels and trends across countries’ trajectories of mean PISA reading literacy scores. Furthermore, all three information criteria showed the smallest values for the CTLCMSR with random effects, indicating the best model fit of all six models. In addition, we also performed chisquare tests for nested models. In all cases the chisquare test was significant at a twotailed significancelevel of 0.05, also indicating the best model fit for the random effects CTLCMSR.^{Footnote 10} Thus, the comparisons of these models provided first support for the existence of trends, differences in trends and initial levels across countries, and a dynamic process.
Results of the CTLCMSR for the PISA countries’ reading literacy scores
The parameter estimates of the random effects CTLCMSR, pooled over the analyses of all five data sets, are shown in Table 3 (parameter estimates of the CTAR(1) and the GCM can be found in the Table 5 in Appendix). The CTLCMSR yielded eight estimated parameters, which were all statistically significantly different from zero. This supported the use of the CTLCMSR, which accounts for dynamics and trends in the data. Additionally, this also supported the use of the random effects CTLCMSR, which allows for differences in terms of initial values and slopes across countries.
Examining trends in PISA countries’ mean reading literacy scores
Five parameters of the CTLCMSR were related to the trends, namely the mean intercept, the intercept standard deviation (SD), the growth rate, the growth rate SD, and the interceptgrowth correlation. The interpretation of these five parameters corresponds with the standard parameters of a GCM with random effects. The estimated mean intercept across all countries was 464.8 PISA points (SE = 7.37), and the corresponding intercept standard deviation was 53.50 (SE = 5.57). The estimated growth rate was 0.43 (SE = 0.20) per year, which implies an average increase of 1.29 PISA points per measurement occasion (3 * 0.43) across all countries. The corresponding standard deviation was SD = 1.06 (SE = 0.32). Thus, for instance, for a country with a growth rate of one SD below the average, a negative trajectory is expected and thus a loss of − 0.63 (0.43–1.06) PISA points per year on average (and − 1.89 for the threeyear interval between adjacent PISA waves). The estimated correlation of − 0.61 (SE = 0.13) between intercept and growth rate indicated that countries with lower mean scores at the initial PISA wave in 2000 tended to make larger gains in mean reading literacy scores on the followup surveys than countries with higher mean scores in 2000.
Thus, with respect to the first research question, we found a significant overall increase of 1.29 PISA points per wave or 0.43 PISA points per year in the sample of 56 PISA countries. However, countries differed significantly in the slope and even the direction of the trend (e.g., a country that is one SD below the average is expected to have a negative trend). In addition, we found that countries also differ substantially in terms of their initial values at the first measurement occasion in 2000, and that countries with lower initial scores tend to achieve larger gains in their mean reading literacy scores across subsequent PISA waves.
Examining the dynamics in the development of PISA countries’ mean reading literacy scores
While the trend parameters of the CTLCMSR describe static change processes (Fig. 3A), the parameters of the dynamic process component describe fluctuations (Fig. 3B). These fluctuations were of substantive interest with respect to the second research question, which concerned the question of how long deviations from the trends (shocks) influence a country’s development of mean PISA reading literacy performance. As mentioned above, such deviations can be caused by shortterm support programs, or temporary investments in the education system as well as a financial crisis or a pandemic.
The five parameters related to the first process component provide information about the linear trends in the data and center the residuals that enter the dynamic process at the withincountry level (Fig. 3A–C). Thus, the dynamic process can be considered as detrended as the growth component accounts for the linear trends in the data (e.g., Usami et al., 2019). The remaining three parameters of the CTLCMSR, namely the autoeffect, the diffusion variance, and the initial residual variance component, are related to the dynamic process. The initial variance component represents the residual variance of the first measurement occasion entering the dynamic process and, thus, yields the initial (predetermined) values of the dynamic process in the CTLCMSR. The diffusion variance represents the CT error variance, which is the part of the variance of the process that is completely random (Oud & Jansen, 2000). The CT autoeffect can be used to study the time course of the impact of temporary deviations (also referred to as shocks) on the process under study and was therefore the relevant parameter to answer the second research question.
