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Circumplex Models with Multivariate Time Series: An Idiographic Approach

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Title: Circumplex Models with Multivariate Time Series: An Idiographic Approach
Language: English
Authors: Dayoung Lee (ORCID 0000-0001-7543-7096), Guangjian Zhang (ORCID 0000-0001-8279-7313), Shanhong Luo (ORCID 0000-0002-0022-8967)
Source: Structural Equation Modeling: A Multidisciplinary Journal. 2024 31(3):498-510.
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: 13
Publication Date: 2024
Document Type: Journal Articles
Reports - Research
Descriptors: Research Methodology, Affective Measures, Family Relationship, Multivariate Analysis, Time Management, Individual Testing
DOI: 10.1080/10705511.2023.2259105
ISSN: 1070-5511
1532-8007
Abstract: The circumplex model posits a circular representation of affect and some personality traits. There is an increasing need to examine the viability of the circumplex model with multivariate time series data collected on the same individuals due to the development of new data collection methods such as smartphone applications and wearable sensors. Estimating the circumplex model with time series data is more complex than with cross-sectional data because scores at nearby time points tend to be correlated. We adapt Browne's circumplex model to accommodate time series data. We illustrate the proposed method with an empirical data set of daily affect ratings of an individual over 70 days. We conducted a simulation study to explore the statistical properties of the proposed method. The results show that the method provides more satisfactory confidence intervals and test statistics than a method that treats time series data as if they were cross-sectional data.
Abstractor: As Provided
Entry Date: 2024
Accession Number: EJ1431593
Database: ERIC
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  Value: <anid>AN0176985635;7mz01may.24;2024May06.01:57;v2.2.500</anid> <title id="AN0176985635-1">Circumplex Models with Multivariate Time Series: An Idiographic Approach </title> <p>The circumplex model posits a circular representation of affect and some personality traits. There is an increasing need to examine the viability of the circumplex model with multivariate time series data collected on the same individuals due to the development of new data collection methods such as smartphone applications and wearable sensors. Estimating the circumplex model with time series data is more complex than with cross-sectional data because scores at nearby time points tend to be correlated. We adapt Browne's circumplex model to accommodate time series data. We illustrate the proposed method with an empirical data set of daily affect ratings of an individual over 70 days. We conducted a simulation study to explore the statistical properties of the proposed method. The results show that the method provides more satisfactory confidence intervals and test statistics than a method that treats time series data as if they were cross-sectional data.</p> <p>Keywords: Circumplex model; multivariate time series; time series</p> <hd id="AN0176985635-2">1. Introduction</hd> <p>The circumplex model explains the correlations between a set of variables by placing these variables around the circumference of a circle. Applications of the model include personality (Evmenenko & Teixeira, [<reflink idref="bib14" id="ref1">14</reflink>]; Fournier et al., [<reflink idref="bib15" id="ref2">15</reflink>]; Wiggins, [<reflink idref="bib46" id="ref3">46</reflink>]), affect (Jonathan et al., [<reflink idref="bib21" id="ref4">21</reflink>]; Russell, [<reflink idref="bib35" id="ref5">35</reflink>]; Watson et al., [<reflink idref="bib44" id="ref6">44</reflink>]), coping with stress (Stanisławski, [<reflink idref="bib37" id="ref7">37</reflink>]), marital and family systems (Olson, [<reflink idref="bib33" id="ref8">33</reflink>]), social behavior in children (Becker & Krug, [<reflink idref="bib2" id="ref9">2</reflink>]), and maternal behavior (Schaefer, [<reflink idref="bib36" id="ref10">36</reflink>]). Although researchers traditionally aim to uncover a universal law that is applicable to all individuals in these studies, the rapid developments in new data collection methods (e.g. smartphone applications and wearable sensors) afford us an opportunity to investigate a circumplex model unique to each individual. Furthermore, psychological states such as affect and mood change from one moment to another; investigating the change pattern requires taking measurements on the same individuals repeatedly over a period of time. For example, an individual rated herself on ten mood adjectives every day for 100 days. We can arrange her ratings into a 100 by 10 matrix. Each row corresponds to the ratings of a certain day, and each column corresponds to the ratings of a certain mood adjective for 100 days. Estimating the circumplex model with this matrix reveals relations among the mood adjectives unique to this particular individual. The circumplex model reflects researchers' hypotheses on the relations among a set of psychological variables (Russell, [<reflink idref="bib35" id="ref11">35</reflink>]; Watson et al., [<reflink idref="bib44" id="ref12">44</reflink>]; Wiggins, [<reflink idref="bib46" id="ref13">46</reflink>]). Thus, it is a confirmatory approach to evaluate the appropriateness of these hypotheses with data. Extending the model to time series data allows us to assess the appropriateness of these hypotheses at the individual level. Researchers recognized the benefits of such an idiographic approach,[<reflink idref="bib1" id="ref14">1</reflink>] but they often found it difficult to collect time series data with many time points. A compromise is to aggregate within-subject data before performing analysis to examine the change patterns. Molenaar ([<reflink idref="bib29" id="ref15">29</reflink>]) argued that the knowledge learned from analyzing the aggregated data may not be directly applicable to understanding particular individuals unless all individual time series are ergodic[<reflink idref="bib2" id="ref16">2</reflink>] (Molenaar, [<reflink idref="bib29" id="ref17">29</reflink>]). Most psychological and behavioral time series are not ergodic because of the heterogeneity among individuals.</p> <p>We propose to adapt Browne ([<reflink idref="bib8" id="ref18">8</reflink>])'s circumplex model to accommodate the time series data of a single individual. This model is a direct assessment of the conceptual circumplex model, and it allows both positive correlations and negative correlations between variables. Its flexibility in handling both positive correlations and negative correlations makes it particularly suitable for modeling effect variables. We obtain point estimates by estimating the circumplex model with correlations computed from the time series. We refer to such correlations as within-subject correlations. Because within-subject correlations are distributed differently from the usual between-subject correlations computed from cross-sectional data with a large number of participants, we need to modify the test statistic and standard error estimates of Browne ([<reflink idref="bib8" id="ref19">8</reflink>]) to reflect the distribution properties of within-subject correlations. More specifically, we adapt standard error estimates and test statistics for the usual structural equation modeling (SEM) models with nonnormal data (Browne, [<reflink idref="bib7" id="ref20">7</reflink>]) for the circumplex model with time series data. Browne ([<reflink idref="bib7" id="ref21">7</reflink>])'s goal was to develop robust methods for structural equation models with nonnormal data, but our goal is to develop robust methods for the circumplex model with time series data.</p> <p>The conventional between-subject approach is inappropriate for producing test statistics and standard error estimates for the P-factor analysis model with time series data (Molenaar & Nesselroade, [<reflink idref="bib30" id="ref22">30</reflink>]). Trichtinger and Zhang ([<reflink idref="bib40" id="ref23">40</reflink>]) developed a proper test statistic presented by adapting the robust test statistic described by Browne ([<reflink idref="bib7" id="ref24">7</reflink>]). Their adaptation of the P-factor analysis models is not readily applicable to the circumplex model because the circumplex model is more complex than a two-factor factor analysis model. Lee and Zhang ([<reflink idref="bib24" id="ref25">24</reflink>]) adapted standard error estimates and test statistics of Browne ([<reflink idref="bib7" id="ref26">7</reflink>]) for the circumplex model with ordinal cross-sectional data.</p> <p>The circumplex model with individual time series is an idiographic approach, and it allows researchers to answer questions about a particular individual. For example, positive affect and negative affect are considered two largely independent dimensions when researchers average their ratings across different individuals (Watson & Clark, [<reflink idref="bib42" id="ref27">42</reflink>]). However, it is unknown whether such a relation holds for a particular individual. We can test the corresponding hypothesis on the particular individual using the proposed method.</p> <p>The rest of the article is organized as follows. We first specify the circumplex model with time series data. We then describe how to accommodate time series data to obtain the appropriate test statistics and standard error estimates. We illustrate the procedure using an empirical data set of daily affect ratings. We then conduct a simulation study to examine the statistical properties of the procedure under a variety of conditions. We conclude the article with several implications of the results.</p> <hd id="AN0176985635-3">2. Model Specification and Model Estimation</hd> <p>Let</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>t</mi></msub></mrow></math> </ephtml> be a <emph>p</emph>-component vector that contains the <emph>p</emph> variables</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>1</mn></mrow></msub><mo>,</mo><mtext /><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>2</mn></mrow></msub><mo>,</mo><mtext /><mo>⋯</mo><mo>,</mo><mtext /><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>p</mi></mrow></msub></mrow></math> </ephtml> measured for an individual at the time point <emph>t</emph>. The collection of</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><mn>1</mn></msub><mo>,</mo><mtext /><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mn>2</mn></msub><mo>,</mo><mtext /><mo>⋯</mo><mo>,</mo><mtext /><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mi>t</mi></msub><mo>,</mo><mtext /><mo>⋯</mo><mo>,</mo><mtext /><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mi>T</mi></msub></mrow></math> </ephtml> form a <emph>p</emph>-variate time series of <emph>T</emph> time points. A defining property of time series is that observations are correlated at nearby time points. Let</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><mn>0</mn></msub></mrow></math> </ephtml> be a <emph>p</emph> ×<emph> p</emph> correlation matrix that contains the concurrent correlations of the <emph>p</emph> components of</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>t</mi></msub></mrow><mo>.</mo></math> </ephtml> We define</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><mn>1</mn></msub></mrow></math> </ephtml> as the lag 1 correlation matrix that contains the correlations between</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><mrow><mi>t</mi><mo>+</mo><mn>1</mn></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 mathvariant="bold-italic">y</mi></mrow><mi>t</mi></msub></mrow><mo>.</mo></math> </ephtml> We use triple subscripts to denote elements of the within-subject (lag 0) correlation matrix and lagged correlation matrices. For example,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></mrow></math> </ephtml> is the within-subject correlation between</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>1</mn></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>y</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>2</mn></mrow></msub></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>r</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></mrow></math> </ephtml> is the lag 1 correlation between</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>1</mn></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>y</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>2</mn></mrow></msub></mrow><mo>.</mo></math> </ephtml> We define</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><mn>2</mn></msub></mrow><mo>,</mo></math> </ephtml> ...,</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><mi>L</mi></msub></mrow><mo>,</mo></math> </ephtml> accordingly. The lagged correlation matrices</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><mn>1</mn></msub><mo>,</mo><mtext /><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mn>2</mn></msub></mrow><mo>,</mo></math> </ephtml> ...,</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><mi>L</mi></msub></mrow></math> </ephtml> reflect the correlations of time series data at nearby time points. In contrast, these lagged correlation matrices are irrelevant in the usual between-subject settings since cross-sectional data are from independent participants.</p> <hd id="AN0176985635-4">2.1. A Circumplex Correlation Structure</hd> <p>The observed scores <emph>y<subs>t</subs></emph> contain measurement error <emph>u<subs>t</subs></emph> in addition to the true score <emph>c<subs>t</subs></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>t</mi></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mi>t</mi></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mi>t</mi></msub><mo>.</mo></mrow></math> </ephtml> (<reflink idref="bib1" id="ref28">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">c</mi></mrow><mi>t</mi></msub></mrow></math> </ephtml> is the common part ("true" scores) 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">u</mi></mrow><mi>t</mi></msub></mrow></math> </ephtml> is the measurement error part. Although we do not directly measure</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mi>t</mi></msub></mrow><mo>,</mo></math> </ephtml> Browne ([<reflink idref="bib8" id="ref29">8</reflink>])'s model allows us to specify a circumplex structure on them. Note that Equation (<reflink idref="bib1" id="ref30">1</reflink>) specifies the conceptual relations between true scores and measurement error and we do not need to estimate the true scores at each time point. We specify the same structure (Browne, [<reflink idref="bib8" id="ref31">8</reflink>], eq. (<reflink idref="bib3" id="ref32">3</reflink>)) for the within-subject variable correlation matrix</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">P</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></mrow><mn>0</mn></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mi>ζ</mi><mo>*</mo></msubsup><mrow><mo stretchy="true">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mi>c</mi></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mi>ν</mi></msub></mrow><mo stretchy="true">)</mo></mrow><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mi>ζ</mi><mo>*</mo></msubsup><mo>.