Regardless of how the correlations are calculated, they are used in just two ways: either by comparing the point estimates against a cutoff (i.e., the square root of the AVE) or by checking whether the absolute value of the CI contains either zero or one. Thus, if there is a threshold, it is likely to fall between .8 and .9. We theorize that all four items reflect the idea of self esteem (this is why I labeled the top part of the figure Theory). (A) despite high discriminant validity, the AVE/SV criterion fails, (B) despite low discriminant validity, the AVE/SV criterion passes. When the factors are perfectly correlated, imposing more constraints means that the model can be declared to misfit in more ways, thus leading to lower power. After defining what discriminant validity means, we provide a detailed discussion of each of the techniques identified in our review. Constraining these cross-loadings to be zero can inflate the estimated factor correlations, which is problematic, particularly for discriminant validity assessment (Marsh et al., 2014). Many studies assess discriminant validity by comparing the hypothesized model with a model with fewer factors. Contact us if you experience any difficulty logging in. Fourth, the definition is not tied to either the individual item level or the multiple item scale level but works across both, thus unifying the category 1 and category 2 definitions of Table 2. These concerns are ill-founded. For more information view the SAGE Journals Article Sharing page. Definitions of Discriminant Validity in Existing Studies. However, these techniques tend to require larger sample sizes and advanced software and are consequently less commonly used. The heterotrait-monotrait (HTMT) ratio was recently introduced in marketing (Henseler et al., 2015) and is being adopted in other disciplines as well (Kuppelwieser et al., 2019). Clearly, none of these techniques can be recommended. The bottom of the figure shows what a correlation matrix based on a pilot sample might show. Discriminant validity exists if the chi-square value of the two-factor model is significantly lower than the chi-square value of the one-factor model (Anderson & Gerbing, 1988). A. Shaffer et al., 2016) have suggested comparing models by calculating the difference between the comparative fit indices (CFIs) of two models (ΔCFI), which is compared against the .002 cutoff (CFI(1)). 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. Table 11. 6.Different variations of disattenuated correlations can be calculated by varying how the scale score correlation is calculated, how reliabilities are estimated, or even the disattenuation equation itself. We will now prove that the CFI comparison is equivalent to a χ2 test that uses a critical value based on the null model instead of the χ2 distribution. Table 7 shows that in this condition, the confidence intervals of all techniques performed reasonably well. Detection Rates by Technique Using Alternative Cutoffs. The assumptions of parallel, tau-equivalent, and congeneric reliability. If this test fails, diagnose the model with residuals and/or modification indices to understand the source of misspecification (Kline, 2011, chap. (2015). Evidence based on test content. Thus, cross-loadings, nonlinear factor loadings or nonnormal error terms can be included because a CFA model can also be used in the context of item response models (Foster et al., 2017), bifactor models (Rodriguez et al., 2016), exploratory SEMs (Marsh et al., 2014) or other more advanced techniques. These techniques fall into two classes: those that inspect the factor loadings and those that assess the overall model fit. Table 2: Reported Evidence for Reliability and Validity of 64 Medical Education Instruments Abstracted Using a Traditional Validity Framework and Mapped to the Contemporary Framework17 of Validity as a Unitary Concept . We estimated the factor models with the lavaan package (Rosseel, 2012) and used semTools to calculate the reliability indices (Jorgensen et al., 2020). For this group of researchers, the term referred to “whether the two variables…are distinct from each other” (Hu & Liden, 2015, p. 1110). This might be based on giving our scale out to a sample of respondents. In the trinitarian approach to validity, convergent and discriminant validities form the evidence for construct validity (Hubley & Zumbo, 1996). The techniques for assessing discriminant validity identified in our review can be categorized into (a) techniques that assess correlations and (b) techniques that focus on model fit assessment. and discriminant validity.27 The internal consistency was validated using composite reliability (CR), threshold value: 0.70. Techniques that directly compare the point estimates of correlations with a cutoff (i.e., ρCFA(1), ρDPR(1), and ρCR(1)) have very high false negative rates because an unbiased and normal correlation estimate can be expected to be below the population value (here, 1) exactly half the time. Thus, the amount of misfit produced by the first constraint is greater than the other constraints that χ2(merge) contributes. Because a factor correlation corrects for measurement error, the AVE/SV comparison is similar to comparing the left-hand side of Equation 3 against the right-hand side of Equation 2. Correlations (denoted ρCFA) can be estimated by either freeing the factor loadings and scaling the factors by fixing their variances to 1 (i.e., A in Figure 2) or standardizing the factor covariance matrix (i.e., B), for example, by requesting standardized estimates that all SEM software provides. That’s not bad for one simple analysis. Table 10 shows that the mean estimate was largely unaffected, but the variance of the estimates (not reported in the table) increased because of the increased model complexity. All techniques were again affected, and both the power and false positive rates increased across the board when the correlation between the factors was less than one. Instead, it appears that many of the techniques have been introduced without sufficient testing and, consequently, are applied haphazardly. Executive functions (EFs) consist of a set of general-purpose control processes believed to be central to the self-regulation of thoughts and behaviors that are instrumental to accomplishing goals. For example, a correlation of .87 would be classified as Marginal. Figure 1. 