SEM Exercises in R: 16 lavaan Practice Problems

These 16 structural equation modeling exercises in R take you through the full lavaan workflow on bundled datasets: writing model syntax, fitting CFA and full SEMs, reading fit indices, interpreting modification indices, estimating indirect effects with bootstrap intervals, testing measurement invariance across groups, and handling ordered-categorical indicators. Solutions are hidden behind a click and explained.

How are these problems structured?

Every problem gives you a task, the exact output your solution must reproduce, and a difficulty tag. You write code in the Your turn block, then expand the solution to verify your approach and read why it works. Code blocks share state across the page like notebook cells, so you only need to load packages once.

The two datasets used throughout are bundled with lavaan itself: PoliticalDemocracy (75 nations, eight indicators of industrialization and political democracy in 1960 and 1965) and HolzingerSwineford1939 (301 schoolchildren, nine cognitive tests grouped into visual, textual, and speed factors).

Section 1. Specifying and fitting your first lavaan model (3 problems)

Exercise 1.1: Fit a two-equation path model on PoliticalDemocracy

Task: A political scientist wants to confirm that 1960 industrialization (x1) predicts 1960 democracy (y1), which in turn predicts 1965 democracy (y5). Use lavaan::sem() to fit a two-equation path model with y1 regressed on x1, and y5 regressed on y1. Save the fitted object to ex_1_1 and print summary(ex_1_1) so the parameter estimates appear.

Expected result:

#> lavaan 0.6 ended normally after 1 iterations
#>   Estimator                                         ML
#>   Number of observations                            75
#> Model Test User Model:
#>   Test statistic                                 0.000
#>   Degrees of freedom                                 0
#> Regressions:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>   y1 ~
#>     x1                1.270    0.131    9.677    0.000
#>   y5 ~
#>     y1                0.838    0.046   18.108    0.000

Difficulty: Beginner

Exercise 1.2: Specify a three-indicator measurement model with the =~ operator

Task: Define a one-factor measurement model where a latent visual factor is measured by three indicators (x1, x2, x3) from HolzingerSwineford1939. Use cfa() to fit it and save the result to ex_1_2. The =~ operator reads as "is measured by" and tells lavaan to estimate factor loadings.

Expected result:

#> lavaan 0.6 ended normally after 19 iterations
#>   Number of observations                           301
#> Model Test User Model:
#>   Test statistic                                 0.000
#>   Degrees of freedom                                 0
#> Latent Variables:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>   visual =~
#>     x1                1.000
#>     x2                0.554    0.100    5.554    0.000
#>     x3                0.729    0.109    6.685    0.000

Difficulty: Beginner

Exercise 1.3: Free a residual covariance with the ~~ operator

Task: Returning to the three-indicator visual factor on HolzingerSwineford1939, suppose x1 and x3 share a residual correlation because both are timed tasks. Modify the model from Exercise 1.2 to also estimate x1 ~~ x3 (free residual covariance) and save the refit to ex_1_3. Use parameterEstimates(ex_1_3) to confirm the new parameter.

Expected result:

#> # The new row in parameterEstimates(ex_1_3):
#>   lhs op rhs   est    se      z pvalue
#>    x1 ~~  x3 0.297 0.198  1.500 0.134

Difficulty: Intermediate

Section 2. Confirmatory factor analysis (3 problems)

Exercise 2.1: Fit a classic three-factor CFA on HolzingerSwineford1939

Task: The canonical Holzinger-Swineford CFA defines three correlated factors: visual measured by x1, x2, x3; textual by x4, x5, x6; and speed by x7, x8, x9. Specify this model, fit it with cfa(), and save the fitted object to ex_2_1. Print only the fit statistics line of summary(ex_2_1, fit.measures = TRUE) (the Test User Model block).

Expected result:

#> lavaan 0.6 ended normally after 35 iterations
#>   Number of observations                           301
#> Model Test User Model:
#>   Test statistic                                85.306
#>   Degrees of freedom                                24
#>   P-value (Chi-square)                           0.000

Difficulty: Intermediate

Exercise 2.2: Extract standardized loadings with the std.all column

Task: Using ex_2_1 from the previous exercise, pull only the factor loadings (rows where op == "=~") with their standardized values (std.all). Save the resulting data frame to ex_2_2. Standardized loadings are easier to interpret than raw because all variables are on the same scale.

