Evaluating Quality of Software Systems by the Confidence and

Prediction Intervals of Regressions for RFC, CBO and WMC Metrics

SERGIY PRYKHODKO

Department of Software for Automated Systems,

Admiral Makarov National University of Shipbuilding,

Heroes of Ukraine Ave., 9, Mykolaiv, 54007,

UKRAINE

Abstract: - We have proposed to apply the confidence and prediction intervals of nonlinear regressions for the

metrics RFC, CBO, and WMC at the app level to evaluate the quality of software systems from the point of

view of their object-oriented design (OOD). A modified technique for evaluating the quality of software

systems has been introduced. We have given the example of using the modified technique to detect the software

quality of open-source Java systems.

Key-Words: - quality, software system, confidence interval, prediction interval, nonlinear regression, software

metric, normalizing transformation.

Received: April 9, 2024. Revised: September 11, 2024. Accepted: October 13, 2024. Published: November 25, 2024.

1 Introduction

As we know, “software quality is given high

priority”, [1]. However, despite the existing

methods of software quality assessment, “there is

still a lack of an effective estimation method for

overall quality”, [2]. Also, the importance of the

problem of evaluating the quality of software

systems is evidenced by publications in recent years,

[3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13],

[14], [15], [16], [17], [18], [19], [20].

At the same time, “The backbone of any

software system is its design” [21], including object-

oriented design (OOD). To analyze the object-

oriented system, special sets of metrics are used, for

instance, CK [22] and MOOD, [23]. However, only

the CK metrics are designed to measure the three

non-implementation steps of OOD in Booch’s

definition, [22].

Nowadays, software metrics, [5], [6], [8], [17],

including RFC (response for a class) at the app level

[5], [6], are used for detecting the quality of

software systems. Also, we know that RFC depends

on the metrics CBO (coupling between object

classes) and WMC (weighted methods per class).

Such dependency in the form of a linear regression

was proposed, [24].

Although machine learning algorithms are

becoming increasingly popular for software quality

evaluation, [11], [12], [13], [14], [15], [16], methods

of regression analysis have not yet reached their full

potential, [1], [5], [6], [20]. In [1], the authors

combined multiple linear regression and a fuzzy

comprehensive evaluation method to build a quality

evaluation algorithm. In [20], the authors used the

linear regression algorithm for predicting the defect

density in software apps and concluded existing

approaches, including Case-Based Reasoning, are

less precise than the Linear Regression

methodology.

There is currently a known use of the technique

[5] based on the confidence and prediction intervals

of nonlinear regression for the RFC metric at the

app level to evaluate the quality of open-source apps

developed in Java [5] and Kotlin [6]. However, the

metrics CBO and WMC, like RFC, also should be

considered as the dependent variables that

characterize the quality of software systems. That is

why we proposed to modify the technique [5] to

evaluate the quality of software systems from the

point of view of their OOD. A modification is based

on the confidence and prediction intervals of

nonlinear regressions for RFC, CBO, and WMC at

the app level. To build the nonlinear regression

models, confidence, and prediction intervals of

nonlinear regressions for RFC, CBO, and WMC, we

apply the appropriate techniques based on

multivariate normalizing transformations, [25].

2 Problem Formulation

Suppose given the original sample as the three-

dimensional non-Gaussian data set: actual values of

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the RFC, CBO, and WMC metrics from N software

systems. Suppose that there are two transformations:

a bijective three-variate normalizing transformation

of a non-Gaussian random vector

 

T

WMCCBORFC ,,P

to a Gaussian random

vector

 

T

WMCCBORFC ZZZ ,,T

that is given by:

 

PψT

(1)

and the inverse transformation for (1):

 

TψP1



,

(2)

ψ

is a vector of normalizing transformation (1),

 

T

WMCCBORFC ψ,ψ,ψψ

.