The interpretation of CT autoeffects is different from autoregressive effects of DTMs. Typical for stable or meanreverting dynamical processes is the negative value of the autoeffect (Ryan et al., 2018). Translated to DTMs, negative CT autoeffects mean that the corresponding DT autoregressive parameters are between 0 and 1. Furthermore, DT autoregressive effects get smaller for longer timeintervals between consecutive measurement occasions in such stable processes. The negative CT autoeffect refers to the fact that if the value of a time series at time \(t\) takes a position far from the process mean the process subsequently tends to return to the process mean with the opposite sign to the deviation (assuming that there are no further shocks afterwards). Furthermore, a negative CT autoeffect indicates that the more distant a deviation from the process mean at a time \(t\), the greater is the speed with which it tends to return to the process mean (assuming the same underlying autoeffect). Likewise, the more negative the autoeffect, the faster the process tends to return to the process mean after a deviation from the trend (for the same timeinterval).
Because DT autoregressive effects for various timeintervals can be derived from the estimated CT autoeffect, this parameter is useful to answer the question of how persistently deviations from the trend affected countries’ development of mean reading literacy achievement. The CT autoeffect was estimated to be − 0.39 (SE = 0.13). It is important to note that the autoeffect was the same for all countries of the sample in this model (no random effects), unlike the (countryspecific) parameters of the trends. To better understand the dynamic effects and find an answer to the second research question, DT autoregressive parameters can be derived from the CT autoeffect for various intervals.^{Footnote 11} DT autoregressive effects can be interpreted as “the amount of withinperson carry over effect” from one measurement occasion to the next (Hamaker et al., 2015, p. 10 l. 13^{Footnote 12}). CT modeling additionally provides information on the dependence of the DT autoregressive effects on the time interval length between measurement occasions.
Deriving discretetime parameters as functions of underlying continuoustime parameters
Using Eq. 3, we can calculate DT parameters for arbitrary timeintervals from the estimates of the underlying CT parameters. In Table 4 we present a range of DT autoregressive coefficients and the related confidence intervals. We can do the same for the diffusion variance and derive DT error variances, using Eq. 5 (Table 4). For longer timeintervals, the autoregressive coefficient becomes smaller, whereas the error variance becomes larger (Oravecz et al., 2018). DT autoregressive effects are usually interpreted in the literature as indicating the stability or the inertia of the construct under study (e.g., Hamaker et al., 2015). Thus, they provide information on how much of the deviation at time \({t}_{i}\) is carried over to \({t}_{i+1}\) (relative to the process mean) on average. To address the second research question, the time course of the autoregressive effects and the associated confidence intervals can be used (Table 4). The confidence interval of the autoregressive effect for the fiveyear interval is the first that includes zero, that is, there is no evidence for a population autoregressive effect larger than zero. An autoregressive effect of zero for a time interval of fiveyears or longer implies that after this period earlier deviations from the trend have no predictive value. To visualize this effect, DT coefficients can be represented as continuous functions of the (time independent) CT parameters and the length of timeinterval. As can be seen in Fig. 5, for increasing length of the timeinterval between measurement occasions, the DT autoregressive effect approaches zero (Fig. 5A), while the error standard deviation approaches about 13 (Fig. 5B).
Representing countriesspecific expected trajectories and making predictions for future states
