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Univariate Autoregressive Structural Equation Models as Mixed-Effects Models

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Title: Univariate Autoregressive Structural Equation Models as Mixed-Effects Models
Language: English
Authors: Steffen Nestler (ORCID 0000-0001-9724-2441), Sarah Humberg (ORCID 0000-0002-7891-3622)
Source: Structural Equation Modeling: A Multidisciplinary Journal. 2024 31(2):357-366.
Availability: Routledge. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
Peer Reviewed: Y
Page Count: 10
Publication Date: 2024
Document Type: Journal Articles
Reports - Descriptive
Descriptors: Structural Equation Models, Computer Software, Models, Measurement, Hierarchical Linear Modeling
DOI: 10.1080/10705511.2023.2212865
ISSN: 1070-5511
1532-8007
Abstract: Several variants of the autoregressive structural equation model were suggested over the past years, including, for example, the random intercept autoregressive panel model, the latent curve model with structured residuals, and the STARTS model. The present work shows how to place these models into a mixed-effects model framework and how to estimate them in mixed-effects model software, namely the R package "nlme." We also show how "nlme" can be used to fit extensions of these models, for example, models that do not assume equally spaced time intervals between measurement occasions (i.e., continuous time models). Overall, our expositions show that autoregressive structural equations models and mixed-effects models are closely related. We think that this insight eases researchers to understand the differences between the variants of the autoregressive structural equation model and also allows them to profitably link the two different modeling perspectives.
Abstractor: As Provided
Entry Date: 2024
Accession Number: EJ1431570
Database: ERIC
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  Value: <anid>AN0176073021;7mz01mar.24;2024Mar19.06:25;v2.2.500</anid> <title id="AN0176073021-1">Univariate Autoregressive Structural Equation Models as Mixed-Effects Models </title> <p>Several variants of the autoregressive structural equation model were suggested over the past years, including, for example, the random intercept autoregressive panel model, the latent curve model with structured residuals, and the STARTS model. The present work shows how to place these models into a mixed-effects model framework and how to estimate them in mixed-effects model software, namely the R package nlme. We also show how nlme can be used to fit extensions of these models, for example, models that do not assume equally spaced time intervals between measurement occasions (i.e., continuous time models). Overall, our expositions show that autoregressive structural equations models and mixed-effects models are closely related. We think that this insight eases researchers to understand the differences between the variants of the autoregressive structural equation model and also allows them to profitably link the two different modeling perspectives.</p> <p>Keywords: Autoregressive models; cross-lagged panel models; mixed-effects models; multilevel models; structural equation models</p> <p>Psychologists increasingly use longitudinal data to examine their substantive research questions. Among the most common models to analyze such longitudinal data are the growth model and the autoregressive model (see Diggle et al., [<reflink idref="bib10" id="ref1">10</reflink>]; Hedeker & Gibbons, [<reflink idref="bib14" id="ref2">14</reflink>]; Little, [<reflink idref="bib19" id="ref3">19</reflink>]). They allow to test hypotheses about the time course of the longitudinally assessed variable, to determine the influence of time-constant and time-varying predictors, and/or to examine between-person differences in the modeled parameters. Growth models are fit either as a structural equation model (see Bollen & Curran, [<reflink idref="bib5" id="ref4">5</reflink>]; Meredith & Tisak, [<reflink idref="bib24" id="ref5">24</reflink>]; Preacher et al., [<reflink idref="bib37" id="ref6">37</reflink>]) or as a mixed-effects model (also called multilevel model, see Bolger & Laurenceau, [<reflink idref="bib3" id="ref7">3</reflink>]; Hoffmann, [<reflink idref="bib16" id="ref8">16</reflink>]; Snijders & Bosker, [<reflink idref="bib42" id="ref9">42</reflink>]), whereby the latter framework is usually preferred in the case of intensive longitudinal data (Bolger & Laurenceau, [<reflink idref="bib3" id="ref10">3</reflink>]). Autoregressive models, by contrast, are typically estimated within a structural equation model framework (see Little, [<reflink idref="bib19" id="ref11">19</reflink>]; Newsom, [<reflink idref="bib33" id="ref12">33</reflink>]).</p> <p>It is well known that under certain conditions, growth models as fitted in the structural equation model framework and in the mixed-effects model framework are equivalent (see Curran, [<reflink idref="bib7" id="ref13">7</reflink>]; Grimm, Ram, & Estabrook, [<reflink idref="bib12" id="ref14">12</reflink>]). However, a fact that is not well represented among psychological researchers is that some of the most frequently used autoregressive structural equation models, such as the random intercept (cross-lagged) panel model (Hamaker et al., [<reflink idref="bib13" id="ref15">13</reflink>]; Newsom, [<reflink idref="bib33" id="ref16">33</reflink>]), the latent curve model with structured residuals (Berry & Willoughby, [<reflink idref="bib2" id="ref17">2</reflink>]), and the STARTS model (Kenny & Zautra, [<reflink idref="bib17" id="ref18">17</reflink>]) can also be specified and estimated as mixed-effects models. In fact, these models are also introduced in mixed-effects model textbooks (see Diggle et al., [<reflink idref="bib10" id="ref19">10</reflink>]; Hedeker & Gibbons, [<reflink idref="bib14" id="ref20">14</reflink>]; Verbeke & Molenberghs, [<reflink idref="bib45" id="ref21">45</reflink>]), problems of these models (e.g., with regard to estimation) are discussed in the respective mixed-effects model literature, and extensions of these models were suggested that might be valuable for applied psychological research.</p> <p>The current manuscript aims to show how to incorporate autoregressive structural equation models into the mixed-effects model framework. We think that this is interesting for several reasons. First, from a conceptual point of view, we believe that a mixed-effects model perspective helps researchers to understand what makes the models different from each other, which in turn could help them to decide which of the models is (more) appropriate for their research. Second, the mixed-effects model has been extended in several ways, for example, the approach allows continuous-time modeling (Pinheiro & Bates, [<reflink idref="bib35" id="ref22">35</reflink>]) or modeling of between-person differences in the Level 1 residual variance and autocorrelation (Hedeker, Mermelstein, & Demirtas, [<reflink idref="bib15" id="ref23">15</reflink>]; Nestler, [<reflink idref="bib30" id="ref24">30</reflink>]). Thus, when the autoregressive structural equation models are incorporated into the mixed-effects model, researchers can use these extensions in their own work, which in turn may allow them to examine interesting new models in their future research. Third, our expositions imply that one can estimate the parameters of the autoregressive structural equation models in standard mixed-effects model software. Here, we use the nlme package (Pinheiro et al., [<reflink idref="bib36" id="ref25">36</reflink>]) in the open-source statistical software R (R Core Team, [<reflink idref="bib38" id="ref26">38</reflink>]). Finally, our explanations enable researchers to fit autoregressive structural equation models to their intensive longitudinal data (as assessed in experience sampling or daily diary studies), because such data is usually analyzed in a mixed-effects model framework.</p> <p>Before we start with our explanations, we note that there is some debate regarding whether the parameters of a model can be used to model a psychological process more appropriately (e.g., in terms of between- or within-person processes; see Orth et al., [<reflink idref="bib34" id="ref27">34</reflink>]) compared to another model, and also regarding the causal interpretation of a model parameter (Lüdtke & Robitzsch, [<reflink idref="bib21" id="ref28">21</reflink>]; Mund & Nestler, [<reflink idref="bib27" id="ref29">27</reflink>]). It is our view that every statistical model makes certain assumptions about the data-generating model and that these assumptions can be better understood by placing them in the mixed-effects model framework. However, whether these assumptions are fulfilled in a particular case, is a substantive question (see Lüdtke & Robitzsch, [<reflink idref="bib21" id="ref30">21</reflink>]). Therefore, we will refrain from discussing these issues in this manuscript. Furthermore, in our explanations we focus on univariate autoregressive path models and do not treat bivariate models. The reason for this is that this focus is sufficient for the conceptual and applied goals of the article, but also that most mixed-effects model literature (software) only explain (allows to fit) univariate models.</p> <p>The present manuscript starts with a brief description of the univariate versions of the random intercept autoregressive panel model, the latent curve model with structured residuals, and the STARTS model. Thereafter, we present a very general mixed-effects model that was introduced by Diggle ([<reflink idref="bib9" id="ref31">9</reflink>]). We then show that this model contains all three autoregressive models as special cases and we describe how to fit the models with the R package nlme (Pinheiro et al., [<reflink idref="bib36" id="ref32">36</reflink>]). Thereafter, we discuss how to fit extensions of these models and some implications of our elaborations for applied and methodological future research.</p> <p>To illustrate our expositions, we use a data set that is provided in the STARTS package (Robitzsch & Lüdtke, [<reflink idref="bib39" id="ref33">39</reflink>]). It contains the extraversion data of 890 individuals measured at five time points. We use the index</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>I</mi></mrow></math> </ephtml> for the individuals and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> for the measurement occasions of individual <emph>i</emph>. In our example, there are thus <emph>I</emph> = 890 individuals each with <emph>n<subs>i</subs></emph> = 5 extraversion measurements</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>y</mi></mrow><mn>1</mn></msub><mo>,</mo><mo>...</mo><mo>,</mo><msub><mrow><mi>y</mi></mrow><mn>5</mn></msub></mrow><mo>.