DMAR (version 1.0.0)

htmt: Heterotrait-Monotrait Ratio of Correlations (HTMT)

Description

Computes the HTMT discriminant-validity index of Henseler, Ringle, and Sarstedt (2015) for every pair of constructs: the average correlation between items of different constructs, divided by the geometric mean of the average correlations among items within each construct. Two constructs whose HTMT approaches 1 are empirically indistinguishable however cleanly the model draws them; the customary red flags are 0.85 (strict) or 0.90 (liberal). An optional bootstrap gives a one-sided upper confidence bound, the quantity actually compared against the cutoff in the validity literature.

Usage

htmt(data, blocks, B = 0, conf_level = 0.95, seed = NULL)

Value

A data.frame (class dmar_tbl) with one row per construct pair: construct_1, construct_2,

htmt, and, when B > 0, upper_limit (the one-sided conf_level bootstrap percentile bound).

Arguments

data

A data.frame of item responses.

blocks

Named list of character vectors: each element names a construct and gives its item columns (two or more per construct, two or more constructs).

B

Number of bootstrap resamples for the upper confidence bound; 0 (default) skips the bootstrap and reports the point estimates only.

conf_level

Confidence level for the one-sided upper bound. Defaults to 0.95.

seed

Optional integer seed for the bootstrap, used locally (the caller's random number generator state is restored on exit).

Author

Ken Kelley [email protected]

Details

All correlations are Pearson, computed on pairwise-complete observations. The statistic uses absolute average heterotrait correlations enter as absolute values, the convention of later implementations (the 2015 proposal used the plain correlations, which can cancel when signs mix; Roemer, Schuberth, & Henseler, 2021, recommend the absolute form for exactly that reason), and values near or above 1 indicating that the two item sets correlate across constructs about as strongly as within them. HTMT is a correlation-based screen, deliberately model-free; the confirmatory companion is the latent correlation between the two factors (see correction_for_attenuation and its factor-model discussion).

The bootstrap, when requested (B > 0), resamples the rows of data with replacement B times and recomputes every pairwise HTMT on each resample; the reported upper_limit is the conf_level empirical quantile of each pair's bootstrap distribution, a one-sided upper percentile bound (Efron & Tibshirani, 1993). That bound is the only interval offered, matching how the validity literature uses HTMT (the question is whether the ratio credibly exceeds the cutoff); no two-sided or bias-corrected and accelerated (BCa) variant is provided. A resample in which some block's average within-construct correlation is not positive leaves HTMT undefined there; such resamples are dropped, a single warning reports how many, and the bound is computed from the resamples that remained (the call stops only when fewer than 100 remain). Bootstrap results vary from run to run; supply seed for reproducibility.

References

Efron, B., & Tibshirani, R. J. (1993). An introduction to the bootstrap. New York, NY: Chapman & Hall/CRC.

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. tools:::Rd_expr_doi("10.1007/s11747-014-0403-8")

See Also

average_variance_extracted for the convergent side of the validity ledger; reliability_omega for the composite reliability of each block; correction_for_attenuation for the latent correlation route.

Other multivariate and latent variable methods: average_variance_extracted(), bifactor_indices(), cfa_1(), cfa_2(), cfa_k(), ci_eigenvalue(), common_method_marker(), common_method_single_factor(), dmacs(), ecvi(), irt_grm(), irt_information(), measurement_alignment(), measurement_invariance(), procrustes_phi(), simple_structure()

Examples

# Two clean constructs and their six items.
set.seed(113)
n <- 300
f1 <- rnorm(n); f2 <- 0.3 * f1 + sqrt(1 - 0.09) * rnorm(n)
d <- data.frame(
  a1 = .8 * f1 + rnorm(n, 0, .6), a2 = .7 * f1 + rnorm(n, 0, .7),
  a3 = .6 * f1 + rnorm(n, 0, .8),
  b1 = .8 * f2 + rnorm(n, 0, .6), b2 = .7 * f2 + rnorm(n, 0, .7),
  b3 = .6 * f2 + rnorm(n, 0, .8))
h <- htmt(d, blocks = list(A = c("a1", "a2", "a3"),
                           B = c("b1", "b2", "b3")))
h

# The broom verbs: one row per construct pair.
generics::tidy(h)
generics::glance(h)

# The upper confidence bound, which is the quantity the validity
# literature compares against 0.85 or 0.90, comes from a bootstrap
# that recomputes every pairwise ratio on each of B resamples; the
# table gains an upper_limit column. B = 1000 keeps the example
# quick; a claim about discriminant validity deserves the bound at
# B = 10000 rather than the point estimate alone.
htmt(d, blocks = list(A = c("a1", "a2", "a3"),
                      B = c("b1", "b2", "b3")),
     B = 1000, seed = 113)