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<!DOCTYPE html>
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<title>Latent Interaction (and Moderation) Analysis in Structural Equation Models (SEM) • modsem</title>
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<meta name="description" content='Estimation of interaction (i.e., moderation) effects between latent variables in structural equation models (SEM). The supported methods are: The constrained approach (Algina & Moulder, 2001). The unconstrained approach (Marsh et al., 2004). The residual centering approach (Little et al., 2006). The double centering approach (Lin et al., 2010). The latent moderated structural equations (LMS) approach (Klein & Moosbrugger, 2000). The quasi-maximum likelihood (QML) approach (Klein & Muthén, 2007) (temporarily unavailable) The constrained- unconstrained, residual- and double centering- approaches are estimated via lavaan (Rosseel, 2012), whilst the LMS- and QML- approaches are estimated via by modsem it self. Alternatively model can be estimated via Mplus (Muthén & Muthén, 1998-2017). References: Algina, J., & Moulder, B. C. (2001). <doi:10.1207/S15328007SEM0801_3>. "A note on estimating the Jöreskog-Yang model for latent variable interaction using LISREL 8.3." Klein, A., & Moosbrugger, H. (2000). <doi:10.1007/BF02296338>. "Maximum likelihood estimation of latent interaction effects with the LMS method." Klein, A. G., & Muthén, B. O. (2007). <doi:10.1080/00273170701710205>. "Quasi-maximum likelihood estimation of structural equation models with multiple interaction and quadratic effects." Lin, G. C., Wen, Z., Marsh, H. W., & Lin, H. S. (2010). <doi:10.1080/10705511.2010.488999>. "Structural equation models of latent interactions: Clarification of orthogonalizing and double-mean-centering strategies." Little, T. D., Bovaird, J. A., & Widaman, K. F. (2006). <doi:10.1207/s15328007sem1304_1>. "On the merits of orthogonalizing powered and product terms: Implications for modeling interactions among latent variables." Marsh, H. W., Wen, Z., & Hau, K. T. (2004). <doi:10.1037/1082-989X.9.3.275>. "Structural equation models of latent interactions: evaluation of alternative estimation strategies and indicator construction." Muthén, L.K. and Muthén, B.O. (1998-2017). "Mplus User’s Guide. Eighth Edition." <https://www.statmodel.com/>. Rosseel Y (2012). <doi:10.18637/jss.v048.i02>. "lavaan: An R Package for Structural Equation Modeling."'>
<meta property="og:description" content='Estimation of interaction (i.e., moderation) effects between latent variables in structural equation models (SEM). The supported methods are: The constrained approach (Algina & Moulder, 2001). The unconstrained approach (Marsh et al., 2004). The residual centering approach (Little et al., 2006). The double centering approach (Lin et al., 2010). The latent moderated structural equations (LMS) approach (Klein & Moosbrugger, 2000). The quasi-maximum likelihood (QML) approach (Klein & Muthén, 2007) (temporarily unavailable) The constrained- unconstrained, residual- and double centering- approaches are estimated via lavaan (Rosseel, 2012), whilst the LMS- and QML- approaches are estimated via by modsem it self. Alternatively model can be estimated via Mplus (Muthén & Muthén, 1998-2017). References: Algina, J., & Moulder, B. C. (2001). <doi:10.1207/S15328007SEM0801_3>. "A note on estimating the Jöreskog-Yang model for latent variable interaction using LISREL 8.3." Klein, A., & Moosbrugger, H. (2000). <doi:10.1007/BF02296338>. "Maximum likelihood estimation of latent interaction effects with the LMS method." Klein, A. G., & Muthén, B. O. (2007). <doi:10.1080/00273170701710205>. "Quasi-maximum likelihood estimation of structural equation models with multiple interaction and quadratic effects." Lin, G. C., Wen, Z., Marsh, H. W., & Lin, H. S. (2010). <doi:10.1080/10705511.2010.488999>. "Structural equation models of latent interactions: Clarification of orthogonalizing