ABSTRACT
BIBLIOGRAPHY
ASTODILLO, LIEZYL T., BUASEN, PAMELA P. and DAO-ANIS,
JOSEPHINE A. March. 2009. Discriminant Analysis on the Performance of
Academic Scholars and Non-Scholars in Benguet State University. Benguet State
University, La Trinidad Benguet
Adviser:
DR. SALVACION Z. BELIGAN
ABSTRACT
This study was conducted to determine the discriminant functions that will
clearly separate the academic scholars and non-academic scholars. Specifically, the
study aimed to: 1) determine the subjects that highly discriminate the academic
scholars and non-academic scholars; 2) estimate the chance of misclassification
given that the derived discriminant function is used as a classification tool; and 3)
identify the non-academic scholars who will be considered academic scholars based
on discriminant analysis.
The variables observed from the 200 non-scholars and 87 scholars were
final grades in Social Science 11, English 11, Math 11 and Filipino 11. The data on
the aforementioned variables were obtained from the Dean’s Office of College of
Arts and Sciences at Benguet State University and analyzed using the SAS
Discriminant Analysis Procedure.
The results showed that Math, English and Filipino 11 grades have high
discriminating powers in differentiating academic scholars and non-scholars. From
the derived discriminant functions, 86.5 percent of the non-scholars and 97.7
percent of the academic scholars were correctly classified. Thus, there was 13.5
percent and 2.3 percent misclassification in the grouping of non-scholars and
academic scholars respectively into its correct classification.
Out of 200 non-academic scholars, 27 could be considered academic
scholars by discriminant analysis.
ii
TABLE OF CONTENTS
Pages
Bibliography ……………………………………………………..........
i
Abstract
……………………………………………………………..
i
Table of Contents
……………………………………………………..
iii
INTRODUCTION
Background of the Study
……………………………………..
1
Statement of the Problem
……………………………………..
2
Objectives of the Study
……………………………………..
3
Significance of the Study …..…..……………………………..
3
Scope and Delimitation of the Study
……………………..
4
REVIEW OF RELATED LITERATURE
Scholarship ……………………………………………………..
5
Studies on Academic Performance ……………………………..
7
Application of Discriminant Analysis …………………...…..
8
Theoretical
Framework …………………………………….. 15
Definition of Terms
……………………………………..
21
iii
METHODOLOGY
Respondents of the Study
……………………………..
23
Sampling
Method
…………………………………………….. 23
Data
Collection …………………………………………….. 22
Statistical
Analysis
…………………………………….. 24
RESULTS AND DISCUSSION
Summary
Statistics
…………………………………….. 26
The Two Group Discriminant Function
……………………..
27
Discriminant Power of Academic Subject
in the Derived Discriminant Function
……….……………. 28
Classification into Group Membership
……………………..
30
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
Summary
…………………………………………………….. 33
Conclusion
…………………………………………………….. 33
Recommendations ……………………………………………..
34
LITERATURE CITED
……………………………………………..
35
APPENDICES
A. Classification Results using Linear Discriminant Function:
Posterior Probability of Membership in Group ……………..
38
B. Output Classification Results of Test Data
……………..
45
C. Discriminant Analysis on Academic Scholars and Non-Scholars:
Classification of Data(SAS)
……………………………..
51
iv
D. Results of Discriminant Analysis (SPSS)
……………..
53
E. Computations …………………………………………….. 58
F. Letter of Permission to the CAS Dean to Gather the Grades
of the Respondents
……………………………………..
59
G. Letter of Permission to the SSGU-OSA to Gather the Grades
of Academic Scholars
………………….………………….
60
H. Application for Oral Defense
...……………………………
61
I. Biographical Sketch
.……………………………………..
62
v
INTRODUCTION
Background of the Study
Institutions of higher learning are engaged in a sustained and continuous
process of their students so as to enhance their readiness for the job market and
further education. Thus, it is important for educational institutions to focus on
improving the aspects of teaching and learning.
One of the schemes in improving the aspect of learning is providing
scholarships to students. In most schools, a unit responsible for providing these
scholarships is usually identified.
At Benguet State University, the Student Scholarship and Grant Unit
(SSGU) is being recognized to administer undergraduate scholarships and awards.
Some of these scholarships are based on leadership abilities, skills and talents, but
mostly, based on academic performance.
The individual grades are used as academic performance indicators. Every
scholar has to maintain a minimum grade requirement set by the school or
university. Some scholarship granting body, an average grade of 75 or 3.0 or higher
is considered as the cut-off point to become a scholar. However, for academic
achiever scholarship award, an average grade of 1.75 or better must be maintained.
To become an academic achiever awardee, he has to prove that he has the
capability to excel in all the subject areas he is enrolled in for a given semester.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
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Based on the university code of Benguet State University, academic achievers must
have an average grade of 1.75 or better with no grade lower than 3.0 in any subject
and no dropped subject(s). Based on this policy code, the researchers of this study
are challenged to determine if an individual who incurred lower than 3.0 because of
some unavoidable circumstances cannot be an academic achiever.
To mitigate the problem of deleting or not an academic scholar from the list
of scholars because of an incurred grade lower than 3.0 in one of the subjects
enrolled through the average is still within the set range, a discriminant analysis
may give solutions to the problem.
Discriminant analysis linearly combines discriminating variables to develop
the basis for assigning an individual of unknown origin to one of the distinct
groups. It is a method of studying group differences on several variables
simultaneously.
Statement of the Problem
This study sought to answer the following problems:
1. Which of the subjects or variables highly discriminate the academic
scholars from the non-academic scholars?
2. What would be the chance of misclassification given that the derived
estimated discriminant function is used as a classification tool?
Discriminant Analysis on the Performance of Academic Scholars and
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3. Who are those non-scholars who can be classified as academic scholars
based on discriminant analysis?
Objectives of the Study
This study aimed to discriminate academic scholars and non-scholars based
on their performance using the grades in the basic subjects.
Specifically, the study aimed to:
1. determine which of the subjects or variables highly discriminate
academic scholars and non-scholars;
2. estimate the chance of misclassification given that the derived estimated
discriminant function is used as a classification tool.
3. identify the non-academic scholars who will be considered academic
scholars based in discriminant analysis.
Significance of the Study
Results of this study can be utilized in future researches, by teachers and
administrators, students – scholars and non-scholars, and even by parents.
The identification of variables that may have a contribution on the academic
performance of the students could be of great help to teachers and administrators.
They would have a better insight on how to motivate their students.
Discriminant Analysis on the Performance of Academic Scholars and
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The results could also help the students to realize and reflect on their study
habits and learning weaknesses, thus, they would be encouraged to find ways to
improve their performance.
The findings could also convince the parents to cooperate and coordinate
with the school in guiding their children on their studies. These could motivate
them to devote more time, attention and guidance needed by their children in order
for their children to improve their performance in school.
Result of this study will serve as guide for administrators and other
sponsoring agencies to revise scholarships’ rules and regulations and for further
improvement in the management of scholarships.
Scope and Delimitation
This study was limited to the academic scholars and non-scholars at
Benguet State University enrolled during the first semester of School Year 2008-
2009. Only the first year and first semester grades for the general education courses
namely: English 11, Filipino 11, Social Science 11 and Math 11 were gathered for
this study.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
REVIEW OF LITERATURE
Scholarship
A scholarship is an award of access to an institution, or a financial aid
award for an individual student called a scholar, for the purpose of furthering their
education. Scholarships are awarded based on a range of criteria which usually
reflect the values and purposes of the donor or founder of the award.
Scholarships may be classified into the following primary groups:
Academic:
A scholarship which is purely based on the academic
performance of the students. Percentage discounts are awarded to the scholars
depending on their grades and their grade point average. This scholarship may be
classified to University or College level.
Merit: A financial aid for which financial need is not used to determine the
recipient. The recipient may be determined by students’ athletic, academic, artistic
or other abilities. The actual monetary value of the scholarship may be negligible,
the scholarship being meant to motivate the student and promote the study of the
subject.
Need: A financial aid for which the student and family’s financial situation
is a primary factor in determining the recipient. Usually, such scholarship will
cover all or part of the tuition and may even cover living costs.
Discriminant Analysis on the Performance of Academic Scholars and
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Sociology: A financial aid where applicants must initially qualify by race,
religion, or national origin. After filtering the applicants based on their ethnicity,
additional factors are taken into consideration to determine the final and deserving
recipients.
Institutional: These are scholarships awarded by a specific college or
university (institution) to a student planning to attend that institution.
General: These are other scholarships which are awarded for a variety of
reasons that do not fall into one of the above categories. These may be for reasons
of the student's association with the objectives of the sponsoring organization or
affiliations.
Some scholarships have a "bond" requirement. Recipients may be required
to work for a particular employer for a specified period of time or to work in rural
or remote areas; otherwise they may be required to repay the value of the support
they received from the scholarship.
It is also typical for persons to find scholarships in their home region.
Information on these can be found by asking local persons and organizations.
Typically, these are less competitive as the eligible population is smaller:
sponsored by Non-profit Organizations or Charitable Trusts, Community
Foundations, Foundations, Labor Unions, Houses of Worship, Chamber of
Commerce and other Volunteer Organizations.
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Academic Performance
Several researches have been made regarding the academic performance of
students, and though many significant findings have been made, it still remains as a
subject of great interest for researchers.
The study of Agustin (2002) aimed to identify the factors affecting the
academic performance of Grade I pupils at Lucban Elementary School. As a result,
six significant factors were extracted, namely: educational and financial support,
sibling number and order, health and geographical location, sex and technology
exposure, parents’ concern, age and guardian. Using regression analysis, it was
found out that the six factors mentioned have only a very small contribution to the
academic performance of the pupils. The R2 indicates only a 4.9% of the pupils’
variations on performance can be explained by the extracted factors.
Milo (2003) conducted a study on determining the indicators of students’
performances and the effects on the performance in school. Factors seen were the
school, facilities, teacher, high school performance, gender and age. It was found
out that these factors contribute highly on the academic performance of students
using regression analysis. The R2 value indicates that 68.6% of the students were
explained by these factors.
Another study conducted by Candiao (2002) determined factors affecting
the academic performance of the SFAO Grantees of BSU under four subjects:
Biology, Communication Arts, Chemistry and Mathematics. Using factor analysis,
Discriminant Analysis on the Performance of Academic Scholars and
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the study revealed two factors having significant relationship in Biology with an R2
of 34.2% and in Communication Arts with 7.7% - father’s educational attainment
and health problems; three factors in Chemistry with an R2 of 49% - average family
income, organization affiliation and serious relationship with someone; and only
the provincial factor was found to have a significant relationship in the performance
in Mathematics with R2 of 11.9%.
