Homogeneity (statistics)


 * 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. Within this framework, the coefficient of homogeneity indicates the degree data approximate the Guttman implicatory scales.

Coefficient of homogeneity
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.