and double-mean-centering strategies." Little, T. D., Bovaird, J. A., & Widaman, K. F. (2006). <doi:10.1207/s15328007sem1304_1>. "On the merits of orthogonalizing powered and product terms: Implications for modeling interactions among latent variables." Marsh, H. W., Wen, Z., & Hau, K. T. (2004). <doi:10.1037/1082-989X.9.3.275>. "Structural equation models of latent interactions: evaluation of alternative estimation strategies and indicator construction." Muthén, L.K. and Muthén, B.O. (1998-2017). "Mplus User’s Guide. Eighth Edition." <https://www.statmodel.com/>. Rosseel Y (2012). <doi:10.18637/jss.v048.i02>. "lavaan: An R Package for Structural Equation Modeling."'>
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<a class="navbar-brand me-2" href="index.html">modsem</a>
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<li><a class="dropdown-item" href="articles/customizing.html">customizing interaction terms</a></li>
<li><a class="dropdown-item" href="articles/interaction_two_etas.html">interaction effects between endogenous variables</a></li>
<li><a class="dropdown-item" href="articles/lavaan.html">using lavaan functions</a></li>
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<main id="main" class="col-md-9"><div class="section level1">
<div class="page-header"><h1 id="modsem-">
<code>modsem</code> <img src="reference/figures/modsem.png" alt="Logo" align="right" height="139" class="logo"><a class="anchor" aria-label="anchor" href="#modsem-"></a>
</h1></div>
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</div>
<div class="section level1">
<h1 id="to-install">To Install<a class="anchor" aria-label="anchor" href="#to-install"></a>
</h1>
<pre><code><span><span class="co"># From CRAN</span></span>
<span><span class="fu"><a href="https://rdrr.io/r/utils/install.packages.html" class="external-link">install.packages</a></span><span class="op">(</span><span class="st">"modsem"</span><span class="op">)</span></span>
<span></span>
<span><span class="co"># Latest version from GitHub</span></span>
<span><span class="fu"><a href="https://rdrr.io/r/utils/install.packages.html" class="external-link">install.packages</a></span><span class="op">(</span><span class="st">"devtools"</span><span class="op">)</span></span>
<span><span class="fu">devtools</span><span class="fu">::</span><span class="fu"><a href="https://remotes.r-lib.org/reference/install_github.html" class="external-link">install_github</a></span><span class="op">(</span><span class="st">"kss2k/modsem"</span>, build_vignettes <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre>
</div>
<div class="section level1">
<h1 id="methodsapproaches">Methods/Approaches<a class="anchor" aria-label="anchor" href="#methodsapproaches"></a>
</h1>
<p>There are a number of approaches for estimating interaction effects in SEM. In <code><a href="reference/modsem.html">modsem()</a></code>, the <code>method = "method"</code> argument allows you to choose which to use. Different approaches can be categorized into two groups: Product Indicator (PI) and Distribution Analytic (DA) approaches.</p>
<div class="section level2">
<h2 id="product-indicator-pi-approaches">Product Indicator (PI) Approaches:<a class="anchor" aria-label="anchor" href="#product-indicator-pi-approaches"></a>
</h2>
<ul>
<li>
<code>"ca"</code> = constrained approach (Algina & Moulder, 2001)
<ul>
<li>Note that constraints can become quite complicated for complex models, particularly when there is an interaction including enodgenous variables. The method can therefore be quite slow.</li>
</ul>
</li>
<li>
<code>"uca"</code> = unconstrained approach (Marsh, 2004)</li>
<li>
<code>"rca"</code> = residual centering approach (Little et al., 2006)</li>
<li>
<code>"dblcent"</code> = double centering approach (Marsh., 2013)
<ul>
<li>default</li>
</ul>
</li>
<li>
<code>"pind"</code> = basic product indicator approach (not recommended)</li>
</ul>
</div>
<div class="section level2">