Application of Discriminant Analysis
Discriminant function analysis is used to classify cases into the values of a
categorical dependent, usually a dichotomy. If discriminant function analysis is
effective for a set of data, the classification table of correct and incorrect estimates
will yield a high percentage correct. This technique has been applied widely in the
field of education, biological and medical sciences. Here are some of its
application:
Cochran (1964) investigated the problem of estimating the discriminating
power of the linear discriminant function from knowledge of the discriminating
powers of the individual variables includes in the linear discriminant function. His
examination suggested that in practice, (a) most correlation is positive; (b) it is
usually safe to exclude from a discriminant before computing it, a group of
variables whose individual discriminatory power are poor except for any such
variable that has negative correlation with most of the good discriminators; (c) the
Discriminant Analysis on the Performance of Academic Scholars and
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performance of the discriminant function can be predicted satisfactorily from a
knowledge of individual powers plus the average correlation coefficient.
Cornelio and Dacus (2007) employed stepwise discriminant analysis in
identifying the variables that discriminate between the sophomore student of BSAS
as below average and above average using the students’ high school grades (Math,
Science, English, General Weighted Average), and their freshmen college grades
(Math 11. Statistics 11, Biology, Chemistry, Physics, English 11). The results
revealed that grades in high school English and Statistics 11 had the highest
discriminating power in classifying the students into below average and above
average.
Bodong (2001), applying the discriminant analysis, was able to classify a
qualified from non-qualified freshmen applicants in Benguet State University. The
variables utilized were the fourth year weighted average, high school Math, Science
and English grades, and IQ scores. Results showed that the IQ score and English
grade in high school gave clear separation between admitted and not admitted
freshmen applicants.
Antiyag and Tognaon (2006) used discriminant analysis to show the
performance indicators of Benguet State University student achievers and non-
achievers to select the ‘best’ from among possible discriminating variables that will
give clear separation between achiever and non-achiever, and estimate the chance
of misclassification tools. The study was therefore concluded that the Physics and
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English grades are the most important variables to consider in the selection of
achievers and non-achievers. The results also indicated that classification of the
subject into group membership is important.
Benito and Cabasoy (2006) applied stepwise discriminant analysis on the
body measurement of graduating student, in determining the set of body
measurement of young male and female adults that allows for the best
discrimination between sizes (small, medium, large) and the accuracy of predicting
cases classified into correct classification. They concluded that the neck, hip, waists
and length of arms are the variables with high discriminating power in the
separation between sizes.
Domiles and Tamayo (2005) also used discriminant analysis in classifying
admitted and not admitted high school freshmen applicants at Benguet State
University-Secondary Laboratory School, and in classifying the admitted high
school freshmen applicants with regard to what section (Science, Vo-Ag and HE)
they will be placed. They concluded that IQ scores and aptitude scores are the most
important variables to be considered in the selection of high school freshmen
applicants. Also, general point average and IQ scores are the most important
variables to be considered in the classification of section.
Inciong (2001) used discriminant analysis to classify the nutritional status
(overweight, normal, underweight) of pre-school age at La Trinidad, Benguet. The
variables considered were the weight, sex, age, date first seen (months), number of
Discriminant Analysis on the Performance of Academic Scholars and
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brothers and sisters, barangay where they belong, parents’ educational attainment
and occupation. It was concluded that the weight of the child has the greatest
contribution in determining the nutritional status of the selected pre-school age
children. The researcher relates this study to find similarities especially in the tool
applied through the variables considered are different.
Castillo (2003), as cited by Cornelio and Dacus (2007), employed
discriminant analysis in classifying food security in selected indicators in certain
regions in the Philippines. A total of 65 variables were considered describing the
1,200 households from the National Capital Region, Region IV and Region VII
with regard to their food consumption, energy and nutrient intake, and other socio-
economic characteristics. These three regions were selected to represent high,
middle and low income regions. Results showed that the three regions with three
different economic conditions have different food security indicators. Household,
thus, be classified according to level of food security in ways unique to each
region.
The business environment of Korean housing industry has changed recently
from a supplier’s market to a buyer’s. Kim, J. (2000) attempted to offer a
characteristics profile and a forecasting model classifying the housing purchase
consumers into three groups: a single-family housing purchase group, an apartment
housing purchase group, and a non-purchase group. These groups were classified
by employing discriminant analysis and were predicted using the discriminant
Discriminant Analysis on the Performance of Academic Scholars and
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function: a linear combination of demographic, socio-economic and residential
characteristics.
Another application of discriminant analysis is the study of Coleman and
Taylor (1996) entitled, “Determination of a Discriminant Function as a Prediction
Model for Effectiveness of Speed Zoning in Urban Areas”. Speed zoning is the
application of a different speed limit to a section of roadway than is applicable to
adjoining highway segments. Speed zoning traditionally has been based on one of
two similar procedures, one relying primarily on the 85th percentile speed and
engineering judgment and the second, which includes 85th percentile speed, with
some form of quantification of environmental and geometric variables to reduce the
speed limit below the 85th percentile speed. Three problems exist with the current
practice of establishing speed zones. First is that traffic engineers have no way of
predicting if their speed zoning actions will result in better compliance with the
speed limit. Second it is unclear whether the section will make the driving
environment safer resulting in fewer accidents. The third problem is that where
states have procedures which quantify and use environmental and geometric
variables, the empirical basis for their exclusion or inclusion has not been
validated. Coleman and Taylor utilized discriminant analysis to determine if a
quantifiable relationship between accident parameters, speed parameters, roadside
friction, and environmental/geometric variables can be used to predict the
effectiveness of proposed speed zoning actions. The findings are that the most
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significant variables identified by the discriminant functions for effective zones
were: skewness index (negatively skewed), signalization, driveway frequency, and
85th percentile speed.
As cited by Bellovary (2000), one of the most well-known bankruptcy
prediction models was developed by Altman using discriminant analysis. Thus,
Bellovary summarized and analyzed existing research on bankruptcy prediction
studies in order to facilitate more productive future researches in this area.
Moreover, analysis of accuracy of the models suggests that multivariate
discriminant analysis and neural networks are the most promising methods for
bankruptcy prediction models.
Discriminant analysis applied to SAR studies using topological descriptors
allowed Galvez (1996) to plot frequency distribution diagrams: a function of the
number of drugs within an interval of values of discriminant function against these
values. Galvez used these representations, pharmacological distribution diagrams
(PDDs), in structurally heterogeneous groups where generally they adopt skewed
Gaussian shapes or present several maxima. The maxima afford intervals of
discriminant function in which existed a good expectancy to find new active drugs.
A set of beta-blockers with contrasted activity had been selected to test the ability
of PDDs as a visualizing technique, for the identification of new beta-blocker
active compounds.
Discriminant Analysis on the Performance of Academic Scholars and
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Alcohol dependence often cannot be diagnosed based on self-report alone.
Various biochemical and haematological parameters to screen alcohol use
disorders. Hence, Vaswani and Rao (2005) a study to develop discriminant
equations based on lipid and liver measures independently for identifying alcohol
dependent and non-dependent subjects. One hundred subjects fulfilling the criteria
of alcohol dependence and seventy healthy controls were included. The socio-
demographic details, caloric intake, height, weight and blood pressure were
recorded, and samples were analyzed for various lipid measures as well as liver
function using diagnostic values such as sensitivity, specificity, positive predictive
value (PV+), negative predictive value (PV-), and discriminant analysis. Two
equations were constructed based on liver and lipid measures independently. 84.7%
of the subjects on the basis of total cholesterol (TC), apolipoprotein B (ApoB) and
low density lipoprotein-cholesterol (LDL?HDL-c) and 89.1% on the basis of
aspartate amino transferase (AST) and gamma glutamyl transferase (GGT) were
correctly classified into their respective groups.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
THEORETICAL FRAMEWORK
Discriminant Analysis
Discriminant analysis is the appropriate statistical technique when the
dependent variable is categorical (nominal or non-metric) and the independent
variables are metric. It is capable of handling either two or multiple groups. When
three or more classifications are involved, the technique is referred to as multiple
discriminant analysis (MDA).
Discriminant analysis involves deriving the linear combination of the two or
more independent variables that will discriminate best between the prior defined
groups. This is achieved by the statistical decision rule of maximizing the between-
group variance relative to the within-group variance; this relationship is expressed
as the ratio of between-group to within-group variance.
Discriminant analysis is appropriate for testing the hypothesis that the group
means of the two or more groups are equal. It multiplies each independent variable
by its corresponding weight and adds these products together.
The discriminant analysis procedure starts from the assignment of the
individual observation to its prior grouping by letting C1 denotes the first
population, C2 the second population, and X = (x1, x2, …, xp) be column vector of
observations in a full set of p measurement that has a multi-normal distribution
with mean μc in the Cth group and common positive covariance matrix Σ.
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The linear discriminant function based on q of the p variables for
discriminating the groups is given by:
DF = a1X1 + a2X2 + … + apXp = a’X
(1)
where: DF = discriminant
score
ai = discriminant
weight
for independent variable
X
=
independent
variable
Such that, the vector of coefficients maximizes a’Σa, where Σ is the
common covariance matrix of the groups and is a vector coefficient which indicates
the contribution of the variables to differentiation along the function. The signs of
Σ are important because it indicates whether the variables are making positive or
negative contribution.
Measure of the Discriminatory Power
of Variables and Functions
The discriminatory power of a discriminant function refers to the distance
of the group centroids relative to the amount of the dispersion within the groups
based on the given variables. If the magnitude is too small, then it is meaningless to
use the variable to discriminate between groups. Wilk’s lambda (λ) statistic is a
generalized likelihood criterion used in determining the variable that contributes
most to the discriminatory power of a model which is a linear combination of p
variables. The formula is given by:
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q
λ
=
∏
1
(2)
i = k+1 (1 + еi)
where: е i
=
eigenvalue of the ith model
k
=
number of derived functions
If
λ approaches 0, this indicates that the two groups are well separated and
if λ is close to 1, this indicates that no group differences exist. This criterion is
based on the overall multivariate F-ratio for the test of differences between the
group centroids. The variable which maximizes the F-ratio also minimizes λ, a
measure of group discrimination. The significance of λ is tested using the formula:
1
–
λ½
=
ms – p (k-1) / 2 + 1
(3)
λ½
p (k-1)
where:
m
=
N – 1 (p + k) / 2
s
=
√ p2 (k-1)2 – 4
√ p2 + (k-1)2 – 5
This is found to be distributed as F with p (k-1) numerator degrees of
freedom and ms – p (k+1) / 2 + 1 denominator degrees of freedom.
Another statistic that can be used to judge the substantive utility of a
discriminant function is the canonical coefficient ri * defined as:
ri * = √ еi / (1 + еi) (4)
where: еi
=
eigenvalue of the ith model
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This coefficient is a measure of association which summarizes the degrees
of relatedness between the group and the discriminant function. A value of zero
denotes no relationship at all, and a value close to 1 represents a high degree of
association. The square of the canonical coefficient is the average squared
canonical coefficient (ASCC). This ASCC is the proportion of variation in the
discriminant function explained by the variable.
Case Classification and Misclassification Probabilities
Classification can be achieved through a series of classification functions,
one for each group. The equation for one group would be of the form:
Ci = Ci0 + Σ CijXj = Ci0 + Ci0 + Ci1X1 + Ci2X2 + … + CipXp
(5)
where:
Ci is the classification score for the group I,
Cij’s are the classification coefficients
Xj’s are the raw scores of the discriminating variables.