</mo></mrow></math> </ephtml> (<reflink idref="bib2" id="ref33">2</reflink>)</p> <p>The common score 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">P</mi></mrow><mi>c</mi></msub></mrow></math> </ephtml> contains correlations of <emph>c<subs>t</subs></emph>, and these correlations indicate relations between constructs without the contamination of measurement error. The diagonal matrix</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>D</mi></mrow><mi>ν</mi></msub></mrow></math> </ephtml> contains variances of measurement error variables <emph>u<subs>t</subs></emph>. Note that a measurement error variable contributes to the variance of the corresponding measured variable and it does not contribute to the covariations of the variable with other variables. Thus, the combination of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mi>c</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">D</mi></mrow><mi>ν</mi></msub></mrow></math> </ephtml> is no longer correlation matrix. We rescale</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mi>c</mi></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mi>ν</mi></msub></mrow></math> </ephtml> with a diagonal matrix</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mi>ζ</mi><mo>*</mo></msubsup></mrow></math> </ephtml> so that it can be a proper model for correlations. Elements of the scaling matrix</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mi>ζ</mi><mo>*</mo></msubsup></mrow></math> </ephtml> are actually the correlations between the measured variables and their respective true scores, and we refer to them as communality indices. The communality index</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>ζ</mi></mrow><mrow><mi>i</mi><mi>i</mi></mrow><mo>*</mo></msubsup></mrow></math> </ephtml> is inversely related to the corresponding measurement error variance <emph>ν<subs>ii</subs></emph>.</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">P</mi></mrow><mi>c</mi></msub></mrow></math> </ephtml> is the common score correlation matrix of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mi>t</mi></msub></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">D</mi></mrow><mi>ν</mi></msub></mrow></math> </ephtml> is a diagonal matrix of variances for measurement errors</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mi>t</mi></msub></mrow><mo>,</mo></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mi>ζ</mi><mo>*</mo></msubsup></mrow></math> </ephtml> is a diagonal scaling matrix. Let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>ζ</mi></mrow><mrow><mi>i</mi><mi>i</mi></mrow><mo>*</mo></msubsup></mrow></math> </ephtml> be the <emph>i</emph>th diagonal element</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi mathvariant="bold-italic">D</mi></mrow><mi>ζ</mi><mo>*</mo></msubsup></mrow></math> </ephtml> and <emph>ν<subs>ii</subs></emph> be the <emph>i</emph>th diagonal element of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mi>ν</mi></msub></mrow><mo>.</mo></math> </ephtml> The element</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>ζ</mi></mrow><mrow><mi>i</mi><mi>i</mi></mrow><mo>*</mo></msubsup></mrow></math> </ephtml> is the correlation between the <emph>i</emph>th variable</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>t</mi></msub></mrow></math> </ephtml> and its common part, and we refer to it as the communality index for the <emph>i</emph>th variable. The communality index</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>ζ</mi></mrow><mrow><mi>i</mi><mi>i</mi></mrow><mo>*</mo></msubsup></mrow></math> </ephtml> is inversely related to the corresponding measurement error variance <emph>ν<subs>ii</subs></emph></p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>ζ</mi></mrow><mrow><mi>i</mi><mi>i</mi></mrow><mo>*</mo></msubsup><mo>=</mo><msup><mrow><mrow><mrow><mo stretchy="true">(</mo><mrow><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>i</mi><mi>i</mi></mrow></msub></mrow></mfrac></mrow><mo stretchy="true">)</mo></mrow></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>.</mo></mrow></math> </ephtml> (<reflink idref="bib3" id="ref34">3</reflink>)</p> <p>Note that we use the notations <emph>i</emph> and <emph>j</emph> to indicate variables and the notation <emph>t</emph> to indicate a time point.</p> <p>A key feature of Browne ([<reflink idref="bib8" id="ref35">8</reflink>], Eq. (<reflink idref="bib34" id="ref36">34</reflink>))'s model is to express the correlation between the common parts (<emph>c<subs>i</subs></emph> and <emph>c<subs>j</subs></emph>) of the two variables <emph>y<subs>i</subs></emph> and <emph>y<subs>j</subs></emph> as a Fourier function.</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ρ</mi><mrow><mo stretchy="true">(</mo><mrow><msub><mrow><mi>c</mi></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mi>c</mi></mrow><mi>j</mi></msub></mrow><mo stretchy="true">)</mo></mrow><mo>=</mo><msub><mrow><mi>β</mi></mrow><mn>0</mn></msub><mo>+</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><mrow><msub><mrow><mi>β</mi></mrow><mi>k</mi></msub></mrow><mo /><mi mathvariant="normal">cos</mi><mo /><mrow><mo stretchy="true">{</mo><mrow><mi>k</mi><mo>×</mo><mrow><mo stretchy="true">(</mo><mrow><msub><mrow><mi>θ</mi></mrow><mi>j</mi></msub><mo>−</mo><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="true">)</mo></mrow></mrow><mo stretchy="true">}</mo></mrow><mo>.</mo></mrow></math> </ephtml> (<reflink idref="bib4" id="ref37">4</reflink>)</p> <p>Here, <emph>θ<subs>i</subs></emph> is the angle between the common parts of <emph>y<subs>i</subs></emph> and the reference variable <emph>y</emph><subs>1</subs> on the circumference of a circle. Note that the common part of a variable is the conceptual psychological construct without the contamination of measurement error. We can choose any variable as the reference variable <emph>y</emph><subs>1</subs> because of the circular placement of the variables. Figure 1 graphically displays Equation (<reflink idref="bib4" id="ref38">4</reflink>). Figure 1 represents the correspondence between the angles and the common score correlations. The more the two variables <emph>y<subs>i</subs></emph> and <emph>y<subs>j</subs></emph> are conceptually close to each other, the closer the model places them together graphically (a smaller angular distance</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>θ</mi></mrow><mi>j</mi></msub><mo>−</mo><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> ). If the angular distance is exactly</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mn>0</mn></mrow><mo>°</mo></msup></mrow><mo>,</mo></math> </ephtml> the two variables are conceptually identical. The more two variables <emph>y<subs>i</subs></emph> and <emph>y<subs>j</subs></emph> are conceptually different from each other, the further away the model places them graphically (a larger angular distance</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>θ</mi></mrow><mi>j</mi></msub><mo>−</mo><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow></math> </ephtml> ). If the angular distance is exactly</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mrow><mn>180</mn></mrow></mrow><mo>°</mo></msup></mrow><mo>,</mo></math> </ephtml> the two variables are conceptually opposite to each other. As shown in Figure 1, the mapping of angles to correlations is more flexible than the usual cosine function. In particular, the angle</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mrow><mn>180</mn></mrow></mrow><mo>°</mo></msup></mrow></math> </ephtml> does not always correspond to the correlation of −1. The circumplex model of Equation (<reflink idref="bib4" id="ref39">4</reflink>) achieves this flexibility by introducing the nonnegative weights <emph>β</emph><subs>1</subs>, <emph>β</emph><subs>2</subs>,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>⋯</mo><mo>,</mo></math> </ephtml> <emph>β<subs>m</subs></emph> on cosine functions. These weights are not conceptually interpretable, but control the flexibility of mapping angles to correlations. The more weights Equation (<reflink idref="bib4" id="ref40">4</reflink>) involves, the more flexible the mapping is. Consequently, the better fit the circumplex model provides. Selecting the number of such weights is conceptually similar to the task of selecting the number of factors in factor analysis. The goal is to find an <emph>m</emph> where <emph>m</emph> − 1 weights fit data poorly, <emph>m</emph> weights fit data well, and <emph>m</emph> + 1 weights do not improve fit much.</p> <p>Graph: Figure 1. The correspondence between the angular distance of two variables (θ) and their common score correlation (ρ).</p> <p>The model involves <emph>p</emph> − 1 angle parameters, <emph>p</emph> error variances, and <emph>m</emph> weights of cosine functions.[<reflink idref="bib3" id="ref41">3</reflink>] We next extend Browne ([<reflink idref="bib8" id="ref42">8</reflink>])'s circumplex model to the within-subject correlations computed with time series data.</p> <hd id="AN0176985635-5">2.2. Model Estimation with Time Series</hd> <p>Let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">γ</mi></math> </ephtml> be a vector that contains all the model parameters. Let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">P</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></mrow><mn>0</mn></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">γ</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> be the model-implied correlation matrix of Equation (<reflink idref="bib2" id="ref43">2</reflink>). The subscript 0 indicates that it is the lag 0 correlation matrix whose elements reflect concurrent relations between variables. Let</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><mn>0</mn></msub></mrow></math> </ephtml> be a vector that contains the nonduplicated elements of the within-subject correlation 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">R</mi></mrow><mn>0</mn></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><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></mrow><mn>0</mn></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">γ</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> be the corresponding vector for</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">P</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></mrow><mn>0</mn></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">γ</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> We estimate</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">γ</mi></math> </ephtml> by minimizing the ordinary least squares (OLS) discrepancy function</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mtext>OLS</mtext></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mn>0</mn></msub><mo>−</mo><msub><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></mrow><mn>0</mn></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">γ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>′</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mn>0</mn></msub><mo>−</mo><msub><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></mrow><mn>0</mn></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">γ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>.</mo></mrow></math> </ephtml> (<reflink idref="bib5" id="ref44">5</reflink>)</p> <p>We consider the OLS estimation because the usual maximum likelihood estimation requires that raw data</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><mn>1</mn></msub><mo>,</mo><mtext /><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mn>2</mn></msub><mo>,</mo><mtext /><mo>⋯</mo><mo>,</mo><mtext /><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mi>t</mi></msub><mo>,</mo><mtext /><mo>⋯</mo><mo>,</mo><mtext /><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mi>T</mi></msub></mrow></math> </ephtml> are from independent participants and the assumption is violated in time series. Molenaar and Nesselroade ([<reflink idref="bib30" id="ref45">30</reflink>]) reported that the usual maximum likelihood estimation did not provide proper test statistics and standard error estimates when structural equation models were estimated with time series data.</p> <p>Let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></math> </ephtml> be the parameter estimates that minimize Equation (<reflink idref="bib5" id="ref46">5</reflink>). We derive their standard errors by adapting the robust standard error estimator of Browne ([<reflink idref="bib7" id="ref47">7</reflink>]). Browne ([<reflink idref="bib7" id="ref48">7</reflink>]) proposed the standard error estimator to accommodate nonnormal cross-sectional data for the usual SEM models, and we adapted it for the circumplex model with time series data.[<reflink idref="bib4" id="ref49">4</reflink>] The adaptation involves the asymptotic covariance matrix (ACM) of within-subject correlations computed with multivariate times series. Because scores at nearby time points in a time series tend to be correlated, the ACM derived for correlations computed from independent individuals (Steiger & Hakstian, [<reflink idref="bib38" id="ref50">38</reflink>]) is no longer appropriate.