\documentclass{article} \newcommand{\titel}{Open-source modern modeling software:\\ the R package lavaan} %\usepackage{Sweave} \usepackage{fancyvrb} \usepackage{graphicx} \usepackage{color} \usepackage{amsmath} \usepackage{times} \usepackage{sectsty} \usepackage{tikz} \tikzstyle{ov}=[shape=rectangle, draw=black!80, minimum height=0.6cm, minimum width=0.6cm, thick] \tikzstyle{av}=[shape=rectangle, draw=black!80, fill=black!10, minimum height=0.6cm, minimum width=0.6cm, thick] \tikzstyle{lv}=[shape=circle,draw=black!80,thick,minimum width=0.7cm] \fvset{fontfamily=courier} \DefineVerbatimEnvironment{Sinput}{Verbatim} {fontseries=b, fontsize=\scriptsize, xleftmargin=0.2cm} \DefineVerbatimEnvironment{Sinput2}{Verbatim} {fontseries=b, fontsize=\scriptsize, xleftmargin=0.2cm} \DefineVerbatimEnvironment{Routput2}{Verbatim} {fontsize=\tiny, xleftmargin=0.2cm} \DefineVerbatimEnvironment{Routput}{Verbatim} {fontseries=b,fontsize=\scriptsize, xleftmargin=0.2cm} \DefineVerbatimEnvironment{Binput}{Verbatim} {fontseries=b, fontsize=\scriptsize,frame=single, label=\fbox{lavaan model syntax}, framesep=2mm} %\DefineShortVerb{\!} \DefineVerbatimEnvironment{Rinput}{Verbatim} {fontseries=b, fontsize=\scriptsize, frame=single, label=\fbox{R code}, framesep=5mm} \DefineVerbatimEnvironment{Link}{Verbatim} {fontseries=b, fontsize=\small, formatcom=\color{darkgreen}, xleftmargin=1.0cm} \newcommand{\pkg}[1]{{\normalfont\fontseries{b}\selectfont #1}} \let\proglang=\textsf \let\code=\texttt \usepackage{bm} \renewcommand{\v}[1]{\ensuremath{\bm{\mathrm{#1}}}} % colors \definecolor{darkred}{rgb}{.4, .0, .0} \definecolor{darkblue}{rgb}{.0, .0, .4} \definecolor{darkgreen}{rgb}{.0, .3, .0} \definecolor{darkgray}{rgb}{.6, .6, .6} \definecolor{brown}{rgb}{.3, .3, .0} \subsectionfont{\color{darkgreen}} \sectionfont{\color{darkblue}} \subsubsectionfont{\sffamily\color{brown}} \usepackage{geometry} \geometry{paperwidth=12.8cm, paperheight=9.6cm, includeheadfoot, scale={0.90,0.95}, headsep=0.3cm,footskip=0.4cm } \usepackage{fancyhdr} \usepackage{lastpage} \pagestyle{fancy} \renewcommand{\headrulewidth}{0.1mm} \renewcommand{\footrulewidth}{0.1mm} \lhead{\tiny \sffamily Department of Data Analysis} \chead{} \rhead{\tiny \sffamily Ghent University} \lfoot{\tiny \sffamily Yves Rosseel} \cfoot{\tiny \sffamily \textcolor{darkred}{\titel}} \rfoot{\tiny \sffamily \thepage\ / {\normalcolor \pageref{LastPage}}} \usepackage{booktabs} \renewcommand{\arraystretch}{1.2} \author{Yves Rosseel\\ Department of Data Analysis\\ Ghent University -- Belgium} \date{\vspace*{1cm} Modern Modeling Methods Conference -- 21 May 2013\\ University of Connecticut} \title{\textcolor{darkred}{\titel}} \newcommand{\sss}[1]{\subsubsection*{#1}} \newcommand{\lijn}{\mbox{}\\*[0.04\textheight]} \begin{document} \maketitle\thispagestyle{fancy}\newpage %\tableofcontents\newpage \newpage \sss{contents} \begin{itemize} \item part I: \begin{itemize} \item software for modern modeling methods \item software for SEM \item lavaan is for statisticians, teachers and applied users \item lavaan features (and missing features) \item lavaan model syntax \end{itemize} \item part II: \begin{itemize} \item lavaan functions and options \item lavaan and the (computational) history of SEM \item future plans \item discussion/questions \end{itemize} \end{itemize} \newpage \sss{software for modern modeling methods: two approaches} \begin{itemize} \item closed-source software \begin{itemize} \item often highly advanced features \item standalone (not integrated in a larger statistical package) \item black box:\\ no access to internal data, no access to internal functions \item commercial (cost, licensing issues, no information about bugs) \end{itemize} \item open-source software \begin{itemize} \item in some areas, less advanced features \item often integrated (R package, Stata module, Matlab toolbox) \item open: full access to internal data, internal functions \item flexibility: can be extended, modified or embedded in a larger pipeline \item (academic) community \end{itemize} \end{itemize} \newpage \sss{software for SEM: commercial -- closed-source} \begin{itemize} \item the big four: \begin{itemize} \item LISREL \item EQS \item AMOS \item Mplus \end{itemize} \item SAS/Stat: proc CALIS, proc TCALIS \item SEPATH (Statistica), RAMONA (Systat), Stata 12 \item Mx (free, closed-source) \end{itemize} \newpage \sss{software for SEM: non-commercial -- open-source} \begin{itemize} \item outside the R ecosystem: gllamm (Stata module), \ldots \item R packages: \begin{itemize} \item sem \item OpenMx \item lavaan \item lava \end{itemize} \item interfaces between R and commercial packages: \begin{itemize} \item REQS \item MplusAutomation \end{itemize} \end{itemize} \newpage \sss{what is \pkg{lavaan}?} \begin{itemize} \item \pkg{lavaan} is an R package for latent variable analysis \begin{center} \includegraphics{narevaraan.jpg} \end{center} \item the long-term goal of \pkg{lavaan} is to implement all the state-of-the-art capabilities that are currently available in commercial packages \end{itemize} \newpage \sss{installing \pkg{lavaan}, finding documentation} \begin{itemize} \item \pkg{lavaan} depends on the R project for statistical computing: \begin{Link} http://www.r-project.org \end{Link} \item to install \pkg{lavaan}, simply start up an R session and type: \begin{Sinput} > install.packages("lavaan") \end{Sinput} \item more information about \pkg{lavaan}: \begin{Link} http://lavaan.org \end{Link} \item the \pkg{lavaan} paper: \begin{quote} Rosseel (2012). lavaan: an R package for structural equation modeling. {\itshape Journal of Statistical Software}, 48(2), 1--36. \end{quote} %\item \pkg{lavaan} development: %\begin{quote} %\Verb!https://github.com/yrosseel/lavaan! %\end{quote} \item \pkg{lavaan} discussion group (mailing list) \begin{Link} https://groups.google.com/d/forum/lavaan \end{Link} \end{itemize} \newpage \sss{why do we need \pkg{lavaan}?} \begin{enumerate} \item \pkg{lavaan} is for statisticians working in the field of SEM \begin{itemize} \item it seems unfortunate that new developments in this field are hindered by the lack of open source software that researchers can use to implement their newest ideas \end{itemize} \item \pkg{lavaan} is for teachers \begin{itemize} \item teaching these techniques to students was often complicated by the forced choice for one of the commercial packages \end{itemize} \item \pkg{lavaan} is for applied researchers \begin{itemize} \item keep it simple, provide all the features they need \end{itemize} \end{enumerate} \newpage \enlargethispage{2cm} \sss{lavaan is for statisticians working in the field of SEM (1)} \begin{itemize} \item case-study: van de Schoot, R., Hoijtink, H. and Dekovi\'{c}, M. (2010). Testing Inequality Constrained Hypotheses in SEM Models. {\itshape Structural Equation Modeling}, 17, 443-463 \end{itemize} \begin{center} \includegraphics[scale=0.30]{rens2.png} \end{center} \newpage \sss{van de Schoot (1)} \begin{itemize} \item the problem: testing {\itshape informative} hypotheses in a SEM model \[ H_0: \beta_{\text{male}} = \beta_{\text{female}} \quad \text{versus} \quad H_c: \beta_{\text{male}} > \beta_{\text{female}} \quad \text{(Type A)}\] \item van de Schoot et.al.