Looking at the Std.all loadings we see that Item 2 loads the weakest onto our SPSS Anxiety factor at -0.23 and Item 4 loads the highest at 0.67. Then the only green paths are $\lambda,\tau$, and among the blue, again $\lambda$ is estimated, as well as $\theta$ and $\psi$. Since we don’t have the population covariances to evaluate, they are estimated by the sample model-implied covariance $\Sigma(\hat{\theta})$ and sample covariance $S$. The residual variance is essentially the variance of $\zeta$, which we classify here as $\psi$. Given that the p-value of the model chi-square was less than 0.05, the CFI = 0.871 and the RMSEA = 0.102, and looking at the standardizes loadings we report to the Principal Investigator that the SAQ-8 as it stands does not possess good psychometric properties. We talk to the Principal Investigator and decide to go with a correlated (oblique) two factor model. Made for Jonathan Butner’s Structural Equation Modeling Class, Fall 2017, University of Utah. This is the confirmatory way of factor analysis where the process is run to confirm with understanding of the data. \end{pmatrix} Due to budget constraints, the lab uses the freely available R statistical programming language, and lavaan as the CFA and structural equation modeling (SEM) package of choice. Suppose the principal investigator thinks that the third, fourth and fifth items of the SAQ are the observed indicators of SPSS Anxiety. Before we move on, let’s understand the confirmatory factor analysis model. In the case of our SAQ-8 factor analysis, $n=2,571$, $df(\mbox{User}) = 20$ and $\delta(\mbox{User} )= 534.191$ which we already known from calculating the CFI. Since we fix one factor variance, and 3 unique residual covariances, the number of free parameters is $10-(1+3)=6$. Overview. 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. Notice that the number of free parameters is now 9 instead of 6, however, our degrees of freedom is still zero. Example 25.18 Confirmatory Factor Analysis: Cognitive Abilities. Since $p < 0.05$, using the model chi-square criteria alone we reject the null hypothesis that the model fits the data. To better interpret the factor loadings, often times you would request the standardized solutions. Featured on Meta Feature Preview: New Review Suspensions Mod UX. For those readers who are more mathematically inclined, the appendix adds additional details. These interrelationships are measured by the covariances. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. SEM is provided in R via the sempackage. Confirmatory factor analysis (CFA) is used to study the relationships between a set of observed variables and a set of continuous latent variables. However, in SPSS a separate program called Amos is needed to run CFA, along with other packages such as Mplus, EQS, SAS PROC CALIS, Stata’s sem and more recently, R’s lavaan. 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. $$. As an exercise, see if you can map the path diagram above to the following regression equations:$$ For the last two decades, the preferred method for such testing has often been confirmatory factor analysis (CFA). 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. 2012) package. The data set is the WISC-R data set that the multivariate statistics textbook by the Tabachnick textbook (Tabachnick et al., 2019) employs for confirmatory factor analysis illustration. The concept of degrees of freedom is essential in CFA. \end{pmatrix} The three-item CFA is saturated (meaning df=0) because we have $3(4)/2=6$ known values and 6 free parameters. \end{pmatrix} If got warning message about non-positive definite (NPD) matrix, this may be due to the linear dependencies among the variables. The term used in the TLI is the relative chi-square (a.k.a. Confirmatory factor analysis (CFA) is a tool that is used to confirm or reject the measurement theory. The solution is to allow for fixed parameters which are parameters that are not estimated and pre-determined to have a specific value. Therefore, our degrees of freedom is zero and we have a saturated or just-identified model! \epsilon_{2} \\ To review, the model to be fit is the following: In this case, you perform factor analysis first and then develop a general idea … David Kenny states that if the CFI is less than one, then the CFI is always greater than the TLI. \end{pmatrix} In the variance standardization method Std.lv, we only standardize by the predictor (the factor, X). Similarly, we can obtain the implied variance from the diagonals of the implied variance-covariance matrix. To convert from Std.lv (which standardizes the X or the latent variable) to Std.allwe need to divide by the implied standard deviation of each corresponding item. T/F The