Problem of Classification
For two groups situation, let Xt = (x1, x2…Xp) denotes the vector of
measurements which are the basis for classifying an individual into one of two
groups C1 and C2. According to Fisher (1936), the p-multivariate variables need to
be transformed into a multivariate variable by finding the linear combination of the
X’s which maximally discriminates the group.
Discriminant Analysis on the Performance of Academic Scholars and
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Let
X1 and X2 denotes the vectors of means for the subjects on the p
variables in group 1 and 2. The location of group 1 on the discriminant function is
then given by Y1 = at x1 and the location of group 2 by y2 = at x2. The midpoint
between the two groups on the discriminant function is the given by:
M
=
½
(y1 + y2) = (½)
(x1 – x2) t S-1 (x1 – x2) (6)
If we let Zi denote the score for the ith subject on the discriminant function,
then the decision rule is as follows:
If
Zi ≥ m, then classify subject in group 1;
If
Zi ≤ m, then classify subject in group 2;
On the classification probability, let f1(x) and f2(x) be the probability
distribution function associated with the p x 1 random vector X for the populations
C1 and C2, respectively. Let Ω be the sample space for x. Let R1 be the set of X
values for which we classify objects as G1 in R2 = Ω – R1 be the remaining values
for which we classify objects to G2. The conditional probability, p (2/1), of
classifying an object in G2 when in fact, it is from G1 is
P (2/1)
=
p (X ε R2 G1) = ∫ R2 = Ω - R1 f1(x) dx
P (1/2)
=
p (X ε R1 G2) = ∫ R2 f1(x) dx
Let
P1 be the prior probability of C1 and P2 be the prior probability of C2;
thus, P1 + P2 = 1. The following are the classification probabilities for X:
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P (correct classifying as C1) = P
(C1/C1) * P1
P (misclassifying as C1) =
P
(C1/C2) * P2
P (correct classifying as C2) = P
(C2/C2) * P2
P (misclassifying as C2) =
P
(C2/C1) * P1
The Huberty one sided Z –test was used to determine the hit-ratio or how
good the discriminant function to correct new cases in the group. The formula is
given as:
Z
=
( O – E ) √ N
(7)
√[ E ( N – E ) ]
where:
O
=
actual frequency of the individuals belonging to the
correct group
E
=
expected frequency of the individuals that should
belong to the group
N = sample
size
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Definition of Terms
Academic Performance. It is the grade point average or the general
weighted average (GWA) of the students at the end of the semester or school year
involving the grades in all the subjects enrolled in.
Centroid. This is the mean value of the discriminant Z-scores of a particular
category or group.
Criterion Variable. This is the dependent variable, also called the grouping
variable. It is the object of classification efforts.
Discriminant Analysis. This is a technique for distinguishing or classifying
observations into groups. It provides sorting procedures into previously chosen
variables, and reveals which combinations of variables discriminate among the
groups.
Discriminant Function. The model of equation formed using discriminant
analysis usually in the form of Y = C1X1 + C2X2 + … + CpXp, where: Y represents
the dependent categorical variable, the C’s are the discriminant weight, and the X’s
are the independent variables.
Discriminant Score. It is the value resulting from applying a discriminant
function formula to the data for a given case.
Discriminating Variables. These are the independent variables, also called
predictors.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
22
Eigenvalue. The characteristic root of each discriminant function reflects
the ratio of importance of the dimensions which classify cases of the dependent
variable.
General Weighted Average. This is the grade point average or the weighted
mean grade of all grade points a student earned by enrolment.
Hit Rate. It is the percentage of group cases correctly classified.
Misclassification Probability. This is the probability of given individual to
be classified in the incorrect group.
Non-Scholars. These are students who are unable to acquire financial aid or
scholarship.
Scholars. These are individuals or students who are awarded scholarship or
financial aid to further their studies.
Scholarship. It is an award usually based on academic achievement,
community involvement, or similar factors. It may be awarded regardless of
financial need. Some scholarships must be applied for, while others are awarded
automatically.
Test. This refers to a systematic procedure for observing a behavior or
describing it with the aid of a numerical scale or category system.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
METHODOLOGY
Respondents of the Study
The respondents of the study consisted of a total of 287 freshmen students
of Benguet State University. Out of the 287 students, 87 are academic scholars and
200 are non-academic scholars.
Sampling Method
The selection of the 87 academic scholars was done by total sampling while
the selection of the 200 non-academic scholars was made using simple random
sampling. This means that each member of the target population has an equal
chance of being included in the sample.
Data Collection
The grades of the non-academic scholars on the following subjects:
English11, Filipino11, Social Science11 and Mathematics11, were obtained from
the College of Arts and Sciences Dean’s Office while the grades of the academic
scholars on the aforementioned general education subjects were obtained at the
Students Scholarship Grant Unit-Office of Student Affairs, Benguet State
University.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
24
Statistical Analysis
The respondents were first classified as academic scholars and non-
academic scholars. The gathered data on the four general education courses were
then subjected for discriminant analysis.
Discriminant analysis usually identifies which subject or variable
contributes most to the discrimination between the scholars and non-scholars, and
also to determine the chances of all the elements to be in the correct classification.
Discriminant analysis works by creating a new variable, which is a
combination of the original variables. This is known as the Discriminant Function
with the formula given as:
DF = W1X1 + W2X2 + … + WpXp.
(DF – score, W – weight, X – independent variable)
The means and standard deviation of the variables under each group were
first obtained. To generate the discriminant function and to estimate the chance of
correct classification, discriminant analysis using the SAS program was employed.
Statistical Packages for Social Sciences or SPSS was also utilized in
obtaining the Covariance Matrices and the Canonical Discriminant Functions.
The flow of the analysis is shown in Figure 1.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
25
1. THE PROBLEM
2. THE DESIGN
- classification of observations
- identification of dependent and
into scholars and non-scholars
independent variables
- identification of sample size
4. THE DISCRIMINANT FUNCTION
3. THE ASSUMPTIONS
- identification of discriminant variable
- normality of the variables
- implication of the discriminant
- linearity of relationships
functions
- equal dispersion of matrices
5. EVALUATION OF FUNCTIONS
- discriminant weights
6. VALIDATION OF RESULT
- Wilk’s Lambda
- Posterior Probability of
- canonical correlations
Classification Cases
- graphical display of the centroids
8. PREDICTIVE ACCURACY
7. CLASSIFICATION MATRICES
- assessing of the significance
- criterion for hit ratio
of accuracy
Figure 1. Flow chart for discriminant analysis
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
RESULTS AND DISCUSSION
Summary Statistics
Table 1 shows the grade point average of the scholars and non-scholars in
the four subject areas under test. Mathematics had the highest mean of 2.50 with a
standard deviation of 0.53 while Filipino had the lowest mean and standard
deviation of 2.02 and 0.51 respectively. For the scholars, Filipino had the highest
mean of 1.80 with standard deviation of 0.42 and Mathematics had the lowest mean
of 1.39 with 0.31 standard deviation.
Generally, Mathematics had the highest mean of 1.95 while Filipino with
the mean of 1.91 was the lowest. For the overall mean, non-scholars had higher
mean of 2.25 while the scholars had lower mean of 1.60.
Table 1. Grade point average of scholars and non-scholars
NON-SCHOLAR SCHOLAR
TOTAL
SUBJECT
Standard
Standard
Mean
Mean
Mean
Deviation
Deviation
SS 2.16
0.67
1.68
0.32
1.92
ENG 2.32
0.90
1.52
0.23
1.92
MATH 2.50
0.53
1.39
0.31
1.95
FIL 2.02
0.51
1.80
0.42
1.91
Overall Mean
2.25 1.60 1.93
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
27
The Two Group Discriminant Function
The derived discriminant functions for the non-scholars and academic
scholars in Benguet State University are shown as equation 1 and equation 2,
respectively.
Y1
=
-22.36 – 2.91SS + 7.29Eng + 9.43Math + 5.39Fil
(1)
Y2
=
-10.51 – 0.06SS + 3.99Eng + 3.34Math + 5.68Fil
(2)
The constant values of -22.36 and -10.51 in equation 1 and equation 2, in
that order, corresponds to the y-intercept of the multiple regression model. The
coefficients for X1, X2, X3 and X4 correspond to the regression coefficients in
Multiple Regression. The computed values of the coefficients for Social Science,
English, Mathematics and Filipino in equation 1 are as follows: -2.90, 7.29, 9.43
and 5.39, respectively.
Moreover, the computed coefficient values for Social Science, English,
Mathematics and Filipino in equation 2 are as follows: -0.059, 3.99, 3.34 and 5.68,
respectively.
In function 1 and 2, the academic subjects English, Math and Filipino were
found to have large weights or large discriminatory power. This result suggests that
English, Math and Filipino grades are the three subjects that contribute highly in
differentiating non-scholars and scholars.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
28
Discriminatory Power of Academic Subjects in
the Derived Discriminant Function
Based on computed Wilk’s Lambda (λ), a criterion used in choosing the
variable contributes most to the discriminatory power of the model, Mathematics
grade had the lowest value of 0.458 and Filipino grade had the highest value of
0.961. Based on the significance of the variables’ discriminatory power in
discriminating academic scholars and non-academic scholars, a computed Wilk’s
lambda approaching zero indicates high separation between the groups. These
findings are supported by the Multivariate F-tests which are all significant at 1%
level of significance (Table 2). From the above results, Math and English grades of
the students are the best indicators for separating academic scholars and non-
academic scholars.
Table 2. Discriminatory power of the different academic subjects in classifying
individuals into group membership
Wilk’s
ACADEMIC SUBJECT
F - value
Significance
Lambda
Social Science 11
0.879
39.379
0.000
English 11
0.630
167.675
0.000
Mathematics 11
0.458
337.296
0.000
Filipino 11
0.961
11.713
0.001
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
29
40
1 1
1
1
30
1
1 1
1 1
1
1 1
1
Misclassified
1
1 1
1
1
1 1
1
1
1
1 11
11 1
1
1
1 1 1
1
1
1
11 11
1 1 111
1 1
1 1
1
2
1 21
1 1 11 1
1
1 1
1
1 1 1 1
11 1
20
2 2
1 111
1
1
1 1 1
1
11
1
2
1
1 1 1 1
11 1
1
1 1 11
1 11
1 1
1 1 1
11
2
1
1
1
1
1 111 1
Non-Scholars
2 2
1 1
1 1
1 1
1
1 1
1
1 1
2
2
1
11
11
1 1 1 1
222
2 21
1
11
22
2 22
21
1 1
11 1
1 1
1
1
2 22
2 21 21
1 1 1
2 22 2
2
2
2
1 1
1
10
2 2
2
2
2 22
2
2
2
2 2 1 11
2
2
22 2
2
2
2
12
2 2
2
2
2
2 2
2
2
2
2
Scholars
DF2
0
-10
0
10
20
30
40
50
DF1
Figure 2. Scatter plot of the derived discriminant functions
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
30
Classification into Group Membership
In the analysis of discriminant functions, 287 samples (200 non-scholars
and 87 academic scholars) were utilized. The number of individuals and percents
classified into groups for the original samples, including the computed value of Z
are presented in Table 3.
