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1. Introduction

Sauer and Reich (2007) stated that having a positive theory would help us to understand project aberrations and improve in getting better results by identifying the root causes. Previously, a new theory of project management was introduced. However, by itself this theory is not sufficient to understand and measure certain phenomenon

within the project. Therefore, we need an applicable theory of measurement that complements the theory of project management.

There are various definitions of measurement. In Stevens (1973), it is defined as “the matching of an aspect of one domain to an aspect of another.” In Sydenham (1982), Fenton, 1994, and Fenton, 1997, it is defined as “measurement is the process by which numbers or symbols are assigned to attributes of entities in the real world in such a way as to describe them according to clearly defined rules.”

There are only a few theories of measurement introduced in the literature: classical, operational and representational theory of measurement (Sarle, 1997; Sydenham, 1982).

The classical theory of measurement assumes that there are only quantitative attributes or qualities that can be measured, and the classical approach only deals with discovering such measures and attributes. In addition, the classical theory of measurement assumes existence of a reality that is being measured. The classical theory of measurement found wide applicability in physics and related areas. However, it was not able to recognize measurement studies in social and behavioral sciences.

The operational theory of measurement deals with the definition and specification of precise measurement operations. On the other hand, it avoids the assumption of the existence of a reality that is being measured. Its concern is limited with the operational aspect of measurement.

The representational theory of measurement handles the limitations posed by both the classical and operational theory of measurement. In this theory, there exists a reality that is being measured and this reality may also be one that is not readily quantitative.

Representational theory of measurement (Pfanzagl, 1968; Krantz et al., 1968; Sydenham, 1982; Fenton, 1994) is found to be applicable. A brief discussion of representational theory is provided as it is pertinent to the study. It is a brief presentation taken from Sydenham (1982).

• An empirical relational system corresponding to a quality.

• A relational system based on a defined symbolism, generally it is numbers.

• A representation condition. • A uniqueness condition.

2. Empirical Relational System

Let q q₁, ₂,...,q_i,... represent the individual manifestations of some quality and define Q as the set of all manifestations:

1 2

{ , ,..., _i,...}

Q= q q q .

Define Ω as the set of all objects that we are interested in measuring:

1 2

{w w, ,...,w_i,...}

Ω = .

There exists a set of R empirical relations r r₁, ,..., ,...,₂ r_i r_n on the defined set Q. Define R as:

1 2

{ , ,..., ,..., }_{i n}

R= r r r r .

Then, the empirical relational system is represented as:

,

L= 〈Q R〉.

3. Numerical Relational System

Define N as a class of numbers and P as a set of relations on N:

1 2

{ , ,..., _i,..., _n}

P= p p p p .

So, a numerical relation system is represented as:

,

4. Representation Condition

The representation condition requires that there exists a correspondence between the set of quality manifestations and the set of numbers in such a way that the relations defined on the set of quality manifestations is preserved on the other set.

Formally, measurement M is defined as an empirical operation:

:

M Q→N,

such that L= 〈Q R, 〉is mapped homomorphically (structure-preserving mapping) onto

,

S= N P by M and F. One-to-one mapping is denoted by F with domain R and range P:

:

F R→P.

Therefore, P_i is denoted as:

( ); ;

i i i i

p =F r p ∈P r∈R,

where p is an n-ary relation if and only if it is the image under F of an n-ary relation. A homomorphic mapping is that for all ri∈Rand all pi∈Pand pi =F r( )i ,

1 1

( ,..., ,..., ) ( ( ),..., ( ),..., ( ))

i i n i i n

r q q q ↔ p M q M q M q .

Measurement M is not a homomorphism (Sydenham, 1982) since, unlike F, M is not a one-to-one mapping. There can be mappings to the same number because there may be multiple but separate qualities corresponding to the same number.

As a result,

, , ,

Y = 〈L S M F〉,

where Y constitutes a scale for ni =M q( )i . The image of qi in N under M is called the

5. Uniqueness Condition

There may be multiple mappings for which the representation condition is valid. It is possible to have transformations from one scale to another as long as the representation condition is valid. The uniqueness condition defines the class of scale transformations to mappings for which the representation condition is valid (Sydenham, 1982).