<h2 id="distribution-analytic-da-approaches">Distribution Analytic (DA) Approaches<a class="anchor" aria-label="anchor" href="#distribution-analytic-da-approaches"></a>
</h2>
<ul>
<li>
<code>"lms"</code> = The Latent Moderated Structural equations (LMS) approach, see the <a href="https://kss2k.github.io/modsem/articles/lms_qml.html" class="external-link">vignette</a>
</li>
<li>
<code>"qml"</code> = The Quasi Maximum Likelihood (QML) approach, see the <a href="https://kss2k.github.io/modsem/articles/lms_qml.html" class="external-link">vignette</a>
</li>
<li>
<code>"mplus"</code>
<ul>
<li>estimates model through Mplus, if it is installed</li>
</ul>
</li>
</ul>
</div>
</div>
<div class="section level1">
<h1 id="examples">Examples<a class="anchor" aria-label="anchor" href="#examples"></a>
</h1>
<div class="section level2">
<h2 id="elementary-interaction-model-kenny--judd-1984">Elementary Interaction Model (Kenny & Judd, 1984)<a class="anchor" aria-label="anchor" href="#elementary-interaction-model-kenny--judd-1984"></a>
</h2>
<pre><code><span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va"><a href="https://github.com/Kss2k/modsem" class="external-link">modsem</a></span><span class="op">)</span></span>
<span><span class="va">m1</span> <span class="op"><-</span> <span class="st">'</span></span>
<span><span class="st"> # Outer Model</span></span>
<span><span class="st"> X =~ x1 + x2 +x3</span></span>
<span><span class="st"> Y =~ y1 + y2 + y3</span></span>
<span><span class="st"> Z =~ z1 + z2 + z3</span></span>
<span><span class="st"></span></span>
<span><span class="st"> # Inner model</span></span>
<span><span class="st"> Y ~ X + Z + X:Z</span></span>
<span><span class="st">'</span></span>
<span></span>
<span><span class="co"># Double centering approach</span></span>
<span><span class="va">est1_dca</span> <span class="op"><-</span> <span class="fu"><a href="reference/modsem.html">modsem</a></span><span class="op">(</span><span class="va">m1</span>, <span class="va">oneInt</span><span class="op">)</span></span>
<span><span class="fu"><a href="https://rdrr.io/r/base/summary.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">est1_dca</span><span class="op">)</span></span>
<span></span>
<span><span class="co"># Constrained approach</span></span>
<span><span class="va">est1_ca</span> <span class="op"><-</span> <span class="fu"><a href="reference/modsem.html">modsem</a></span><span class="op">(</span><span class="va">m1</span>, <span class="va">oneInt</span>, method <span class="op">=</span> <span class="st">"ca"</span><span class="op">)</span></span>
<span><span class="fu"><a href="https://rdrr.io/r/base/summary.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">est1_ca</span><span class="op">)</span></span>
<span></span>
<span><span class="co"># QML approach</span></span>
<span><span class="va">est1_qml</span> <span class="op"><-</span> <span class="fu"><a href="reference/modsem.html">modsem</a></span><span class="op">(</span><span class="va">m1</span>, <span class="va">oneInt</span>, method <span class="op">=</span> <span class="st">"qml"</span><span class="op">)</span></span>
<span><span class="fu"><a href="https://rdrr.io/r/base/summary.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">est1_qml</span>, standardized <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
<span></span>
<span><span class="co"># LMS approach</span></span>
<span><span class="va">est1_lms</span> <span class="op"><-</span> <span class="fu"><a href="reference/modsem.html">modsem</a></span><span class="op">(</span><span class="va">m1</span>, <span class="va">oneInt</span>, method <span class="op">=</span> <span class="st">"lms"</span><span class="op">)</span></span>
<span><span class="fu"><a href="https://rdrr.io/r/base/summary.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">est1_lms</span><span class="op">)</span></span></code></pre>
</div>
<div class="section level2">
<h2 id="theory-of-planned-behavior">Theory Of Planned Behavior<a class="anchor" aria-label="anchor" href="#theory-of-planned-behavior"></a>