\ (2010) present a frequentist solution: parametric bootstrap of the LRT to obtain a plug-in p-value \item seven steps: \begin{enumerate} \item estimate model under $H_0$ ($\beta_{\text{male}} = \beta_{\text{female}}$), save results \item estimate model under $H_c$ ($\beta_{\text{male}} > \beta_{\text{female}}$), save results \item (manually) compute LRT.obs $(\text{logL}_c - \text{logL}_0)$ \item generate (R=1000) datasets under $H_0$ \item for each generated dataset, estimate $H_0$ and $H_c$ model \item (manually) compute (R=1000) LRT values, \item compute plug-in p-value (proportion of LRTs $\geq$ LRT.obs) \end{enumerate} \end{itemize} \newpage \sss{van de Schoot (2)} \begin{itemize} \item but under the null, the plug-in p-values should be (asymptotically) uniformly distributed; unfortunately, in this setting, this is not the case \item van de Schoot et.al.\ (2010) use a `double bootstrap' procedure to calibrate the alpha ($\alpha$) level: \begin{itemize} \item each `bootstrap sample' is again `bootstrapped' (R2=1000) \item we end up with R plug-in p-values, and the 5th percentile provides $\alpha^{\star}$ (the calibrated alpha) \item we should use $\alpha^{\star}$ instead of the regular $\alpha = 0.05$ \end{itemize} \item challenge: try to do this using black box commercial SEM software \ldots \begin{itemize} \item van de Schoot et.al.\ (2010) used Mplus, combined with a handful of custom R scripts (MplusAutomation was not available yet) \end{itemize} \end{itemize} \newpage \sss{van de Schoot (3)} \begin{itemize} \item what if we could do this in R? \item Leonard Vanbrabant wrote a high-level function to do all this in one simple step (available in \pkg{lavaan}) \item code snippet: \begin{Sinput} # simple regression model model <- ' anti ~ b1*pos + b2*neg + b3*dis ' # inequality constraints (Hc) constraints <- ' b1 < b3 b2 < b3 ' # compute plug-in p-value based on the double-bootstrap procedure out <- InformativeTesting(model, data = social, verbose = TRUE, constraints = constraints, R = 1000, type = "parametric", double.bootstrap = "standard") \end{Sinput} \end{itemize} \newpage \sss{informativeTesting output} \begin{Routput} Variable names in model : anti pos neg dis Number of variables : 4 Number of groups : 1 Used sample size per group : 596 Used sample size : 596 Total sample size : 603 Estimator : ML Missing data : listwise Bootstrap method : parametric Double bootstrap method : standard Order Constrained Hypothesis Testing: Type A Type B ------------------------------------------------- LR statistic 47.98 00.05 P-value 0.00 0.43 Adjusted alpha 0.05 0.04 Adjusted p-value 0.00 0.79 Alpha 0.05 0.05 Significance Significant Non-significant \end{Routput} \newpage \enlargethispage{2cm} \sss{informativeTesting plot} \vspace*{-0.5cm} \begin{center} \includegraphics[scale=0.45]{leonard.pdf} \end{center} \newpage \enlargethispage{2cm} \sss{lavaan is for statisticians working in the field of SEM (2)} \begin{itemize} \item case-study: Myrsini Katsikatsou (2012), Ph.D.\ dissertation, Paper 4 \begin{center} \vspace*{-0.3cm} \includegraphics[scale=0.23]{myrsini.png} \vspace*{-0.3cm} \end{center} \item a lot of the infrastructure for the PML estimator is already present in commercial software (but inaccessible) \end{itemize} \newpage \enlargethispage{2cm} \sss{lavaan is for statisticians working in the field of SEM (3)} \begin{itemize} \item case-study: Merkle, E.C.\ and Zeileis, A. (2013). Tests of measurment invariance without subgroups: a generalization of classical methods. {\itshape Psychometrika, 78}, 59--82 \begin{center} \includegraphics[scale=0.18]{merkle2.png} \end{center} \item again, computations were needed (eg.\ case-wise scores) that are commonly computed by commercial packages (but we can not extract the results) \end{itemize} %\newpage %\enlargethispage{2cm} %\sss{lavaan is for statisticians working in the field of SEM (3)} %\begin{itemize} %\item case-study: Merkle, E.C.\ and Zeileis, A. (2013). Tests of measurment %invariance without subgroups: a generalization of classical methods. %{\itshape Psychometrika, 78}, 59--82 %\end{itemize} %\begin{center} %\includegraphics[scale=0.26]{merkle2.png} %\end{center} \newpage \sss{lavaan is for teachers} \begin{itemize} \item case-study: \begin{quote} {\itshape doctoral school SEM course in a Faculty of Psychology and Education Sciences; 40 Ph.D.\ students from 7 different departments; 5 departments had previously bought several licenses for either LISREL(2), EQS(1), AMOS(2) } \end{quote} \begin{itemize} \item which software package should we choose? \item what is the cost? (we prefer the full version) \end{itemize} \item the lavaan syntax is easy to learn \item learning an R package will increase general R skills \item the `mimic' option in lavaan makes a smooth transition possible from lavaan to one of the major commercial programs (and back) \item share your teaching materials: \begin{Link} http://lavaan.ugent.be/resources/teaching.html/ \end{Link} \end{itemize} \newpage \sss{lavaan is for applied researchers} \begin{itemize} \item what matters most to many applied researchers: \begin{itemize} \item the software is easy and intuitive to use \item the software has all the features they want \item results of lavaan are very close, if not identical, to those reported by their current commercial program \end{itemize} \item a 37-pages tutorial should get you started right away \begin{Link} http://lavaan.ugent.be/tutorial/ \end{Link} \item you will be able to replicate your results (say, after ten years) \begin{Link} http://lavaan.ugent.be/history/ \end{Link} \end{itemize} \newpage \sss{features of \pkg{lavaan}} \begin{itemize} \item \pkg{lavaan} is well-tested \item user-friendly fitting functions (cfa, sem, growth) \item power-user fitting function (lavaan) \item support for non-normal continuous data: \begin{itemize} \item robust standard errors, Satorra-Bentler correction, ADF estimation, bootstrapping \end{itemize} \item support for categorical (binary/ordinal) data \begin{itemize} \item \pkg{lavaan} has implemented the three-stage WLS approach as developed by Bengt Muth\'{e}n (1984); including robust variants (aka WLSMV) \end{itemize} \item full support for missing data (fiml), meanstructures, and multiple groups \item linear and non-linear equality and inequality constraints \end{itemize} \newpage \sss{unique features} \begin{itemize} \item default model specification: lavaan model syntax \begin{itemize} \item mplus2lavaan (Michael Hallquist, included in lavaan 0.5-13) \item lisrel2lavaan (Corbin Quick, included in the semTools package) \item graphical (via Onyx) \item \ldots \end{itemize} \item mimic the (numerical) results of commercial packages: \begin{itemize} \item \verb!mimic="Mplus"! \item \verb!mimic="EQS"! \end{itemize} \item new technical features: \begin{itemize} \item informative hypothesis testing (Leonard Vanbrabant) \item pairwise ML for binary/ordinal data (Myrsini Katsikatsou) \item fraction of missing information (Mijke Rhemtulla) \item \ldots \end{itemize} \end{itemize} \newpage \sss{features NOT in \pkg{lavaan} (yet)} \begin{itemize} \item multilevel SEM \item mixture (latent class) SEM \item \ldots \end{itemize} \sss{features we are working on} \begin{itemize} \item Bayesian SEM (BUGS interface, stan interface, native) \item small-sample corrections \item causal inference \item standard errors for standardized parameters \item ML