</p> <p>Zhang et al. ([<reflink idref="bib47" id="ref51">47</reflink>], eq. (<reflink idref="bib14" id="ref52">14</reflink>)) derived the ACM of within-subject correlations. We denote the ACM of within-subject correlations using</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="italic">ϒ</mi><mo>,</mo></math> </ephtml> and it involves both the within-subject correlations and lagged correlations that reflect the correlations of variables of nearby time points. A typical element of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="italic">ϒ</mi></math> </ephtml> is</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="left"><mtr><mtd><mi>T</mi><mo>·</mo><mtext>Cov</mtext><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>k</mi><mo>,</mo><mi>l</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>=</mo><munderover><mo>∑</mo><mrow><mi>u</mi><mo>=</mo><mo>−</mo><mi>∞</mi></mrow><mi>∞</mi></munderover><mrow><mrow><mo stretchy="true">{</mo><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msub><mrow><mi>ρ</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>k</mi><mo>,</mo><mi>l</mi></mrow></msub><mrow><mo stretchy="true">[</mo><mrow><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>i</mi><mo>,</mo><mi>k</mi></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>i</mi><mo>,</mo><mi>l</mi></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow><mn>2</mn></msubsup></mrow><mo stretchy="true">]</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mo>−</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>k</mi><mo>,</mo><mi>l</mi></mrow></msub><mrow><mo stretchy="true">[</mo><mrow><msub><mrow><mi>ρ</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>i</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>+</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>i</mi><mo>,</mo><mi>l</mi></mrow></msub></mrow><mo stretchy="true">]</mo></mrow></mtd></mtr><mtr><mtd><mo>−</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mrow><mo stretchy="true">[</mo><mrow><msub><mrow><mi>ρ</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>i</mi><mo>,</mo><mi>l</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><mi>u</mi><mo>−</mo><mi>n</mi><mo>,</mo><mi>i</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>+</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></msub></mrow><mo stretchy="true">]</mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mo>+</mo><mrow><mo stretchy="true">[</mo><mrow><msub><mrow><mi>ρ</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>l</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>i</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>+</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></msub><msub><mrow><mi>ρ</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>i</mi><mo>,</mo><mi>l</mi></mrow></msub></mrow><mo stretchy="true">]</mo></mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow><mo>}</mo></mrow><mo>.</mo></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib6" id="ref53">6</reflink>)</p> <p>Here, <emph>i</emph>, <emph>j</emph>, <emph>k</emph>, <emph>l</emph>, and <emph>u</emph> are integers. They satisfy the following requirements:</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>p</mi><mo>,</mo><mtext /><mn>1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mi>p</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><mn>1</mn><mo>≤</mo><mi>l</mi><mo>≤</mo><mi>p</mi></mrow><mo>.</mo></math> </ephtml> The symbol</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>ρ</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow></math> </ephtml> denotes the population counterpart of the lagged correlation</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow><mo>.</mo></math> </ephtml> Our adaptation of Browne ([<reflink idref="bib7" id="ref54">7</reflink>])'s robust standard error estimator leads to the following sandwich estimator for the large sample covariance matrix for</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>cov</mtext><mrow><mo stretchy="true">(</mo><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow><mo stretchy="true">)</mo></mrow><mo>=</mo><msup><mrow><mi>T</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mrow><mrow><mrow><mo stretchy="true">[</mo><mrow><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow><mo>′</mo></msubsup><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow></msub></mrow><mo stretchy="true">]</mo></mrow></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo stretchy="true">[</mo><mrow><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow><mo>′</mo></msubsup><mi mathvariant="bold">Υ</mi><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow></msub></mrow><mo stretchy="true">]</mo></mrow><msup><mrow><mrow><mrow><mo stretchy="true">[</mo><mrow><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow><mo>′</mo></msubsup><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow></msub></mrow><mo stretchy="true">]</mo></mrow></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>.</mo></mrow></math> </ephtml> (<reflink idref="bib7" id="ref55">7</reflink>)</p> <p>Here,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Υ</mi></math> </ephtml> is the ACM of the within-subject correlations, <emph>T</emph> is the time length, and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow></msub></mrow></math> </ephtml> is a</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>/</mo><mn>2</mn></mrow></math> </ephtml> by <emph>q</emph> matrix of derivatives of the circumplex correlation structure with respect to the model parameters[<reflink idref="bib5" id="ref56">5</reflink>]</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow></msub><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">γ</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">γ</mi><mo>′</mo></mrow></mfrac><msub><mrow><mo>|</mo></mrow><mrow><mi mathvariant="bold-italic">γ</mi><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow></msub><mo>.</mo></mrow></math> </ephtml> (<reflink idref="bib8" id="ref57">8</reflink>)</p> <p>These derivatives are evaluated at the population value</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow><mo>,</mo></math> </ephtml> which minimizes the OLS discrepancy function of Equation (<reflink idref="bib5" id="ref58">5</reflink>) when the sample correlations are replaced by the corresponding population correlations. When the circumplex model perfectly satisfies the population correlation matrix, the population OLS discrepancy function value is zero. The ideal condition of perfect population models is often unrealistic in practice. Box ([<reflink idref="bib5" id="ref59">5</reflink>]) pointed out that some degree of model error always exists even in the population. MacCallum ([<reflink idref="bib25" id="ref60">25</reflink>]) made a similar argument for factor analysis models. Thus, we allow model error in the circumplex model so that it represents the real world more truthfully. When the circumplex model does not perfectly satisfy the population correlation matrix, the population OLS discrepancy function value is greater than zero.</p> <p>We need to replace the population values of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Υ</mi></math> </ephtml> by their sample estimates</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold">Υ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></math> </ephtml> when using Equation (<reflink idref="bib7" id="ref61">7</reflink>) in practice. Taking the square root of the diagonal elements of Equation (<reflink idref="bib7" id="ref62">7</reflink>) produces the standard error estimates</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow><mo>.</mo></math> </ephtml> We can then conduct tests or construct confidence intervals for a parameter using its point estimate and standard error estimate.</p> <p>We assess the adequacy of the circumplex model by adapting Browne ([<reflink idref="bib7" id="ref63">7</reflink>])'s robust test statistic.[<reflink idref="bib6" id="ref64">6</reflink>] Let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">e</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></math> </ephtml> be a vector of the sample correlation residuals</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">e</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mn>0</mn></msub><mo>−</mo><msub><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></mrow><mn>0</mn></msub><mo stretchy="false">(</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>.</mo></math> </ephtml> The test statistic is</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>f</mi><mo stretchy="false">(</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>T</mi><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">e</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>′</mo><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>,</mo><mi>c</mi></mrow></msub><msup><mrow><mrow><mrow><mo stretchy="true">{</mo><mrow><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>,</mo><mi>c</mi></mrow><mo>′</mo></msubsup><mrow><mrow><mover accent="true"><mi mathvariant="bold">Υ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>,</mo><mi>c</mi></mrow></msub></mrow><mo stretchy="true">}</mo></mrow></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>,</mo><mi>c</mi></mrow><mo>′</mo></msubsup><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">e</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>.</mo></mrow></math> </ephtml> (<reflink idref="bib9" id="ref65">9</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">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>,</mo><mi>c</mi></mrow></msub></mrow></math> </ephtml> is a matrix of the null space of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></msub></mrow></math> </ephtml> such that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow><mo>′</mo></msubsup><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>,</mo><mi>c</mi></mrow></msub><mo>=</mo><mi mathvariant="bold-italic">0</mi></mrow><mo>.</mo></math> </ephtml> Let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>p</mi></mrow><mo>*</mo></msup><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>/</mo><mn>2</mn></mrow></math> </ephtml> be the number of non-duplicated elements of the variable correlation 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">R</mi></mrow><mi mathvariant="bold-italic">0</mi></msub></mrow></math> </ephtml> and let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>q</mi><mo>=</mo><mn>2</mn><mi>p</mi><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow></math> </ephtml> be the number of parameters of the circumplex model. The distribution of the test statistic is a central</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>χ</mi></mrow><mn>2</mn></msup></mrow></math> </ephtml> distribution with</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>p</mi></mrow><mo>*</mo></msup><mo>−</mo><mi>q</mi></mrow></math> </ephtml> degrees of freedom with a large <emph>T</emph> when the circumplex model does not contain model specification error. When the model contains model specification error, the distribution becomes a noncentral</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>χ</mi></mrow><mn>2</mn></msup></mrow></math> </ephtml> distribution with</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>p</mi></mrow><mo>*</mo></msup><mo>−</mo><mi>q</mi></mrow></math> </ephtml> degrees of freedom and the noncentral parameter</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>T</mi><mi mathvariant="bold-italic">e</mi><mo>′</mo><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup><mo>,</mo><mi>c</mi></mrow></msub><msup><mrow><mrow><mrow><mo stretchy="true">{</mo><mrow><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup><mo>,</mo><mi>c</mi></mrow><mo>′</mo></msubsup><mi mathvariant="bold">Υ</mi><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup><mo>,</mo><mi>c</mi></mrow></msub></mrow><mo stretchy="true">}</mo></mrow></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup><mo>,</mo><mi>c</mi></mrow><mo>′</mo></msubsup><mi mathvariant="bold-italic">e</mi></mrow><mo>.</mo></math> </ephtml> </p> <p>A transformation of the noncentrality parameter produces a widely used model fit measure, the root mean square error of approximation (Steiger & Lind, [<reflink idref="bib39" id="ref66">39</reflink>], RMSEA). Browne and Cudeck ([<reflink idref="bib9" id="ref67">9</reflink>]) described how to construct confidence intervals for the RMSEA. Additionally, we can conduct the test of perfect fit (RMSEA</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>=</mo><mn>0</mn></mrow></math> </ephtml> ) and test of close fit (RMSEA</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>≤</mo><mn>0.05</mn></mrow></math> </ephtml> ) for the circumplex model with multivariate time series.</p> <hd id="AN0176985635-6">3. Comparisons with Other Models</hd> <p></p> <hd id="AN0176985635-7">3.1. Multivariate Multilevel Models</hd> <p>Multilevel models (Bryk & Raudenbush, [<reflink idref="bib12" id="ref68">12</reflink>]; Goldstein, [<reflink idref="bib16" id="ref69">16</reflink>]) were developed to account for the lack of independence introduced by observations made on groups of individuals, for example, students are nested within a class and classes are nested within a school. Although multilevel models were traditionally univariate model that accommodated the nested data structure, researchers have extended the models to multiple variables (Blozis, [<reflink idref="bib3" id="ref70">3</reflink>]; MacCallum et al., [<reflink idref="bib26" id="ref71">26</reflink>]), with longitudinal data. Although both the circumplex model with time series data and the multilevel models involve repeated measurements at multiple time points, the number of time points of time series data (from fifties to hundreds, or even thousands) is much larger than that of the traditional longitudinal data (from several to ten). Their primary research goals are also different. Researchers often aim to uncover a nomothetic law that governs the change patterns of all individuals in traditional longitudinal data analysis. The circumplex model with time series data aims to uncover an idiographic law that governs the change of a unique individual.</p> <hd id="AN0176985635-8">Random-Intercept Cross Lagged Panel Models</hd> <p>The random-intercept cross-lagged panel model (Hamaker et al., [<reflink idref="bib19" id="ref72">19</reflink>]) is a model for traditional longitudinal data with several time points. The primary motivation for the development was to move away from a pure nomothetic approach to allow an idiographic component. Participant's raw score at time <emph>t</emph> is decomposed into three parts. The first part is the between-subject mean at time <emph>t</emph>, the second part is the within-subject mean for this participant, and the third part is the residual for Participant <emph>i</emph> at time <emph>t</emph>. The model specifies a vector autoregressive (AR) process for the residuals. The vector AR process is a nomothetic model that is applicable to all individuals. In particular, the AR weights are used to summarize the lagged relations of the residuals for each and every individual. The AR weights are, in general, different at different time points, however. The model specifies the within-subject means (intercepts) as random components. Thus, adding the random intercepts makes the otherwise pure nomothetic model a hybrid one that includes both a nomothetic part and an idiographic part. In comparison, we use the circumplex model to study within-subject correlations computed from multiple time series of individuals. This approach is a pure idiographic approach with the goal of uncovering a pattern unique to each individual. Although both this approach and the random intercept cross-lagged panel models include a part for modeling individual specific features of the data, such a part plays a different role in the two models. Uncovering individual specific features is the primary goal of the circumplex model with time series data, but adding random intercepts to cross-lagged panel models is used to improve a nomothetic model. Furthermore, the random intercept cross-lagged panel model is appropriate for intermediate- and long-term relations since the time intervals in the traditional longitudinal study are often months or even years, but the circumplex model with time series data is appropriate for short-term relations since behavioral time series data are collected daily or even hourly.