Apply Path analysis in R with Lavaan
Osama Alwusaidi
02/11/2020
Contents
- Overview
- Firstly, Model specification | بناء النموذج النظري وتوصيفه
- Secondly, Model Identification | تحديد النموذج وتعيينه
- Thirdly, Model Estimation | تقدير معالم النموذج
- Fourthly, Model Modification | تعديل النموذج المفترض في ضوء مؤشرات التعديل
- Fifthly, Path diagrams | الرسم التخطيطي للنموذج المقترح
Overview
There are two basic functions that allow you to run path analysis in Lavaan: the ‘sem’ and the ‘lavaan’ functions.This video will demonstrate how to specify a path model involving only manifest variables and how to estimate model parameters using the ‘lavaan’ function.
Reference | المرجع
- Crowson, M. (2019). Multivariate statistics for the real world - Structural equation modeling [webpage]. https://qrgo.page.link/etYmMt.
Firstly, Model specification in ‘lavaan’ function | بناء النموذج النظري وتوصيفه
Step 1: Use lavaan model syntax to specify path model and have it stored in an R object. In our model, we will treat ses, mastery goals, and performance goals as predictors of student achievement. The effect of mastery on achievement will be both direct and indirect (via interest and anxiety. The effects of ses and performance goals will be treated as being fully mediated through anxiety and interest.
When specifying predictive relationships in the model, we use the tilde sign (‘~’), which separates thedependent variable in each equation from its predictors. Predictors are separated in each equation by ‘+’ sign. In our model, we will also allow the residuals for anxiety and interest to correlate (see ‘~~’ in syntax below).
Model specification
model<-'
#equation where interest is predicted by ses & mastery and
#performance goals
interest~mastery+perfgoal+ses
#equation where achieve is predicted by interest and anxiety
achieve~anxiety+interest+mastery
#equation where anxiety is predicted by mastery and performance goals
anxiety~perfgoal+mastery
#estimtating the variances of the exogenous variables (ses, mastery,performance)
mastery~~mastery
perfgoal~~perfgoal
ses~~ses
#estimtating the covariances of the exogenous variables (ses, mastery,performance)
mastery~~perfgoal+ses
perfgoal~~ses
#estimating the residual variances for endogenous variables (interest, anxiety, achieve)
interest~~interest
anxiety~~anxiety
achieve~~achieve
#estimating the covariance of residuals for interest and anxiety
interest~~anxiety'Secondly, Model Identification | تحديد النموذج وتعيينه
\[ \begin{array}{l}\text { Parameters being estimated } \\ \text { Variances of three exogenous variables } \\ \text { Residual variances associated with three endogenous variables } \\ 3 \text { covariances among exogenous variables } \\ 1 \text { residual covariance between endogenous variables } \\ 8 \text { path coefficients } \\ \qquad \text { Total }\end{array} \]
\[ M=\frac{V(V+1)}{2}=\frac{6(6+1)}{2}=21 \] \[ d f=M-P=21-18=3 \]
Thirdly, Model Estimation | تقدير معالم النموذج
Step 2: Use ‘lavaan’ function to run analysis. Here, I will be saving the results in an R object called ‘fit’ (arbitrarily named). Inside the parenthesis are arguments separated by commas. The first argument contains the name of the object containing the model syntax (see above). The object is named ‘model’ (again, arbitrarily named above). Next, we have the ‘data’ argument. This identifies the object (i.e., data frame) containing the raw data.