Expected result:

#>      lhs op rhs   est std.all
#> 1 visual =~  x1 1.000   0.772
#> 2 visual =~  x2 0.554   0.424
#> 3 visual =~  x3 0.729   0.581
#> 4 textual =~ x4 1.000   0.852
#> 5 textual =~ x5 1.113   0.855
#> 6 textual =~ x6 0.926   0.838
#> 7 speed =~  x7 1.000   0.570
#> 8 speed =~  x8 1.180   0.723
#> 9 speed =~  x9 1.082   0.665

Difficulty: Intermediate

Exercise 2.3: Inspect estimates with the standardizedSolution helper

Task: Instead of slicing parameterEstimates() by hand, use standardizedSolution(ex_2_1) to get a clean table of standardized estimates with confidence intervals. Filter the result to only the factor covariance rows (where both sides are latent factors and op == "~~") and save it to ex_2_3. This gives you the inter-factor correlations.

Expected result:

#>       lhs op     rhs est.std    se      z pvalue ci.lower ci.upper
#> 1  visual ~~  textual   0.459 0.064  7.189  0.000    0.334    0.585
#> 2  visual ~~    speed   0.471 0.073  6.461  0.000    0.328    0.613
#> 3 textual ~~    speed   0.283 0.069  4.117  0.000    0.149    0.418

Difficulty: Intermediate

Section 3. Model fit and respecification (3 problems)

Exercise 3.1: Pull CFI, TLI, RMSEA, and SRMR from fitMeasures

Task: Global fit of a CFA is judged by a handful of indices. Pull just cfi, tli, rmsea, and srmr from ex_2_1 using fitMeasures() with the index names passed as a character vector. Save the named numeric to ex_3_1 and round it to three decimals. Common thresholds for acceptable fit are CFI/TLI > .90, RMSEA < .08, SRMR < .08.

Expected result:

#>   cfi   tli rmsea  srmr
#> 0.931 0.896 0.092 0.065

Difficulty: Intermediate

Exercise 3.2: Use modification indices to find a missing parameter

Task: The borderline RMSEA in ex_2_1 suggests one or two paths are missing. Run modindices(ex_2_1, sort. = TRUE) and save the top 5 rows (largest mi) to ex_3_2. Each row tells you the expected drop in chi-square (mi) and the standardized parameter value (sepc.all) if the suggested path were freed.

Expected result:

#>       lhs op rhs     mi     epc sepc.all
#> 1  visual =~  x9 36.411   0.577    0.519
#> 2      x7 ~~  x8 34.145   0.536    0.488
#> 3      x8 ~~  x9 14.946  -0.423   -0.415
#> 4  visual =~  x7  18.631  -0.422   -0.380
#> 5     x2 ~~  x7   8.918   0.213    0.157

Difficulty: Advanced

Exercise 3.3: Compare nested models with anova

Task: Test whether freeing the visual =~ x9 cross-loading from Exercise 3.2 produces a significantly better fit than the original three-factor model. Fit the augmented model, then call anova(ex_2_1, ex_3_3_alt) and save the comparison table to ex_3_3. A significant chi-square difference (p < .05) supports the more complex model.

Expected result:

#> Chi-Squared Difference Test
#>             Df    AIC    BIC Chisq Chisq diff Df diff Pr(>Chisq)
#> ex_2_1      24   7517   7596 85.31
#> ex_3_3_alt  23   7484   7567 50.34      34.97       1  3.34e-09

Difficulty: Advanced

Section 4. Full SEM with structural paths (3 problems)

Exercise 4.1: Fit the classic Bollen industrialization-democracy SEM

Task: The hallmark SEM on PoliticalDemocracy defines three latent variables: ind60 (industrialization 1960) measured by x1, x2, x3; dem60 (democracy 1960) measured by y1, y2, y3, y4; and dem65 (democracy 1965) measured by y5, y6, y7, y8. The structural part regresses both dem60 and dem65 on ind60, plus dem65 on dem60. Fit this model and save it to ex_4_1.

Expected result:

#> Model Test User Model:
#>   Test statistic                                72.462
#>   Degrees of freedom                                51
#>   P-value (Chi-square)                           0.026
#> Regressions:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>   dem60 ~
#>     ind60             1.483    0.399    3.715    0.000
#>   dem65 ~
#>     ind60             0.572    0.221    2.586    0.010
#>     dem60             0.837    0.098    8.514    0.000

Difficulty: Advanced

Exercise 4.2: Define an indirect effect with the := operator

Task: In the model from Exercise 4.1, the indirect effect of ind60 on dem65 flows through dem60. Label the two structural coefficients (a for dem60 ~ ind60, b for dem65 ~ dem60) and define indirect := a * b and total := a * b + c (where c is the direct path). Save the refit to ex_4_2 and print the "Defined Parameters" block from summary().