</mo></math> </ephtml> Table 1 shows an excerpt of the data in the wide and the long format. The former is used in most structural equation modeling software packages, while the latter is standard in mixed-effects model software. Furthermore, Table 2 gives some summary statistics for the five variables and the R codes to fit the models and to reproduce the results are available in an OSF project accompanying this manuscript (see https://osf.io/ztcge/). When not stated otherwise, we use the lme function of the nlme package (Pinheiro et al., [<reflink idref="bib36" id="ref34">36</reflink>]) to fit the mixed-effects models and the sem function in the lavaan package (Rosseel, [<reflink idref="bib40" id="ref35">40</reflink>]) to fit the structural equation models.[<reflink idref="bib1" id="ref36">1</reflink>]</p> <p>Table 1. Excerpt of the illustrative data.</p> <p> <ephtml> <table><thead><tr><td>Panel A: Wide format</td></tr><tr><td>ID</td><td><italic>y</italic><sub>1</sub></td><td><italic>y</italic><sub>2</sub></td><td><italic>y</italic><sub>3</sub></td><td><italic>y</italic><sub>4</sub></td><td><italic>y</italic><sub>5</sub></td></tr></thead><tbody valign="top"><tr><td>100006</td><td char=".">−0.280</td><td char=".">0.120</td><td char=".">1.083</td><td char=".">0.495</td><td char=".">−0.092</td></tr><tr><td>100008 ...</td><td char=".">1.482</td><td char=".">−0.092</td><td char=".">0.120</td><td char=".">0.120</td><td char=".">0.707</td></tr><tr><td><p id="ilm0069"><graphic href="hsem_a_2212865_ilm0069.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo xmlns="" /></math></p></td><td /><td /><td /><td /><td /></tr></tbody></table> </ephtml> </p> <p>Table 1. Excerpt of the illustrative data.</p> <p> <ephtml> <table><thead><tr><td>Panel B: Long format</td></tr><tr><td>ID</td><td><italic>Y</italic></td><td>TIME</td></tr></thead><tbody valign="top"><tr><td>100006</td><td char=".">−0.280</td><td char=".">1</td></tr><tr><td>100006</td><td char=".">0.120</td><td char=".">2</td></tr><tr><td>100006</td><td char=".">1.083</td><td char=".">3</td></tr><tr><td>100006</td><td char=".">0.495</td><td char=".">4</td></tr><tr><td>100006</td><td char=".">−0.092</td><td char=".">5</td></tr><tr><td>100008</td><td char=".">1.482</td><td char=".">1</td></tr><tr><td>100008</td><td char=".">−0.092</td><td char=".">2</td></tr><tr><td>100008</td><td char=".">0.120</td><td char=".">3</td></tr><tr><td>100008</td><td char=".">0.120</td><td char=".">4</td></tr><tr><td>100008</td><td char=".">0.707</td><td char=".">5</td></tr><tr><td><p id="ilm0070"><graphic href="hsem_a_2212865_ilm0070.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo xmlns="">⋯</mo></math></p></td><td><p id="ilm0071"><graphic href="hsem_a_2212865_ilm0071.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo xmlns="">⋯</mo></math></p></td><td><p id="ilm0072"><graphic href="hsem_a_2212865_ilm0072.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo xmlns="">⋯</mo></math></p></td></tr></tbody></table> </ephtml> </p> <p>Table 2. Descriptive statistics for the illustrative data.</p> <p> <ephtml> <table><thead><tr><td>Variable</td><td><italic>M</italic></td><td><italic>SD</italic></td><td char=".">1.</td><td char=".">2.</td><td char=".">3.</td><td char=".">4.</td><td char=".">5.</td></tr></thead><tbody valign="top"><tr><td>1. <italic>y</italic><sub>1</sub></td><td char=".">−0.09</td><td char=".">0.99</td><td char=".">1.00</td><td char=".">0.71</td><td char=".">0.67</td><td char=".">0.63</td><td char=".">0.62</td></tr><tr><td>2. <italic>y</italic><sub>2</sub></td><td char=".">−0.03</td><td char=".">0.98</td><td char=".">0.71</td><td char=".">1.00</td><td char=".">0.76</td><td char=".">0.72</td><td char=".">0.72</td></tr><tr><td>3. <italic>y</italic><sub>3</sub></td><td char=".">−0.02</td><td char=".">0.99</td><td char=".">0.67</td><td char=".">0.76</td><td char=".">1.00</td><td char=".">0.76</td><td char=".">0.73</td></tr><tr><td>4. <italic>y</italic><sub>4</sub></td><td char=".">0.12</td><td char=".">1.02</td><td char=".">0.63</td><td char=".">0.72</td><td char=".">0.76</td><td char=".">1.00</td><td char=".">0.79</td></tr><tr><td>5. <italic>y</italic><sub>5</sub></td><td char=".">0.02</td><td char=".">1.01</td><td char=".">0.62</td><td char=".">0.72</td><td char=".">0.73</td><td char=".">0.79</td><td char=".">1.00</td></tr></tbody></table> </ephtml> </p> <p>1 <emph>Note</emph>. M = Mean, SD = Standard deviation, Columns 4 to 8 contain the correlations between the five variables.</p> <hd id="AN0176073021-2">1. Three Autoregressive Path Models</hd> <p>We focus on three univariate structural equation models (see Mund et al., [<reflink idref="bib26" id="ref37">26</reflink>]; Mund & Nestler, [<reflink idref="bib27" id="ref38">27</reflink>]; Orth et al., [<reflink idref="bib34" id="ref39">34</reflink>]; Usami et al., [<reflink idref="bib43" id="ref40">43</reflink>], for more detailed explanations of the models), all of which are extensions of the standard autoregressive model (i.e., a univariate version of the cross-lagged panel model; see Finkel, [<reflink idref="bib11" id="ref41">11</reflink>]; Little, [<reflink idref="bib19" id="ref42">19</reflink>]; Newsom, [<reflink idref="bib33" id="ref43">33</reflink>]): a univariate version of the random intercept cross-lagged panel model that we call <emph>random intercept autoregressive panel model</emph> (RI-APM, Hamaker et al., [<reflink idref="bib13" id="ref44">13</reflink>]), the <emph>latent curve model with structured residuals</emph> (LCM-SR, Berry & Willoughby, [<reflink idref="bib2" id="ref45">2</reflink>]; Curran et al., [<reflink idref="bib8" id="ref46">8</reflink>]), and the <emph>STARTS model</emph> (Kenny & Zautra, [<reflink idref="bib17" id="ref47">17</reflink>], [<reflink idref="bib18" id="ref48">18</reflink>]). Figure 1 presents graphical illustrations of the three models for three time points. As indicated in the figures, all three models assume an autoregressive process that is modeled on the level of the residualized scores (<emph>ly</emph><subs>1</subs> to <emph>ly</emph><subs>3</subs> in the Figure) and not on the level of the observed scores (i.e., <emph>y</emph><subs>1</subs> to <emph>y</emph><subs>3</subs>, e.g., the observed extraversion scores). In all three models, a person's residualized score represents a time point-specific deviation from the person's expected score at that time point. The residual variance—the variance of the residualized scores—indicates the magnitude of these deviations. When an error term is included in the model (i.e., an <emph>e</emph> term), it reflects the assumption that the residualized scores contain measurement error, in which case the respective error variance indicates the amount of this unreliability. Finally, we assume in the following that the autoregressive parameter, the error variance terms (when present), and the residual variance terms from the second time point onward are constrained to be constant across time points.</p> <p>Graph: Figure 1. Path diagrams of the random intercept autoregressive panel model (A), the latent curve model with structured residuals (B), and the STARTS model (C).</p> <p>Figure 1A displays the path diagram of the RI-APM. The model includes a random intercept factor (<emph>η<subs>I</subs></emph> in the Figure) that captures stable, trait-like variance in the longitudinally assessed variable <emph>y</emph> (e.g., mean-level extraversion). Not shown in the figure is that the mean of the intercept factor is set to zero and that the time point specific means of the observed variables <emph>y<subs>j</subs></emph> are estimated with the data. Furthermore, the model includes an autoregressive parameter <emph>a</emph> that is modeled on the level of the residualized scores. Therefore, <emph>a</emph> is a measure of the rank-order stability <emph>within</emph> a person, that is, it indicates how strong a person's deviations from her/his mean (or trait level; e.g. mean-level extraversion) are correlated across time. The RI-APM further assumes that the observed variables are measured without error, that is, the error variances of the observed indicators are constrained to be zero. Finally, the model includes variance terms <emph>u</emph> for the residualized scores which reflect variations of a person's scores not explained by the autoregressive part of the model.</p> <p>The LCM-SR (Figure 1B) is similar to the RI-APM in that it models a random intercept factor <emph>η<subs>I</subs></emph>, an autoregressive effect on the level of the residualized scores, and that it assumes that the observed variables are measured without error. However, different from the RI-APM, the LCM-SR contains a random intercept factor <emph>η<subs>I</subs> and</emph> a random slope factor <emph>η<subs>S</subs></emph> (see Figure 1B again), where <emph>η<subs>I</subs></emph> captures between-person differences in initial levels (e.g., predicted extraversion at the first occasion) and <emph>η<subs>S</subs></emph> captures linear developments across time (e.g., predicted linear changes in extraversion). Typically, the two latent factors are allowed to correlate and it is assumed, as in Figure 1B, that the trait-based changes follow a <emph>linear</emph> trend. More complex growth trends can also be fit to the data, at least when the number of assessed time points is sufficient (e.g., a quadratic growth model). A second difference between the RI-APM and the LCM-SR is that in the LCM-SR, the means of the observed variables are constrained to be zero and that the means of the intercept and the slope factor are freely estimated. The observed means are thus modeled as (linear) functions of the intercept and the slope factor means.</p> <p>The STARTS model (Figure 1C) is similar to the RI-APM in that it assumes that the variance of the observed scores is explained by an intercept factor and that it includes an autoregressive parameter on the level of the residualized scores. Furthermore, the mean of the intercept factor is set to zero and a mean is estimated for each time point <emph>j</emph>. However, different from the RI-APM, the STARTS model assumes that the variables are measured with error, hence the model includes variance terms for the observed scores (i.e., the variance of the <emph>e</emph> terms in Figure 1C).</p> <hd id="AN0176073021-3">2. A General Mixed-Effects Model</hd> <p>The general mixed-effects model (GMEM) that we consider here was first suggested by Diggle ([<reflink idref="bib9" id="ref49">9</reflink>]; see also Diggle et al., [<reflink idref="bib10" id="ref50">10</reflink>]; Liu, [<reflink idref="bib20" id="ref51">20</reflink>]; Verbeke & Molenberghs, [<reflink idref="bib45" id="ref52">45</reflink>]). In this model, the response <emph>y<subs>ij</subs></emph> of person <emph>i</emph> on time point <emph>j</emph> (e.g., the extraversion rating of person <emph>i</emph> on day <emph>j</emph>) is given by:</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mi mathvariant="bold-italic">β</mi><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">b</mi></mrow><mi>i</mi></msub><mo>+</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><msub><mrow><mi>W</mi></mrow><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mo>.