0 & 0 & \theta_{33} \\ Confirmatory factor analysis As discussed above (background section), to begin the confirmatory facto r analysis, the researcher should have a model in mind. The model chi-square is a meaningful test only when you have an over-identified model (i.e., there are still degrees of freedom left over after accounting for all the free parameters in your model). An absolute fit index on the other hand, does not compare the user model against a baseline model, but instead compares it to the observed data. (1) By the end of this training, you should be able to understand enough of these concepts to run your own confirmatory factor analysis in lavaan. You either have to assume The variance standardization method assumes that the residual variance of the two first order factors is one which means that you assume homogeneous residual variance. As a simple analogy, suppose you have a data set with observed outcomes $y = 13, 14, 15$, then the mean parameter, $\mu$, the estimate of this parameter is called “mu-hat” denoted $\hat{\mu}=\bar{y}=\frac{1}{n}\sum y_i$. Table of Contents Data Input Confirmatory Factor Analysis Using lavaan: Factor variance identification Model Comparison Using lavaan Calculating Cronbach’s Alpha Using psych Made for Jonathan Butner’s Structural Equation Modeling Class, Fall 2017, University of Utah. Finally, pass this object into summary but specify fit.measures=TRUE to obtain additional fit measures and standardized=TRUE to obtain both Std.lv and Std.all solutions. Can you think of other ways? In this case, you perform factor analysis first and then develop a general idea … The second argument is the dataset that contains the observed variables. Confirmatory Factor Analysis - Basic. \lambda_{2} \\ y_3 = \tau_3 + \lambda_{3}\eta_{1} + \epsilon_{3} (Answer: 10), The number of free parameters is defined as, $$\mbox{number of free parameters} = \mbox{number of (unique) model parameters } – \mbox{number of fixed parameters}.$$, How many free parameters have we obtained after fixing 10 (unique) model parameters? & = & \mathbf{\Lambda} \Psi \mathbf{\Lambda}’ + \Theta_{\epsilon} \\ To make sure you fit an equivalent method though, the degrees of freedom for the User model must be the same. \begin{pmatrix} \theta_{11} &  \theta_{12} & \theta_{13} \\ Factors are correlated (conceptually useful to have correlated factors). Suppose the principal investigator thinks that the third, fourth and fifth items of the SAQ are the observed indicators of SPSS Anxiety. The parameters coming from the model are called model parameters. Related. I am using AMOS for Confirmatory Factor Analysis (CFA) and factor loadings are calculated to be more than 1 is some cases. David Kenny states that if the CFI is less than one, then the CFI is always greater than the TLI. Then pass this object into the wrapper function cfa and store the lavaan-method object into onefac8items but specify std.lv=TRUE to automatically use variance standardization. This chapter will cover conducting CFAs with the sem package. The option to.data.frame ensures the data imported is a data frame and not an R list, and use.value.labels = FALSE converts categorical variables to numeric values rather than factors. The more similar the deviation from the baseline model, the closer the ratio to one. Circles represent latent variables, squares represent observed indicators, triangles represent intercept or means, one-way arrows represent paths and two-way arrows represent either variances or covariances. \Sigma(\theta)= Since we have 6 known values, our degrees of freedom is $6-6=0$, which is defined to be saturated. In traditional confirmatory factor analysis or structural equation modeling, the. Overview. Recall from the CFI that $\delta=\chi^2 – df$ where $df$ is the degrees of freedom for that particular model. Our sample of $n=2,571$ is considered relatively large, hence our conclusion may be supplemented with other fit indices. In order to undrestand the model, we have to understand endogenous and exogenous factors. Recall that in the model-implied covariance matrix we have the following model parameters: $$ To begin, first count the number of known values in your observed population variance-covariance matrix $\Sigma$, given by the formula $p(p+1)/2$ where $p$ is the number of items in your survey. The observed population covariance matrix $\Sigma$ is a matrix of bivariate covariances that determines how many total parameters can be estimated in the model. Explain why fixing $\lambda_1=1$ and setting the unique residual covariances to zero (e.g., $\theta_{12}=\theta_{21}=0$, $\theta_{13}=\theta_{31}=0$, and $\theta_{23}=\theta_{32}=0$) results in a just-identified model. This seminar will show you how to perform a confirmatory factor analysis using lavaan in the R statistical programming language. The first line is the model statement. Given the eight-item one factor model: $$TLI= \frac{4164.572/28-554.191/20}{4164.572/28-1} =0.819.