Optimizing transformation for linearity between online

Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.9, 2014
Optimizing Transformation for Linearity between Online
Software Repository Variables.
Ogunyinka, Peter I1*and Badmus, Nofiu Idowu2
1. Department of Mathematical Sciences, Olabisi Onabanjo University, Ago-Iwoye, Nigeria
2. Department of Statistics, Abraham Adesanya Polytechnic, Ijebu-Igbo, Nigeria
* E-mail of the corresponding author: pixelgoldprod@yahoo.com
Abstract
Online Software Repositories (OSR) like sourceforge.net and google code contain a wealth of valuable data
about software projects but these data violate the linearity and normal assumptions, hence making the data
impossible for use in most statistical data analysis. To prepare these data for statistical data analysis, the data
were non-linearly transformed, hence, these research established the best twelve (12) transformed model that
obey linearity assumptions, higher coefficient of determination (푅2), positive and negative relationship and
gained variable significance over the original data. Similarly, the back transformation or interpretation was
provided about each of these twelve (12) best ranked linear models to solve the challenges of data transformation
encountered by researchers.
Keywords: Data transformation, Linear regression model, OkikiSoft, Online Software Repository and
Sourceforge.net.
1. Introduction
Online Software repository (OSR) is a web based storage of Computer software. OSR contains a wealth of
valuable information about software projects. Ahmed (2008) gave types of software repository as historical
repositories, run-time repositories and Code repositories. This research focuses on the code repositories (CR).
CR, such as www.sourceforge.net and google code (www.code.google.com), host the source codes of various
applications developed by several developers. According to Ahmed (2008), very often, data available on OSR
exhibit large amount of noise and skew. The use of such data may lead to incorrect results and conclusions.
Ahmed (2008) recommends that software repository researchers should closely study the noise and skew in the
data and better understand the effect on the analysis. Statistical visualization is essential to spot the noise and
skewness. He concluded in his recommendation that OSR researchers should provide guidelines and tools to
improve the quality of repository data. This research uses mining software repository (MSR) software called
OKIKISOFT. Okikisoft is the authors developed artificial intelligence software for automatic mining of data on
the webpages of sourceforge.net. The statistical analysis of the data mined on the repository revealed the
violation of linearity assumption between the repository variables. This violation of statistical assumption can
lead to type-I or type-II error, hence, calling for data transformation for the improvement of the quality of the
repository data for subsequent analysis.
Why Sourceforge.net?
Wikipedia (2014) established sourceforge.net as the first free web based source code repository that hosts free
and open source software. Among its competitors are Github (www.github.com), Google Code
(www.code.google.com) and Javaforge (www.javaforge.com). Alexa (2014) rates sourceforge as 162nd world
best website and the first best repository among the aforementioned competitors. These characteristics have
called the attention of this research to take a study of the repository for the hundreds of researchers and millions
of visitors visiting the website.
2. Methodology
Regression analysis requires the satisfaction of linearity, normality, homoscedasticity and independence.
Osborne (2002) established that the violation of the conditions can increase the probability of committing type-I
or type-II error. Over decades, data transformation has been recommended as the solution to non-linearity,
outliers, among others. Data transformation involves using a mathematical operation to change the measurement
scale of variable(s). Linear and nonlinear data transformations are the types of data transformation available.
Linear transformation retains the relationship between variables while non-linear transformation changes
(increases or decreases) the integrity of the relationship between variables. Tabanick and Fidell (2007)
205