</mo></math> </ephtml> (<reflink idref="bib1" id="ref53">1</reflink>)</p> <p>Here,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></math> </ephtml> is a row vector containing explanatory variables (e.g., a time-constant variable such as gender or time-varying variables such as a variable coding the day) and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">β</mi></math> </ephtml> contains the corresponding regression weights.</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">b</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> is a vector of random effects for the elements in</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">β</mi></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></math> </ephtml> is a design vector containing 0s and 1s that codes the presence/absence of the random effects in</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">b</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> for <emph>y<subs>ij</subs>. H<subs>ij</subs></emph> is a random effect term that represents measurement error and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>W</mi></mrow><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> is a residual term that is used to model serial correlations and that can depend on the time point <emph>j</emph> (see below). Finally, the model can also be specified for the vector of all measurements of person <emph>i</emph>:</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mi>i</mi></msub><mo>=</mo><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub></mrow></msub></mrow></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo /><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub><mi mathvariant="bold-italic">β</mi><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><msub><mrow><mi mathvariant="bold-italic">b</mi></mrow><mi>i</mi></msub><mo>+</mo><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub></mrow></msub></mrow></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo>+</mo><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mrow><msub><mrow><mi>W</mi></mrow><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>W</mi></mrow><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>W</mi></mrow><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub></mrow></msub><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo>,</mo></mrow></math> </ephtml> (<reflink idref="bib2" id="ref54">2</reflink>)</p> <p>where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> is now a matrix containing the values of the explanatory variables and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> is a design matrix. The number of rows of both matrices is <emph>n<subs>i</subs></emph> and the number of columns depends on the number of included explanatory variables and random effects, respectively.</p> <p>It is typically assumed (see Diggle, [<reflink idref="bib9" id="ref55">9</reflink>]; Diggle et al., [<reflink idref="bib10" id="ref56">10</reflink>]; Verbeke & Molenberghs, [<reflink idref="bib45" id="ref57">45</reflink>]) that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">b</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> is a vector of normally distributed variables with expectation zero and covariance matrix <bold><emph>G</emph></bold>, that the <emph>H<subs>ij</subs></emph> terms are also normally distributed with expectation zero and variance</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>ν</mi></mrow><mn>2</mn></msup></mrow><mo>,</mo></math> </ephtml> and that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>W</mi></mrow><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> is a Gaussian process with expectation zero and covariance matrix</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi mathvariant="bold-italic">σ</mi></mrow><mi mathvariant="bold-italic">2</mi></msup><mi mathvariant="bold-italic">R</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">t</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mo>.</mo></math> </ephtml> Here,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi mathvariant="bold-italic">σ</mi></mrow><mi mathvariant="bold-italic">2</mi></msup></mrow></math> </ephtml> is a</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub><mo>×</mo><mn>1</mn></mrow></math> </ephtml> vector of variance terms, that is,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi mathvariant="bold-italic">σ</mi></mrow><mi mathvariant="bold-italic">2</mi></msup><mo>=</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>σ</mi></mrow><mn>1</mn><mn>2</mn></msubsup><mo>,</mo><mo>...</mo><mo>,</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub></mrow><mn>2</mn></msubsup></mrow></math> </ephtml> ), and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">R</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">t</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> is a</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub><mo>×</mo><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> symmetric matrix whose entries may depend on the specific time codings of person <emph>i</emph>, that are collected in the vector</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">t</mi></mrow><mi>i</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn><mi>i</mi></mrow></msub><mo>,</mo><mo>...</mo><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mo>.</mo></math> </ephtml> These assumptions entail that the model-implied covariance matrix</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> for the data vector</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> of person <emph>i</emph> is</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mi>i</mi></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mi mathvariant="bold-italic">G</mi><msubsup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi><mi>T</mi></msubsup><mo>+</mo><msup><mrow><mi>ν</mi></mrow><mn>2</mn></msup><mi mathvariant="bold-italic">I</mi><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">σ</mi></mrow><mi mathvariant="bold-italic">2</mi></msup><mo>·</mo><mi mathvariant="bold-italic">R</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">t</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mo>,</mo></math> </ephtml> (<reflink idref="bib3" id="ref58">3</reflink>)</p> <p>where <bold><emph>I</emph></bold> is an</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub><mo>×</mo><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> identity matrix.</p> <p>The GMEM contains some well-known multilevel models as special cases. For example, the multilevel random intercept model (Bolger & Laurenceau, [<reflink idref="bib3" id="ref59">3</reflink>]; Hoffmann, [<reflink idref="bib16" id="ref60">16</reflink>]) is usually written as</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mi>μ</mi><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>0</mn><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mo>,</mo></math> </ephtml> (<reflink idref="bib4" id="ref61">4</reflink>)</p> <p>where <emph>μ</emph> is the grand mean across all persons and days (e.g., the average of all extraversion ratings),</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn>0</mn><mi>i</mi></mrow></msub></mrow></math> </ephtml> is person <emph>i</emph>'s deviation from this grand mean, and <emph>ϵ<subs>ij</subs></emph> is the Level 1 residual term that reflects the deviation of the extraversion measurement at time point <emph>j</emph> from person <emph>i</emph>'s average level of extraversion. Writing all measurements of person <emph>i</emph> as a vector, the model in Equation (<reflink idref="bib4" id="ref62">4</reflink>) is equivalent to</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub></mrow></msub></mrow></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo /><mo>=</mo><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo>·</mo><mi>μ</mi><mo>+</mo><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo>·</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>0</mn><mi>i</mi></mrow></msub><mo>+</mo><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo>+</mo><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mrow><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>i</mi><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub></mrow></msub></mrow></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow></mrow><mo>.</mo></math> </ephtml> (<reflink idref="bib5" id="ref63">5</reflink>)</p> <p>That is, the model is a GMEM with</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> being vectors containing just 1s,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">β</mi><mo>=</mo><mi>μ</mi></mrow><mo>,</mo></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">b</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> =</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn>0</mn><mi>i</mi></mrow></msub></mrow><mo>.</mo></math> </ephtml> Finally, <emph>H<subs>ij</subs></emph> = 0 for each time point <emph>j</emph> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>W</mi></mrow><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mo>.</mo></math> </ephtml> </p> <p>Since the multilevel random intercept model includes only one random effect</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn>0</mn><mi>i</mi></mrow></msub></mrow><mo>,</mo></math> </ephtml> the covariance matrix <bold><emph>G</emph></bold> of these effects is a 1 × 1 matrix containing the random intercept variance</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>τ</mi></mrow><mn>0</mn><mn>2</mn></msubsup></mrow><mo>.</mo></math> </ephtml> Furthermore, because the measurement error terms <emph>H<subs>ij</subs></emph> are assumed to be zero,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>ν</mi></mrow><mn>2</mn></msup></mrow></math> </ephtml> is also zero (i.e., there is no measurement variance). The multilevel random intercept model also assumes that the variance of the residuals <emph>ϵ<subs>ij</subs></emph> is constant across time, that is,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>σ</mi></mrow><mn>1</mn><mn>2</mn></msubsup><mo>=</mo><msubsup><mrow><mi>σ</mi></mrow><mn>2</mn><mn>2</mn></msubsup><mo>=</mo><mo>⋯</mo><mo>=</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub></mrow><mn>2</mn></msubsup></mrow><mo>.</mo></math> </ephtml> Finally, it is assumed that their are <emph>no</emph> serial correlations between these residuals, that is,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">R</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">t</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="bold-italic">I</mi></mrow><mo>.