$$, We can confirm our answers for both the TLI and CFI which are reported together in lavaan. Recall that the magnitude of a correlation $|r|$ is determined by the absolute value of the correlation. \begin{matrix} The off-diagonal cells in $S$ correspond to bivariate sample covariances between two pairs of items; and the diagonal cells in $S$ correspond to the sample variance of each item (hence the term “variance-covariance matrix“). Notice that the number of free parameters is now 9 instead of 6, however, our degrees of freedom is still zero. Note that the loadings $\lambda$ are the same parameters shared between the measurement model and the model-implied covariance model. This means that $\theta$ is composed of the parameters $\Lambda, \Psi, \Theta_{\epsilon}$, which correspond to the loadings, the covariances of the latent variables and the covariance of the residual errors. A perfect fitting model which generate a TLI which equals 1. Recall that =~ represents the indicator equation where the latent variable is on the left and the indicators (or observed variables) are to the right the symbol. \end{pmatrix} The syntax NA*f1 means to free the first loading because by default the marker method fixes the loading to 1, and equal("f3=~f1")*f2 fixes the loading of the second factor on the third to be the same as the first factor. <> The concept of degrees of freedom is essential in CFA. Outline. Featured on Meta Feature Preview: New Review Suspensions Mod UX. An incremental fit index (a.k.a. \begin{pmatrix} Why do we care so much about the variance-covariance matrix of the items? Even though this is an SPSS file, R can translate this file directly to an R object through the function read.spss via the library foreign. Examples of incremental fit indexes are the CFI and TLI. The specification cov.ov stands for “observed covariance”. Typically, rejecting the null hypothesis is a good thing, but if we reject the CFA null hypothesis then we would reject our user model (which is bad). It belongs to the family of structural equation modeling techniques that allow for the investigation of causal relations among latent and observed variables in a priori specified, theory-derived models. It is used to test whether measures of a construct are consistent with a researcher's understanding of the nature of that construct (or factor). With the full data, the total number of model parameters is calculated accordingly: $$ \mbox{number of model parameters} = \mbox{intercepts from the measurement model} + \mbox{ unique parameters in the model-implied covariance}$$. e.g., five factor uncorrelated; five factor correlated. To review, the model to be fit is the following: Recall that the model implied covariance matrix is defined as, $$ The formula for the CFI is: $$CFI= \frac{\delta(\mbox{Baseline}) – \delta(\mbox{User})}{\delta(\mbox{Baseline})} $$. The root mean square error of approximation is an absolute measure of fit because it does not compare the discrepancy of the user model relative to a baseline model like the CFI or TLI. To request additional fit statistics you add the fit.measures=TRUE option to summary, passing in the lavaan object onefac8items_a. What would be the acceptable range of chi-square values based on the criteria that the relative chi-square greater than 2 indicates poor fit? Though several books have documented how to perform factor analysis using R (e.g.,Beaujean2014;Finch and French2015), procedures for conducting a MCFA are not readily available and as of yet are not built-in lavaan. The term used in the TLI is the relative chi-square (a.k.a. This is the confirmatory way of factor analysis where the process is run to confirm with understanding of the data. \theta_{31} & \theta_{32} & \theta_{33} \\ \lambda_{1} & \lambda_{2} & \lambda_{3} \\ Proceed through the seminar in order or click on the hyperlinks below to go to a particular section: Before beginning the seminar, please make sure you have R and RStudio installed. %PDF-1.5 Then the only green paths are $\lambda,\tau$, and among the blue, again $\lambda$ is estimated, as well as $\theta$ and $\psi$. We proceed with a correlated two-factor CFA. In order to identify a factor in a CFA model with three or more items, there are two options known respectively as the marker method and the variance standardization method. See the optional section Degrees of freedom with means for the more technically accurate explanation. There are three main differences between the factor analysis model and linear regression: We can represent this multivariate model (i.e., multiple outcomes, items, or indicators) as a matrix equation: $$ $$, In matrix notation, the variance standardization method (Option 2) looks like, $$ The marker method assumes that both loadings from the second order factor to the first factor is 1. y_1 = \tau_1 + \lambda_{1}\eta_{1} + \epsilon_{1} \\ NOTE: changing the standardization method should not change the degrees of freedom and chi-square value. Since we fix one loading, and 3 unique residual covariances, the number of free parameters is $10-(1+3)=6$. The fa function includes ve methods of factor analysis (minimum residual, principal axis, weighted least squares, generalized