Introductory SEM using lavaan
Michael Hallquist
15 Aug 2018
1 Introduction
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. Although OpenMX provides a broader set of functions, the learning curve is steeper.
Here, we will consider models in which all variables are observed/manifest, as well as models with latent variables. The first is sometimes called ‘path analysis’, whereas the latter is sometimes called a ‘measurement model.’
2 Use lavaan for simple multiple regression
SEM is largely a multivariate extension of regression in which we can examine many predictors and outcomes at once. SEM also provides the innovation of examining latent structure (i.e., where some variables are not observed). More specifically, the idea of ‘structural equations’ refers to the fact that we have more than one equation representing a model of covariance structure in which we (usually) have multiple criterion variables and multiple predictors.
Let’s start with the simple demonstration that a path model in SEM can recapitulate simple single predictor-single outcome regression. We’ll examine housing price data from Boston’s 1970 census to review important concepts in correlation and regression. This is a nice dataset for regression because there are many interdependent variables: crime, pollutants, age of properties, etc.
We can estimate the same multiple regression models using the lavaan package that we’ll be using for more complex SEMs. And the syntax even has many similarities with lm().
Here’s the single-predictor regression from above, run as a path model in lavaan:
#example dataset from mlbench package with home prices in Boston by census tract
data(BostonHousing2)
BostonSmall <- BostonHousing2 %>% dplyr::select(
cmedv, #median value of home in 1000s
crim, #per capita crime by town
nox, #nitric oxide concentration
lstat, #proportion of lower status
rad #proximity to radial highways
) %>% mutate(log_crim = log2(crim))
lavaan_m <- 'cmedv ~ log_crim'
mlav <- sem(lavaan_m, data=BostonSmall)
summary(mlav)## lavaan 0.6-2 ended normally after 11 iterations
##
## Optimization method NLMINB
## Number of free parameters 2
##
## Number of observations 506
##
## Estimator ML
## Model Fit Test Statistic 0.000
## Degrees of freedom 0
##
## Parameter Estimates:
##
## Information Expected
## Information saturated (h1) model Structured
## Standard Errors Standard
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## cmedv ~
## log_crim -1.346 0.116 -11.567 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .cmedv 66.549 4.184 15.906 0.000And for comparison, the output of lm()
summary(lm(cmedv ~ log_crim, BostonSmall))##
## Call:
## lm(formula = cmedv ~ log_crim, data = BostonSmall)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.31 -5.17 -2.48 2.66 33.30
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 21.013 0.386 54.4 <2e-16 ***
## log_crim -1.346 0.117 -11.5 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.17 on 504 degrees of freedom
## Multiple R-squared: 0.209, Adjusted R-squared: 0.208
## F-statistic: 133 on 1 and 504 DF, p-value: <2e-16The regression coefficient is identical (good!). One thing to note is that we don’t have an intercept in the lavaan output. This highlights an important difference that basic SEM often focuses on the covariance structure of the data. We can include means as well, but typically only when it’s relevant to our scientific questions. For example, do males and females differ on mean level of a depression latent factor?
2.1 Mean structure
We can ask lavaan to include the mean (intercept) in the model in this case using meanstructure=TRUE:
mlav_w_intercept <- sem(lavaan_m, data=BostonSmall, meanstructure=TRUE)
summary(mlav_w_intercept)## lavaan 0.6-2 ended normally after 12 iterations
##
## Optimization method NLMINB
## Number of free parameters 3
##
## Number of observations 506
##
## Estimator ML
## Model Fit Test Statistic 0.000
## Degrees of freedom 0
##
## Parameter Estimates:
##
## Information Expected
## Information saturated (h1) model Structured
## Standard Errors Standard
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## cmedv ~
## log_crim -1.346 0.116 -11.567 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .cmedv 21.013 0.386 54.494 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .cmedv 66.549 4.184 15.906 0.0002.2 Details about parameters in the model
It’s good to get in the habit of examining the ‘parameter’ table in lavaan, which provides an important summary of what parameters are free in the model (i.e., have to be estimated), and what parameters were requested by the user (you!) in the model syntax.
parTable(mlav)| id | lhs | op | rhs | user | block | group | free | ustart | exo | label | plabel | start | est | se |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | cmedv | ~ | log_crim | 1 | 1 | 1 | 1 | NA | 0 | .p1. | 0.00 | -1.35 | 0.116 | |
| 2 | cmedv | ~~ | cmedv | 0 | 1 | 1 | 2 | NA | 0 | .p2. | 42.07 | 66.55 | 4.184 | |
| 3 | log_crim | ~~ | log_crim | 0 | 1 | 1 | 0 | NA | 1 | .p3. | 9.71 | 9.71 | 0.000 |
Here, ‘user’ refers to a parameter we’ve request explicitly in the syntax, and non-zero values for the ‘free’ column denote parameters that are freely estimated by the model.
