Identification, Analysis & Empirical Validation (IAV) of Object Oriented Design (OO) Metrics as Quality Indicators

International Journal on Recent and Innovation Trends in Computing and Communication ISSN: 2321-8169
Volume: 5 Issue: 8 31 – 40
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IJRITCC | August 2017, Available @ http://www.ijritcc.org
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Identification, Analysis & Empirical Validation (IAV) of Object Oriented Design (OO)
Metrics as Quality Indicators
Dr. Brajesh Kochar 1
Shailendra Singh Gaur 2
Dinesh Kumar Bhardwaj3
1
Department of C.S.E.,Bhagwan Parshuram Institute of Technology, G.G.S.I.P.U.,Delhi, India
2
Department of I.T.,Bhagwan Parshuram Institute of Technology, G.G.S.I.P.U.,Delhi, India
3
Department of C.S.E.,Bhagwan Parshuram Institute of Technology, G.G.S.I.P.U.,Delhi, India
Abstract : Metrics and Measure are closely inter-related to each other. Measure is defined as way of defining amount, dimension, capacity or
size of some attribute of a product in quantitative manner while Metric is unit used for measuring attribute. Software quality is one of the major
concerns that need to be addressed and measured. Object oriented (OO) systems require effective metrics to assess quality of software. The
paper is designed to identify attributes and measures that can help in determining and affecting quality attributes.
The paper conducts empirical study by taking public dataset KC1 from NASA project database. It is validated by applying statistical techniques
like correlation analysis and regression analysis. After analysis of data, it is found that metrics SLOC, RFC, WMC and CBO are significant and
treated as quality indicators while metrics DIT and NOC are not significant. The results produced from them throws significant impact on
improving software quality.
Keywords:- CK object oriented (OO) design metrics , Software quality, C++, Internal and External attributes.
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1. INTRODUCTION
The metrics used in software engineering are being used by
every industry to enhance, develop and maintain given
software. Software metrics are magnanimous in number and
are dependent on type of software attributes that user wants
to estimate. There are mainly two categories of software
metrics viz process metrics and product metrics. Process
metrics deals with how much effort is required (Person-
Months), quantity of resources and methodology involved in
it. Product metrics deals with specifications of project like
requirements, complexity, cohesion, coupling, reliability and
maintenance. Software product involves finding number of
lines of code (LOC), complexity of code and test cases
generated during testing process.
It is not feasible to design full fault prone system due to
changing business requirements and complexity of software.
A magnanimous amount of research is being done by
researchers to improve software quality by innovating novel
techniques. Object oriented (OO) metrics are commonly
used for quality estimation. As the name suggests, OO
metrics are related to measurement of design characteristics
like encapsulation, inheritance, information hiding and
message passing. Software quality is measured in terms of
metrics. It is measured value that is assigned to product for
measuring its quality. So, it can be said that it is possible to
predict software processes by relying on software metrics.
Software metrics acts as crucial source of information for
decision making [22]. Testing and validation of all parts of
software is time consuming and cost effective. So, it is
essential to identify attributes that can predict fault
proneness and acts as measurable quality attributes. It
requires proper selection of product design metrics.
The objective of study defines exploring various metrics and
validating them by taking public KC1 dataset under NASA
project database. The remainder of paper is organized as
follows: Section 2 makes readers aware with the
methodology involved in study. Section 3 deals with some
terminologies and background of conducted studies by
various researchers. Section 4 conducts empirical study by
collecting, processing and analyzing data using various
statistical techniques. Section 5 provides answers to research
questions. Section 6 concludes the given paper.
2. METHODOLOGY
It is inevitable that research elements are not limited to
review literature on the analysis of metrics and measures of
quality factors being studied under Kitchenham and Charters
[1] unified approach.
Methodology covers following points:
 To explore various studies conducted by
researchers in context of OO metrics and software
quality.
 To make users aware of software related
terminologies by providing suitable examples.
 To identify OO metrics that can act as quality
indicators by predicting fault prone classes.
 To validate metrics by taking some dataset and
evaluate them through statistical techniques.
2.1 Research Questions that needs to be answered at the
end of study.
RQ1. How to estimate effort using Lines of Code (LOC)
measure?

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RQ2: How to measure % error in length of program and
estimated program level if operands and operators are
given? Does error in length affects quality of software?
