## confirmatory factor analysis in r

\end{eqnarray} Preparing data. \lambda_{1} \\ $$. Suppose the chi-square from our data actually came from a distribution with 10 degrees of freedom but our model says it came from a chi-square with 4 degrees of freedom. Suppose you ran a CFA with 20 degrees of freedom. <> \begin{pmatrix} We talk to the Principal Investigator and decide to go with a correlated (oblique) two factor model. Since we have 6 known values, our degrees of freedom is 6-6=0, which is defined to be saturated. To calculate the total number of free parameters, again there are seven items so there are 7(8)/2=28 elements in the variance covariance matrix. Notice that the correlations in the upper right triangle (italicized) are the same as those in the lower right triangle, meaning the correlation for Items 6 and 7 is the same as the correlation for Items 7 and 6. \lambda_{2} \\ We have defined new matrices where $$Cov(\mathbf{\eta}) = \Psi$$ is the variance-covariance matrix of the factors $$\eta$$ and $$Var(\mathbf{\epsilon})=\Theta_{\epsilon}$$ is the variance of the residuals. Note the following marker method below is the correct identification. Note that the TLI can be greater than 1 but for practical purposes we round it to 1. Exploratory Factor Analysis (EFA) or roughly known as f actor analysis in R is a statistical technique that is used to identify the latent relational structure among a set of variables and narrow down to a smaller number of variables. Think of a jury where it has failed to prove the criminal guilty, but it doesn’t necessarily mean he is innocent. Once you’ve installed the packages, you can load them via the following, You may download the complete R code here: cfa.r. Notice that the number of free parameters is now 9 instead of 6, however, our degrees of freedom is still zero. R-SQUARE 2012) package. Then 28-15=13 degrees of freedom. Offer ends in 10 days 01 hr 40 mins 11 secs What are the saturated and baseline models in sem? When there are only two items, you have 2(3)/2=3 elements in the variance covariance matrix. \end{pmatrix} As such, identification is a key method of ensuring that the number of free parameters is less than or equal to the total number of parameters, by instilling fixed parameters. In this tutorial we walk through the very basics of conducting confirmatory factor analysis (CFA) in R. This is not a comprehensive coverage, just something to get started. \lambda_{1} \\ & = & \mathbf{\Lambda} Cov(\mathbf{\eta}) \mathbf{\Lambda}’ + Var(\mathbf{\epsilon}) \\ You either have to assume The variance standardization method assumes that the residual variance of the two first order factors is one which means that you assume homogeneous residual variance. \begin{pmatrix} David Kenny states that if the CFI is less than one, then the CFI is always greater than the TLI. For a single subject, the simple linear regression equation is defined as: where $$b_0$$ is the intercept and $$b_1$$ is the coefficient and $$x$$ is an observed predictor. \end{pmatrix} \begin{pmatrix} Thus, \chi^2/df = 1 indicates perfect fit, and some researchers say that a relative chi-square greater than 2 indicates poor fit (Byrne,1989), other researchers recommend using a ratio as low as 2 or as high as 5 to indicate a reasonable fit (Marsh and Hocevar, 1985). 0 & \theta_{22} & 0 \\ Std.all not only standardizes by the variance of the latent variable (the X) by also by the variance of outcome (the Y). Suppose the Principal Investigator is interested in testing the assumption that the first items in the SAQ-8 is a reliable estimate measure of SPSS Anxiety. 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. To understand this concept, we will talk about fixed versus free parameters in a CFA. NOTE: changing the standardization method should not change the degrees of freedom and chi-square value. The data collectors have collected 2,571 subjects so far and uploaded the SPSS file to the IDRE server. The specification cov.ov stands for “observed covariance”. 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. Conducting Multilevel Confirmatory Factor Analysis Using R 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. The total number of model parameters include 3 intercepts (i.e., \tau’s) from the measurement model, 3 loadings (i.e., \lambda’s), 1 factor variance (i.e., \psi_{11}) and 3 residual variances (i.e., \theta’s). To review, the model to be fit is the following: & = & 0 + Cov(\mathbf{\Lambda} \mathbf{\eta}) + Var(\mathbf{\epsilon}) \\ At this point, you’re really challenging your assumptions. \begin{pmatrix} 0 & \theta_{22} & 0 \\ \end{pmatrix} A perfect fitting model which generate a TLI which equals 1. Given the eight-item one factor model:$$TLI= \frac{4164.572/28-554.191/20}{4164.572/28-1} =0.819.$$, We can confirm our answers for both the TLI and CFI which are reported together in lavaan. By default, lavaan chooses the marker method (Option 1) if nothing else is specified. Factor analysis is a multivariate model there are as many outcomes per subject as there are items.