</p> <hd id="AN0176985635-9">3.2. Dynamic Factor Analysis Models</hd> <p>Dynamic factor analysis models (Browne & Zhang, [<reflink idref="bib11" id="ref73">11</reflink>]; Molenaar, [<reflink idref="bib28" id="ref74">28</reflink>]; Nesselroade et al., [<reflink idref="bib32" id="ref75">32</reflink>]) are also an idiographic method for analyzing multivariate time series. It consists of two parts: a factor analysis part and a time series part. The factor analysis part reduces the dimension from variables to factors, and the time series part specifies how factors affect each other over time. The circumplex model with time series can be regarded as a special case of a dynamic factor analysis model because the circumplex model accounts for the concurrent relations (within-subject correlations) among variables but dynamic factor analysis accounts for both concurrent relations and lagged relations among variables. However, estimating the circumplex model as a constrained dynamic factor analysis model is cumbersome and infeasible because the correspondence of the circumplex model and a factor analysis model is complicated (Browne, [<reflink idref="bib8" id="ref76">8</reflink>], eq. (<reflink idref="bib41" id="ref77">41</reflink>)). In addition, we interpret these two models differently. In a dynamic factor analysis model, we compare factor loading matrices with Thurstone's simple structure to better understand factors; in a circumplex model, we arrange variables along a circle, and Thurstone's simple structure is irrelevant.</p> <hd id="AN0176985635-10">3.4. The Unique Feature of the Circumplex Model with Multivariate Time Series</hd> <p>The circumplex model reflects the expectations of the researchers about the relationships among a set of variables, and the model has been supported in the between-subject setting. Between-subject correlations contain the covariation of both stable traits and variable states, but some of these expectations are more suitable for variable states (e.g. mood and affect). Thus, it is of interest to estimate the circumplex model with within-subject correlations that contain the covariations of variable states exclusively.</p> <hd id="AN0176985635-11">4. An Empirical Illustration</hd> <p>We illustrate the circumplex model with multivariate time series using an affect data set.[<reflink idref="bib7" id="ref78">7</reflink>] The data set was part of a larger study (Watson & Clark, [<reflink idref="bib43" id="ref79">43</reflink>]) on the development of a questionnaire to measure effect. The illustration data set includes a participant's daily ratings on 60 affect items for 70 days. The ratings were five-point Likert variables (1: very slightly or not at all, 5: extremely). Watson et al. ([<reflink idref="bib44" id="ref80">44</reflink>], p. 824) suggested that one compute seven composite scores out of the 60 items before estimating a circumplex model. These seven composite scores are: "high positive affect", "high negative affect," "low positive affect," "low negative affect," "pleasantness," "unpleasantness," and "engagement."[<reflink idref="bib8" id="ref81">8</reflink>] These composite scores form a 7-variate time series of 70 time points.[<reflink idref="bib9" id="ref82">9</reflink>]</p> <p>We first computed the 7 × 7 correlation matrix from the time series. We estimated the circumplex model by minimizing the discrepancy function value of Equation (<reflink idref="bib5" id="ref83">5</reflink>). We obtained the standard error estimates and the test statistic using Equations (<reflink idref="bib7" id="ref84">7</reflink>) and (<reflink idref="bib9" id="ref85">9</reflink>), respectively. To accommodate time series data, we considered lagged correlations while computing the ACM</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold">Υ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></math> </ephtml> in Equations (<reflink idref="bib7" id="ref86">7</reflink>) and (<reflink idref="bib9" id="ref87">9</reflink>). For comparison purpose, we also obtain standard error estimates and the test statistic while ignoring lagged correlations of time series data. We refer to this approach as a "cross-sectional" approach because it treats time series data as if they were cross-sectional data with many independent participants. More specifically, we computed</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold">Υ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></math> </ephtml> Equations (<reflink idref="bib7" id="ref88">7</reflink>) and (<reflink idref="bib9" id="ref89">9</reflink>) using a method that is appropriate for cross-sectional data (Browne & Shapiro, [<reflink idref="bib10" id="ref90">10</reflink>]). Note that the time series approach and the cross-sectional approach will produce the same point estimates for all model parameters but their respective test statistics and standard error estimates tend to be different due to different estimates for the ACM of correlations. When the correlations are computed from time series data, the ACM estimate obtained using the cross-sectional approach can be substantially different from the ACM estimate properly obtained using the time series approach.</p> <p>Table 1 presents the estimates for six angle parameters, seven communality indices, and one cosine weight parameter. The variable "high positive affect" is the reference variable and its angle parameter is fixed as</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mn>0</mn></mrow><mo>°</mo></msup></mrow><mo>.</mo></math> </ephtml> The standard error estimates produced with the time series approach are close to those produced with the cross-sectional approach.</p> <p>Table 1. Parameter estimates and two types of standard error estimates of a circumplex model with a time series of daily affect ratings.</p> <p> <ephtml> <table><thead><tr><td /><td>PE</td><td><p id="ilm0178"><graphic href="hsem_a_2259105_ilm0178.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>S</mi><msub><mrow><mi>E</mi></mrow><mrow><mi>T</mi><mi>S</mi></mrow></msub></mrow></math></p></td><td><p id="ilm0179"><graphic href="hsem_a_2259105_ilm0179.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>S</mi><msub><mrow><mi>E</mi></mrow><mrow><mi>C</mi><mi>S</mi></mrow></msub></mrow></math></p></td></tr></thead><tbody valign="top"><tr><td><p id="ilm0180"><graphic href="hsem_a_2259105_ilm0180.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mrow><mrow><mover accent="true"><mi>θ</mi><mo>̂</mo></mover></mrow></mrow></mrow><mrow><mtext>HP</mtext></mrow></msub></mrow></math></p></td><td><p id="ilm0181"><graphic href="hsem_a_2259105_ilm0181.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msup><mrow><mn>0</mn></mrow><mo>°</mo></msup></mrow></math></p></td><td>– –</td><td>– –</td></tr><tr><td><p id="ilm0182"><graphic href="hsem_a_2259105_ilm0182.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mrow><mrow><mover accent="true"><mi>θ</mi><mo>̂</mo></mover></mrow></mrow></mrow><mrow><mtext>LP</mtext></mrow></msub></mrow></math></p></td><td><p id="ilm0183"><graphic href="hsem_a_2259105_ilm0183.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msup><mrow><mrow><mn>201</mn></mrow></mrow><mo>°</mo></msup></mrow></math></p></td><td char=".">6.54</td><td char=".">6.31</td></tr><tr><td><p id="ilm0184"><graphic href="hsem_a_2259105_ilm0184.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mrow><mrow><mover accent="true"><mi>θ</mi><mo>̂</mo></mover></mrow></mrow></mrow><mrow><mtext>HN</mtext></mrow></msub></mrow></math></p></td><td><p id="ilm0185"><graphic href="hsem_a_2259105_ilm0185.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msup><mrow><mrow><mn>121</mn></mrow></mrow><mo>°</mo></msup></mrow></math></p></td><td char=".">7.35</td><td char=".">8.11</td></tr><tr><td><p id="ilm0186"><graphic href="hsem_a_2259105_ilm0186.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mrow><mrow><mover accent="true"><mi>θ</mi><mo>̂</mo></mover></mrow></mrow></mrow><mrow><mo /><mi mathvariant="normal">LN</mi><mo /></mrow></msub></mrow></math></p></td><td><p id="ilm0187"><graphic href="hsem_a_2259105_ilm0187.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msup><mrow><mrow><mn>309</mn></mrow></mrow><mo>°</mo></msup></mrow></math></p></td><td char=".">8.13</td><td char=".">8.01</td></tr><tr><td><p id="ilm0188"><graphic href="hsem_a_2259105_ilm0188.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mrow><mrow><mover accent="true"><mi>θ</mi><mo>̂</mo></mover></mrow></mrow></mrow><mrow><mtext>UP</mtext></mrow></msub></mrow></math></p></td><td><p id="ilm0189"><graphic href="hsem_a_2259105_ilm0189.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msup><mrow><mrow><mn>146</mn></mrow></mrow><mo>°</mo></msup></mrow></math></p></td><td char=".">7.90</td><td char=".">8.35</td></tr><tr><td><p id="ilm0190"><graphic href="hsem_a_2259105_ilm0190.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mrow><mrow><mover accent="true"><mi>θ</mi><mo>̂</mo></mover></mrow></mrow></mrow><mrow><mtext>EG</mtext></mrow></msub></mrow></math></p></td><td><p id="ilm0191"><graphic href="hsem_a_2259105_ilm0191.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msup><mrow><mrow><mn>358</mn></mrow></mrow><mo>°</mo></msup></mrow></math></p></td><td char=".">6.46</td><td char=".">7.61</td></tr><tr><td><p id="ilm0192"><graphic href="hsem_a_2259105_ilm0192.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mrow><mrow><mover accent="true"><mi>θ</mi><mo>̂</mo></mover></mrow></mrow></mrow><mrow><mtext>PL</mtext></mrow></msub></mrow></math></p></td><td><p id="ilm0193"><graphic href="hsem_a_2259105_ilm0193.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msup><mrow><mrow><mn>341</mn></mrow></mrow><mo>°</mo></msup></mrow></math></p></td><td char=".">4.12</td><td char=".">4.72</td></tr><tr><td><p id="ilm0194"><graphic href="hsem_a_2259105_ilm0194.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mrow><mrow><mover accent="true"><mi>β</mi><mo>̂</mo></mover></mrow></mrow></mrow><mn>1</mn></msub></mrow></math></p></td><td char=".">0.93</td><td char=".">0.02</td><td char=".">0.02</td></tr><tr><td><p id="ilm0195"><graphic href="hsem_a_2259105_ilm0195.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mrow><mrow><mover accent="true"><mi>ζ</mi><mo>̂</mo></mover></mrow></mrow></mrow><mrow><mtext>HP</mtext></mrow></msub></mrow></math></p></td><td char=".">0.94</td><td char=".">0.03</td><td char=".">0.03</td></tr><tr><td><p id="ilm0196"><graphic href="hsem_a_2259105_ilm0196.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mrow><mrow><mover accent="true"><mi>ζ</mi><mo>̂</mo></mover></mrow></mrow></mrow><mrow><mtext>LP</mtext></mrow></msub></mrow></math></p></td><td char=".">0.97</td><td char=".">0.07</td><td char=".">0.07</td></tr><tr><td><p id="ilm0197"><graphic href="hsem_a_2259105_ilm0197.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mrow><mrow><mover accent="true"><mi>ζ</mi><mo>̂</mo></mover></mrow></mrow></mrow><mrow><mtext>HN</mtext></mrow></msub></mrow></math></p></td><td char=".">0.93</td><td char=".">0.06</td><td char=".">0.06</td></tr><tr><td><p id="ilm0198"><graphic href="hsem_a_2259105_ilm0198.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mrow><mrow><mover accent="true"><mi>ζ</mi><mo>̂</mo></mover></mrow></mrow></mrow><mrow><mo /><mi mathvariant="normal">LN</mi><mo /></mrow></msub></mrow></math></p></td><td char=".">0.85</td><td char=".">0.06</td><td char=".">0.05</td></tr><tr><td><p id="ilm0199"><graphic href="hsem_a_2259105_ilm0199.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mrow><mrow><mover accent="true"><mi>ζ</mi><mo>̂</mo></mover></mrow></mrow></mrow><mrow><mtext>UP</mtext></mrow></msub></mrow></math></p></td><td char=".">0.80</td><td char=".">0.05</td><td char=".">0.06</td></tr><tr><td><p id="ilm0200"><graphic href="hsem_a_2259105_ilm0200.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mrow><mrow><mover accent="true"><mi>ζ</mi><mo>̂</mo></mover></mrow></mrow></mrow><mrow><mtext>EG</mtext></mrow></msub></mrow></math></p></td><td char=".">0.72</td><td char=".">0.06</td><td char=".">0.06</td></tr><tr><td><p id="ilm0201"><graphic href="hsem_a_2259105_ilm0201.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mrow><mrow><mover accent="true"><mi>ζ</mi><mo>̂</mo></mover></mrow></mrow></mrow><mrow><mtext>PL</mtext></mrow></msub></mrow></math></p></td><td char=".">0.94</td><td char=".">0.02</td><td char=".">0.02</td></tr></tbody></table> </ephtml> </p> <p>1 Note: The time series includes seven variables: "PE": point estimate; "</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>S</mi><msub><mrow><mi>E</mi></mrow><mrow><mi>T</mi><mi>S</mi></mrow></msub></mrow></math> </ephtml> ": standard errors computed with the time series approach; '</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>S</mi><msub><mrow><mi>E</mi></mrow><mrow><mi>C</mi><mi>S</mi></mrow></msub></mrow></math> </ephtml> ": standard errors computed with the cross-sectional approach; "HP": high positive affect; "LP": low positive affect; "HN": high negative affect; "LN": low negative affect; "UP": unpleasantness; "EG": engagement; "PL": pleasantness.</p> <p>Figure 2 displays the estimated circumplex structure of the seven effects for this particular individual. These seven effects form two clusters. One cluster consists of high positive affect, low negative affect, pleasantness, and engagement; the other cluster consists of high negative affect, low positive Affect, and unpleasantness. The angle between high positive affect and engagement is only</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mn>2</mn></mrow><mo>°</mo></msup></mrow></math> </ephtml> (</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mrow><mn>358</mn></mrow></mrow><mo>°</mo></msup></mrow></math> </ephtml> is equivalent to</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mn>2</mn></mrow><mo>°</mo></msup></mrow></math> </ephtml> ), which corresponds to a model-implied correlation of 1.00 with 95% confidence interval of (0.97 and 1.00).</p> <p>Graph: Figure 2. The circumplex model with daily affect ratings of an individual for 70 days (Watson & Clark, [<reflink idref="bib43" id="ref91">43</reflink>]). The lines represent angular placements of seven variables (1: High Positive Affect, 2: Low Positive Affect, 3: High Negative Affect, 4: Low Negative Affect, 5: Unpleasantness, 6: Engagement, 7: Pleasantness). The distance between a triangle symbol and the center indicates the communality index of a variable.