fit<-lavaan(model,data=Data_OAlwusaidi)
summary(fit,fit.measures=TRUE)## lavaan 0.6-7 ended normally after 34 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 18
##
## Number of observations 140
##
## Model Test User Model:
##
## Test statistic 29.231
## Degrees of freedom 3
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 202.409
## Degrees of freedom 15
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.860
## Tucker-Lewis Index (TLI) 0.300
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -1391.274
## Loglikelihood unrestricted model (H1) -1376.659
##
## Akaike (AIC) 2818.548
## Bayesian (BIC) 2871.498
## Sample-size adjusted Bayesian (BIC) 2814.548
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.250
## 90 Percent confidence interval - lower 0.172
## 90 Percent confidence interval - upper 0.336
## P-value RMSEA <= 0.05 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.074
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## interest ~
## mastery 0.708 0.088 8.066 0.000
## perfgoal -0.035 0.040 -0.879 0.380
## ses 0.520 0.242 2.154 0.031
## achieve ~
## anxiety -0.040 0.054 -0.747 0.455
## interest 0.211 0.060 3.527 0.000
## mastery 0.345 0.079 4.358 0.000
## anxiety ~
## perfgoal 0.025 0.045 0.556 0.578
## mastery -0.387 0.097 -4.009 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## mastery ~~
## perfgoal -0.935 0.361 -2.590 0.010
## ses 0.170 0.061 2.805 0.005
## perfgoal ~~
## ses -0.226 0.128 -1.768 0.077
## .interest ~~
## .anxiety 0.059 0.181 0.329 0.742
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## mastery 1.944 0.232 8.367 0.000
## perfgoal 8.936 1.068 8.367 0.000
## ses 0.249 0.030 8.367 0.000
## .interest 1.895 0.227 8.367 0.000
## .anxiety 2.410 0.288 8.367 0.000
## .achieve 0.988 0.118 8.367 0.000summary(fit,fit.measures=TRUE,standardized=TRUE,rsquare=TRUE)## lavaan 0.6-7 ended normally after 34 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 18
##
## Number of observations 140
##
## Model Test User Model:
##
## Test statistic 29.231
## Degrees of freedom 3
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 202.409
## Degrees of freedom 15
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.860
## Tucker-Lewis Index (TLI) 0.300
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -1391.274
## Loglikelihood unrestricted model (H1) -1376.659
##
## Akaike (AIC) 2818.548
## Bayesian (BIC) 2871.498
## Sample-size adjusted Bayesian (BIC) 2814.548
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.250
## 90 Percent confidence interval - lower 0.172
## 90 Percent confidence interval - upper 0.336
## P-value RMSEA <= 0.05 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.074
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## interest ~
## mastery 0.708 0.088 8.066 0.000 0.708 0.558
## perfgoal -0.035 0.040 -0.879 0.380 -0.035 -0.060
## ses 0.520 0.242 2.154 0.031 0.520 0.147
## achieve ~
## anxiety -0.040 0.054 -0.747 0.455 -0.040 -0.053
## interest 0.211 0.060 3.527 0.000 0.211 0.294
## mastery 0.345 0.079 4.358 0.000 0.345 0.379
## anxiety ~
## perfgoal 0.025 0.045 0.556 0.578 0.025 0.045
## mastery -0.387 0.097 -4.009 0.000 -0.387 -0.327
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## mastery ~~
## perfgoal -0.935 0.361 -2.590 0.010 -0.935 -0.224
## ses 0.170 0.061 2.805 0.005 0.170 0.244
## perfgoal ~~
## ses -0.226 0.128 -1.768 0.077 -0.226 -0.151
## .interest ~~
## .anxiety 0.059 0.181 0.329 0.742 0.059 0.028
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## mastery 1.944 0.232 8.367 0.000 1.944 1.000
