Quality & Quantity 37: 1–19, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
The Use of Indices in Surveys
P. E. MARAVELAKIS, M. PERAKIS, S. PSARAKIS and J. PANARETOS
Department of Statistics, Athens University of Economics and Business, Greece
Abstract. The paper deals with some new indices for ordinal data that arise from sample surveys.
Their aim is to measure the degree of concentration to the “positive” or “negative” answers in a
given question. The properties of these indices are examined. Moreover, methods for constructing
conﬁdence limits for the indices are discussed and their performance is evaluated through an ex-
tensive simulation study. Finally, the values of the indices deﬁned and their conﬁdence intervals are
calculated for an example with real data.
Key words: multinomial proportions, ordinal data, indices, conﬁdence intervals, sample surveys
Various types of indices are widely used in real world applications. Some ﬁelds
where the use of indices is widespread are index numbers (see e.g., Mudgett
(1951)), statistical quality control (see e.g., Kotz and Lovelace (1998) and Mont-
gomery (1997)), economics (see e.g., Cowell (1995)), fundamental analysis (see
e.g., Ritchie (1996)) and sample surveys (see e.g., Bnerjee et al. (1999)).
In the area of sample surveys, questions requiring answers that have a somewhat
natural ordering are frequently included. A common example of such type of an-
swers is “Very Good”, “Good”, “Moderate”, “Bad” and “Very Bad”. In practice, the
presentation of the observed proportions of the possible answers of such questions
is restricted to frequency tables, graphs (bar and pie charts) and some coefﬁcients
such as Cohen’s (1960) Kappa and its modiﬁcations (see e.g., Bnerjee et al. (1999)
and Doner (1999)). A detailed presentation of categorical data analysis can be
found in Agresti (1990). However, no measure of the potential concentration of
the positive or negative answers is used.
In this paper we introduce some indices that can be used to measure this con-
centration, based on the observed proportions of the answers. In Section 2 we
deﬁne three alternative indices, we examine the properties of these indices and
compare their behavior. The third section deals with the construction of conﬁdence
intervals for the true values of the indices. In particular, some methods for assessing
simultaneous conﬁdence intervals for multinomial proportions are reviewed brieﬂy.
These methods can be implemented for constructing conﬁdence intervals for one
of the indices deﬁned in Section 2. Furthermore, three bootstrap methods applied
to these indices (standard, percentile and bias corrected percentile) are illustrated