estimation for categorical data (IRT) \item \ldots \item better (technical) documentation \end{itemize} \newpage \sss{the \pkg{lavaan} ecosystem} \begin{itemize} \item \pkg{lavaan.survey} (Daniel Oberski) \begin{quote} survey weights, clustering, strata, and finite sampling corrections in SEM (including support for replicate weights, eg. PISA) \end{quote} \item \pkg{Onyx} (Timo von Oertzen, Andreas M. Brandmaier, Siny Tsang) \begin{quote} interactive graphical interface for SEM (written in Java) \end{quote} \item \pkg{semTools} (Sunthud Pornprasertmanit and many others) \begin{quote} collection of useful functions for SEM \end{quote} \item \pkg{simsem} (Sunthud Pornprasertmanit and many others) \begin{quote} simulation of SEM models \end{quote} \item \pkg{semPlot} (Sacha Epskamp) \begin{quote} visualizations of SEM models \end{quote} \end{itemize} \newpage \enlargethispage{2cm} \sss{semPlot} \vspace*{-0.8cm} \begin{center} \includegraphics[scale=0.49]{semplot.pdf} \end{center} \newpage \sss{a simple regression analysis in R} {\scriptsize \begin{center} \begin{tabular}{p{0.35\textwidth}p{0.59\textwidth}} \hline \begin{center} \begin{tikzpicture}[>=stealth,semithick,scale=0.5] \node[ov] (x1) at (0,0) {$x_{1}$}; \node[ov] (x2) at (0,-2) {$x_{2}$}; \node[ov] (x3) at (0,-4) {$x_{3}$}; \node[ov] (x4) at (0,-6) {$x_{4}$}; \node[av] (y) at (4,-3) {$y$}; \path[->] (x1) edge node[above,scale=0.6] {} (y) (x2) edge node[above,scale=0.6] {} (y) (x3) edge node[above,scale=0.6] {} (y) (x4) edge node[above,scale=0.6] {} (y); \end{tikzpicture} \end{center} & \begin{Sinput} # read in your data myData <- read.csv("c:/temp/myData.csv") # fit model using lm fit <- lm(formula = y ~ x1 + x2 + x3 + x4, data = myData) # show results summary(fit) \end{Sinput} \end{tabular} \end{center} } The standard linear model: \begin{itemize} \item $y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \beta_3 x_{i3} + \beta_4 x_{i4} + \epsilon_i \quad (i=1,2,\ldots,n)$ \end{itemize} \newpage \sss{lm() output artificial data (N=100)} \begin{Routput} Call: lm(formula = y ~ x1 + x2 + x3 + x4, data = myData) Residuals: Min 1Q Median 3Q Max -102.372 -29.458 -3.658 27.275 148.404 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 97.7210 4.7200 20.704 <2e-16 *** x1 5.7733 0.5238 11.022 <2e-16 *** x2 -1.3214 0.4917 -2.688 0.0085 ** x3 1.1350 0.4575 2.481 0.0149 * x4 0.2707 0.4779 0.566 0.5724 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 46.74 on 95 degrees of freedom Multiple R-squared: 0.5911, Adjusted R-squared: 0.5738 F-statistic: 34.33 on 4 and 95 DF, p-value: < 2.2e-16 \end{Routput} {\tiny \begin{Verbatim} # create artificial data set.seed(1) x1 <- rnorm(100) * 10; x2 <- rnorm(100) * 10 x3 <- rnorm(100) * 10; x4 <- rnorm(100) * 10 y <- 100 + 5*x1 + (-2)*x2 + 1*x3 + 0.1*x4 + rnorm(100, sd=40) myData <- data.frame(y,x1,x2,x3,x4) \end{Verbatim} } \newpage \sss{the lavaan model syntax -- a simple regression} {\scriptsize \begin{center} \begin{tabular}{p{0.35\textwidth}p{0.59\textwidth}} \hline \begin{center} \begin{tikzpicture}[>=stealth,semithick,scale=0.5] \node[ov] (x1) at (0,0) {$x_{1}$}; \node[ov] (x2) at (0,-2) {$x_{2}$}; \node[ov] (x3) at (0,-4) {$x_{3}$}; \node[ov] (x4) at (0,-6) {$x_{4}$}; \node[av] (y) at (4,-3) {$y$}; \path[->] (x1) edge node[above,scale=0.6] {} (y) (x2) edge node[above,scale=0.6] {} (y) (x3) edge node[above,scale=0.6] {} (y) (x4) edge node[above,scale=0.6] {} (y); \end{tikzpicture} \end{center} & \begin{Sinput} library(lavaan) myData <- read.csv("c:/temp/myData.csv") myModel <- ' y ~ x1 + x2 + x3 + x4 ' # fit model fit <- sem(model = myModel, data = myData) # show results summary(fit) \end{Sinput} \end{tabular} \end{center} } \begin{itemize} \item to `see' the intercept, use either \begin{Sinput} fit <- sem(model=myModel, data=myData, meanstructure=TRUE) \end{Sinput} or include it explicitly in the syntax: \begin{Sinput} myModel <- ' y ~ 1 + x1 + x2 + x3 + x4 ' \end{Sinput} \end{itemize} \newpage \sss{output (artificial data, N=100)} \begin{Routput} lavaan (0.5-13) converged normally after 1 iterations Number of observations 100 Estimator ML Minimum Function Test Statistic 0.000 Degrees of freedom 0 P-value (Chi-square) 1.000 Parameter estimates: Information Expected Standard Errors Standard Estimate Std.err Z-value P(>|z|) Regressions: y ~ x1 5.773 0.511 11.309 0.000 x2 -1.321 0.479 -2.757 0.006 x3 1.135 0.446 2.545 0.011 x4 0.271 0.466 0.581 0.561 Variances: y 2075.100 293.463 \end{Routput} \newpage \sss{the lavaan model syntax -- multivariate regression} {\scriptsize \begin{center} \begin{tabular}{p{0.35\textwidth}p{0.59\textwidth}} \hline \begin{center} \begin{tikzpicture}[>=stealth,semithick] % x \node[ov] (x1) {$x_1$}; \node[ov] (x2) [below of=x1] {$x_2$}; \node[ov] (x3) [below of=x2] {$x_3$}; \node[ov] (x4) [below of=x3] {$x_4$}; % y \begin{scope}[xshift=2cm] \node[av] (y1) at (0,-1) {$y_1$}; \node[av] (y2) [below of=y1] {$y_2$}; \end{scope} % arrows \path[->] (x1) edge node[above,scale=0.6] {} (y1) (x2) edge node[above,scale=0.6] {} (y1) (x3) edge node[above,scale=0.6] {} (y1) (x4) edge node[above,scale=0.6] {} (y1); \path[->] (x1) edge node[above,scale=0.6] {} (y2) (x2) edge node[above,scale=0.6] {} (y2) (x3) edge node[above,scale=0.6] {} (y2) (x4) edge node[above,scale=0.6] {} (y2); \end{tikzpicture} \end{center} & \begin{Sinput} myModel <- ' y1 ~ x1 + x2 + x3 + x4 y2 ~ x1 + x2 + x3 + x4 ' \end{Sinput} \end{tabular} \end{center} } \newpage \sss{the lavaan model syntax -- path analysis} {\scriptsize \begin{center} \begin{tabular}{p{0.35\textwidth}p{0.59\textwidth}} \hline \begin{center} \begin{tikzpicture}[>=stealth,semithick] % x \node[ov] (x1) {$x_1$}; \node[ov] (x2) [below of=x1] {$x_2$}; \node[ov] (x3) [below of=x2] {$x_3$}; \node[ov] (x4) [below of=x3] {$x_4$}; % y \begin{scope}[xshift=2cm] \node[av] (x5) at (0,0) {$x_5$}; \node[av] (x6) at (0,-3) {$x_6$}; \end{scope} \begin{scope}[xshift=2cm] \node[av] (x7) at (0,-5) {$x_7$}; \end{scope} \path[->] (x1) edge node[above,scale=0.6] {} (x5) (x2) edge node[above,scale=0.6] {} (x5) (x3) edge node[above,scale=0.6] {} (x5) (x4) edge node[above,scale=0.6] {} (x6) (x5) edge node[above,scale=0.6] {} (x6) (x6) edge node[above,scale=0.6] {} (x7); \end{tikzpicture} \end{center} & \begin{Sinput} myModel <- ' x5 ~ x1 + x2 + x3 x6 ~ x4 + x5 x7 ~ x6 ' \end{Sinput} \end{tabular} \end{center} } \newpage \sss{the lavaan model syntax -- mediation analysis} {\scriptsize \begin{center} \begin{tabular}{p{0.35\textwidth}p{0.59\textwidth}} \hline \begin{center} \begin{tikzpicture}[>=stealth,semithick,scale=0.5] \node[ov] (x) at (0,-4) {$X$}; \node[ov] (m) at (3, 0) {$M$}; \node[ov] (y) at (6,-4) {$Y$}; \path[->] (x) edge node[above,scale=1.2] {a} (m) (x) edge node[above,scale=1.2] {c} (y) (m) edge node[above,scale=1.2] {b} (y); \end{tikzpicture} \end{center} & \begin{Sinput} model <- ' Y ~ b*M + c*X M ~ a*X indirect := a*b total := c + (a*b) ' fit <- sem(model, data = myData, se = "bootstrap") summary(fit) \end{Sinput} \end{tabular} \end{center} } \newpage \sss{output} \begin{Routput} ... Parameter estimates: Information Observed Standard Errors Bootstrap Number of requested bootstrap draws 1000 Number of successful bootstrap draws 1000 Estimate Std.err Z-value P(>|z|) Regressions: Y ~ M (b) 0.597 0.098 6.068 0.000 X (c) 2.594 1.210 2.145 0.032 M ~ X (a) 2.739 0.999 2.741 0.006 Variances: Y 108.700 17.747 M 105.408 16.556 Defined parameters: indirect 1.636 0.645 2.535 0.011 total 4.230 1.383 3.059 0.002 \end{Routput} \newpage \sss{the lavaan model syntax -- using cfa() or sem()} {\scriptsize \begin{center} \begin{tabular}{p{0.35\textwidth}p{0.59\textwidth}} \hline \begin{center} \begin{tikzpicture}[>=stealth,semithick,scale=0.5] \node[ov] (y1) at (0,0) {\code{x1}}; \node[ov] (y2) at (0,-1.4) {\code{x2}}; \node[ov] (y3) at (0,-2.8) {\code{x3}}; \node[ov] (y4) at (0,-4.2) {\code{x4}}; \node[ov] (y5) at (0,-5.6) {\code{x5}}; \node[ov] (y6) at (0,-7) {\code{x6}}; \node[ov] (y7) at (0,-8.4) {\code{x7}}; \node[ov] (y8) at (0,-9.8) {\code{x8}}; \node[ov] (y9) at (0,-11.2) {\code{x9}}; \node[lv] (f1) at (4, -1.4) {\code{visual}}; \node[lv] (f2) at (4, -5.6) {\code{textual}}; \node[lv] (f3) at (4, -9.8) {\code{speed}}; \path[->] (f1) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y1.east); \path[->] (f1) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y2.east); \path[->] (f1) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y3.east); \path[->] (f2) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y4.east); \path[->] (f2) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y5.east); \path[->] (f2) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y6.east); \path[->] (f3) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y7.east); \path[->] (f3) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y8.east); \path[->] (f3) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y9.east); \path[<->] (f1.south east) edge [bend left=45] node[left,scale=0.8] {} (f2.north east); \path[<->] (f1.east) edge [bend left=45] node[left,scale=0.8] {} (f3.east); \path[<->] (f2.south east) edge [bend left=45] node[left,scale=0.8] {} (f3.north east); \end{tikzpicture} \end{center} & \begin{Sinput} HS.model <- ' visual =~ x1 + x2 + x3 textual =~ x4 + x5 + x6 speed =~ x7 + x8 + x9 ' fit <- cfa(model = HS.model, data = HolzingerSwineford1939) summary(fit, fit.measures = TRUE, standardized = TRUE) \end{Sinput} \end{tabular} \end{center} } \newpage \sss{output} \begin{Routput} lavaan (0.5-13) converged normally after 35 iterations Number of observations 301 Estimator ML Minimum Function Test Statistic 85.306 Degrees of freedom 24 P-value (Chi-square) 0.000 Model test baseline model: Minimum Function Test Statistic 918.852 Degrees of freedom 36 P-value 0.000 Full model versus baseline model: Comparative Fit Index (CFI) 0.931 Tucker-Lewis Index (TLI) 0.896 Loglikelihood and Information Criteria: Loglikelihood user model (H0) -3737.745 Loglikelihood unrestricted model (H1) -3695.092 Number of free parameters 21 Akaike (AIC) 7517.490 Bayesian (BIC) 7595.339 Sample-size adjusted Bayesian (BIC) 7528.739 Root Mean Square Error of Approximation: RMSEA 0.092 90 Percent Confidence Interval 0.071 0.114 P-value RMSEA <= 0.05 0.001 Standardized Root Mean Square Residual: SRMR 0.065 Parameter estimates: Information Expected Standard Errors Standard Estimate Std.err Z-value P(>|z|) Std.lv Std.all Latent variables: visual =~ x1 1.000 0.900 0.772 x2 0.554 0.100 5.554 0.000 0.498 0.424 x3 0.729 0.109 6.685 0.000 0.656 0.581 textual =~ x4 1.000 0.990 0.852 x5 1.113 0.065 17.014 0.000 1.102 0.855 x6 0.926 0.055 16.703 0.000 0.917 0.838 speed =~ x7 1.000 0.619 0.570 x8 1.180 0.165 7.152 0.000 0.731 0.723 x9 1.082 0.151 7.155 0.000 0.670 0.665 Covariances: visual ~~ textual 0.408 0.074 5.552 0.000 0.459 0.459 speed 0.262 0.056 4.660 0.000 0.471 0.471 textual ~~ speed 0.173 0.049 3.518 0.000 0.283 0.283 Variances: x1 0.549 0.114 0.549 0.404 x2 1.134 0.102 1.134 0.821 x3 0.844 0.091 0.844 0.662 x4 0.371 0.048 0.371 0.275 x5 0.446 0.058 0.446 0.269 x6 0.356 0.043 0.356 0.298 x7 0.799 0.081 0.799 0.676 x8 0.488 0.074 0.488 0.477 x9 0.566 0.071 0.566 0.558 visual 0.809 0.145 1.000 1.000 textual 0.979 0.112 1.000 1.000 speed 0.384 0.086 1.000 1.000 \end{Routput} \newpage \enlargethispage{1cm} \sss{the lavaan model syntax -- using lavaan()} {\scriptsize \begin{center} \begin{tabular}{p{0.30\textwidth}p{0.69\textwidth}} \hline \begin{center} \begin{tikzpicture}[>=stealth,semithick,scale=0.5] \node[ov] (y1) at (0,0) {\code{x1}}; \node[ov] (y2) at (0,-1.4) {\code{x2}}; \node[ov] (y3) at (0,-2.8) {\code{x3}}; \node[ov] (y4) at (0,-4.2) {\code{x4}}; \node[ov] (y5) at (0,-5.6) {\code{x5}}; \node[ov] (y6) at (0,-7) {\code{x6}}; \node[ov] (y7) at (0,-8.4) {\code{x7}}; \node[ov] (y8) at (0,-9.8) {\code{x8}}; \node[ov] (y9) at (0,-11.2) {\code{x9}}; \node[lv] (f1) at (4, -1.4) {\code{visual}}; \node[lv] (f2) at (4, -5.6) {\code{textual}}; \node[lv] (f3) at (4, -9.8) {\code{speed}}; \path[->] (f1) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y1.east); \path[->] (f1) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y2.east); \path[->] (f1) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y3.east); \path[->] (f2) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y4.east); \path[->] (f2) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y5.east); \path[->] (f2) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y6.east); \path[->] (f3) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y7.east); \path[->] (f3) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y8.east); \path[->] (f3) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y9.east); \path[<->] (f1.south east) edge [bend left=45] node[left,scale=0.8] {} (f2.north east); \path[<->] (f1.east) edge [bend left=45] node[left,scale=0.8] {} (f3.east); \path[<->] (f2.south east) edge [bend left=45] node[left,scale=0.8] {} (f3.north east); \end{tikzpicture} \end{center} & \begin{Sinput} HS.model <- ' # latent variables visual =~ 1*x1 + x2 + x3 textual =~ 1*x4 + x5 + x6 speed =~ 1*x7 + x8 + x9 # factor (co)variances visual ~~ visual; visual ~~ textual visual ~~ speed; textual ~~ textual textual ~~ speed; speed ~~ speed # residual variances x1 ~~ x1; x2 ~~ x2; x3 ~~ x3 x4 ~~ x4; x5 ~~ x5; x6 ~~ x6 x7 ~~ x7; x8 ~~ x8; x9 ~~ x9 ' fit <- lavaan(model = HS.model, data = HolzingerSwineford1939) \end{Sinput} \end{tabular} \end{center} } \newpage \enlargethispage{1cm} \sss{lavaan model syntax: full sem} {\scriptsize \begin{center} \begin{tabular}{p{0.54\textwidth}p{0.45\textwidth}} \hline \begin{center} \begin{tikzpicture}[>=stealth,semithick,scale=0.5] \node[ov] (y1) at (0,0) {\code{y1}}; \node[ov] (y2) at (0,-1.4) {\code{y2}}; \node[ov] (y3) at (0,-2.8) {\code{y3}}; \node[ov] (y4) at (0,-4.2) {\code{y4}}; \node[ov] (y5) at (0,-5.6) {\code{y5}}; \node[ov] (y6) at (0,-7) {\code{y6}}; \node[ov] (y7) at (0,-8.4) {\code{y7}}; \node[ov] (y8) at (0,-9.8) {\code{y8}}; \node[ov] (x1) at ( 6,0) {\code{x1}}; \node[ov] (x2) at ( 8,0) {\code{x2}}; \node[ov] (x3) at (10,0) {\code{x3}}; \node[lv] (f1) at (4, -2.8) {\code{dem60}}; \node[lv] (f2) at (4, -7.6) {\code{dem65}}; \node[lv] (f3) at (8, -2.8) {\code{ind60}}; \path[->] (f1) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y1.east); \path[->] (f1) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y2.east); \path[->] (f1) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y3.east); \path[->] (f1) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y4.east); \path[->] (f2) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y5.east); \path[->] (f2) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y6.east); \path[->] (f2) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y7.east); \path[->] (f2) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (y8.east); \path[->] (f3) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (x1.south); \path[->] (f3) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (x2.south); \path[->] (f3) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (x3.south); \path[->] (f1) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (f2); \path[->] (f3) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (f1.east); \path[->] (f3) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (f2.north east); \path[<->] (y1.west) edge [bend right=45] node[left,scale=0.8] {} (y5.west); \path[<->] (y2.west) edge [bend right=45] node[left,scale=0.8] {} (y6.west); \path[<->] (y3.west) edge [bend right=45] node[left,scale=0.8] {} (y7.west); \path[<->] (y4.west) edge [bend right=45] node[left,scale=0.8] {} (y8.west); \path[<->] (y2.south west) edge [bend right=45] node[left,scale=0.8] {} (y4.north west); \path[<->] (y6.south west) edge [bend right=45] node[left,scale=0.8] {} (y8.north west); \end{tikzpicture} \end{center} & \begin{Sinput} myModel <- ' # latent variable definitions ind60 =~ x1 + x2 + x3 dem60 =~ y1 + a*y2 + b*y3 + c*y4 dem65 =~ y5 + a*y6 + b*y7 + c*y8 # regressions dem60 ~ ind60 dem65 ~ ind60 + dem60 # residual covariances y1 ~~ y5 y2 ~~ y4 y2 ~~ y6 y3 ~~ y7 y4 ~~ y8 y6 ~~ y8 ' fit <- sem(model = myModel, data = ...) \end{Sinput} \end{tabular} \end{center} } \newpage \sss{output} \begin{Routput} lavaan (0.5-13) converged normally after 61 iterations Number of observations 75 Estimator ML Minimum Function Test Statistic 40.179 Degrees of freedom 38 P-value (Chi-square) 0.374 Parameter estimates: Information Expected Standard Errors Standard Estimate Std.err Z-value P(>|z|) Latent variables: ind60 =~ x1 1.000 x2 2.180 0.138 15.751 0.000 x3 1.818 0.152 11.971 0.000 dem60 =~ y1 1.000 y2 (a) 1.191 0.139 8.551 0.000 y3 (b) 1.175 0.120 9.755 0.000 y4 (c) 1.251 0.117 10.712 0.000 dem65 =~ y5 1.000 y6 (a) 1.191 0.139 8.551 0.000 y7 (b) 1.175 0.120 9.755 0.000 y8 (c) 1.251 0.117 10.712 0.000 Regressions: dem60 ~ ind60 1.471 0.392 3.750 0.000 dem65 ~ ind60 0.600 0.226 2.660 0.008 dem60 0.865 0.075 11.554 0.000 Covariances: y1 ~~ y5 0.583 0.356 1.637 0.102 y2 ~~ y4 1.440 0.689 2.092 0.036 y6 2.183 0.737 2.960 0.003 y3 ~~ y7 0.712 0.611 1.165 0.244 y4 ~~ y8 0.363 0.444 0.817 0.414 y6 ~~ y8 1.372 0.577 2.378 0.017 Variances: x1 0.081 0.019 x2 0.120 0.070 x3 0.467 0.090 y1 1.855 0.433 y2 7.581 1.366 y3 4.956 0.956 y4 3.225 0.723 y5 2.313 0.479 y6 4.968 0.921 y7 3.560 0.710 y8 3.308 0.704 ind60 0.449 0.087 dem60 3.875 0.866 dem65 0.164 0.227 \end{Routput} \newpage \sss{a simple growth curve model with time-varying covariates} \begin{center} \begin{tabular}{p{0.45\textwidth}p{0.45\textwidth}} \hline \begin{center} \begin{tikzpicture}[>=stealth,semithick,scale=0.6] \node[ov] (c1) at (0,-2) {\code{c1}}; \node[ov] (c2) at (0,-3.5) {\code{c2}}; \node[ov] (c3) at (0,-5) {\code{c3}}; \node[ov] (c4) at (0,-6.5) {\code{c4}}; \node[ov] (t1) at (2,0) {\code{t1}}; \node[ov] (t2) at (4,0) {\code{t2}}; \node[ov] (t3) at (6,0) {\code{t3}}; \node[ov] (t4) at (8,0) {\code{t4}}; \node[lv] (i) at (3, -6) {\code{i}}; \node[lv] (s) at (7, -6) {\code{s}}; \node[ov] (x1) at (3, -9) {\code{x1}}; \node[ov] (x2) at (7, -9) {\code{x2}}; \path[->] (i) edge node[above=0.08cm,scale=0.8,pos=0.2] {} (t1); \path[->] (i) edge node[above=0.08cm,scale=0.8,pos=0.2] {} (t2); \path[->] (i) edge node[above=0.08cm,scale=0.8,pos=0.2] {} (t3); \path[->] (i) edge node[above=0.08cm,scale=0.8,pos=0.2] {} (t4); \path[->] (s) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (t1); \path[->] (s) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (t2); \path[->] (s) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (t3); \path[->] (s) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (t4); \path[->] (c1) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (t1); \path[->] (c2) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (t2); \path[->] (c3) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (t3); \path[->] (c4) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (t4); \path[->] (x1) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (i); \path[->] (x1) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (s); \path[->] (x2) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (i); \path[->] (x2) edge node[above=0.08cm,scale=0.8,pos=0.7] {} (s); \end{tikzpicture} \end{center} & \begin{Sinput} model <- ' # random intercept and slope i =~ 1*t1 + 1*t2 + 1*t3 + 1*t4 s =~ 0*t1 + 1*t2 + 2*t3 + 3*t4 # regressions i ~ x1 + x2 s ~ x1 + x2 # time-varying covariates t1 ~ c1 t2 ~ c2 t3 ~ c3 t4 ~ c4 ' fit <- growth(model, data=Demo.growth) summary(fit) \end{Sinput} \end{tabular} \end{center} \newpage \sss{further syntax} \begin{itemize} \item fixing parameters, and overriding auto-fixed parameters \begin{Sinput} HS.model.bis <- ' visual =~ NA*x1 + x2 + x3 textual =~ NA*x4 + x5 + x6 speed =~ NA*x7 + x8 + x9 visual ~~ 1*visual textual ~~ 1*textual speed ~~ 1*speed ' \end{Sinput} \item linear and nonlinear equality and inequality constraints \begin{Sinput} model.constr <- ' # model with labeled parameters y ~ b1*x1 + b2*x2 + b3*x3 # constraints b1 == (b2 + b3)^2 b1 > exp(b2 + b3) ' \end{Sinput} \item several modifiers (eg.\ fix and label) \begin{Sinput} myModel <- ' y ~ 0.5*x1 + x2 + x3 + b1*x1 ' \end{Sinput} \end{itemize} \newpage \sss{user-friendly fitting functions} \begin{itemize} \item \Verb!cfa()! for confirmatory factor analysis \item \Verb!sem()! for path analysis and SEM \item \Verb!growth()! for growth curve modeling \end{itemize} \sss{arguments of the cfa() and sem() fitting functions} \begin{Sinput} sem(model = NULL, meanstructure = "default", fixed.x = "default", orthogonal = FALSE, std.lv = FALSE, data = NULL, std.ov = FALSE, missing = "default", sample.cov = NULL, sample.mean = NULL, sample.nobs = NULL, group = NULL, group.equal = "", group.partial = "", constraints = '', estimator = "default", likelihood = "default", information = "default", se = "default", test = "default", bootstrap = 1000L, mimic = "default", representation = "default", do.fit = TRUE, control = list(), start = "default", verbose = FALSE, warn = TRUE, debug = FALSE) \end{Sinput} \newpage \sss{power-user fitting functions} \begin{itemize} \item \Verb!lavaan()! by default, no model parameters are added automagically to the parameter table \item several \verb!auto.*! arguments are available to \begin{itemize} \item automatically add a set of parameters (e.g.\ all (residual) variances) \item take actions to make the model identifiable (e.g.\ set the metric of the latent variables) \end{itemize} \end{itemize} \sss{example using lavaan with an auto.* argument} \begin{Sinput} HS.model.mixed <- ' # latent variables visual =~ 1*x1 + x2 + x3 textual =~ 1*x4 + x5 + x6 speed =~ 1*x7 + x8 + x9 # factor covariances visual ~~ textual + speed textual ~~ speed ' fit <- lavaan(HS.model.mixed, data=HolzingerSwineford1939, auto.var=TRUE) \end{Sinput} \newpage \sss{auto.* flags} {\footnotesize \begin{center} \begin{tabular}{p{0.20\textwidth}p{0.10\textwidth}p{0.60\textwidth}} \toprule keyword & operator & parameter set\\ \midrule \Verb!auto.var! & \Verb!~~! & (residual) variances observed and latent variables\\ \Verb!auto.cov.y! & \Verb!