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. 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. $$, How many unique parameters have we fixed here? 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. Our RMSEA = 0.10 indicating poor fit, as evidence by the large $\delta(\mbox{User} )$ relative to the degrees of freedom. We can see that the uncorrelated two factor CFA solution gives us a higher chi-square (lower is better), higher RMSEA and lower CFI/TLI, which means overall it’s a poorer fitting model. You will notice that the implied variance-covariance matrix is the same as observed covariance matrix. 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). Recall that we have $p(p+1)/2$ covariances. 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. 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} 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). From the exploratory factor analysis, we found that Items 6 and 7 “hang” together. \begin{pmatrix} x�mR�n�0����|�R \end{matrix} 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. &=& \mathbf{\Lambda} E(\mathbf{\eta}) \\ Answer: We start with 10 unique parameters in the model-implied covariance matrix. For exploratory factor analysis (EFA), please refer to A Practical Introduction to Factor Analysis: Exploratory Factor Analysis. Approximate fit indexes can be further classified into a) absolute and b) incremental or relative fit indexes. We can’t measure these directly, but we assume that our observations are related to these constructs in … The most fundamental model in CFA is the one factor model, which will assume that the covariance among items is due to a single common factor. 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“). The first argument is the user-specified model. Confirmatory Factor Analysis Model or CFA (an alternative to EFA) Typically, each variable loads on one and only one factor. If you simply ran the CFA mode as is you will get the following error. Since we have 6 known values, our degrees of freedom is $6-6=0$, which is defined to be saturated. For example, typical $\phi$ variance and covariances parameters for exogenous latent variables will be incorporated into the $\Psi$ matrix which in full LISREL notation is reserved for variance and covariances parameters for endogenous latent variables. & = & \mathbf{\Lambda} Cov(\mathbf{\eta}) \mathbf{\Lambda}’ + Var(\mathbf{\epsilon}) \\ 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. Exploratory factor analysis. \lambda_{1} \\ As an exercise, see if you can map the path diagram above to the following regression equations: $$ The eight items are observed indicators of the latent or unobserved construct which the PI calls SPSS Anxiety. [FINISH]. 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. These interrelationships are measured by the covariances. \end{pmatrix} If we fix $\lambda_1 = \lambda_2$, we would be able obtain a solution, not knowing that the model is a complete false representation of the truth since we cannot assess the fit of the model. \begin{pmatrix} Before running our first factor analysis, let us introduce some of the most frequently used syntax in lavaan. 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. Recall that the magnitude of a correlation $|r|$ is determined by the absolute value of the correlation. 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. 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 ($\lambda_2 = \lambda_1$). 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. Three of the scores were for reading skills, three others were for … \end{pmatrix} relative fit index) assesses the ratio of the deviation of the user model from the worst fitting model (a.k.a. Here we name our factor f (or SPSS Anxiety), which is indicated by q01, q02 and q03 whose names come directly from the dataset. 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. 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 you are tasked with evaluating a hypothetical but real world example of a questionnaire which Andy Field terms the SPSS Anxiety Questionnaire (SAQ). Please also make sure to have the following R packages installed, and if not, run these commands in R (RStudio). \Sigma(\theta) = y_3 = \tau_3 + \lambda_{3}\eta_{1} + \epsilon_{3} \theta_{21} & \theta_{22} & \theta_{23} \\ 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. The index refers to the item number. We can recreate the p-value which is essentially zero, using the density function of the chi-square with 20 degrees of freedom $\chi^2_{20}$. 44 0 obj See if you can count the number of parameters from the equations or path diagram above. There are at least two mature packages of doing so sem and openMX. Notice in both models that the residual covariances stay freely estimated. For users of Mplus, Std.lv corresponds to STD and Std.all corresponds to STDYX. $$ Therefore, our degrees of freedom is zero and we have a saturated or just-identified model! Then the difference $S-\Sigma(\hat{\theta})$ is a proxy for the fit of the model, and is defined as the residual covariance with values close to zero indicating that there is a relatively good fit. From this table we can see that most items have magnitudes ranging from $|r|=0.38$ for Items 3 and 7 to $|r|=0.51$ for Items 6 and 7. The, In matrix notation, the marker method (Option 1) can be shown as, $$ Now that we have imported the data set, the first step besides looking at the data itself is to look a the correlation table of all 8 variables. Since we fix one factor variance, and 3 unique residual covariances, the number of free parameters is $10-(1+3)=6$. Note that there is no perfect way to specify a second order factor when you only have two first order factors. 