</p> <p>Of particular interest are the relations between positive affect and negative affect. Some affect researchers (Green et al., [<reflink idref="bib18" id="ref92">18</reflink>]) argued that positive and negative effects are the two poles of a single dimension, while other researchers (Watson et al., [<reflink idref="bib44" id="ref93">44</reflink>]) argued that positive and negative effects are relatively independent dimensions. We can examine the relationship between positive and negative effects for a particular individual. The estimated angle between high positive affect and high negative affect is</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mrow><mn>121</mn></mrow></mrow><mo>°</mo></msup></mrow></math> </ephtml> with a 95% confidence interval</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msup><mrow><mrow><mn>106</mn></mrow></mrow><mo>°</mo></msup><mo>,</mo><mo /><msup><mrow><mrow><mn>135</mn></mrow></mrow><mo>°</mo></msup><mo stretchy="false">)</mo></mrow><mo>,</mo></math> </ephtml> which corresponds to a model-implied correlation of −0.40 with a 95% confidence interval</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>.59</mn><mo>,</mo><mo /><mo>−</mo><mn>.19</mn><mo stretchy="false">)</mo></mrow><mo>.</mo></math> </ephtml> These two effects have a low to medium level of negative correlation for this particular individual.</p> <p>The angle between low positive affect and low negative affect is</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mrow><mn>108</mn></mrow></mrow><mo>°</mo></msup></mrow></math> </ephtml> with a 95% confidence interval</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msup><mrow><mrow><mn>88</mn></mrow></mrow><mo>°</mo></msup><mo>,</mo><mo /><msup><mrow><mrow><mn>127</mn></mrow></mrow><mo>°</mo></msup><mo stretchy="false">)</mo></mrow><mo>,</mo></math> </ephtml> which corresponds to a model-implied correlation of −0.22 with a 95% confidence interval</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>.49</mn><mo>,</mo><mo /><mn>.10</mn><mo stretchy="false">)</mo></mrow><mo>.</mo></math> </ephtml> These two effects have a low level of correlation for this individual. Because the confidence interval includes a correlation value of 0, we cannot rule out the possibility of low positive affect and low negative affect being orthogonal.</p> <p>Equation (<reflink idref="bib9" id="ref94">9</reflink>) produced the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>χ</mi></mrow><mn>2</mn></msup></mrow></math> </ephtml> statistic of 3.40 with 7 degrees of freedom. The corresponding root mean square error of approximation (RMSEA) (Browne & Cudeck, [<reflink idref="bib9" id="ref95">9</reflink>]) is 0.00 with a 90% confidence interval</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mn>0.000</mn><mo>,</mo><mn>0.083</mn><mo stretchy="false">)</mo></mrow><mo>.</mo></math> </ephtml> The test statistic and the RMSEA indicate a perfect to mediocre fit of the model. The cross-sectional approach produced a similar test statistic (4.29) and the RMSEA (0.00 with a 90% confidence interval</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mn>0.000</mn><mo>,</mo><mn>0.105</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> ), which indicates a perfect to unacceptable fit of the model. Although the test of perfect fit (RMSEA</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>=</mo><mn>0.00</mn></mrow></math> </ephtml> ) is not rejected, we interpret it as the amount of model error being comparable to the amount of sampling error. A longer time series will allow us to detect the model error for this participant.</p> <p>The circumplex structure of the seven effects reveals how these effects co-vary over time for this particular individual. Other individuals may have a similar or entirely different pattern. The method developed in the present article is an idiographic method, and our goal is to uncover the pattern unique to an individual. In comparison, a nomothetic approach will estimate the circumplex model with a correlation matrix that is an average of many participants. Unless the multivariate time series are ergodic in these different individuals, averaging them together can produce misleading results (Molenaar, [<reflink idref="bib29" id="ref96">29</reflink>]). The results of such a nomothetic approach may not be appropriate for any of the individuals from which the mean was computed.</p> <hd id="AN0176985635-12">5. A Simulation Study</hd> <p>Although the empirical illustration demonstrated the usefulness of the newly developed method, we need to assess its performance under a variety of conditions. We next conduct a simulation study to further explore the statistical properties of the new method. Because the "true" parameter values are known in a simulation study, we can compare the parameter estimates against them for a better understanding of the statistical properties. We manipulated three factors in the simulation study: the population model (Models I and II), the level of communality (low, high, and wide), and the time series length (<reflink idref="bib50" id="ref97">50</reflink>, 100, 200, and 500). Crossing these three factors results in 24 conditions. We generated 1000 multivariate time series in each condition.</p> <hd id="AN0176985635-13">5.1. The Design of the Simulation Study and Parameter Values</hd> <p>We considered two models in the simulation study. Model I involves seven variables and Model II involves eight variables. Model I is actually the model of the empirical illustration. Model II is a theoretical model proposed by affect researchers (Russell, [<reflink idref="bib35" id="ref98">35</reflink>]; Watson et al., [<reflink idref="bib44" id="ref99">44</reflink>]) and personality researchers (Conte & Plutchik, [<reflink idref="bib13" id="ref100">13</reflink>]; Wiggins, [<reflink idref="bib45" id="ref101">45</reflink>]). Figure 2 displays Model I and Figure 3 displays Model II. Models I and II represent a realistic situation and an ideal situation, respectively. The "population" values of angle parameters and the cosine weight in Model I are the same as the estimates reported in Table 1. The population angle parameters of Model II are</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mn>0</mn></mrow><mo>°</mo></msup><mo>,</mo><mtext /><msup><mrow><mrow><mn>45</mn></mrow></mrow><mo>°</mo></msup><mo>,</mo><mtext /><msup><mrow><mrow><mn>90</mn></mrow></mrow><mo>°</mo></msup><mo>,</mo><mtext /><msup><mrow><mrow><mn>135</mn></mrow></mrow><mo>°</mo></msup><mo>,</mo><mtext /><msup><mrow><mrow><mn>180</mn></mrow></mrow><mo>°</mo></msup><mo>,</mo><mtext /><msup><mrow><mrow><mn>225</mn></mrow></mrow><mo>°</mo></msup><mo>,</mo><mtext /><msup><mrow><mrow><mn>270</mn></mrow></mrow><mo>°</mo></msup></mrow><mo>,</mo></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mrow><mn>315</mn></mrow></mrow><mo>°</mo></msup></mrow><mo>.</mo></math> </ephtml> The cosine weights of Model II are the same as the one in Model I.[<reflink idref="bib10" id="ref102">10</reflink>]</p> <p>Graph: Figure 3. An ideal circumplex model (Model II) of eight variables that are evenly distributed around the circle. The distance between a triangle symbol and the center indicates the communality index of a variable.</p> <p>We considered two models in the simulation study. Model I is the same as the one considered in the empirical illustration. The "population" values of angle parameters and the cosine weight in Model I are the same as the estimates reported in Table 1. Model II is a theoretical model proposed by affect researchers (Russell, [<reflink idref="bib35" id="ref103">35</reflink>]; Watson et al., [<reflink idref="bib44" id="ref104">44</reflink>]) and personality researchers (Conte & Plutchik, [<reflink idref="bib13" id="ref105">13</reflink>]; Wiggins, [<reflink idref="bib45" id="ref106">45</reflink>]). Model II involves eight variables, and these eight variables are evenly spaced around the circumference of a circle. The values of the eight angles are then</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mn>0</mn></mrow><mo>°</mo></msup><mo>,</mo><mtext /><msup><mrow><mrow><mn>45</mn></mrow></mrow><mo>°</mo></msup><mo>,</mo><mtext /><msup><mrow><mrow><mn>90</mn></mrow></mrow><mo>°</mo></msup><mo>,</mo><mtext /><msup><mrow><mrow><mn>135</mn></mrow></mrow><mo>°</mo></msup><mo>,</mo><mtext /><msup><mrow><mrow><mn>180</mn></mrow></mrow><mo>°</mo></msup><mo>,</mo><mtext /><msup><mrow><mrow><mn>225</mn></mrow></mrow><mo>°</mo></msup><mo>,</mo><mtext /><msup><mrow><mrow><mn>270</mn></mrow></mrow><mo>°</mo></msup></mrow><mo>,</mo></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mrow><mn>315</mn></mrow></mrow><mo>°</mo></msup></mrow><mo>.</mo></math> </ephtml> The cosine weight is the same as the one in Model I.</p> <p>We considered three levels of commonality indices: low (0.2–0.4), high (0.6–0.8), and wide (0.2–0.8). These three levels of commonality indices correspond to the high, low, and wide levels of measurement error variances. The influences of measurement error on variable correlations were debated in affect research (Green et al., [<reflink idref="bib18" id="ref107">18</reflink>]; Watson & Clark, [<reflink idref="bib42" id="ref108">42</reflink>]). In addition, these three levels of communality are commonly used in simulation studies of factor analysis (MacCallum et al., [<reflink idref="bib27" id="ref109">27</reflink>]).</p> <p>We considered four time series lengths (<reflink idref="bib50" id="ref110">50</reflink>, 100, 200, and 500). The time series lengths of 50 and 100 represent the typical time series lengths of the studies that require participants to respond to survey items (Borkenau & Ostendorf, [<reflink idref="bib4" id="ref111">4</reflink>]; Lebo & Nesselroade, [<reflink idref="bib22" id="ref112">22</reflink>]; Watson & Clark, [<reflink idref="bib43" id="ref113">43</reflink>]). The time series lengths of 200 and 500 are included for comparison purposes. Although the time series length of 500 seems long in a typical intensive longitudinal study, it could be easily achievable with new data collection methods such as smartphone applications and wearable sensors. These new data collection methods often continually monitor certain behavior without the need for participants' active involvement.</p> <hd id="AN0176985635-14">5.2. Generating Multivariate Time Series</hd> <p>We generate the multivariate time series using a vector AR time series model.[<reflink idref="bib11" id="ref114">11</reflink>]</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><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi mathvariant="bold-italic">A</mi><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mi>t</mi></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">z</mi></mrow><mi>t</mi></msub><mo>.</mo></mrow></math> </ephtml> (<reflink idref="bib10" id="ref115">10</reflink>)</p> <p>Here <bold><emph>A</emph></bold> is a <emph>p</emph> ×<emph> p</emph> AR weight matrix, 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>t</mi></msub></mrow></math> </ephtml> is a <emph>p</emph>-component vector of shock variables. The shock variables</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>t</mi></msub></mrow></math> </ephtml> are not correlated 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">y</mi></mrow><mi>t</mi></msub></mrow><mo>.</mo></math> </ephtml> The shock variables are normally distributed with a null mean vector and a covariance matrix</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Ψ</mi><mo>.</mo></math> </ephtml> Generating the multivariate time series requires the specification of the two matrices <bold><emph>A</emph></bold> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Ψ</mi><mo>.</mo></math> </ephtml> </p> <p>The AR weight matrix <bold><emph>A</emph></bold> is a diagonal matrix, and we generated its diagonal elements from a uniform distribution (0.4, 0.6). These diagonal elements are</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mn>0.45</mn><mo>,</mo><mn>0.47</mn><mo>,</mo><mn>0.51</mn><mo>,</mo><mn>0.58</mn><mo>,</mo><mn>0.44</mn><mo>,</mo><mn>0.58</mn><mo>,</mo><mn>0.59</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> in Model I; and they are</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mn>0.45</mn><mo>,</mo><mn>0.47</mn><mo>,</mo><mn>0.51</mn><mo>,</mo><mn>0.58</mn><mo>,</mo><mn>0.44</mn><mo>,</mo><mn>0.58</mn><mo>,</mo><mn>0.59</mn><mo>,</mo><mn>0.53</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> in Model II. The shock variable covariance matrix</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Ψ</mi></math> </ephtml> is a function of the AR weight matrix <bold><emph>A</emph></bold> and the correlation structure <bold><emph>P</emph></bold> given by the circumplex model of Equation (<reflink idref="bib2" id="ref116">2</reflink>)</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Ψ</mi><mo>=</mo><mi mathvariant="bold-italic">P</mi><mo>−</mo><mi mathvariant="bold-italic">APA</mi><mo>′</mo><mo>.</mo></mrow></math> </ephtml> (<reflink idref="bib11" id="ref117">11</reflink>)</p> <p>Note that Equation (<reflink idref="bib11" id="ref118">11</reflink>) is a solution to the multivariate version of Yule–Walker Equation (Brockwell & Davis, [<reflink idref="bib6" id="ref119">6</reflink>], p.420).</p> <p>To simulate a multivariate time series of Equation (<reflink idref="bib10" id="ref120">10</reflink>), we start 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">y</mi></mrow><mi>t</mi></msub><mo>=</mo><mi mathvariant="bold-italic">0</mi></mrow><mo>.</mo></math> </ephtml> The shock variables</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><mn>1</mn></msub><mo>,</mo><mtext /><msub><mrow><mi mathvariant="bold-italic">z</mi></mrow><mn>2</mn></msub><mo>,</mo><mtext /><mo>⋯</mo></mrow></math> </ephtml> are generated from a multivariate normal distribution with a null vector and the covariance matrix</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Ψ</mi><mo>.</mo></math> </ephtml> To ensure the stationarity of the generated multivariate time series, we discarded the first 100 time points as a burn-in period.</p> <hd id="AN0176985635-15">5.3. Model Estimation</hd> <p>We estimated the circumplex model with each simulated time series by the OLS estimation of Equation (<reflink idref="bib5" id="ref121">5</reflink>). We computed the corresponding standard error estimates and test statistics using Equation (<reflink idref="bib7" id="ref122">7</reflink>) and (<reflink idref="bib9" id="ref123">9</reflink>), respectively. For comparison purposes, we also computed standard error estimates and the test statistic using the cross-sectional approach, which ignores the dependence of observations at nearby time points.