</h2>
<pre><code><span><span class="va">tpb</span> <span class="op"><-</span> <span class="st">"</span></span>
<span><span class="st"># Outer Model (Based on Hagger et al., 2007)</span></span>
<span><span class="st"> ATT =~ att1 + att2 + att3 + att4 + att5</span></span>
<span><span class="st"> SN =~ sn1 + sn2</span></span>
<span><span class="st"> PBC =~ pbc1 + pbc2 + pbc3</span></span>
<span><span class="st"> INT =~ int1 + int2 + int3</span></span>
<span><span class="st"> BEH =~ b1 + b2</span></span>
<span><span class="st"></span></span>
<span><span class="st"># Inner Model (Based on Steinmetz et al., 2011)</span></span>
<span><span class="st"> INT ~ ATT + SN + PBC</span></span>
<span><span class="st"> BEH ~ INT + PBC</span></span>
<span><span class="st"> BEH ~ PBC:INT</span></span>
<span><span class="st">"</span></span>
<span></span>
<span><span class="co"># double centering approach</span></span>
<span><span class="va">est_tpb_dca</span> <span class="op"><-</span> <span class="fu"><a href="reference/modsem.html">modsem</a></span><span class="op">(</span><span class="va">tpb</span>, data <span class="op">=</span> <span class="va">TPB</span>, method <span class="op">=</span> <span class="st">"dblcent"</span><span class="op">)</span></span>
<span><span class="fu"><a href="https://rdrr.io/r/base/summary.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">est_tpb_dca</span><span class="op">)</span></span>
<span></span>
<span><span class="co"># Constrained approach using Wrigths path tracing rules for generating</span></span>
<span><span class="co"># the appropriate constraints</span></span>
<span><span class="va">est_tpb_ca</span> <span class="op"><-</span> <span class="fu"><a href="reference/modsem.html">modsem</a></span><span class="op">(</span><span class="va">tpb</span>, data <span class="op">=</span> <span class="va">TPB</span>, method <span class="op">=</span> <span class="st">"ca"</span><span class="op">)</span></span>
<span><span class="fu"><a href="https://rdrr.io/r/base/summary.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">est_tpb_ca</span><span class="op">)</span></span>
<span></span>
<span><span class="co"># LMS approach</span></span>
<span><span class="va">est_tpb_lms</span> <span class="op"><-</span> <span class="fu"><a href="reference/modsem.html">modsem</a></span><span class="op">(</span><span class="va">tpb</span>, data <span class="op">=</span> <span class="va">TPB</span>, method <span class="op">=</span> <span class="st">"lms"</span><span class="op">)</span></span>
<span><span class="fu"><a href="https://rdrr.io/r/base/summary.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">est_tpb_lms</span>, standardized <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
<span></span>
<span><span class="co"># QML approach</span></span>
<span><span class="va">est_tpb_qml</span> <span class="op"><-</span> <span class="fu"><a href="reference/modsem.html">modsem</a></span><span class="op">(</span><span class="va">tpb</span>, data <span class="op">=</span> <span class="va">TPB</span>, method <span class="op">=</span> <span class="st">"qml"</span><span class="op">)</span></span>
<span><span class="fu"><a href="https://rdrr.io/r/base/summary.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">est_tpb_qml</span>, standardized <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span></code></pre>
</div>
<div class="section level2">
<h2 id="interactions-between-two-observed-variables">Interactions between two observed variables<a class="anchor" aria-label="anchor" href="#interactions-between-two-observed-variables"></a>
</h2>
<pre><code><span><span class="va">est2</span> <span class="op"><-</span> <span class="fu"><a href="reference/modsem.html">modsem</a></span><span class="op">(</span><span class="st">'y1 ~ x1 + z1 + x1:z1'</span>, data <span class="op">=</span> <span class="va">oneInt</span>, method <span class="op">=</span> <span class="st">"pind"</span><span class="op">)</span></span>