It is required to build three nonlinear regression

models in the form

 

11 ε,,WMCCBOFRFC 

,

 

22 ε,,WMCRFCFCBO 

, and

 

33 ε,,CBORFCFWMC 

, respectively, using

transformations (1) and (2). Here

j

ε

is the error

term that is the Gaussian random variable to

describe residuals,

j

ε



 

2

ε

σ,0 j

N

,

j

ε

σ

is the

standard deviation,

3,2,1j

.

Also, it is required to build the confidence and

prediction intervals for the above three nonlinear

regressions for the RFC, CBO, and WMC metrics to

evaluate the quality of software systems from the

point of view of their OOD.

3 Problem Solution

To evaluate the quality of software systems, we

modify the technique for detecting software quality

based on the confidence and prediction intervals of

nonlinear regression for the RFC metric at the app

level, [5]. The need for a modification is primarily

due to that the other two metrics CBO and WMC,

like RFC, should also be considered as dependent

variables. Before using a modified technique, it is

necessary to build nonlinear regression models,

confidence, and prediction intervals. To construct

them, you can use the appropriate techniques based

on multivariate normalizing transformations, [25].

The modified technique follows six steps.

Step 1. Normalize the RFC, CBO, and WMC

values (three-dimensional data point i) for the

software system (system i) by the three-variate

normalizing transformation, which has been used

for finding the confidence and prediction intervals

of nonlinear regressions for the metrics RFC, CBO,

and WMC at the system level (app level).

Step 2. Calculate the squared Mahalanobis

distance (SMD) for the three-dimensional

normalized data point (point i).

Step 3. Check whether the SMD test statistic for

the three-dimensional normalized data point (point

i) is greater than a quantile of the corresponding

distribution for this statistic. If yes then stop and go

away (we cannot use the modified technique for

point i) else go to step 4.

Step 4. Calculate borders of the confidence and

prediction intervals of nonlinear regressions for the

RFC, CBO, and WMC metrics at the system level

for the three-dimensional data point (point i).

Step 5. Detect where the three-dimensional data

point (point i) falls. If the data point (point i) for the

software system is inside all confidence intervals of

nonlinear regressions for the metrics RFC, CBO,

and WMC, then stop (the software system has

medium quality) else go to step 6.

Step 6. If the data point (point i) for the software

system is between the upper borders of confidence

intervals and the lower borders of prediction

intervals for all three metrics, then the software

system has high quality else the software system has

low quality.

In step 1, we recommend using multivariate

transformations, for instance, the Box-Cox [26] or

Johnson, [27]. The choice of the multivariate

transformation will depend on the data set for the

metrics RFC, CBO, and WMC.

To calculate the SMD for a three-dimensional

normalized data point (point i) in step 2, we apply

the following formula:

   

TTSTT  iN

T

ii

d12

,

(3)

where

T

is the sample mean vector,

 

T

WMCCBORFC ZZZ ,,T

;

N

S

is the sample

covariance matrix:

  





 N

i

T

ii

NN1

1TTTTS

.

(4)

In step 3, we apply a test statistic for value

2

i

d

as

follows, [28]:

 

133 22  NdNN i

,

(5)

which has an approximate F distribution with a 3

and

3N

degrees of freedom and

α

significance

level. According to [29], we take

α

as 0.005. We

use the F distribution quantile

005.0,3,3 N

F

with a 3

and

3N

degrees of freedom and 0.005

significance level to compare with (5).

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We use the appropriate techniques based on

multivariate normalizing transformations [25] to

build models and intervals (confidence and

prediction) of nonlinear regressions for the metrics

RFC, CBO, and WMC. According to [25], we can

build the confidence intervals of nonlinear

regressions for the metrics RFC, CBO, and WMC

as:

   























  21

1

ν,2α

11

ˆ

ψXZ

T

XZYY N

StZ YzSz

,

(6)

where

Y

ψ

is the normalizing transformation

component for dependent variable Y;

Y

Z

ˆ

is a

prediction result by a linear regression equation

22110 ˆˆˆ

ˆZbZbbZY

dependent on predictors Z1

and Z2 for the normalized data, which are

transformed by the three-variate normalizing

transformation;

