Journal of Software Engineering and Applications, 2011, 4, 123-128
doi:10.4236/jsea.2011.42013 Published Online February 2011 (http://www.SciRP.org/journal/jsea)
Analysis of Inconsistencies in Object Oriented
Metrics
Ahmed M. Salem1, Abrar A. Qureshi2
1Department of Computer Science, California State University, Sacramento, USA; 2Department of Mathematics and Computer
Science, University of Virginia’s College at Wise, Virginia, USA.
Received December 17th, 2010; revised January 25th, 2011; accepted January 28th, 2011.
ABSTRACT
Software Metrics have been proposed for procedural and object oriented paradigms to measure various attributes like
complexity, cohesion, software quality, and productivity. Among all of these, “Complexity” and “Cohesion” are consi-
dered to be the most important attributes. As object oriented analysis and design appears to be at the forefront of soft-
ware engineering technologies, many different object-oriented complexity and cohesion metrics have been developed.
The aim of the paper is to compare some of the complexity and cohesion metrics and to analyze these metrics and ex-
pose their inconsistencies. The paper provides a brief introduction of CK and Morris’s metrics for calculating the com-
plexity and cohesion of a software. The inconsistencies in these methods are exposed by providing various examples.
The paper concludes by proving inconsistencies in CK’s cohesion matrices and Morris’s complexity matrices.
Keywords: Software Metrics, Object Orient Systems
1. Introduction
Object oriented design and development has been the
cornerstone for many development projects. With the in-
crease in the use of object oriented methods there is a
growing need to improve current practices in object deve-
lopment and in particular software measurement and me-
trics. A great deal of research work has been done with
respect to object oriented measurement and metrics.
The focus has been on developing various object-ori-
ented metrics to measure software quality, productivity,
complexity, coupling, coh esion. More research and atten-
tion has been given to complexity and cohesion which
constitute two of the most important attributes. Complex-
ity of a class can be defined as the number of methods
used in a class whereas cohesion can be defined as the
measure of the degree to which elements of a module
relates to one another. Cohesion implies the tightness
among the components of a software module.
Approaches of how to measure complexity and cohe-
sion of the object oriented programs has become an im-
portant research area. Many object oriented complexity
and cohesion metrics have been proposed in [1-5]. Vari-
ous studies have been performed on these metrics but not
many object oriented metrics has achieved a wide spread
use or standard. Even in [6-8] inconsistencies have been
shown in various object oriented metrics. In this research
work some of complexity and cohesion metrics proposed
for object oriented systems have been compared and their
inconsistencies were discussed. The first section of this
paper introduces the basic concepts used in object orien-
ted metrics and provides a description of some existing
complexity and cohesion measures. The second section
addresses the complexity and cohesion metrics proposed
by CK and Morris. In the third section, an analysis of the
complexity and cohesion metrics is provided. The paper
concludes with a conclusion and final remarks.
2. Related Work
Measurement is the process by which numbers or sym-
bols are assigned to the attributes of entities in the real
world to describe them according to clearly defined rules.
Software metrics are units of measurement and play an
important r ol e in the so ftware enginee ri n g pr ocess.
A number of traditional metrics were proposed for
structural techniques, but with the advancement of Object
Oriented Methodology (inheritance, polymorphism etc)
the traditional metrics were no longer useful to measure
Object Oriented systems. In an attempt to measure the
attributes of object oriented systems many researchers
have proposed various metrics like CK, MOOD, Morris,

