Procedures Used for Assessing the Reliability and Validity of Formative

Although traditional methods for assessing construct reliability and validity are not appropriate for formative constructs, "it is bad practice to… claim that one's measures are formative, and do nothing more."451 _{Therefore, this dissertation uses some of the}

procedures recently suggested by Diamantopoulos and Winklhofer, which should assist researchers in evaluating latent variable constructs with formative indicators.452

According to Diamantopoulos and Winklhofer, the first critical issue, when adopting a formative measurement perspective, is the content specification of the latent construct. Content specification relates to the definition of the scope of the latent variable, i.e., the domain of content that the construct is intended to capture. Breadth of definition is

extremely important under formative measurement because failure to consider all facets of the construct will lead to an exclusion of relevant indicators (and thus exclude part of the construct itself).453 _{Thus, for a formative construct to be reliable and valid, its defined} content domain needs to include all possible dimensions.454

Because under formative measurement the latent variable is determined by its indicators rather than vice versa, content specification is inextricably linked with indicator

specification. Proper indicator specification requires that the formative indicators used to measure the construct cover the entire scope (dimensions) of the latent variable as

described under the content specification. Thus, the selected indicators need to fully capture the construct's content domain. However, instead of requiring a census455 _of indicators, this dissertation follows Rossiter's recommendation to only include the construct's main indicators. The aim of using a census of indicators (i.e., every possible indicator) is practically not possible because it would lead to an infinite search for low- incidence indicators that most raters would not include in the construct definition.456 However, the main indicators selected to measure the latent construct still need to fully capture the construct's content domain, i.e. all of its dimensions.

451 Edwards and Bagozzi (2000), p. 171.

452 Cf. Diamantopoulos and Winklhofer (2001), pp. 269-277.

453 See the conceptual differences between reflective and formative constructs outlined in chapter 6.1. 454 Cf. Diamantopoulos and Winklhofer (2001), p. 271.

455 Cf. Bollen and Lennox (1991), p. 308; Diamantopoulos and Winklhofer (2001), p. 271. 456 Cf. Rossiter (2002), pp. 314-315.

6.1 Conceptual Differences between Formative and Reflective Constructs 145 A potential issue also relates to the collinearity among formative indicators. As previously discussed, formative indicators jointly determine the conceptual and empirical meaning of the construct, with each indicator contributing a unique facet. Thus, although the formative measurement perspective does not explicitly assume or require a specific pattern of signs or magnitude of the indicator correlations457_{, formative indicators should tend to show} rather low intercorrelations.458 _{In contrast, excessive collinearity among formative} indicators would make it difficult to separate the distinct influence of the individual

indicators (xs) on the latent variable. Moreover, if a particular formative indicator (xi) turns out to be almost an exact linear combination of the other indicators (xs), it is likely to provide redundant information and is therefore less critical.459 _{Therefore, it is prudent to} examine indicator collinearity when assessing the reliability and validity of formative constructs. Following the suggestion by Diamantopoulos and Winklhofer, collinearity among the indicators of a formative construct is assessed by calculating variance inflation factors (VIF) using the SPSS software package. In line with literature, this dissertation uses a maximum VIF greater than 10 as cut-off threshold for high (multi)collinearity among formative indicators.460 _{While low collinearity among formative indicators is certainly} desirable, it is important to note that high collinearity alone does not justify the elimination of individual indicators. Under formative measurement, conceptual considerations play a dominant role because failing to include or dropping one indicator may change the meaning of the latent variable. Therefore, Albers and Hildebrandt suggest treating multicollinearity among formative indicators by constructing indices rather than eliminating individual indicators.461

In order to further validate formatively specified constructs, Diamantopoulos and Winklhofer suggest including some reflective indicators and estimating a multiple

indicators and multiple causes (MIMIC) model. In this model, the formative indicators (xi) act as direct causes of the latent variable, which is indicated by one or more reflective measures.462 _{However, this approach has been criticized for its limited practicability and}

457 Indeed, the correlations among formative indicators can take all values within the permitted interval [-1; +1], (Eberl (2006), p. 652).

458 Cf. Eberl (2006), pp. 661-662; Rossiter (2002), p. 315. 459 Cf. Bollen and Lennox (1991), p. 308.

460 Cf. Diamantopoulos and Winklhofer (2001), p. 272; Mason and Perreault (1991), p. 270; Belsley (1991), p. 28. 461 Cf. Albers and Hildebrandt (2006), p. 13.

its intention to delete indicators from formative scales.463 _{Therefore, this dissertation} conducts a confirmatory tetrad analysis (CTA) to validate the formative specification of latent variable constructs. A "tetrad" refers to the difference between the product of a pair of covariances and the product of another pair among four random variables.464 _{Thus, for a} latent variable construct with four indicator variables, the six covariances between the indicators can be arranged into three tetrads: 1234= ^12^34- ^13^24, 1342= ^13^42- ^14^32, and

1423= ^14^23- ^12^43.