Unlike DTMs, in which processes can only be represented with respect to the investigated interval (see Fig. 1B), CTMs such as the CTLCMSR allow to represent the time course of developmental trajectories. Combining trends and dynamics, Fig. 6 shows the countryspecific trajectories of four PISA countries in the sample. One major advantage of CTMs like the CTLCMSR is that predictions can be made with respect to any timeinterval. If PISA reading literacy scores were analyzed using a DTM, the predictions of future states would be limited to the threeyear intervals (and their multiples), because of the timeinterval dependency of DTMs. Moreover, unlike predictions made by a simple GCM, the CTLCMSR uses the information of the residuals, which are fluctuating around the static trends (the grey dotted lines in Fig. 6). Predictions of a GCM would always lie exactly on the static trend and would not take into account the dynamics of the residuals. Figure 6 illustrates the mean reverting dynamic process fluctuating around the countryspecific static trends. The size of the autoregressive effect decreases as the length of the timeintervals increases, representing the impact of deviations (shocks) on later states over time. The autoregressive effect reaches zero after a certain length of the timeinterval (see also Fig. 5A) indicating that there is no further influence of the previous deviation. For long timeintervals the autoregressive effect is zero and the best prediction lies on the linear trend. Thus, the point predictions return to the static linear trend process after a certain interval (Fig. 6).
Discussion
The aim of the present study was to discuss advantages of CTMs, especially the newly proposed CTLCMSR, for longitudinal data analysis in the education sciences and to provide the reader with an easytofollow introduction to specifying and interpreting CTMs with the help of an illustrative example using PISA data.
In the first section, we discussed that CTMs are often better suited to capture the continuous nature and development of constructs in educational research than DTMs. Using CTMs, we can thus better align our statistical models with our theories of change that are the subject of educational research. This includes not only variables at the level of individuals but also variables that describe characteristics of entire education systems as in the exemplary application with PISA data. In addition, we outlined other advantages of CTMs over DTMs associated with the timeinterval dependency of DTMs, such as limitations in crossstudy comparisons, limitations in the procedures of data collection, and the lack of relevant information regarding the dynamic processes.
Based on this, we developed the CTLCMSR, a CTM variant that is suitable for a range of typical longitudinal research questions. The CTLCMSR takes into account both the dynamic process as well as trends in the data. Trends are often of substantive interest in the education sciences and are also relevant for dynamic models because unaccounted trends can lead to biases in parameter estimates (Asparouhov et al., 2018; NúñezRegueiro et al., 2021; Walls & Schäfer, 2006). The process’ dynamics can provide information about the stability or variability of constructs, their selfrelated structure, and about reciprocal effects of multiple variables in multivariate models. In addition to the advantage that (1) the CTLCMSR provides information on both trends and dynamics, the use of the CTLCM has further advantages. (2) Trends and dynamic processes are clearly separated in the CTLCMSR, meaning that trends and dynamics cannot influence or “compete” with each other. (3) The interpretation of the parameters follows the interpretation of CTMs and GCMs. (4) All parameters of the CTLCMSR describe the process under study independently of the length of timeintervals in a given study (Delsing & Oud, 2008). Finally, (5) both the standard linear GCM and the CTAR(1) model are nested within the CTLCMSR, and thus, these models can be tested against the CTLCMSR using chisquared difference testing. In addition, with the CTLCMSR, we can take advantage of all the other benefits associated with CTMs.
In the second part, we demonstrated the application of the CTLCMSR with the R package ctsem (Driver & Voelkle, 2017, 2021). The analysis was guided by two exemplary research questions. First, we wanted to examine whether there is a significant trend in the mean PISA reading literacy scores of countries over the period from 2000 to 2018, and whether countries differ in the slope and direction of trends. Second, we aimed to study the stability of education systems when deviations from the trend occur. Examining the persistence of deviations is relevant to study the system's resilience against disturbances caused by different influences.
Regarding the first research question, we found an overall increase in PISA countries’ mean reading literacy scores. However, countries differed considerably in the slope and the direction of trends. Moreover, countries also differed with respect to the initial levels at the first measurement occasion in 2000. We found that countries with lower scores at the first measurement occasion tended to achieve larger gains in mean reading literacy scores on the followup surveys than countries with higher initial scores.