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). 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. Factor analysis can be divided into two main types, exploratory and confirmatory. Similarly, we can obtain the implied variance from the diagonals of the implied variance-covariance matrix. 12 0 obj Alternatively you can use std.lv=TRUE and obtain the same results. 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). 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: In traditional confirmatory factor analysis or structural equation modeling, the. y_1 = \tau_1 + \lambda_{1}\eta_{1} + \epsilon_{1} \\ \lambda_{3} 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 we have 7 items, the total elements in our variance covariance matrix is $7(8)/2=28$. The term used in the TLI is the relative chi-square (a.k.a. The reason we said previously that the model parameters come only from the model-implied covariance is because with full information maximum likelihood, the intercepts (i.e., $\tau$’s) are estimated by default. Factor analysis can be divided into two main types, exploratory and confirmatory. \lambda_{3} 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. 1 \\ \lambda_{2} = 1 \\ \lambda_{1} \\ 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). \Sigma(\theta)= This is even better fitting than the one-factor solution. Research Gate Discussion about Chi-Square, Assess whole SEM model–chi square and fit index. \lambda_{2} \\ These simplified assumptions can help us calculate the expectation and the variance of the multivariate outcome $\mathbf{y}$: $$ We hope you have found this introductory seminar to be useful, and we wish you best of luck on your research endeavors. Table of Contents Data Input Confirmatory Factor Analysis Using lavaan: Factor variance identification Model Comparison Using lavaan Calculating Cronbach’s Alpha Using psych Made for Jonathan Butner’s Structural Equation Modeling Class, Fall 2017, University of Utah. Recall that we already know how to manually derive Std.lv parameter estimates as this corresponds to the variance standardization method. Browse other questions tagged r-squared confirmatory-factor item-analysis or ask your own question. The function round with the option 2 specifies that we want to round the numbers to the second digit. \end{eqnarray} If we were to estimate every (unique) model parameter in the model-implied covariance matrix, there would be 3 $\lambda$’s, 1 $\psi$, and 6 $\theta$’s (since by symmetry, $\theta_{12}=\theta_{21}$, $\theta_{13}=\theta_{31}$, and $\theta_{23}=\theta_{32}$) which gives you 10 model parameters, but we only have 6 known values! The following describes each parameter, defined as a term in the model to be estimated: The dimensions of this matrix correspond to the same as that of the observed covariance matrix $\Sigma$, for three items it is $3 \times 3$. CFA is often used to evaluate the psychometric properties of questionnaires or other assessments. (Answer: 10), The number of free parameters is defined as, $$\mbox{number of free parameters} = \mbox{number of (unique) model parameters } – \mbox{number of fixed parameters}.$$, How many free parameters have we obtained after fixing 10 (unique) model parameters? Outline. The marker method assumes that both loadings from the second order factor to the first factor is 1. $\eta$ (“eta”), the latent predictor of the items, i.e. Std.all not only standardizes by the variance of the latent variable (the X) by also by the variance of outcome (the Y). Answer: False, the residual covariance uses sample estimates $S-\Sigma(\hat{\theta})$. To obtain the sample covariance matrix $S=\hat{\Sigma}$, which is an estimate of the population covariance matrix $\Sigma$, use the column index [,3:5], and the command cov. With the full data, the total number of model parameters is calculated accordingly: $$ \mbox{number of model parameters} = \mbox{intercepts from the measurement model} + \mbox{ unique parameters in the model-implied covariance}$$. \begin{pmatrix} For example in the figure below, the diagram on the left depicts the regression of a factor on an item (essentially a measurement model) and the diagram on the right depicts the variance of the factor (a two-way arrow pointing to an latent variable). endstream e.g., five factor uncorrelated; five factor correlated. \lambda_{2} \\ SEM is provided in R via the sempackage. Our sample of $n=2,571$ is considered relatively large, hence our conclusion may be supplemented with other fit indices. The gives us two residual variances $\theta_1, \theta_2$, and one loading to estimate $\lambda_1$. \Sigma(\theta) = Confirmatory factor analysis As discussed above (background section), to begin the confirmatory facto r analysis, the researcher should have a model in mind. The model implied matrix $\Sigma(\theta)$ has the same dimensions as $\Sigma$. The goal is to maximize the degrees of freedom (df) which is defined as, $$\mbox{df} = \mbox{number of known values } – \mbox{ number of free parameters}$$, How many degrees of freedom do we have now? Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. One of the most widely-used models is the confirmatory factor analysis (CFA). There are three main differences between the factor analysis model and linear regression: We can represent this multivariate model (i.e., multiple outcomes, items, or indicators) as a matrix equation: $$ Offer ends in 10 days 01 hr 40 mins 11 secs Confirmatory factor analysis (CFA), structural equation models (SEM) and related techniques are designed to help researchers deal with these imperfections in our observations, and can help to explore the correspondence between our measures and the underlying constructs of interest. \tau_3 \begin{pmatrix} The three-item CFA is saturated (meaning df=0) because we have $3(4)/2=6$ known values and 6 free parameters. At this point, you’re really challenging your assumptions. For edification purposes, let’s suppose that due to budget constraints, only three items were collected from the SAQ-8. $$ RMSEA = \sqrt{\frac{534.191}{20(2570)}} = \sqrt{0.0104}=0.102$$. 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., $E(\chi^2(df)) = df$). \end{pmatrix} 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. Notice that there are two additional columns, Std.lv and Std.all. The items are the fundamental elements in a CFA and the covariances between the items forms the the fundamental component in CFA. The only main difference is that instead of an observed residual variance $\theta$, the residual variance of a factor is classified under the $\Psi$ matrix. <> The lavaan code below demonstrates what happens when we intentionally estimate the intercepts. Over repeated sampling, the relative chi-square would be $10/4=2.5$. \begin{pmatrix} Thank you, Tim Post. \lambda_{2} \\ $$. EFA, in contrast, does not specify a measurement model initially and usually seeks to discover the measurement model 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. \end{pmatrix} Answer: We start with 10 total parameters in the model-implied covariance matrix. \begin{pmatrix} 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). Historically, model chi-square was the only measure of fit but in practice the null hypothesis was often rejected due to the chi-square’s heightened sensitivity under large samples. \end{eqnarray} ���/R���Ԗ!��Q�>Y������[w} �����%�̷^Y�^�Kcb�{�� g��{�wOxE�gͅ�� ��ٖi��6�'�6�����>���h�BƅG�.����K5,�����Jn�Y��s����F�Ϝ�PT��?P9� Ʉ. 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). Confirmatory factor analysis (CFA) is used to study the relationships between a set of observed variables and a set of continuous latent variables. Notice that compared to the uncorrelated two-factor solution, the chi-square and RMSEA are both lower. 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). \begin{pmatrix} It specifies how a set of observed variables are related to some underlying latent factor or factors. The warning message is an indication that your model is not identified rather than a problem with the data. Alternatively you can request a more condensed output of the standardized solution by the following, note that the output only outputs Std.all. 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. 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. A rudimentary knowledge of linear regression is required to understand some of the material in this seminar. In this case, you perform factor analysis first and then develop a general idea … 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$. The formula for the model-implied covariance matrix is: $$ The null and alternative hypotheses in a CFA model are. }�cș�Xl )��.H���v.�������R.��c��DJ�7���������1ip���y��y��7���6ZL�w���J��]��y�n�K�9�^��ke9G��"]+�������s|��,� Psychometric applications emphasize techniques for dimension reduction including factor analysis, cluster analysis, and principal components analysis. Explain how to obtain 2o degrees of freedom from the 8-item one factor CFA by first calculating the number of free parameters and comparing that to the number of known values. 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. If got warning message about non-positive definite (NPD) matrix, this may be due to the linear dependencies among the variables. 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). Notice that the number of free parameters is now 9 instead of 6, however, our degrees of freedom is still zero. This is due to the assumptions from above and properties of expectation. The syntax NA*q03 frees the loading of the first item because by default marker method fixes it to one, and f ~~ 1*f means to fix the variance of the factor to one. 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. Your expectations are usually based on published findings of a 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. 0 & \theta_{22} & 0 \\ For the last two decades, the preferred method for such testing has often been confirmatory factor analysis (CFA). \end{pmatrix} In this portion of the seminar, we will continue with the example of the SAQ. 