3. STATE OF ART & TERMINOLOGY
3.1 State of Art
A continuous efforts and studies have been conducted in
finding and measuring metrics for Object oriented design
(OO) systems. The external metrics deal with features like
portability, reliability, usability etc while internal factors
deal with factors that lead to internal movement of software
modules. They are cohesion and coupling.
Anna et. Al proposed framework for measuring reusability
by refining CK metrics [4,5]. Zhao and Xu [6, 7] defined
cohesion and coupling metrics that works on dependency
graphs between software modules and dependencies.
Coupling is implemented with the help of Aspect J. The
coupling measures are identified by dependencies between
aspects and classes only. Kumar et. al [8] designed
framework for measuring complexity of software but model
failed to measure interaction among various components of
software.
Several OO metrics have been proposed in order to predict
software quality with the help of classification and
prediction models. It is important to know the metrics that
needs to be measured like effort, productivity, fault
tolerance etc. Use of OO methodology is widespread in all
applications that need development of software. Previous
research studies are being referenced that tells about
empirical validation of various product metrics like NOC,
LCOM, McCabe metric [23] [24], [25]. Li and Henry [12]
have proposed prediction model consisting of ten OO
metrics using statistical analysis technique in order to derive
relationship between maintenance and metrics. Khoshgaftar
et .al [13] used NN to estimate software quality. They
compared parametric model and ANN model to estimate
accuracy. Giovani [14] maintains relationship between static
metrics and software fault proneness by computing static
metrics (Cyclomatic complexity) and dynamic metrics
(dataflow coverage). Emam et. al [15] devised model to
predict faulty classes in java application. K, Rakesh., Kaur,
Gurvinder 2011 [16] studied on Comparing Complexity in
Accordance with Object Oriented Metrics. The study
highlighted the object-oriented software metrics proposed in
90s‟ by Chidamber, Kemerer and several studies were
conducted to validate the metrics and discovered several
deficiencies. Gyimothy, T., Ferenc, R., & Siket, I. 2005
[17] conducted a study on Empirical Validation of Object-
Oriented Metrics on Open Source Software for Fault
Prediction. The study is based on source code of the well-
known open source Web and e-mail suite called Mozilla.
The study also used these modified metrics and added one
more object-oriented metric i.e. Lack of cohesion on
methods (LCOM) and the well-known lines of code metric
(LOC). The study used logistic regression and machine
learning methods to predict the fault proneness of the code.
Few software organizations like TRW [26] and SEI [27]
have designed their own product metrics to build cost and
fault prone models.
3.2 Terminology
Before going to literature study, it is mandatory to make
users aware of some software related terminologies.
Software Complexity: - According to Basili [2], complexity
is defined as measure of resources needed by system while
performing given task. Complexity has no units as it is
measured as function of time and space. It is measured on
basis of number of inputs. There is quite difference between
terms complicated and complex. Complicated means the
solution is not known presently but it has possibility to get
solved later. Complex specifies interactions between
software modules are difficult to understand.
Entity and Attributes:-
Measurement is done on entities and attributed. Entity is
defined as an object, event or action over specific time. An
attribute is characteristic of an entity like size of program.
Both entities and attributes are dependent on each other i.e.
saying only measure a program is vague sentence because
its attribute is not specified. “Measure size of program” is
valid sentence holding both entity and attribute.
Table 1: Types of Attributes
Internal Attributes External Attributes
1. These attributes depend only on entity. 1. They depend both on entity as well as
context of entity.
2. It is easy to measure.
Examples: Size, cost, effort etc.
2. It is difficult to measure because it requires
several other factors and parameters to be
tested under consideration.
Examples: Maintenance, reliability etc.

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Table 2: Definition of External Attributes
Attributes Definition
1. Effectiveness & Extendibility Allows adaptive changes in system design as
per user requirements. It covers inheritance
metric.
2. Functionality & Reusability Allows redefining existing component in
software system and reusing it in order to
achieve higher outcome. It relates to
complexity metric.
3. Understandability Measures the degree of complexity and level
of programmer to understand design of
software. It relates to coupling and cohesion
metric.
Types of Measures
Measures depend on size of program, structure and nature of
modules. Few measures that are commonly used in software
industry:
(a) Lines of Code (LOC):- It is simplest and well understood
measure that is used for prediction of effort and fault
proneness. LOC helps in finding number of executable lines
in given code.