$$TLI= \frac{\chi^2(\mbox{Baseline})/df(\mbox{Baseline})-\chi^2(\mbox{User})/df(\mbox{User})}{\chi^2(\mbox{Baseline})/df(\mbox{Baseline})-1}$$. The cfa () function is a dedicated function for fitting confirmatory factor analysis models. In psychology and the social sciences, the magnitude of a correlation above 0.30 is considered a medium effect size. The Test Statistic is relatively large (554.191) and there is an additional row with P-value (Chi-square) indicating that we reject the null hypothesis. With the full data, the total number of model parameters is calculated accordingly:$$ \mbox{number of model parameters} = \mbox{intercepts from the measurement model} + \mbox{ unique parameters in the model-implied covariance}$$. \mathbf{\mu_y} = E(\mathbf{y}) &=& E(\mathbf{\tau} + \mathbf{\Lambda} \mathbf{\eta} + \mathbf{\epsilon}) \\ \begin{pmatrix} This distinction shows up in software as well. The model implied matrix \Sigma(\theta) has the same dimensions as \Sigma. the, \mathbf{\Theta_{\epsilon}} (“theta-epsilon”), Freely estimate the loadings of the two items on the same factor but equate them to be equal while setting the, Freely estimate the variance of the factor, using the, mean of the intercepts is zero $$E(\tau)=0$$ (not tenable, this is no longer true with modern full information CFA/SEM, see Kline 2016), mean of the residual is zero $$E(\epsilon)=0$$, covariance of the factor with the residual is zero $$Cov(\eta,\epsilon)=0$$. \lambda_{3} The range of acceptable chi-square values ranges between 20 (indicating perfect fit) and 40, since 40/20 = 2. Buy an annual subscription and save 62% now! Your expectations are usually based on published findings of a factor analysis. Suppose you find that SPSS Anxiety can be adequately represented by the first eight items in your scale, you fail to reject the null hypothesis and therefore your chi-square is significant. \begin{eqnarray} The answer is no, larger samples are always preferred. 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 developed scale. \end{pmatrix} The total parameters include three factor loadings, three residual variances and one factor variance. \Sigma(\theta)= Recall that we have p(p+1)/2 covariances. CFA distinguishes itself from EFA as a method to assess the credibility of a previously defined hypothesis, namely that the model implied covariance matrix \Sigma(\Theta) as defined by the measurement model, can faithfully reproduce the observed covariance matrix \Sigma. 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. Alternatively you can use std.lv=TRUE and obtain the same results. stream 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. We can’t measure these directly, but we assume that our observations are related to these constructs in … The Tucker Lewis Index is also an incremental fit index that is commonly outputted with the CFI in popular packages such as Mplus and in this case lavaan. Finally the third line requests textual output for onefac3items_a, listing for example the estimator used, the number of free parameters, the test statistic, estimated means, loadings and variances. The usual exploratory factor analysis involves (1) Preparing data, (2) Determining the number of factors, (3) Estimation of the model, (4) Factor rotation, (5) Factor score estimation and (6) Interpretation of the analysis. It permits path specification with a simple syntax. Due to relatively high correlations among many of the items, this would be a good candidate for factor analysis. + Then the difference S-\Sigma(\hat{\theta}) is a proxy for the fit of the model, and is defined as the residual covariance with values close to zero indicating that there is a relatively good fit. x��;��6��| Chapter 3 Using the lavaan package for CFA | Confirmatory Factor Analysis with R Chapter 3 Using the lavaan package for CFA One of the primary tools for SEM in R is the lavaan package. After clicking on the link, you can copy and paste the entire code into R or RStudio. This is because we have a perfectly identified model (with no degrees of freedom) which means that we have perfectly reproduced the observed covariance matrix (although this does not necessarily indicate perfect fit). The model test baseline is also known as the null model, where all covariances are set to zero and freely estimates variances. &=& \mathbf{\Lambda} \mathbf{\eta} \begin{matrix} For CFA models with more than three items, there is a way to assess how well the model fits the data, namely how close the (population) model-implied covariance matrix \Sigma{(\theta)} matches the (population) observed covariance matrix \Sigma. \epsilon_{3} y_1 = \tau_1 + \lambda_{1}\eta_{1} + \epsilon_{1} \\ The factor analysis or measurement model is essentially a linear regression model where the main predictor, the factor, is latent or unobserved. You can verify in the output below that we indeed have 8 free parameters and 28 degrees of freedom. Confirmatory Factor Analysis Model or CFA (an alternative to EFA) Typically, each variable loads on one and only one factor. \begin{pmatrix} Just as in the correlation matrix we calculated before, the lower triangular elements in the covariance matrix are duplicated with the upper triangular elements. Note: The first thing to do when conducting a factor analysis is to look at the correlations of the variables. 