</p> <hd id="AN0176985635-16">5.4. Results of the Simulation Study</hd> <p></p> <hd id="AN0176985635-17">5.4.1. Clean Convergence Rates</hd> <p>We first examine the convergence rates under different conditions. We define the estimation as a clean convergence if the iterative optimization algorithm converges and none of the parameter estimates are on the boundary (e.g. a communality estimate of 1.00). Table 2 presents the clean convergence rates. We make three observations. First, Model II has higher convergence rates than Model I. Model II represents an ideal model in which variables are evenly spaced around a circle, but Model I is a more realistic representation of the real-world phenomenon. Second, longer time series lead to higher convergence rates. Third, higher levels of communality lead to higher convergence rates. Note that higher levels of communality correspond to lower levels of measurement error, which are expected to produce more satisfactory results. We remove a time series from the further analysis if it does not produce a clean convergence.</p> <p>Table 2. The clean convergence rates of Model I and Model II.</p> <p> <ephtml> <table><thead><tr><td>Communality</td><td>T</td><td>Model I</td><td>Model II</td></tr></thead><tbody valign="top"><tr><td>Low</td><td char=".">50</td><td char=".">.514</td><td char=".">.785</td></tr><tr><td char=".">100</td><td char=".">.624</td><td char=".">.953</td></tr><tr><td char=".">200</td><td char=".">.790</td><td char=".">.997</td></tr><tr><td char=".">500</td><td char=".">.956</td><td char=".">1.000</td></tr><tr><td>High</td><td char=".">50</td><td char=".">.744</td><td char=".">.915</td></tr><tr><td char=".">100</td><td char=".">.897</td><td char=".">.991</td></tr><tr><td char=".">200</td><td char=".">.991</td><td char=".">1.000</td></tr><tr><td char=".">500</td><td char=".">1.000</td><td char=".">1.000</td></tr><tr><td>Wide</td><td char=".">50</td><td char=".">.598</td><td char=".">.838</td></tr><tr><td char=".">100</td><td char=".">.794</td><td char=".">.959</td></tr><tr><td char=".">200</td><td char=".">.956</td><td char=".">.997</td></tr><tr><td char=".">500</td><td char=".">.998</td><td char=".">1.000</td></tr></tbody></table> </ephtml> </p> <p>2 Note: "T": the time series length.</p> <hd id="AN0176985635-18">5.4.2. The Coverage Performances of Confidence Intervals</hd> <p>We examine empirical coverage rates of the confidence intervals to assess the combined statistical properties of point estimates and standard error estimates. The empirical coverage rate of confidence intervals is the proportion of confidence intervals computed with simulated multivariate time series that cover the "true" parameter value. Figure 4 displays the empirical coverage rates of 95% confidence intervals of all model parameters under different conditions. The most striking finding is that the empirical coverage rates of the confidence intervals constructed with the time series approach are much closer to the nominal level of 95% than those constructed with the cross-sectional approach. Note that the time series approach and the cross-sectional approach have the same point estimates, but different standard error estimates. Thus, ignoring the dependence of observations at nearby points leads to less satisfactory standard error estimates of the cross-sectional approach. In addition, increasing the time series length substantially improves the empirical coverage rates of the time series approach, but it does not improve the empirical coverage rates of the cross-sectional approach. Furthermore, satisfactory empirical coverage rates require a time series length of 100 and longer. The influences of communality on the empirical coverage rates are less obvious. We observe the same pattern of the influences of the two estimation methods, the four-time series lengths, and the three communality levels in Model I and Model II, and the coverage rates are slightly better in Model II, particularly with shorter time series.</p> <p>Graph: Figure 4. Empirical coverage rates of 95% confidence intervals of Model I and Model II. Model I includes 13 parameters (six angles, one cosine weight, and six communality indices) and Model II includes 15 parameters (seven angles, one cosine weight, and seven communality indices). The horizontal lines represent the nominal level, 0.95. "TS" stands for the time series approach and "CS" stands for the cross-sectional approach. "Low," "High," and "Wide" stand for the three communality conditions. "T" stands for a time series length.</p> <hd id="AN0176985635-19">5.4.3. The Test Statistics and Their Reference Distributions</hd> <p>Figures 5 and 6 display the empirical cumulative distribution functions (CDF) of the test statistics of Model I and Model II, respectively. We also include the reference distributions of the test statistics. It is the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>χ</mi></mrow><mn>2</mn></msup></mrow></math> </ephtml> distribution with 7 degrees of freedom in Model I and the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>χ</mi></mrow><mn>2</mn></msup></mrow></math> </ephtml> distribution with 12 degrees of freedom in Model II. We make three observations. First, the empirical CDFs of the time series approach are closer to their reference distributions than those of the cross-sectional approach when time series lengths are 100 or longer. Second, increasing the time series length makes the CDFs of the time series approach move closer to their reference distributions, but it does not move the CDFs of the cross-sectional approach closer to their reference distributions. These two observations highlight the importance of properly accounting for the dependence of variables at nearby time points in the circumplex model with multivariate time series data. Third, the influences of communality on the test statistics are less obvious. Although the three observations are largely the same in both Model I and Model II, the empirical CDFs of the time series approach seem closer to their reference distributions in Model I than in Model II, particularly at the time series length of 100.</p> <p>Graph: Figure 5. Cumulative distribution functions of test statistics, Model I. The plots compare cumulative distribution functions (CDF) of test statistics obtained with the time series approach (TS, the dotted line) and the cross-sectional approach (CS, the dashed line) against their theoretical χ2 distributions (the solid line). The columns represent time series length conditions (<reflink idref="bib50" id="ref124">50</reflink>, 100, 200, and 500). the rows represent communality conditions ("Low", "High", and "Wide").</p> <p>Graph: Figure 6. Cumulative distribution functions of test statistics, Model II. The plots compare cumulative distribution functions (CDF) of test statistics obtained with the time series approach (TS, the dotted line) and the cross-sectional approach (CS, the dashed line) against their theoretical χ2 distributions (the solid line). the columns represent time series length conditions (<reflink idref="bib50" id="ref125">50</reflink>, 100, 200, and 500). The rows represent communality conditions ("Low", "High", and "Wide").</p> <hd id="AN0176985635-20">5.4.4. Empirical Type I Error Rates</hd> <p>We can perform the goodness-of-fit test with test statistics. We can compute the <emph>p</emph>-value for each test statistic, and the test statistic is significant if the <emph>p</emph>-value is less than 0.05. The empirical type I error rates are the proportions of simulated time series with a significant test statistic. Table 3 presents the empirical Type I error rates under all conditions. The results of empirical type I error rates are largely consistent with the results of the CDFs of the test statistics presented earlier. We make four observations. First, the empirical Type I error rates of the time series approach are closer to 0.05 (the nominal level) than those of the cross-sectional approach at the time series length of 100 and longer. Second, the empirical Type I error rates of the time series approach move closer to 0.05 as the time series lengths increase. In contrast, the empirical Type I error rates of the cross-sectional approach do not move closer to 0.05 as the time series length increases. These two observations highlight the necessity of accounting for the dependence of variables at nearby time points in analyzing time series data. We then focus on the time series approach. The third observation is that the empirical type I error rate of Model I is closer to 0.05 than that of Model II for short and medium-length time series. A possible reason is that Model II has more variables and more degrees of freedom. The asymptotic properties of a larger model require more time points. Fourth, the empirical type I error rates at the high and wide levels of communality are closer to 0.05 than those at the low level of communality. The influences of communality are particularly evident in Model II.</p> <p>Table 3. Empirical Type I error rates of the goodness-of-fit test of Model I and Model II.</p> <p> <ephtml> <table><thead><tr><td /><td /><td>Model I</td><td>Model II</td></tr><tr><td>Communality</td><td>T</td><td>TS</td><td>CS</td><td>TS</td><td>CS</td></tr></thead><tbody valign="top"><tr><td>Low</td><td char=".">50</td><td char=".">0.24</td><td char=".">0.14</td><td char=".">0.62</td><td char=".">0.29</td></tr><tr><td char=".">100</td><td char=".">0.07</td><td char=".">0.15</td><td char=".">0.22</td><td char=".">0.30</td></tr><tr><td char=".">200</td><td char=".">0.03</td><td char=".">0.16</td><td char=".">0.11</td><td char=".">0.42</td></tr><tr><td char=".">500</td><td char=".">0.03</td><td char=".">0.28</td><td char=".">0.07</td><td char=".">0.42</td></tr><tr><td>High</td><td char=".">50</td><td char=".">0.22</td><td char=".">0.10</td><td char=".">0.49</td><td char=".">0.13</td></tr><tr><td char=".">100</td><td char=".">0.05</td><td char=".">0.18</td><td char=".">0.14</td><td char=".">0.27</td></tr><tr><td char=".">200</td><td char=".">0.04</td><td char=".">0.24</td><td char=".">0.06</td><td char=".">0.38</td></tr><tr><td char=".">500</td><td char=".">0.05</td><td char=".">0.30</td><td char=".">0.05</td><td char=".">0.40</td></tr><tr><td>Wide</td><td char=".">50</td><td char=".">0.27</td><td char=".">0.12</td><td char=".">0.55</td><td char=".">0.22</td></tr><tr><td char=".">100</td><td char=".">0.07</td><td char=".">0.19</td><td char=".">0.18</td><td char=".">0.32</td></tr><tr><td char=".">200</td><td char=".">0.05</td><td char=".">0.23</td><td char=".">0.08</td><td char=".">0.40</td></tr><tr><td char=".">500</td><td char=".">0.04</td><td char=".">0.29</td><td char=".">0.06</td><td char=".">0.40</td></tr></tbody></table> </ephtml> </p> <p>3 Note: "T": the time series length; "TS": the time series approach; "CS": the cross-sectional approach.</p> <hd id="AN0176985635-21">6. Discussion</hd> <p>Social and behavioral scientists collect multivariate time series data to study rapidly changing constructs like mood, affect, and personality states. The circumplex model posits a circular representation for these constructs. We extended a flexible circumplex model (Browne, [<reflink idref="bib8" id="ref126">8</reflink>]), originally developed for modeling inter-individual differences in stable personality traits, to multivariate time series with the goal of modeling intra-individual differences in rapidly changing psychological states. We presented standard error estimates and test statistics that are appropriate for the circumplex model with multivariate time series. The simulation study showed that (<reflink idref="bib1" id="ref127">1</reflink>) statistical properties of the newly developed method are satisfactory when the time series length is 100 or longer and (<reflink idref="bib2" id="ref128">2</reflink>) ignoring the cross-lag correlations in multivariate time series resulted in lower coverage rates of confidence intervals and higher type I error rates of the test statistic.</p> <p>Although the estimation of the circumplex model appears to involve only the within-subject correlations of variables that were measured at the same time, the standard error estimates and the test statistic require the distribution of these correlations. We adapted robust standard error estimates and test statistic of the usual SEM model with nonnormal cross-sectional data (Browne, [<reflink idref="bib7" id="ref129">7</reflink>]) for circumplex models with time series data. An important part of the adaptation was to include the ACM of the within-subject correlations. The ACM is a function of both the within-subject correlations between variables measured at the same time and the cross-lag correlations between variables measured at different time points (Zhang et al., [<reflink idref="bib47" id="ref130">47</reflink>]). These cross-lag correlations reflect a defining feature of multivariate time series data: variables are correlated with each other when they are at nearby time points. When the time series consists of completely independent data, all the cross-lag correlations are zero. The corresponding ACM is thus the same as the one for the usual cross-sectional data with many participants (Nel, [<reflink idref="bib31" id="ref131">31</reflink>]). Our adaptation is most needed when there are substantial cross-lag correlations in the multivariate time series.</p> <p>Both the standard error estimates and the test statistic are asymptotic methods, and their performances are more satisfactory with longer time series. The confidence intervals had satisfactory empirical coverage rates for time series of 100-time points or more, and the test statistics were close to their reference</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>χ</mi></mrow><mn>2</mn></msup></mrow></math> </ephtml> distributions for time series of 200-time points or more. Collecting long time series can be challenging if participants need to respond repeatedly to a survey, but new data collection methods can help collect certain types of time series. For example, Ram et al. ([<reflink idref="bib34" id="ref132">34</reflink>]) took a screenshot of participants' smart phones every 5 s for between 34 and 105 days, which corresponds to half a million to one and half million-time points.</p> <p>There was a debate on how the amounts of measurement error influenced the interpretation of the circumplex model in affect research (Green et al., [<reflink idref="bib17" id="ref133">17</reflink>]; Watson & Clark, [<reflink idref="bib42" id="ref134">42</reflink>]). The presence of measurement error attenuates the correlation between certain affects, which could lead to different theories of affect. In the simulation study, we considered three levels of measurement error (1 – communality). The empirical coverage rates of the confidence intervals are close to 95% at all three levels of measurement error when the time series length is 100 or longer. Because the circumplex model explicitly accounts for the influence of measurement error on variable correlations, the correlations between the common parts of the variables provide 'error-free' measures of how these constructs are related. The influences of measurement error on the overall quality of the inferences made with the circumplex model are similar to the influences of measurement error in factor analysis (MacCallum et al., [<reflink idref="bib27" id="ref135">27</reflink>]; Tucker et al., [<reflink idref="bib41" id="ref136">41</reflink>]). Higher levels of measurement error correspond to less information about the common parts. Thus, larger samples are needed to achieve the same amount of information about the common parts. The simulation study confirmed this conjecture. The empirical coverage rates of confidence intervals at the low communality conditions (high levels of measurement error) deviate from 95% more substantially than those at the high and wide communality conditions when the time series length is 50 and 100. Similarly, the test statistics at the low communality conditions are further away from the reference distributions than those at the high and wide communality conditions when the time series length is 200 or shorter.