Convergent and discriminant construct-validity evidence

nomo_validity() summarizes several complementary forms of measurement evidence from a fitted first-order CFA. Convergent evidence includes standardized loadings and average variance extracted (AVE). Discriminant evidence includes latent-factor correlations and, where appropriate, HTMT2 as the primary heterotrait-monotrait summary, with the original HTMT available as a sensitivity/teaching comparison.

Usage

nomo_validity(
  fit,
  ave_obs_var = TRUE,
  htmt = c("both", "htmt2", "htmt", "none"),
  htmt_missing = "default",
  fornell_larcker = FALSE,
  guidance = nomo_defaults()
)

Arguments

fit: A fitted nomo_cfa() object or fitted lavaan CFA model.
ave_obs_var: Logical passed to semTools::AVE(). TRUE (default) uses observed variances in the denominator; FALSE uses model-implied variances. For ordinal indicators, semTools calculates AVE using the polychoric correlation structure.
htmt: Which heterotrait-monotrait summaries to request: "both" (default), "htmt2", "htmt", or "none". HTMT2 is prioritized because its geometric-mean formulation is designed for congeneric indicators; original HTMT assumes tau-equivalence.
htmt_missing: Missing-data option passed to semTools::htmt(). The default "default" delegates the unrestricted-correlation missing-data handling to lavaan::lavCor() rather than silently imposing listwise deletion. Other supported values are "listwise", "pairwise", "direct", "ml", and "fiml".
fornell_larcker: Logical. If TRUE, also produce a legacy Fornell-Larcker matrix and pairwise comparison where available. It is not used as the primary discriminant-validity criterion.
guidance: Guidance settings from nomo_defaults(). The AVE and HTMT references are review prompts rather than universal pass/fail cutoffs.

Value

A nomo_validity object. The fields to read are:

ave: average variance extracted per construct, as convergent evidence.
latent_correlations: construct correlations with intervals.
htmt2 and htmt: heterotrait-monotrait ratios per pair.
discriminant: each pair's separation evidence with its reference and interpretation.
htmt_status: which HTMT variants were computed, and why any was not.
fornell_larcker_pairs: the historical comparison, when requested.
standardized_loadings, references, 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 function deliberately avoids a binary declaration that a construct is "valid" or "invalid". Numerical references trigger inspection and explanation. HTMT-family results are considered together with theory, indicator content, CFA fit, and latent-factor overlap. The Fornell-Larcker comparison is available only by explicit request and is labeled legacy/ supporting evidence because simulation work has shown that it can miss discriminant-validity problems that HTMT detects.

References

Historical context:

Campbell, D. T., & Fiske, D. W. (1959). Convergent and discriminant validation by the multitrait-multimethod matrix. Psychological Bulletin, 56(2), 81-105. doi:10.1037/h0046016

Fornell, C., & Larcker, D. F. (1981). Evaluating structural equation models with unobservable variables and measurement error. Journal of Marketing Research, 18(1), 39-50. doi:10.2307/3151312

Contemporary construct-separation evidence:

Henseler, J., Ringle, C. M., & Sarstedt, M. (2015). A new criterion for assessing discriminant validity in variance-based structural equation modeling. Journal of the Academy of Marketing Science, 43(1), 115-135. doi:10.1007/s11747-014-0403-8

Roemer, E., Schuberth, F., & Henseler, J. (2021). HTMT2–An improved criterion for assessing discriminant validity in structural equation modeling. Industrial Management & Data Systems, 121(12), 2637-2650. doi:10.1108/IMDS-02-2021-0082

Rönkkö, M., & Cho, E. (2022). An updated guideline for assessing discriminant validity. Organizational Research Methods, 25(1). doi:10.1177/1094428120968614

Voorhees, C. M., Brady, M. K., Calantone, R., & Ramirez, E. (2016). Discriminant validity testing in marketing: An analysis, causes for concern, and proposed remedies. Journal of the Academy of Marketing Science, 44(1), 119-134. doi:10.1007/s11747-015-0455-4

Examples

model <- '
  visual  =~ x1 + x2 + x3
  textual =~ x4 + x5 + x6
  speed   =~ x7 + x8 + x9
'
cfa <- nomo_cfa(model, data = lavaan::HolzingerSwineford1939)
val <- nomo_validity(cfa)
summary(val)
#> <nomo_validity summary> Convergent and discriminant evidence
#> 
#> Convergent evidence by construct
#>   Construct    AVE  Min |loading|  Median |loading|  Loadings flagged  Flag
#>   visual     0.371          0.424             0.581                 1  review
#>   textual    0.721          0.838             0.852                 0
#>   speed      0.424          0.570             0.665                 0  review
#> 
#> Construct separation
#>   Construct 1  Construct 2  Latent r  95% CI          HTMT2   HTMT
#>   visual       textual         0.459  [0.334, 0.584]  0.384  0.424
#>   visual       speed           0.471  [0.328, 0.613]  0.387  0.467
#>   textual      speed           0.283  [0.148, 0.418]  0.280  0.290
#> 
#> Standardized loadings and AVE address convergent evidence; latent correlations
#> and HTMT-family statistics address construct separation. These are
#> complementary questions, not interchangeable pass/fail tests.
val$decision_log
#> # A tibble: 11 × 10
#>    stage    object   metric  value reference severity observation recommendation
#>    <chr>    <chr>    <chr>   <dbl> <chr>     <chr>    <chr>       <chr>         
#>  1 validity measure… evide… NA     Fornell … info     Convergent… Interpret num…
#>  2 validity x2       stand…  0.424 configur… review   Absolute s… Inspect item …
#>  3 validity visual   AVE     0.371 configur… review   AVE is bel… Inspect stand…
#>  4 validity textual  AVE     0.721 configur… info     AVE is at … Carry AVE for…
#>  5 validity speed    AVE     0.424 configur… review   AVE is bel… Inspect stand…
#>  6 validity textual… HTMT2   0.280 configur… info     HTMT2 does… Interpret thi…
#>  7 validity visual … HTMT2   0.387 configur… info     HTMT2 does… Interpret thi…
#>  8 validity visual … HTMT2   0.384 configur… info     HTMT2 does… Interpret thi…
#>  9 validity textual… HTMT    0.290 configur… info     HTMT does … Interpret thi…
#> 10 validity visual … HTMT    0.467 configur… info     HTMT does … Interpret thi…
#> 11 validity visual … HTMT    0.424 configur… info     HTMT does … Interpret thi…
#> # ℹ 2 more variables: decision <chr>, rationale <chr>