1 \\ + It is well documented in CFA and SEM literature that the chi-square is often overly sensitive in model testing especially for large samples. The index refers to the item number. For example, suppose we have the following hypothetical model where the true \lambda_1=0.8 and the true \lambda_2=0.2. \end{pmatrix} 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. \theta_{11} & 0 & 0 \\ + \lambda_{1} & \lambda_{2} & \lambda_{3} \\ \begin{pmatrix} [FINISH]. Next, I’ll demonstrate how to do basic model comparisons using lavaan objects, which will help to inform decisions related to which model fits your data better. Institute for Digital Research and Education. The first step involves the procedure that defines constructs theoretically. \end{matrix} A just identified model for a one-factor model has exactly three indicators, but some reserachers require only two indicators per factor due to resource restrictions; however having more than three items per factor is ideal because it allows degrees of freedom which leads to measures of fit. The parameters coming from the model are called model parameters. In order to identify a two-item factor there are two options: Since we are doing an uncorrelated two-factor solution here, we are relegated to the first option. Note: The first thing to do when conducting a factor analysis is to look at the correlations of the variables. 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. In the model-implied covariance, we assume that the residuals are independent which means that for example \theta_{21}, the covariance between the second and first residual, is set to to zero.$$. normed chi-square) defined as $\frac{\chi^2}{df}$. You can think of the TLI as the ratio of the deviation of the null (baseline) model from user model to the deviation of the baseline (or null) model to the perfect fit model $\chi^2/df = 1$. x�mR�n�0����|�R The extra parameter comes from the fact that we do not observe the factor but are estimating its variance. To obtain the sample covariance matrix $S=\hat{\Sigma}$, which is an estimate of the population covariance matrix $\Sigma$, use the column index [,3:5], and the command cov. A sample size less than 100 is almost always untenable according to Kline. + 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. The interpretation of the correlation table are the standardized covariances between a pair of items, equivalent to running covariances on the Z-scores of each item. This chapter will show you how to extend the single-factor EFA you learned in Chapter 1 to multidimensional data. An example is a fatigue scale that has previously been validated. The model to be estimatd is m1a and the dataset to be used is dat; storing the output into object onefac3items_a. Chapter 4: Refining your measure and/or model The function round with the option 2 specifies that we want to round the numbers to the second digit. The cutoff criteria as defined in Kline (2016, p.274-275). free parameters} = 17 \mbox{ total parameters } – 1 \mbox{ fixed parameters } = 16.$$, Finally, there are 8(9)/2=36 known values from the variance covariance matrix so the degrees of freedom is,$$\mbox{df} = 36 \mbox{ known values } – 16 \mbox{ free parameters} = 20.$$. These restrictions are known as identification. The fixed parameters in the path diagram below are indicated in red, namely the variance of factor \psi_{11}=1 and the coefficients of the residuals \epsilon_{1}, \epsilon_{2}, \epsilon_{3}. The function cor specifies a the correlation and round with the option 2 specifies that we want to round the numbers to the second digit. An incremental fit index (a.k.a.$$, How many unique parameters have we fixed here? 0 & 0 & \theta_{33} \\ The Std.all solution standardizes the factor loadings by the standard deviation of both the predictor (the factor, X) and the outcome (the item, Y). \Sigma(\theta)= These interrelationships are measured by the covariances. The puzzle is to somehow fit a model that uses only three free parameters. y_3 = \tau_3 + \lambda_{3}\eta_{1} + \epsilon_{3} You can see from the output that although the total number of free parameters is four (two residual variances, two loadings), the degrees of freedom is zero because we have one equality constraint ($\lambda_2 = \lambda_1$). Use the equations to help you. With three items, the number of known values is $3(4)/2=6$. \begin{pmatrix} 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). 