Regarding the second research question, we found a significant continuoustime dynamic process of fluctuations around the trends. From this dynamic process it can be inferred how long temporary deviations from the trends are associated with later states of the variable(s) under study and how much of the deviation is carried over to future states depending on the time that has passed. Using the CT autoeffect, we derived DT autoregressive effects for various timeintervals and found that carryover effects (represented by DT autoregressions) for fiveyear or longer timeintervals were not significantly different from zero. This means that both positive and negative influences on countries’ mean reading literacy scores that led to temporary deviations from the trends disappeared on average after about five years.
Limitations and future directions
By introducing the CTLCMSR, the current article suggests and illustrates an advanced method with several benefits and potentials for longitudinal studies in the education sciences. However, the presented work is not without limitations.
First, it is necessary to consider that the application presented is a simplified example chosen in order to provide an easy introduction to the potential benefits, practical modeling, and interpretation of CTMs. Therefore, the substantive value of the results is limited. Only a single variable was analyzed, the mean PISA reading scores of countries. For some applications, such as predictions of future states, univariate models can already provide useful information (e.g., Bulteel et al., 2018). Of course, however, research questions are often more complex. In the present application, for example, it is only possible to speculate on why temporary deviations from trends occur, as we are studying shocks of unknown cause. In practical applications, it might be of interest to examine the impact of a specific intervention (or crises). This requires the inclusion of an input that occurs at specific points in time in the model (see Driver & Voelkle, 2018b for the implementation of such models in ctsem). Such models make it possible to examine, for instance, when the input reaches the maximum effect on the process under study and after what period of time the effect of an input vanishes.
Moreover, the dynamic interplay between two or more parallel processes might be of researchers’ interest (see, e.g., Jindra et al., under review, to be published in the same special issue applying a bivariate CTLCMSR). Such models allow, for instance, to explore the time interval in which one process has a maximum effect on the other and vice versa (e.g., Hecht & Zitzmann, 2021b). A typical question for educational research might be after how much time a change in a student's achievement does develop its maximum effect on his or her selfconcept. R code for studying the reciprocal effects of two variables over time in a bivariate CTLCMSR is provided in the Additional file 1.
In addition, the use of covariates plays a crucial role in practical research, including the study of trends in countries’ mean PISA scores, for which socalled adjusted trends have been reported meanwhile (OECD, 2014). Such more complex CTMs with predictors and covariates can be implemented with ctsem (e.g., Driver & Voelkle, 2018b).
It is also important to keep in mind, that development at the countrylevel was examined, not at the student level. While shortterm interventions might not change the performance of the entire education system permanently, they might have effects on the students’ cohort. However, statements concerning developmental trajectories of students require repeated measurements of the same students. A first application of a bivariate CTLCMSR on the studentlevel is provided by Jindra et al. (under review) to be published in the same special issue.
Another possibility in ctsem is specifying measurement models (e.g., Hecht et al., 2019). In the presented application with PISA data, a typical threestep procedure was chosen to address measurement error (cf. OECD, 2009). We first calculated aggregate country means for each of the five plausible values and for all seven available PISA waves and created five data sets (so that all plausible values could be considered). These data sets were then used to conduct all analyses separately. Finally, the obtained results were pooled across the five separate analyses (Rubin, 2004).