Expected result:

#> Defined Parameters:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>     indirect          1.241    0.350    3.548    0.000
#>     total             1.813    0.421    4.304    0.000

Difficulty: Advanced

Exercise 4.3: Bootstrap a confidence interval for the indirect effect

Task: Delta-method standard errors assume the sampling distribution of the indirect effect is normal, which is wrong when sample size is small or the product is skewed. Refit ex_4_2 with se = "bootstrap" and bootstrap = 200 (use a small number so the page runs fast), then use parameterEstimates(ex_4_3, boot.ci.type = "perc") to pull the percentile bootstrap CI for the indirect row. Save the row to ex_4_3.

Expected result:

#>        lhs op rhs    label   est    se     z pvalue ci.lower ci.upper
#> 1 indirect :=  a*b indirect 1.241 0.317 3.913  0.000    0.711    1.916

Difficulty: Advanced

Section 5. Multi-group, invariance, and categorical SEM (3 problems)

Exercise 5.1: Fit the Holzinger-Swineford CFA separately for two schools

Task: The HolzingerSwineford1939 dataset has a school variable with two levels: Pasteur and Grant-White. Fit the three-factor CFA from Exercise 2.1 separately to each school using the group = "school" argument and save the fit to ex_5_1. Inspect summary(ex_5_1, fit.measures = TRUE) to see per-group estimates and the combined fit.

Expected result:

#> Number of observations per group:
#>   Pasteur                                          156
#>   Grant-White                                      145
#> Model Test User Model:
#>   Test statistic                               115.851
#>   Degrees of freedom                                48
#>   P-value (Chi-square)                           0.000

Difficulty: Advanced

Exercise 5.2: Test metric invariance by constraining loadings equal across groups

Task: To test whether factor loadings are equivalent across schools (metric invariance), refit the model with group.equal = "loadings", then compare to ex_5_1 with anova(). Save the comparison table to ex_5_2. A non-significant chi-square difference supports metric invariance, which is the prerequisite for comparing factor means or regression coefficients across groups.

Expected result:

#> Chi-Squared Difference Test
#>            Df    AIC    BIC Chisq Chisq diff Df diff Pr(>Chisq)
#> ex_5_1     48   7484   7715 115.9
#> ex_5_2_alt 54   7483   7691 124.0       8.19       6     0.224

Difficulty: Advanced

Exercise 5.3: Fit an ordered-categorical CFA with the WLSMV estimator

Task: Treat indicators y1 through y4 from PoliticalDemocracy as ordered-categorical (they are 4-point democracy ratings). Fit a one-factor CFA on them with ordered = c("y1","y2","y3","y4"), which switches lavaan to the WLSMV estimator and tetrachoric/polychoric correlations. Save the fit to ex_5_3 and print the Estimator line plus standardized loadings.

Expected result:

#> lavaan 0.6 ended normally after 22 iterations
#>   Estimator                                       DWLS
#>   Number of observations                            75
#> Latent Variables:
#>                    Estimate  Std.Err  z-value  P(>|z|)   Std.all
#>   dem60 =~
#>     y1                1.000                              0.838
#>     y2                1.302    0.171    7.617    0.000   0.866
#>     y3                1.230    0.171    7.207    0.000   0.815
#>     y4                1.305    0.158    8.275    0.000   0.876

Difficulty: Advanced

Section 6. Reporting (1 problem)

Exercise 6.1: Build a publication-ready loadings table

Task: For the three-factor CFA fit ex_2_1, build a tidy data frame with one row per indicator, columns factor, indicator, loading (raw est), std_loading (std.all), and pvalue, rounded to three decimals. Sort by factor and then by descending standardized loading. Save the table to ex_6_1. This is the format you would paste into a paper or report.

Expected result:

#>    factor indicator loading std_loading pvalue
#> 1 textual        x5   1.113       0.855  0.000
#> 2 textual        x4   1.000       0.852  0.000
#> 3 textual        x6   0.926       0.838  0.000
#> 4  visual        x1   1.000       0.772  0.000
#> 5  visual        x3   0.729       0.581  0.000
#> 6  visual        x2   0.554       0.424  0.000
#> 7   speed        x8   1.180       0.723  0.000
#> 8   speed        x9   1.082       0.665  0.000
#> 9   speed        x7   1.000       0.570  0.000

Difficulty: Intermediate

What to do next

You have written lavaan syntax for path models, CFAs, and full SEMs; pulled fit indices; tested nested models; bootstrapped indirect effects; and handled multi-group and ordered-categorical data. The next steps:

• Revisit the parent tutorial, CFA and Structural Equation Modeling in R, for theory and figure-by-figure walkthroughs of every model used above.
• Practise the regression building blocks behind path models in the Linear Regression Exercises in R.
• Move into latent-variable mixture and growth models with the Factor Analysis Exercises in R.
• Use the ggplot2 Exercises in R to plot path diagrams and residual matrices for your fits.