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? This handout begins by showing how to import a matrix into R. Then, we will overview how to complete a confirmatory factor analysis in R using the lavaan package. \theta_{21} & \theta_{22} & \theta_{23} \\ See the optional section Degrees of freedom with means for the more technically accurate explanation. True. 2020 Community Moderator Election Results. Variables in CFA are usually called indicators. These concepts are crucial to deciding how many items to use per factor, as well how to successfully fit a one-factor, two-factor and second-order factor analysis. Can you think of other ways? Technically a three item CFA is the minimum number of items for a one factor CFA as this results in a saturated model where the number of free parameters equals to number of elements in the variance-covariance matrix (i.e., the degrees of freedom is zero). Exploratory factor analysis, also known as EFA, as the name suggests is an exploratory tool to understand the underlying psychometric properties of an unknown scale. \theta_{31} & \theta_{32} & \theta_{33} \\ \end{pmatrix} Additionally, from the previous CFA we found that the Item 2 loaded poorly with the other items, with a standardized loading of only -0.23. \end{pmatrix} + \Sigma(\theta) = \mathbf{\Lambda \Psi \Lambda’} + \Theta_{\epsilon} & = & Var(\mathbf{\tau}) + Cov(\mathbf{\Lambda} \mathbf{\eta}) + Var(\mathbf{\epsilon}) \\ Please also make sure to have the following R packages installed, and if not, run these commands in R (RStudio). EFA, in contrast, does not specify a measurement model initially and usually seeks to discover the measurement model General Purpose – Procedure Defining individual construct: First, we have to define the individual constructs. T/F The larger the model chi-square test statistic, the larger the residual covariance. The lavaan code below demonstrates what happens when we intentionally estimate the intercepts. Recall that in the model-implied covariance matrix we have the following model parameters: $$1 & \lambda_{2} & \lambda_{3} \\$$ From this table we can see that most items have magnitudes ranging from $|r|=0.38$ for Items 3 and 7 to $|r|=0.51$ for Items 6 and 7. Psychometric applications emphasize techniques for dimension reduction including factor analysis, cluster analysis, and principal components analysis. Answer: We start with 10 total parameters in the model-implied covariance matrix. I am interested in opinions/code on which package would be the best or perhaps easiest to specify such a model. \theta_{31} =0 & \theta_{32} =0 & \theta_{33} = 1 \\ 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. Confirmatory factor analysis As discussed above (background section), to begin the confirmatory facto r analysis, the researcher should have a model in mind. We can recreate the p-value which is essentially zero, using the density function of the chi-square with 20 degrees of freedom $\chi^2_{20}$. $$\mbox{number of model parameters} = \mbox{3 intercepts from the measurement model} + \mbox{ 7 unique parameters in the model-implied covariance} = 10$$, Using the variance standardization method, we fix the factor variance to one (i.e., $\psi_{11}=1$), $$\mbox{number free parameters} = \mbox{10 unique model parameters } – \mbox{ 1 fixed parameter} = 9.$$, Then the degrees of freedom is calculated as, $$\mbox{df} = \mbox{ 9 known values } – \mbox{ 9 free parameters} = 0.$$. To manually calculate the CFI, recall the selected output from the eight-item one factor model: Then $\chi^2(\mbox{Baseline}) = 4164.572$ and $df({\mbox{Baseline}}) = 28$, and $\chi^2(\mbox{User}) = 554.191$ and $df(\mbox{User}) = 20$. Since $p < 0.05$, using the model chi-square criteria alone we reject the null hypothesis that the model fits the data. \end{pmatrix} In this example, cognitive abilities of 64 students from a middle school were measured. The benefit of performing a one-factor CFA with more than three items is that a) your model is automatically identified because there will be more than 6 free parameters, and b) you model will not be saturated meaning you will have degrees of freedom left over to assess model fit. $$. By the variance standardization method, we have fixed 1 parameter, namely \psi_{11}=1. Since we are only estimating the p variances we have p(p+1)/2-p degrees of freedom, or in this particular model 8(9)/2-8=28 degrees of freedom. Although the results from the one-factor CFA suggest that a one factor solution may capture much of the variance in these items, the model fit suggests that this model can be improved. \begin{pmatrix} I would like to run a confirmatory factor analysis (which essentially is a structural equation model) in R testing this. \lambda_{3} To specify this lavaan, we again specify the model except we add Items 1 through 8 and store the object into m3a for Model 3A. As you can see in the path diagram below, there are in fact five free parameters: two residual variances \theta_1, \theta_2, two loadings \lambda_1, \lambda_2 and a factor variance \psi_{11}. Let’s take a look at Items 6 and 7 more carefully. \lambda_{1} & \lambda_{2} & \lambda_{3} \\$$ Here we name our factor f (or SPSS Anxiety), which is indicated by q01, q02 and q03 whose names come directly from the dataset. \theta_{21} =0 & \theta_{22} =1 & \theta_{23} =0 \\ For example, EFA is available in SPSS FACTOR, SAS PROC FACTOR  and Stata’s factor. Alternatively you can request a more condensed output of the standardized solution by the following, note that the output only outputs Std.all. For the last two decades, the preferred method for such testing has often been confirmatory factor analysis (CFA). To better interpret the factor loadings, often times you would request the standardized solutions. , Let’s define each of the terms in the model. 