larger the model chi-square test statistic, the larger the residual covariance. The model, which consists of two latent variables and eight manifest variables, is described in our previous post which sets up a running CFA and SEM example. Explore latent variables, such as personality using exploratory and confirmatory factor analyses. If $\delta(\mbox{User})=0$, then it means that the user model is not misspecified, so the numerator becomes $\delta(\mbox{Baseline})$ and the ratio becomes 1. The larger the chi-square value the larger the difference between the sample implied covariance matrix $\Sigma{(\hat{\theta})}$ and the sample observed covariance matrix  $S$, and the more likely you will reject your model. To see internally how lavaan stores the parameters, you can inspect your model output and request a partable or parameter table. + \end{pmatrix} The model test baseline is also known as the null model, where all covariances are set to zero and freely estimates variances. ��v� A�� �gи�U��9;+�M�έ��WP?VYZ�;�U��a5K��w���(���T��>����P@�U��A�X�ՁP�W(�Y�t�v-#�L��j�D�{h^�%����"/7"��z������G5H'��uޅ�S�6�-�֣хec��s�E����}�w�X�n0�JR����$]��6t:�'�c ��V�/'���zKu�)�ƨ̸"j�T�Q�1[1+SX���c;ڗ��� ��- The fixed parameters in the path diagram below are indicated in red, namely the variance of factor$\psi_{11}=1$and the coefficients of the residuals$\epsilon_{1}, \epsilon_{2}, \epsilon_{3}$. Without going into the technical details (see optional section), you can think of the factor residual variance as another variance parameter. \begin{pmatrix} First calculate the number of total parameters, which are 8 loadings$\lambda_1, \cdots, \lambda_8$, 8 residual variances$\theta_1, \cdots, \theta_8$and 1 variance of the factor$\psi_{11}$. \end{pmatrix} Over repeated sampling, the relative chi-square would be$10/4=2.5$. The index refers to the item number. Then$28-15=13$degrees of freedom. However if the correlations between factors are represented as regression paths, then we move beyond the scope of this seminar into what is known as structural equation modeling. The SPSS file can be download through the following link: SAQ.sav. The null and alternative hypotheses in a CFA model are. \lambda_{1} \\ Please also make sure to have the following R packages installed, and if not, run these commands in R (RStudio). \lambda_{2} \\ \begin{pmatrix} Preparing data. The goal of this document is to outline rudiments of Confirmatory Factor Analysis strategies implmented with three different packages in R. The predictor or factor, $$\eta$$ (“eta”), is unobserved whereas in a linear regression the predictors are observed. 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. More recent work by Asparouhov and Muthén (2009) blurs the boundaries between EFA and CFA, but traditionally the two methods have been distinct. 1 \\ 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$df(\mbox{User}) = 20$. Traditionally, the$\tau$is not estimated, which means that all the parameters we need can come directly from the covariance model. Taking the implied variance of Item 3, 1.155, obtain the standard deviation by square rooting$\sqrt{1.155}=1.075$we can divide the Std.lv loading of Item 3, 0.583 by 1.075 which equals 0.542 matching our results for Std.all given rounding error. \lambda_{1} & \lambda_{2} & \lambda_{3} If we were to estimate every (unique) model parameter in the model-implied covariance matrix, there would be 3$\lambda$’s, 1$\psi$, and 6$\theta$’s (since by symmetry,$\theta_{12}=\theta_{21}$,$\theta_{13}=\theta_{31}$, and$\theta_{23}=\theta_{32}$) which gives you 10 model parameters, but we only have 6 known values! By the variance standardization method, we have fixed 1 parameter, namely$\psi_{11}=1$. $$. 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). NOTE: changing the standardization method should not change the degrees of freedom and chi-square value. \end{pmatrix} Chapter 4: Refining your measure and/or model 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. \Sigma(\theta) = Cov(\mathbf{y}) = {\Lambda} \Psi \mathbf{\Lambda}’ + \Theta_{\epsilon} \begin{pmatrix} 0 & 0 & \theta_{33} \\ \lambda_{2} \\ The first argument is the user-specified model. The path diagram can assist us in understanding our CFA model because it is a symbolic one-to-one visualization of the measurement model and the model-implied covariance. \psi_{11} (Answer: 6 – 0 = 6), Models that are just-identified or saturated have df = 0, which means that the number of free parameters equals the number of known values in \Sigma. Now that we are familiar with some syntax rules, let’s see how we can run a one-factor CFA in lavaan with Items 3, 4 and 5 as indicators of your SPSS Anxiety factor. 