(b) Halstead Software:- It is used to identify basic elements
of program and measures them to predict their attributes. It
states that program is combination of operators and
operands.
(c) Cyclomatic Complexity:- It measures complexity of
given code from control flow graph. It counts base paths that
start from one point and end till the program is finished.
This approach is used to find number of independent paths
through a program. An independent path is any path through
the program that contains at least one new condition. The
calculation of paths is given by Mc Cabe‟s Cyclomatic
Metric [3] by using formula:
V (g) = e-n+2p (Equation 1)
Where v= vertices of graph, e = edges, n = number of nodes
in graph and p= connected components.
Fig 1: Flow Graph
Each node in a graph represent block of code in a program
and arcs represent branches in program. In flow graph, the
flow is sequential and it is based on assumption that each
node can be reached by starting node and ending node.
From above graph, the value of Cyclomatic complexity is
given as:
V (G) = 9 – 6 + 2 = 5 where e=9, n=6 and p=1
There are 5 independent paths in flow graph as:
Path 1 : acf
Path 2: abef
Path 3: adcf
Path 4: abeacf
Path 5: abebef
There are other two methods for calculating complexity as
follows:
(1) It is equal to number of predicate (decision) nodes plus
one V (G) = P + 1 where P = predicate nodes in graph.
(2) It is equal to number of regions of flow graph.
Object Oriented (OO) Metrics
(a) CK Metrics
The word CK stands for Chidamber and Kemerer [21].
They are of six types:
(1) Depth of Inheritance Tree (DIT):- This metric is defined
as length of root to the deepest leaf node in tree. In this
metric, the class becomes more complex on going from top
to bottom. Depth of tree is also called as height of tree.
(2) Number of Children (NOC):- This metric calculates
number of classes in inheritance tree down from class. It
relates to reusability attribute that gets affected. If NOC
increases, then amount of effort required in testing also
increases.
(3) Coupling between objects (CBO):- It relates the number
of other modules that are coupled to the current module
either as a client or supplier. Increase in CBO will decrease
the usability. It is used to measure complexity, reusability
and quality.
(4) Response for a class (RFC):- It defines number of
methods that are executed in context of messages received
by objects of class and that methods are called as local
methods. Greater number of methods more will be
complexity of class.
a
e
c
b
d
f

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(5) Lack of Cohesion on Methods (LCOM):- LCOM is the
difference between the number of methods whose similarity
is zero and the methods whose similarity is not zero. It is not
a good metric of quality.
(6) Weighted Methods per Class (WMC): - WMC is the
number of all member functions and operators defined in
each class. It is used to measure the understandability,
reusability, maintainability and complexity and quality.
(b) MOOD Metrics
Srinivasan et.al [9] defined object oriented design metrics
commonly known as MOOD metrics. MOOD deals with
structural phases like polymorphism, inheritance, cohesion
and coupling. They are defined as below:
(1) Method Inheritance Factor (MIF):- This metric is
defined as ratio of number of inherited methods (NIM) to
the sum of number of inherited methods and number of
defined methods (NDM) in the class.
Mathematically,
MIF = NIM / NIM + NDM (Eq. 2)
(2) Attribute Inheritance Factor (AIF):- This metric is
similar to MIF except the methods are replaced by
attributes. It is defined as ratio of number of inherited
attributes (NIA) to the sum of number of inherited attributes
and number of defined attributes (NDA) in the class.
Mathematically,
AIF = NIA / NIA + NDA (Eq. 3)
(3) Polymorphism Factor (PF):- This metric is defined as
ratio of number of overriding methods to total possible
number of overridden methods in given class. Its value
ranges from (0 – 100) %.
(4) Coupling factor (CF):- Coupling means inter-relatedness
among modules. This metric is defined as ratio of number of
possible couplings (NPC) of given class with other classes
to the number of actual classes (NAC) minus 1.
Mathematically,
CF = NPC / (NAC -1)
Table 3: Quality factors belonging to various metrics and
measures
4. EMPIRICAL STUDY
4.1. Constituents of empirical study
The first step is to select dependent and independent
variables that needs to be measures and validated using OO
metrics. Prediction of fault prone classes in system can be
treated as good indicator to tell about quality of software.
