Create a link to share a read only version of this article with your colleagues and friends. We prove that the HTMT index is simply a scale score correlation disattenuated with parallel reliability (i.e., the standardized alpha) and thus should not be expected to outperform modern CFA techniques, which our simulation demonstrates. Our main results concern inference against a cutoff and are relevant when a researcher wants to make a yes/no decision about discriminant validity. Indeed, the definitions shown in Table 2 show little connections to the original MTMM matrices. For more information view the SAGE Journals Sharing page. Overall, χ2(cut) and CICFA (cut) can be recommended as general solutions because they meet the definition of discriminant validity, have the flexibility to adapt to various levels of cutoffs, and can be extended to more complex scenarios such as nonlinear measurement models (Foster et al., 2017), scales with minor dimensions (Rodriguez et al., 2016), or cases in which factorial validity is violated because of cross-loadings. The assumption appears to be invalid as it ignores an important difference between these uses: Whereas the degrees of freedom of an invariance test scale roughly linearly with the number of indicators, the degrees of freedom in CFI(1) are always one. It does show that, as you predicted, the three self esteem measures seem to reflect the same construct (whatever that might be), the three locus of control measures also seem to reflect the same construct (again, whatever that is) and that the two sets of measures seem to be reflecting two different constructs (whatever they are). The full factorial (6 × 3 × 5 × 3 × 4) simulation was implemented with the R statistical programming environment using 1,000 replications for each cell. In the trinitarian approach to validity, convergent and discriminant validities form the evidence for construct validity (Hubley & Zumbo, 1996). When the loadings varied, ρDTR and ρDPR became positively biased. Fifth, the definition does not confound the conceptually different questions of whether two measures measure different things (discriminant validity) and whether the items measure what they are supposed to measure and not something else (i.e., lack of cross-loadings in Λ, factorial validity),3 which some of the earlier definitions (categories 3 and 4 in Table 2) do. These two variables also have different causes and consequences (American Psychological Association, 2015), so studies that attempt to measure both can lead to useful policy implications. Table 9 considers cutoffs other than 1, using values of .85, .90, and .95 that are sometimes recommended in the literature, showing results that are consistent with those of the previous tables. Second, many techniques were used differently than originally presented. 14.We thank Terrence Jorgensen for pointing this out. The term “discriminant validity” was typically used without a definition or a citation, giving the impression that there is a well-known and widely accepted definition of the term. The current state of the discriminant validity literature and research practice suggests that this is not the case. However, this also has the disadvantage that it steers a researcher toward making yes/no decisions instead of assessing the degree to which discriminant validity holds in the data. (2016) further claim that the common omission of the correction is “the most troublesome issue with the [χ2(1)] approach” (p. 123). This effect is seen in Table 11, where the pattern of results for CFA models was largely similar between the cross-loading conditions, but the presence of cross-loadings increased the false positive rate. 5.The disattenuation equation shows that the scale score correlation is constrained to be no greater than at the geometric mean of the two reliabilities. In contrast, when the correlation between the factors is less than 1, the additional constraints are somewhat redundant because constraining the focal correlation to 1 will also bias all other correlations involving the focal variables. Because direct criticism of existing techniques is often avoided, there appears to be a tendency in which new techniques continue to be added without clarifying the problems of previously used techniques. We demonstrate this problem in Online Supplement 1. The full simulation code is available in Online Supplement 2, and the full set of simulation results at the design level can be found in Online Supplement 3. Paradoxically, this power to reject the null hypothesis has been interpreted as a lack of power to detect discriminant validity (Voorhees et al., 2016). 4.Of the AMJ and JAP articles reviewed, most reported a correlation table (AMJ 96.9%, JAP 89.3%), but most did not specify whether the reported correlations were scale score correlations or factor correlations (AMJ 100%, JAP 98.5%). If the pattern coefficients have no cross-loadings, this condition is equivalent to say that the factor correlation is greater than 1 (see Table 5). (2017) criticized the conceptual redundancy between grit and conscientiousness based on a disattenuated correlation of .84 (ρSS=.66). While the difference was small, it is surprising that χ2(1) was strictly superior to χ2(merge), having both more power and a smaller false positive rate. We start by reviewing articles in leading organizational research journals and demonstrating that the concept of discriminant validity is understood in at least two different ways; consequently, empirical procedures vary widely. 