Relationship between quadratic component of optimal scores and individual consistency in a questionnaire answers

The plot of the first two components issued from a categorical factor analysis (or correspondences analysis) have the typical horseshoe pattern, known as Guttman effect. Submitting the n responses to k items of a unidimensional questionnaire always yields such an effect. With high reliable questionnaires (i.e. high alpha coefficient) the spread around the curb representing the quadratic effect, is small, whereas unreliable questionnaires scores have a very dispersed pattern. However, reliability is a property of individual respondents: those respondents who have a congruent (or consistent ) pattern of responses have a low quadratic component, compared to the first dimension, whereas respondents who give their answers in an inconsistent pattern, have a high quadratic component. If we discard some 10-15 % of unreliable scores, the purified scores display a higher Cronbach Alpha. Some empirical examples with real date will be presented to illustrate the method.