So, it is treated as dependent variable of our study. The
relationship between CK OO design metrics and this
dependent variable is also being presented in paper.
Now, its turn to measure size of classes involved in project.
It is known that various metrics like LOC, FPA are language
dependent [28, 29]. So, we have used concepts of C++
programming language among these metrics. They are
defined as follows:
Table 4: Definitions of OO metrics in context of C++
CK OO design metrics Definition in context of C++
WMC (Weighted methods per class) Number of member functions and operators defined in each class. It is
invalid if we take member functions and operators inherited from super
class.
DIT (Depth of Inheritance Tree) Classes are organized into directed acyclic graphs (DAG) instead of trees
because C++ allows multiple inheritances. It is inheritance related OO
metric.
NOC (Number of children) Number of direct children of each class. It is inheritance related OO
metric.
CBO (Coupling between objects) If a class uses member functions of other class, it is said to be coupled.
RFC (Response for Class) Number of C++ functions directly called by member functions or
operators of given class.
LCOM (Lack of Cohesion on Methods) It is given as member functions (x,y) without variables – member
functions (x,y) with variables.
4.2. Hypothesis
It is performed in order to validate and maintain relationship
with OO metrics and fault proneness dependent variable.
The results later on are used to describe above metrics as
quality indicators. For each metric, hypothesis is given

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below that is validated by taking public data set in further
section.
HWMC- A class with more member functions is more
complex and thus more fault prone.
HDIT- The class located in deepest level hierarchy is treated
as more fault prone.
HCBO- Highly coupled classes are more fault prone
because they depend more on methods and objects of other
class.
HNOC- Classes with large number of children are more
faults prone and requires more testing.
HRFC- Classes having larger response sets in response to
member functions are likely to more fault prone.
HLCOM- Classes that binds member functions and
variables are more likely to fault prone i.e. low cohesion.
4.3. Data Collection
The following components that need to be collected are:
 LOC of C++ programs at end of implementation
phase related to developed project
 Data about C++ programs
 List of erroneous data found in testing phase
 Replaced source code of C++ programs at
maintenance phase.
Source of data:
The study involves usage of public dataset KC1 from NASA
IV and V Facility Metrics Data Program Repository (MDP)
[30]. The dataset consists of 43 KSLOC of C++ code with
145 classes and 2107 instances.
Data processing:
Empirical analysis of OOCK metrics and code metric
(SLOC) has been done in order to predict number of faults
associated with different severity levels by making use of
public data set KC1 (NASA program metric data program
database). The dataset contains faulty data at method level
(faulty classes according to severity level) while metric
information is at class level.
Different severity levels represent impacts of faults on
performance of system. In KC1 NASA dataset, severity of
faults decreases from 1st
to 5th
level. But for simplicity,
faults that have similar impacts on system can be treated as
single fault. According to [31], fault data is categorized into
three levels-high (severity 1), medium (severity 2) and low
(severity 3,4,5).
Table 5: Distribution of classes among three severity levels
[31]
Level No. of
faulty
classes
% of
faulty
classes
No. of
faults
% of faults
High 24 16.55 49 7.63
Medium 58 40 449 69.94
Low 38 26.20 144 22.43
The intention of this study is to compare and validate effect
of OO metrics on quality of software.
4.4. Data Analysis
This section presents statistical analysis of six OO metrics
and 1 code metric (SLOC) using descriptive method,
correlation analysis and regression analysis.
(a) Descriptive Analysis
It includes different categories of values for all seven
metrics namely Minimum (Min), Maximum (Max), Mean,
Mode, Standard Deviation (SD) and Median.
Below table shows statistics of 145 classes from KC1
dataset.
Table 6: Descriptive statistics of 145 studied C++ classes of
KC1 [32]
Metric
s
Mi
n
Ma
x
Mea
n
Mod
e
S.D. Media
n
WMC 0 110 17.42 8 17.4
5
12
RFC 0 222 34.38 7 36.2
0
28
CBO 0 24 8.32 0 6.34 8
DIT 1 7 2 1 1.26 2
NOC 0 5 0.20 0 0.70 0
LCOM 0 100 68.71 1 36.8
9
84
SLOC 0 231
3
211.1 0 345.
6
108
In above table, high mean value of LCOM indicates classes
are less cohesive.