<span><span class="fu"><a href="https://rdrr.io/r/base/summary.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">est2</span><span class="op">)</span></span></code></pre>
</div>
<div class="section level2">
<h2 id="interaction-between-an-obsereved-and-a-latent-variable">Interaction between an obsereved and a latent variable<a class="anchor" aria-label="anchor" href="#interaction-between-an-obsereved-and-a-latent-variable"></a>
</h2>
<pre><code><span><span class="va">m3</span> <span class="op"><-</span> <span class="st">'</span></span>
<span><span class="st"> # Outer Model</span></span>
<span><span class="st"> X =~ x1 + x2 +x3</span></span>
<span><span class="st"> Y =~ y1 + y2 + y3</span></span>
<span><span class="st"></span></span>
<span><span class="st"> # Inner model</span></span>
<span><span class="st"> Y ~ X + z1 + X:z1</span></span>
<span><span class="st">'</span></span>
<span></span>
<span><span class="va">est3</span> <span class="op"><-</span> <span class="fu"><a href="reference/modsem.html">modsem</a></span><span class="op">(</span><span class="va">m3</span>, <span class="va">oneInt</span>, method <span class="op">=</span> <span class="st">"pind"</span><span class="op">)</span></span>
<span><span class="fu"><a href="https://rdrr.io/r/base/summary.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">est3</span><span class="op">)</span></span></code></pre>
</div>
</div>
</main><aside class="col-md-3"><div class="links">
<h2 data-toc-skip>Links</h2>
<ul class="list-unstyled">
<li><a href="https://cloud.r-project.org/package=modsem" class="external-link">View on CRAN</a></li>
<li><a href="https://github.com/Kss2k/modsem/" class="external-link">Browse source code</a></li>
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<div class="license">
<h2 data-toc-skip>License</h2>
<ul class="list-unstyled">
<li><a href="LICENSE.html">Full license</a></li>
<li><small><a href="https://opensource.org/licenses/mit-license.php" class="external-link">MIT</a> + file <a href="LICENSE-text.html">LICENSE</a></small></li>
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<div class="community">
<h2 data-toc-skip>Community</h2>
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<li><a href="CONTRIBUTING.html">Contributing guide</a></li>
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<div class="citation">
<h2 data-toc-skip>Citation</h2>
<ul class="list-unstyled">
<li><a href="authors.html#citation">Citing modsem</a></li>
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<div class="developers">
<h2 data-toc-skip>Developers</h2>
<ul class="list-unstyled">
<li>Kjell Solem Slupphaug <br><small class="roles"> Author, maintainer </small> <a href="https://orcid.org/0009-0005-8324-2834" target="orcid.widget" aria-label="ORCID" class="external-link"><span class="fab fa-orcid orcid" aria-hidden="true"></span></a> </li>
<li><a href="authors.html">More about authors...</a></li>
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<div class="dev-status">
<h2 data-toc-skip>Dev status</h2>
<ul class="list-unstyled">
<li><a href="https://github.com/kss2k/modsem/actions/workflows/checks.yml" class="external-link"><img src="https://github.com/kss2k/modsem/actions/workflows/checks.yml/badge.svg" alt="R-CMD-check"></a></li>
<li><a href="https://github.com/kss2k/modsem/actions/workflows/tests.yml" class="external-link"><img src="https://github.com/kss2k/modsem/actions/workflows/tests.yml/badge.svg" alt="Tests"></a></li>
<li><a href="https://cran.r-project.org/package=modsem" class="external-link"><img src="https://www.r-pkg.org/badges/version/modsem" alt="CRAN"></a></li>
<li><a href="https://github.com/kss2k/modsem/actions/workflows/pkgdown.yml" class="external-link"><img src="https://github.com/kss2k/modsem/actions/workflows/pkgdown.yml/badge.svg" alt="PKGDOWN-Build"></a></li>
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<p>Developed by Kjell Solem Slupphaug.</p>
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