## perfgoal 8.936 1.068 8.367 0.000 8.936 1.000
## ses 0.249 0.030 8.367 0.000 0.249 1.000
## .interest 1.895 0.227 8.367 0.000 1.895 0.606
## .anxiety 2.410 0.288 8.367 0.000 2.410 0.884
## .achieve 0.988 0.118 8.367 0.000 0.988 0.613
##
## R-Square:
## Estimate
## interest 0.394
## anxiety 0.116
## achieve 0.387parameterEstimates(fit)## lhs op rhs est se z pvalue ci.lower ci.upper
## 1 interest ~ mastery 0.708 0.088 8.066 0.000 0.536 0.880
## 2 interest ~ perfgoal -0.035 0.040 -0.879 0.380 -0.114 0.043
## 3 interest ~ ses 0.520 0.242 2.154 0.031 0.047 0.994
## 4 achieve ~ anxiety -0.040 0.054 -0.747 0.455 -0.146 0.066
## 5 achieve ~ interest 0.211 0.060 3.527 0.000 0.094 0.328
## 6 achieve ~ mastery 0.345 0.079 4.358 0.000 0.190 0.500
## 7 anxiety ~ perfgoal 0.025 0.045 0.556 0.578 -0.063 0.113
## 8 anxiety ~ mastery -0.387 0.097 -4.009 0.000 -0.576 -0.198
## 9 mastery ~~ mastery 1.944 0.232 8.367 0.000 1.488 2.399
## 10 perfgoal ~~ perfgoal 8.936 1.068 8.367 0.000 6.842 11.029
## 11 ses ~~ ses 0.249 0.030 8.367 0.000 0.191 0.308
## 12 mastery ~~ perfgoal -0.935 0.361 -2.590 0.010 -1.642 -0.227
## 13 mastery ~~ ses 0.170 0.061 2.805 0.005 0.051 0.288
## 14 perfgoal ~~ ses -0.226 0.128 -1.768 0.077 -0.476 0.024
## 15 interest ~~ interest 1.895 0.227 8.367 0.000 1.451 2.339
## 16 anxiety ~~ anxiety 2.410 0.288 8.367 0.000 1.845 2.974
## 17 achieve ~~ achieve 0.988 0.118 8.367 0.000 0.757 1.220
## 18 interest ~~ anxiety 0.059 0.181 0.329 0.742 -0.295 0.414fitMeasures(fit)## npar fmin chisq df
## 18.000 0.104 29.231 3.000
## pvalue baseline.chisq baseline.df baseline.pvalue
## 0.000 202.409 15.000 0.000
## cfi tli nnfi rfi
## 0.860 0.300 0.300 0.278
## nfi pnfi ifi rni
## 0.856 0.171 0.868 0.860
## logl unrestricted.logl aic bic
## -1391.274 -1376.659 2818.548 2871.498
## ntotal bic2 rmsea rmsea.ci.lower
## 140.000 2814.548 0.250 0.172
## rmsea.ci.upper rmsea.pvalue rmr rmr_nomean
## 0.336 0.000 0.122 0.122
## srmr srmr_bentler srmr_bentler_nomean crmr
## 0.074 0.074 0.074 0.088
## crmr_nomean srmr_mplus srmr_mplus_nomean cn_05
## 0.088 0.074 0.074 38.428
## cn_01 gfi agfi pgfi
## 55.335 0.941 0.587 0.134
## mfi ecvi
## 0.911 0.466Fourthly, Model Modification | تعديل النموذج المفترض في ضوء مؤشرات التعديل
modificationIndices(fit)## lhs op rhs mi epc sepc.lv sepc.all sepc.nox
## 19 interest ~~ achieve 25.396 -2.899 -2.899 -2.118 -2.118
## 23 achieve ~~ anxiety 6.668 6.803 6.803 4.408 4.408
## 24 achieve ~~ mastery 22.476 -1.743 -1.743 -1.257 -1.257
## 25 achieve ~~ perfgoal 2.762 -0.406 -0.406 -0.137 -0.137
## 26 achieve ~~ ses 20.541 0.186 0.186 0.376 0.376
## 27 anxiety ~~ mastery 0.921 0.765 0.765 0.354 0.354
## 28 anxiety ~~ perfgoal 0.921 -3.576 -3.576 -0.771 -0.771
## 29 anxiety ~~ ses 0.921 -0.061 -0.061 -0.078 -0.078
## 30 interest ~ achieve 25.396 -2.933 -2.933 -2.106 -2.106
## 32 achieve ~ perfgoal 4.551 -0.062 -0.062 -0.146 -0.146
## 33 achieve ~ ses 22.431 0.837 0.837 0.329 0.329
## 34 anxiety ~ interest 0.921 -0.502 -0.502 -0.538 -0.538
## 35 anxiety ~ achieve 1.119 1.922 1.922 1.478 1.478
## 36 anxiety ~ ses 0.921 -0.261 -0.261 -0.079 -0.079
## 37 mastery ~ interest 0.921 12.878 12.878 16.335 16.335
## 38 mastery ~ achieve 22.801 -1.774 -1.774 -1.616 -1.616
## 39 mastery ~ anxiety 0.921 0.318 0.318 0.376 0.376
## 42 perfgoal ~ interest 0.920 -60.126 -60.126 -35.570 -35.570
## 43 perfgoal ~ achieve 2.744 -0.410 -0.410 -0.174 -0.174
## 44 perfgoal ~ anxiety 0.921 -1.484 -1.484 -0.819 -0.819
## 47 ses ~ interest 0.921 -1.020 -1.020 -3.614 -3.614
## 48 ses ~ achieve 20.964 0.190 0.190 0.484 0.484
## 49 ses ~ anxiety 0.921 -0.025 -0.025 -0.083 -0.083# Specification of model using auto.var argument...
model<-'