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. \Sigma(\theta)= For simplicity, let’s assume that the known values come only from the model-implied covariance matrix. endstream The following simplified path diagram depicts the SPSS Anxiety factor indicated by Items 3, 4 and 5 (note what’s missing from the complex diagram introduced in previous sections). 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. As a general rule only paths (\lambda,\tau) and bidirectional arrows (\psi) are estimated, not circles or squares (i.e., y, \epsilon, \eta). + Because the TLI and CFI are highly correlated, only one of the two should be reported. This variance-covariance matrix can be described using the model-implied covariance matrix \Sigma(\Theta). 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. One of the most widely-used models is the confirmatory factor analysis (CFA). \theta_{11} & 0 & 0 \\ 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. 17. 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. Approximate fit indexes can be further classified into a) absolute and b) incremental or relative fit indexes. Historically, factor analysis is used to answer the question, how much common variance is shared among the items. An incremental fit index (a.k.a. General Purpose – Procedure Defining individual construct: First, we have to define the individual constructs. EFA has a longer historical precedence, dating back to the era of Spearman (1904) whereas CFA became more popular after a breakthrough in both computing technology and an estimation method developed by Jöreskog (1969). 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. These are referred to as Heywood cases and explained beautifully here (even though the linked documentation is from SAS it applies to any confirmatory factor analysis). Notice that compared to the uncorrelated two-factor solution, the chi-square and RMSEA are both lower. When fit measures are requested, lavaan outputs a plethora of statistics, but we will focus on the four commonly used ones: The model chi-square is defined as either nF_{ML} or (n-1)(F_{ML}) depending on the statistical package where n is the sample size and F_{ML} is the fit function from maximum likelihood, which is a statistical method used to estimate the parameters in your model. In a linear regression, there is only one outcome per subject. However, if the chi-square is significant, it may be possible that the rejection is due to the sensitivity of the chi-square to large samples rather than a true rejection of the model. After talking with the Principal Investigator, we choose the final two correlated factor CFA model as shown below. Note that this is in contrast to the observed population covariance matrix \Sigma which comes only from the data. Note that based on the logic of hypothesis testing, failing to reject the null hypothesis does not prove that our model is the true model, nor can we say it is the best model, as there may be many other competing models that can also fail to reject the null hypothesis. Compared to the model chi-square, relative chi-square is less sensitive to sample size. Additionally the CFI and TLI are both higher and pass the 0.95 threshold. \lambda_{1} & \lambda_{2} & \lambda_{3} \\ Conducting Multilevel Confirmatory Factor Analysis Using R Even though the chi-square fit is the same however, you will get different standardized variances and loadings depending on the the assumptions you make (to set the loadings to 1 for the two first order factors and freely estimate the variance or to freely estimate but equate the loadings and set the residual variance of the first order factors to 1). 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. Similarly, for a single item, the factor analysis model is:$$y_{1} = \tau_1 + \lambda_1 \eta + \epsilon_{1} $$. Note the following marker method below is the correct identification. This is due to the assumptions from above and properties of expectation. \begin{pmatrix} \theta_{11} & \theta_{12} & \theta_{13} \\ In this chapter, we use the sem package to implement the same two CFA analyses that we produced with lavaan in chapter 3. sem provides an equally simple way to obtain the models and only the basics are shown here. See the optional section Degrees of freedom with means for the more technically accurate explanation. Generally errors (or uniquenesses) across variables are uncorrelated. \lambda_{2} = 1 \\ An example is a fatigue scale that has previously been validated. The path diagram and the equations quickly inform us of the parameters coming either from the measurement model or model-implied covariance; and knowing how to count parameters is an essential. Some criteria claims 0.90 to 0.95 as a good cutoff for good fit [citation needed]. What would be the acceptable range of chi-square values based on the criteria that the relative chi-square greater than 2 indicates poor fit? This distinction shows up in software as well. The gives us two residual variances \theta_1, \theta_2, and one loading to estimate \lambda_1. In psychology and the social sciences, the magnitude of a correlation above 0.30 is considered a medium effect size. The lavaan code below demonstrates what happens when we intentionally estimate the intercepts. 