Quality & Quantity 32: 247–255, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
A Multivariate Two-Group Effect Size Measure
& SPIRIDON PENEV
University of Melbourne;
University of New South Wales, Sydney
Abstract. This article describes a multivariate effect size measure useful in social and behavioral re-
search, which complements existing descriptive indicators of group differences. The measure reﬂects
probabilistic aspects of the disparity of two multivariate distributions, and its interpretation does not
require advanced training in statistics. Estimation of the discussed effect size indicator is illustrated
on data from a two-group cognitive intervention study (Baltes, Dittmann-Kohli, & Kliegl, 1986).
Key words: effect size, multivariate distribution, probability.
Quantitative social and behavioral research, within a hypothesis testing framework,
is frequently interested in the extent to which the analyzed data violates a stipulated
assumption of ‘no effect’, as reﬂected in the so-called “effect size” measure (e.g.,
Cohen, 1988). Indicators of the degree to which a phenomenon under investigation
is present in studied populations in these sciences have received a considerable
amount of interest over the past 20–30 years (e.g., Cohen, 1988; Dunlap, 1994;
Friedman, 1968; Levy, 1967; McGraw & Wong, 1992; Rosenthal & Rubin, 1982).
Thereby, an emphasis has been placed on construction of such measures that are
easily comprehended by a researcher without advanced training in statistics. Fol-
lowing this lead, the aim of the present article is to describe a multivariate effect
size measure useful in two-group social and behavioral contexts, which is read-
ily understood. It capitalizes on the comparison idea underlying the approach of
McGraw and Wong.
1. An Indicator of Multivariate Distribution Difference
By their nature, social and behavioral research are multivariate because most stud-
ied phenomena are multifactorially determined. It is therefore of interest, in empir-
ical settings with p ≥ 1 interrelated (continuous) variables measured on samples
from two distinct populations, to have a multidimensional effect size measure that
(a) reﬂects the disparity of the two underlying multi-variable distributions; (b) takes
into account all existing interrelationships among the variables; and (c) is easy to
comprehend by a statistically unsophisticated researcher.
To exemplify the scope of applicability, such a desired measure will be useful
when the studied variables represent p (p ≥ 1) scores on a test-battery in an
experimental and control group (e.g., last section). Similarly, this indicator will
also be informative when the variables are scores on a test-battery administered