ν,2α

t

is a student's t-distribution

quantile with a

2α

significance level and

ν

degrees of freedom;

3ν  N

;



X

z

is a vector with

components

11 ZZ i

,

22 ZZ i

for i-row;





N

ijj i

Z

N

Z

1

,

2,1j

;

 

2

1

2ˆ

ν

1



 N

iYYZ iiY ZZS

;

Z

S

is the

22

matrix:















2221

2111

ZZZZ

ZSS

SS

S

,

(7)

where

  

rr

N

iqqZZ ZZZZS iirq  

1

,

2,1, rq

.

To build the confidence interval of nonlinear

regression for the metric RFC by (6), we need to

substitute RFC,

RFC

ψ

,

RFC

Z

ˆ

,

CBO

Z

,

WMC

Z

,

CBO

Z

, and

WMC

Z

instead of Y,

Y

ψ

,

Y

Z

ˆ

,

1

Z

,

2

Z

,

1

Z

, and

2

Z

, respectively. To construct the

confidence interval of nonlinear regression for the

metric CBO by (6), we need to substitute CBO,

CBO

ψ

,

CBO

Z

ˆ

,

RFC

Z

,

WMC

Z

,

RFC

Z

, and

WMC

Z

instead of Y,

Y

ψ

,

Y

Z

ˆ

,

1

Z

,

2

Z

,

1

Z

, and

2

Z

,

respectively. To build the confidence interval of

nonlinear regression for the metric WMC by (6), we

need to substitute WMC,

WMC

ψ

,

WMC

Z

ˆ

,

RFC

Z

,

CBO

Z

,

RFC

Z

, and

CBO

Z

instead of Y,

Y

ψ

,

Y

Z

ˆ

,

1

Z

,

2

Z

,

1

Z

, and

2

Z

, respectively.

The prediction interval of the nonlinear

regression is constructed analogously (6) with the

only difference that 1 more must be added to the

sum in curly brackets (6).

4 An Example of Problem Solution

We give an example of using the modified

technique to detect the software quality of open-

source Java systems. To build models, the

confidence and prediction intervals of nonlinear

regressions for the metrics RFC, CBO, and WMC at

the system level by (6) for our example, we use the

data of RFC, CBO, and WMC of 46 open-source

Java-systems hosted on GitHub from [5]. In [5], the

data was obtained using the CK tool and cleaned

from the three-variate outliers. The data of RFC,

CBO, and WMC metrics of 46 open-source Java

systems was supplemented by others of the same

metrics from [30] for three popular open-source

Java systems of some versions: FreeMind

0.9.0Beta17, jEdit (2.6final and 3.0final), and

TuxGuitar 1.3.0. Also, we added the data other three

versions of the above systems hosted on GitHub:

FreeMind 1.1.0Beta2, jEdit 5.5.0, and TuxGuitar

1.5.2src. Thus, we had the data of the metrics RFC,

CBO, and WMC from 53 software systems. Like

[5], the data was cleaned from two three-variate

outliers (FreeMind 1.1.0Beta2 and TuxGuitar 1.3.0).

In the following, we used 51 data points.

As in [5], to normalize the data, we applied the

three-variate Box-Cox transformation (BCT) with

components:



 

















.0λif,ln

;0λif,λ1

λ

jj

jjj

jX

X

Z

j

(8)

Here

j

Z

is the Gaussian variable and

j

λ

is a

parameter of BCT,

3,2,1j

. The variable

RFC

Z

is

defined analogously (8) with the only difference that

instead of

j

Z

,

j

X

, and

j

λ

should be put

RFC

Z

,

RFC, and

RFC

λ

, respectively. The variables

CBO

Z

and

WMC

Z

are defined similarly. The parameter

estimates of the three-variate BCT for the data are

calculated by the maximum likelihood method

according to [29] and are

0.194965λ

ˆ

RFC

,

0.851253λ

ˆ

CBO

,

0.567096λ

ˆ

WMC

.

