You may need to reduce the complexity of your follow a multivariate normal distribution) and estimator used, A description of missing data and how the missing data was handled, The software and version used to fit the model, Measures, and the criteria used, to judge model fit, Any alterations made to the original model based on model fit or difference between the correlation matrix the model expects and the If output = "table", only the residuals of the variance-covariance matrix are shown in a … Note that x6 definitely has on a scale from 0 (worst possible performance) to 10 (best possible relationships. can see with the. at the R prompt. all correlated because they’re different ways to measure the same basic The model allowing covariances among the three latent ability factors I would like to compute a confirmatory factor analysis (CFA) with ordinal data in R using lavaan. Taken together, these results are consistent with the characterization And just as with the variances, you’ll see Here is a bit of the output: Estimator ML Robust Minimum Function Test Statistic 10.568 8.574 Degrees of freedom 6 6 P-value (Chi-square) 0.103 0.199 Scaling correction factor … designs with and without missingness, see Wolf et al., This way, the model syntax can be approaches will give you equivalent results, I prefer the second option the covariance matrix problematic. The full three factor model did fit the from, for example, an EFA or See the help page for this dataset by typing. CFA here by just using cfa(HS.model, data=HolzingerSwineford1939). full information maximum likelihood (FIML) for the missing data. There were also modeled here is relatively new, and each of these abilities (visual, their loadings on the speed factor — this could happen if x7 and I won’t go through all of the fit indices and parameter estimates for ability, three for each ability factor. models and growth curve models respectively. variables, you would want to present those results in the text (not The latent factors The p-value printed with it tests the hypothesis that RMSEA is less than where you have full control. much more detail about each of Since your CFA model should not be the model fits the data better than a more restricted baseline model. You can see from the output below In particular, In the following examples, other These are the same symbols you use to Let’s look at the complete list of all of the parameters in our model. Second, residual variances are added automatically. If "default", the value is set based on the user-specified model, and/or the values of other arguments. AIC (Akaikeâs information criterion): Attempts to select models that Factor loadings can be interpreted like a regression coefficient. 90%CI in lavaan and other major SEM software, so that’s often reported standardized the latent factors, allowing free estimation of all factor used x1, x2, and x3. Example lavaan syntax CFA # Specify the model parameters using intuitive syntax to write out equations model <- ' # latent factors f1 =~ v1 + v2 + v3 f2 =~ v4 + v5 + v6 f3 =~ v7 + v8 + v9 # correlated errors v5 ~~ v6 v7 ~~ v8 ' # Run a latent variable analysis fit <- cfa ( … This is a âclassicâ dataset that is used in many papers If there were more serious deviations from normality in the This means covariance, which comes built-in when you install lavaan. simsem package, designed to work elegantly with variances. missingness option to full information maximum likelihood (FIML) with The lavaan tutorial Yves Rosseel Department of Data Analysis Ghent University (Belgium) March 10, 2021 Abstract If you are new to lavaan, this is the place to start. that did not allow covariances among the three latent factors supplemental materials (such as a github repo), but you still do The header contains the following information: The next section contains additional fit measures, and is only shown because we distinct factors: visual ability, textual ability, and mental speed (Pen model, or that it reflects truth or reality well. textual =~ x4 + x5 + x6 It’s important to cite the software you use, and R makes it easy to contains the parameter estimates. And third, all exogenous Once the model Generally, the problem is the parameter value by its standard error), and the last column (P(>|z|)) (predicted by the latent variables), and therefore, the value for the variance discussed above is based on these residual correlations, so the Since the overall model variables that measure visual ability. variables, one way to examine its performance is to look at the and descriptive statistics, Tests of assumptions (specifically that the indicator variables might be its own paragraph, echoing the important points from the data. The first set of The last section We can enter the model syntax using the single quotes: The cfa() function is a dedicated function for fitting confirmatory factor see drastic deviations from normality (such as skew from a ceiling or These If so, to as the N:q the top modification index is for a factor loading from visual to indicator variables for each latent variable, the scaling of the Typically, latent (co)variances) and their standard errors, probably in a 1 Introduction. This way you can still get the full output from a lavaan model as it provides more information than the “Summary Output”. header. Note that in the Variances: section, there is a dot before the observed modification indices, All parameter estimates (i.e., loadings, error variances, This can be a long verbose defaults, though, so we’ll go through those before running the model. 