</mo></math> </ephtml> According to Equation (<reflink idref="bib3" id="ref64">3</reflink>), the model-implied covariance matrix for the data vector</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> of person <emph>i</emph> thus is:</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mi>i</mi></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mo>·</mo><msubsup><mrow><mi>τ</mi></mrow><mn>0</mn><mn>2</mn></msubsup><mo>·</mo><msubsup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi><mi>T</mi></msubsup><mo>+</mo><mn>0</mn><mo>·</mo><mi mathvariant="bold-italic">I</mi><mo>+</mo><msup><mrow><mi>σ</mi></mrow><mn>2</mn></msup><mo>·</mo><mi mathvariant="bold-italic">I</mi><mo>=</mo><msubsup><mrow><mi>τ</mi></mrow><mn>0</mn><mn>2</mn></msubsup><mo>·</mo><mi mathvariant="bold-italic">J</mi><mo>+</mo><msup><mrow><mi>σ</mi></mrow><mn>2</mn></msup><mo>·</mo><mi mathvariant="bold-italic">I</mi></mrow><mo>,</mo></math> </ephtml> (<reflink idref="bib6" id="ref65">6</reflink>)</p> <p>where <bold><emph>J</emph></bold> is a</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub><mo>×</mo><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> matrix with all its elements 1. For the example data, the intercept variance is</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>τ</mi></mrow><mn>0</mn><mn>2</mn></msubsup></mrow></math> </ephtml> = 0.71 and it reflects the amount of between-person differences in mean-level extraversion. The variance of the residual terms is</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>σ</mi></mrow><mn>2</mn></msup></mrow></math> </ephtml> = 0.29. It measures the extent to which individuals time-point specific values fluctuate around their specific means.</p> <p>The assumption of no serial correlations is often not plausible in the case of intensive longitudinal data, where the data is collected only a few hours or days apart. Therefore, researchers analyzing such data often assume that the residuals follow an autoregressive process of order 1 (AR-1 process, see Hedeker & Gibbons, [<reflink idref="bib14" id="ref66">14</reflink>]). When we presume a constant time lag between measurement occasions (see below for alternative specifications) and a constant variance term</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>σ</mi></mrow><mi>j</mi><mn>2</mn></msubsup><mo>=</mo><msup><mrow><mi>σ</mi></mrow><mn>2</mn></msup></mrow></math> </ephtml> across time, the AR-1 process implies the serial correlation matrix:</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>σ</mi></mrow><mn>2</mn></msup><mi mathvariant="bold-italic">R</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">t</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>σ</mi></mrow><mn>2</mn></msup><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>i</mi><mi>ρ</mi></mrow></msub><mo>=</mo><msup><mrow><mi>σ</mi></mrow><mn>2</mn></msup><mrow><mo stretchy="true">(</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mi>ρ</mi></mtd><mtd><mrow><msup><mrow><mi>ρ</mi></mrow><mn>2</mn></msup></mrow></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mrow><msup><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub></mrow></msup></mrow></mtd></mtr><mtr><mtd><mi>ρ</mi></mtd><mtd><mn>1</mn></mtd><mtd><mi>ρ</mi></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mrow><msup><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup></mrow></mtd></mtr><mtr><mtd><mrow><msup><mrow><mi>ρ</mi></mrow><mn>2</mn></msup></mrow></mtd><mtd><mi>ρ</mi></mtd><mtd><mn>1</mn></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mi>ρ</mi></mtd></mtr><mtr><mtd><mrow><msup><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub></mrow></msup></mrow></mtd><mtd><mrow><msup><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup></mrow></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mrow><mo stretchy="true">)</mo></mrow></mrow><mo>,</mo></math> </ephtml> (<reflink idref="bib7" id="ref67">7</reflink>)</p> <p>where <emph>ρ</emph> is the autocorrelation with</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>|</mo><mi>ρ</mi><mo>|</mo><mo><</mo><mn>1</mn></mrow></math> </ephtml> (see Hedeker & Gibbons, [<reflink idref="bib14" id="ref68">14</reflink>]). For the illustrative data, the residual variance is</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>σ</mi></mrow><mn>2</mn></msup></mrow></math> </ephtml> =.26 and the autocorrelation is</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ρ</mi><mo>=</mo><mn>.26</mn></mrow><mo>.</mo></math> </ephtml> The latter indicates that the extraversion ratings are autocorrelated to a certain degree.</p> <p>Under an AR-1 process, the variance of the errors at a particular time point equals</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Var</mtext><mo stretchy="false">(</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>ρ</mi></mrow><mn>2</mn></msup><mtext>Var</mtext><mo stretchy="false">(</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>i</mi><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>ϵ</mi><mo>,</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>=</mo><msup><mrow><mi>ρ</mi></mrow><mn>2</mn></msup><mo /><mtext>Var</mtext><mo stretchy="false">(</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>ϵ</mi><mo>,</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>,</mo></math> </ephtml> (<reflink idref="bib8" id="ref69">8</reflink>)</p> <p>where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>ϵ</mi><mo>,</mo><mn>1</mn></mrow><mn>2</mn></msubsup></mrow></math> </ephtml> is the residual variance for the first time point. Above we assumed that the variance and covariance terms of the errors for person <emph>i</emph> are independent of the time point <emph>j</emph>. This is called <emph>stationarity</emph> (see Hedeker & Gibbons, [<reflink idref="bib14" id="ref70">14</reflink>]; Pinheiro & Bates, [<reflink idref="bib35" id="ref71">35</reflink>]) and it means that the variance of the errors is constant across time and that the correlations are the same within a time lag. Thus, we can write Equation (<reflink idref="bib8" id="ref72">8</reflink>) as</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>ϵ</mi><mo>,</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>ρ</mi></mrow><mn>2</mn></msup><mo stretchy="false">)</mo><msup><mrow><mi>σ</mi></mrow><mn>2</mn></msup></mrow><mo>,</mo></math> </ephtml> (<reflink idref="bib9" id="ref73">9</reflink>)</p> <p>where we set Var(<emph>ϵ<subs>ij</subs></emph>) =</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>σ</mi></mrow><mn>2</mn></msup></mrow></math> </ephtml> to denote the variance at the later time points. Equation (<reflink idref="bib9" id="ref74">9</reflink>) implies that when estimating a multilevel model with a stationary AR-1 process, one <emph>either</emph> needs to estimate <emph>ρ</emph> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>σ</mi></mrow><mn>2</mn></msup></mrow></math> </ephtml> <emph>or ρ</emph> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>ϵ</mi><mo>,</mo><mn>1</mn></mrow><mn>2</mn></msubsup></mrow><mo>,</mo></math> </ephtml> and the formula given in Equation (<reflink idref="bib9" id="ref75">9</reflink>) can be used to convert one variance parameter into the other variance parameter. Put differently, the multilevel model with an stationary AR-1 process is typically estimated with the constraint given in Equation (<reflink idref="bib9" id="ref76">9</reflink>) and this is important to consider when comparing the results of a mixed-effects model software with the results of a structural equation model software, because the latter does not (per default) use the constraint when estimating the model's parameters. Rather one can fit a model to the data, in which the residual variance of the first time point can differ from the residual variance parameters of the subsequent time points. Confusingly, it is also possible to relax the constrain in nlme (see below) and this is the reason why we report the results of models that were fitted either with or without the constraint in the next section.</p> <hd id="AN0176073021-4">3. Autoregressive Structural Equation Models as GMEMs</hd> <p>We now explain how to place the three autoregressive structural equation models (RI-APM, LCM-SR, STARTs) into the GMEM and how to fit the models with the lme function of the nlme package in R (Pinheiro et al., [<reflink idref="bib36" id="ref77">36</reflink>]). In fact, as we will demonstrate, these models are extensions of the multilevel random intercept model with an AR-1 process just introduced.</p> <hd id="AN0176073021-5">3.1. RI-APM</hd> <p>The RI-APM is a multilevel random intercept model with an AR-1 error structure and time-point specific means. To see this first note that in the multilevel random intercept model with an AR-1 process only the grand mean <emph>μ</emph> is contained in the regression weight vector</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">β</mi></math> </ephtml> (see Equation (<reflink idref="bib5" id="ref78">5</reflink>)). However, one can easily extend this model so that it equals the RI-APM by including <emph>time-point specific</emph> means (see Skrondal & Rabe-Hesketh, [<reflink idref="bib41" id="ref79">41</reflink>]):</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub></mrow></msub></mrow></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo>=</mo><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo>·</mo><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mrow><msub><mrow><mi>μ</mi></mrow><mn>1</mn></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>μ</mi></mrow><mn>2</mn></msub></mrow></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>μ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub></mrow></msub></mrow></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo>+</mo><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo>·</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>0</mn><mi>i</mi></mrow></msub><mo>+</mo><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo>+</mo><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mrow><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>i</mi><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub></mrow></msub></mrow></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo>,</mo></math> </ephtml> (<reflink idref="bib10" id="ref80">10</reflink>)</p> <p>where the <emph>j</emph>th element in</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">β</mi><mo>,</mo></math> </ephtml> that is, <emph>μ<subs>j</subs></emph>, represents the average rating across persons for a <emph>specific</emph> time point <emph>j</emph>. By setting</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mi mathvariant="bold-italic">I</mi></mrow><mo>,</mo></math> </ephtml> each mean <emph>μ<subs>j</subs></emph> affects only the measurement <emph>y<subs>ij</subs></emph> at the respective time point <emph>j</emph>.