For the purpose of demonstration, we retain the raw data. The next stage is to draw the measurement model. This can be done using the Indicator Icon. After selecting the Indicator Icon, move to the blank Path Diagram page. Holding the click, draw a medium-sized ellipse on the page using the mouse. Next, click the ellipse shape the same number of times as you have observed variables.
Starting with the animosity latent factor, click 5 times to represent its five observed variables. You now have one latent factor ready to populate.
Now add the remaining three latent factors with the following number of observed variables: ethnocentrism three variables , brand attitude 2 items , and perceived fit 2 items. You can move or rotate the factor using the lorry icon or the rotate icon.
Between all of the ellipses, add a double-headed covariance line from the icon screen. Your model should approximately look like the one in Figure 3. The next task is to provide a name for the latent factors ellipses and errors small circles. Hovering over one of the latent factors, right click and select the following:. In the Variable Name box, insert the latent variable name i. Do the same for all latent factors. Clicking on the Variable List Icon see Figure 5 , drag the relevant observed variable to the rectangular observed variable boxes in the model.
Each variable should occupy its own box. Having done this for all four latent factors, your model should look something like the one in Figure 6. Note : If the full label appears for each variable, follow this sequence:. Having finished the specification , you can now estimate the model. Click the Calculate Estimates icon piano keys.
Marginal analysis. Contrasts for generalized SEM. Pairwise comparisons for generalized SEM. Explore more about SEM in Stata. Checkout Continue shopping. Stata: Data Analysis and Statistical Software. Go Stata. Purchase Products Training Support Company. ORDER STATA Structural equation modeling SEM Estimate mediation effects, analyze the relationship between an unobserved latent concept such as depression and the observed variables that measure depression, model a system with many endogenous variables and correlated errors, or fit a model with complex relationships among both latent and observed variables.
Fit models with continuous, binary, count, ordinal, fractional, and survival outcomes. Even fit multilevel models with groups of correlated observations such as children within the same schools. Evaluate model fit. Compute indirect and total effects. Fit models by drawing a path diagram or using the straightforward command syntax.
Model specification Use the SEM Builder or command language SEM Builder uses standard path diagrams Command language is a natural variation on path diagrams Group estimation is as easy as adding group sex. Also easily add or relax constraints— ginvariant mcoef constrains all coefficients in the measurement model to be equal across groups. Or add paths for some groups but not others. Watch Multiple-group generalized SEM. This is because by adding a regression path from Reading to Arithmetic, we presume that Reading accounts for all the variance in Arithmetic hence a residual covariance is unnecessary.
Try making the path analysis model above a saturated model by adding a covariance between the residuals of the endogenous variables. How do you interpret this covariance? We see that the path analysis Model 4A as well as the multivariate regressions Models 3A and 3D are over-identified models which means that their degrees of freedom is greater than zero. Over-identified models allow flexibility in modeling the remaining degrees of freedom. For example in Model 4A, we can add an additional path between ppsych and read but we can also add a covariance between.
Adding either of these parameters results in a fully saturated model. Without a strong a priori hypothesis, it may be difficult ascertain the best parameter to estimate. One solution is to use the modification index, which is a one degree of freedom chi-square test that assesses how the model chi-square will change as a result of including the parameter in the model.
The higher the chi-square change, the bigger the impact of adding the additional parameter. To implement the modification index in lavaan , we must input into the modindices function a previously estimated lavaan model, which in this case is fit4a. The modification index is suggesting that we regard Motivation as an endogenous predictor and Arithmetic as its exogenous predictor. The expected change from zero in this regression coefficient would be 8.
Although this sounds promosing, not all modification index suggestions make sense. Recall that the chi-square for Model 4A is 4. Additionally, Rows 17, 20, 11, 19, 10 and 22 have a chi-square change of zero which would make their modifications unnecessary. For this example, we decide to look at the second row which suggests that we maintain Arithmetic as an endogenous variable but add Negative Parental Psychology as an exogenous variable. Based on our hypothesis, it makes sense that Negative Parental Psychology predicts both Arithmetic and Reading.
Not all modifications to the model make sense. However, by adding residual covariances, you are modeling unexplained covariance between variables that by definition are not modeled by your hypothesized model. Although modeling these covariances artificially improves fit, they say nothing about the casual mechanisms your model hypothesizes. Each modification index is a 1 degree of freedom chi-square difference test which means that it only tests one parameter at a time, although the algorithm simultaneously estimates all parameter changes.
Two fully saturated models must have exactly the same estimated parameters. There are many ways to specify a saturated model that results in the same zero degrees of freedom. Just because a model is saturated does not mean it is the best model because there may be many more equivalently saturated models.
Why is there no modification index left in fit4b? Is the chi-square you obtain the same or different from the modification index? Answer: There is no modification index because it is a saturated model. The modification index is close to the Test User chi-square but slightly bigger since it is an approximation. In Model 4B, we see that the degrees of freedom is zero which is what we expected from the modification index of 4. Recall that for Model 4A, the chi-square was 4. Additionally, the coefficient estimate is 0.
In general, the modification index is just an approximation of the chi-square based on the Lagrange multiplier and may not match the results of actually re-running the model and obtaining the difference of the chi-square values.