Table 3. Classification matrix for the discriminant function
NON-
FROM GROUP
SCHOLARS TOTAL
SCHOLARS
Non-Scholars
173* 27 200
Percent
86.5 13.5 100
Scholars
2 85* 87
Percent
2.3 97.7 100
* Correctly classified cases
Hit Rate:
89.90 %
Z: 13.46
Significance:
0.000
The table shows the result of classification computed from derived
discriminant functions. Out of 287, 89.90 percent of the sample size was correctly
classified and 10.10 percent of the samples were misclassified.
Among the non-scholars, 173 or 86.50 percent of the individuals were
classified correctly, while 27 or 13.50 percent of the original 200 samples were
incorrectly classified as non-scholars. For the group of academic scholars, 85 or
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
31
97.70 percent of the students were in the correct classification and two or 2.30
percent of the individuals originally classified as scholars were assigned to non-
academic scholars.
A test on the 95 percent hit-rate ratio obtained a Z-value of -4.11 (p<.01)
suggests that the hit-rate ratio of 90% is significantly lower than the assumed hit-
rate of 95%. This result indicates that the derived function can classify the
individuals into its correct classification by less than 95% at a time.
Presented in Table 4 is the list of misclassified non-academic scholars to
academic scholars with corresponding probabilities of misclassification. In this
study, 27 out of 200 non-academic scholars could probably be academic scholars
by discriminant analysis.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
32
Table 4. Posterior probability of membership in each group
POSTERIOR PROBABILITY
MISCLASSIFIED
OF MEMBERSHIP
OBSERVATION
Classified Non-Academic
Academic
From Group into Group
Scholar
Scholar
14
NS AS 0.2402 0.7598
25
NS AS 0.4602 0.5398
36
NS AS 0.2558 0.7442
38
NS AS 0.4088 0.2912
51
NS AS 0.2722 0.7278
63
NS AS 0.0670 0.9330
65
NS AS 0.2402 0.7598
71
NS AS 0.0123 0.9877
76
NS AS 0.0143 0.9857
85
NS AS 0.2639 0.7361
88
NS AS 0.3791 0.6209
92
NS AS 0.1240 0.8760
98
NS AS 0.4189 0.5811
115
NS AS 0.4946 0.5054
131
NS AS 0.4395 0.5605
139
NS AS 0.3912 0.6088
143
NS AS 0.2499 0.7501
144
NS AS 0.3134 0.6866
170
NS AS 0.4114 0.5886
174
NS AS 0.1275 0.8725
176
NS AS 0.0670 0.0167
179
NS AS 0.2700 0.0167
180
NS AS 0.4166 0.0167
183
NS AS 0.0619 0.0167
185
NS AS 0.0724 0.0167
193
NS AS 0.2442 0.0167
194
NS AS 0.4088 0.0167
AS – Academic Scholars
NS – Non-Academic Scholars
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
SUMMARY, CONCLUSION AND RECOMMENDATIONS
Summary
The variables observed from the 200 non-scholars and 87 scholars were
final grades in Social Science 11, English 11, Math 11 and Filipino 11. The data on
the aforementioned variables were obtained from the Dean’s Office of College of
Arts and Sciences at Benguet State University and analyzed using the SAS
Discriminant Analysis Procedure.
The results showed that Math, English and Filipino 11 grades have high
discriminating powers in differentiating academic scholars and non-scholars. From
the derived discriminant functions, 86.5 percent of the non-scholars and 97.7
percent of the academic scholars were correctly classified. Thus, there was 13.5
percent and 2.3 percent misclassification in the grouping of non-scholars and
academic scholars respectively into its correct classification.
Out of 200 non-academic scholars, 27 could become academic scholars by
discriminant analysis.
Conclusion
From the above findings, it can be concluded that Mathematics, English and
Filipino are the three subjects that can be used to differentiate an academic scholars
from non-academic scholars.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
34
The derived discriminant function was able to classify individuals into its
correct grouping by only 90%.
Out of the 200 non-academic scholars, 27 of them could be classified as
academic scholars.
Recommendation
Based on the above results, the researchers have come up with the following
recommendations:
1. Include other performance related variables to come up with a good
discriminant function with 100% hit-rate.
2. The derived function may be utilized by the registrar’s office in
determining academic scholars.
3. Revisions of the guidelines on scholarship in the Benguet State
University code is suggested.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
35
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http://safa.uwaterloo.ca/scholarships/scholarship.html
www.arts.yorku.ca/advising/handbook2002/glossary.html
en.wikipedia.org/wiki/Scholarship
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
38
APPENDIX A.
DISCRIMINANT ANALYSIS OF SCHOLARS AND NON-SCHOLARS
CLASSIFICATION OF TEST DATA
Discriminant Analysis Classification Results for Test Data: WORK.TEST
Classification Results using Linear Discriminant Function
Posterior Probability of Membership in GRP:
Obs. Classified into GRP 1 2
1 1 0.9377 0.0623
2 1 0.9696 0.0304
3 1 0.8993 0.1007
4 1 0.9943 0.0057
5 1 0.9934 0.0066
6 1 0.9997 0.0003
7 1 0.9866 0.0134
8 1 0.8647 0.1353
9 1 0.9994 0.0006
10 1 0.9875 0.0125
11 1 0.9899 0.0101
12 1 0.9987 0.0013
13 1 0.9884 0.0116
14 2 0.2402 0.7598
15 1 0.7897 0.2103
16 1 0.6121 0.3879
17 1 0.9997 0.0003
18 1 0.9973 0.0027
19 1 0.9970 0.0030
20 1 0.9977 0.0023
21 1 0.9971 0.0029
22 1 0.9853 0.0147
23 1 0.8925 0.1075
24 1 0.9962 0.0038
25 2 0.4602 0.5398
26 1 0.9973 0.0027
27 1 0.9874 0.0126
28 1 0.9931 0.0069
29 1 0.9952 0.0048
30 1 0.9940 0.0060
31 1 0.7586 0.2414
32 1 0.5999 0.4001
33 1 0.9987 0.0013
34 1 0.9871 0.0129
35 1 0.9714 0.0286
36 2 0.2558 0.7442
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
39
Classification Results using Linear Discriminant Function
Posterior Probability of Membership in GRP:
Obs. Classified into GRP 1 2
37 1 0.9988 0.0012
38 2 0.4088 0.5912
39 1 0.9851 0.0149
40 1 0.9867 0.0133
41 1 0.9447 0.0553
42 1 0.9838 0.0162
43 1 0.9997 0.0003
44 1 0.9986 0.0014
45 1 0.9750 0.0250
46 1 0.9974 0.0026
47 1 0.9977 0.0023
48 1 0.9785 0.0215
49 1 0.9395 0.0605
50 1 0.9988 0.0012
51 2 0.2722 0.7278
52 1 0.7683 0.2317
53 1 0.8787 0.1213
54 1 0.9988 0.0012
55 1 0.7772 0.2228
56 1 0.7791 0.2209
57 1 0.9988 0.0012
58 1 0.9755 0.0245
59 1 0.6486 0.3514
60 1 0.9971 0.0029
61 1 0.9869 0.0131
62 1 0.9771 0.0229
63 2 0.0670 0.9330
64 1 0.9860 0.0140
65 2 0.2402 0.7598
66 1 0.9974 0.0026
67 1 0.9892 0.0108
68 1 0.9943 0.0057
69 1 0.9984 0.0016
70 1 0.9999 0.0001
71 2 0.0123 0.9877
72 1 0.9513 0.0487
73 1 0.9941 0.0059
74 1 0.9677 0.0323
75 1 0.6121 0.3879
76 2 0.0143 0.9857
77 1 0.9941 0.0059
78 1 0.9918 0.0082
79 1 0.9677 0.0323
80 1 0.6440 0.3560
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
40
Classification Results using Linear Discriminant Function
Posterior Probability of Membership in GRP:
Obs. Classified into GRP 1 2
81 1 0.6440 0.3560
82 1 0.9972 0.0028
83 1 0.9962 0.0038
84 1 0.9867 0.0133
85 2 0.2639 0.7361
86 1 0.9657 0.0343
87 1 0.9997 0.0003
88 2 0.3791 0.6209
89 1 0.9989 0.0011
90 1 0.9991 0.0009
91 1 0.9999 0.0001
92 2 0.1240 0.8760
93 1 0.9974 0.0026
94 1 0.9995 0.0005
95 1 0.9649 0.0351
96 1 0.9995 0.0005
97 1 0.9967 0.0033
98 2 0.4189 0.5811
99 1 0.9886 0.0114
100 1 0.9999 0.0001
101 1 0.8741 0.1259
102 1 0.5670 0.4330
103 1 0.9489 0.0511
104 1 0.9998 0.0002
105 1 0.9992 0.0008
106 1 0.9938 0.0062
107 1 1.0000 0.0000
108 1 1.0000 0.0000
109 1 0.9974 0.0026
110 1 0.9869 0.0131
111 1 0.9937 0.0063
112 1 0.9504 0.0496
113 1 0.9991 0.0009
114 1 0.9972 0.0028
115 2 0.4946 0.5054