We built the nonlinear regression models for the

metrics RFC, CBO, and WMC based on the three-

variate BCT in the form [5]:

 

 

Y

YY ZY λ

ˆ

1

1ε

ˆ

λ

ˆ

,

(9)

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where

Y

Z

ˆ

is a prediction result by the linear

regression equation

22110 ˆˆˆ

ˆZbZbbZY

dependent

on predictors Z1 and Z2 for the normalized data,

which are transformed by the three-variate

normalizing transformation;

ε

is a Gaussian random

variable,

ε



 

2

ε

σ,0N

.

To build the nonlinear regression model for the

metric RFC by (9), we need to substitute RFC,

RFC

λ

ˆ

,

RFC

Z

ˆ

,

CBO

Z

,

WMC

Z

,

1

ε

, and

1

ε

σ

instead of

Y,

Y

λ

ˆ

Y

Z

ˆ

,

1

Z

,

2

Z

,

ε

, and

ε

σ

, respectively. To

build the nonlinear regression model for the metric

CBO by (9), we need to substitute CBO,

CBO

λ

ˆ

,

CBO

Z

ˆ

,

RFC

Z

,

WMC

Z

,

2

ε

, and

2

ε

σ

instead of Y,

Y

λ

ˆ

,

Y

Z

ˆ

,

1

Z

,

2

Z

,

ε

, and

ε

σ

, respectively. To build

the nonlinear regression model for the metric WMC

by (9), we need to substitute WMC,

WMC

λ

ˆ

,

WMC

Z

ˆ

,

RFC

Z

,

CBO

Z

,

3

ε

, and

3

ε

σ

instead of Y,

Y

λ

ˆ

,

Y

Z

ˆ

,

1

Z

,

2

Z

,

ε

, and

ε

σ

, respectively. The parameter

estimates of the nonlinear regression models for the

metrics RFC, CBO, and WMC are shown in Table

1.

Table 1. The parameter estimates of the nonlinear

regression models

No

Y

b0

b1

b2

ε

σ

MMRE

PRED

1

RFC

-3.69701

0.11637

4.59287

0.2743

0.1346

0.8627

2

CBO

8.09952

4.37867

-11.4975

1.6823

0.1974

0.7451

3

WMC

0.99535

0.12980

-0.00864

0.0461

0.1949

0.7059

To assess the predictive accuracy of nonlinear

regression models for RFC, CBO, and WMC in the

form (9), we utilized standard metrics, namely

MMRE and PRED(0.25). The acceptable values of

MMRE and PRED(0.25) are not more than 0.25 and

not less than 0.75, respectively. Table 1 contains the

MMRE and PRED(0.25) values for the above

models. These values indicate the satisfactory

quality of the models.

To calculate SMD for the three-dimensional

normalized data point (point i) in step 2 of the

considered example of the modified technique, we

need to use the following values in (3):

731.3

RFC

Z

244.8

CBO

Z

,

409.1

WMC

Z

, and

the matrix inverse (4)





















825.479144.4283.62

144.43604.0578.1

283.62578.1561.13

1

N

S

.

In step 3, we use the F distribution quantile with

3 and 48 degrees of freedom and 0.005 significance

level

85.4

005.0,48,3 F

.

To calculate the borders of the confidence

interval of nonlinear regression for the RFC metric,

we need to use the following values in (6) and (7):

2799.0

Y

Z

S

,

244.8

1Z

,

409.1

2Z

, and



















79940.306087.0

06087.0003466.0

1

Z

S

.

To calculate the borders of the confidence

interval of nonlinear regression for the metric CBO,

we need to use the following values in (6) and (7):

7170.1

Y

Z

S

,

731.3

1Z

,

409.1

2Z

, and



















47418.886547.0

86547.013041.0

1

Z

S

.

To calculate the borders of the confidence

interval of nonlinear regression for the metric

WMC, we need to use the following values in (6)

and (7):

0471.0

Y

Z

S

,

731.3

1Z

,

244.8

2Z

, and



















006365.002040.0

02040.010738.0

1

Z

S

.