Analysis of Inconsistencies in Object Oriented Metrics
124
Chen and Li etc. Reference [9] presents a survey of ob-
ject oriented design metrics and identifies a set o f metrics
that have effect on design quality attributes. In [10] the
authors investigate and explain 22 different object orien-
ted metrics proposed by various researchers. However,
none of these metrics have been able to acquire an accep-
ted standard. Furthermore, some of these proposed metri-
cs are only useful and applicable to measure only some
types of object oriented systems. This has been and re-
mains to be a challenge in software engineering.
Complexity and Cohesion are two important attributes
in object oriented systems and brief description of each is
outlined below:
2.1. Complexity Measures
The complexity of an object class can be defined as the
cardinality of its set of properties and it can be measured
in terms of the depth of inheritance. In object-oriented
design there is a need to develop new metrics for class of
objects and their associated children of the class. This
concept relates to the complexity as the depth of inherit-
ance indicates the extent to which the class is influenced
by the property of its ancestor and number of children
indicates the potential impact on descendant. The depth
of inheritance and number of children collectively indi-
cate the genealogy of a class. Various other complexity
metrics have been proposed and two complexity metrics,
proposed by CK and Morris, are discussed and compared.
2.2. Cohesion Measures
Cohesion can be defined as the interamodular functional
relatedness of a software module. Cohesion is commonly
categorized into seven levels (ranging from low cohesion
to high cohesion): coincidental, logical, temporal, proce-
dural, communicational, sequential and functional. Low
cohesion modules for example are modules with coinci-
dental cohesion, are indicative of a module that performs
two or more basic functions. High cohesion modules
have functional cohesion which indicates that modules
perform only one basic function. In such modules all
module components need to perform the same task which
makes functional cohesion is the most desirable among
all of them. Many researchers have proposed their own
cohesion metrics and some of which are CK’s lack of
cohesion, Montazeri’s connectivity Metrics, Chen and
Lu’s cohesion measures, Bieman/Kang’s class cohesion
measures and H. Morris’s Degree of cohesion of objects.
Some research has been done on existing object orien-
ted metrics and studies found inconsistencies among
them. Some inconsistencies have been found in the most
popular object oriented metrics CK which wads proposed
by Chidamber and Kemerer. There are also inconsisten-
cies in the cohesion metrics which have been shown in
[6,8]. In [6] Bindu Gupta has compared the existing co-
hesion measures with the properties of cohesion to show
the inconsistencies whereas in [7,8] they showed that
Lack Of cohesion Metric is inconsistent as it gives the
same value of LCOM to both less cohesive and highly
cohesive class. In the next section CK’S and Morris’s
complexity & cohesion metrics are discussed.
3. CK Metrics
CK’s metrics have been most readily used. CK’s metrics
contain fewer flaws than any other metrics. The follow-
ing is a brief discussion on CK’s metrics.
3.1. CK’s Depth of Inheritance
CK’s metrics have used McCabe’s complexity to meas-
ure the complexity of an object oriented system. CK’s
Depth of Inheritance metrics is directly related with the
complexity of an object. It relates to the notion of scope
of properties, i t i s a measur e of how many ancestor classes
can potentially affect this class. The deeper the class is in
its hierarchy, the greater the number of methods it is likely
to inherit, making it more complex. Deeper trees consti-
tute greater design complexity since more methods and
classes are involved. C&K used McCabe’s Cyclomatic
number to measure the complexity of a class. As detailed
in [7,8] McCabe’s cyclomatic number can be calculated
by subtracting total number of nodes of a tree from total
number of edges of a tree and by adding 2 to th e result of
subtraction.
So McCabe’s Cyclomatic Number = E – N + 2 {where
E is total no of edges, N total no of nodes}. Then com-
plexity = McCabe’s cyclomatic number/total no of ob-
jects.
Consider the inheritance tree in Figure 1. The Cyclo-
matic Number for the tree would be calculated as fol-
lows,
A
BCD E
FGH
Figure 1. Inheritance tree of objects with complexity 1/8.

Analysis of Inconsistencies in Object Oriented Metrics
125
Cyclomatic Number = E – N + 2
= 7 – 8 + 2
= 1
Thus the complexity would be,
= Cyclomatic number/total number of objects
= 1/8.
3.2. CK’s Lack of Cohesion Measures
CK’s Lack of Cohesion metrics are used to measure the
cohesion of an object. It uses the concept that if an object
has more number of pairs of methods who shares their
instances than the number of method pairs who don’t
share each other’s instances then the object is classified
as cohesive. The following example will provide a more
clear view of Lack of Cohesion metrics:
Suppose there are three methods in an object called M1,
M2, M3 and I1, I2, I3 are instance variables respectively.
Then there is P and Q as
 



PIi,IjIiJj empty
QIi,IjIiJi nonempty
LCOMPQPQ0 otherwise


 
Let suppose



I1 = a, b, c, d, e
I2 = b, c, d
I3 = f, g
Now
 
I1I2 = b, c, d, non empty

I1I3 = empty

I2I3 = empty

PTotal no of empty sets2

QTota l no. of no n empty sets1
LCOM = 1
More similarities mean lower value of LCOM. Con-
sider the following example with fewer similarities be-
tween the objects.