</p> <p>The model-implied covariance matrix for the data vector of person <emph>i</emph> then is</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mi>i</mi></msub><mo>=</mo><msubsup><mrow><mi>τ</mi></mrow><mn>0</mn><mn>2</mn></msubsup><mi mathvariant="bold-italic">J</mi><mo>+</mo><msup><mrow><mi>σ</mi></mrow><mn>2</mn></msup><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>i</mi><mi>ρ</mi></mrow></msub></mrow></math> </ephtml> (<reflink idref="bib11" id="ref81">11</reflink>)</p> <p>and when we presume the constraint mentioned in Equation (<reflink idref="bib8" id="ref82">8</reflink>) for the RI-APM, the model specified in Equations (<reflink idref="bib10" id="ref83">10</reflink>) and (<reflink idref="bib11" id="ref84">11</reflink>) is equivalent to the RI-APM: The intercept variance</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>τ</mi></mrow><mn>0</mn><mn>2</mn></msubsup></mrow></math> </ephtml> equals the variance of the latent intercept factor</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>Var</mtext><mo stretchy="false">(</mo><msub><mrow><mi>η</mi></mrow><mi>I</mi></msub><mo stretchy="false">)</mo></mrow><mo>,</mo></math> </ephtml> the autocorrelation <emph>ρ</emph> in</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>i</mi><mi>ρ</mi></mrow></msub></mrow></math> </ephtml> equals the autoregressive parameter <emph>a</emph>, and the residual variance</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>σ</mi></mrow><mn>2</mn></msup></mrow></math> </ephtml> equals the variance of the latent variables representing the observed ratings for the second and all subsequent time points</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>Var</mtext><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> (see Figure 1A).</p> <p>The upper part of Table 3 shows the results of the RI-APM, when we fit the model with the constraint in lavaan (Rosseel, [<reflink idref="bib40" id="ref85">40</reflink>]; see Hamaker et al., [<reflink idref="bib13" id="ref86">13</reflink>] for the Mplus codes and https://osf.io/ztcge/ for the codes used here). As a special case of the GMEM, the model can also be fit with the lme function:</p> <p>fit_riapm <- lme(y ∼ as.factor(TIME),</p> <p>random =∼1|ID,</p> <p>correlation = corAR1(form = ∼1),</p> <p>data = mydata, method = "ML")</p> <p>and a comparison of the results of the lme fit with the results of the lavaan fit (see Table 3) shows that they are almost identical.</p> <p>Table 3. Results of the three models fit as a structural equation model or as a mixed-effects model.</p> <p> <ephtml> <table><thead><tr><td /><td /><td>With constraint</td><td>Without constraint</td></tr><tr><td>Software</td><td>Parameter</td><td>RI-APM</td><td>LCM-SR</td><td>STARTS</td><td>RI-APM</td><td>LCM-SR</td><td>STARTS</td></tr></thead><tbody valign="top"><tr><td>lavaan</td><td><p id="ilm0073"><graphic href="hsem_a_2212865_ilm0073.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mtext>Var</mtext><mo stretchy="false">(</mo><msub><mrow><mi>η</mi></mrow><mi>I</mi></msub><mo stretchy="false">)</mo></mrow></math></p></td><td char=".">0.67</td><td char=".">0.68</td><td char=".">0.29</td><td char=".">0.69</td><td char=".">0.64</td><td>–</td></tr><tr><td><p id="ilm0074"><graphic href="hsem_a_2212865_ilm0074.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mtext>Var</mtext><mo stretchy="false">(</mo><msub><mrow><mi>η</mi></mrow><mi>S</mi></msub><mo stretchy="false">)</mo></mrow></math></p></td><td>–</td><td char=".">0.01</td><td>–</td><td>–</td><td char=".">0.01</td><td>–</td></tr><tr><td><p id="ilm0075"><graphic href="hsem_a_2212865_ilm0075.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mtext>Var</mtext><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></math></p></td><td>–</td><td>–</td><td char=".">0.20</td><td>–</td><td>–</td><td>–</td></tr><tr><td><p id="ilm0076"><graphic href="hsem_a_2212865_ilm0076.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mtext>Var</mtext><mo stretchy="false">(</mo><msub><mrow><mi>u</mi></mrow><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></math></p></td><td char=".">0.33</td><td char=".">0.28</td><td char=".">0.51</td><td char=".">0.42</td><td char=".">0.38</td><td>–</td></tr><tr><td><p id="ilm0077"><graphic href="hsem_a_2212865_ilm0077.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mtext>Var</mtext><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow></math></p></td><td char=".">0.31</td><td char=".">0.27</td><td char=".">0.10</td><td char=".">0.28</td><td char=".">0.26</td><td>–</td></tr><tr><td><italic>a</italic></td><td char=".">0.25</td><td char=".">0.13</td><td char=".">0.90</td><td char=".">0.19</td><td char=".">0.13</td><td>–</td></tr><tr><td>nlme</td><td><p id="ilm0078"><graphic href="hsem_a_2212865_ilm0078.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>τ</mi></mrow><mn>0</mn><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.67</td><td char=".">0.68</td><td char=".">0.29</td><td char=".">0.69</td><td char=".">0.65</td><td>–</td></tr><tr><td><p id="ilm0079"><graphic href="hsem_a_2212865_ilm0079.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>τ</mi></mrow><mn>1</mn><mn>2</mn></msubsup></mrow></math></p></td><td>–</td><td char=".">0.01</td><td>–</td><td>–</td><td char=".">0.01</td><td>–</td></tr><tr><td><p id="ilm0080"><graphic href="hsem_a_2212865_ilm0080.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msup><mrow><mi>ν</mi></mrow><mn>2</mn></msup></mrow></math></p></td><td>–</td><td>–</td><td char=".">0.20</td><td>–</td><td>–</td><td>–</td></tr><tr><td><p id="ilm0081"><graphic href="hsem_a_2212865_ilm0081.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>ϵ</mi><mo>,</mo><mn>1</mn></mrow><mn>2</mn></msubsup></mrow></math></p></td><td char=".">0.33</td><td char=".">0.28</td><td char=".">0.51</td><td char=".">0.42</td><td char=".">0.37</td><td>–</td></tr><tr><td><p id="ilm0082"><graphic href="hsem_a_2212865_ilm0082.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msup><mrow><mi>σ</mi></mrow><mn>2</mn></msup></mrow></math></p></td><td char=".">0.31</td><td char=".">0.27</td><td char=".">0.10</td><td char=".">0.29</td><td char=".">0.28</td><td>–</td></tr><tr><td><italic>ρ</italic></td><td char=".">0.25</td><td char=".">0.12</td><td char=".">0.90</td><td char=".">0.21</td><td char=".">0.13</td><td>–</td></tr></tbody></table> </ephtml> </p> <p>2 <emph>Note.</emph>lavaan (Rosseel, [<reflink idref="bib40" id="ref87">40</reflink>]) was used to fit a model as a structural equation model and nlme (Pinheiro et al., [<reflink idref="bib36" id="ref88">36</reflink>]) was used to fit it as a mixed-effects model. When we estimated the mixed-effects model with the stationarity assumption, we calculated the missing variance parameter using Equation (<reflink idref="bib9" id="ref89">9</reflink>).</p> <p>As mentioned, in the structural equation model framework, the RI-APM is typically estimated without the constraint on the residual variance parameters. Rather the model is estimated without the constraint, so that the residual variance at the first time point differs from the variance terms at the subsequent time points, but that these terms are the same (i.e.,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>σ</mi></mrow><mn>2</mn><mn>2</mn></msubsup><mo>=</mo><mo>⋯</mo><mo>=</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub></mrow><mn>2</mn></msubsup></mrow></math> </ephtml> ). Table 3 displays the results when this model is estimated with lavaan. The same model can be estimated with lme using the varIdent-specification:</p> <p>fit_riapm_het <- lme(y ∼ as.factor(TIME),</p> <p>random =∼ 1|ID,</p> <p>weights = varIdent(form = ∼ 1| TIME1vsREST)</p> <p>correlation = corAR1(form = ∼1),</p> <p>data = mydata, method = "ML"),</p> <p>where TIME1vsREST is a dummy variable that is 0 for the first and 1 for all subsequent time points. The resulting variance estimates have to be transformed (see the R code accompanying this manuscript), because they are provided on a log-scale. Table 3 shows the estimates after transformation, which are almost identical to the estimates obtained with lavaan.</p> <hd id="AN0176073021-6">3.2. LCM-SR</hd> <p>The LCM-SR differs from the RI-APM in that the time-point specific means are modeled with a linear growth model (see Figure 1B). Specifically, the intercept factor of the RI-APM is complemented with a slope factor coding linear time. To write the LCM-SR as a GMEM, we include the growth trend in the model for the data of person <emph>i</emph></p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub></mrow></msub></mrow></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo>=</mo><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mrow><msub><mrow><mi>β</mi></mrow><mn>0</mn></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>β</mi></mrow><mn>1</mn></msub></mrow></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo>+</mo><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn>0</mn><mi>i</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn><mi>i</mi></mrow></msub></mrow></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo>+</mo><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo>+</mo><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mrow><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>i</mi><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub></mrow></msub></mrow></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo>,</mo></math> </ephtml> (<reflink idref="bib12" id="ref90">12</reflink>)</p> <p>where β<subs>0</subs> is the predicted value of the variable (e.g., extraversion in our example) at the first time point and <emph>β</emph><subs>1</subs> is an estimate of the linear rate of change.</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn>0</mn><mi>i</mi></mrow></msub></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub></mrow></math> </ephtml> are the corresponding random effects for the intercept and for the slope, representing the deviation of individual <emph>i</emph> from <emph>β</emph><subs>0</subs> (i.e., the intercept) and from <emph>β</emph><subs>1</subs> (i.e., the linear growth trend), respectively. Finally, the measurement error term <emph>H<subs>ij</subs></emph> is assumed to be zero for each time point <emph>j</emph>.</p> <p>The model-implied covariance matrix for the data vector of person <emph>i</emph> is more complex than the covariance matrix for the RI-APM, because it depends on the number of time points considered in the design matrix</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> and on the coding of the time variable. Assuming constant residual variance terms, the model-implied covariance matrix is</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mi>i</mi></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi></msub><mi mathvariant="bold-italic">G</mi><msubsup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mi>i</mi><mi>T</mi></msubsup><mo>+</mo><msup><mrow><mi>σ</mi></mrow><mn>2</mn></msup><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>i</mi><mi>ρ</mi></mrow></msub><mo>.