11.A full discriminant validity analysis requires the pairwise comparisons of all possible factor pairs. These techniques could also be used in multiple item scenarios, if a researcher does not have access to SEM software, or in some small sample scenarios (Rosseel, 2020). Relationship between correlation values and the problem of discriminant validity. The estimation of factor correlations in a CFA is complicated by the fact by default latent variables are scaled by fixing the first indicator loadings, which produces covariances that are not correlations. To fill this gap, various less-demanding techniques have been proposed, but few of these techniques have been thoroughly scrutinized. However, the use of uncorrelated factors can rarely be justified (Fabrigar et al., 1999), which means that, in most cases, pattern and structure coefficients are not equal. (2015) explain that ρDPR is a factor correlation estimate, they do not compare it against other factor correlation estimation techniques. This definition also supports a broad range of empirical practice: If considered on the scale level, the definition is compatible with the current tests, including the original MTMM approach (Campbell & Fiske, 1959). Because ρDPR and HTMT were proven equivalent and always produced identical results, we report only the former. Even if the latent variable correlation is only slightly different from 1 (e.g., .98), such small differences will be detected as statistically significant if the sample size is sufficiently large. (B) Fixing one of the loadings to unity (i.e., using the default option). This inconsistency might be an outcome of researchers favoring cutoffs for their simplicity, or it may reflect the fact that after calculating a discriminant validity statistic, researchers must decide whether further analysis and interpretation is required. If significantly different, the correlation is classified into the current section. The same results are mirrored in the second set of rows in Table 7; both CIDPR and CIDTR produced positively biased CIs with poor coverage and balance. 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The techniques and the symbols that we use for them are summarized in Table 4. Thus, the term “HTMT” is misleading, giving the false impression that HTMT is related to MTMM and obscuring the fact that it is simply a variant of disattenuated correlation. ORCID iDMikko Rönkkö https://orcid.org/0000-0001-7988-7609, Eunseong Cho https://orcid.org/0000-0003-1818-0532. There is Fisher’s (1936) classic example o… 16.For example, Henseler et al. This alternative form shows that AVE is actually an item-variance weighted average of item reliabilities. The results shown in Table 10 show that all estimates become biased toward 1. ρDCR was slightly most robust to these misspecifications, but the differences between the techniques were not large. This mathematical fact is why the cross-loading technique produced strange results in their simulation, which was not explained in the original paper. Following Cho’s (2016) suggestion and including the assumption of each reliability coefficient in the name will hopefully also reduce the chronic misuse of these reliability coefficients. Because datasets used by applied researchers rarely lend themselves to MTMM analysis, the need to assess discriminant validity in empirical research has led to the introduction of numerous techniques, some of which have been introduced in an ad hoc manner and without rigorous methodological support. We will next address the various techniques in more detail. By continuing to browse Finally, the AVE statistic is sometimes calculated from a partial least squares analysis (AVEPLS), which overestimates indicator reliabilities and thus cannot detect even the most serious problems (Rönkkö & Evermann, 2013). We present a definition that does not depend on a particular model and makes it explicit that discriminant validity is a feature of a measure instead of a construct:2 Two measures intended to measure distinct constructs have discriminant validity if the absolute value of the correlation between the measures after correcting for measurement error is low enough for the measures to be regarded as measuring distinct constructs. Instead, it appears that many of the techniques have been introduced without sufficient testing and, consequently, are applied haphazardly. (2016) and Podsakoff et al. Model comparison techniques involve comparing the original model against a model where a factor correlation is fixed to a value high enough to be considered a discriminant validity problem. To verify this assumption, we took a random sample of 49 studies out of the 199 studies that applied SEMs and emailed the authors to ask for the type of correlation used in the study. But as I said at the outset, in order to argue for construct validity we really need to be able to show that both of these types of validity are supported. First, researchers should clearly indicate what they are assessing when assessing discriminant validity by stating, for example, that “We addressed discriminant validity (whether two scales are empirically distinct).” Second, the correlation tables, which are ubiquitous in organizational research, are in most cases calculated with scale scores or other observed variables. This technique proliferation causes confusion and misuse. All factors had unit variances in the population, and we scaled the error variances so that the population variances of the items were one. Furthermore, convergent validity coefficients (shown in bold in Tables 1, 2 and 3) should be large enough to encourage further examination of discriminant validity. The original meaning of the term “discriminant validity” was tied to MTMM matrices, but the term has since evolved to mean a lack of a perfect or excessively high correlation between two measures after considering measurement error. The definition can also be applied on both the scale level and the scale-item level. Across many theoretical frameworks these functions include planning, organizing, sequencing, problem solving, decision-making, goal selection, switching between task sets, monitoring for conflict, monitoring for task-relevant information, monitoring performance levels, updating working memory, interference suppressio… Detection Rates by Technique Using Alternative Cutoffs. take a good look at the table and you will see that in this example the convergent correlations are always higher than the discriminant ones. If the estimate falls outside the interval (e.g., less than .9), then the correlation is constrained to be at the endpoint of the interval, and the model is re-estimated. (2010) diagnosed a discriminant validity problem between job satisfaction and organizational commitment based on a correlation of .91, and Mathieu and Farr (1991) declared no problem of discriminant validity between the same variables on the basis of a correlation of .78. In summary, Table 8 supports the use of CICFA(1) and χ2(1). This result is easiest to understand in the context of CICFA(cut); when estimates became less precise, this also widened the confidence intervals and, consequently, increased the frequency of results where the cutoff fell within the interval. In a CFA, the model parameters are pattern coefficients, and these are also more commonly reported in EFA applications (Henson & Roberts, 2006). Table 1. 