116 1 0.9220 0.0780
117 1 0.7754 0.2246
118 1 0.9988 0.0012
119 1 0.9997 0.0003
120 1 0.9952 0.0048
121 1 0.9873 0.0127
122 1 0.9473 0.0527
123 1 0.9776 0.0224
124 1 0.9853 0.0147
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
41
Classification Results using Linear Discriminant Function
Posterior Probability of Membership in GRP:
Obs. Classified into GRP 1 2
125 1 1.0000 0.0000
126 1 0.9987 0.0013
127 1 0.9970 0.0030
128 1 0.9345 0.0655
129 1 0.7472 0.2528
130 1 0.9994 0.0006
131 2 0.4395 0.5605
132 1 0.9673 0.0327
133 1 0.9938 0.0062
134 1 0.9997 0.0003
135 1 0.9975 0.0025
136 1 0.9986 0.0014
137 1 0.8694 0.1306
138 1 0.9990 0.0010
139 2 0.3912 0.6088
140 1 0.9968 0.0032
141 1 0.9967 0.0033
142 1 0.9551 0.0449
143 2 0.2499 0.7501
144 2 0.3134 0.6866
145 1 0.9723 0.0277
146 1 0.9702 0.0298
147 1 0.9833 0.0167
148 1 0.9943 0.0057
149 1 0.9436 0.0564
150 1 0.9970 0.0030
151 1 0.9031 0.0969
152 1 0.9995 0.0005
153 1 0.9997 0.0003
154 1 0.9686 0.0314
155 1 0.9973 0.0027
156 1 0.8694 0.1306
157 1 0.9636 0.0364
158 1 0.9985 0.0015
159 1 0.9871 0.0129
160 1 0.9986 0.0014
161 1 0.8873 0.1127
162 1 0.9999 0.0001
163 1 0.9988 0.0012
164 1 0.9985 0.0015
165 1 0.5946 0.4054
166 1 0.9970 0.0030
167 1 0.9975 0.0025
168 1 0.9699 0.0301
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Classification Results using Linear Discriminant Function
Posterior Probability of Membership in GRP:
Obs. Classified into GRP 1 2
169 1 0.8478 0.1522
170 2 0.4114 0.5886
171 1 0.9925 0.0075
172 1 0.6046 0.3954
173 1 0.7624 0.2376
174 2 0.1275 0.8725
175 1 0.8925 0.1075
176 2 0.0670 0.9330
177 1 0.9499 0.0501
178 1 0.9988 0.0012
179 2 0.2700 0.7300
180 2 0.4166 0.5834
181 1 0.8993 0.1007
182 1 0.9736 0.0264
183 2 0.0619 0.9381
184 1 0.8694 0.1306
185 2 0.0724 0.9276
186 1 0.9988 0.0012
187 1 0.8787 0.1213
188 1 0.9986 0.0014
189 1 0.9948 0.0052
190 1 0.6676 0.3324
191 1 0.9974 0.0026
192 1 0.9941 0.0059
193 2 0.2442 0.7558
194 2 0.4088 0.5912
195 1 0.8532 0.1468
196 1 0.8895 0.1105
197 1 0.9457 0.0543
198 1 0.9997 0.0003
199 1 0.9931 0.0069
200 1 0.9968 0.0032
201 2 0.0299 0.9701
202 2 0.0035 0.9965
203 2 0.0074 0.9926
204 2 0.4292 0.5708
205 2 0.0992 0.9008
206 1 0.5512 0.4488
207 2 0.0128 0.9872
208 2 0.0034 0.9966
209 2 0.0055 0.9945
210 2 0.0328 0.9672
211 2 0.1357 0.8643
212 2 0.0139 0.9861
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Classification Results using Linear Discriminant Function
Posterior Probability of Membership in GRP:
Obs. Classified into GRP 1 2
213 2 0.0035 0.9965
214 2 0.0545 0.9455
215 2 0.0139 0.9861
216 2 0.0128 0.9872
217 2 0.0154 0.9846
218 2 0.0626 0.9374
219 2 0.0034 0.9966
220 2 0.0265 0.9735
221 2 0.0034 0.9966
222 2 0.0032 0.9968
223 2 0.3765 0.6235
224 2 0.0074 0.9926
225 2 0.2271 0.7729
226 2 0.0078 0.9922
227 2 0.0139 0.9861
228 2 0.0066 0.9934
229 2 0.0143 0.9857
230 2 0.0167 0.9833
231 2 0.0066 0.9934
232 2 0.0146 0.9854
233 2 0.0034 0.9966
234 2 0.1206 0.8794
235 2 0.0072 0.9928
236 2 0.0008 0.9992
237 2 0.0034 0.9966
238 2 0.0083 0.9917
239 2 0.0499 0.9501
240 2 0.2761 0.7239
241 2 0.0312 0.9688
242 2 0.0341 0.9659
243 2 0.0030 0.9970
244 2 0.0154 0.9846
245 2 0.0007 0.9993
246 2 0.0065 0.9935
247 2 0.0008 0.9992
248 2 0.1499 0.8501
249 2 0.0167 0.9833
250 2 0.0154 0.9846
251 2 0.0124 0.9876
252 2 0.0626 0.9374
253 2 0.0060 0.9940
254 2 0.0345 0.9655
255 2 0.0578 0.9422
256 2 0.0071 0.9929
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Classification Results using Linear Discriminant Function
Posterior Probability of Membership in GRP:
Obs. Classified into GRP 1 2
257 2 0.0312 0.9688
258 2 0.0013 0.9987
259 2 0.0076 0.9924
260 2 0.0118 0.9882
261 2 0.0129 0.9871
262 2 0.4039 0.5961
263 2 0.2458 0.7542
264 2 0.0632 0.9368
265 2 0.0059 0.9941
266 2 0.0007 0.9993
267 2 0.0299 0.9701
268 1 0.9833 0.0167
269 2 0.0017 0.9983
270 2 0.0016 0.9984
271 2 0.0535 0.9465
272 2 0.4472 0.5528
273 2 0.0234 0.9766
274 2 0.0137 0.9863
275 2 0.0070 0.9930
276 2 0.0632 0.9368
277 2 0.0026 0.9974
278 2 0.4395 0.5605
279 2 0.0014 0.9986
280 2 0.0318 0.9682
281 2 0.0029 0.9971
282 2 0.0545 0.9455
283 2 0.0055 0.9945
284 2 0.0658 0.9342
285 2 0.1228 0.8772
286 2 0.0149 0.9851
287 2 0.1395 0.8605
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APPENDIX B.
DISCRIMINANT ANALYSIS OF SCHOLARS AND NON-SCHOLARS
OUTPUT CLASSIFICATION RESULTS OF TEST DATA
OBS SS ENG MATH FIL 1 2 INTO
1 2.75 2.00 2.75 2.25 0.93767 0.06233 1
2 2.75 2.25 2.75 2.50 0.96962 0.03038 1
3 2.25 2.25 2.25 1.50 0.89933 0.10067 1
4 1.50 2.50 2.25 1.75 0.99432 0.00568 1
5 2.00 2.00 2.75 2.00 0.99335 0.00665 1
6 2.00 2.50 3.00 2.25 0.99970 0.00030 1
7 1.75 2.00 2.50 1.75 0.98658 0.01342 1
8 2.50 2.00 2.50 2.50 0.86472 0.13528 1
9 2.25 3.00 2.75 2.50 0.99940 0.00060 1
10 2.50 1.75 3.00 1.50 0.98751 0.01249 1
11 2.75 2.50 2.75 1.50 0.98985 0.01015 1
12 2.50 2.50 3.00 2.25 0.99868 0.00132 1
13 2.25 3.00 2.25 2.25 0.98841 0.01159 1
14 1.00 1.50 1.50 1.50 0.24015 0.75985 2
15 1.75 2.00 2.00 1.50 0.78967 0.21033 1
16 1.25 1.75 1.75 1.50 0.61208 0.38792 1
17 1.75 2.75 2.75 2.75 0.99967 0.00033 1
18 2.25 2.50 2.75 2.00 0.99731 0.00269 1
19 2.50 2.25 3.00 2.25 0.99698 0.00302 1
20 2.00 2.25 2.75 1.25 0.99765 0.00235 1
21 2.25 3.00 2.50 2.75 0.99705 0.00295 1
22 2.50 2.25 2.75 2.50 0.98528 0.01472 1
23 2.25 2.25 2.25 1.75 0.89252 0.10748 1
24 2.00 1.75 3.00 2.50 0.99616 0.00384 1
25 3.00 1.75 2.50 1.50 0.46017 0.53983 2
26 2.50 2.75 2.75 2.25 0.99734 0.00266 1
27 2.00 2.75 2.25 2.25 0.98740 0.01260 1
28 1.50 2.00 2.50 2.00 0.99307 0.00693 1
29 2.75 2.75 2.75 1.75 0.99519 0.00481 1
30 2.75 2.75 2.75 2.50 0.99402 0.00598 1
31 3.00 1.75 2.75 2.25 0.75856 0.24144 1
32 1.75 2.25 1.75 2.25 0.59988 0.40012 1
33 2.00 3.00 2.50 2.50 0.99869 0.00131 1
34 2.25 2.00 2.75 1.75 0.98712 0.01288 1
35 2.50 2.00 2.75 2.00 0.97139 0.02861 1
36 1.25 1.75 1.50 1.50 0.25583 0.74417 2
37 1.75 2.75 2.50 2.00 0.99877 0.00123 1
38 1.25 1.50 1.75 1.50 0.40877 0.59123 2
39 2.25 2.00 2.75 2.25 0.98512 0.01488 1
40 2.00 2.25 2.50 2.00 0.98672 0.01328 1
41 2.75 2.50 2.50 2.25 0.94466 0.05534 1
42 2.00 1.75 2.75 2.25 0.98384 0.01616 1
43 2.50 3.00 3.00 2.50 0.99973 0.00027 1
44 2.25 2.25 3.00 2.25 0.99856 0.00144 1
45 2.75 2.75 2.50 2.25 0.97497 0.02503 1
46 1.75 2.50 2.50 1.75 0.99739 0.00261 1
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OUTPUT CLASSIFICATION RESULTS OF TEST DATA
OBS SS ENG MATH FIL 1 2 INTO
47 2.75 3.00 2.75 2.00 0.99773 0.00227 1
48 3.00 3.00 2.50 2.00 0.97854 0.02146 1
49 2.25 2.00 2.50 2.00 0.93948 0.06052 1
50 2.25 2.75 2.75 2.00 0.99882 0.00118 1
51 1.50 2.00 1.50 1.50 0.27217 0.72783 2
52 2.25 2.50 2.00 2.50 0.76830 0.23170 1
53 1.25 1.75 2.00 1.50 0.87867 0.12133 1
54 1.75 2.75 2.50 2.00 0.99877 0.00123 1
55 2.50 1.75 2.50 1.75 0.77717 0.22283 1
56 2.75 2.00 2.50 2.00 0.77907 0.22093 1
57 2.00 2.50 2.75 1.75 0.99881 0.00119 1
58 2.00 2.50 2.25 1.75 0.97547 0.02453 1
59 2.25 1.75 2.25 1.25 0.64855 0.35145 1
60 2.00 1.75 3.00 1.50 0.99713 0.00287 1
61 3.00 2.25 3.00 2.25 0.98686 0.01314 1
62 2.75 2.25 2.75 1.50 0.97714 0.02286 1
63 1.50 1.50 1.50 1.50 0.06699 0.93301 2
64 1.75 3.00 2.00 2.75 0.98605 0.01395 1
65 1.00 1.50 1.50 1.50 0.24015 0.75985 2
66 2.75 3.00 2.75 2.50 0.99737 0.00263 1
67 3.00 2.75 2.75 2.00 0.98921 0.01079 1
68 1.50 2.50 2.25 1.75 0.99432 0.00568 1
69 1.50 2.00 2.75 2.25 0.99837 0.00163 1
70 1.75 3.00 2.75 1.50 0.99990 0.00010 1
71 2.50 2.00 1.50 3.00 0.01229 0.98771 2
72 1.75 2.50 2.00 1.50 0.95135 0.04865 1
73 2.00 2.50 2.50 2.00 0.99414 0.00586 1
