Evaluate measurement invariance across groups

nomo_invariance() evaluates increasingly constrained multi-group CFA models while retaining every generated semTools::measEq.syntax() object and every fitted lavaan model.

Usage

nomo_invariance(
  model,
  data,
  group,
  ordered = NULL,
  levels = NULL,
  partial = NULL,
  localize = TRUE,
  estimator = NULL,
  missing = NULL,
  ID.fac = "std.lv",
  ID.cat = "Wu.Estabrook.2016",
  parameterization = "theta",
  guidance = nomo_defaults()
)

Arguments

model

A researcher-specified lavaan measurement-model syntax string or an object created by nomo_model().

data

A non-empty data frame.

group

Character scalar naming the grouping variable in data.

ordered

Optional character vector naming ordered indicators.

levels

Optional invariance levels. NULL uses the sequence implied by the indicator category structure.

partial

Optional researcher-specified partial-invariance releases from nomo_partial().

localize

Logical. If TRUE, retain equality-constraint score tests as diagnostic evidence. Diagnostics never trigger automatic release/refitting.

estimator

Optional lavaan estimator. Ordered models default to WLSMV.

missing

Optional lavaan missing-data option.

ID.fac

Factor-identification method passed to semTools::measEq.syntax(). "std.lv" is the default.

ID.cat

Ordered-indicator identification method passed to semTools::measEq.syntax(). Wu-Estabrook is the default.

parameterization

Lavaan categorical parameterization. "theta" is the default for ordered indicators.

guidance

Guidance settings from nomo_defaults().

Value

A nomo_invariance object. The fields to read are:

• groups, requested_levels, and completed_levels.

• fit_evidence: one row per level, with its fit, its change from the level before, the likelihood-ratio test, and any warning or error.

• local_strain: score diagnostics for each equality constraint, which localize strain without releasing anything.

• partial: the researcher-specified releases, when given.

• fits: the fitted lavaan model at each level.

• syntax_text: the model syntax at each level.

• engine_warnings: warnings lavaan raised at each level.

• indicator_type, identification_note, and decision_log.

Other fields record the call, the settings used, and intermediate engine results. They may change between releases and are not part of the stable interface (see ?nomologR).

Details

The sequence is adapted to the observed category structure. Continuous indicators use configural -> metric -> scalar -> strict. Ordered indicators with four or more categories permit a separate threshold step. Three-category indicators require threshold equality as part of the metric step. When any binary indicator is present, threshold, loading, and intercept restrictions are imposed together at the first equality step (strong) because those restrictions cannot be treated as independent tests under Wu-Estabrook identification.

Partial invariance is researcher controlled. Supply an object from nomo_partial() to request specific equality-constraint releases. Releases are carried forward to more restrictive levels and their rationales are retained. nomo_invariance() never searches for a combination of releases that makes a fit rule pass.

When localize = TRUE, univariate score tests for equality constraints are retained as diagnostic evidence. They are explicitly not used to modify the fitted model.

References

Historical foundations:

Jöreskog, K. G. (1971). Simultaneous factor analysis in several populations. Psychometrika, 36(4), 409-426. doi:10.1007/BF02291366

Meredith, W. (1993). Measurement invariance, factor analysis and factorial invariance. Psychometrika, 58(4), 525-543. doi:10.1007/BF02294825

Vandenberg, R. J., & Lance, C. E. (2000). A review and synthesis of the measurement invariance literature: Suggestions, practices, and recommendations for organizational research. Organizational Research Methods, 3(1), 4-70. doi:10.1177/109442810031002

Change-in-fit evidence:

Chen, F. F. (2007). Sensitivity of goodness of fit indexes to lack of measurement invariance. Structural Equation Modeling, 14(3), 464-504. doi:10.1080/10705510701301834

Cheung, G. W., & Rensvold, R. B. (2002). Evaluating goodness-of-fit indexes for testing measurement invariance. Structural Equation Modeling, 9(2), 233-255. doi:10.1207/S15328007SEM0902_5

Putnick, D. L., & Bornstein, M. H. (2016). Measurement invariance conventions and reporting: The state of the art and future directions for psychological research. Developmental Review, 41, 71-90. doi:10.1016/j.dr.2016.06.004

Ordered-categorical indicators:

Svetina, D., Rutkowski, L., & Rutkowski, D. (2020). Multiple-group invariance with categorical outcomes using updated guidelines: An illustration using Mplus and the lavaan/semTools packages. Structural Equation Modeling, 27(1), 111-130. doi:10.1080/10705511.2019.1602776

Wu, H., & Estabrook, R. (2016). Identification of confirmatory factor analysis models of different levels of invariance for ordered categorical outcomes. Psychometrika, 81(4), 1014-1045. doi:10.1007/s11336-016-9506-0

See also

nomo_partial() for researcher-specified releases.