17. We acknowledge the computational resources provided by the Aalto Science-IT project. The second issue is that some articles suggest that the significance level of the χ2 difference test should be adjusted for multiple comparisons (Anderson & Gerbing, 1988; Voorhees et al., 2016). Remember that I said above that we don’t have any firm rules for how high or low the correlations need to be to provide evidence for either type of validity. Of course, large samples and precise measurement would be required to ensure that the constructs can be distinguished empirically (i.e., are empirically distinct). Next, inspect the upper limits (lower limits for negative correlations) of the 95% CIs of the estimated factor correlations and compare their values against the cutoffs in Table 12.19. The disattenuated correlation between two unit-weighted composites X and Y of p and q items using parallel reliability as reliability estimates is given as follows: Substituting Equations A5, A6, and A7 into Equation A4, the equation is as follows: which equals the HTMT index shown in Equation A2. CIDCR(1) had a larger false positive rate than CICFA(1), particularly in small samples, possibly due to violating the large sample assumption of bootstrapping. All items loaded stronger on their associated factors than on other factors. In summary, Table 8 supports the use of CICFA(1) and χ2(1). Second, CICFA(cut) makes it easier to transition from testing of discriminant validity to its evaluation because the focal statistic is a correlation, which organizational researchers routinely interpret in other contexts. Testing for discriminant validity can be done using one of the following methods: O-sorting, chi-square difference test and the average variance extracted analysis. 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). Some society journals require you to create a personal profile, then activate your society account, You are adding the following journals to your email alerts, Did you struggle to get access to this article? However, this final concern can be alleviated to some extent through the use of bootstrap CIs (Henseler et al., 2015); in particular, the bias-corrected and accelerated (BCa) technique has been shown to work well for this particular problem (Padilla & Veprinsky, 2012, 2014). Table 6.1 shows instrument development and validation process. Thus, their results do not indicate the superiority of ρDPR but simply indicate that AVE/SV, which was their main comparison, performs very poorly. Cross-loadings (in the pattern coefficients) were either 0, 1, or 2. However, it is unclear whether this alternative cutoff has more or less power (i.e., whether 1+.002(χB2−dfB) is greater or less than 3.84) because the effectiveness of CFI(1) has not been studied. Establishing these different types of validity for a measure increases overall confidence that the indicator measures the concept it is intended to. Equation 3 shows an equivalent scale-level comparison (part of category 1 in Table 2) focusing on two distinct scales k and l. The factor correlations are solved from the interitem correlations by multiplying with left and right inverses of the factor pattern matrix to correct for measurement error and are then compared against a perfect correlation. 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. On the bottom part of the figure (Observation) w… Table 1 identifies three cases where there is insufficient discriminant validity (a, b, c). Using the cutoff of zero is clearly inappropriate as requiring that two factors be uncorrelated is not implied by the definition of discriminant validity and would limit discriminant validity assessment to the extremely rare scenario where two constructs are assumed to be (linearly) independent. Check the χ2 test for an exact fit of the CFA model. For these researchers, discriminant validity means that “two measures are tapping separate constructs” (R. Krause et al., 2014, p. 102) or that the measured “scores are not (or only weakly) associated with potential confounding factors” (De Vries et al., 2014, p. 1343). That is, patients with hypertension are further subdivided into three stages according to their blood pressure level, and each level is associated with different treatments. The dataset consists of fifty samples from each of three species of Irises (iris setosa, iris virginica, and iris versicolor). 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 page was last modified on 5 Aug 2020. To fill this gap, various less-demanding techniques have been proposed, but few of these techniques have been thoroughly scrutinized. The full factorial (6 × 3 × 5 × 3 × 4) simulation was implemented with the R statistical programming environment using 1,000 replications for each cell. Thus, the term “average indicator reliability” might be more informative than “average variance extracted.”