least squares and maximum likelihood factor analysis). The first eight items consist of the following (note the actual items have been modified slightly from the original data set): Throughout the seminar we will use the terms items and indicators interchangeably, with the latter emphasizing the relationship of these items to a latent variable. \begin{pmatrix} The figure below represents the same model above as a path diagram. + So how big of a sample do we need? A more common approach is to understand the data using factor analysis. + stream ), Department of Statistics Consulting Center, Department of Biomathematics Consulting Clinic, I dream that Pearson is attacking me with correlation coefficients, Computers are useful only for playing games, My friends are better at statistics than me, Item 6: My friends are better at statistics than me, A Practical Introduction to Factor Analysis: Exploratory Factor Analysis, Motivating example SPSS Anxiety Questionairre, Known values, parameters, and degrees of freedom, Identification of a three-item one factor CFA, (Optional) How to manually obtain the standardized solution, One factor CFA with more than three items (SAQ-8), (Optional) Model test of the baseline or null model, (Optional) Warning message with second-order CFA, Inspect or extract information from a fitted lavaan object. Here’s what the model looks like graphically: Since we picked Option 1, we set the loadings to be equal to each other: We know the factors are uncorrelated because the estimate of f1 ~~ f2 is zero under the Covariances, which is what we expect. Model chi-square is sensitive to large sample sizes, but does that mean we stick with small samples? Approximate fit indexes can be further classified into a) absolute and b) incremental or relative fit indexes. \end{pmatrix} As you can see in the path diagram below, there are in fact five free parameters: two residual variances $\theta_1, \theta_2$, two loadings $\lambda_1, \lambda_2$ and a factor variance $\psi_{11}$. Before we move on, let’s understand the confirmatory factor analysis model. For simplicity, let’s assume that the known values come only from the model-implied covariance matrix. The closer the CFI is to 1, the better the fit of the model; with the maximum being 1. However, if theory is that the correlation between these two constructs is caused by a third factor, then these two first-order factors can serve as latent indicators of the underlying second order factor. Because the basic assumption of factor analysis is that for a collection of observed variables there are a set of underlying factors (smaller than the observed variables, i.e., the \(\eta\)s), that can explain the interrelationships among those variables. Exploratory Factor Analysis (EFA) or roughly known as f actor analysis in R is a statistical technique that is used to identify the latent relational structure among a set of variables and narrow down to a smaller number of variables. The first step involves the procedure that defines constructs theoretically. $$. In this portion of the seminar, we will continue with the example of the SAQ. Alternatively you can request a more condensed output of the standardized solution by the following, note that the output only outputs Std.all. \end{pmatrix} For those readers who are more mathematically inclined, the appendix adds additional details. Recall that we already know how to manually derive Std.lv parameter estimates as this corresponds to the variance standardization method. The range of acceptable chi-square values ranges between 20 (indicating perfect fit) and 40, since 40/20 = 2. From talking to the Principal Investigator, we decide the use only Items 1, 3, 4, 5, and 8 as indicators of SPSS Anxiety and Items 6 and 7 as indicators of Attribution Bias. Similarly, for a single item, the factor analysis model is: $$y_{1} = \tau_1 + \lambda_1 \eta + \epsilon_{1} $$. Confirmatory factor analysis borrows many of the same concepts from exploratory factor analysis except that instead of letting the data tell us the factor structure, we pre-determine the factor structure and verify the psychometric structure of a previously de… ���/R���Ԗ!��Q�>Y������[w} \lambda_{3} \lambda_{3} Answer: False, the residual covariance uses sample estimates $S-\Sigma(\hat{\theta})$. \theta_{21} & \theta_{22} & \theta_{23} \\ An under-identified model means that the number known values is less than the number of free parameters and an over-identified model means that the number of known values is greater than the number of free parameters. \begin{pmatrix} Note that  $\Sigma -\Sigma{(\theta)}=0$ is always true under the null hypothesis. \begin{pmatrix} 0 & 0 & \theta_{33} \\ Just as in our exploratory factor analysis our Principal Investigator would like to evaluate the psychometric properties of our proposed 8-item SPSS Anxiety Questionnaire “SAQ-8”, proposed as a shortened version of the original SAQ in order to shorten the time commitment for participants while maintaining internal consistency and validity. 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