Structural equation models with unobservable variables and measurement error: Algebra and statistics
Abstract
Several issues relating to goodness of fit in structural equations are examined. The convergence and differentiation criteria, as applied by Bagozzi, are shown not to stand up under mathematical or statistical analysis. The authors argue that the choice of interpretative statistic must be based on the research objective. They demonstrate that when this is done the Fornell-Larcker testing system is internally consistent and that it conforms to the rules of correspondence for relating data to abstract variables.
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All measures showed satisfactory reliability with a threshold of 0.7, except perceived physical conditions, which is greater than 0.6.
“Results revealed satisfactory reliability of all the measures with a threshold of 0.7, except for the measure of perceived physical conditions, which is greater than 0.6 [100].”
From The Investigation of Adoption of Voice-User Interface (VUI) in Smart Home Systems among Chinese Older Adults (2022) · cites this paper as [100]
All discriminant validity requirements were met, since each scale's average variance extracted exceeded multiple squared correlations.
“all discriminant validity requirements (Fornell and Larcker 1981) were met, since each scale ' s average variance extracted exceeded multiple squared correlations.”
From AI-based chatbots in customer service and their effects on user compliance (2021) · cites this paper as Fornell and Larcker 1981
The measurement model has good discriminant validity.
“The analysis showed that the square roots of the AVE of all constructs are higher than the correlation coefficients between constructs, indicating good discriminant validity.”
From People's Intentions to Use Metaverse Technology: Investigating the Role of Gratifications and Perceptions (2024) · cites this paper as Fornell & Larcker, 1981
The discriminant validity of the measurement model was good.
“There was a higher discriminant validity in this study, and a path analysis of the structural equation model can be applied (Table 4).”
From Health and Wellness Tourists’ Motivation and Behavior Intention: The Role of Perceived Value (2023) · cites this paper as [75]
The five constructs in this research met the criteria that CR should not be below 0.70 and AVE should exceed 0.50.
“From the research by Fornell and Larcker (1981), for all the constructs, the CR should not be below 0.70 and the AVE should exceed 0.50, and the five constructs in this research met the two criteria (Table 3).”
From A study on the influence of rural tourism’s perceived destination restorative qualities on loyalty based on SOR model (2025) · cites this paper as Fornell and Larcker (1981)
Discriminant validity was supported, as the inter-factor correlations were less than the square root of the factors' average variances extracted.
“The discriminant validity was also supported, considering that the inter-factor correlations were less than the square root of the factor's AVEs, as advocated by Fornell and Larcker and depicted in Table 2 [33].”
The reliability and convergent validity of the constructs were confirmed, as all five factors reached construct reliabilities exceeding 0.70 and average variances extracted exceeding 0.50.
“The reliability and the convergent validity were confirmed, as all five factors reached construct reliabilities (CRs) exceeding the generally recommended 0.70 threshold [31] and (b) the average variances extracted (AVE) exceeding the suggested 0.50 threshold [33], indicating that more than 50% of a factor's variance is due to its measurement items.”
The squared root of the AVE for each construct exceeded the correlation of that construct with other constructs, establishing discriminant validity of all constructs.
“As shown in Table 4, the squared root of the AVE for each construct (the value along the diagonal presented in bold) exceeded the correlation of that construct with other constructs, thereby establishing discriminant validity of all constructs.”
From Is it harmful or helpful? Examining the causes and consequences of generative AI usage among university students (2024) · cites this paper as Fornell and Larcker's (1981)
Discriminant validity is established if the squared root of the AVE for each construct is larger than the correlation of that construct with other constructs.
“As per Fornell and Larcker's (1981) criteria, discriminant validity is established if the squared root of the AVE for each construct is larger than the correlation of that construct with other constructs.”
From Is it harmful or helpful? Examining the causes and consequences of generative AI usage among university students (2024) · cites this paper as Fornell and Larcker's (1981)
All constructs have CR above 0.50 and rho_A above 0.70, with the lowest CR 0.889 and the lowest rho_A 0.853.