In all cases of the considered example for

calculating borders of the confidence intervals we

need to use the following values in (6):

011.2

48,205.0 t

, N=51, and

48ν

.

We consider the examples of evaluating the

software quality of open-source Java systems by the

proposed technique. We took the values of the RFC,

CBO, and WMC metrics at the app level from three

popular open-source Java systems [30]: FreeMind,

jEdit, and TuxGuitar. Also, we obtained the values

of these metrics using the CK tool for the above

software systems. In addition, we took the values of

the RFC, CBO, and WMC metrics at the app level

from four other open-source Java systems hosted on

GitHub: Apache Commons Lang, Hosebird Client,

gwt-bootstrap, and itcoinj. Apache Commons Lang

(commons-lang) is a package of Java utility classes

for the classes that are in Java.lang's hierarchy.

Hosebird Client (HBC) is a Java HTTP client for

consuming Twitter's real-time Streaming API. Gwt-

bootstrap is a GWT Library that provides the

widgets of Bootstrap, from Twitter. The bitcoinj

library is a Java implementation of the Bitcoin

protocol, which allows it to maintain a wallet and

send/receive transactions without needing a local

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copy of Bitcoin Core. Table 2 shows the metrics

RFC, CBO, and WMC from the above apps, and test

statistic (TS) (5).

The quality evaluation results from Table 2 are

slightly different from the results from [5]. This can

be explained primarily by the modified technique

(unlike the technique from [5]) considers the other

two metrics CBO and WMC, like RFC, as the

dependent variables.

Table 2. The quality evaluation results

i

App name

RFC

CBO

WMC

TS (5)

quality

1

TuxGuitar 1.5.2-src

15,45

8,54

14,52

0.52

low

2

jEdit 5.5.0

26,49

7,91

39,03

3.38

low

3

jEdit 3.0final

10.38

4.29

14.00

1.28

high

4

jEdit 2.6final

8.84

4.24

9.27

1.68

low

5

FreeMind 0.9.0Beta17

13.29

5.31

12.16

1.88

low

6

commons-lang 4x

25.39

13.80

42.11

0.97

low

7

bitcoinj

37.16

17.81

65.84

1.43

medium

8

gwt-bootstrap

10.62

7.52

12.95

0.55

high

9

HBC

13.10

10.53

13.74

0.13

high

Also, we tried to use the modified technique

example to evaluate the quality of three software

systems (A, B, and C), for which the quality is

classified in NASA's research as low, high, and

medium, respectively [24]. We could not evaluate

the quality of these systems by the modified

technique example since their relevant values of the

test statistic (5) for the normalized metrics RFC,

CBO, and WMC are greater than 4.85. These results

may be explained by the system A is commercial

software, system B is NASA software, and system C

is developed in C++.

5 Discussion

To evaluate the quality of software systems, we

propose the modified technique based on the

confidence and prediction intervals of nonlinear

regressions for the metrics RFC, CBO, and WMC.

This choice is due to the following. Firstly,

according to [22], the CK metrics are designed to

measure the three non-implementation steps in

Booch’s definition of OOD. These are the metrics

WMC, DIT, NOC, RFC, CBO, and LCOM, which

define the OOD complexity in the above steps. In

particular, the metrics RFC and CBO define the

OOD complexity due to the relationships between

classes, [31]. And, as we know, the OOD

complexity affects the quality of software systems.

Finally, the above metrics together characterize

the OOD complexity and quality of software

systems that require the use of multivariate analysis

methods, such as multivariate statistical analysis.

One of them is regression analysis. In this case, as a

rule, nonlinear regression analysis should be used

since only in special cases can the use of a linear

regression model be theoretically justified for

estimating software metrics.