#equation where interest is predicted by ses & mastery and
#performance goals
interest~mastery+perfgoal+ses
#equation where achieve is predicted by interest and anxiety
achieve~anxiety+interest+mastery
#equation where anxiety is predicted by mastery and performance goals
anxiety~perfgoal+mastery
#estimtating the variances of the exogenous variables (ses, mastery,performance)
mastery~~mastery
perfgoal~~perfgoal
ses~~ses
#estimtating the covariances of the exogenous variables (ses, mastery,performance)
mastery~~perfgoal+ses
perfgoal~~ses
#The auto.var argument when fitting the model can be used so that
#you do not have to directly request estimation of residual variances
#Estimating the covariance of residuals for interest and anxiety
interest~~anxiety'
fit<-lavaan(model,data=Data_OAlwusaidi,auto.var=TRUE)
summary(fit,fit.measures=TRUE,standardized=TRUE,rsquare=TRUE)## lavaan 0.6-7 ended normally after 34 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 18
##
## Number of observations 140
##
## Model Test User Model:
##
## Test statistic 29.231
## Degrees of freedom 3
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 202.409
## Degrees of freedom 15
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.860
## Tucker-Lewis Index (TLI) 0.300
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -1391.274
## Loglikelihood unrestricted model (H1) -1376.659
##
## Akaike (AIC) 2818.548
## Bayesian (BIC) 2871.498
## Sample-size adjusted Bayesian (BIC) 2814.548
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.250
## 90 Percent confidence interval - lower 0.172
## 90 Percent confidence interval - upper 0.336
## P-value RMSEA <= 0.05 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.074
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## interest ~
## mastery 0.708 0.088 8.066 0.000 0.708 0.558
## perfgoal -0.035 0.040 -0.879 0.380 -0.035 -0.060
## ses 0.520 0.242 2.154 0.031 0.520 0.147
## achieve ~
## anxiety -0.040 0.054 -0.747 0.455 -0.040 -0.053
## interest 0.211 0.060 3.527 0.000 0.211 0.294
## mastery 0.345 0.079 4.358 0.000 0.345 0.379
## anxiety ~
## perfgoal 0.025 0.045 0.556 0.578 0.025 0.045
## mastery -0.387 0.097 -4.009 0.000 -0.387 -0.327
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## mastery ~~
## perfgoal -0.935 0.361 -2.590 0.010 -0.935 -0.224
## ses 0.170 0.061 2.805 0.005 0.170 0.244
## perfgoal ~~
## ses -0.226 0.128 -1.768 0.077 -0.226 -0.151
## .interest ~~
## .anxiety 0.059 0.181 0.329 0.742 0.059 0.028
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## mastery 1.944 0.232 8.367 0.000 1.944 1.000
## perfgoal 8.936 1.068 8.367 0.000 8.936 1.000
## ses 0.249 0.030 8.367 0.000 0.249 1.000
## .interest 1.895 0.227 8.367 0.000 1.895 0.606
## .achieve 0.988 0.118 8.367 0.000 0.988 0.613
## .anxiety 2.410 0.288 8.367 0.000 2.410 0.884
##
## R-Square:
## Estimate
## interest 0.394
## achieve 0.387
## anxiety 0.116Fifthly, Path diagrams | الرسم التخطيطي للنموذج المقترح
# There are a couple of ways you can obtain path diagrams (although they can be somewhat tricky to implement.
# One approach is to use the 'semPaths' function
library("semPlot")## Registered S3 methods overwritten by 'huge':
## method from
## plot.sim BDgraph
## print.sim BDgraphsemPaths(fit,what="paths",whatLabels="par",style="lisrel",layout="tree",
rotation=2)
# A second approach is to use the 'lavaanPlot" function
library(lavaanPlot)
lavaanPlot(model = fit, node_options = list(shape = "box", fontname =
"Helvetica"), edge_options = list(color = "grey"), coefs = TRUE,covs=
TRUE,stars = c("regress"))