</p> <p>The circumplex model with multivariate time series developed in the present article is an idiographic method to study the pattern unique to each individual. After the establishment of these personal circumplex models for multiple individuals, researchers can examine inter-individual differences in the parameters that characterize the intra-individual circumplex structure of the personal multivariate time series. One could relate these personal parameters to some stable traits. For example, one could investigate how extraversion or neuroticism relates to the parameters that characterize an individual's personal circumplex structure of affect. The efforts to extend this personal circumplex model to multiple individuals represent the integration of an idiographic method and a nomothetic method. When estimating the circumplex model with multiple individuals, we need to require that the cosine weights be invariant to make the angle parameters comparable across different individuals.</p> <p>There are many psychological theories that propose a circular representation of important constructs (Conte & Plutchik, [<reflink idref="bib13" id="ref137">13</reflink>]; Watson & Clark, [<reflink idref="bib43" id="ref138">43</reflink>]; Wiggins, [<reflink idref="bib45" id="ref139">45</reflink>]). Although researchers are interested in studying rapidly changing intra-individual variability within the framework of these theories, collecting multivariate time series data suitable for addressing these questions has been difficult until recently. The time is ripe for taking this idiographic or person-centered perspective due to the availability of new and affordable methods (e.g. smartphone applications and wearable sensors) for collecting multivariate time series data. We develop the circumplex model with time series data to contribute to this research endeavor. The circumplex model with multivariate time series has been implemented in a user-friendly R package (Lee, [<reflink idref="bib23" id="ref140">23</reflink>]).</p> <hd id="AN0176985635-22">Appendix A</hd> <p></p> <hd id="AN0176985635-23">R Code for Empirical Illustration</hd> <p># Install the R package "devtools" to download "CircumO" from GitHub.</p> <p>install.packages("devtools"); library(devtools)</p> <p># Download the R package "CircumO" from GitHub and load it.</p> <p>install_github("Dayoung-L/CircumO"); library(CircumO)</p> <p># A public data set included in the package</p> <p>data("affect") # Time series data from Watson and Clark ([<reflink idref="bib43" id="ref141">43</reflink>]), T = 70</p> <p>mod = CircumO(affect, m = 1, type = "TS")</p> <p># Find more information of the package</p> <p>?CircumO</p> <p>#obtains parameter estimates and standard error estimates</p> <p>mod$coefficients</p> <p>#obtains a p-value for a perfect fit test</p> <p>mod$test.stat$perfect.fit</p> <p>#obtains a p-value for a close fit test</p> <p>mod$test.stat$close.fit</p> <hd id="AN0176985635-24">Appendix B.</hd> <p></p> <hd id="AN0176985635-25">Standard Error and Test Statistics</hd> <p>Browne ([<reflink idref="bib7" id="ref142">7</reflink>]) described a robust standard error estimator (Proposition 2) and a test statistic (Proposition 4) for the usual SEM model with nonnormal cross-sectional data. We now adapt them for the cicumplex models with time series data. The proofs of the adaptations are very similar to those presented in Browne ([<reflink idref="bib7" id="ref143">7</reflink>]). We also assume that (<reflink idref="bib1" id="ref144">1</reflink>) the model is identified, (<reflink idref="bib2" id="ref145">2</reflink>) none of the population parameters are on the boundary, and (<reflink idref="bib3" id="ref146">3</reflink>) the level of model error is comparable to the level of random sampling error.</p> <hd id="AN0176985635-26">B.1 Standard Error</hd> <p>Let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></math> </ephtml> be the parameter estimate that minimizes the OLS discrepancy function of Equation (<reflink idref="bib5" id="ref147">5</reflink>). The gradient function of the OLS discrepancy function is then</p> <p>(<reflink idref="bib12" id="ref148">12</reflink>)</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">g</mi><mo stretchy="false">(</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo>̂</mo></mover></mrow></mrow><mo stretchy="false">)</mo><mo>=</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi mathvariant="bold-italic">γ</mi></mrow></mfrac><msub><mrow><mi>f</mi></mrow><mrow><mtext>OLS</mtext></mrow></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow><mo>′</mo></msubsup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mn>0</mn></msub><mo>−</mo><msub><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></mrow><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="bold-italic">0</mi><mo>.</mo></mrow></math> </ephtml> </p> <p>Here</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></mrow><mn>0</mn></msub></mrow></math> </ephtml> contains the non-duplicated elements of the model-implied variable correlations</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">P</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>.</mo></math> </ephtml> We can express the gradient function as</p> <p>(<reflink idref="bib13" id="ref149">13</reflink>)</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">g</mi><mo stretchy="false">(</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo>̂</mo></mover></mrow></mrow><mo stretchy="false">)</mo><mo>=</mo><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow><mo>′</mo></msubsup><mrow><mo stretchy="true">(</mo><mrow><mo stretchy="true">[</mo><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mn>0</mn></msub><mo>−</mo><msub><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn></msub><mo stretchy="true">]</mo><mo>+</mo><mo stretchy="true">[</mo><msub><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn></msub><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn><mo>*</mo></msubsup><mo stretchy="true">]</mo><mo>−</mo><mo stretchy="true">[</mo><msub><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></mrow><mn>0</mn></msub><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn><mo>*</mo></msubsup><mo stretchy="true">]</mo></mrow><mo stretchy="true">)</mo></mrow><mo>=</mo><mi mathvariant="bold-italic">0</mi><mo>.</mo></mrow></math> </ephtml> </p> <p>Here</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn><mo>*</mo></msubsup></mrow></math> </ephtml> contains the non-duplicated elements of the variable correlation matrix</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">P</mi><mrow><mo stretchy="false">(</mo><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow><mo stretchy="false">)</mo></mrow></mrow></math> </ephtml> that bests approximate</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mn>0</mn></msub></mrow></math> </ephtml> according to the OLS discrepancy function.</p> <p>We multiply Equation (<reflink idref="bib13" id="ref150">13</reflink>) by the square root of the time series length</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>T</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></math> </ephtml> and examine how the three terms change as the time series length</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> </ephtml> increases to a large number. The first term approaches</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow><mo>′</mo></msubsup><msup><mrow><mi>T</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mn>0</mn></msub><mo>−</mo><msub><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> because the parameter estimates</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></math> </ephtml> are consistent estimates of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow><mo>.</mo></math> </ephtml> The second term approaches</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow><mo>′</mo></msubsup><msup><mrow><mi>T</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn></msub><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn><mo>*</mo></msubsup><mo stretchy="false">)</mo></mrow></math> </ephtml> because the parameter estimates</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></math> </ephtml> are consistent estimates of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow><mo>.</mo></math> </ephtml> When the level of model error is comparable to the level of random sampling error,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>T</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn></msub><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn><mo>*</mo></msubsup><mo stretchy="false">)</mo></mrow></math> </ephtml> is bounded. Because</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow></math> </ephtml> minimizes the OLS discrepancy function between</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">P</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></mrow><mn>0</mn></msub><mtext> and </mtext><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mn>0</mn></msub></mrow><mo>,</mo></math> </ephtml> the second term corresponds to the gradient function and it approaches <bold>0</bold>. We apply the mean value theorem to</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msub><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></mrow><mn>0</mn></msub><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn><mo>*</mo></msubsup><mo stretchy="false">)</mo></mrow></math> </ephtml> </p> <p>(<reflink idref="bib14" id="ref151">14</reflink>)</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></mrow><mn>0</mn></msub><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn><mo>*</mo></msubsup><mo>=</mo><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></msub><mo stretchy="false">(</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>−</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup><mo stretchy="false">)</mo><mo>.</mo></mrow></math> </ephtml> </p> <p>Here</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></math> </ephtml> is a point that is in the line segment connecting</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mtext> and </mtext><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow><mo>.</mo></math> </ephtml> The third term approaches</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow><mo>′</mo></msubsup><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow></msub><mo stretchy="false">(</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>−</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup><mo stretchy="false">)</mo></mrow><mo>,</mo></math> </ephtml> because the parameter estimates</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></math> </ephtml> are consistent estimates of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow><mo>.</mo></math> </ephtml> </p> <p>We rearrange the first and the third term to</p> <p>(<reflink idref="bib15" id="ref152">15</reflink>)</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow><mo>′</mo></msubsup><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow></msub><msup><mrow><mi>T</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo stretchy="false">(</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>−</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup><mo stretchy="false">)</mo><mo>=</mo><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow><mo>′</mo></msubsup><msup><mrow><mi>T</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mn>0</mn></msub><mo>−</mo><msub><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn></msub><mo stretchy="false">)</mo><mo>.</mo></mrow></math> </ephtml> </p> <p>Note that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>T</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo stretchy="false">(</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>−</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup><mo stretchy="false">)</mo></mrow></math> </ephtml> is the asymptotic distribution of the parameter estimates</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mtext> and </mtext><msup><mrow><mi>T</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mn>0</mn></msub><mo>−</mo><msub><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> is the asymptotic distribution of sample correlations.</p> <p>An algebra manipulation of Equation (<reflink idref="bib15" id="ref153">15</reflink>) leads to Equation (<reflink idref="bib7" id="ref154">7</reflink>), which is the asymptotic covariance matrix of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow><mo>.</mo></math> </ephtml> </p> <hd id="AN0176985635-27">The Test Statistic</hd> <p>To adapt Proposition 4 of Browne ([<reflink idref="bib7" id="ref155">7</reflink>]) for the circumplex model with time series data, we define two additional matrices</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></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 mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></msub></mrow></math> </ephtml> in a way similar to</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow></msub></mrow></math> </ephtml> in Equation (<reflink idref="bib8" id="ref156">8</reflink>). We use</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>,</mo><mi>c</mi></mrow></msub><mo>,</mo><mtext /><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow><mo>,</mo><mi>c</mi></mrow></msub></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">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup><mo>,</mo><mi>c</mi></mrow></msub></mrow></math> </ephtml> to denote the matrices that spann the null spaces of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></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 mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></msub></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">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow></msub></mrow><mo>,</mo></math> </ephtml> respectively.