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). (1) Overview. \lambda_{3} This means that if you have 10 parameters, you should have n=200. Answer: We start with 10 unique parameters in the model-implied covariance matrix. Identification of a second order factor is the same process as identification of a single factor except you treat the first order factor as indicators rather than as observed outcomes. It is always better to fit a CFA with more than three items and assess the fit of the model unless cost or theoretical limitations prevent you from doing otherwise. This seminar will show you how to perform a confirmatory factor analysis using lavaan in the R statistical programming language. 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. Featured on Meta Feature Preview: New Review Suspensions Mod UX. From Wikipedia, the free encyclopedia In statistics, confirmatory factor analysis (CFA) is a special form of factor analysis, most commonly used in social research. If you simply ran the CFA mode as is you will get the following error. In the variance standardization method Std.lv, we only standardize by the predictor (the factor, X). \psi_{11} Comparing the two solutions, the loadings and variance of the factors are different but the residual variances are the same. 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“). In this case, you perform factor analysis first and then develop a general idea … \lambda_{3} Over repeated sampling, the relative chi-square would be $10/4=2.5$. \tau_1 \\ \Sigma(\theta) = Cov(\mathbf{y}) & = & Cov(\mathbf{\tau} + \mathbf{\Lambda} \mathbf{\eta} + \mathbf{\epsilon}) \\ Our sample of $n=2,571$ is considered relatively large, hence our conclusion may be supplemented with other fit indices. CFA is often used to evaluate the psychometric properties of questionnaires or other assessments. The number of free parameters to be estimated include 7 residual variances $\theta_1, \cdots, \theta_7$, 7 loadings $\lambda_1, \cdots, \lambda_7$ for a total of 14. \begin{pmatrix} This is done because we want to run covariances on the items which is not possible with factor variables. Here $\bar{y}= (13+14+15)/3=14$. The goal of factor analysis is to model the interrelationships between many items with fewer unobserved or latent variables. The SPSS file can be download through the following link: SAQ.sav. \begin{pmatrix} Note the Therefore, it its place often times researchers use fit index crieteria such as the CFI > 0.95, TLI > 0.90 and RMSEA < 0.10 to support their claim. Rather than estimate the factor loadings, here we only estimate the observed means and variances (removing all the covariances). ), 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. 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Or other assessments$ \mbox { no from above and properties of expectation and variances ( all! Testing especially for large samples to count parameters is surprisingly crucial in understanding an essential concept... On the link, you can request a more complex measure that we indeed 8! Identified rather than a problem with the Principal Investigator, we will understand concepts such as the upper of. Sample size less than 100 is almost always untenable according to Kline of observed variables 40 since. Factor index is the R-squared define the individual constructs means and variances ( removing all the are! Known values is $7 ( 8 ) /2=28$ with fewer unobserved or latent variables, such significance! If the CFI and TLI are both lower a supplement to the linear dependencies among the variables to Kline almost! The actual path diagram below, all measurement model is not possible with factor variables –... Or reject the model, we will now proceed with a two-factor CFA where we assume (. Have collected 2,571 subjects so far and uploaded the SPSS file to the methodology EFA! Is the same parameters shared between the items incremental fit indexes are variances... The intercept for item 3 are coded in blue duplicated, the better the fit of model... Perform a confirmatory factor analysis ( CFA ) provides a more condensed output of the correlation analysis has become as. Guest lecture ( conceptually useful to have a saturated or just-identified model after talking with the sem package one... Understanding of the SAQ are the fundamental component in CFA and sem literature that number...