Note that we can get standardized estimates in lavaan as well. This is a more complicated topic in SEM because we can standardize with respect to the latent variables alone (std.lv) or both the observed and latent variables (std.all). The latter is usually what is reported as standardized estimates in SEM papers.
2.3 Standardized estimates
standardizedSolution(mlav, type="std.all")| lhs | op | rhs | est.std | se | z | pvalue | ci.lower | ci.upper |
|---|---|---|---|---|---|---|---|---|
| cmedv | ~ | log_crim | -0.457 | 0.033 | -13.7 | 0 | -0.523 | -0.392 |
| cmedv | ~~ | cmedv | 0.791 | 0.030 | 26.0 | 0 | 0.731 | 0.851 |
| log_crim | ~~ | log_crim | 1.000 | 0.000 | NA | NA | 1.000 | 1.000 |
3 Path analysis on housing data
Let’s look at something more interesting. What if we believe that the level nitric oxides (nox) also predicts home prices alongside crime? We can add this as a predictor as in standard multiple regression.
Furthermore, we hypothesize that the proximity of a home to large highways (rad) predicts the concentration of nitric oxides, which predicts lower home prices?
The model syntax could be specified as:
lavaan_m2 <- '
cmedv ~ log_crim + nox #crime and nox predict lower home prices
nox ~ rad #proximity to highways predicts nox
'
mlav2 <- sem(lavaan_m2, data=BostonSmall)## Warning in lav_data_full(data = data, group = group, cluster = cluster, :
## lavaan WARNING: some observed variances are (at least) a factor 1000 times
## larger than others; use varTable(fit) to investigateThe model looks like this (using the handy semPaths function from semPlot):
semPaths(mlav2, what='std', nCharNodes=6, sizeMan=10,
edge.label.cex=1.25, curvePivot = TRUE, fade=FALSE)
Here’s the text output:
summary(mlav2)## lavaan 0.6-2 ended normally after 33 iterations
##
## Optimization method NLMINB
## Number of free parameters 5
##
## Number of observations 506
##
## Estimator ML
## Model Fit Test Statistic 274.360
## Degrees of freedom 2
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Information Expected
## Information saturated (h1) model Structured
## Standard Errors Standard
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## cmedv ~
## log_crim -0.925 0.135 -6.831 0.000
## nox -14.391 3.643 -3.950 0.000
## nox ~
## rad 0.008 0.000 17.382 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .cmedv 65.499 4.118 15.906 0.000
## .nox 0.008 0.001 15.906 0.000A few things to note:
- Note the warning: ‘some observed variances are (at least) a factor 1000 times larger than others.’ This is called ill conditioning.
- Our hypotheses all seem to be supported.
- The model chi-square is highly significant, suggesting poor global model fit.
3.1 Ill conditioning
Parameter estimation can be hampered when the variances of variables in the model differ substantially (orders of magnitude). Given the above warning, let’s take a look.
varTable(mlav2)| name | idx | nobs | type | exo | user | mean | var | nlev | lnam |
|---|---|---|---|---|---|---|---|---|---|
| cmedv | 1 | 506 | numeric | 0 | 0 | 22.529 | 84.312 | 0 | |
| nox | 3 | 506 | numeric | 0 | 0 | 0.555 | 0.013 | 0 | |
| log_crim | 6 | 506 | numeric | 1 | 0 | -1.126 | 9.729 | 0 | |
| rad | 5 | 506 | numeric | 1 | 0 | 9.549 | 75.816 | 0 |
Looks like the scale of nox is much smaller than any other predictor, likely because it’s in parts per 10 million! We can rescale variables in this case by multiplying by a constant. This has no effect on the fit or interpretation of the model – we just have to recall what the new units represent. Also, you can always divide out the constant from the parameter estimate to recover the original units, if important.