</mo></mrow></math> </ephtml> (<reflink idref="bib13" id="ref91">13</reflink>)</p> <p>The diagonal elements of <bold><emph>G</emph></bold> contain the intercept variance and the slope variance and the off-diagonal element is the covariance between the intercepts and slopes.</p> <p>The R code to fit the LCM-SR in Equations (<reflink idref="bib12" id="ref92">12</reflink>) and (<reflink idref="bib13" id="ref93">13</reflink>) with lme is:</p> <p>fit_lcmsr <- lme(y ∼ 1 + TIME,</p> <p>random =∼1 + TIME|ID,</p> <p>correlation = corAR1(form = ∼1),</p> <p>data = mydata, method = "ML").</p> <p>When we estimate the LCM-SR in lavaan with the constraint and compare the results with the parameters obtained by lme, the estimates of the two models are very similar (see Table 3). The intercept variance is</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>τ</mi></mrow><mn>0</mn><mn>2</mn></msubsup></mrow></math> </ephtml> =</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>Var</mtext><mo stretchy="false">(</mo><msub><mrow><mi>η</mi></mrow><mi>I</mi></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> = 0.68 and there is a tiny slope variance</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>τ</mi></mrow><mn>1</mn><mn>2</mn></msubsup></mrow></math> </ephtml> =</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>Var</mtext><mo stretchy="false">(</mo><msub><mrow><mi>η</mi></mrow><mi>S</mi></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> = 0.01. The autocorrelation is also very similar to the autoregressive parameter, indicating that within a person, subsequent ratings are correlated to a small degree even if we control for between-person differences in (linear) time trends. Finally, when we fit the models without the constraint (in lme using the varIdent-specification shown above), the results obtained with lavaan and lme are again almost identical (see Table 3).</p> <hd id="AN0176073021-7">3.3. STARTS Model</hd> <p>The difference between the STARTS model (see Figure 1C) and the RI-APM is that the model includes time point specific measurement errors. The STARTS model can thus be obtained by considering the measurement error variable <emph>H<subs>ij</subs></emph> in the GMEM specification of the RI-APM:</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub></mrow></msub></mrow></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo>=</mo><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mrow><msub><mrow><mi>μ</mi></mrow><mn>1</mn></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>μ</mi></mrow><mn>2</mn></msub></mrow></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>μ</mi></mrow><mi>j</mi></msub></mrow></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo>+</mo><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><msub><mrow><mi>b</mi></mrow><mrow><mn>0</mn><mi>i</mi></mrow></msub><mo>+</mo><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo>+</mo><mrow><mo stretchy="true">[</mo><mrow><mtable><mtr><mtd><mrow><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>i</mi><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>n</mi></mrow><mi>i</mi></msub></mrow></msub></mrow></mtd></mtr></mtable></mrow><mo stretchy="true">]</mo></mrow><mo>.</mo></math> </ephtml> (<reflink idref="bib14" id="ref94">14</reflink>)</p> <p>We note that the STARTS model is the only model in which the stationarity assumption is explicitly discussed in the respective literature (Kenny & Zautra, [<reflink idref="bib17" id="ref95">17</reflink>], [<reflink idref="bib18" id="ref96">18</reflink>]; Lüdtke, Robitzsch, & Wagner, [<reflink idref="bib22" id="ref97">22</reflink>]) and also imposed during estimation. Therefore, the model-implied covariance matrix for the data vector</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> of person <emph>i</emph> is</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mi>i</mi></msub><mo>=</mo><msubsup><mrow><mi>τ</mi></mrow><mn>0</mn><mn>2</mn></msubsup><mi mathvariant="bold-italic">J</mi><mo>+</mo><msup><mrow><mi>ν</mi></mrow><mn>2</mn></msup><mi mathvariant="bold-italic">I</mi><mo>+</mo><msup><mrow><mi>σ</mi></mrow><mn>2</mn></msup><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>i</mi><mi>ρ</mi></mrow></msub></mrow><mo>,</mo></math> </ephtml> (<reflink idref="bib15" id="ref98">15</reflink>)</p> <p>indicating that the only difference between model-implied covariance matrix of the STARTS model and the RI-APM with the constrain is the inclusion of the measurement error variance</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>ν</mi></mrow><mn>2</mn></msup></mrow><mo>.</mo></math> </ephtml> Thus, from the GMEM perspective, the STARTS model is a multilevel random intercept model with an stationary AR-1 process in which measurement error is additionally considered.</p> <p>Fitting the STARTS model with the lme function is more challenging than fitting the RI-APM and the LCM-SR. The reason is that lme does not allow to fit a random effect term reflecting measurement error. However, the estimation framework behind lme is very flexible with regard to the definition of a customized random effect structure (i.e., using the pdMat class; see Pinheiro & Bates, [<reflink idref="bib35" id="ref99">35</reflink>], for an introduction). The following R code can be used to fit the model:</p> <p>fit_starts <- lme(y ∼ 0 + factor(TIME),</p> <p>random = list(ID = pdDiag(∼1),</p> <p>ID = pdIdent(∼0 + factor(TIME))),</p> <p>correlation = corAR1(form = ∼1),</p> <p>data = mydata, method = "ML"),</p> <p>where we define random slopes for each dummy variable coding a specific measurement occasion and constrain the variance of these terms to the same value and their covariance to zero (ID = pdIdent(∼0 + factor(TIME))). The code 'ID = pdDiag(∼1)' is used to represent the random intercept. As can be seen in Table 3, the lme estimates are again identical to the estimates obtained with lavaan.</p> <p>To summarize, the foregoing explanations showed how to incorporate three autoregressive structural equation models into the GMEM and how to fit the models using the nlme package. From the perspective of the GMEM, the RI-APM is a multilevel random-intercept model with an AR-1 process and time point specific means. Since the autocorrelation/autoregressive parameter is defined for residual terms at Level 1 (time points within person), the model provides a 'within-person' estimate of the autoregressive process. The LCM-SR extend the RI-APM by including a linear time variable whose slope can differ between persons. From the perspective of the GMEM, the LCM-SR is thus a random intercept-random slope model with an AR-1 process. Finally, The STARTS model adds measurement errors on the level of the observed variables to the RI-APM.</p> <hd id="AN0176073021-8">4. Discussion</hd> <p>We showed that the three autoregressive models (i.e., the RI-APM, the LCM-SR, and the STARTS model) are special cases of the GMEM. We now discuss some extensions of our expositions including questions for future methodological research, focusing on two aspects: Estimation and convergence issues and further model specifications that might be of practical interest.</p> <hd id="AN0176073021-9">4.1. Estimation and Convergence Issues</hd> <p>The parameters of the univariate autoregressive structural equation models (and of structural equation models more generally) are typically estimated with a maximum likelihood approach. By contrast, most mixed-effects models are usually estimated with a restricted maximum likelihood approach, because it leads to more accurate estimates of the variance parameters when the sample sizes are small on both levels (i.e., data with few persons and few time points; see Pinheiro & Bates, [<reflink idref="bib35" id="ref100">35</reflink>]; Verbeke & Molenberghs, [<reflink idref="bib45" id="ref101">45</reflink>]) and/or models containing many fixed effects (i.e., the elements in <emph>β</emph>). The differences between the two estimation methods should be negligible when the two approaches are used to analyze data-sets in which many individuals are examined across a few waves (i.e., few time points). In our example, the data comprised about 900 individuals measure at five time points and when we fit the RI-APM, we estimated five fixed effects (i.e., the time point specific means). However, in the context of experience sampling studies, one often has fewer subjects available, but they are asked to provide information at many more time points (e.g., 30 measurements per subject). In such data situations, we would recommend the use of restricted maximum likelihood, at least when one needs to estimate many time point specific means (as in case of the RI-APM).</p> <p>Furthermore, one often faces convergence problems when fitting univariate structural equation models. In most cases, these problems refer to the optimization of the maximum likelihood function, where most problems occur when the variance terms are very small. Such convergence problems are relevant from a practical point of view as can best be seen in the article of Orth et al. ([<reflink idref="bib34" id="ref102">34</reflink>]) that compared the results of different (bivariate) autoregressive structural equation models. Here, convergence issues occurred in 40–70% of the samples (even when equality constraints were used), complicating a complete comparison of the models. Similarly, Lüdtke et al. ([<reflink idref="bib22" id="ref103">22</reflink>]) recently published a simulation study on the univariate STARTS model in which estimation problems occurred in 62.9% of the replications when unconstrained maximum likelihood estimation was used, the measurement error variance was small</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>ν</mi></mrow><mn>2</mn></msup></mrow></math> </ephtml> = 0.10, and the number of participants (<emph>I</emph> = 200) and the number of time points was also small (<emph>n<subs>i</subs></emph> = 4).