74 1.75 1.75 2.50 2.00 0.96768 0.03232 1
75 1.25 1.75 1.75 1.50 0.61208 0.38792 1
76 2.25 2.00 1.50 5.00 0.01434 0.98566 2
77 2.50 2.00 3.00 1.75 0.99407 0.00593 1
78 1.00 1.50 2.50 2.00 0.99181 0.00819 1
79 1.00 2.00 2.00 2.00 0.96770 0.03230 1
80 2.25 2.25 2.00 1.75 0.64404 0.35596 1
81 2.25 2.25 2.00 1.75 0.64404 0.35596 1
82 2.25 2.00 3.00 1.75 0.99717 0.00283 1
83 2.00 1.75 3.00 2.50 0.99616 0.00384 1
84 2.00 2.25 2.50 2.00 0.98672 0.01328 1
85 1.75 1.75 1.75 1.50 0.26387 0.73613 2
86 2.00 2.00 2.50 2.50 0.96568 0.03432 1
87 2.50 3.00 3.00 2.75 0.99971 0.00029 1
88 1.75 2.00 1.75 2.50 0.37912 0.62088 2
89 3.00 3.00 3.00 2.25 0.99888 0.00112 1
90 3.00 3.00 3.00 1.50 0.99910 0.00090 1
91 2.25 3.00 3.00 2.25 0.99988 0.00012 1
92 1.50 1.75 1.50 2.00 0.12401 0.87599 2
93 3.00 2.75 3.00 2.25 0.99745 0.00255 1
94 2.00 2.75 2.75 1.75 0.99948 0.00052 1
95 1.50 1.50 2.50 2.00 0.96494 0.03506 1
96 2.50 2.75 3.00 1.75 0.99950 0.00050 1
97 2.00 1.75 3.00 2.00 0.99668 0.00332 1
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OUTPUT CLASSIFICATION RESULTS OF TEST DATA
OBS SS ENG MATH FIL 1 2 INTO
98 1.75 1.50 2.00 1.50 0.41890 0.58110 2
99 3.00 2.25 3.00 1.75 0.98863 0.01137 1
100 2.00 3.00 3.00 2.25 0.99994 0.00006 1
101 1.50 1.50 2.25 1.50 0.87408 0.12592 1
102 2.00 2.50 1.75 3.00 0.56703 0.43297 1
103 3.00 2.75 2.50 2.25 0.94890 0.05110 1
104 2.25 2.75 3.00 1.50 0.99978 0.00022 1
105 1.00 1.75 2.75 2.00 0.99921 0.00079 1
106 1.00 2.00 2.25 1.50 0.99376 0.00624 1
107 5.00 3.00 5.00 2.00 1.00000 0.00000 1
108 5.00 3.00 5.00 2.50 1.00000 0.00000 1
109 1.75 2.50 2.50 1.75 0.99739 0.00261 1
110 2.25 2.50 2.50 2.25 0.98687 0.01313 1
111 2.75 2.25 3.00 2.25 0.99369 0.00631 1
112 2.50 2.75 2.25 2.00 0.95040 0.04960 1
113 3.00 3.00 3.00 1.50 0.99910 0.00090 1
114 2.25 2.00 3.00 1.75 0.99717 0.00283 1
115 2.50 2.75 1.75 1.75 0.49459 0.50541 2
116 2.25 1.50 2.75 2.50 0.92198 0.07802 1
117 1.50 1.75 2.00 1.50 0.77536 0.22464 1
118 2.75 2.75 3.00 2.25 0.99878 0.00122 1
119 2.00 3.00 2.75 2.25 0.99973 0.00027 1
120 2.75 2.75 2.75 1.75 0.99519 0.00481 1
121 2.50 2.25 2.75 2.00 0.98726 0.01274 1
122 2.25 2.00 2.50 1.50 0.94727 0.05273 1
123 2.50 3.00 2.25 2.00 0.97764 0.02236 1
124 2.50 2.25 2.75 2.50 0.98528 0.01472 1
125 1.75 3.00 3.00 2.00 0.99997 0.00003 1
126 2.00 2.50 2.75 2.00 0.99872 0.00128 1
127 2.50 2.25 3.00 2.25 0.99698 0.00302 1
128 2.00 1.75 2.50 2.00 0.93451 0.06549 1
129 1.75 2.50 1.75 2.75 0.74722 0.25278 1
130 1.50 2.25 2.75 1.50 0.99942 0.00058 1
131 2.00 1.75 2.00 1.50 0.43950 0.56050 2
132 1.50 1.50 2.50 1.75 0.96733 0.03267 1
133 2.25 2.75 2.50 2.50 0.99377 0.00623 1
134 2.25 2.75 3.00 2.00 0.99974 0.00026 1
135 2.50 2.75 2.75 2.00 0.99753 0.00247 1
136 2.25 2.25 3.00 2.25 0.99856 0.00144 1
137 1.00 1.50 2.00 1.50 0.86941 0.13059 1
138 2.00 2.50 2.75 1.25 0.99897 0.00103 1
139 1.25 1.50 1.75 1.75 0.39122 0.60878 2
140 2.25 3.00 2.50 3.00 0.99683 0.00317 1
141 5.00 5.00 2.75 3.00 0.99674 0.00326 1
142 2.75 2.50 2.50 1.50 0.95507 0.04493 1
143 1.75 1.75 1.75 1.75 0.24991 0.75009 2
144 5.00 2.50 3.00 2.25 0.31343 0.68657 2
145 2.00 2.00 2.50 1.75 0.97225 0.02775 1
146 2.00 2.00 2.50 2.00 0.97021 0.02979 1
147 2.50 1.75 3.00 2.50 0.98333 0.01667 1
148 2.25 2.25 2.75 1.75 0.99432 0.00568 1
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OUTPUT CLASSIFICATION RESULTS OF TEST DATA
OBS SS ENG MATH FIL 1 2 INTO
149 2.25 2.75 2.25 5.00 0.94357 0.05643 1
150 2.25 2.00 3.00 2.00 0.99695 0.00305 1
151 2.75 2.25 2.50 1.50 0.90305 0.09695 1
152 2.75 3.00 3.00 2.00 0.99950 0.00050 1
153 2.50 3.00 3.00 2.50 0.99973 0.00027 1
154 1.25 1.75 2.25 1.75 0.96865 0.03135 1
155 2.25 3.00 2.50 2.50 0.99726 0.00274 1
156 1.00 1.50 2.00 1.50 0.86941 0.13059 1
157 2.25 2.25 2.50 3.00 0.96356 0.03644 1
158 2.00 2.00 3.00 2.00 0.99854 0.00146 1
159 1.50 2.25 2.25 1.75 0.98713 0.01287 1
160 2.00 2.00 3.00 1.75 0.99865 0.00135 1
161 2.25 1.75 2.50 1.50 0.88729 0.11271 1
162 1.75 2.50 3.00 2.00 0.99987 0.00013 1
163 2.50 2.50 3.00 2.00 0.99877 0.00123 1
164 1.75 2.25 2.75 2.25 0.99850 0.00150 1
165 1.25 1.75 1.75 1.75 0.59458 0.40542 1
166 2.50 2.25 3.00 2.25 0.99698 0.00302 1
167 2.50 2.75 2.75 2.00 0.99753 0.00247 1
168 3.00 2.50 2.75 2.75 0.96994 0.03006 1
169 2.25 1.25 2.75 2.25 0.84782 0.15218 1
170 1.50 1.75 1.75 1.75 0.41143 0.58857 2
171 2.25 1.75 3.00 2.25 0.99255 0.00745 1
172 1.75 1.75 2.00 1.75 0.60461 0.39539 1
173 1.50 1.75 2.00 1.75 0.76237 0.23763 1
174 1.00 1.75 1.25 1.75 0.12746 0.87254 2
175 2.25 2.25 2.25 1.75 0.89252 0.10748 1
176 1.50 1.50 1.50 1.50 0.06699 0.93301 2
177 2.25 2.50 2.25 1.75 0.94988 0.05012 1
178 2.50 3.00 2.75 2.25 0.99883 0.00117 1
179 1.25 1.75 1.50 1.25 0.27000 0.73000 2
180 2.75 2.00 2.25 2.25 0.41662 0.58338 2
181 2.25 2.25 2.25 1.50 0.89933 0.10067 1
182 2.75 2.25 2.75 2.00 0.97364 0.02636 1
183 1.25 1.25 1.50 1.50 0.06192 0.93808 2
184 1.00 1.50 2.00 1.50 0.86941 0.13059 1
185 1.75 1.75 1.50 1.50 0.07244 0.92756 2
186 1.50 2.50 2.50 1.75 0.99876 0.00124 1
187 1.25 1.75 2.00 1.50 0.87867 0.12133 1
188 2.00 3.00 2.50 2.75 0.99859 0.00141 1
189 2.50 2.50 2.75 1.75 0.99477 0.00523 1
190 1.75 2.25 1.75 1.25 0.66761 0.33239 1
191 3.00 2.75 3.00 2.25 0.99745 0.00255 1
192 2.00 2.50 2.50 2.00 0.99414 0.00586 1
193 1.50 2.00 1.50 2.00 0.24419 0.75581 2
194 1.25 1.50 1.75 1.50 0.40877 0.59123 2
195 2.00 1.50 2.50 2.25 0.85320 0.14680 1
196 2.00 2.50 2.00 2.00 0.88953 0.11047 1
197 2.75 2.00 2.75 1.75 0.94569 0.05431 1
198 2.25 2.75 3.00 3.00 0.99966 0.00034 1
199 2.50 2.00 3.00 2.25 0.99314 0.00686 1
Discriminant Analysis on the Performance of Academic Scholars and
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OUTPUT CLASSIFICATION RESULTS OF TEST DATA
OBS SS ENG MATH FIL 1 2 INTO
200 3.00 2.75 3.00 3.00 0.99683 0.00317 1
201 1.50 1.75 1.25 2.00 0.02992 0.97008 2
202 2.00 1.50 1.25 1.50 0.00354 0.99646 2
203 1.75 1.50 1.25 1.50 0.00740 0.99260 2
204 1.50 1.75 1.75 1.50 0.42924 0.57076 2
205 2.75 1.50 2.25 3.00 0.09919 0.90081 2
206 1.75 1.75 2.00 2.50 0.55117 0.44883 1
207 1.75 1.25 1.50 2.00 0.01279 0.98721 2
208 1.50 1.50 1.00 1.50 0.00340 0.99660 2
209 1.50 1.25 1.25 2.25 0.00547 0.99453 2
210 2.75 2.00 1.75 2.25 0.03279 0.96721 2
211 1.75 1.50 1.75 1.50 0.13574 0.86426 2
212 2.00 1.50 1.50 2.00 0.01390 0.98610 2
213 2.00 1.50 1.25 1.50 0.00354 0.99646 2
214 1.50 1.50 1.50 2.25 0.05452 0.94548 2
215 2.00 1.50 1.50 2.00 0.01390 0.98610 2
216 1.75 1.25 1.50 2.00 0.01279 0.98721 2
217 1.50 1.50 1.25 1.50 0.01540 0.98460 2
218 1.50 1.50 1.50 1.75 0.06256 0.93744 2
219 1.50 1.50 1.00 1.50 0.00340 0.99660 2
220 1.50 1.25 1.50 2.00 0.02646 0.97354 2
221 1.50 1.50 1.00 1.50 0.00340 0.99660 2
222 1.50 1.50 1.00 1.75 0.00316 0.99684 2
223 1.50 1.75 1.75 2.25 0.37654 0.62346 2
224 1.75 1.50 1.25 1.50 0.00740 0.99260 2
225 1.00 1.50 1.50 1.75 0.22707 0.77293 2
226 1.75 2.00 1.00 1.75 0.00780 0.99220 2
227 2.00 1.50 1.50 2.00 0.01390 0.98610 2
228 1.25 1.50 1.00 1.75 0.00660 0.99340 2
229 1.50 1.50 1.25 1.75 0.01433 0.98567 2
230 1.75 1.75 1.25 1.50 0.01673 0.98327 2
231 1.25 1.50 1.00 1.75 0.00660 0.99340 2
232 1.50 1.00 1.50 1.25 0.01461 0.98539 2
233 1.50 1.50 1.00 1.50 0.00340 0.99660 2
234 2.00 1.75 1.75 2.25 0.12065 0.87935 2
235 1.50 1.75 1.00 1.75 0.00718 0.99282 2
236 2.25 1.75 1.00 1.75 0.00078 0.99922 2
237 1.50 1.50 1.00 1.50 0.00340 0.99660 2
238 1.50 1.75 1.00 1.25 0.00830 0.99170 2
239 2.25 1.75 1.75 3.00 0.04990 0.95010 2
240 1.50 1.50 1.75 1.00 0.27610 0.72390 2
241 2.00 1.75 1.50 2.00 0.03116 0.96884 2
242 1.25 1.50 1.25 1.25 0.03411 0.96589 2
