\documentclass{article} \newcommand{\titel}{\pkg{lavaan}: an \proglang{R} package for structural equation modeling} %\usepackage{Sweave} \usepackage{fancyvrb} \usepackage{graphicx} \usepackage{color} \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=\footnotesize, xleftmargin=0.2cm} \DefineVerbatimEnvironment{Routput}{Verbatim} {fontsize=\footnotesize, xleftmargin=1cm} \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} \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}Utrecht -- April 24, 2012} \title{\textcolor{darkred}{\titel}} \newcommand{\sss}[1]{\subsubsection*{#1}} \newcommand{\lijn}{\mbox{}\\*[0.04\textheight]} \begin{document} \maketitle\thispagestyle{fancy}\newpage %\tableofcontents\newpage \subsection*{Overview} \begin{enumerate} \item software for SEM \item what is lavaan? why do we need lavaan? \item features of lavaan \item lavaan and the (computational) history of SEM \end{enumerate} %\newpage %\subsection*{Structural Equation Modeling} %\begin{itemize} %\item path analysis with latent variables %\end{itemize} %\begin{center} %\begin{tikzpicture}[>=stealth,semithick] %\node[ov] (y1) at (0,-1) {$y_1$}; %\node[ov] (y2) [below of=y1] {$y_2$}; %\node[ov] (y3) [below of=y2] {$y_3$}; %\node[ov] (y4) [below of=y3] {$y_4$}; %\node[ov] (y5) [below of=y4] {$y_5$}; %\node[ov] (y6) [below of=y5] {$y_6$}; % %\node[lv] (f1) at (1.5, -2) {$\eta_1$}; %\node[lv] (f2) at (1.5, -5) {$\eta_2$}; % %\node[ov] (y7) at (3,0) {$y_7$}; %\node[ov] (y8) [right of=y7] {$y_8$}; %\node[ov] (y9) [right of=y8] {$y_9$}; %\node[ov] (y10) [right of=y9] {$y_{10}$}; %\node[ov] (y11) [right of=y10] {$y_{11}$}; %\node[ov] (y12) [right of=y11] {$y_{12}$}; % %\node[lv] (f3) at (4, -2) {$\eta_3$}; %\node[lv] (f4) at (7, -2) {$\eta_4$}; % %\path[->] (f1) edge node[above,scale=0.6] {} (y1) % (f1) edge node[above,scale=0.6] {} (y2) % (f1) edge node[above,scale=0.6] {} (y3) % (f2) edge node[above,scale=0.6] {} (y4) % (f2) edge node[above,scale=0.6] {} (y5) % (f2) edge node[above,scale=0.6] {} (y6) % % (f3) edge node[above,scale=0.6] {} (y7) % (f3) edge node[above,scale=0.6] {} (y8) % (f3) edge node[above,scale=0.6] {} (y9) % (f4) edge node[above,scale=0.6] {} (y10) % (f4) edge node[above,scale=0.6] {} (y11) % (f4) edge node[above,scale=0.6] {} (y12) % % (f1) edge node[above,scale=0.6] {} (f3) % (f2) edge node[above,scale=0.6] {} (f3) % (f3) edge node[above,scale=0.6] {} (f4); %\end{tikzpicture} %\end{center} \newpage \subsection*{Software for SEM (commercial)} \sss{The big four} \begin{itemize} \item LISREL \item EQS \item AMOS \item MPLUS \end{itemize} \sss{Others} \begin{itemize} \item CALIS/TCALIS (SAS/Stat) \item SEPATH (Statistica) \item RAMONA (Systat) \item Stata 12 \item \ldots \end{itemize} \newpage \subsection*{Software for SEM: non-commercial} \begin{itemize} \item outside the R ecosystem: Mx, gllamm (Stata), \ldots \item three R packages: \begin{enumerate} \item sem \begin{itemize} \item developer: John Fox (since 2001) \item for a long time the only option in R \end{itemize} \item OpenMx \begin{itemize} \item Mx reborn -- website: \verb!http://openmx.psyc.virginia.edu/! \item free, but the solver (NPSOL) is (currently) not open-source \end{itemize} \item lavaan \begin{itemize} \item first public release: May 2010 (version 0.3-1) \item current version: 0.4-13 \end{itemize} \end{enumerate} \item interfaces between R and commercial packages: \begin{itemize} \item \code{REQS}, \code{MplusAutomation} \end{itemize} \end{itemize} \newpage \subsection*{What is lavaan?} \begin{itemize} \item \pkg{lavaan} is an R package for latent variable analysis: \begin{itemize} \item confirmatory factor analysis: function \code{cfa()} \item structural equation modeling: function \code{sem()} \item latent curve analysis / growth modeling: function \code{growth()} \item general mean/covariance structure modeling: function \code{lavaan()} \item (item response theory (IRT) models) \item (latent class + mixture models) \item (multilevel models) \end{itemize} \item the \pkg{lavaan} package is developed to provide useRs, researchers and teachers a free, open-source, but commercial-quality package for latent variable modeling \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 \subsection*{Installing \pkg{lavaan}, finding documentation} \begin{itemize} \item \pkg{lavaan} depends on the R project for statistical computing: \begin{quote} \Verb!http://www.r-project.org! \end{quote} \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{quote} \Verb!http://lavaan.org! \end{quote} \item journal paper: \begin{quote} Rosseel (2012). lavaan: an R package for structural equation modeling. {\itshape Journal of Statistical Software}, (??), ??