The degrees of freedom however, match what we expect, which is a one degree of freedom change resulting in our saturated path analysis model Model 4B. Just because modification indexes present us with suggestions for improving our model fit does not mean as a researcher we can freely alter our model.
Note that Type I error is the probability of finding a false positive, and altering our model based on modification indexes can have a grave impact on our Type I error.
This means that you may be finding many statistically significant relationships that fail to be replicated in another sample.
See our page on Retiring Statistical Significance for more information about the consequences of tweaking your hypothesis after your hypothesis has been made. Modification indexes gives suggestions about ways to improve model fit, but it is helpful to assess the model fit of your current model to see if improvements are necessary. As we have seen, multivariate regression and path analysis models are not always saturated, meaning the degrees of freedom is not zero.
SEM is also known as covariance structure analysis, which means the hypothesis of interest is regarding the covariance matrix. The null and alternative hypotheses in an SEM model are. Typically, rejecting the null hypothesis is a good thing, but if we reject the SEM null hypothesis then we would reject our user model which is bad. Failing to reject the model is good for our model because we have failed to disprove that our model is bad.
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. By default, lavaan outputs the model chi-square a.
To request additional fit statistics you add the fit. When fit measures are requested, lavaan outputs a plethora of statistics, but we will focus on the four commonly used ones:.
Then depending on the software,. The Test Statistic is 4. It is well documented in CFA and SEM literature that the chi-square is often overly sensitive in model testing especially for large samples. David Kenny states that for models with 75 to cases chi-square is a reasonable measure of fit, but for cases or more it is nearly almost always significant.
Model chi-square is sensitive to large sample sizes, but does that mean we stick with small samples? The answer is no, larger samples are always preferred. CFA and the general class of structural equation model are actually large sample techniques and much of the theory is based on the premise that your sample size is as large as possible.
So how big of a sample do we need? A sample size less than is almost always untenable according to Kline. Suppose we modified Model 4A to become a baseline model, we would take out all paths and covariances; essentially estimating only the variances. To model this in lavaan fit a model that simply estimates the variances of every variable in your model. To confirm whether we have truly generated the baseline model, we compare our model to the Model Test Baseline Model in lavaan.
We see that the User Model chi-square is We will see in the next section how baseline models are used in testing model fit. For over-identified models, there are many types of fit indexes available to the researcher. To resolve this problem, approximate fit indexes that were not based on accepting or rejecting the null hypothesis were developed.
Approximate fit indexes can be further classified into a absolute and b incremental or relative fit indexes. An incremental fit index a. Conceptually, if the deviation of the user model is the same as the deviation of the saturated model a.
Alternatively, the more discrepant the two deviations, the closer the ratio is to 0 see figure below. An absolute fit index on the other hand, does not compare the user model against a baseline model, but instead compares it to the observed data. The CFI or confirmatory factor index is a popular fit index as a supplement to the model chi-square. The formula for the CFI is:.
We can plug all of this into the following equation:. Verify that the calculations match lavaan output. The closer the CFI is to 1, the better the fit of the model; with the maximum being 1. Some criteria claims 0. 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. The term used in the TLI is the relative chi-square a. Compared to the model chi-square, relative chi-square is less sensitive to sample size.
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. 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.
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. 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?
Note that the TLI can be greater than 1 but for practical purposes we round it to 1. Given the eight-item one factor model:. The more similar the deviation from the baseline model, the closer the ratio to one. A perfect fitting model which generate a TLI which equals 1. CFI pays a penalty of one for every parameter estimated. 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.
The cutoff criteria as defined in Kline , p. Given that the p-value of the model chi-square was less than 0. We have talked so far about how to model structural relationships between observed variables. A measurement model is essentially a multivariate regression model where the predictor is an exogenous or endogenous latent variable a.
The model is defined as. Then the path diagram Model 5A for our factor model looks like the following. The extra parameter comes from the fact that we do not observe the factor but are estimating its variance. In order to identify a factor model with three or more items, there are two options known as the marker method and the variance standardization method. For the variance standardization method, go through the process of calculating the degrees of freedom. If we have six known values is this model just-identified, over-identified or under-identified?
Answer: We start with 10 unique parameters in the model-implied covariance matrix. The first line is the model statement. Here we name our factor risk , which is indicated by verbal , ses and ppsych note the names must match the variable names in the dataset. Then store the model into object m5a for Model 5A. The second line is where we specify that we want to run the analysis using the sem function, which is actually a wrapper for the lavaan function. The model to be estimated is m5a and the dataset to be used is dat ; storing the output into object fit5a.
By default, lavaan chooses the marker method Option 1 if nothing else is specified. To better interpret the factor loadings, often times you would request the standardized solutions. Notice two additional columns in the output, Std. For users of Mplus, Std. In the variance standardization method Std. Answer: : Refer to Running a one-factor CFA in lavaan for more details on how perform variance standardization in lavaan.
Not all latent variables are exogenous. Note that the direction of the arrows face the right. In LISREL path diagram notation, exogenous latent variables have measurement arrows pointing to the left and endogenous latent variables have measurement arrows pointing to the right.
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