243 1.50 1.00 1.25 1.50 0.00299 0.99701 2
244 1.50 1.50 1.25 1.50 0.01540 0.98460 2
245 1.75 1.25 1.00 1.75 0.00066 0.99934 2
246 1.75 1.00 1.50 1.50 0.00653 0.99347 2
247 2.00 1.50 1.00 1.50 0.00077 0.99923 2
248 1.50 1.75 1.50 1.25 0.14987 0.85013 2
249 1.75 1.75 1.25 1.50 0.01673 0.98327 2
250 1.50 1.50 1.25 1.50 0.01540 0.98460 2
Discriminant Analysis on the Performance of Academic Scholars and
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OUTPUT CLASSIFICATION RESULTS OF TEST DATA
OBS SS ENG MATH FIL 1 2 INTO
251 1.50 1.50 1.25 2.25 0.01241 0.98759 2
252 1.50 1.50 1.50 1.75 0.06256 0.93744 2
253 1.75 1.50 1.25 2.25 0.00595 0.99405 2
254 1.50 1.75 1.25 1.50 0.03447 0.96553 2
255 1.25 1.25 1.50 1.75 0.05781 0.94219 2
256 1.25 1.50 1.00 1.50 0.00710 0.99290 2
257 2.00 1.75 1.50 2.00 0.03116 0.96884 2
258 1.75 1.00 1.25 1.75 0.00133 0.99867 2
259 1.25 1.50 1.00 1.25 0.00764 0.99236 2
260 1.50 1.00 1.50 2.00 0.01177 0.98823 2
261 2.00 1.50 1.50 2.25 0.01293 0.98707 2
262 2.00 1.75 2.00 2.00 0.40386 0.59614 2
263 1.25 1.25 1.75 1.25 0.24581 0.75419 2
264 1.75 1.75 1.50 2.00 0.06321 0.93679 2
265 1.50 1.25 1.25 2.00 0.00589 0.99411 2
266 1.75 1.25 1.00 1.50 0.00071 0.99929 2
267 2.25 1.50 1.75 2.00 0.02991 0.97009 2
268 1.75 2.00 2.50 2.50 0.98334 0.01666 1
269 2.00 1.25 1.25 1.25 0.00167 0.99833 2
270 1.75 1.50 1.00 1.50 0.00162 0.99838 2
271 2.25 1.75 1.75 2.75 0.05348 0.94652 2
272 1.50 1.75 1.75 1.25 0.44723 0.55277 2
273 2.00 1.75 1.50 3.00 0.02344 0.97656 2
274 1.75 1.25 1.50 1.75 0.01375 0.98625 2
275 2.00 1.75 1.25 2.00 0.00696 0.99304 2
276 1.75 1.75 1.50 2.00 0.06321 0.93679 2
277 2.00 1.50 1.25 2.50 0.00265 0.99735 2
278 2.00 1.75 2.00 1.50 0.43950 0.56050 2
279 1.75 1.00 1.25 1.50 0.00143 0.99857 2
280 1.25 1.50 1.25 1.50 0.03178 0.96822 2
281 1.50 1.50 1.00 2.00 0.00294 0.99706 2
282 1.50 1.50 1.50 2.25 0.05452 0.94548 2
283 1.50 1.25 1.25 2.25 0.00547 0.99453 2
284 1.50 2.00 1.25 2.00 0.06577 0.93423 2
285 1.25 1.50 1.50 1.75 0.12282 0.87718 2
286 2.00 1.50 1.50 1.75 0.01493 0.98507 2
287 1.25 1.50 1.50 1.25 0.13946 0.86054 2
Discriminant Analysis on the Performance of Academic Scholars and
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APPENDIX C.
Discriminant Analysis of Scholars and Non-Scholars
Classification of Test Data
Discriminant Analysis
287 Observations 286 DF Total
4 Variables 285 DF Within Classes
2 Classes 1 DF Between Classes
Class Level Information
Output Prior
GRP SAS Name Frequency Weight Proportion Probability
1 _1 200 200.0000 0.696864 0.500000
2 _2 87 87.0000 0.303136 0.500000
Discriminant Analysis Pooled Covariance Matrix Information
Covariance Natural Log of the Determinant
Matrix Rank of the Covariance Matrix
4 -6.6095354
Discriminant Analysis Pairwise Generalized Squared Distances Between
Groups
2 _ _ -1 _ _
D (i|j) = (X - X )' COV (X - X )
i j i j
Generalized Squared Distance to GRP
From GRP 1 2
1 0 7.80453
2 7.80453 0
Discriminant Analysis Linear Discriminant Function
_ -1 _ -1 _
Constant = -.5 X' COV X Coefficient Vector = COV X
j j j
GRP
1 2
CONSTANT -22.35626 -10.51363
SS -2.90547 0.05859
ENG 7.28604 3.98555
MATH 9.43072 3.33555
FIL 5.38761 5.68003
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Discriminant Analysis Classification Summary for
Calibration Data: WORK.GRADE
Resubstitution Summary using Linear Discriminant Function
Generalized Squared Distance Posterior Probability of Membership in each
Function:
GRP:
2 _ -1 _ 2 2
D (X) = (X-X )' COV (X-X ) Pr(j|X) = exp(-.5 D (X)) / SUM exp(-.5 D (X))
j j j j k k
Number of Observations and Percent Classified into GRP:
From GRP 1 2 Total
1 173 27 200
86.50 13.50 100.00
2 2 85 87
2.30 97.70 100.00
Total 175 112 287
Percent 60.98 39.02 100.00
Priors 0.5000 0.5000
Error Count Estimates for GRP:
1 2 Total
Rate 0.1350 0.0230 0.0790
Priors 0.5000 0.5000
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APPENDIX D.
TWO-GROUP DISCRIMINANT ANALYSIS
Group Statistics
Valid N (listwise)
GROUP
Mean
Std. Deviation
Unweighted
Weighted
1.00
SS
2.1550
.6671
200
200.000
ENG
2.2700
.5175
200
200.000
MATH
2.4987
.5272
200
200.000
FIL
2.0175
.5096
200
200.000
2.00
SS
1.6839
.3200
87
87.000
ENG
1.5201
.2326
87
87.000
MATH
1.3851
.3093
87
87.000
FIL
1.8046
.4201
87
87.000
Total
SS
2.0122
.6225
287
287.000
ENG
2.0427
.5673
287
287.000
MATH
2.1611
.6965
287
287.000
FIL
1.9530
.4934
287
287.000
Tests of Equality of Group Means
Wilks'
Lambda
F
df1
df2
Sig.
SS
.879
39.379
1
285
.000
ENG
.630
167.675
1
285
.000
MATH
.458
337.296
1
285
.000
FIL
.961
11.713
1
285
.001
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54
Analysis 1
Box's Test of Equality of Covariance Matrices
Log Determinants
Log
GROUP
Rank
Determinant
1.00
4
-5.862
2.00
4
-9.877
Pooled within-groups
4
-6.610
The ranks and natural logarithms of determinants
printed are those of the group covariance matrices.
Test Results
Box's M
132.136
F
Approx.
12.964
df1
10
df2
134083.8
Sig.
.000
Tests null hypothesis of equal population covariance matrices.
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55
Summary of Canonical Discriminant Functions
Eigenvalues
Canonical
Function Eigenvalue % of Variance Cumulative % Correlation
1
1.660a
100.0
100.0
.790
First 1 canonical discriminant functions were used in the
a.
analysis.
Wilks' Lambda
Wilks'
Test of Function(s) Lambda
Chi-square
df
Sig.
1
.376
276.891
4
.000
Standardized Canonical Discriminant Function Coefficients
Function
1
SS
-.620
ENG
.533
MATH
1.030
FIL
-.051
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56
Structure Matrix
Function
1
MATH
.844
ENG
.595
SS
.288
FIL
.157
Pooled within-groups correlations between discriminating
variables and standardized canonical discriminant functions
Variables ordered by absolute size of correlation within function.
Functions at Group Centroids
Function
GROUP
1
1.00
.847
2.00
-1.947
Unstandardized canonical discriminant
functions evaluated at group means
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57
Classification Statistics
Classification Processing Summary
Processed
287
Excluded
Missing or out-of-range
0
group codes
At least one missing
0
discriminating variable
Used in Output
287
Prior Probabilities for Groups
Cases Used in Analysis
GROUP
Prior
Unweighted
Weighted
1.00
.500
200
200.000
2.00
.500
87
87.000
Total
1.000
287
287.000
Classification Function Coefficients
GROUP
1.00
2.00
SS
-2.905
5.859E-02
ENG
7.286
3.986
MATH
9.431
3.336
FIL
5.388
5.680
(Constant)
-23.049
-11.207
Fisher's linear discriminant functions
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58
APPENDIX E.
COMPUTATIONS
Z- test:
Z
= (258-273)√287
√273 (287-273)
= 254.1
61.82
= -4.11
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
ASTODILLO, LIEZYL T., BUASEN, PAMELA P. and DAO-ANIS,
JOSEPHINE A. March. 2009. Discriminant Analysis on the Performance of
Academic Scholars and Non-Scholars in Benguet State University. Benguet State
University, La Trinidad Benguet
Adviser:
DR. SALVACION Z. BELIGAN
ABSTRACT
This study was conducted to determine the discriminant functions that will
clearly separate the academic scholars and non-academic scholars. Specifically, the
study aimed to: 1) determine the subjects that highly discriminate the academic
scholars and non-academic scholars; 2) estimate the chance of misclassification
given that the derived discriminant function is used as a classification tool; and 3)
identify the non-academic scholars who will be considered academic scholars based
on discriminant analysis.