Vol.4, No.9, 2014
acknowledging the importance of transformation stated that it may not be generally acceptable by authors as a
results of difficulty of interpreting the transformed variables but remains a legitimate statistical tool for the
realization of linear assumption. This research intends to apply transformation technique to online software
repository variables to establish relationship between these variables which was found absent in the original data.
2.1 Linear Regression and Data Transformation
The interpretation of data based on the analysis of variance (ANOVA) is valid if the assumptions of normality,
homogeneity, independence and addition assumptions are satisfied. Similarly, regression analysis acknowledges
linearity and the first three aforementioned assumptions for implementation. O’Hara and Hotze (2010) emphases
that the main purpose of data transformation is to get a sample data to conform with the assumptions of
parametric statistics such as ANOVA, t-test and linear regression or to manage outliers in a dataset. Marija (2004)
established that data transformation technique is neither a cheating technique nor distortion of the true picture of
the data under consideration, rather, it is a legitimate statistical tool. Literatures have established that results
interpretation of transformed data analysis is the major challenge facing this statistical technique. However, one
added benefit about most transformation technique is that when data are transformed to meet a certain
assumption, we often come closer to satisfy another assumptions as well. For instance, square root
transformation may help to equate group variances by compressing the upper of the distribution end more than it
compresses the lower end. It may also have effect of making a positively skewed distribution more nearly
normal in shape. Howel (2007) recommended, as a solution to interpretation problem of data transformation, that
researchers should look at both the transformed and original data means and make sure that they are telling the
same basic story. Table 1 presents the common ways to transform variables to achieve literatures for regression
analysis.
Table 1: The common statistical transformation techniques
SN Method Transformed Variable Regression
206
Equation
Predicted/Back
transformation value (풀̂
)
01 Standard linear
regression
None
푌 = 푏0 + 푏1푋
푌̂
= 푏0 + 푏1푋
02 Exponential
transformation
Dependent variable
(푙표푔10푌)
푙표푔10푌 = 푏0 + 푏1푋 푌̂
= 10(푏0+푏1푋)
03 Quadratic
transformation
Dependent variable
(푆푞푟푡(푌))
푆푞푟푡(푌) = 푏0 + 푏1푋 푌̂
= (푏0 + 푏1푋)2
04 Reciprocal
transformation
Dependent variable
(푦−1)
푦−1 = 푏0 + 푏1푋 푌̂
= 1⁄(푏0 + 푏1푋)
05 Logarithm
transformation
Independent variable
(푙표푔10푋)
푌 = 푏0 + 푏1푙표푔10푋 푌̂
= 푏0 + 푏1푙표푔10푋
06 Power
transformation
Dependent variable 푙표푔10푌
and independent variable
푙표푔10푋
푙표푔10푌
= 푏0 + 푏1푙표푔10푋
푌̂
= 10(푏0+푏1푙표푔10푋)
SN Method Transformed Variable Regression
Equation
Predicted/Back
transformation value (풀̂
)
07 Square
transformation
Independent variable (푋2) 푌 = 푏0 + 푏1푋2 푌̂
= 푏0 + 푏1푋2
In practise, these methods need to be tested on the data to which they are applied for the confirmation that they
increase rather than decrease the linearity strength of the relationship. Among the methods to detect the
efficiency of the transformed data are to establish linearity, obtain the coefficient of determination (푅2) and to
conduct a significant test of the independent variable on the response variable. It is expected that 푅2 of the
transformed variables will be higher than the non-transformed variables and that the independent variables will
be significant to the response variable. Back transformation is used to return a transformed predicted value to its
original scale. Back transformation predicted values give values for the medium response but not the mean
response as it is expected. Miller (1984) established that back transformation on the mean of the dependent
variable results to serious bias. He, further, established a solution that minimizes the bias. Jia and Rathi (2008),

Vol.4, No.9, 2014
confirming the bias, established a more efficient solution that almost removed the bias. Researchers are advised
to consult this aforementioned literature for implementation.
2.2 Mining of data
Visitors to sourceforge have the privilege to find, create, publish, rate and download free and open source
software from the repository. Rating of software is done based on two criteria viz. Stars rating and laid down
criteria rating. The user can rate 5,4,3,2 or 1 star(s) after which the average stars rating is computed automatically.
Computation of the average rating is shown in table 2 below.
Table 2: Average rating computation for filezilla on sourceforge.net as at April 26, 2014.
207
(a)
Rating Star
(b)
Number of Raters
(c=a*b)
Total
5 843 4215
4 11 44
3 6 18
2 4 8
1 113 113
Total 977 4398
Average (
ퟓ풊
Σ 풄풊
=ퟏ
ퟓ풊
Σ 풃풊
=ퟏ
) 4.5
Similarly, laid-down criteria including design, ease, feature and support are considered in rating software by the
user. The repository takes account of total number of people that rate software and download software on it.
Perhaps, knowing to the visitors, the repository system collects some information about the users’ operating
system maker and users’ country locations.
Manually collecting data on sourceforge can be a tedious task spanning through days and weeks depending on
the volume of the data under concern. This research uses mining software repository (MSR) software named
Okikisoft which was specially developed for this research purpose. Okikisoft automatically and invisibly visits
the pages of sourceforge to extract the required data. It compiles the mined data into CSV files and save it in the
application folder. Okikisoft can act as a server-side or client-side mining system.
3. Presentation and Analysis of the Mined Data
The extracted data variables are represented as follows: D represents Download total for software, F represents
Filesize of the software , R represents Average Rating for the software and T represents Total number of visitors
that rate the software. Okikisoft mined 1802 software data ranging between February 1 through February 28,
2014. However, a sample size of 푛 = 50 was randomly selected for this research. The scatter plots matrix of the
original data is presented in figure 1.