10.Notably, Bagozzi (1981) wrote a critical commentary, to which Fornell and Larcker (1981b) published a rejoinder, but neither of these articles addressed the issues that we raise in this article. This effect and the general undercoverage of the CIs were most pronounced in small samples. (2014) note that in psychology research, the symptoms or characteristics of different disorders commonly overlap, producing nonnegligible cross-loadings in the population. This product could help you, Accessing resources off campus can be a challenge. I find it easiest to think about convergent and discriminant validity as two inter-locking propositions. The reliability coefficients presented above make a unidimensionality assumption, which may not be realistic in all empirical research. This study draws an unambiguous conclusion about which method is best for assessing discriminant validity and which methods are inappropriate. The CFA comparison was most commonly used to assess whether two factors could be merged, but a range of other comparisons were also presented. We also omit the two low correlation conditions (i.e., .5, .6) because the false positive rates are already clear in the .7 condition. Declaration of Conflicting InterestsThe author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. (2015) defined a cross-loading when the loading (i.e., structure coefficient) between an item and its unintended factor is greater than the loading between the item and its intended factor. But, at the very least, we can assume from the pattern of correlations that the four items are converging on the same thing, whatever we might call it. First, CFA correlations estimated with maximum likelihood (ML) can be expected to be more efficient than multistep techniques that rely on corrections for attenuation (Charles, 2005; Muchinsky, 1996). The goal of discriminant validity evidence is to be able to discriminate between measures of dissimilar constructs. For instance, Item 1 might be the statement “I feel good about myself” rated using a 1-to-5 Likert-type response format. For example, defining discriminant validity in terms of a (true) correlation between constructs implies that a discriminant validity problem cannot be addressed with better measures. Table 7. Scale score correlation ρSS was always negatively biased due to the well-known attenuation effect. The number of required model comparisons is the number of unique correlations between the variables, given by k(k−1)/2, where k is the number of factors. If this is the case, the typical discriminant validity assessment techniques that are the focus of our article are not directly applicable, but other techniques are needed (Tay & Jebb, 2018). In empirical applications, the correlation level of .9 was nearly universally interpreted as a problem, and we therefore use this level as a cutoff between the Marginal and Moderate cases. The covariances between factors obtained in the latter way equal the correlations; alternatively, when using CICFA(sys), the standardized factor solution can be inspected. The balance statistics were all negative, indicating that when the population value was outside the CI, it was generally more frequently below the lower limit of the interval than above the upper limit. While simple to use, the disattenuation correction is not without problems. There’s a number of things we can do to address that question. However, there are two problems. We theorize that all four items reflect the idea of self esteem (this is why I labeled the top part of the figure Theory). This simpler form makes it clear that HTMT is related to the disattenuation formula (Equation 4). We then review techniques that have been proposed for discriminant validity assessment, demonstrating some problems and equivalencies of these techniques that have gone unnoticed by prior research. We theorize that all four items reflect the idea of self esteem (this is why I labeled the top part of the figure Theory). Even the two relatively low correlations (between informant-report and Time 1 The e-mail addresses that you supply to use this service will not be used for any other purpose without your consent. This example demonstrates that researchers who use systematically biased measures cannot accurately assess discriminant validity. There are four correlations between measures that reflect different constructs, and these are shown on the bottom of the figure (Observation). The term “ The TTM holds that individuals progress through qualitatively distinct stages when changing be-haviors such as smoking cessation (Prochaska & Velicer, 1997). Mikko Rönkkö is associate professor of entrepreneurship at Jyväskylä University School of Business and Economics (JSBE) and a docent at Aalto University School of Science. However, mixed judgments were made about the correlation values between .8 and .9. 20.The software is available at https://github.com/eunscho/MQAssessor. Coverage and Balance of 95% Confidence Intervals by Loadings and Sample Size. While commonly used, the AVE statistic has been rarely discussed by methodological research and, consequently, is poorly understood.10 One source of confusion is the similarity between the formula for AVE and that of congeneric reliability (Fornell & Larcker, 1981a): The meaning of AVE becomes more apparent if we rewrite the original equation as: where ρi= λi2σxi2 is the reliability and σxi2= λi2+σei2 is the variance of item i. This page was last modified on 5 Aug 2020. Table 10. Thus, while marketed as a new technique, the HTMT index has actually been used for decades; parallel reliability is the oldest reliability coefficient (Brown, 1910), and disattenuated correlations have been used to assess discriminant validity for decades (Schmitt, 1996). However, in this case, it is difficult to interpret the latent variables as representing distinct concepts. Thus, we favor a more transparent term “disattenuated correlation using parallel reliability” (denoted ρDPR) because this more systematic name tells what the coefficient is (i.e., correlation), how it is obtained (i.e., disattenuated), and under what conditions it can be used (i.e., parallel reliability). Definitions were also implicitly present in other fields to characterize essentially continuous phenomena: Consider a ’. Definitions were also implicitly present in other fields to characterize essentially continuous phenomena: Consider a doctor ’ diagnosis... 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