Measurement ESEM: target and geomin rotation

Use with your data

Replace MY_DATA with your data frame. This template selects ordinal items in columns 2:13 and excludes column 1, such as a demographic variable.

result <- PsychoMatic::esem_auto(
  data = MY_DATA,
  n_factors = 3,
  items = 2:13,
  rotation = "geomin",
  ordered = TRUE,
  language = "esp"
)
result$manuscript_text

items selects columns: use 2:13 or names(MY_DATA)[2:13], not MY_DATA[, 2:13], which returns the data themselves. Raw geomin data always require an explicit selection. Use items = names(MY_DATA) only when every column is an item. Use ordered = FALSE for continuous numeric indicators. The installed examples/EXAMPLE_ESEM_AUTO.R contains reusable geomin and target templates. The executable examples below generate data only to make this methodological walkthrough reproducible.

Choosing the rotation

Both rotations are oblique. Target zeros are soft targets, not zero constraints; primary entries of 1 in the stored target matrix mean that these positions are untargeted, not fixed to one. The supplied primary membership is an assumption, not a finding. A target constructed from this same sample remains exploratory. A theoretically motivated target also needs independent evidence before stronger generalization.

For the same unrestricted measurement block, changing the rotation preserves the reproduced covariance and global fit. CFI/RMSEA cannot determine whether target or geomin is preferable. Inspect pattern coefficients (not the oblique structure matrix), factor correlations, interpretability and replication. With oblique factors, an item’s largest pattern coefficient is a descriptive diagnostic; it does not automatically assign the item or warrant deletion.

Geomin epsilon defaults to 0.001. It affects the rotation criterion and can change the loading pattern; this value is not a universal optimum. Report the chosen epsilon and examine justified alternatives. The default 30 random starts reduce dependence on initial values, without proving a global optimum or sampling stability. The default seed = NULL leaves RNG management to lavaan without imposing a wrapper seed. Set seed explicitly to make starts reproducible and restore the incoming random-number state after fitting.

Reproducible target example

population <- '
 A =~ .75*x1 + .70*x2 + .65*x3
 B =~ .75*x4 + .70*x5 + .65*x6
 C =~ .75*x7 + .70*x8 + .65*x9
 A =~ .12*x4
 B =~ .12*x7
 C =~ .12*x1
'
d <- lavaan::simulateData(population, sample.nobs = 400, seed = 101)
keys <- list(A = paste0("x", 1:3), B = paste0("x", 4:6), C = paste0("x", 7:9))
ordinal <- as.data.frame(lapply(d, function(v) {
  cut(v, unique(stats::quantile(v, seq(0, 1, length.out = 6))),
      include.lowest = TRUE, labels = FALSE)
}))
target <- PsychoMatic::esem_auto(ordinal, factors = keys, ordered = TRUE,
  compare_cfa = TRUE, seed = 2026, language = "esp", verbose = FALSE)
target$results$fit_indices[, c("metric", "value", "variant")]
#>           metric       value variant
#> 1          chisq 11.02910652  scaled
#> 2             df 12.00000000  scaled
#> 3         pvalue  0.52642591  scaled
#> 4            cfi  1.00000000  scaled
#> 5            tli  1.00921903  scaled
#> 6          rmsea  0.00000000  scaled
#> 7 rmsea.ci.lower  0.00000000  scaled
#> 8 rmsea.ci.upper  0.04764217  scaled
#> 9           srmr  0.01958798    base
target$results$item_diagnostics
#>   item largest_factor largest_loading second_factor second_loading difference
#> 1   x1              A       0.4870955             C     0.06474756  0.4223479
#> 2   x2              A       0.5572156             B    -0.03394696  0.5232686
#> 3   x3              A       0.5338453             C    -0.05674009  0.4771052
#> 4   x4              B       0.5616007             A     0.10882890  0.4527718
#> 5   x5              B       0.6219951             A    -0.02974065  0.5922544
#> 6   x6              B       0.6261329             C     0.06182033  0.5643126
#> 7   x7              C       0.5330818             A     0.05913692  0.4739449
#> 8   x8              C       0.5332534             B    -0.15071381  0.3825396
#> 9   x9              C       0.5770832             B     0.12486968  0.4522135
#>   expected_factor expected_loading expected_is_largest
#> 1               A        0.4870955                TRUE
#> 2               A        0.5572156                TRUE
#> 3               A        0.5338453                TRUE
#> 4               B        0.5616007                TRUE
#> 5               B        0.6219951                TRUE
#> 6               B        0.6261329                TRUE
#> 7               C        0.5330818                TRUE
#> 8               C        0.5332534                TRUE
#> 9               C        0.5770832                TRUE

Reproducible geomin example

geomin <- PsychoMatic::esem_auto(d, items = names(d), n_factors = 3,
  rotation = "geomin", ordered = FALSE,
  rotation_args = list(geomin_epsilon = .01), seed = 2026, verbose = FALSE)
summary(geomin)$loadings[, c("factor", "item", "standardized")]
#>    factor item  standardized
#> 1      F1   x1  0.5310693711
#> 2      F1   x2  0.6415773092
#> 3      F1   x3  0.5024999957
#> 4      F1   x4  0.1620236155
#> 5      F1   x5 -0.0306772542
#> 6      F1   x6 -0.0291787356
#> 7      F1   x7  0.0444202548
#> 8      F1   x8  0.0153696262
#> 9      F1   x9 -0.0294351877
#> 10     F2   x1 -0.0085659819
#> 11     F2   x2 -0.0208539074
#> 12     F2   x3  0.0512450294
#> 13     F2   x4  0.5447452198
#> 14     F2   x5  0.6273212113
#> 15     F2   x6  0.5446770989
#> 16     F2   x7  0.0274126021
#> 17     F2   x8 -0.1334954679
#> 18     F2   x9  0.0434392005
#> 19     F3   x1  0.0767923665
#> 20     F3   x2 -0.0049153050
#> 21     F3   x3 -0.0005462775
#> 22     F3   x4 -0.0864654722
#> 23     F3   x5  0.0299277521
#> 24     F3   x6  0.0708322525
#> 25     F3   x7  0.5488330298
#> 26     F3   x8  0.5441075938
#> 27     F3   x9  0.5385514824

