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Ultra-short scales are increasingly popular in surveys. Congeneric model fit of a three-item scale cannot be tested with Confirmatory Factor Analysis (CFA) without additional assumptions because the number of degrees of freedom is equal to zero. A more rigorous tau-equivalent model, assuming equality of factor loadings can be tested instead. The objective of this study was to demonstrate this approach with an example of the psychometric study of the Polish version of the Three-Item Loneliness Scale (TILS), and to discuss the arising problems and possible solutions. There seems to be a high need for such analysis because currently, some properties of CFA make it an approach still predominant over Item Response Theory (IRT) models in the quality of life research. A sample of 3510 students completed TILS together with the questionnaires measuring a variety of indicators of well-being. The results provided evidence for a good fit of a tau-equivalent model. Furthermore, multi-group CFAs provided support for strict measurement invariance of this model. To the Authors' knowledge, it is the first practical application of a tau-equivalent model to testing the factorial validity of an ultra-short scale and probably the first empirical case of tau-equivalent measurement invariance in psychological literature in general. TILS showed good criterion validity and satisfactory reliability. Unidimensionality of three-item scales can be examined with a tau-equivalent model that has some favorable psychometric properties. However, it might be exceedingly restrictive in certain practical cases. When developing a new short scale, it is recommended to maintain at least four items.
Czerwiński et al. (Sun,) studied this question.