0 & \theta_{22} & 0 \\ To understand relative chi-square, we need to know that the expected value or mean of a chi-square is its degrees of freedom (i.e., E(\chi^2(df)) = df).$$. 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. Once you’ve installed the packages, you can load them via the following, You may download the complete R code here: cfa.r. We hope you have found this introductory seminar to be useful, and we wish you best of luck on your research endeavors. 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. Since we have 7 items, the total elements in our variance covariance matrix is$7(8)/2=28$. A more common approach is to understand the data using factor analysis. \theta_{11} & 0 & 0 \\ The goal is to maximize the degrees of freedom (df) which is defined as, $$\mbox{df} = \mbox{number of known values } – \mbox{ number of free parameters}$$, How many degrees of freedom do we have now? The following describes each parameter, defined as a term in the model to be estimated: The dimensions of this matrix correspond to the same as that of the observed covariance matrix$\Sigma$, for three items it is$3 \times 3$. \begin{pmatrix} The total parameters include three factor loadings, three residual variances and one factor variance. \tau_3 + Since the focus of this seminar is CFA and R, we will focus on lavaan. Taking advantage of our correlated factors, let’s use the second option. 0 & \theta_{22} & 0 \\ Due to its goal of reproducing the observed covariance matrix, its free parameters are completely determined by the dimensions of$\Sigma$. \end{matrix} Suppose that one of the data collectors accidentally lost part of the survey and we are left with only Items 4 and 5 from the SAQ-8. &=& 0 + E( \mathbf{\Lambda} \mathbf{\eta}) + 0 \\ Can you think of other ways? \end{matrix} We can plug all of this into the following equation: $$CFI= \frac{4136.572- 534.191}{4136.572}=\frac{3602.381}{4136.572}=0.871$$. Suppose the Principal Investigator believes that the correlation between SPSS Anxiety and Attribution Bias are first-order factors is caused more by the second-order factor, overall Anxiety. Here$\bar{y}= (13+14+15)/3=14$. \tau_1 \\ \lambda_{2} \\ This is even better fitting than the one-factor solution. The goal of factor analysis is to model the interrelationships between many items with fewer unobserved or latent variables. The test of RMSEA is not significant which means that we do not reject the null hypothesis that the RMSEA is less than or equal to 0.05. \theta_{31} & \theta_{32} & \theta_{33} \\ \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$\psi_{11}$. Perhaps SPSS Anxiety is a more complex measure that we first assume. For example, given that the test statistic truly came from a chi-square distribution with 4 degrees of freedom, we would expect the average chi-square value across repeated samples would also be 4. Notice that the only parameters estimated are$\theta_1, \cdots, \theta_8$. Factor analysis is a multivariate model there are as many outcomes per subject as there are items. Conducting Multilevel Confirmatory Factor Analysis Using R Anxiety, working memory. \lambda_{3} &=& \mathbf{\Lambda} \mathbf{\eta} \begin{eqnarray} \lambda_{1} = 1 \\ For edification purposes, let’s suppose that due to budget constraints, only three items were collected from the SAQ-8. The known values serve as the upper limit of the number of parameters you can possibly estimate in your model. $$. \epsilon_{3} \theta_{11} & 0 & 0 \\ &=& \mathbf{\Lambda} E(\mathbf{\eta}) \\ & = & Var(\mathbf{\tau}) + Cov(\mathbf{\Lambda} \mathbf{\eta}) + Var(\mathbf{\epsilon}) \\ y_{2} \\ It is well documented in CFA and SEM literature that the chi-square is often overly sensitive in model testing especially for large samples. Although the results from the one-factor CFA suggest that a one factor solution may capture much of the variance in these items, the model fit suggests that this model can be improved. \end{pmatrix} y_{3} \lambda_{2} \\ There are seven residual variances \theta_1, \cdots, \theta_7, seven loadings \lambda_1, \cdots \lambda_7. There are three main differences between the factor analysis model and linear regression: We can represent this multivariate model (i.e., multiple outcomes, items, or indicators) as a matrix equation:$$ When there are only two items, you have$2(3)/2=3$elements in the variance covariance matrix. 