Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Heterotrait-monotrait ratio of indicator correlations for discriminant validity

htmt(R, K_i, K_j) = HTMT_ij

Maps the correlation matrix of the indicators of two or more multi-item constructs, with each indicator assigned to one construct, to the heterotrait-monotrait ratio HTMT_ij for each pair of constructs, the mean correlation between the indicators of constructs i and j divided by the geometric mean of the mean correlations among the indicators within each construct, for comparison with a cut-off of 0.85 or 0.90. HE-FM-DISV-001 writes the ratio for two indicators per construct, the case of the article's worked example, and HE-CF-DISV-001 computes it for any numbers of indicators and every pair of constructs. The article's known-groups step is the live HE-FM-CD-001 applied to the mild and moderate groups (pooled standard deviations of 0.2102 and 17.03, standardised differences of 0.24 and 0.94), its QALY gains of 0.10 and 0.24 are HE-FM-QALY-001, the sum of utility times duration, for two constant-utility states over 2 years differenced, and its ratios of £30,000 and £12,500 per QALY are HE-FM-ICER-001, so none of them is restated. The factor of 2.4 by which the ratio would be overstated is two HE-FM-ICER-001 ratios with the same incremental cost divided, which equals the inverse ratio of the two QALY gains. The multitrait-multimethod checks in the article's steps 1 to 3 are differences of correlations set against the analyst's prespecified cut-offs and have no formula record. Correcting prevalence and cost for misclassification against a gold standard is HE-FN-CRV-001, a criterion-validity function and a different property. Notation follows the Discriminant Validity article.

  • Heterotrait-monotrait ratio for two constructs with two indicators each

    rbar_ij = (r_i1j1 + r_i1j2 + r_i2j1 + r_i2j2) / 4; HTMT_ij = rbar_ij / sqrt(r_i1i2 * r_j1j2)

    With two indicators per construct each within-construct mean is a single correlation, r_i1i2 or r_j1j2, and the between-construct mean rbar_ij has four terms, so the heterotrait-monotrait ratio is rbar_ij divided by the square root of r_i1i2 times r_j1j2. This is Eq. 6 of Henseler, Ringle and Sarstedt with K_i and K_j both equal to 2, and HE-CF-DISV-001 takes any numbers of indicators. The authors describe the ratio as an estimate of the correlation between the two constructs, so a value clearly below 1 indicates constructs that differ. Used as a criterion, a value above a cut-off of 0.85 or 0.90 points to a lack of discriminant validity, a level the authors call debatable. In the article's example the questionnaire and the clinician rating serve as the two indicators of physical functioning and of mood.

Discriminant Validity — Functions & Formulae | HealthEconomics.wiki