</p> <p>Convergence problems also occur when fitting mixed-effects models. Interestingly, the mixed-effects model literature discuss these problems more prominently than the literature on autoregressive structural equation models (e.g., most introductory mixed-effects model books have entire sections devoted to this issue; see e.g. the book by Verbeke & Molenberghs, [<reflink idref="bib45" id="ref104">45</reflink>] or Hedeker & Gibbons, [<reflink idref="bib14" id="ref105">14</reflink>]) and a number of solutions have been proposed on how to deal with them. Suggestions range from rescaling the outcome variable and predictors (e.g., if one wants to study quadratic time effects; e.g., Hedeker & Gibbons, [<reflink idref="bib14" id="ref106">14</reflink>]), algorithmic suggestions such as using the Cholesky transformation during optimization (McNeish & Bauer, [<reflink idref="bib23" id="ref107">23</reflink>]) to re-parametrizations of the model. Diggle ([<reflink idref="bib9" id="ref108">9</reflink>]), for example, specifically discussed the STARTS model in his article. However, he used a different parametrization of the likelihood function for his GMEM version of the STARTS model and states that this model can be 'easily' fit. The nlme package uses the aforementioned Cholesky transformation and when we simulate 1,000 samples in the setting of Lüdtke et al. ([<reflink idref="bib22" id="ref109">22</reflink>]) described above (with non-convergence in 62.9% of the replications), then the model converges for both estimators (i.e., maximum and restricted maximum likelihood) in all 1,000 replications,[<reflink idref="bib2" id="ref110">2</reflink>] suggesting that this approach–similar to the constrained maximum likelihood approach described in Lüdtke et al. ([<reflink idref="bib22" id="ref111">22</reflink>])–circumvents convergence problems well. We think that it is an interesting task for future research to more thoroughly examine, how often convergence problems reported for the autoregressive models occur in the GMEM framework and whether they can be reduced by using the suggestions in the GMEM literature (see McNeish & Bauer, [<reflink idref="bib23" id="ref112">23</reflink>], for first work in this direction).</p> <hd id="AN0176073021-10">4.2. Alternative Model Specifications</hd> <p>Above we described how to fit the models in nlme that include time point specific residual variance terms with the varIdent-specification. It is possible to further extend the definition of the random option to examine whether the intercept variance differs between the levels of a grouping variable (e.g., gender):</p> <p>fit_riapm_het <- lme(y ∼ as.factor(TIME),</p> <p>random = list(ID = pdDiag(∼1|grouping_var)),</p> <p>correlation = corAR1(form = ∼1),</p> <p>data = mydata, method = "ML").</p> <p>The effect of grouping variables can also be examined for the autocorrelation and residual variance parameters,[<reflink idref="bib3" id="ref113">3</reflink>] allowing researchers to estimate what is called a 'multiple group model' in the structural equation modeling literature (Bollen, [<reflink idref="bib4" id="ref114">4</reflink>]; Little, [<reflink idref="bib19" id="ref115">19</reflink>]). Moreover, one can easily include time-constant and time-varying predictors in nlme by extending the fixed-component of the model definition (e.g., the E ∼ as.factor(TIME) part in the above model; see Mulder & Hamaker, [<reflink idref="bib25" id="ref116">25</reflink>], for respective extensions of the RI-APM for structural equation modeling software). This also allows to fit a combination of the LCM-SR with the STARTS model (see Diggle, [<reflink idref="bib9" id="ref117">9</reflink>], and also Lüdtke et al., [<reflink idref="bib22" id="ref118">22</reflink>]), a model that is–at least to our knowledge–not used in the psychological literature so far.</p> <p>In all three models used here, an AR-1 is assumed. However, in the mixed-effects model literature, there are a number of other ways in which the autocorrelation between measurements can be modeled. The most prominent alternative is the moving-average (MA) process. For example, for an MA-process of order 1 with constant variance terms across time, the corresponding covariance matrix is:</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>σ</mi></mrow><mn>2</mn></msup><mi mathvariant="bold-italic">R</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">t</mi></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>σ</mi></mrow><mn>2</mn></msup><mrow><mo stretchy="true">(</mo><mrow><mtable><mtr><mtd><mrow><mn>1</mn><mo>+</mo><msup><mrow><mi>θ</mi></mrow><mn>2</mn></msup></mrow></mtd><mtd><mrow><mo>−</mo><mi>θ</mi></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mo>−</mo><mi>θ</mi></mrow></mtd><mtd><mrow><mn>1</mn><mo>+</mo><msup><mrow><mi>θ</mi></mrow><mn>2</mn></msup></mrow></mtd><mtd><mrow><mo>−</mo><mi>θ</mi></mrow></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow><mo>−</mo><mi>θ</mi></mrow></mtd><mtd><mrow><mn>1</mn><mo>+</mo><msup><mrow><mi>θ</mi></mrow><mn>2</mn></msup></mrow></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mrow><mo>...</mo></mrow></mtd><mtd><mrow><mo>−</mo><mi>θ</mi></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mo>−</mo><mi>θ</mi></mrow></mtd><mtd><mrow><mn>1</mn><mo>+</mo><msup><mrow><mi>θ</mi></mrow><mn>2</mn></msup></mrow></mtd></mtr></mtable></mrow><mo stretchy="true">)</mo></mrow><mo>,</mo></mrow></math> </ephtml> (<reflink idref="bib16" id="ref119">16</reflink>)</p> <p>where <emph>θ</emph> is the autocorrelation parameter. In contrast to the AR-1 process, a MA process posits the errors at a particular time point to be correlated with the errors of the <emph>immediate</emph> subsequent time point, but to be independent of the errors of all other time points.</p> <p>Alternatively, one can also combine the AR-process with a MA-process, resulting in a ARMA-process (see Hedeker & Gibbons, [<reflink idref="bib14" id="ref120">14</reflink>]). Both approaches–of any order–can be easily fit with the nlme package using the corARMA argument. For instance, a RI-APM with an AR process of order <emph>p</emph> = 1 <emph>and</emph> a MA process of order <emph>q</emph> = 1, can be estimated with</p> <p>fit_riapm <- lme(y ∼ as.factor(TIME),</p> <p>random =∼1|ID,</p> <p>correlation = corARMA(form = ∼1, p = 1, q = 1),</p> <p>data = mydata, method = "ML").</p> <p>Of note, Zyphur et al. ([<reflink idref="bib47" id="ref121">47</reflink>]) recently introduced the 'general cross-lagged panel model' that included an autoregressive and a moving average component.</p> <p>In deriving the covariance matrix for the autoregressive process of order 1 (see Equation (<reflink idref="bib5" id="ref122">5</reflink>)), we assumed that the time points are equally spaced and that the spacing is the same for all individuals. For experience sampling data, however, this assumption is often not fulfilled, because people are surveyed several times a day across several days. This can have the consequence, for example, that the time interval between the last evening survey and the next morning survey is greater than the time interval between surveys within a day. In this case, it is useful to take the true time intervals between the individual measurements into account and a number of suggestions have been made in the mixed-effects literature for this purpose. lme, for example, allows to fit a AR process of order 1 whose covariance matrix is time point and person specific. For three time points this matrix is:</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>σ</mi></mrow><mn>2</mn></msup><mi mathvariant="bold-italic">R</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">t</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>σ</mi></mrow><mn>2</mn></msup><mrow><mo stretchy="true">(</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msup><mrow><mi>ρ</mi></mrow><mrow><mo>|</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mn>2</mn></mrow></msub><mo>−</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub><mo>|</mo></mrow></msup></mrow></mtd><mtd><mrow><msup><mrow><mi>ρ</mi></mrow><mrow><mo>|</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mn>3</mn></mrow></msub><mo>−</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub><mo>|</mo></mrow></msup></mrow></mtd></mtr><mtr><mtd><mrow><msup><mrow><mi>ρ</mi></mrow><mrow><mo>|</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mn>2</mn></mrow></msub><mo>−</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub><mo>|</mo></mrow></msup></mrow></mtd><mtd><mn>1</mn></mtd><mtd><mrow><msup><mrow><mi>ρ</mi></mrow><mrow><mo>|</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mn>3</mn></mrow></msub><mo>−</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mn>2</mn></mrow></msub><mo>|</mo></mrow></msup></mrow></mtd></mtr><mtr><mtd><mrow><msup><mrow><mi>ρ</mi></mrow><mrow><mo>|</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mn>3</mn></mrow></msub><mo>−</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub><mo>|</mo></mrow></msup></mrow></mtd><mtd><mrow><msup><mrow><mi>ρ</mi></mrow><mrow><mo>|</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mn>3</mn></mrow></msub><mo>−</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mn>2</mn></mrow></msub><mo>|</mo></mrow></msup></mrow></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mrow><mo stretchy="true">)</mo></mrow><mo>,</mo></mrow></math> </ephtml> (<reflink idref="bib17" id="ref123">17</reflink>)</p> <p>and the model can be estimated by specifying the corCAR1 argument:</p> <p>fit_riapm <- lme(y ∼ as.factor(TIME),</p> <p>random =∼1|ID,</p> <p>correlation = corCAR1(form = ∼SubjectsTimePoints),</p> <p>data = mydata, method = "ML"),</p> <p>whereby the user has to deliver a variable that provides participants specific time points (i.e., saved in the variable 'SubjectsTimePoints'). Note that this is just one way to model the data continuously (see Diggle, [<reflink idref="bib9" id="ref124">9</reflink>] or Verbeke & Molenberghs, [<reflink idref="bib45" id="ref125">45</reflink>], for alternatives such as using an exponential function) and we think that it might be interesting to compare the results of these models with the continuous time structural equation model (see e.g., Voelkle, Oud, Davidov, & Schmidt, [<reflink idref="bib46" id="ref126">46</reflink>]). Furthermore, nlme also contains correlation structures that allow to model spatial dependencies, a feature that might be interesting for the analysis of mobile sensing data.</p> <p>In this article, we assumed that the outcome variable is continuous and this allowed us to use nlme to fit the GMEM. When the variable has a categorical response format or when a researcher wants to employ more complex measurement models (e.g., more complex IRT models), nlme cannot be used. In lavaan, by contrast, it is possible to estimate models in which variables have a categorical response format, allowing, for example, to estimate a categorical RI-APM. For the GMEM, packages for the generalized linear mixed effects model have to be used that allow modeling autoregressive structures, such as, for example, the glmmPQL function in the MASS package (Venables & Ripley, [<reflink idref="bib44" id="ref127">44</reflink>]) or the glmmTMB package (Brooks et al., [<reflink idref="bib6" id="ref128">6</reflink>]). However, when fitting the models to non-continuous outcomes, the statistical approach used to model these outcomes implies certain constraints concerning the model's parameters. For example, in the case of the RI-APM, the variances of the residualized scores are fixed to 1, when a probit link function is employed, as is the case in lavaan.