~~! & (residual) covariances observed and latent endogenous variables\\ \Verb!auto.cov.lv.x! & \Verb!~~! & covariances among exogenous latent variables\\ \midrule keyword & default & action\\ \midrule \Verb!auto.fix.first! & TRUE & fix the factor loading of the first indicator to 1\\ \Verb!auto.fix.single! & TRUE & fix the residual variance of a single indicator to 1\\ \Verb!int.ov.free! & TRUE & freely estimate the intercepts of the observed variables (only if a mean structure is included)\\ \Verb!int.lv.free! & FALSE & freely estimate the intercepts of the latent variables (only if a mean structure is included)\\ \bottomrule \end{tabular} \end{center} } \newpage \sss{extractor functions: inspecting fitted models} {\footnotesize \begin{center} \begin{tabular}{p{0.15\textwidth}p{0.80\textwidth}} \toprule Method & Description \\ \midrule \Verb!summary()! & print a long summary of the model results\\ \Verb!show()! & print a short summary of the model results\\ \Verb!coef()! & returns the estimates of the free parameters in the model as a named numeric vector\\ %\Verb!parameterEstimates()! & returns the parameter estimates, including confidence intervals, as a data frame\\ %\Verb!standardizedSolution()! & returns one of three types of standardized parameter estimates, as a data frame\\ %\Verb!modindices()!& computes modification \Verb!fitted()! & returns the implied moments (covariance matrix and mean vector) of the model\\ \Verb!resid()! & returns the raw, normalized or standardized residuals (difference between implied and observed moments)\\ \Verb!vcov()! & returns the covariance matrix of the estimated parameters\\ \Verb!predict()! & compute factor scores\\ \Verb!logLik()! & returns the log-likelihood of the fitted model (if maximum likelihood estimation was used)\\ \Verb!AIC()!, \Verb!BIC()! & compute information criteria (if maximum likelihood estimation was used)\\ \Verb!update()! & update a fitted lavaan object\\ \Verb!inspect()! & peek into the internal representation of the model; by default, it returns a list of model matrices counting the free parameters in the model; can also be used to extract starting values, gradient values, and much more\\ \bottomrule \end{tabular} \end{center} } \newpage \sss{other functions} {\footnotesize \begin{center} \begin{tabular}{p{0.35\textwidth}p{0.60\textwidth}} \toprule Function & Description \\ \midrule \Verb!lavaanify()! & converts a lavaan model syntax to a parameter table\\ \Verb!parameterTable()! & returns the parameter table\\ \Verb!parameterEstimates()! & returns the parameter estimates, including confidence intervals, as a data frame\\ \Verb!standardizedSolution()! & returns one of three types of standardized parameter estimates, as a data frame\\ \Verb!modindices()!& computes modification indices and expected parameter changes\\ \Verb!bootstrapLavaan()! & bootstrap any arbitrary statistic that can be extracted from a fitted lavaan object\\ \Verb!bootstrapLRT()! & bootstrap a chi-square difference test for comparing to alternative models\\ \bottomrule \end{tabular} \end{center} } \newpage \sss{extractor examples (1)} \begin{Routput} > fit <- cfa(HS.model, data=HolzingerSwineford1939) > fitted(fit) $cov x1 x2 x3 x4 x5 x6 x7 x8 x9 x1 1.358 x2 0.448 1.382 x3 0.590 0.327 1.275 x4 0.408 0.226 0.298 1.351 x5 0.454 0.252 0.331 1.090 1.660 x6 0.378 0.209 0.276 0.907 1.010 1.196 x7 0.262 0.145 0.191 0.173 0.193 0.161 1.183 x8 0.309 0.171 0.226 0.205 0.228 0.190 0.453 1.022 x9 0.284 0.157 0.207 0.188 0.209 0.174 0.415 0.490 1.015 $mean x1 x2 x3 x4 x5 x6 x7 x8 x9 0 0 0 0 0 0 0 0 0 > predict(fit) visual textul speed [1,] -0.818 -0.138 0.062 [2,] 0.050 -1.013 0.625 [3,] -0.761 -1.872 -0.841 [4,] 0.419 0.018 -0.271 [5,] -0.416 -0.122 0.194 ... \end{Routput} \newpage \sss{extractor examples (2)} \begin{Routput} > inspect(fit) # matrix representation $lambda visual textul speed x1 0 0 0 x2 1 0 0 x3 2 0 0 x4 0 0 0 x5 0 3 0 x6 0 4 0 x7 0 0 0 x8 0 0 5 x9 0 0 6 $theta x1 x2 x3 x4 x5 x6 x7 x8 x9 x1 7 x2 0 8 x3 0 0 9 x4 0 0 0 10 x5 0 0 0 0 11 x6 0 0 0 0 0 12 x7 0 0 0 0 0 0 13 x8 0 0 0 0 0 0 0 14 x9 0 0 0 0 0 0 0 0 15 $psi visual textul speed visual 16 textual 19 17 speed 20 21 18 > inspect(fit, "sampstat") $cov x1 x2 x3 x4 x5 x6 x7 x8 x9 x1 1.358 x2 0.407 1.382 x3 0.580 0.451 1.275 x4 0.505 0.209 0.208 1.351 x5 0.441 0.211 0.112 1.098 1.660 x6 0.455 0.248 0.244 0.896 1.015 1.196 x7 0.085 -0.097 0.088 0.220 0.143 0.144 1.183 x8 0.264 0.110 0.212 0.126 0.181 0.165 0.535 1.022 x9 0.458 0.244 0.374 0.243 0.295 0.236 0.373 0.457 1.015 $mean x1 x2 x3 x4 x5 x6 x7 x8 x9 4.936 6.088 2.250 3.061 4.341 2.186 4.186 5.527 5.374 \end{Routput} \newpage \sss{fitMeasures} \begin{Routput} > fitMeasures(fit) fmin chisq df pvalue 0.142 85.306 24.000 0.000 baseline.chisq baseline.df baseline.pvalue cfi 918.852 36.000 0.000 0.931 tli nnfi rfi nfi 0.896 0.896 0.861 0.907 pnfi ifi rni logl 0.605 0.931 0.931 -3737.745 unrestricted.logl npar aic bic -3695.092 21.000 7517.490 7595.339 ntotal bic2 rmsea rmsea.ci.lower 301.000 7528.739 0.092 0.071 rmsea.ci.upper rmsea.pvalue rmr rmr_nomean 0.114 0.001 0.082 0.082 srmr srmr_nomean cn_05 cn_01 0.065 0.065 129.490 152.654 gfi agfi pgfi mfi 0.943 0.894 0.503 0.903 ecvi 0.423 \end{Routput} \newpage \sss{parameterTable} \begin{Routput} > parameterTable(fit) id lhs op rhs user group free ustart exo label eq.id unco 1 1 visual =~ x1 1 1 0 1 0 0 0 2 2 visual =~ x2 1 1 1 NA 0 0 1 3 3 visual =~ x3 1 1 2 NA 0 0 2 4 4 textual =~ x4 1 1 0 1 0 0 0 5 5 textual =~ x5 1 1 3 NA 0 0 3 6 6 textual =~ x6 1 1 4 NA 0 0 4 7 7 speed =~ x7 1 1 0 1 0 0 0 8 8 speed =~ x8 1 1 5 NA 0 0 5 9 9 speed =~ x9 1 1 6 NA 0 0 6 10 10 x1 ~~ x1 0 1 7 NA 0 0 7 11 11 x2 ~~ x2 0 1 8 NA 0 0 8 12 12 x3 ~~ x3 0 1 9 NA 0 0 9 13 13 x4 ~~ x4 0 1 10 NA 0 0 10 14 14 x5 ~~ x5 0 1 11 NA 0 0 11 15 15 x6 ~~ x6 0 1 12 NA 0 0 12 16 16 x7 ~~ x7 0 1 13 NA 0 0 13 17 17 x8 ~~ x8 0 1 14 NA 0 0 14 18 18 x9 ~~ x9 0 1 15 NA 0 0 15 19 19 visual ~~ visual 0 1 16 NA 0 0 16 20 20 textual ~~ textual 0 1 17 NA 0 0 17 21 21 speed ~~ speed 0 1 18 NA 0 0 18 22 22 visual ~~ textual 0 1 19 NA 0 0 19 23 23 visual ~~ speed 0 1 20 NA 0 0 20 24 24 textual ~~ speed 0 1 21 NA 0 0 21 \end{Routput} \newpage \sss{parameterEstimates} \begin{Routput} > parameterEstimates(fit) lhs op rhs est se z pvalue ci.lower ci.upper 1 visual =~ x1 1.000 0.000 NA NA 1.000 1.000 2 visual =~ x2 0.554 0.100 5.554 0 0.358 0.749 3 visual =~ x3 0.729 0.109 6.685 0 0.516 0.943 4 textual =~ x4 1.000 0.000 NA NA 1.000 1.000 5 textual =~ x5 1.113 0.065 17.014 0 0.985 1.241 6 textual =~ x6 0.926 0.055 16.703 0 0.817 1.035 7 speed =~ x7 1.000 0.000 NA NA 1.000 1.000 8 speed =~ x8 1.180 0.165 7.152 0 0.857 1.503 9 speed =~ x9 1.082 0.151 7.155 0 0.785 1.378 10 x1 ~~ x1 0.549 0.114 4.833 0 0.326 0.772 11 x2 ~~ x2 1.134 0.102 11.146 0 0.934 1.333 12 x3 ~~ x3 0.844 0.091 9.317 0 0.667 1.022 13 x4 ~~ x4 0.371 0.048 7.779 0 0.278 0.465 14 x5 ~~ x5 0.446 0.058 7.642 0 0.332 0.561 15 x6 ~~ x6 0.356 0.043 8.277 0 0.272 0.441 16 x7 ~~ x7 0.799 0.081 9.823 0 0.640 0.959 17 x8 ~~ x8 0.488 0.074 6.573 0 0.342 0.633 18 x9 ~~ x9 0.566 0.071 8.003 0 0.427 0.705 19 visual ~~ visual 0.809 0.145 5.564 0 0.524 1.094 20 textual ~~ textual 0.979 0.112 8.737 0 0.760 1.199 21 speed ~~ speed 0.384 0.086 4.451 0 0.215 0.553 22 visual ~~ textual 0.408 0.074 5.552 0 0.264 0.552 23 visual ~~ speed 0.262 0.056 