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    Area of Science:

    • Psychometrics
    • Quantitative Psychology
    • Factor Analysis

    Background:

    • The G index by Holley and Guilford was developed for Q-type factor analysis with arbitrary scoring directions.
    • The G index is specifically designed for dichotomous data.
    • Limitations exist in applying the G index to broader data types and scenarios.

    Purpose of the Study:

    • To derive a generalized measure applicable beyond dichotomous data.
    • To extend the utility of the G index in factor analysis.
    • To explore connections with component analysis and proportionality.

    Main Methods:

    • Mathematical derivation of a generalized measure.
    • Comparison of the generalized measure with the G index for dichotomous data.
    • Analysis of the measure's relationship with inter-person covariance and proportionality.

    Main Results:

    • A general measure was derived that is identical to the G index for dichotomous data.
    • The generalized measure approximates the mean inter-person covariance across all scoring direction permutations.
    • The measure demonstrates close connections to component analysis of individual differences and proportionality.

    Conclusions:

    • The derived general measure offers a more versatile alternative to the G index.
    • This measure enhances the analysis of individual differences in psychometrics.
    • The findings link factor analysis with concepts of proportionality and covariance structures.