</p> <p>Finally, we would like to mention some extensions of the GMEM that are not implemented in nlme but are discussed in the current mixed-effects model literature. One of these extensions is the mixed-effects location scale model (Hedeker et al., [<reflink idref="bib15" id="ref129">15</reflink>]) that allows to model differences between individuals in the Level 1 residual variance in a multilevel model. Recently, the model has been extended to estimate differences between individuals in the autocorrelation (Nestler, [<reflink idref="bib30" id="ref130">30</reflink>]; Nestler & Humberg, [<reflink idref="bib32" id="ref131">32</reflink>], for multivariate versions see Asparouhov, Hamaker, & Muthén, [<reflink idref="bib1" id="ref132">1</reflink>]; Nestler, [<reflink idref="bib28" id="ref133">28</reflink>], [<reflink idref="bib29" id="ref134">29</reflink>], in press). Thus, from the perspective of the descriptions made here, the model allows to model interindividual differences between the intercepts, the residual variance term, and the autoregressive parameter (i.e., a fully multilevel RI-APM). In addition, person variables can be used to predict the interindividual differences and the model is implemented in such a way that it allows person-specific time lags during estimation. Currently, it is not possible to include measurement error as in the STARTS model, but this is a model extension that will certainly be developed in future research.</p> <hd id="AN0176073021-11">5. Conclusion</hd> <p>To conclude, the aim of the current manuscript was to show how to incorporate some well-known autoregressive structural equation models into the mixed-effects model and how to estimate the models using mixed-effects model software. We think that our expositions are interesting from a conceptual point of view, as all three models are different types of multilevel random intercepts models assuming an AR process of order 1. 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Psychological Methods, 17, 176 – 192. https://doi.org/10.1037/a0027543</bibtext> </blist> <blist> <bibtext> Zyphur, M. J., Allison, P. D., Tay, L., Voelkle, M. C., Preacher, K. J., Zhang, Z., Hamaker, E. L., Shamsollahi, A., Pierides, D. C., Koval, P., & Diener, E. (2020). From data to causes I: Building a general cross-lagged panel model (GCLM). Organizational Research Methods, 23, 651 – 687. https://doi.org/10.1177/1094428119847278</bibtext> </blist> </ref> <ref id="AN0176073021-13"> <title> Footnotes </title> <blist> <bibtext> Differences between the lavaan and the nlme estimates can arise, because lavaan uses a maximum likelihood estimator whereas nlme employs a restricted maximum likelihood approach. In all fitted models, we therefore use the option method ="ML" in the lme function for consistency with the lavaan results. Nevertheless, there may still be small differences between the estimates of the two packages, since they differ in several further factors. For example, nlme uses the raw data for parameter estimation, while lavaan (typically) uses sufficient statistics (i.e., the vector of observed means and the covariance matrix of the observed variables). Furthermore, nlme optimizes the entries of the Cholesky decomposition of the random effect covariance matrix, while lavaan uses the model-implied covariance matrix.</bibtext> </blist> <blist> <bibtext> Note that this result does not invalidate the arguments and findings reported in Lüdtke et al. ([22]) that Bayes estimates are more stable. In fact, the nlme estimates for the measurement error variance are very unstable and the estimate of the autocorrelation is substantially biased (i.e., the average estimate is.29 while the true value is.20).</bibtext> </blist> <blist> <bibtext> There is very little literature on how to use the full flexibility of nlme to fit complex mixed-effects models. One web page that we found very helpful is: https://quantdev.ssri.psu.edu/sites/qdev/files/ILD_Ch06_2017_MLMwithHeterogeneousVariance.html</bibtext> </blist> </ref> <aug> <p>By Steffen Nestler and Sarah Humberg</p> <p>Reported by Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib10" firstref="ref1"></nolink> <nolink nlid="nl2" bibid="bib14" firstref="ref2"></nolink> <nolink nlid="nl3" bibid="bib19" firstref="ref3"></nolink> <nolink nlid="nl4" bibid="bib24" firstref="ref5"></nolink> <nolink nlid="nl5" bibid="bib37" firstref="ref6"></nolink> <nolink nlid="nl6" bibid="bib16" firstref="ref8"></nolink> <nolink nlid="nl7" bibid="bib42" firstref="ref9"></nolink> <nolink nlid="nl8" bibid="bib33" firstref="ref12"></nolink> <nolink nlid="nl9" bibid="bib12" firstref="ref14"></nolink> <nolink nlid="nl10" bibid="bib13" firstref="ref15"></nolink> <nolink nlid="nl11" bibid="bib17" firstref="ref18"></nolink> <nolink nlid="nl12" bibid="bib45" firstref="ref21"></nolink> <nolink nlid="nl13" bibid="bib35" firstref="ref22"></nolink> <nolink nlid="nl14" bibid="bib15" firstref="ref23"></nolink> <nolink nlid="nl15" bibid="bib30" firstref="ref24"></nolink> <nolink nlid="nl16" bibid="bib36" firstref="ref25"></nolink> <nolink nlid="nl17" bibid="bib38" firstref="ref26"></nolink> <nolink nlid="nl18" bibid="bib34" firstref="ref27"></nolink> <nolink nlid="nl19" bibid="bib21" firstref="ref28"></nolink> <nolink nlid="nl20" bibid="bib27" firstref="ref29"></nolink> <nolink nlid="nl21" bibid="bib39" firstref="ref33"></nolink> <nolink nlid="nl22" bibid="bib40" firstref="ref35"></nolink> <nolink nlid="nl23" bibid="bib26" firstref="ref37"></nolink> <nolink nlid="nl24" bibid="bib43" firstref="ref40"></nolink> <nolink nlid="nl25" bibid="bib11" firstref="ref41"></nolink> <nolink nlid="nl26" bibid="bib18" firstref="ref48"></nolink> <nolink nlid="nl27" bibid="bib20" firstref="ref51"></nolink> <nolink nlid="nl28" bibid="bib41" firstref="ref79"></nolink> <nolink nlid="nl29" bibid="bib22" firstref="ref97"></nolink> <nolink nlid="nl30" bibid="bib23" firstref="ref107"></nolink> <nolink nlid="nl31" bibid="bib25" firstref="ref116"></nolink> <nolink nlid="nl32" bibid="bib47" firstref="ref121"></nolink> <nolink nlid="nl33" bibid="bib46" firstref="ref126"></nolink> <nolink nlid="nl34" bibid="bib44" firstref="ref127"></nolink> <nolink nlid="nl35" bibid="bib32" firstref="ref131"></nolink> <nolink nlid="nl36" bibid="bib28" firstref="ref133"></nolink> <nolink nlid="nl37" bibid="bib29" firstref="ref134"></nolink>
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Items – Name: Title
  Label: Title
  Group: Ti
  Data: Univariate Autoregressive Structural Equation Models as Mixed-Effects Models
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Steffen+Nestler%22">Steffen Nestler</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0001-9724-2441">0000-0001-9724-2441</externalLink>)<br /><searchLink fieldCode="AR" term="%22Sarah+Humberg%22">Sarah Humberg</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-7891-3622">0000-0002-7891-3622</externalLink>)
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="SO" term="%22Structural+Equation+Modeling%3A+A+Multidisciplinary+Journal%22"><i>Structural Equation Modeling: A Multidisciplinary Journal</i></searchLink>. 2024 31(2):357-366.
– Name: Avail
  Label: Availability
  Group: Avail
  Data: Routledge. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
– Name: PeerReviewed
  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 10
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2024
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Descriptive
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Structural+Equation+Models%22">Structural Equation Models</searchLink><br /><searchLink fieldCode="DE" term="%22Computer+Software%22">Computer Software</searchLink><br /><searchLink fieldCode="DE" term="%22Models%22">Models</searchLink><br /><searchLink fieldCode="DE" term="%22Measurement%22">Measurement</searchLink><br /><searchLink fieldCode="DE" term="%22Hierarchical+Linear+Modeling%22">Hierarchical Linear Modeling</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1080/10705511.2023.2212865
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 1070-5511<br />1532-8007
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Several variants of the autoregressive structural equation model were suggested over the past years, including, for example, the random intercept autoregressive panel model, the latent curve model with structured residuals, and the STARTS model. The present work shows how to place these models into a mixed-effects model framework and how to estimate them in mixed-effects model software, namely the R package "nlme." We also show how "nlme" can be used to fit extensions of these models, for example, models that do not assume equally spaced time intervals between measurement occasions (i.e., continuous time models). Overall, our expositions show that autoregressive structural equations models and mixed-effects models are closely related. We think that this insight eases researchers to understand the differences between the variants of the autoregressive structural equation model and also allows them to profitably link the two different modeling perspectives.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2024
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1431570
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1431570
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1080/10705511.2023.2212865
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 10
        StartPage: 357
    Subjects:
      – SubjectFull: Structural Equation Models
        Type: general
      – SubjectFull: Computer Software
        Type: general
      – SubjectFull: Models
        Type: general
      – SubjectFull: Measurement
        Type: general
      – SubjectFull: Hierarchical Linear Modeling
        Type: general
    Titles:
      – TitleFull: Univariate Autoregressive Structural Equation Models as Mixed-Effects Models
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Steffen Nestler
      – PersonEntity:
          Name:
            NameFull: Sarah Humberg
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 01
              Type: published
              Y: 2024
          Identifiers:
            – Type: issn-print
              Value: 1070-5511
            – Type: issn-electronic
              Value: 1532-8007
          Numbering:
            – Type: volume
              Value: 31
            – Type: issue
              Value: 2
          Titles:
            – TitleFull: Structural Equation Modeling: A Multidisciplinary Journal
              Type: main
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