We apply the three-variate Box-Cox

normalizing transformation to build the nonlinear

regression models, the confidence and prediction

intervals for the nonlinear regressions for the

metrics RFC, CBO, and WMC by [25] since, firstly,

according to the Mardia test [32], the distribution of

the three-dimensional normalized data is Gaussian

and, secondly, the residuals distribution of

corresponding linear regression models for

normalized data is Gaussian.

To build the confidence and prediction intervals

for the nonlinear regressions for the metrics RFC,

CBO, and WMC for evaluating the quality of

software systems, we used a 0.05 significance level,

as the appointed one usually, although this value

may be discussed.

Preliminary, we have studied the stability of the

quality evaluation results dependent on a

significance level value. We evaluated the quality of

software systems from Table 2 for two values of

significance level: 0.04 and 0.06. The results are the

same as for a 0.05 significance level. That indicates

the stability of the quality evaluation results at least

within a 20 percent change in a significance level.

Concerning the example of using the modified

technique to detect the software quality of open-

source Java systems two limitations should be

acknowledged and addressed concerning the data

sample from 51 open-source apps in Java. The first

limitation concerns the estimation of the data

sample for open-source apps developed in Java

only. The evaluation of other data samples, for

instance, the industrial systems in Java, may affect

the bounds of the confidence and prediction

intervals of the nonlinear regressions for the metrics

RFC, CBO, and WMC. In such cases, the proposed

bounds of the confidence and prediction intervals of

the nonlinear regressions for the metrics RFC, CBO,

and WMC remain to be confirmed or changed.

The second limitation concerns the sample size,

which equals 51. This value cannot be

unambiguously considered as the lower size limit of

the large sample. Larger sample sizes may lead to a

reduction of the widths of the confidence and

prediction intervals of nonlinear regressions for the

metrics RFC, CBO, and WMC.

The quality of some Java systems from Table 2

is rated as low because the upper bound of the

prediction interval is exceeded for only one metric.

For instance, TuxGuitar 1.5.2-src, jEdit 5.5.0, and

jEdit 2.6final have low quality since the RFC values

of these systems exceed the upper bounds of the

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prediction intervals on 11, 18, and 6 percent,

respectively. These results can be explained by the

above systems having such classes for which the

RFC values are greater than 100 (an acceptable

maximum limit [33]). For instance, there are two

such classes in TuxGuitar 1.5.2-src:

org.herac.tuxguitar.app.action.installer.TGActionIns

taller and

org.herac.tuxguitar.android.action.installer.TGActio

nInstaller for which the RFC values equal 252 and

178, respectively. In this case, it is necessary to

decide whether these classes are difficult to

understand due to the large number of methods in

every class's response set, and if necessary, reduce

their number.

Also, the quality of the above systems can be

improved by improving the relationships between

classes. In our opinion, the modified technique

allows us to assess how balanced OOD of a

software system using the metrics RFC, CBO, and

WMC. And, as we know [34], “Systems engineering

seeks a safe and balanced design in the face of

opposing interests and multiple, sometimes

conflicting constraints.”

The given example of the modified technique

use is illustrative and demonstrates its capabilities.

In the future, it is necessary to build corresponding

models, the confidence, and prediction intervals of

nonlinear regressions for the metrics RFC, CBO,

and WMC based on various data sets.

6 Conclusion

We have proposed to apply the confidence and

prediction intervals of nonlinear regressions for the

RFC, CBO, and WMC metrics for evaluating the

quality of software systems from the point of view

of their OOD. To estimate the confidence and

prediction intervals of nonlinear regressions for the

RFC, CBO, and WMC metrics need to use

multivariate normalizing transformations. In this

case, we have used the three-variate Box-Cox

transformations.

We have introduced the modified technique for

evaluating the quality of software systems. We have

given the example of using the modified technique

to detect the software quality of open-source Java

systems.

Moving forward, we plan to develop examples

of applying the modified technique that does not

have the above limitations due to the programming

language and the sample size.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

Prof. Sergiy Prykhodko is the only author of this

article. He made an alone contribution to this

research at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The author has no conflicts of interest to declare that

are relevant to the content of this article.

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