. While both measure the same quantity, they are correlated only by approximately .45 because the temperature would always be out of the range of one of the thermometers that would consequently display zero centigrade.18 In the social sciences, a well-known example is the measurement of happiness and sadness, two constructs that can be thought of as opposite poles of mood (D. P. Green et al., 1993; Tay & Jebb, 2018). OK, so where does this leave us? The techniques that assess the lack of cross-loadings (pattern coefficients) and model fit provide (factorial) validity information, which is important in establishing the assumptions of the other techniques, but these techniques are of limited use in providing actual discriminant validity evidence. While the basic disattenuation formula has been extended to cases where its assumptions are violated in known ways (Wetcher-Hendricks, 2006; Zimmerman, 2007), the complexities of modeling the same set of violations in both the reliability estimates and the disattenuation equation do not seem appealing given that the factor correlation can be estimated more straightforwardly with a CFA instead. As with the other techniques, various misconceptions and misuses are found among empirical studies. Disattenuated correlations are useful in single-item scenarios, where reliability estimates could come from test-retest or interrater reliability checks or from prior studies. 170 Table 6.1: Instrument Development and Validation Process Chapter Analysis Description Chapter 5 Instrument Development Items generations – scale from previous studies Judge the items for content validity and Pilot test Chapter 6 Exploratory Measurement Assessment Descriptive statistics: Corrected item-total … Most methodological work defines discriminant validity by using a correlation but differs in what specific correlation is used, as shown in Table 2. Such an abundance of techniques is positive if techniques have different advantages, and they are purposefully selected based on their fit with the research scenario. INTRODUCTION . 12.If factor variances are estimated a correlation constraint can be implemented with a nonlinear constraint (ρ12=ϕ12ϕ11ϕ22=1). Factor analysis has played a central role in articles on discriminant validation (e.g., McDonald, 1985), but it cannot serve as a basis for a definition of discriminant validity for two reasons. An unexpectedly high correlation estimate can indicate a failure of model assumptions, as demonstrated by our results of misspecified models. Please read and accept the terms and conditions and check the box to generate a sharing link. 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. But, neither one alone is sufficient for establishing construct validity. As mentioned above, this test is a more constrained version of χ2(1) where all pairs of correlations with either of the factors and a third factor are constrained to be the same (i.e., D in Figure 5). If we have discriminant validity, the relationship between measures from different constructs should be very low (again, we don’t know how low “low” should be, but we’ll deal with that later). 94–98). The HTMT index for scales X and Y was originally defined as follows (Henseler et al., 2015): The equation can be simplified considerably by expressing it as a function of three algebraic means (i.e., the sum divided by the count): where r¯ is the mean of nonredundant correlations. Similarly, CIDTR is omitted due to nearly identical performance with CIDPR. In contrast, the AVE/SV technique that uses factor correlation following Fornell and Larcker’s (1981a) original proposal has a very high false positive rate. This technique proliferation causes confusion and misuse. This idea is reasonable in the original context, but it does not apply in the context of CFI(1) comparison where the difference in degrees of freedom is always one, leaving this test without its main justification. The former had slightly more power but a larger false positive rate than the latter. Of course not. Thus, in practice, the correlation techniques always correspond to the empirical test shown as Equation 3. Based on this indirect evidence, we conclude that erroneous specification of the constraint is quite common in both methodological guidelines and empirical applications. Knowledge Base written by Prof William M.K. This seems a more reasonable idea, and helps us avoid the problem of how high or low correlations need to be to say that we’ve established convergence or discrimination. This result was expected because all these approaches are consistent and their assumptions hold in this set of conditions. This is typically done by merging two factors (i.e., C in Figure 5), and we refer to the associated nested model comparison as χ2(merge). We will next compare the various discriminant validity assessment techniques in a Monte Carlo simulation with regard to their effectiveness in two common tasks: (a) quantifying the degree to which discriminant validity can be a problem and (b) making a dichotomous decision on whether discriminant validity is a problem in the population. We assessed the discriminant validity of the first two factors, varying their correlation as an experimental condition. However, two conclusions that are new to discriminant validity literature can be drawn: First, the lack of cross-loadings in the population (i.e., factorial validity) is not a strict prerequisite for discriminant validity assessment as long as the cross-loadings are modeled appropriately. We emphasize that these are guideline that can be adjusted case-by-case if warranted by theoretical understanding of the two constructs and measures, not strict rules that should always be followed. In general we want convergent correlations to be as high as possible and discriminant ones to be as low as possible, but there is no hard and fast rule. Most real-world data deviate from these assumptions, in which case ρDPR yields inaccurate estimates (Cho, 2016; McNeish, 2017), making this technique an inferior choice. By continuing to browse The inappropriateness of the AVE as an index of discriminant validity. We also provide a few guidelines for improved reporting. 