Vol.4, No.9, 2014
1 푅 푇 푅 = 훼 + 훽푇 + 0.03% 푃 = 0.976: 푥 푖푛푠푖푔푛푖푓푖푐푎푛푡
2 푅 퐹 푅 = 훼 + 훽퐹 + 0.02% 푃 = 0.879: 푥 푖푛푠푖푔푛푖푓푖푐푎푛푡
3 푅 퐷 푅 = 훼 + 훽퐷 - 0.02% 푃 = 0.880: 푥 푖푛푠푖푔푛푖푓푖푐푎푛푡
4 푇 퐹 푇 = 훼 + 훽퐹 + 0.0% 푃 = 0.992: 푥 푖푛푠푖푔푛푖푓푖푐푎푛푡
5 (*) 푇 퐷 푇 = 훼 + 훽퐷 - 23.6% 푃 = 0.000: 푥 푠푖푔푛푖푓푖푐푎푛푡 (*)
6 (*) 퐹 퐷 퐹 = 훼 + 훽퐷 - 21.4% 푃 = 0.001: 푥 푠푖푔푛푖푓푖푐푎푛푡 (*)
208
Figure 1: Scatter plot matrix of the original data.
Figure 1 reveals that none of the plots in the scatter plots matrix can be assumed to be linear. Similarly table 3
shows the analysis results done on the original data. Very low coefficient of determination (푅2) and
insignificance of the independent variable were experienced between (푅 and 푇), (푅 and 퐹), (푅 and 퐷) and (푇
and 퐹) and negative relationship was experienced between (푇 and 퐷) and (퐹 and 퐷). This result coincides
with Ahmed (2008) that the original online repository data may violate fundamental assumptions required by
researchers, hence, we recommend that researchers should study the data before further analysis.
Table 3: Analysis results of original data.
Sn.
푽풂풓ퟏ
= 풚
푽풂풓ퟐ
= 풙
Linear Regression
Model
+/
- r
푹ퟐ
푯ퟎ: 휷 = ퟎ
휶 = ퟎ. ퟎퟓ
3.1 Data transformation and analysis
To prevent values between 0 and 1 in the original data (Osborne (2002)), a linear transformation was done by
adding 2 to all data in the four variables. The linearly transformed variables were further nonlinearly transformed
with the seven tools viz: 푙표푔2푦, 푙표푔10푦, √푦, 푦2, √푦 3 , 푦3 and 푦−1. Table 4 shows the results of the data
transformation and figure 2 shows the scatter plot matrix for the data.