--??. \end{quote} \item \pkg{lavaan} development: \begin{quote} \Verb!https://github.com/yrosseel/lavaan! \end{quote} \end{itemize} \newpage \subsection*{Why do we need \pkg{lavaan}?} \begin{enumerate} \item 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 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 lavaan is for applied researchers \begin{itemize} \item keep it simple, provide all the features they need \end{itemize} \end{enumerate} \newpage \subsection*{Features of lavaan} \vspace*{1cm} \sss{\pkg{lavaan} is reliable and robust} \begin{itemize} \item extensive testing before a `public' release on CRAN \item no convergence problems (for admissible models) \item numerical results are very close (if not identical) to commercial packages: \begin{itemize} \item Mplus (if \code{mimic="Mplus"}, default) \item EQS (if \code{mimic="EQS"}) \end{itemize} \end{itemize} \newpage \sss{the lavaan model syntax} {\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} \\ \hline \end{tabular} \end{center} } \newpage \sss{lavaan output} \begin{Routput} lavaan (0.4-13) converged normally after 41 iterations Number of observations 301 Estimator ML Minimum Function Chi-square 85.306 Degrees of freedom 24 P-value 0.000 Chi-square test baseline model: Minimum Function Chi-square 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 0.553 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 \sss{lavaan model syntax (2)} {\scriptsize \begin{center} \begin{tabular}{p{0.55\textwidth}p{0.40\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{Binput} # latent variable definitions ind60 =~ x1 + x2 + x3 dem60 =~ y1 + y2 + y3 + y4 dem65 =~ y5 + y6 + y7 + y8 # regressions dem60 ~ ind60 dem65 ~ ind60 + dem60 # residual covariances y1 ~~ y5 y2 ~~ y4 y2 ~~ y6 y3 ~~ y7 y4 ~~ y8 y6 ~~ y8 \end{Binput} \\ \end{tabular} \end{center} } \newpage \sss{lavaan features (0.4)} \sss{support for non-normal continuous data} \begin{itemize} \item asymptotically distribution-free (ADF) estimation (Browne 1984) \item Satorra-Bentler scaled test statistic and robust standard errors \item Yuan-Bentler scaled test statistic and robust standard errors when data are both non-normal and missing (at random) \item bootstrapping: the na\"{i}ve bootstrap and the Bollen-Stine bootstrap \end{itemize} \sss{support for missing data (fiml)} \sss{multiple groups and measurement invariance} \sss{linear and nonlinear equality and inequality constraints} \sss{defined parameters and mediation analysis} \sss{bootstrapping} \newpage \sss{what to expect in lavaan 0.5} \begin{itemize} \item expected release: 11 May 2012 \item support for categorical data (three-step/WLS approach) \item modularity + access to all lavaan internals \end{itemize} \sss{what to expect in lavaan 0.6-0.7} \begin{itemize} \item Bayesian estimation \item support for multilevel data \item speed! \item \ldots \item technical documentation \end{itemize} \newpage \subsection*{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 GLS-ML (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 \subsection*{lavaan and the history of SEM} \begin{itemize} \item lavaan is in many areas still trying to catch up with commercial software; but instead of trying to implement one tradition (based on one program), lavaan tries to implement several traditions \item all fitting functions in lavaan have a \code{mimic} argument which can be set to \code{"EQS"} or \code{"Mplus"} respectively; \code{"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 \end{itemize} \subsection*{lavaan and the future of SEM?} \begin{itemize} \item we need to (re)evaluate old/new/unexplored computational methods in many areas (optimization, constrained inference, Bayesian techniques, limited information estimation, \ldots) \item lavaan should `by default' implement best practices in all areas \end{itemize} \newpage \vspace*{2cm} \begin{center} \subsection*{End of part I} \end{center} \end{document}