Examples

inv <- nomo_invariance(
  "Agency =~ ag1 + ag2 + ag3 + ag4",
  data = nomo_demo_network,
  group = "group",
  levels = c("configural", "metric", "scalar")
)
inv
#> <nomo_invariance> Measurement invariance
#> Grouping variable: group (2 groups: online, paper) | Indicators: continuous
#> Requested: configural -> metric -> scalar
#> Completed: configural -> metric -> scalar
#> 
#>   Level         CFI  RMSEA   SRMR  CFI change  RMSEA change   LRT p
#>   configural  1.000  0.000  0.002           -             -       -
#>   metric      1.000  0.000  0.028      +0.000        +0.000    .146
#>   scalar      0.954  0.121  0.065      -0.046        +0.121  < .001
#> Localized equality-constraint diagnostics retained: 12
#> 
#> Fit changes and score diagnostics are evidence. They are not pass/fail rules,
#> and nomologR never frees a parameter because of them.
nomo_table(inv, "fit")
#> # A tibble: 3 × 19
#>   level     constraints partial_requested status converged  chisq    df   pvalue
#>   <chr>     <chr>       <chr>             <chr>  <lgl>      <dbl> <dbl>    <dbl>
#> 1 configur… none        ""                estim… TRUE       0.327     4 9.88e- 1
#> 2 metric    loadings    ""                estim… TRUE       5.71      7 5.75e- 1
#> 3 scalar    loadings, … ""                estim… TRUE      68.9      10 7.12e-11
#> # ℹ 11 more variables: cfi <dbl>, rmsea <dbl>, srmr <dbl>, delta_cfi <dbl>,
#> #   delta_rmsea <dbl>, delta_srmr <dbl>, lrt_chisq <dbl>, lrt_df <dbl>,
#> #   lrt_p <dbl>, warnings <chr>, error <chr>
head(nomo_table(inv, "local_strain"))
#> # A tibble: 6 × 8
#>   level  constraint_index constraint    score_x2    df  p_value diagnostic_only
#>   <chr>             <int> <chr>            <dbl> <dbl>    <dbl> <lgl>          
#> 1 metric                3 .p3. == .p17.   4.92       1 2.66e- 2 TRUE           
#> 2 metric                2 .p2. == .p16.   1.03       1 3.10e- 1 TRUE           
#> 3 metric                4 .p4. == .p18.   0.690      1 4.06e- 1 TRUE           
#> 4 metric                1 .p1. == .p15.   0.0273     1 8.69e- 1 TRUE           
#> 5 scalar                7 .p7. == .p21.  61.1        1 5.33e-15 TRUE           
#> 6 scalar                5 .p5. == .p19.  11.3        1 7.68e- 4 TRUE           
#> # ℹ 1 more variable: constraint_display <chr>

# \donttest{
# A release is a documented researcher decision, never an automatic search.
release <- nomo_partial(
  level = "scalar",
  syntax = "ag3 ~ 1",
  rationale = "Prior evidence anticipated a mode difference in ag3 wording."
)
inv_partial <- nomo_invariance(
  "Agency =~ ag1 + ag2 + ag3 + ag4",
  data = nomo_demo_network,
  group = "group",
  levels = c("configural", "metric", "scalar"),
  partial = release
)
nomo_table(inv_partial, "fit")
#> # A tibble: 3 × 19
#>   level  constraints partial_requested status converged chisq    df pvalue   cfi
#>   <chr>  <chr>       <chr>             <chr>  <lgl>     <dbl> <dbl>  <dbl> <dbl>
#> 1 confi… none        ""                estim… TRUE      0.327     4  0.988     1
#> 2 metric loadings    ""                estim… TRUE      5.71      7  0.575     1
#> 3 scalar loadings, … "ag3 ~ 1"         estim… TRUE      6.79      9  0.659     1
#> # ℹ 10 more variables: rmsea <dbl>, srmr <dbl>, delta_cfi <dbl>,
#> #   delta_rmsea <dbl>, delta_srmr <dbl>, lrt_chisq <dbl>, lrt_df <dbl>,
#> #   lrt_p <dbl>, warnings <chr>, error <chr>
nomo_table(inv_partial, "partial")
#> # A tibble: 1 × 4
#>   release_id level  syntax  rationale                                           
#>   <chr>      <chr>  <chr>   <chr>                                               
#> 1 P1         scalar ag3 ~ 1 Prior evidence anticipated a mode difference in ag3…
# }