Neither example chooses the number of factors automatically. The target list fixes the requested dimensionality, while a geomin model without a list requires n_factors. Retention evidence and theory should inform that decision before running the model.

Estimation, uncertainty and failed solutions

ordered = NULL uses the data specification or heuristically inferred levels. Inspect effective_settings$ordered; integer response scales can be mistaken for ordinal indicators. Explicit ordered = TRUE, FALSE, or selected column names are available. Nominal factors require explicit category mapping through psychomatic_spec(). Specifications preserve declared reverse scoring and category order. Continuous treatment of ordered factors requires deliberate numeric coding rather than silent conversion.

Ordinal/mixed indicators default to WLSMV; continuous indicators to MLR. Explicit alternatives are ULSMV or ML, respectively. Default deletion is listwise; ordinal models support pairwise treatment and continuous ML/MLR supports FIML. No estimator, imputation or missing-data substitution is made silently.

The rotated native fit is available as $fit. Safe item aliases avoid syntax ambiguity; $results$item_map links them to original names. Use lavaan::standardizedSolution(result$fit) for native standardized uncertainty. The reported loading intervals are 95%; RMSEA intervals are native 90% intervals. Original-scale and standardized loading intervals have separate columns and reader-facing tables. effective_settings$preparation retains declared category order, reverse scoring and effective reversal bounds; inferred bounds are recorded explicitly. Input encoding is displayed before reversal, which is documented separately. Reader reports also include target, maps and versions. The fit_variant argument selects scaled, robust or base CFI/TLI/RMSEA; a missing requested index remains NA with a reason. Corrected χ²/df/p use scaled values unless base is requested. SRMR always uses the base value.

Convergence alone is insufficient: post-check admissibility, finite parameter inference and rotation warnings are inspected. Failed, inadmissible or unavailable inference produces a limitation statement instead of a substantive manuscript interpretation. Saturated models are flagged because perfect global fit cannot evaluate restrictions. A descriptive CFA comparison checks the same cases, ordinal items and estimation options; it neither ranks models nor runs an automatic likelihood-ratio test. Comparison failure does not invalidate an otherwise interpretable ESEM fit.

Final interpretation and reports

References precede the final Interpretation section in console, Word and Excel. It contains a numerical draft describing the observed sample size, estimation, rotation, fit, loading ranges and factor correlations. It does not invent substantive meanings for factors, apply a universal fit cutoff or infer construct validity from the rotation choice. Review and adapt it to your study’s theory and reporting conventions. Preparation, item/factor maps and versions remain in the returned object and Word/Excel reports but are omitted from the console.

cat(target$manuscript_text)
#> Se estimó un modelo ESEM de 3 factores con 9 ítems (N = 400), mediante WLSMV y rotación oblicua target. El patrón teórico orientó la rotación target; las cargas secundarias se estimaron libremente y la rotación buscó aproximarlas a cero, sin fijarlas a ese valor. Los índices de ajuste fueron: χ²(12) = 11.029, p = 0.526; CFI = 1.000; TLI = 1.009; RMSEA = 0.000 (IC del 90% [0.000, 0.048]); SRMR = 0.020. Se solicitó la variante scaled para CFI, TLI y RMSEA; χ² utiliza la variante scaled y SRMR la base. Las cargas estandarizadas en los factores previstos oscilaron, en valor absoluto, entre 0.487 y 0.626; la mayor carga secundaria absoluta fue 0.151. En 9 de 9 ítems, la carga de mayor magnitud correspondió al factor previsto. Las correlaciones entre factores variaron entre -0.115 y 0.217. La comparación descriptiva con el CFA de referencia, sobre las mismas observaciones, arrojó ΔCFI (ESEM − CFA) = 0.023 y ΔRMSEA = -0.028; estas diferencias no constituyen una prueba de selección entre modelos.

PsychoMatic::psychomatic_report(target, format = c("word", "excel"),
  file_name = "esem_results", language = "esp")

Files use getwd() when no path is supplied and existing paths are protected unless overwrite = TRUE. Reader reports contain aggregate results, not rows of participant responses. Detailed audit output records settings, conditions and decisions. The native fit object should be handled as study data because lavaan itself retains input data internally.

References

Asparouhov, T., & Muthén, B. (2009). Exploratory structural equation modeling. Structural Equation Modeling: A Multidisciplinary Journal, 16(3), 397–438. https://doi.org/10.1080/10705510903008204

Marsh, H., & Alamer, A. (2024). When and how to use set-exploratory structural equation modelling to test structural models: A tutorial using the R package lavaan. British Journal of Mathematical and Statistical Psychology, 77(3), 459–476. https://doi.org/10.1111/bmsp.12336

The lavaan project. ESEM and EFA (official tutorial source). https://github.com/yrosseel/lavaan.org/blob/master/tutorial/efa.qmd