The CTMs presented must also be considered with some limitations and model assumptions in mind. In the present article, we have restricted ourselves to univariate stable dynamic processes, also called meanreverting processes (Ryan et al., 2018). That implies that the corresponding DT autoregressive effects are between 0 and 1 for any timeinterval. In principle, however, CTMs for explosive processes, meaning DT autoregressions greater than 1 (e.g., Driver & Voelkle, 2021), or processes with negative DT autoregressions, may be of interest as well (Fisher, 2001). In this article, we furthermore restricted ourselves to the representation of firstorder CTMs, which should be suitable for representing many processes studied in educational research (Sivo & Fan, 2008). However, modeling higherorder models is also possible and might be useful in certain applications (e.g., Lüdtke & Robitzsch, 2021; Oud, 2010; Oud et al., 2018). Furthermore, in the models presented, the dynamic effects were assumed to be invariant over the period studied. This means, for example, that the CT autoeffect does not change and the DT autoregressive effects change only as a function of the length of the timeinterval but not with time itself. However, more complex models with timevarying dynamic effects are also possible (e.g., Driver & Voelkle, 2021; Oud & Jansen, 2000). In addition, in the application presented, the dynamic effects were assumed to be the same across all observation units. Random effects were only included in the trend components but not in the dynamic process. This is a limitation that can be overcome within ctsem because fully hierarchical models can be specified in which all parameters vary across units (Driver & Voelkle, 2018a). Future simulation studies are needed to further investigate the estimation performance of CTMs in general and the CTLCMSR in particular with respect to different influencing factors, such as sample size and length of individual time series (for first samplesize recommendations and a discussion of the compensating effects of sample size N and length of time series T in CTMs, see Hecht & Zitzmann, 2021b).
In the CTLCMSR, we proposed to consider linear trends. The use of linear trends is not without criticism and can usually only be justified for the duration of specific time periods, as they always tend toward infinity in the long run (e.g., Oud, 2010). However, linear trends have a clear and wellknown interpretation, which can be an advantage. Nevertheless, other types of trends might be considered and in future research, continuoustime models that incorporate other than linear trends could be developed or further refined (e.g., Oud, 2010; Voelkle & Oud, 2015). For example, in the PISA example, we also tested a GCM with nonlinear (quadratic) components, which did however not reach significance. We provide example code for setting up GCMs and the CTLCMSR with quadratic trend components in ctsem (similar code can be found on the GitHub page of ctsem; Driver, 2022).
In the CTLCMSR as applied in the present article, it remains uncertain how a given shock affects the trend. Models, which allow a shock to influence the direction and shape of the trend, are also possible (Driver & Voelkle, 2018b).
In educational research, specific data situations may occur which demand appropriate analysis strategies like resampling and missing data imputation techniques (e.g., Weirich et al., 2021). Future research should generate guidance on how to address such educational research specific challenges when applying CTMs. Furthermore, estimation of CTMs might become very timeconsuming, especially in educational largescale assessment contexts. One promising optimization approach to reduce run times for continuoustime models has been proposed by Hecht and Zitzmann (2020) based on the work of Hecht et al. (2020).
Finally, although CTMs provide information about the development of dynamic effects, interpolations to timeintervals that have not been investigated should be considered with caution. While it may be useful to explore intervals of maximum effects based on CTMs (Hecht & Zitzmann, 2021a), it is not advisable to draw conclusions about change processes that occur at much smaller intervals, for example days or hours, based on annual surveys, for instance. Thus, even when using CTMs, the underlying theory of change should still guide decisions about the design of longitudinal studies. However, equal timeintervals between subjects and measurement occasions are not necessary when using CTMs, which allows for much greater flexibility in data collection than when using DTMs (see Voelkle & Oud, 2013, for an argument why it can even be beneficial to choose timeintervals of different length between individuals and measurement occasions).
Conclusion
In the present article, we presented continuoustime modeling as a suitable approach to align statistical models for longitudinal data analysis with theories of change, especially to represent the continuous nature of variables and observation units. It is essential to align theories and statistical models to obtain valid answers to the research questions. In addition, the use of CTMs comes with other considerable advantages which were mentioned. Based on this, we proposed the CTLCMSR, which is a CTM variant and a suitable model to address many research questions in the education sciences. The CTLCMSR yields parameters, which are independent of specific timeintervals, thus represents developmental processes on a continuoustime scale. We illustrated the implementation and interpretation of the CTLCMSR with PISA data and, thus, gave the reader an introduction to the use of CTMs and the advantages and functionality of the newly proposed CTLCMSR.