I1 = a, b, c
I2 = d, e, f
I3 = i, k
I4 = a, g, h
Now



I1I2 = empty
I1I3 = emp ty
I1I4 = a







I2I3 = empty
I2I4 = empty
I3I4 = empty






PTotal no. of empty sets5
QTotal no. of non empty sets1



Thus LCOM = 4.
4. Morris’s Metrics
Morris proposed a metric suite for object oriented sys-
tems in terms of tree structures. And following are Mor-
ris’s complexity and cohesion metrics:
4.1. Morris’s Complexity Metrics
Morris defined complexity of an object oriented systems
in terms of depth of a tree. Depth of a tree can be meas-
ured in terms of number of sub nodes of a tree. The more
the number of sub nodes of tree the more complex the
system. So complexity of an object is equal to the depth
of tree or total number of sub nodes .
The depth of tree displayed in Figure 2 is 3. Thus ac-
cording to Morris’s Complexity Metric the co mplexity of
objects of the tree is equal to 3.
The depth of the inheritance tree in Figure 3 is equal
to 1. Thus the complexity of objects on tree is also equal
to 1.
A
BCD
EFG
Figure 2. Inheritance tree of objects with depth 3.
A
BCD
Figure 3. Inheritance tree of objects with depth 1.

Analysis of Inconsistencies in Object Oriented Metrics
126
4.2. Morris’s Degree of Cohesion
CK’s proposed metrics for cohesion only gives the cohe-
siveness of a class as either low or high. Morris proposed
a cohesion metrics which is not bound to only two values—
high and low. Morris defined a new cohesion metrics in
his thesis in terms of Fan In. This can be defined as:
“Fan-In” of a module M is the number of local flows
that terminate at M, plus the number of data structures
from which information is retrieved by M [11]. In simple
words Fan In is the total number of inputs received by a
method.
Degree of cohesion of objects = total number of Fan-
In/total number of objects. For example consid er the data
structure displayed in Figure 4. The arrow heads show
the fan-ins to the object.
Thus according to Morris’s,
Degree of cohesion = total number of Fan-In/total
number of objects.
Degree of cohesion = 11/7.
5. Inconsistencies
As we analyze CK’S and Morris’s Metrics, inconsisten-
cies have been found in their metrics. CK’s Lack Of co-
hesion Metrics and Morris’s complexity metrics are in-
consistent and these inconsistencies are discussed in this
section.
5.1. CK’s Lack of Cohesion Metrics
In software metric’s CK’S metrics are most widely used
but these metrics have some inconsistencies, and accord-
ing to [7,8] LCOM doesn’t distinguish two dissimilar
entities as its shows same value of LCOM for two dif-
ferent classes with different cohesiveness. LCOM only
shows low cohesion or a high cohesion for objects even if
some objects have medium cohesiveness.
A
BCD E
F G
Figure 4. Data structure describing fan-ins to various ob-
jects with Degree of Cohesion = 11/7.
In Figure 5(a) and 5(b) two different classes are shown
with their methods {Ii, Ji}. In Figure 5(a) according to
CK, value of LCOM will be 8 as |P| = 9 and |Q| = 1 and
|P| – |Q| = 8
Only






QI4,J4I5,J5 is a non-empty set and all
other are empty pairs.
I1
I2
I3
I4
I5
J1
J2
J3
J4
J5
(a)
I1
I2
I3
I4
I5
J1
J2
J3
J4
J5
I6
I7
I8
J6
J7
J8
(b)
Figure 5. (a) Mapping between methods (Ii) and instance
variables (Ji) with LCOM = 8; (b) Mapping between Me-
thods (Ii) and Instance variables (Ji) with LCOM = 8.