4.660 0 0.152 0.373 24 textual ~~ speed 0.173 0.049 3.518 0 0.077 0.270 \end{Routput} \newpage \sss{varTable} \begin{Routput} > varTable(fit) name idx nobs type exo user mean var nlev lnam 1 x1 7 301 numeric 0 0 4.936 1.363 0 2 x2 8 301 numeric 0 0 6.088 1.386 0 3 x3 9 301 numeric 0 0 2.250 1.279 0 4 x4 10 301 numeric 0 0 3.061 1.355 0 5 x5 11 301 numeric 0 0 4.341 1.665 0 6 x6 12 301 numeric 0 0 2.186 1.200 0 7 x7 13 301 numeric 0 0 4.186 1.187 0 8 x8 14 301 numeric 0 0 5.527 1.025 0 9 x9 15 301 numeric 0 0 5.374 1.018 0 \end{Routput} \newpage \sss{modification indices} \begin{Routput} > subset(modindices(fit), mi > 5) lhs op rhs mi epc sepc.lv sepc.all sepc.nox 1 visual =~ x5 7.441 -0.210 -0.189 -0.147 -0.147 2 visual =~ x7 18.631 -0.422 -0.380 -0.349 -0.349 3 visual =~ x9 36.411 0.577 0.519 0.515 0.515 4 textual =~ x1 8.903 0.350 0.347 0.297 0.297 5 textual =~ x3 9.151 -0.272 -0.269 -0.238 -0.238 6 x1 ~~ x7 5.420 -0.129 -0.129 -0.102 -0.102 7 x1 ~~ x9 7.335 0.138 0.138 0.117 0.117 8 x2 ~~ x3 8.532 0.218 0.218 0.164 0.164 9 x2 ~~ x7 8.918 -0.183 -0.183 -0.143 -0.143 10 x3 ~~ x5 7.858 -0.130 -0.130 -0.089 -0.089 11 x4 ~~ x6 6.220 -0.235 -0.235 -0.185 -0.185 12 x4 ~~ x7 5.920 0.098 0.098 0.078 0.078 13 x7 ~~ x8 34.145 0.536 0.536 0.488 0.488 14 x7 ~~ x9 5.183 -0.187 -0.187 -0.170 -0.170 15 x8 ~~ x9 14.946 -0.423 -0.423 -0.415 -0.415 \end{Routput} \newpage \sss{bootstrapLavaan} \begin{Routput} fit <- cfa(HS.model, data=HolzingerSwineford1939, se="none") # get the test statistic for the original sample T.orig <- fitMeasures(fit, "chisq") # bootstrap to get bootstrap test statistics # we only generate 10 bootstrap sample in this example; in practice # you may wish to use a much higher number T.boot <- bootstrapLavaan(fit, R = 10, type = "bollen.stine", FUN = fitMeasures, fit.measures = "chisq") # compute a bootstrap based p-value pvalue.boot <- length(which(T.boot > T.orig))/length(T.boot) \end{Routput} \newpage \sss{testing for measurement invariance} \begin{Sinput} # model 1: configural invariance fit1 <- cfa(HS.model, data=HolzingerSwineford1939, group="school") # model 2: weak invariance fit2 <- cfa(HS.model, data=HolzingerSwineford1939, group="school", group.equal="loadings") # model 3: strong invariance fit3 <- cfa(HS.model, data=HolzingerSwineford1939, group="school", group.equal=c("loadings", "intercepts")) \end{Sinput} \sss{comparing two (nested) models: the anova() function} \begin{Sinput} anova(fit1, fit2) \end{Sinput} \begin{Routput} Chi Square Difference Test Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq) fit1 48 7484.4 7706.8 115.85 fit2 54 7480.6 7680.8 124.04 8.1922 6 0.2244 \end{Routput} \begin{itemize} \item this also works correctly if the chi-square test statistics are scaled (robust) \end{itemize} \newpage \sss{measurement invariance tests -- all together} \begin{Sinput} > library(semTools) > measurementInvariance(HS.model, data=HolzingerSwineford1939, group="school", strict=FALSE) \end{Sinput} \begin{Routput} Measurement invariance tests: Model 1: configural invariance: chisq df pvalue cfi rmsea bic 115.851 48.000 0.000 0.923 0.097 7706.822 Model 2: weak invariance (equal loadings): chisq df pvalue cfi rmsea bic 124.044 54.000 0.000 0.921 0.093 7680.771 [Model 1 versus model 2] delta.chisq delta.df delta.p.value delta.cfi 8.192 6.000 0.224 0.002 Model 3: strong invariance (equal loadings + intercepts): chisq df pvalue cfi rmsea bic 164.103 60.000 0.000 0.882 0.107 7686.588 [Model 1 versus model 3] delta.chisq delta.df delta.p.value delta.cfi 48.251 12.000 0.000 0.041 [Model 2 versus model 3] delta.chisq delta.df delta.p.value delta.cfi 40.059 6.000 0.000 0.038 Model 4: equal loadings + intercepts + means: chisq df pvalue cfi rmsea bic 204.605 63.000 0.000 0.840 0.122 7709.969 [Model 1 versus model 4] delta.chisq delta.df delta.p.value delta.cfi 88.754 15.000 0.000 0.083 [Model 3 versus model 4] delta.chisq delta.df delta.p.value delta.cfi 40.502 3.000 0.000 0.042 \end{Routput} \newpage \sss{missing data: fiml} \begin{Sinput} fit <- cfa(HS.model, data = HolzingerSwineford1939, missing = "ml") \end{Sinput} \sss{alternative estimators} \begin{Sinput} fit <- cfa(HS.model, data = HolzingerSwineford1939, estimator = "GLS") \end{Sinput} \sss{robust standard errors} \begin{Sinput} fit <- cfa(HS.model, data = HolzingerSwineford1939, se = "robust") \end{Sinput} \sss{Satorra-Bentler scaled test statistic} \begin{Sinput} fit <- cfa(HS.model, data = HolzingerSwineford1939, test = "Satorra-Bentler") \end{Sinput} \newpage \sss{shortcut: robust standard errors and scaled test statistic} \begin{Sinput} > fit <- cfa(HS.model, data = HolzingerSwineford1939, estimator = "MLM") > summary(fit, fit.measures=TRUE) lavaan (0.5-13) converged normally after 35 iterations Number of observations 301 Estimator ML Robust Minimum Function Test Statistic 85.306 80.872 Degrees of freedom 24 24 P-value (Chi-square) 0.000 0.000 Scaling correction factor 1.055 for the Satorra-Bentler correction Model test baseline model: Minimum Function Test Statistic 918.852 789.298 Degrees of freedom 36 36 P-value 0.000 0.000 Full model versus baseline model: Comparative Fit Index (CFI) 0.931 0.925 Tucker-Lewis Index (TLI) 0.896 0.887 ... \end{Sinput} \sss{mimic option} \begin{Sinput} > cfa(HS.model, data = HolzingerSwineford1939, estimator = "MLM", mimic = "EQS") ... Estimator ML Robust Minimum Function Test Statistic 85.022 81.141 ... > cfa(HS.model, data = HolzingerSwineford1939, estimator = "MLM", mimic = "Mplus") ... Estimator ML Robust Minimum Function Test Statistic 85.306 81.908 ... > cfa(HS.model, data = HolzingerSwineford1939, estimator = "MLM", mimic = "lavaan") ... Estimator ML Robust Minimum Function Test Statistic 85.306 80.872 ... \end{Sinput} \newpage \sss{binary and ordered categorical data} \begin{Sinput} # binary version of Holzinger & Swineford HS9 <- HolzingerSwineford1939[,c("x1","x2","x3","x4","x5", "x6","x7","x8","x9")] HSbinary <- as.data.frame( lapply(HS9, cut, 2, labels=FALSE) ) # single factor model model <- ' visual =~ x1 + x2 + x3 textual =~ x4 + x5 + x6 speed =~ x7 + x8 + x9 ' # binary CFA fit <- cfa(model, data=HSbinary, ordered=names(HSbinary)) # summary summary(fit, fit.measures=TRUE) \end{Sinput} \newpage \sss{output} \begin{Routput} lavaan (0.5-13) converged normally after 36 iterations Number of observations 301 Estimator DWLS Robust Minimum Function Test Statistic 30.918 38.546 Degrees of freedom 24 24 P-value (Chi-square) 0.156 0.030 Scaling correction factor 0.866 Shift parameter 2.861 for simple second-order correction (Mplus variant) Model test baseline model: Minimum Function Test Statistic 582.533 469.769 Degrees of freedom 36 36 P-value 0.000 0.000 Full model versus baseline model: Comparative Fit Index (CFI) 0.987 0.966 Tucker-Lewis Index (TLI) 0.981 0.950 Root Mean Square Error of Approximation: RMSEA 0.031 0.045 90 Percent Confidence Interval 0.000 0.059 0.014 0.070 P-value RMSEA <= 0.05 0.848 0.598 Parameter estimates: Information Expected Standard Errors Robust.sem Estimate Std.err Z-value P(>|z|) Latent variables: visual =~ x1 1.000 x2 0.900 0.188 4.788 0.000 x3 0.939 0.197 4.766 0.000 textual =~ x4 1.000 x5 0.976 0.118 8.241 0.000 x6 1.078 0.125 8.601 0.000 speed =~ x7 1.000 x8 1.569 0.461 3.403 0.001 x9 1.449 0.409 3.541 0.000 Covariances: visual ~~ textual 0.303 0.061 4.981 0.000 speed 