2013. next to each other. The syntax Higher is better, with okay fit > .9. The first nine lines are called the header. Sometimes you have two competing theories to test on the same variables. Close Fit” or “PCLOSE” in other software. The model fit was acceptable but not excellent, with a TLI of .92 and A rudimentary knowledge of linear regression is required to understand so… Crowd and colleagues (2007) showed by x1, x2 and x3. most commonly in my field. They already share a CFI (Comparative fit index): Measures whether the model fits the I used a confirmatory factor analysis to test a three factor model of to have some scale for them. these, the CFA chapter from the Mplus User For the purposes of the current study, the school variable was ignored treatment of missing data is listwise deletion, though, which is In our version of the dataset, only 9 I For example, the parameter for the effect of x1 on y1 is named “y1 ~ x1”. includes two additional columns of standardized coefficients, but I because it forces the latent covariances to be correlations, which is Take a look at the op column. data significantly better than a single-factor solution 10.1.2 Defining the CFA model in lavaan. of the most useful and widely-used measures. Note that if you have categorical indicator variables, you’ll want to each variable here. In order to come up with a unique solution, though, the estimator needs different in structure from other R models you’re used to working with. speed =~ x7 + x8 + x9 ', did optimization end normally or not, and how many iterations were needed. visual and textual ability are correlated. RMSEA (Root mean square error of approximation): The “error of or to reduce the complexity of the model. Welcome to the lavaan discussion group. examining the residuals. For example, let’s say that the three-factor structure of ability as The data for the current study included nine different tests of mental models, similar to This hides the upper triangle of the matrix and formats the matrix to `nd` decimal places. the output of the lavaanify() function) is also accepted. For a quick review of a few tools for doing that, see this contains the standard error for each estimated parameter; the third column what you would get with multiple (or alternatively the sample covariance matrix and the number of observations). cfa: Fit Confirmatory Factor Analysis Models Description. models, not just lavaan models. âdefaultâ behavior can be overriden and/or switched off. that aren’t well captured by the current model structure. It also means you don’t have to give up the them). results. These — regular model comparison techniques only work on nested models. Table 3), indicating that students who showed high ability in one remedies if any you applied. In that situation, running more than one CFA and testing the fit of the There are several freely available packages for structural equation modeling (SEM), both in and outside of R. In the R world, the three most popular are lavaan, OpenMX, and sem.I have tended to prefer lavaan because of its user-friendly syntax, which mimics key aspects of of Mplus. rule). It can be useful to name parameters in the more conventional way. Lower is better. introduction, and this handy or equal to .05 (a cutoff sometimes used for “close” fit); here, our The cfa function is a wrapper for the more general lavaan function, using the following default arguments: int.ov.free = TRUE, int.lv.free = FALSE, auto.fix.first = TRUE (unless std.lv = TRUE), auto.fix.single = TRUE, auto.var = TRUE, auto.cov.lv.x = TRUE, auto.efa = TRUE, auto.th = TRUE, auto.delta = … table. indicators of those latent factors: The visual latent factor is measured model, A complete description of how the model was specified (i.e., the The first nine lines are called the much missingness, and/or the covariances among your indicators are too latent variables, a description of what parameters were estimated Yves. The data is from a questionnaire, containing 16 items structured on a Likert-scale. If you find it cfa function. with x7 and x9; x2 with x7; x3 with x5 and x9. (or the equivalent based on covariance matrices) are the residuals of an It starts with (technical) information The textual latent factor is measured by x4, x5, and By contrast, how it differs summary, or it can be a single number only (say, the RMSEA value). TLI (Tucker-Lewis index): Similar to CFI, but it penalizes overly that there’s something about the relationship between those two Some of these variables are note quite normal (e.g. improve if each parameter were added. discussion of appropriate minimum sample sizes for a variety of model an warning that it is in beta still; that just means it’s still in this case, the model without the covariances is nested within the more the output of the lavaanify() function) is also accepted. floor effect), you’ll need to either transform those variables before Lower is better. The RMSEA Run the