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. 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. Note that the loadings$\lambda$are the same parameters shared between the measurement model and the model-implied covariance model. This is done because we want to run covariances on the items which is not possible with factor variables. I am using AMOS for Confirmatory Factor Analysis (CFA) and factor loadings are calculated to be more than 1 is some cases. ), 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. $$An under-identified model means that the number known values is less than the number of free parameters and an over-identified model means that the number of known values is greater than the number of free parameters. The model, which consists of two latent variables and eight manifest variables, is described in our previous post which sets up a running CFA and SEM example.To review, the model to be fit is the following: y_{1} \\ Notice that the correlations in the upper right triangle (italicized) are the same as those in the lower right triangle, meaning the correlation for Items 6 and 7 is the same as the correlation for Items 7 and 6. Suppose the Principal Investigator is interested in testing the assumption that the first items in the SAQ-8 is a reliable estimate measure of SPSS Anxiety. In the model-implied covariance, we assume that the residuals are independent which means that for example \theta_{21}, the covariance between the second and first residual, is set to to zero.$$, How many unique parameters have we fixed here? 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. \end{pmatrix} relative fit index) assesses the ratio of the deviation of the user model from the worst fitting model (a.k.a. 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. Traditionally, CFA was only concerned with the covariance matrix and only the summary statistic in the form of the covariance matrix was supplied as the raw data due to computer memory constraints. Osx� �9��y �F��DL1C Additionally, since we have two endogenous factors which have their own residual variances$\psi_{11}, \psi_{22}$. \theta_{21} & \theta_{22} & \theta_{23} \\ Outline. The cfa() function is a dedicated function for fitting confirmatory factor analysis models. The model to be estimatd is m1a and the dataset to be used is dat; storing the output into object onefac3items_a. Just as in the correlation matrix we calculated before, the lower triangular elements in the covariance matrix are duplicated with the upper triangular elements. Factor analysis can be divided into two main types, exploratory and confirmatory. The limitation of doing this is that there is no way to assess the fit of this model. Recall that the model covariance matrix can be defined by the following:$$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. So for example$\tau_1$means the intercept of the first item,$\lambda_2$is the loading of the second item with the factor and$\epsilon_{3}$is the residual of the third item, after accounting for the factor. This also makes no sense. Note that scientific notation of$1.25 \times 10^{-104}$means$125/10^{102}$which is a really small number. So how big of a sample do we need? \begin{pmatrix} \end{pmatrix} In order to undrestand the model, we have to understand endogenous and exogenous factors. %���� For example, the covariance of Item 3 with Item 4 is -0.39, which is the same as the covariance of Item 4 and Item 3 (recall the property of symmetry). 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. e.g., five factor uncorrelated; five factor correlated. }�cș�Xl )��.H���v.�������R.��c��DJ�7���������1ip���y��y��7���6ZL�w���J��]��y�n�K�9�^��ke9G��"]+�������s|��,� For over-identified models, there are many types of fit indexes available to the researcher. + With the full data available, the number of known values becomes$p(p+1)/2 + p$where$p$is the number of items. I would like to run a confirmatory factor analysis (which essentially is a structural equation model) in R testing this. 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. The figure below represents the same model above as a path diagram. This chapter will cover conducting CFAs with the sem package. 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). Even if we used the marker method, which the default, that leaves us with one less parameter,$\lambda_1$resulting in four free parameters when we only have three to work with. Two deviations, the magnitude of a guest lecture symbols we will focus on lavaan six values! 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