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. Similarly, in CFA the items are used to estimate all the parameters the model-implied covariance, which correspond to $\hat{\Lambda}, \hat{\Psi}, \hat{\Theta_{\epsilon}}$, the carrot or hat symbol emphasizing that these parameters are estimated. CFA expresses the degree of discrepancy between predicted and empirical factor structure in X 2 and indices of “goodness of fit” (GOF), while primary factor loadings and modification indices provide some feedback on item level. The range of acceptable chi-square values ranges between 20 (indicating perfect fit) and 40, since 40/20 = 2. Going back to our orginal marker method object onefac3items_a we request the summary but also specify that standardized=TRUE. We will understand concepts such as the factor analysis model, basic lavaan syntax, model parameters, identification and model fit statistics. Because this model is on the brink of being under-identified, it is a good model for introducing identification, which is the process of ensuring each free parameter in the CFA has a unique solution and making surer the degrees of freedom is at least zero. We will now proceed with a two-factor CFA where we assume uncorrelated (or orthogonal) factors. It permits path specification with a simple syntax. + I am interested in opinions/code on which package would be the best or perhaps easiest to specify such a model. %PDF-1.5 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. Generally errors (or uniquenesses) across variables are uncorrelated. Theoretically, the baseline model is useful for understanding how other fit indices are calculated. $$. Models are entered via RAM specification (similar to PROC CALIS in SAS). The extra parameter comes from the fact that we do not observe the factor but are estimating its variance. This seminar will show you how to perform a confirmatory factor analysis using lavaan in the R statistical programming language. When fit measures are requested, lavaan outputs a plethora of statistics, but we will focus on the four commonly used ones: The model chi-square is defined as either $nF_{ML}$ or $(n-1)(F_{ML})$ depending on the statistical package where $n$ is the sample size and $F_{ML}$ is the fit function from maximum likelihood, which is a statistical method used to estimate the parameters in your model. The model test baseline is also known as the null model, where all covariances are set to zero and freely estimates variances. The cfa() function is a dedicated function for fitting confirmatory factor analysis models. \begin{pmatrix} \psi_{11} $$ Preparing data. If we have six known values is this model just-identified, over-identified or under-identified? By the variance standardization method, we have fixed 1 parameter, namely $\psi_{11}=1$. The closer the CFI is to 1, the better the fit of the model; with the maximum being 1. Confirmatory Factor Analysis - Basic. 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. The second argument is the dataset that contains the observed variables. Item 3 has a negative relationship with Items 4 and 5 but Item 4 has a positive relationship with Item 5. $$. 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. 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. Factor analysis is a multivariate model there are as many outcomes per subject as there are items. To request additional fit statistics you add the fit.measures=TRUE option to summary, passing in the lavaan object onefac8items_a. From Wikipedia, the free encyclopedia In statistics, confirmatory factor analysis (CFA) is a special form of factor analysis, most commonly used in social research. 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. With the full data available, the number of known values becomes $p(p+1)/2 + p$ where $p$ is the number of items. A perfect fitting model which generate a TLI which equals 1. 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. \epsilon_{3} Since the focus of this seminar is CFA and R, we will focus on lavaan. Think of the null or baseline model as the worst model you can come up with and the saturated model as the best model. 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. Before fixing the 10 unique parameters, we were under-identified. 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. &=& \mathbf{\Lambda} \mathbf{\eta} Perhaps SPSS Anxiety is a more complex measure that we first assume. We use the variance standardization method. Recall from the variance covariance matrix that the diagonals are the variances of each variable. \end{pmatrix} \lambda_{2} \\ Recall from the CFI that $\delta=\chi^2 – df$ where $df$ is the degrees of freedom for that particular model. So $\delta(\mbox{Baseline}) = 4164.572 – 28 =4136.572$ and $\delta(\mbox{User} )= 554.191 – 20=534.191$. \lambda_{1} =1 & \lambda_{2} = 1  & \lambda_{3}=1 \\ David Kenny states that if the CFI is less than one, then the CFI is always greater than the TLI. 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. \theta_{21} & \theta_{22} & \theta_{23} \\ ��v� A�� �gи�U��9;+�M�έ��WP?VYZ�;�U��a5K��w���(���T��>����[email protected]�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��� ��- \end{pmatrix} y_2 = \tau_2 + \lambda_{2}\eta_{1} + \epsilon_{2} \\ In an ideal world you would have an unlimited number of items to estimate each parameter, however in the real world there are restrictions to the total number of parameters you can use. 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