BostonSmall <- BostonSmall %>% mutate(nox = nox*100) #not parts per 100,000 not 10 million
mlav2 <- sem(lavaan_m2, data=BostonSmall)
summary(mlav2)## lavaan 0.6-2 ended normally after 18 iterations
##
## Optimization method NLMINB
## Number of free parameters 5
##
## Number of observations 506
##
## Estimator ML
## Model Fit Test Statistic 274.360
## Degrees of freedom 2
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Information Expected
## Information saturated (h1) model Structured
## Standard Errors Standard
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## cmedv ~
## log_crim -0.925 0.135 -6.831 0.000
## nox -0.144 0.036 -3.950 0.000
## nox ~
## rad 0.814 0.047 17.382 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .cmedv 65.499 4.118 15.906 0.000
## .nox 83.910 5.275 15.906 0.0003.2 Model fit indices
You can request more detailed global fit indices from lavaan in the model summary output using fit.measures=TRUE.
summary(mlav2, fit.measures=TRUE)## lavaan 0.6-2 ended normally after 18 iterations
##
## Optimization method NLMINB
## Number of free parameters 5
##
## Number of observations 506
##
## Estimator ML
## Model Fit Test Statistic 274.360
## Degrees of freedom 2
## P-value (Chi-square) 0.000
##
## Model test baseline model:
##
## Minimum Function Test Statistic 638.018
## Degrees of freedom 5
## P-value 0.000
##
## User model versus baseline model:
##
## Comparative Fit Index (CFI) 0.570
## Tucker-Lewis Index (TLI) -0.076
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -3614.747
## Loglikelihood unrestricted model (H1) -3477.566
##
## Number of free parameters 5
## Akaike (AIC) 7239.493
## Bayesian (BIC) 7260.626
## Sample-size adjusted Bayesian (BIC) 7244.755
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.519
## 90 Percent Confidence Interval 0.468 0.571
## P-value RMSEA <= 0.05 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.090
##
## Parameter Estimates:
##
## Information Expected
## Information saturated (h1) model Structured
## Standard Errors Standard
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## cmedv ~
## log_crim -0.925 0.135 -6.831 0.000
## nox -0.144 0.036 -3.950 0.000
## nox ~
## rad 0.814 0.047 17.382 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .cmedv 65.499 4.118 15.906 0.000
## .nox 83.910 5.275 15.906 0.000You can also get just the fit measures (including additional statistics) using fitmeasures():
fitmeasures(mlav2)## npar fmin chisq
## 5.000 0.271 274.360
## df pvalue baseline.chisq
## 2.000 0.000 638.018
## baseline.df baseline.pvalue cfi
## 5.000 0.000 0.570
## tli nnfi rfi
## -0.076 -0.076 1.000
## nfi pnfi ifi
## 0.570 0.228 0.572
## rni logl unrestricted.logl
## 0.570 -3614.747 -3477.566
## aic bic ntotal
## 7239.493 7260.626 506.000
## bic2 rmsea rmsea.ci.lower
## 7244.755 0.519 0.468
## rmsea.ci.upper rmsea.pvalue rmr
## 0.571 0.000 4.249
## rmr_nomean srmr srmr_bentler
## 4.249 0.090 0.090
## srmr_bentler_nomean srmr_bollen srmr_bollen_nomean
## 0.090 0.089 0.089
## srmr_mplus srmr_mplus_nomean cn_05
## 0.089 0.089 12.050
## cn_01 gfi agfi
## 17.987 0.748 -0.259
## pgfi mfi ecvi
## 0.150 0.764 0.562These look atrocious: CFI is < .95 (and much less than even .9), and RMSEA is much greater than the .08 level that we would consider just ‘okay.’
3.3 Model diagnostics
This suggests the need to examine the fit in more detail. First, we can look at the mismatch between the model-implied and observed covariance matrices.
Conceptually, the goal of structural equation modeling (SEM) is to test whether a theoretically motivated model of the covariance among variables provides a good approximation of the data.
More specifically, we are trying to test how well a parsimonious model (composed of measurement and/or structural components) reproduces the observed covariance matrix. Formally, we are seeking to develop a model whose model-implied covariance matrix approaches the sample (observed) covariance matrix.
\[ \mathbf{S_{XX}} \approx \mathbf{\Sigma}(\hat{\theta}) \]
We can obtain these from lavaan for further diagnosis of model misfit.
