The Manifest Fornell-Larcker criterion

Assessing Distinctiveness in Multidimensional Instruments without Access to Raw Data – A Manifest Fornell-Larcker Criterion

This calculator helps estimating the distinctiveness between two dimensions of an already published multidimensional instrument using Cronbach's alpha and the manifest correlation between the two dimensions. The calculator computes both the manifest Fornell-Larcker criterion with double correction and the Fornell-Larcker criterion with single correction. When both criteria point into the same direction one can be sure that the result is correct. Only when the double-correction does indicate violations and the single correction does not, it is uncertain if there is any violation of distinctiveness between the two dimensions.

The Manifest Fornell-Larcker criterion calculator

Input

α_X	i.e. Cronbach's alpha for your first latent variable X
K_X	i.e. the number of items of your first latent variable X
α_Y	i.e. Cronbach's alpha for your second latent variable Y
K_Y	i.e. the number of items of your second latent variable Y
r_XY	i.e. the Manifest correlation between the composite scores of your two variables (dimensions) X and Y

Output

Manifest Fornell-Larcker criterion with single correction	-
Manifest Fornell-Larcker criterion with double correction	-

Intermediate steps

αAVE_X	0.65
√αAVE_X	0.81
αAVE_Y	0.38
√αAVE_Y	0.62
r_XY with single correction using α_min	0.57
r_XY with double correction	0.62

An empirical example: Let’s suppose one is interested in the influence of the personality trait “openness” on political attitudes and voting behaviour. A likely choice to measure personality traits is to use the NEO-PI-R (Costa & McCrae, 1992). In this inventory, “openness” is made up of 6 facets (or sub traits), and each facet contains 8 items on the questionnaire.

Inputting the values of the correlation matrix and Cronbach’s alpha provided by the test manual (Costa & McCrae, 1992) into this calculator reveals that not all of these six facets are distinct from each other (i.e. they violate the manifest Fornell-Larcker criterion both with single- and double correction).

If one is not interested in using or interpreting these facets of “openness” separately (which is the case in most research endeavours), one can go on and use the NEO-PI-R. However, if interpretation of these facets is vital to the research question at hand, it would be best to search for another measurement instrument. For instance, the IPIP-NEO-120 (Johnson, 2014) inventory offers to measure constructs similar to those assessed by the 30 facet scales in the NEO-PI-R. These items are not only public domain and therefore accessible from the IPIP webseite https://ipip.ori.org/, but there is also the norm-dataset freely available at https://osf.io/tbmh5/ . This means, to assess distincitiveness of the IPIP-NEO-120, one does not need this auxiliary calculator, but can do so oneself based on the available raw-data using the original, latent Fornell-Larcker criterion. Fortunately, all 6 facets of “openness” in the IPIP-NEO-120 are indeed distinct from each other, so one can use this instrument in the desired way.

References:

Costa, P. T. & McCrae, R. R. (1992). Revised NEO Personality Inventory (NEO-PI-R) and NEO Five Factor Inventory. Professional Manual. Odessa: Psychological Assessment Resources.

Johnson, J. A. (2014). Measuring thirty facets of the five factor model with a 120-item public domain inventory: Development of the IPIP-NEO-120. Journal of Research in Personality, 51, 78-89.