Vol.4, No.9, 2014
-/+
r
1 T 푙표푔10퐷 = 퐷∗ 푇 = 훼 + 훽퐷∗ ++ 푇̂
2 푙표푔10퐷 = 퐷∗ √푇 3 = 푇∗ 퐷∗ = 훼 + 훽푇∗ ++ 퐷̂
3 푙표푔10푇 = 푇∗ 푙표푔10퐷 = 퐷∗ 푇∗ = 훼 + 훽퐷∗ ++ 푇̂
4 √퐷 = 퐷∗ √푇 3 = 푇∗ 퐷∗ = 훼 + 훽푇∗ ++ 퐷̂
5 푙표푔2푇 = 푇∗ 푙표푔10퐷 = 퐷∗ 푇∗ = 훼 + 훽퐷∗ ++ 푇̂
6 √퐷 = 퐷∗ 푇2 = 푇∗ 퐷∗ = 훼 + 훽푇∗ ++ 퐷̂
7 T √퐷 = 퐷∗ 푇 = 훼 + 훽퐷∗ ++ 푇̂
8 √푇 = 푇∗ √퐷 = 퐷∗ 푇∗ = 훼 + 훽퐷∗ ++ 푇̂
9 푙표푔10푇 = 푇∗ 퐷−1 = 퐷∗ 푇∗ = 훼 + 훽퐷∗ - - 푇̂
10 푇3 = 푇∗ 퐷 푇∗ = 훼 + 훽퐷 ++ 푇̂
= √(훼 + 훽퐷) 3
11 D 푇2 = 푇∗ 퐷 = 훼 + 훽푇∗ ++ 퐷̂
= 훼 + 훽푇∗ 29.0% 11th
12 푇3 = 푇∗ √퐷 3 = 퐷∗ 푇∗ = 훼 + 훽퐷∗ ++ 푇̂
= √(훼 + 훽퐷∗) 3
13 푙표푔10퐷 = 퐷∗ 푇−1 = 푇∗ 퐷∗ = 훼 + 훽푇∗ - - 퐷̂
14 푇−1 = 푇∗ 퐷−1 = 퐷∗ 푇∗ = 훼 + 훽퐷∗ ++ 푇̂
= (훼 + 훽퐷∗)−1 13.9%
209
Table 4: Analysis results of transformed data.
Sn 푽풂풓ퟏ = 풚 푽풂풓ퟐ = 풙
Linear Regre-ssion
Model
Back
Transformation
푹ퟐ
= 훼 + 훽퐷∗ 55.7% 1st
= 10(훼+훽푇∗) 52.2% 2nd
= 10(훼+훽퐷∗) 46.4% 3rd
= (훼 + 훽푇∗)2 41.6% 4th
= 2(훼+훽퐷∗) 41.2% 5th
= (훼 + 훽푇∗)2 40.6% 6th
= 훼 + 훽퐷∗ 40.5% 7th
= (훼 + 훽퐷∗)2 35.5% 8th
= 10(훼+훽퐷∗) 31.1% 9th
31.0% 10th
27.1% 12th
= 10(훼+훽푇∗) 17.5%
Figure 2: Scatter plot matrix of the first 3 rated models.
Ran-king
4. Discussion of findings
Linearity or almost linearity was ascertained between transformed variables. Table 4 shows 푅2 values, linear
regression models and the back transformation or interpretation between the respective variables that proved to
have linear relationship after transformation was successfully executed. Similarly, twelve (12) models out of the
fourteen transformed linear regression models claim to have positive relationship between variables while the
independent variables proved to be significant (at 5% significant level) to the corresponding dependent variable.
These transformations only discovered linearity between T and D variables while relationship between other
variables failed to claim linearity. We hope attention will be focused on this in future research. Since the result in
table 4 are linear and significant, hence, ranking of the result was done using the values of the 푅2. Column 7 of
table 4 shows the ranking result. The first 12 ranked models proved better with 푅2 > 23.6% which was the
highest 푅2 value obtained between T and D in table 3. Fig. 2 shows the scatter plot matrix of the first 3 ranked
models. This research only uses 푦 to represent the dependent variable and 푥 for the independent variable in
table 4, it is important for researchers to note that these variables can be interchanged but with consequences on
the back transformation of the model. To prevent the aforementioned problem of bias (Miller (1984)) from back
transformation on the dependent variables, we recommend models 1, 7 and 11 since they do not include the
transformation of the dependent variable.

Vol.4, No.9, 2014
5. Conclusion
In this research, we have used data transformation statistical tools in preparing mined data from online software
repository, a case study of sourceforge.net, for researchers for subsequent analysis in improving the efficiency of
repositories. It was established that data from online repositories disobey linearity assumptions and may not be
significant as required for regression analysis. Hence, we recommend that researchers should study the mined
data from online repositories before utilizing them for analysis. Combination of non-linear transformation tools
proved to be effective in establishing linearity between repository variables. It was also established that after
transformation, only Download total and total number of visitors that rate software on sourceforge can be linear.
This research has provided list, in rank, of the first best 12 linear regression models for the direct use of
researchers in future analysis.
References
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Doi:10.1109/FOSM.2008.4659248, Pp. 48-57.
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Cleveland, W.S. (1984), “Graphical methods for data presentation. Full Scale breaks dot charts multibased
logging”, The American Statistician, Vol.38(4), Pp. 270-280.
Howel, D.C. (2007), “Statistical methods for psychology” Belmont, C.A. Thomson Wadsworth. 6th Edition.
Jia, Siwei and Rathi, Sarika (2008), “On predicting log-transformed linear models with heteroscedaticity”, SAS
Global Forum, Paper 370-2008.
Manikandan, S. (2010), “Data transformation”, J. Pharmacol Pharmocother. Jul.-Dec, Vol.1(2),
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Marija, J. Norusis (2004), “SPSS 12.0 Guide to Data Analysis”, Prentice hall Inc., ISBN 0-13-147886-9.
Miller, D. (1984), “Reducing transformation bias in Curve fitting”, The American Statistician, 30(2), 124-126.
O’Hara, Robert B. And Hotze, Johan D. (2010), “Do not log-transform Count data”, Methods in Ecology and
Evolution, Doi:10.1111/j.2041-210x.2010.00021.x.
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210

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