0.132 0.049 2.700 0.007 textual ~~ speed 0.076 0.046 1.656 0.098 Intercepts: visual 0.000 textual 0.000 speed 0.000 Thresholds: x1|t1 -0.388 0.074 -5.223 0.000 x2|t1 -0.054 0.072 -0.748 0.454 x3|t1 0.318 0.074 4.309 0.000 x4|t1 0.180 0.073 2.473 0.013 x5|t1 -0.257 0.073 -3.506 0.000 x6|t1 1.024 0.088 11.641 0.000 x7|t1 0.231 0.073 3.162 0.002 x8|t1 1.128 0.092 12.284 0.000 x9|t1 0.626 0.078 8.047 0.000 Variances: x1 0.592 x2 0.670 x3 0.640 x4 0.303 x5 0.336 x6 0.191 x7 0.778 x8 0.453 x9 0.534 visual 0.408 0.112 textual 0.697 0.101 speed 0.222 0.094 > inspect(fit, "sampstat") $cov x1 x2 x3 x4 x5 x6 x7 x8 x9 x1 1.000 x2 0.284 1.000 x3 0.415 0.389 1.000 x4 0.364 0.328 0.232 1.000 x5 0.319 0.268 0.138 0.688 1.000 x6 0.422 0.322 0.206 0.720 0.761 1.000 x7 -0.048 0.061 0.041 0.200 0.023 -0.029 1.000 x8 0.159 0.105 0.439 -0.029 -0.059 0.183 0.464 1.000 x9 0.165 0.210 0.258 0.146 0.183 0.230 0.335 0.403 1.000 $mean x1 x2 x3 x4 x5 x6 x7 x8 x9 0 0 0 0 0 0 0 0 0 $th x1|t1 x2|t1 x3|t1 x4|t1 x5|t1 x6|t1 x7|t1 x8|t1 x9|t1 -0.388 -0.054 0.318 0.180 -0.257 1.024 0.231 1.128 0.626 \end{Routput} \newpage \sss{the history of SEM, from a computational point of view} \begin{itemize} \item several traditions in the SEM (software) world: \begin{itemize} \item LISREL (Karl J\"{o}reskog) \item EQS (Peter Bentler) \item Mplus (Bengt Muth\'{e}n) \item RAM-based approaches (AMOS, Mx, sem, OpenMx, \ldots) \end{itemize} \item superficially, all SEM software packages produce the same results \item there are some subtle (and less subtle) differences in the output \item looking deeper, there are many computational differences \end{itemize} \newpage \sss{some differences} \begin{itemize} \item matrix representation \begin{itemize} \item standard number of matrices: LISREL: 8; Mplus: 4, EQS: 3, RAM: 2 \end{itemize} \item optimization algorithm \begin{itemize} \item quasi-Newton, gradient-only + quasi-Newton, Gauss-Newton, \ldots \end{itemize} \item variances constrained (strictly positive) versus unrestricted \item constrained optimization algorithm \begin{itemize} \item mostly undocumented \item a Lagrangian-multiplier variant, simple slacks, \ldots \end{itemize} \item normal likelihood versus Wishart likelihood, ML versus RLS \begin{itemize} \item $N$ versus $N-1$ \item GLS-ML based chi-square test statistic influences fit measures (CFI!) \end{itemize} \end{itemize} \newpage \sss{some differences (2)} \begin{itemize} \item Satorra-Bentler/Yuan-Bentler scaled test statistic \begin{itemize} \item each program seems to use a different implementation \item often asymptotically equivalent; but large differences in small samples \end{itemize} \item categorical data using the limited information approach \begin{itemize} \item Muth\'{e}n 1984; J\"{o}reskog 1994; Lee, Poon, Bentler (1992) \item many ways to compute the asymptotic covariance matrix (needed for WLS) \end{itemize} \item naive bootstrapping, Bollen-Stine bootstrapping \begin{itemize} \item mostly undocumented; one-iteration bootstrap? \item Bollen-Stine with missing data \end{itemize} \ldots \end{itemize} \newpage \sss{lavaan and the history of SEM} \begin{itemize} \item lavaan tries to preserve the many differences from the several traditions \item all fitting functions in lavaan have a \code{mimic} argument: \begin{itemize} \item \code{mimic="EQS"}, \code{mimic="Mplus"} \item \code{mimic="LISREL"} is under development \item this was originally intended to convince users that lavaan could produce `identical' results as the (commercial) competition \item it is now one of the main design goals of lavaan \item for reproducibility, we plan to make `mimic' version-specific:\\ \code{mimic="Mplus4"}, \code{mimic="LISREL8"}, \ldots \end{itemize} \item we need to (re)evaluate old/new/unexplored computational methods in many areas (optimization, constrained inference, Bayesian techniques, limited information estimation, \ldots) \end{itemize} \newpage \sss{version differences: Mplus 6/7 versus Mplus $<$6 (HSbinary example)} \begin{Routput} Mplus VERSION 7 (Linux) ... MODEL FIT INFORMATION Number of Free Parameters 21 Chi-Square Test of Model Fit Value 38.546* Degrees of Freedom 24 P-Value 0.0305 ... \end{Routput} \begin{Routput} Mplus VERSION 4.1 ... TESTS OF MODEL FIT Chi-Square Test of Model Fit Value 32.074* Degrees of Freedom 19** P-Value 0.0307 ... \end{Routput} \newpage \sss{Satterthwaite versus simple second-order correction} \begin{Routput} > cfa(model, data=HSbinary, ordered=names(HSbinary), mimic="Mplus") ... Estimator DWLS Robust Minimum Function Test Statistic 30.918 38.546 Degrees of freedom 24 24 P-value (Chi-square) 0.156 0.030 Scaling correction factor 0.866 Shift parameter 2.861 for simple second-order correction (WLSMV) > cfa(model, data=HSbinary, ordered=names(HSbinary), mimic="Mplus", estimator="WLSMVS") ... Estimator DWLS Robust Minimum Function Test Statistic 30.918 32.074 Degrees of freedom 24 19 P-value (Chi-square) 0.156 0.031 Scaling correction factor 0.964 for the mean and variance adjusted correction (WLSMV) \end{Routput} \newpage \sss{future plans (1)} \begin{itemize} \item lavaan should `by default' implement best practices in all areas \begin{itemize} \item we welcome advice from scholars \item more simulations studies may be needed \item `defaults' settings should be flexible (depending on the model, depending on the data) \end{itemize} \item lavaan should be 100\% documented \begin{itemize} \item every single number computed by lavaan should be documented \item this includes the `mimic' variations \item all documentation should be available online \end{itemize} \item lavaan should become the golden standard \begin{itemize} \item all computations can be checked at the source code level \item there should be a one-to-one mapping between the actual computations (at the source level) and the technical documentation \end{itemize} \end{itemize} \newpage \sss{future plans (2)} \begin{itemize} \item lavaan should become a computational toolbox \begin{itemize} \item statisticians should be able to quickly (and reliably) implement their new statistical (modern) modeling ideas \item modular, self-contained building blocks \item application programming interface (API) \item \ldots \end{itemize} \item lavaan should be extremely fast \begin{itemize} \item we hope to write a kernel in \code{C++} soon \end{itemize} \item lavaan should be available in other languages \begin{itemize} \item python \item Julia \item Matlab \item \ldots \end{itemize} \end{itemize} \newpage \sss{future plans (3)} \begin{itemize} \item lavaan should become the Rosetta stone for SEM: \begin{itemize} \item read/parse syntax from many external programs\\ (we already have \code{mplus2lavaan()} and \code{lisrel2lavaan()}) \item \code{lavExport()}: write/export syntax to external programs (we can already write Mplus input files) \end{itemize} \item lavaan should be able to detect potential problems in your data and/or model automatically \begin{itemize} \item warning and error messages should be informative \item computing on the model (as represented by the parameter table) \end{itemize} \item lavaan version 1.0 \end{itemize} \newpage \vspace*{2cm} \begin{center} \subsection*{Thank you!\\*[1cm] (questions?)} \end{center} \vspace*{1cm} \begin{center} {\bfseries \large \verb!http://lavaan.org!} \end{center} \end{document}