Analysis of Inconsistencies in Object Oriented Metrics
127
In Figure 5(b) there are 18 empty pairs (|P| = 18) and
10 non empty pairs (|Q| = 10) of (Ii, Ji) so |P| – |Q| = 8.
Hence both classes have same cohesion value. But from
the image it can be seen that class (b) is more cohesive
than class (a). Hence LCOM doesn’t distinguish between
two dissimilar entities or we can say LCOM does not
represent a precise value of cohesiveness of a class.
In [6], LCOM measure is compared with various basic
properties of cohesion measure, LCOM satisfies many
properties but not all, as minimum value of LCOM is
zero but maximum value is not defined.
5.2. Morris Complexity Metrics
Morris’s complexity metrics measure the complexity of
an object as the depth of its tree structure. To show in-
consistencies in this metrics we have taken the measure-
ment of two classes by using CK’s complexity metrics
and Morris’s Cohesion metrics. Basically a class’s cohe-
siveness and complexity are interrelated. If a class is
highly cohesive then it is obviou s it will have more com-
plexity. But the Morris’s complexity metrics doesn’t obey
this basic law. Let’s have two modules as shown in Fig-
ure 6(a) and Figure 6(b).
By measuring the complexity of modules shown in
Figure 6(a) and Figure 6(b) using Morris’s Complexity
Metrics, it can be seen that module given in Figure 6(b)
is more complex than module given in Figure 6(a).
For example, Number of sub nodes in Figure 6(a) is 1
and in Figure 6(b) the number of sub nodes is 5. Hence
module in Figure 6(b) has more complexity than module
given in Figure 6(a).
Let us suppose that these results are true. The cohe-
siveness of module given in Figure 6(a) using Morris’s
Cohesion Metrics, is measured as the degree of cohesion
= 10/6, where to tal nu mber o f fa n- in for module is 10 an d
total number of objects are 6 {fan-in for A = 5 and for B,
C, D, E, F fan-in = 1, total fan-in = 5 + 1 + 1 + 1 + 1 + 1
= 10}. Now if we measure the cohesiveness for second
module given in Figure 6(b), results are same and for
Figure 6(b) Degree of cohesion of objects= 10/6.
Now both modules have same cohesiveness but dif-
ferent complexities. So either Morris’s cohesion metrics
is inconsistent or complexity metrics is inconsistent.
A
B CDE F
(a)
A
B
C
D
E
F
(b)
Figure 6. (a) Fan-ins between modules with degree of cohe-
sion 10/6 and complexity 1; (b) Fan-ins between modules
with degree of cohesion 10/6 and complexity 5.
In the next step we will now measure the complexity
of those modules using CK’s complexity metrics. CK’s
complexity measures complexity in terms of McCabe’s
Cyclomatic number and therefore complexity of a module
given in Figure 6(a) = (McCabe’s Cyclomatic No/Total
number of Metho ds ) = (5 – 6 + 2)/6 = 1/6
Where McCabe’s cyclomatic number = e – n + 2 {where
e total no of edges, n total no of nodes}
Now if we measure the CK’s complexity for module
given in Figure 6(b), then the Complexity = (5 – 6 + 2)/6

Analysis of Inconsistencies in Object Oriented Metrics
128
= 1/6.
So both classes have the same complexity values, which
in turn show that Morris’s complexity metrics is incon-
sistent.
6. Conclusions
CK’s cohesion metric doesn’t distinguish between two
classes which have different cohesiveness, it shows both
classes have LCOM value =8. The problem is that LCOM
value can be either a non-zero value if |P| > |Q| or 0 oth-
erwise, which means low cohesion or high cohesion. This
clearly shows that CK’s is not able to distinguish a value
of cohesion between 0 and any other non-zero value or
between a class with high cohesion and a class with me-
dium cohesion, where as Morris’s cohesion metrics
doesn’t bound to only two values of cohesiveness. They
can provide a precise value of class cohesiveness.
Morris’s complexity metrics shows that the class in
Figure 6(b) is more complex than the class given by
Figure 6(b). However, both classes have the same cohe-
siveness and so should be equally complex.
This paper documented that coh esion metrics proposed
by CK and complexity metrics proposed by Morris both
have some inconsistencies. We feel that more it will be
more beneficial to use CK’S complexity metrics and
Morris’s cohesion metrics. The paper also demonstrated
the need for refining these metrics and developing new
object oriented metrics.
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