First, the model-implied covariance matrix:
fitted(mlav2)## $cov
## cmedv nox lg_crm rad
## cmedv 81.58
## nox -36.69 134.01
## log_crim -11.69 18.82 9.71
## rad -30.25 61.57 23.13 75.67
##
## $mean
## cmedv nox log_crim rad
## 0 0 0 0We might be able to interpret this more easily in correlational (standardized) units. That is, what’s the model-implied correlation among variables? The inspect function in lavaan gives access to a number of model details, including this:
inspect(mlav2, what="cor.all")## cmedv nox lg_crm rad
## cmedv 1.000
## nox -0.351 1.000
## log_crim -0.415 0.522 1.000
## rad -0.385 0.611 0.853 1.000How does this compare to the observed correlations?
lavCor(mlav2)## cmedv nox lg_crm rad
## cmedv 1.000
## nox -0.429 1.000
## log_crim -0.457 0.789 1.000
## rad -0.385 0.611 0.853 1.000In particular, getting the misfit of the bivariate associations is very helpful. Here, we ask for residuals in correlational units, which can be more intuitive than dealing with covariances that are unstandardized. Note that this is the subtraction of the observed - model-implied matrices above. Large positive values indicate the model underpredicts the correlation; large negative values suggest overprediction of correlation. Usually values \(|r > .1|\) are worth closer consideration.
resid(mlav2, "cor")## $type
## [1] "cor.bollen"
##
## $cor
## cmedv nox lg_crm rad
## cmedv 0.000
## nox -0.078 0.000
## log_crim -0.042 0.267 0.000
## rad 0.000 0.000 0.000 0.000
##
## $mean
## cmedv nox log_crim rad
## 0 0 0 0So the model significantly underpredicts the association between nox and log_crim.
We could visualize the problems, too:
plot_matrix <- function(matrix_toplot){
corrplot::corrplot(matrix_toplot, is.corr = FALSE,
type = 'lower',
order = "original",
tl.col='black', tl.cex=.75)
}
plot_matrix(residuals(mlav2, type="cor")$cor)
3.4 Modification indices
Let’s take a look at the modification indices to see if we can fix the misfit by freeing one or more paths, particularly the relationship between nox and log_crim.
modificationindices(mlav2, minimum.value = 20) #only print MIs > 20 | lhs | op | rhs | mi | epc | sepc.lv | sepc.all | sepc.nox | |
|---|---|---|---|---|---|---|---|---|
| 11 | nox | ~ | cmedv | 38.5 | -0.748 | -0.748 | -0.584 | -0.584 |
| 12 | nox | ~ | log_crim | 211.7 | 3.648 | 3.648 | 0.982 | 0.315 |
| 14 | log_crim | ~ | nox | 82.8 | 0.045 | 0.045 | 0.167 | 0.167 |
| 17 | rad | ~ | nox | 80.2 | -0.142 | -0.142 | -0.189 | -0.189 |
Here, we see that model fit would improve massively if we allowed log_crim to predict nox. Whether this makes theoretical sense is a different (and likely more important) matter. For demonstration purposes, let’s accept that this path needs to be freely estimated.
#we can use the add parameter to add a path, while leaving all other model elements the same
mlav3 <- update(mlav2, add="nox ~ log_crim")
summary(mlav3, fit.measures=TRUE)## lavaan 0.6-2 ended normally after 25 iterations
##
## Optimization method NLMINB
## Number of free parameters 6
##
## Number of observations 506
##
## Estimator ML
## Model Fit Test Statistic 0.080
## Degrees of freedom 1
## P-value (Chi-square) 0.777
##
## Model test baseline model:
##
## Minimum Function Test Statistic 638.018
## Degrees of freedom 5
## P-value 0.000
##
## User model versus baseline model:
##
## Comparative Fit Index (CFI) 1.000
## Tucker-Lewis Index (TLI) 1.007
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -3477.606
## Loglikelihood unrestricted model (H1) -3477.566
##
## Number of free parameters 6
## Akaike (AIC) 6967.213
## Bayesian (BIC) 6992.572
## Sample-size adjusted Bayesian (BIC) 6973.527
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.000
## 90 Percent Confidence Interval 0.000 0.078
## P-value RMSEA <= 0.05 0.880
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.002
##
## Parameter Estimates:
##
## Information Expected
## Information saturated (h1) model Structured
## Standard Errors Standard
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## cmedv ~
## log_crim -0.925 0.188 -4.924 0.000
## nox -0.144 0.051 -2.847 0.004
## nox ~
## rad -0.302 0.068 -4.403 0.000
## log_crim 3.648 0.191 19.081 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .cmedv 65.499 4.118 15.906 0.000
## .nox 48.798 3.068 15.906 0.000This looks way better in terms of fit. There is a strong positive association between crime and nox levels that we missed before. Conceptually, this suggests that the relationship between crime and home prices is partially mediated by crime’s effect on pollutant levels. By contrast, the effect of proximity to highways on home prices appears to be fully mediated by pollutant levels (as indicated by the absence of a large modification index for this path).
semPaths(mlav3, what='std', nCharNodes=6, sizeMan=10,
edge.label.cex=1.25, curvePivot = TRUE, fade=FALSE)