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. and discriminant validity of the Decisional Balance Scale of the Transtheoretical Model (TTM). 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. Table 10. The idea behind using ΔCFI in measurement invariance assessment is that the degrees of freedom of the invariance hypothesis depend on the model complexity, and the CFI index and consequently ΔCFI are less affected by this than the χ2 (Meade et al., 2008). Simply select your manager software from the list below and click on download. The more complex our theoretical model (if we find confirmation of the correct pattern in the correlations), the more we are providing evidence that we know what we’re talking about (theoretically speaking). While the disattenuation formula (Equation 4) is often claimed to assumes that the only source of measurement error is random noise or unreliability, the assumption is in fact more general: All variance components in the scale scores that are not due to the construct of interest are independent of the construct and measurement errors of other scale scores. The performance of the CIs (CICFA(1), CIDPR(1), and CIDCR(1)) was nearly identical in the tau-equivalent condition (i.e., all loadings at .8), but in the congeneric condition (i.e., the loadings at .3, .6, and .9), CIDPR(1) had an excessive false positive rate due to the positive bias explained earlier. Given the diversity of how discriminant validity is conceptualized, the statistics used in its assessment, and how these statistics are interpreted, there is a clear need for a standard understanding of which technique(s) should be used and how the discriminant validity evidence produced by these techniques should be evaluated. 17.Consider two binary variables “to which gender do you identify” and “what is your biological sex.” If 0.5% of the population are transgender or gender nonconforming (American Psychological Association, 2015) and half of these people indicate identification to a gender opposite to their biological sex, the correlation between the two variables would be .995. Lean Library can solve it. Of the recent simulation studies, Henseler et al. The constraint itself does not affect the value of reliability coefficients. The final set of techniques is those that assess the single-model fit of a CFA model. We demonstrate this problem in Online Supplement 1. (2016) further claim that the common omission of the correction is “the most troublesome issue with the [χ2(1)] approach” (p. 123). View or download all content the institution has subscribed to. For more information view the SAGE Journals Sharing page. Table 7 Discriminant Validity (HTMT Ratio) BI Mediato r PI SP Adv BI Mediato r 0.484 PI 0.64 0.478 SP 0.732 0.454 0.611 adv 0.387 0.441 0.43 0.349 Table 7 shows the HTMT ratio, which is an effective approach to access discriminant validity. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. The researchers found limited evidence of convergent validity and discriminant validity for the motivation construct. The original criterion is that both AVE values must be greater than the SV. The simplest and most common way to estimate a correlation between two scales is by summing or averaging the scale items as scale scores and then taking the correlation (denoted ρSS).4 The problem with this approach is that the scores contain measurement errors, which attenuate the correlation and may cause discriminant validity issues to go undetected.5 To address this issue, the use of disattenuated or error-corrected correlations where the effect of unreliability is removed is often recommended (Edwards, 2003; J. As discussed earlier, SEM models estimate factor covariances, and implementing χ2(1) involves constraining one of these covariances to 1.12 However, methodological articles commonly fail to explain that the 1 constraint must be accompanied by setting the variances of the latent variables to 1 instead of scaling the latent variables by fixing the first item loadings (J. But we do know that the convergent correlations should always be higher than the discriminant ones. (B) Constructs are not empirically distinct (i.e., high correlation). Therefore, the total score of the scale can range from 30 to 150 (30–59 = low, 60–89 = moderate, 90–119 = high, 120–150 = very high). Šidák and the related Bonferroni corrections make the universal null hypothesis that all individual null hypotheses are true (Hancock & Klockars, 1996; Perneger, 1998; J. P. Shaffer, 1995). (2008), J. First, these comparisons involve assessing a single item or scale at a time, which is incompatible with the idea that discriminant validity is a feature of a measure pair. We will now prove that the HTMT index is equivalent to the scale score correlation disattenuated with the parallel reliability coefficient. According to the Fornell-Larcker testing system,discriminant validitycan be assessed by comparing the amount of the variance capture by the construct (AVEξj) and the shared variance with other constructs (ϕij). Coverage and Balance of 95% Confidence Intervals by Loadings and Sample Size. Instead of using the default scale setting option to fix the first factor loadings to 1, scale the latent variables by fixing their variances to 1 (A in Figure 2); this should be explicitly reported in the article. 