Availability of data and materials
The data analyzed in this article are available at the PISA website: https://www.oecd.org/pisa/data/.
The R code for all models run for the present analyses and an additional script for a bivariate CTLCMSR are available at the following link: https://doi.org/10.6084/m9.figshare.19850926
Notes
The term AR or VAR model stems from the tradition of time series analysis. In classical time series analyses, only a single unit (N = 1) is often examined (e.g., Ryan et al., 2018). The term CLPM can be understood as a label for a version of the VAR model for panel data (cf. Kuiper & Ryan, 2018), with the caveat that sometimes, the lagged coefficients are not equated across measurement occasions in CLPMs (Sivo & Fan, 2008). Because panel data are the norm in educational research, the term VAR model is used in the following to refer to the panel VAR model (Hsiao, 2014; Sivo & Fan, 2008).
One possible solution to address unequally spaced measurement occasions using DTMs is the socalled phantomvariable approach, but this approach is limited to few unequal intervals. Furthermore, other drawbacks associated with the intervaldependency of DTMs remain (see Oud & Voelkle, 2014).
Figure 3B shows a so called meanreverting process, which means, that the CT autoeffect is negative (i.e., discretetime autoregressive coefficients are between 0 and 1 for all time intervals). Other types of dynamics are also possible, but they are rare in educational and psychological research (cf. Hamaker et al., 2015), which is why we are limiting ourselves to this type of dynamics in the present article.
Since PISA 2015, there is a shift to including ten PVs. Due to the smaller number of five corresponding PVs in the previous waves, only the first five PVs were used in the data preparation for the 2015 and 2018 waves in the present study.
The MANIFESTVAR matrix in ctsem was used to define the error term parameters of the GCM (see also Driver, 2020, and the specification in the Supplemental Material). Another more complicated alternative to estimate the standard GCM's homoscedastic error term (Fig. 3D) would be to constrain the variance of \({T0dyn}_{SD}\) to \({q}_{\mathrm{\Delta t}}^{*}\)(Fig. 3D). While the latter approach better captures the nested structure of GCM and CTLCMSR, it requires changes in the source code of ctsem, which is why using MANIFESTVAR to do this is much easier to implement in ctsem.
To install the required ctsem package, we run the usual procedure in R (R Core Team, 2021):
install.packages( "ctsem").
library( ctsem).
DTMs are also possible via type = ”standt” (Driver & Voelkle, 2018a).
This specification is only possible in the case of linear trends. We provide an example code in the supplemental material how to include quadratic trends in ctsem (and examples can also be found on GitHub; Driver, 2022). In the case of quadratic effects, the PARS argument has to be used and the LAMBDA matrix must be employed in a different way.
Moreover, we tested a GCM with additional quadratic effects. The quadratic effects did not have a statistically significant contribution and were therefore not considered in further analysis (e.g., Voelkle, 2008).
In multivariate models the same can be done for all lagged parameters including crosseffects (e.g., Ryan et al., 2018).
Recognize the difference in interpretation between fixed and random effects models discussed by Hamaker et al. (2015) in the same article.
Abbreviations
 AIC:

Akaike Information Criterion
 ALT:

Autoregressive latent trajectory (model)
 AR (model):

Autoregressive model
 BIC:

Bayesian Information Criterion
 CALT:

Continuoustime autoregressive latent trajectory (model)
 CTAR (model):

Continuoustime autoregressive (model)
 CTLCMSR:

Continuoustime latent curve model with structured residuals
 CTVAR (model):

Continuoustime vector autoregressive (model)
 CTM:

Continuoustime model
 DTM:

Discretetime model
 GCM:

Growthcurve model
 LCMSR:

(Discretetime) latent curve model with structured residuals
 VAR (model):

Vector autoregressive model
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Lohmann, J.F., Zitzmann, S., Voelkle, M.C. et al. A primer on continuoustime modeling in educational research: an exemplary application of a continuoustime latent curve model with structured residuals (CTLCMSR) to PISA Data. Largescale Assess Educ 10, 5 (2022). https://doi.org/10.1186/s40536022001268
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DOI: https://doi.org/10.1186/s40536022001268
Keywords
 Longitudinal data analysis
 Continuoustime
 Dynamic model
 Latent curve models
 PISA