Moreover, Bollen and Ting show that a reflective measurement specification implies that all tetrads equal zero. Thus, in the case of a reflective construct with four indicators 1234= 1342= 1423 = 0. In the case of formative indicators, however, the products of the pairs of covariances do not all have to be of equal value because the indicators are exogenous. Therefore, the tetrads of a formative construct do not all have to equal zero, i.e., they do not all have to "vanish". 465 _{The CTA developed by Bollen and Ting provides a}

simultaneous test of the model implied vanishing tetrads against the null hypothesis that the tetrad values equal zero (H0: = 0). Therefore, it simultaneously provides a statistical test for the validation of a latent construct's measurement specification (H0: »construct is reflective«). Thus, if the CTA test statistic is significant (p-value < 0.05), it lends support to a formative specification of a latent variable construct.466 _{Because the CTA test is able} to validate the formative specification of latent variable constructs, it replaces the first three steps of the traditional reliability and validity analysis used for reflective latent variables (internal consistency reliability, unidimensionality, convergent validity).467 The CTA test statistic asymptotically follows a chi-square distribution under the null hypothesis. However, Bollen and Ting find that the test statistic can deviate significantly from its asymptotic distribution with sample sizes that are small to moderate and with models that have a large number of parameters (indicators).468 _{This usually causes the} tetrad test to be conservative. To remedy this problem, Bollen and Ting propose a nonparametric bootstrap tetrad test, which "generally is more accurate than using the chi- square distribution to compute the p-value of the test statistic in small to moderate sample

463 Cf. Eberl (2004), p. 9; Albers and Hildebrandt (2006), p. 25; Rossiter (2002), p. 315. 464 Cf. Bollen and Ting (1993), p. 147; Bollen and Ting (2000), p. 5.

465 Cf. Bollen and Ting (2000), p. 7; Eberl (2006), p. 660. 466 Cf. Bollen and Ting (2000), pp. 13-15.

467 Cf. Cadieux et al. (2006), pp. 417. 468 Cf. Bollen and Ting (1998), pp. 77-102.

6.1 Conceptual Differences between Formative and Reflective Constructs 147 sizes."469 _{Because the size of the sample available in this study meets Bollen and Ting's} definition of small to moderate sample sizes, this dissertation chooses to conduct the confirmatory tetrad analysis (CTA) by applying a nonparametric bootstrap tetrad test that was developed by Johnson and Bodner.470 _{This test is a modified version of the original} bootstrap tetrad test and offers several advantages. First, for a given number of indicators, it is more powerful than the original bootstrap test statistic. Second, in contrast to the original version, the modified bootstrap tetrad test increases in power with an increase in the number of indicators. Third, the modified bootstrap tetrad test is computationally more feasible.471 _{The Johnson and Bodner bootstrap tetrad test is implemented using the Ox} software version 4.1472_{, selecting 1,000 bootstrap replications. In addition, the results of} the CTA are supplemented with the results of the Bollen-Stine bootstrap test available in AMOS.473 _{The Bollen-Stine bootstrap test is employed to assess the fit of a reflectively} specified one-factor model of the latent variable constructs used in this study. Thus, a significant p-value (p < 0.05) of the Bollen-Stine test statistic indicates that a reflective specification does not fit the data well and therefore leads to the rejection of a reflective construct specification in favor of a formative one.

Although traditional approaches to assessing construct reliability and validity generally do not apply to formative constructs, some researchers maintain that discriminant validity still needs to be demonstrated.474 _{This step is necessary to ensure that the formative latent} constructs are sufficiently different from each other to be considered legitimate. However, in the case of formative indicators, discriminant validity between two variables cannot be assessed using their latent form. That is, it is not possible to compute the average variance shared between the constructs and their respective indicators (AVE) and use the

Fornell/Larcker criterion to assess the discriminant validity between the latent variables.475 Thus, prior to assessing the discriminant validity between formative latent constructs, composite scores of their respective indicators need to be calculated in order to create new "observed" variables.476 _{The actual assessment of discriminant validity then comes down} to examining the correlation between all pairs of these composite variables. In line with 469 Bollen and Ting (1998), p. 77.

470 See Johnson and Bodner (2007), pp. 113-124. 471 Cf. Johnson and Bodner (2007), p. 121. 472 See Doornik (2002).

473 Cf. Bollen and Stine (1992), pp. 205-229. 474 Cf. Cadieux et al. (2006), p. 417. 475 See chapter 6.1.1.

literature, this study considers correlation coefficients below 0.8 as indicative of the absence of strong linear associations between the composite formative variables, thus suggesting a sufficient level of discriminant validity.477