0
500
1000
1 2 3 4 5 6
WMC distribution
RFC No. of faulty classes % of faulty classes No. of faults % of faults

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Fig 2: WMC metric distribution among faulty classes
Fig 3: Combined distribution results of all metrics
From above graph, DIT, NIC metrics states that inheritance
is not well properly suited to KC1 project. SLOC shows
classes are larger in size. Values of DIT, NOC are very low
so it is not observable in various classes of KC1 dataset.
Rest of features like abstraction, encapsulation, and
complexity are observable.
(b) Correlation Analysis
This technique is used to find dependency among dependent
and independent variables. Metrics are taken as independent
while fault prone is dependent. The formula is given by
Spearman‟s rank function:
ρ=1-6∑di2
/n (n2
-1) [33]
Where di=xi-yi, n is sample size of project
Table 7: Correlation results after Spearman‟s rank
Metric WMC RFC CBO DIT NOC LCOM SLOC
WMC 1
RFC 0.528 1
CBO 0.234 0.379 1
DIT 0.134 0.654 0.460 1
NOC 0.026 0.015 --0.001 --0.0.32 1
LCOM 0.218 0.30 0.217 0.217 --0.28 1
SLOC 0.625 0.509 0.572 0.345 --0.034 0.217 1
Categorization of correlation values according to Hopkins
[34] is shown in table
Table 8: Category wise distribution of values
Range of values category
<0.1 Trivial
0.1-0.3 minor
0.3-0.5 moderate
0.5-0.7 large
0.7-0.9 very large
0.9-1.0 perfect
From table 7, it is seen that SLOC, CBO, WMC and RFC
are strongly connected with each other. So, these metrics are
not independent and they lead to redundancy.
(c) Logistic Regression Analysis
It is most widely statistical technique for predicting faulty
classes in system. It is of two types- univariate and
multivariate.
Univariate logistic regression- It is used to analyze
individual effect of each independent variables and
dependent variables. It is given by equation:
P(X1, X2,…. ,Xn)= e(A0+A1X
)/ 1+e(A0+A1X)
[33]
Where P is probability that fault was found in system and
series of A are regression coefficients.
Multivariate logistic regression- It is used to find combined
effect of both variables.
P(X1,X2,…., Xn)=e(A0+A1X1+….+AnXn
)/1+e(A0+A1X1+….+AnXn)
[33]
The detail of regression model includes coefficient (coeff),
constant, pvalue(statistical effect), R2
value (coefficient of
determination) and std err for estimation.
0
500
1000
1500
2000
2500
SLOC RFC CBO DIT NOC LCOM WMC No. of
faulty
classes
% of
faulty
classes
No. of
faults
% of
faults
Series1
Series2
Series3
Series4
Series5
Series6

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Table 9: Univariate regression analysis for severity fault as high
WMC RFC CBO DIT NOC LCOM SLOC
Coeff 0.12 0.0004 0.016 0.010 --0.39 0.002 0.001
Constant 0.085 0.118 --0.13 0.301 0.371 0.105 0.101
R2
0.61 0.051 0.117 0.0 0.02 0.012 0.131
p value 0.001 0.002 0.0 0.694 0.117 0.121 0
std err 0.87 0.98 0.95 1.01 1 1.004 0.93
From above table, four metrics SLOC, CBO, RFC and WMC have less p values thus they are significant. SLOC has maximum R2
value and NOC has negative coefficient which means large number of children leads to less faults. DIT, NOC, LCOM is non
significant metrics.
Table 10: Univariate regression analysis for severity fault as medium
Coeff 0.200 0.051 0.412 0.116 --1.11 0.013 0.16
Constant --0.500 1.140 --0.290 2.71 2.91 1.190 0.17
R2
0.116 0.071 0.112 0.0 0.09 0.012 0.391
p value 0 0.02 0.0 0.82 0.193 0.14 0
std err 5.91 7.62 7.12 6.91 7.81 7.82 06.87
Table 11: Univariate regression analysis for severity fault as low
Coeff 0.031 0.032 0.13 0.16 --0.17 0.005 0.004
Constant 0.210 0.510 --0.18 0.65 1.1 0.512 0.100
R2
0.110 0.031 0.147 0.007 0.002 0.008 0.421
p value 0 0.012 0.0 0.2 0.5 0.34 0
std err 2.14 2.18 1.98 2.27 2.17 2.27 1.95
The results show that overall four metrics (SLOC, RFC, CBO and WMC) are significant.