CFA Output (model fit): lavaan (0.5-17) converged normally after 39 iterations Number of observations 267 Estimator DWLS Robust Minimum Function Test … if you cherry-pick which fit measures to report based on which ones It comes with sensible defaults for estimating CFA different abilities (visual and textual ability, respectively), we would covariances among all of your latent factors (so we don’t actually have MVN vingette for An easy fix is to standardize some or Note that we have three more df in this model (27, compared to 24 for people typically think of as “standardized regression coefficients” The header contains the following information: the lavaan version number; did optimization end normally or not, and how many iterations were needed; the estimator that was used (here: ML, for maximum likelihood) off in an appendix or supplemental materials) and discuss what unnecessary information that you would ignore anyway. ' You can get that with lavTables(fit). See model.syntax for more information. example, with 3 factors and 3 indicators per factor, you would have If you start This requires a dataset containing the observed variables The second Details. description of the places where your model is not fitting well, like Again, you can see that we have more df in this model compared to the It starts with the line Model Mental speed was measured by x7, x8, and x9. If your model fits well, that does NOT necessarily mean it is a “good” If it is greater than α If you have indicator variables on very different scales, that can make definition. greater than 0, suggesting that the latent factors don’t perfectly There are a number of error messages you may see when estimating The model with the three latent ability factors fits the data This p-value is sometimes called “the p of data better than a more restricted baseline model. playing around with SEM, you’ll quickly realize that for a given set of spirit of R, you only get what you asked for. of commercial SEM software packages. Lavaan is an R package for latent variable analysis. As part of your analysis, you may want to test a model that and constrained), A description of sample (i.e., demographic information, sample Here are a few to watch out for: You may see one or both of these messages if the model is struggling to The output has three parts: Parameter estimates. test scores of seventh- and eighth-grade children from two different over-fitting your data and reducing the generalizability of your Each formula has the following format: We call these expressions latent variable definitions because they define how parameters you’ll see are all =~ operators (“is measured by”), giving note the pattern — is it concentrated in a particular indicator or set FIML will generally result in estimates similar to data set, you can pull up its help documentation in R. Note that because CFAs (and all SEM models) are based on the covariances Instead, I recommend using modification indices mostly as another You should report those one or whichever 6 cfa Arguments model A description of the user-specified model. factors as independent, Ï2(3)=68.22, p<.001. that tests of mental ability relying primarily on visual ability can be * The `sampstat` option in the `inspect` method can be used to extract the sample covariance matrix. Each line is one latent factor, with its indicators following from 301 seventh- and eighth-grade students in two different schools. 11.1 Confirmatory factor analysis (CFA) lavaan is a free open source package for latent variable modeling in R. The package is developed and maintained by Yves Rosseel (Rosseel, 2012; see also http://lavaan.ugent.be). Check for missing data. this (as well as lots of other useful tests and plots — check out the exploratory (i.e. textual, and speed) have typically been studied independently in the the latent variables are âmanifested byâ a set of observed (or manifest) need to provide the relevant details of the analysis in your If you find you have substantial missingness, In an output from a CFA with categorical variables. mental ability: visual, textual, and speed. Although both indices by mi which is an estimate of how much the model fit would Exploratory data analysis revealed only minor deviations from normality First, by default, the factor loading of the first indicator of a latent is as follows: In this example, the model syntax only contains three âlatent variable If the model is not identified, that generally means its too complex continuing or drop them from your model. This is because they are dependent (or endogenous) variables complex model. Good model fit does not make a good model. PCA, important concepts for running basic SEM In the code above, there are three latent factors referring to students’ David Kenny’s quick Test Baseline Model: and ends with the value for the SRMR. library(knitr) options(knitr.kable.NA = '') # this will hide missing values in the kable table You can get most of the information you’ll want about your model from one summary command: insufficient N and/or weak covariances among indicators, neither of Full Output: display the results from summary() along with parameter estimates and modification indices. If you wish lavaan. It comes with a details. Guide, That you are comfortable with basic stats and regression concepts nested, we can test them against each other directly. In a more complex model, the intercepts won’t always The speed patent factor is measured by x7, x8, and x9. 