# Homogeneity (statistics)

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For homogeneity of variance see homoscedasticity.

Homogeneity in statistics and data analysis pertains to properties of logically consistent data matrices. In meta-analysis, which combines the data from several studies, homogeneity measures the differences or similarities between the several studies.[1] Within this framework, the coefficient of homogeneity indicates the degree data approximate the Guttman implicatory scales.

## Coefficient of homogeneityEdit

The original coefficient of homogeneity, wrapped in complex algebraic considerations, was introduced in 1948 by Loevinger. Interest in homogeneity of data was revived during the closing decades of the last century by Cliff (1977), and by Krus and Blackman (1988). On the basis of theoretical analysis outlined above, Krus and Blackman defined the coefficient of homogeneity as

$h_{xx} =\frac{MS_I - MS_{RES}}{MS^*_I - MS^*_{RES}}$

where MS stands for mean square, I for individuals and RES for residual terms of the analysis of variance. The * indicates that these indices were obtained from the data matrix where the variance of the variables was maximized. This coefficient of homogeneity is numerically equivalent with both the Loevinger's and Cliff's conceptualizations of the coefficient of homogeneity. As the Hoyt's (1941) formula for the internal consistency reliability is

$r_{xx} =\frac{MS_I - MS_{RES}}{MS_I}$

the Krus and Blackman formulation of the coefficient of homogeneity brings both the coefficient of internal consistency reliability and the coefficient of homogeneity within the framework of the analysis of variance.

## ReferencesEdit

• Hoyt, C. (1941). Test reliability estimated by analysis of variance. Psychometrika, 6, 153-160
• Cliff, N. (1977). A theory of consistency of ordering generalizable to tailored testing. Psychometrika, 42, 375-399.
• Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297-333.
• Kuder, G. & Richardson, M. (1937). The theory of estimation of test reliability. Psychometrika, 2, 151-160.
• Krus, D.J., & Blackman, H.S. (1988).Test reliability and homogeneity from perspective of the ordinal test theory. Applied Measurement in Education, 1, 79-88 (Request reprint).
• Loevinger, J. (1948). The technic of homogeneous tests compared with some aspects of scale analysis and factor analysis. Psychological Bulletin, 45, 507-529.