Table 12: Multivariate regression analysis for severity fault as high
WMC RFC CBO DIT SLOC
Coeff --0.007 0.001 0.021 --0.32 0.001
std err 0.007 0.04 0.12 0.07 0.0
p value 0.201 0.16 0.12 0.001 0.07
From above table, WMC, DIT are negative here that are positive in univariate analysis. It occurs due to interaction between
various metrics included in regression model.
Table 13: Multivariate regression analysis for severity fault as medium
NOC DIT SLOC
Coeff --0.9 --1.40 0.12
std err 0.59 0.41 0.01
p value 0.15 0.41 0.0
NOC is still negative but DIT is also negative here that leads them as non quality indicators.
5. ANSWERING RESEARCH QUESTIONS
RQ1. How to estimate effort using Lines of Code (LOC) measure?
Boehm devised a formula for estimating maintenance costs and uses a quantity called as Annual Change Traffic (ACT) which is
related to number of times the requests are handled to perform changes in software.
It is given as:
ACT = KLOCadd + KLOCdel / KLOCtot

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Then Annual Maintenance Effort is being computed on basis of ACT as:
AME (person –months) = ACT * SDE
where SDE is software development effort (person-months)
Justification of above formula:
It is given that ACT in software system is 25% per year. The initial development cost was Rs 20 lacs. Total lifetime for the
software is 10 years. What is total cost of software system?
Total cost= software development cost + 0.10 * (software development cost * 0.25)
= 20 + 0.10 (20 * 0.25)
= 20.5 lacs
RQ2. How to measure % error in length of program and estimated program level if operands and operators are given? Does error
in length affects quality of software?
The table consisting of operands and operators are given below:
Operators Occurrences Operands occurrences
main () 1 - -
; 1 extern variable 1
for 2 main function 3
== 3 found 2
!= 4 lim 3
getchar 1 -
() 1 -
&& 3 e 4
return 1 t 2
++ 4 i 1
printf 6 -
if 1 k 3
getline 1 0 4
while 3 MAXLINE 2
Total (n1 = 14) Total (N1= 32) Total (n2=10) total (N2= 25)
Program vocabulary is given by n (n1 +n2) = 24
Program length N = N1+N2 = 57
Estimated Length = NL = n1 log n1 + n2 log n2
= 14 log 14 + 10 log 10
= 25.4
% error = (57 -25.4) / 57 = 0.55 * 100 = 55%
Yes error in length affects the quality of software as less
error means there is less improvement of making changes in
software design.
6. CONCLUSIONS & FUTURE WORKS
Software developers and researchers have acknowledged
that software quality prediction plays vital role in Object
oriented design metrics. Metrics play an important role in
software engineering. There are mainly two categories of
software metrics viz process metrics and product metrics.
Process metrics deals with how much effort is required
(Person-Months), quantity of resources and methodology
involved in it. Product metrics deals with specifications of
project like requirements, complexity, cohesion, coupling,
reliability and maintenance. Categorization of faults on basis
of their severity can be helpful in performing validation of
OO design metrics to predict number of faults in system.
The paper validates the CK OO design metrics by
employing statistical methods like correlation, regression
(univariate, multivariate) under study from KC1 dataset by
NASA MDP.
After applying statistical techniques, it is observed that
metrics SLOC, WMC, RFC and CBO are significant and
related to each other in low, medium and high severity
levels. DIT and NOC are non significant metrics due to their
inheritance concept and low p values. The SLOC metric is
most significant that is termed as quality attribute. The study
also concludes that metrics (SLOC, CBO, WMC, RFC)
measuring class size, complexity, cohesion, coupling are
linked more tightly rather than inheritance metrics (DIT,
NOC).
Future Works
As future work, this study can be replicated by taking
industrial project database in C++, Ada95 and Java. We can
extend this empirical investigation to some soft computing
methods like applying machine learning techniques (neuro
fuzzy, neural networks) in order to produce more refined
results. It will lead to better understanding of prediction
capabilities of suite of CK OO metrics. The best subsets of
classes in multivariate regression model can be used to
produce more significant values of metrics. It is called
introducing Mallow CP in multivariate analysis.

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