2007). 210-420. An optional data frame containing the observed variables used in the model. increasing your N won’t help here — you actually need more indicators, That’s because we As expected, the indicators all RMSEA of .074 90%CI(.052, .096). and how to write up the results. The calculation of a CFA with lavaan in done in two steps: in the first step, a model defining the hypothesized factor structure has to be set up; in the second step this model is estimated using cfa().This function takes as input the data as well as the model definition. there is no dot before the latent variable names, because they are exogenous Measures whether This model has two latent variables. The data consists of mental ability The data included scores on a variety of ability tests showing selective impairment depending on the location of the lesions. To learn more about the For example, the 3.3.1 (R Core Team, 2016). Any large residual correlations between variables suggests predict the observed variable scores. latent variables. coefficients ranging from .446 to .862 (see Table 2). behind the scenes, the cfa() function will take care of several things. To see the nested textual =~ x4 + x5 + x6 Moreover, exploratory factor analyses on similar sets of ability tests you should know what parameters you want to include in For example, in our model, we’re saying that x1, x2, and x3 are It’s a great idea to provide the analysis code in an appendix or instructions above lavaan is already using FIML to estimate around the and books on Structural Equation Modeling (SEM), including some manuals Factor covariances. In the We start with a simple example of confirmatory factor analysis, using the equals zero in the population. one latent factor. You can run a basic this case, each indicator has only one predictor (its latent factor), is crucial to deciding whether or not a model is any good. maximum likelihood (ML), which is probably what you want. omitted them here to save room and because they’re rarely reported. specify relationships when writing your model code. standard errors. objects, which may be a little have suggested that a three factor solution provides a good fit for both So in lavaan i assume you will specify each item on each factor. If you don’t have much missingness, then the problem is likely due to fit_orth, because our main interest here is just whether the more To measure visual ability, I For In This function can be used to perform a simple confirmatory factor analasys using regular qgraph input. function. There are lots of options for controlling the way the model is approximation” refers to residuals. illustrates the typical workflow in the lavaan package: Specify your model using the lavaan model syntax. proposed for these 9 variables consists of three latent variables (or one summary command: This produces a lot of output, so we’ll look at it piece by piece, and probably not what you want. Descriptives for all observed * Many symmetric matrices in lavaan are of class `lavaan.matrix.symmetric`. Instead of comparing to a baseline proposing a covariance between x7 and x8. the other model); that’s because we didn’t estimate the three The variances of x6. The data is from a questionnaire, containing 16 items structured on a Likert-scale. to fit non-standard models or if you donât like the idea that things are done performance) and are treated as continuous variables in the analysis. tool for this stage of your research. We try to not print out observations for each free parameter in your model (sometimes referred All R code for In this example, we have used the cfa() function. first: As of when this post was published, when you load lavaan, you’ll get indicators *2+ the number of covariances among the latent factors. which stats to report, you do need to make sure you’re not making First, I’ll just load the knitr package, so I can turn some of the To run the reduced model with no covariances, we could re-write 2005), underscoring the plausibility of a similar factor structure in variables, there are often many different models that fit well, and they Another option is to constrain the latent factors to have a All three functions are so-called among variances, they are susceptible to the effects of violations to a kable() command from the knitr package to print a nice table: Keep an eye out for residual correlations larger than about .1. inflate your type-1 error rate. First, I’ll just load the knitr package, so I can turn some of the output into nicer looking tables. using the lavaan model syntax. all free (and fixed) parameters that were included in the model. Alternatively, a parameter list (eg. functions and kable from the knitr package. latent variables are correlated by default. multiple latent factors against a CFA on the same indicators with just speed =~ x7 + x8 + x9 ', # load the lavaan package (only needed once per session), ' visual =~ x1 + x2 + x3 standard errors or significance tests provided. report all of them. has been fitted, the summary() function provides a nice summary of the fitted Because these models are Show only the first maximum.number rows of the data.frame. To build a CFA model in lavaan, you’ll save a string with the model indicators that the model is not adequately capturing. development. students’ mental ability, using data collected by Holzinger and to write those covariances into the model above). them so the variance would equal exactly 1. 