</p> <p>Similar to Equation (<reflink idref="bib13" id="ref157">13</reflink>), we express</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">e</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mn>0</mn></msub><mo>−</mo><msub><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></mrow><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> in the following equation</p> <p>(<reflink idref="bib16" id="ref158">16</reflink>)</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>T</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>,</mo><mi>c</mi></mrow><mo>′</mo></msubsup><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">e</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>=</mo><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>,</mo><mi>c</mi></mrow><mo>′</mo></msubsup><mrow><mo stretchy="true">(</mo><mrow><msup><mrow><mi>T</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo stretchy="true">[</mo><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mn>0</mn></msub><mo>−</mo><msub><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn></msub><mo stretchy="true">]</mo><mo>+</mo><msup><mrow><mi>T</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo stretchy="true">[</mo><msub><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn></msub><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn><mo>*</mo></msubsup><mo stretchy="true">]</mo><mo>−</mo><msup><mrow><mi>T</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo stretchy="true">[</mo><msub><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></mrow><mn>0</mn></msub><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn><mo>*</mo></msubsup><mo stretchy="true">]</mo></mrow><mo stretchy="true">)</mo></mrow><mo>.</mo></mrow></math> </ephtml> </p> <p>We again examine the three terms as</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> </ephtml> increases to a large number. The first term</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>T</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mn>0</mn></msub><mo>−</mo><msub><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn></msub><mo stretchy="false">]</mo></mrow></math> </ephtml> is the asymptotic distribution of lag 0 correlations, which are multivariate normal with a null mean vector and the covariance matrix of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Υ</mi><mo>.</mo></math> </ephtml> The second term</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>T</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn></msub><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn><mo>*</mo></msubsup><mo stretchy="false">]</mo></mrow></math> </ephtml> is the population model error, which is bounded according to the assumption that the level of model error is comparable to the level of random sampling error. We denote the population error using</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">e</mi><mo>.</mo></math> </ephtml> We again apply the mean value theorem to the third term,</p> <p>(<reflink idref="bib17" id="ref159">17</reflink>)</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>T</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo stretchy="true">[</mo><msub><mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">ρ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></mrow><mn>0</mn></msub><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">ρ</mi></mrow><mn>0</mn><mo>*</mo></msubsup><mo stretchy="true">]</mo><mo>=</mo><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></msub><mrow><mo stretchy="true">(</mo><mrow><msup><mrow><mi>T</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo stretchy="true">[</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>−</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup><mo stretchy="true">]</mo></mrow><mo stretchy="true">)</mo></mrow></mrow></math> </ephtml> </p> <p>Note that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>T</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo stretchy="false">[</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>−</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup><mo stretchy="false">]</mo></mrow></math> </ephtml> is the asymptotic covariance matrix of the parameter estimates. It is bounded according to Equation (<reflink idref="bib7" id="ref160">7</reflink>) as the time series length increases. Therefore, the third term vanishes</p> <p>(<reflink idref="bib18" id="ref161">18</reflink>)</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>,</mo><mi>c</mi></mrow><mo>′</mo></msubsup><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></msub><mrow><mo stretchy="true">(</mo><mrow><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo stretchy="true">[</mo><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>−</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup><mo stretchy="true">]</mo></mrow><mo stretchy="true">)</mo></mrow><mo>→</mo><mi mathvariant="bold-italic">0</mi><mo>,</mo></mrow></math> </ephtml> </p> <p>since</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>,</mo><mi>c</mi></mrow></msub><mtext> approaches </mtext><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup><mo>,</mo><mi>c</mi></mrow></msub><mtext> and </mtext><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">˜</mo></mover></mrow></mrow></mrow></msub></mrow></math> </ephtml> goes to</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup></mrow></msub></mrow><mo>.</mo></math> </ephtml> </p> <p>The asymptotic distribution of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>T</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo stretchy="true">̂</mo></mover></mrow></mrow><mo>,</mo><mi>c</mi></mrow><mo>′</mo></msubsup><mrow><mrow><mover accent="true"><mi mathvariant="bold-italic">e</mi><mo stretchy="true">̂</mo></mover></mrow></mrow></mrow></math> </ephtml> of Equation (<reflink idref="bib16" id="ref162">16</reflink>) is multivariate normal with the mean vector</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>T</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup><mo>,</mo><mi>c</mi></mrow></msub><mi mathvariant="bold-italic">e</mi></mrow></math> </ephtml> and the covariance matrix</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup><mo>,</mo><mi>c</mi></mrow><mo>′</mo></msubsup><mi mathvariant="bold">Υ</mi><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup><mo>,</mo><mi>c</mi></mrow></msub></mrow><mo>.</mo></math> </ephtml> Thus, the test statistic of Equation (<reflink idref="bib9" id="ref163">9</reflink>) is a non-central chisquare distribution with the degrees of freedom as the rank of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup><mo>,</mo><mi>c</mi></mrow></msub></mrow></math> </ephtml> and the noncentrality parameter of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>T</mi><mi mathvariant="bold-italic">e</mi><mo>′</mo><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup><mo>,</mo><mi>c</mi></mrow></msub><msup><mrow><mrow><mrow><mo stretchy="true">{</mo><mrow><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup><mo>,</mo><mi>c</mi></mrow><mo>′</mo></msubsup><mi mathvariant="bold">Υ</mi><msub><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup><mo>,</mo><mi>c</mi></mrow></msub></mrow><mo stretchy="true">}</mo></mrow></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msubsup><mrow><mi mathvariant="bold">Δ</mi></mrow><mrow><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mo>*</mo></msup><mo>,</mo><mi>c</mi></mrow><mo>′</mo></msubsup><mi mathvariant="bold-italic">e</mi></mrow><mo>.</mo></math> </ephtml> </p> <ref id="AN0176985635-28"> <title> Footnotes </title> <blist> <bibl id="bib1" idref="ref14" type="bt">1</bibl> <bibtext> An idiographic approach is defined to "involve the thorough, intensive study of a single person or case in order to obtain an in-depth understanding of that person or case, as contrasted with a study of the universal aspects of groups of people or cases." (APA Dictionary of Psychology, n.[1].)</bibtext> </blist> <blist> <bibl id="bib2" idref="ref9" type="bt">2</bibl> <bibtext> Molenaar ([29]) defined ergodic process as "a process in which the structures of intraindividual variation and interindividual variation are (asymptotically) equivalent."</bibtext> </blist> <blist> <bibl id="bib3" idref="ref32" type="bt">3</bibl> <bibtext> Because one variable is chosen as the reference variable, its angle is fixed as</bibtext> </blist> <blist> <bibtext>Graph</bibtext> </blist> <blist> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mn>0</mn></mrow><mo>°</mo></msup></mrow><mo>.</mo></math> </ephtml> Thus, the model involves only <emph>p</emph> − 1 angles. Because</bibtext> </blist> <blist> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib4" idref="ref37" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>θ</mi></mrow><mi>j</mi></msub><mo>−</mo><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub><mo>=</mo><mn>0</mn></mrow></math> </ephtml> implies a correlation of 1,</bibtext> </blist> <blist> <bibl id="bib5" idref="ref44" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib6" idref="ref53" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>β</mi></mrow><mn>0</mn></msub><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><mrow><msub><mrow><mi>β</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math> </ephtml> We can compute <emph>β</emph><subs>0</subs> from other weights.</bibtext> </blist> <blist> <bibtext> We present a sketch of the proof for the adaptation in Appendix B.</bibtext> </blist> <blist> <bibtext> Details of the derivatives were described by Lee and Zhang ([24]).</bibtext> </blist> <blist> <bibtext> We present a sketch of the proof for the adaptation in Appendix B.</bibtext> </blist> <blist> <bibl id="bib7" idref="ref20" type="bt">7</bibl> <bibtext> We thank David Watson for sharing the data.</bibtext> </blist> <blist> <bibl id="bib8" idref="ref18" type="bt">8</bibl> <bibtext> Watson et al. ([44], p. 824) originally designed the 60 items to measure 8 affects, but "disengagement" was not assessed in the within-subject situations. Indicators of high positive affect are enthusiastic, interested, determined, excited, inspired, alert, active, strong, proud, and attentive; indicators of high negative affect are scared, afraid, upset, distressed, jittery, nervous, ashamed, guilty, irritable, and hostile; indicators of low positive affect are sleepy, tired, sluggish, and drowsy; indicators of low negative affect are calm, relaxed, and at ease; indicators of pleasantness are happy, joyful, cheerful, and delighted; indicators of unpleasantness are sad, blue, downhearted, alone, and lonely; and indicators of engagement are surprised, amazed, and astonished.</bibtext> </blist> <blist> <bibl id="bib9" idref="ref65" type="bt">9</bibl> <bibtext> The appendix contains R code for the illustration.</bibtext> </blist> <blist> <bibtext> We present common score correlations (</bibtext> </blist> <blist> <bibtext>Graph</bibtext> </blist> <blist> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mi>c</mi></msub></mrow></math> </ephtml> ) of both models in an online support file (Figures A1 and A2).</bibtext> </blist> <blist> <bibtext> We assume that the time series is weakly stationary (Brockwell & Davis, [6], Definition (1.3.3)). Thus, the within-subject correlations are invariant across different time points. More sophisticated methods (Hamilton, [20]) are needed if the stationarity assumption seems inappropriate. The vector AR process is a simple way to simulate a stationary time series. Because the proposed method is valid for any stationary process, we use the simulation study to confirm a theoretical expectation. We expect that the general results will hold if we simulate stationary time series with other methods (e.g. more complex AR weight matrices, higher AR orders, with a moving average process).</bibtext> </blist> <blist> <bibtext> Supplemental data for this article can be accessed online at https://doi.org/10.1080/10705511.2023.2259105.</bibtext> </blist> </ref> <ref id="AN0176985635-29"> <title> References </title> <blist> <bibtext> APA dictionary of psychology. (n.d.). Retrieved from https://dictionary.apa.org/idiographic</bibtext> </blist> <blist> <bibtext> Becker, W. C., & Krug, R. S. (1964). 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  Data: Circumplex Models with Multivariate Time Series: An Idiographic Approach
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  Data: <searchLink fieldCode="AR" term="%22Dayoung+Lee%22">Dayoung Lee</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0001-7543-7096">0000-0001-7543-7096</externalLink>)<br /><searchLink fieldCode="AR" term="%22Guangjian+Zhang%22">Guangjian Zhang</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0001-8279-7313">0000-0001-8279-7313</externalLink>)<br /><searchLink fieldCode="AR" term="%22Shanhong+Luo%22">Shanhong Luo</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-0022-8967">0000-0002-0022-8967</externalLink>)
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  Data: <searchLink fieldCode="SO" term="%22Structural+Equation+Modeling%3A+A+Multidisciplinary+Journal%22"><i>Structural Equation Modeling: A Multidisciplinary Journal</i></searchLink>. 2024 31(3):498-510.
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  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
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  Data: 13
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  Data: <searchLink fieldCode="DE" term="%22Research+Methodology%22">Research Methodology</searchLink><br /><searchLink fieldCode="DE" term="%22Affective+Measures%22">Affective Measures</searchLink><br /><searchLink fieldCode="DE" term="%22Family+Relationship%22">Family Relationship</searchLink><br /><searchLink fieldCode="DE" term="%22Multivariate+Analysis%22">Multivariate Analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Time+Management%22">Time Management</searchLink><br /><searchLink fieldCode="DE" term="%22Individual+Testing%22">Individual Testing</searchLink>
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  Data: 10.1080/10705511.2023.2259105
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  Data: 1070-5511<br />1532-8007
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  Data: The circumplex model posits a circular representation of affect and some personality traits. There is an increasing need to examine the viability of the circumplex model with multivariate time series data collected on the same individuals due to the development of new data collection methods such as smartphone applications and wearable sensors. Estimating the circumplex model with time series data is more complex than with cross-sectional data because scores at nearby time points tend to be correlated. We adapt Browne's circumplex model to accommodate time series data. We illustrate the proposed method with an empirical data set of daily affect ratings of an individual over 70 days. We conducted a simulation study to explore the statistical properties of the proposed method. The results show that the method provides more satisfactory confidence intervals and test statistics than a method that treats time series data as if they were cross-sectional data.
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