“In the same vein, CR and rho_A were higher than the recommended thresholds 0.50 (Fornell and Larcker 1981) and 0.70 (Camilleri 2022;Ravand and Firoozi 2016) respectively, with the lowest CR equal to 0.889 and the lowest rho_A equal to 0.853.”
From Does Greenwashing Wash Away Gen Z's Green Purchase Intention? (2025) · cites this paper as Fornell and Larcker 1981
All constructs exhibit AVE values higher than 0.5, with the lowest AVE 0.618.
“All the constructs exhibit AVE values higher than 0.5 (Fornell and Larcker 1981) with the lowest AVE equal to 0.618.”
From Does Greenwashing Wash Away Gen Z's Green Purchase Intention? (2025) · cites this paper as Fornell and Larcker 1981
All AVE values ranged from 0.604 to 0.698, surpassing 0.50, confirming adequate convergent validity.
“all AVE values ranged from 0.604 to 0.698, surpassing the threshold of 0.50, which confirms adequate convergent validity”
From Understanding consumer purchase intention in social media influencer marketing: the mediating role of parasocial relationship and trust in the Moroccan context (2026) · cites this paper as Fornell and Larcker, 1981
The square root of the AVE for each construct exceeds its highest correlation with any other construct, providing additional evidence of discriminant validity at the construct level.
“the square root of the AVE for each construct exceeds its highest correlation with any other construct, providing additional evidence of discriminant validity at the construct level.”
From Understanding consumer purchase intention in social media influencer marketing: the mediating role of parasocial relationship and trust in the Moroccan context (2026) · cites this paper as Fornell and Larcker, 1981
The measurement model had discriminant validity.
“In addition, we measured discriminant validity by comparing the AVE and correlation efficiency of the constructs, and more specifically, if the AVE of each construct was greater than the square of the correlation coefficient of two constructs (AVE > ∅2), the measurement model had discriminant validity (Fornell & Larcker, 1981).”
From What influences tourist behaviors during and after the COVID-19 pandemic? Focusing on theories of risk, coping, and resilience (2022) · cites this paper as Fornell & Larcker, 1981
Almost all AVE values are above the suggested threshold; values above 0.7 are considered good, while a value of 0.5 is acceptable.
“Almost all AVE values are above the suggested threshold (Fornell and Larcker, 1981); values above 0.7 are considered good, while a value of 0.5 is acceptable.”
From Completing the food waste management loop: Is there market potential for value-added surplus products (VASP)? (2020) · cites this paper as Fornell and Larcker, 1981
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- Authors
- Fornell, C., & Larcker, D. F.
- Affiliations
- University of Michigan (US)
- Northwestern University (US)
- ORCID
- Larcker, David F.
- Venue
- Journal of Marketing Research
- Publisher
- SAGE Publications
- Published
- 1981-08
- Volume
- 18
- Issue
- 3
- Pages
- 382-388
- ISSN
- 0022-2437 (ISSN-L)
- 1547-7193
- OpenAlex
- W4230637740
- Cited by
- 13,287
- References
- 25
- Citations by year
- 2026: 951 · 2025: 1433 · 2024: 1735 · 2023: 2006 · 2022: 1837 · 2021: 1445 · 2020: 1189 · 2019: 998 · 2018: 643 · 2017: 296 · 2016: 177 · 2015: 130 · 2014: 113 · 2013: 79 · 2012: 68
- FWCI
- 5.74
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of Marketing Research (publishedVersion)
- Topics
- Social Sciences › Decision Sciences › Management Science and Operations Research › Psychometric Methodologies and Testing
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Applications
- Social Sciences › Decision Sciences › Management Science and Operations Research › Business Strategies and Management Research
- Keywords
- structural equation modeling, goodness of fit, latent variables, measurement error, correspondence rules
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