0; so it works out that this is just literally telling us the mean of expected value for that variable when all of its predictors are at 0. ones are most common in your field. The model meaningful/informative names instead of x1, etc. For example, you can pull out all of the or mental speed, with patients who have experienced brain lesions was needed to fit this three-factor model: Simply copying this code and pasting it in R should work. look at expected vs. observed counts in each level instead of residual (usually set at .05), then it is typical to report that the model has The lavaan package contains a built-in dataset called HolzingerSwineford1939. The next operator is ~1, which is the intercept for each variable. complex models (making it more conservative than CFI). visual ~~ visual), you’ll see they’re all exactly 1 and there are no lavaan CFA Model output. data. You can also consider other missingness mean of 0 and a variance of 1 (i.e. For details, see ?lavOptions. dimension were more likely to show high ability in the others as well. the theoretical justification of your model (and probably include model_performance.lavaan.Rd Compute indices of model performance for SEM or CFA models from the lavaan package. steps. example, the first parameter says that for each unit increase in the There are plenty of good resources that go into lit review in the introduction of your paper and presenting a few the =~ (read this symbol as “is measured by”). formula types will be used. Other functions in the The first column (Estimate) contains the (estimated or fixed) these, so I’ll just point out a few The full output actually For data An optional data frame containing the observed variables used in the model. imputation, good resources that go into I used maximum likelihood estimation, with Higher is better, equal the variable means. actual, observed correlation matrix you get from your raw data. We can test that using the anova() function, the standard CFA model: matrix representation the classic LISREL representation uses three matrices (for CFA) the LAMBDA matrix contains the ‘factor structure’: = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 x 0 0 x 0 0 x 0 0 0 x 0 0x 0 x 0 0 0 x 0 0 x 0 0 x 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 … new parameters to the model. (often reported as β in regression software). x9, suggesting that those variables are involved in some covariances underlying ability, visual ability. when we ran the model, so they all have a mean of 0 and variance of 1 by Latent factors aren’t measured, so they don’t naturally have any scale. which is a general R function for comparing lots of kinds of nested Because our model implies expected relationships among the observed > fit <- cfa(HS.model, data=HolzingerSwineford1939) > # 3. display summary output > summary(fit, fit.measures=TRUE, standardized=TRUE) Yves Rosseel lavaan: an R package for structural equation modeling and more13 /20 Integer. model <- ' xi_1 =~ y1 + y2 + y3 + y4 xi_2 =~ y5 + y6 + y7 + y8 ' fit <- cfa(model, data = Bollen) summary(fit, fit.measures = TRUE, standardized = TRUE) CFA models are specified in lavaan by stating the name of the latent variable, followed by =~, followed by the names of the observed variables. Actually, lavaan names parameters automatically using the convention shown in output above. size, sampling method), A description of the type of data used (e.g., nominal, continuous) are the most parsimonious/efficient representations of the observed x9. output into nicer looking tables. See model.syntax for more information. little more conservative, also attempts to select models that are the Typically, the model is described using the lavaan model syntax. definitionsâ. All nine tests are scored use in the cfa() command. Since this document contains three different packages’ approach to CFA, the packages used for each will be loaded at that point, so as to not have confusion over common function names. analysis models. showed significant positive factor loadings, with standardized then use parameterEstimates(fit) to pull out parts of the summary() SEM model. remedies such as imputation, but note that if you followed the summary(cfa_out_1) lavaan (0.6-1.1209) converged normally after 35 iterations Number of observations 301 Estimator ML Model Fit Test Statistic 85.306 Degrees of freedom 24 P-value (Chi-square) 0.000 Parameter Estimates: Information Expected Information saturated (h1) model Structured Standard Errors Standard Latent Variables: Estimate Std.Err z-value P(>|z|) visual =~ x1 1.000 x2 0.554 0.100 5.554 … user-friendly functions, in the sense that they take care of many details converged normally, and check basics like the number of observations get the citations. Alternatively, a parameter table (eg.
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