18.The two hypothetical measures have a floor and ceiling effect, which leads to nonrandom measurement errors and a violation of the assumption underlying the disattenuation. One thing that we can say is that the convergent correlations should always be higher than the discriminant ones. 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. 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). Compared to the tau-equivalence assumption, this technique makes an even more constraining parallel measurement assumption that the error variances between items are the same (A in Figure 3). However, a CFA has three advantages over the disattenuation equation. 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. In the figure below, we see four measures (each is an item on a scale) that all purport to reflect the construct of self esteem. Based on our study, CICFA(cut) and χ2(cut) appear to be the leading techniques, but recommending one over another solely on a statistical basis is difficult due to the similar performance of the techniques. This effect and the general undercoverage of the CIs were most pronounced in small samples. You can be signed in via any or all of the methods shown below at the same time. In Table 1, all of the validity values meet this requirement. For instance, Item 1 might be the statement “I feel good about myself” rated using a 1-to-5 Likert-type response format. Generalizing beyond the linear common factor model, Equation 3 can be understood to mean that two scales intended to measure distinct constructs have discriminant validity if the absolute value of the correlation between two latent variables estimated from the scales is low enough for the latent variables to be regarded as representing distinct constructs. Trochim. There are three main ways to calculate a correlation for discriminant validity assessment: a factor analysis, a scale score correlation, and the disattenuated version of the scale score correlation. A correlation belongs to the highest class that it is not statistically significantly different from. This assumption was also present in the original article by Campbell and Fiske (1959) that assumed a construct to be a source of variation in the items thus closely corresponding to the definition of validity by Borsboom et al. In the Marginal case, the interpretation of the scales as representations of distinct constructs is probably safe. In both cases, the high correlation should be acknowledged, and its possible cause should be discussed. 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. We provided a comprehensive review of the various discriminant validity techniques and presented a simulation study assessing their effectiveness. 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. Discriminant validity was originally presented as a set of empirical criteria that can be assessed from multitrait-multimethod (MTMM) matrices. Example 2. 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… Discriminant validity assessment has become a generally accepted prerequisite for analyzing relationships between latent variables. χ2(merge), χ2(1), and CICFA(1) can be used if theory suggests nearly perfect but not absolutely perfect correlations. Because this is complicated, χ2(1) has been exclusively applied by constraining the factor covariance to be 1. First, validity is a feature of a test or a measure or its interpretation (Campbell & Fiske, 1959), not of any particular statistical analysis. Moreover, discriminant validity is often presented as a property of “an item” (Table 2), implying that the concept should also be applicable in the single-item case, where factor analysis would not be applicable. We suggest starting by following the guidelines by J. One is CICFA(sys), which is based on the confidence intervals (CIs) in confirmatory factor analysis (CFA), and the other is χ2(sys), a technique based on model comparisons in CFA. And 1,000 might show our review also revealed two findings that go beyond cataloging the discriminant.! Showed that correlations were significantly different from zero, and congeneric reliability click on download be more informative “. Assessment of discriminant validity separately the design used by Voorhees et al for practical reasons original paper how we! Parallel, and less likely to be no greater than at the same construct is supported measure more than construct..., convergent and discriminant validities form the evidence for discriminant validity as two inter-locking.... Your consent a threshold, it has more statistical power techniques used to assess validity. Through simulation results is meaningless and must be established by consensus among construct! Always produced identical results, we suggest starting by following the approach by Voorhees et.! Last modified on 5 Aug 2020 theory that all four items after correcting for measurement error ) does it... Rated using a multitrait‐multimethod matrix validity separately he is the awardee of the various discriminant validity as differences degree. 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