The variables observed from the 200 non-scholars and 87 scholars were
final grades in Social Science 11, English 11, Math 11 and Filipino 11. The data on
the aforementioned variables were obtained from the Dean’s Office of College of
Arts and Sciences at Benguet State University and analyzed using the SAS
Discriminant Analysis Procedure.
The results showed that Math, English and Filipino 11 grades have high
discriminating powers in differentiating academic scholars and non-scholars. From
the derived discriminant functions, 86.5 percent of the non-scholars and 97.7
percent of the academic scholars were correctly classified. Thus, there was 13.5
percent and 2.3 percent misclassification in the grouping of non-scholars and
academic scholars respectively into its correct classification.
Out of 200 non-academic scholars, 27 could be considered academic
scholars by discriminant analysis.
ii
TABLE OF CONTENTS
Pages
Bibliography ……………………………………………………..........
i
Abstract
……………………………………………………………..
i
Table of Contents
……………………………………………………..
iii
INTRODUCTION
Background of the Study
……………………………………..
1
Statement of the Problem
……………………………………..
2
Objectives of the Study
……………………………………..
3
Significance of the Study …..…..……………………………..
3
Scope and Delimitation of the Study
……………………..
4
REVIEW OF RELATED LITERATURE
Scholarship ……………………………………………………..
5
Studies on Academic Performance ……………………………..
7
Application of Discriminant Analysis …………………...…..
8
Theoretical
Framework …………………………………….. 15
Definition of Terms
……………………………………..
21
iii
METHODOLOGY
Respondents of the Study
……………………………..
23
Sampling
Method
…………………………………………….. 23
Data
Collection …………………………………………….. 22
Statistical
Analysis
…………………………………….. 24
RESULTS AND DISCUSSION
Summary
Statistics
…………………………………….. 26
The Two Group Discriminant Function
……………………..
27
Discriminant Power of Academic Subject
in the Derived Discriminant Function
……….……………. 28
Classification into Group Membership
……………………..
30
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
Summary
…………………………………………………….. 33
Conclusion
…………………………………………………….. 33
Recommendations ……………………………………………..
34
LITERATURE CITED
……………………………………………..
35
APPENDICES
A. Classification Results using Linear Discriminant Function:
Posterior Probability of Membership in Group ……………..
38
B. Output Classification Results of Test Data
……………..
45
C. Discriminant Analysis on Academic Scholars and Non-Scholars:
Classification of Data(SAS)
……………………………..
51
iv
D. Results of Discriminant Analysis (SPSS)
……………..
53
E. Computations …………………………………………….. 58
F. Letter of Permission to the CAS Dean to Gather the Grades
of the Respondents
……………………………………..
59
G. Letter of Permission to the SSGU-OSA to Gather the Grades
of Academic Scholars
………………….………………….
60
H. Application for Oral Defense
...……………………………
61
I. Biographical Sketch
.……………………………………..
62
v
INTRODUCTION
Background of the Study
Institutions of higher learning are engaged in a sustained and continuous
process of their students so as to enhance their readiness for the job market and
further education. Thus, it is important for educational institutions to focus on
improving the aspects of teaching and learning.
One of the schemes in improving the aspect of learning is providing
scholarships to students. In most schools, a unit responsible for providing these
scholarships is usually identified.
At Benguet State University, the Student Scholarship and Grant Unit
(SSGU) is being recognized to administer undergraduate scholarships and awards.
Some of these scholarships are based on leadership abilities, skills and talents, but
mostly, based on academic performance.
The individual grades are used as academic performance indicators. Every
scholar has to maintain a minimum grade requirement set by the school or
university. Some scholarship granting body, an average grade of 75 or 3.0 or higher
is considered as the cut-off point to become a scholar. However, for academic
achiever scholarship award, an average grade of 1.75 or better must be maintained.
To become an academic achiever awardee, he has to prove that he has the
capability to excel in all the subject areas he is enrolled in for a given semester.
Discriminant Analysis on the Performance of Academic Scholars and
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Based on the university code of Benguet State University, academic achievers must
have an average grade of 1.75 or better with no grade lower than 3.0 in any subject
and no dropped subject(s). Based on this policy code, the researchers of this study
are challenged to determine if an individual who incurred lower than 3.0 because of
some unavoidable circumstances cannot be an academic achiever.
To mitigate the problem of deleting or not an academic scholar from the list
of scholars because of an incurred grade lower than 3.0 in one of the subjects
enrolled through the average is still within the set range, a discriminant analysis
may give solutions to the problem.
Discriminant analysis linearly combines discriminating variables to develop
the basis for assigning an individual of unknown origin to one of the distinct
groups. It is a method of studying group differences on several variables
simultaneously.
Statement of the Problem
This study sought to answer the following problems:
1. Which of the subjects or variables highly discriminate the academic
scholars from the non-academic scholars?
2. What would be the chance of misclassification given that the derived
estimated discriminant function is used as a classification tool?
Discriminant Analysis on the Performance of Academic Scholars and
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3
3. Who are those non-scholars who can be classified as academic scholars
based on discriminant analysis?
Objectives of the Study
This study aimed to discriminate academic scholars and non-scholars based
on their performance using the grades in the basic subjects.
Specifically, the study aimed to:
1. determine which of the subjects or variables highly discriminate
academic scholars and non-scholars;
2. estimate the chance of misclassification given that the derived estimated
discriminant function is used as a classification tool.
3. identify the non-academic scholars who will be considered academic
scholars based in discriminant analysis.
Significance of the Study
Results of this study can be utilized in future researches, by teachers and
administrators, students – scholars and non-scholars, and even by parents.
The identification of variables that may have a contribution on the academic
performance of the students could be of great help to teachers and administrators.
They would have a better insight on how to motivate their students.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
4
The results could also help the students to realize and reflect on their study
habits and learning weaknesses, thus, they would be encouraged to find ways to
improve their performance.
The findings could also convince the parents to cooperate and coordinate
with the school in guiding their children on their studies. These could motivate
them to devote more time, attention and guidance needed by their children in order
for their children to improve their performance in school.
Result of this study will serve as guide for administrators and other
sponsoring agencies to revise scholarships’ rules and regulations and for further
improvement in the management of scholarships.
Scope and Delimitation
This study was limited to the academic scholars and non-scholars at
Benguet State University enrolled during the first semester of School Year 2008-
2009. Only the first year and first semester grades for the general education courses
namely: English 11, Filipino 11, Social Science 11 and Math 11 were gathered for
this study.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
REVIEW OF LITERATURE
Scholarship
A scholarship is an award of access to an institution, or a financial aid
award for an individual student called a scholar, for the purpose of furthering their
education. Scholarships are awarded based on a range of criteria which usually
reflect the values and purposes of the donor or founder of the award.
Scholarships may be classified into the following primary groups:
Academic:
A scholarship which is purely based on the academic
performance of the students. Percentage discounts are awarded to the scholars
depending on their grades and their grade point average. This scholarship may be
classified to University or College level.
Merit: A financial aid for which financial need is not used to determine the
recipient. The recipient may be determined by students’ athletic, academic, artistic
or other abilities. The actual monetary value of the scholarship may be negligible,
the scholarship being meant to motivate the student and promote the study of the
subject.
Need: A financial aid for which the student and family’s financial situation
is a primary factor in determining the recipient. Usually, such scholarship will
cover all or part of the tuition and may even cover living costs.
Discriminant Analysis on the Performance of Academic Scholars and
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6
Sociology: A financial aid where applicants must initially qualify by race,
religion, or national origin. After filtering the applicants based on their ethnicity,
additional factors are taken into consideration to determine the final and deserving
recipients.
Institutional: These are scholarships awarded by a specific college or
university (institution) to a student planning to attend that institution.
General: These are other scholarships which are awarded for a variety of
reasons that do not fall into one of the above categories. These may be for reasons
of the student's association with the objectives of the sponsoring organization or
affiliations.
Some scholarships have a "bond" requirement. Recipients may be required
to work for a particular employer for a specified period of time or to work in rural
or remote areas; otherwise they may be required to repay the value of the support
they received from the scholarship.
It is also typical for persons to find scholarships in their home region.
Information on these can be found by asking local persons and organizations.
Typically, these are less competitive as the eligible population is smaller:
sponsored by Non-profit Organizations or Charitable Trusts, Community
Foundations, Foundations, Labor Unions, Houses of Worship, Chamber of
Commerce and other Volunteer Organizations.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
7
Academic Performance
Several researches have been made regarding the academic performance of
students, and though many significant findings have been made, it still remains as a
subject of great interest for researchers.
The study of Agustin (2002) aimed to identify the factors affecting the
academic performance of Grade I pupils at Lucban Elementary School. As a result,
six significant factors were extracted, namely: educational and financial support,
sibling number and order, health and geographical location, sex and technology
exposure, parents’ concern, age and guardian. Using regression analysis, it was
found out that the six factors mentioned have only a very small contribution to the
academic performance of the pupils. The R2 indicates only a 4.9% of the pupils’
variations on performance can be explained by the extracted factors.
Milo (2003) conducted a study on determining the indicators of students’
performances and the effects on the performance in school. Factors seen were the
school, facilities, teacher, high school performance, gender and age. It was found
out that these factors contribute highly on the academic performance of students
using regression analysis. The R2 value indicates that 68.6% of the students were
explained by these factors.
Another study conducted by Candiao (2002) determined factors affecting
the academic performance of the SFAO Grantees of BSU under four subjects:
Biology, Communication Arts, Chemistry and Mathematics. Using factor analysis,
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
8
the study revealed two factors having significant relationship in Biology with an R2
of 34.2% and in Communication Arts with 7.7% - father’s educational attainment
and health problems; three factors in Chemistry with an R2 of 49% - average family
income, organization affiliation and serious relationship with someone; and only
the provincial factor was found to have a significant relationship in the performance
in Mathematics with R2 of 11.9%.
Application of Discriminant Analysis
Discriminant function analysis is used to classify cases into the values of a
categorical dependent, usually a dichotomy. If discriminant function analysis is
effective for a set of data, the classification table of correct and incorrect estimates
will yield a high percentage correct. This technique has been applied widely in the
field of education, biological and medical sciences. Here are some of its
application:
Cochran (1964) investigated the problem of estimating the discriminating
power of the linear discriminant function from knowledge of the discriminating
powers of the individual variables includes in the linear discriminant function. His
examination suggested that in practice, (a) most correlation is positive; (b) it is
usually safe to exclude from a discriminant before computing it, a group of
variables whose individual discriminatory power are poor except for any such
variable that has negative correlation with most of the good discriminators; (c) the
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
9
performance of the discriminant function can be predicted satisfactorily from a
knowledge of individual powers plus the average correlation coefficient.
Cornelio and Dacus (2007) employed stepwise discriminant analysis in
identifying the variables that discriminate between the sophomore student of BSAS
as below average and above average using the students’ high school grades (Math,
Science, English, General Weighted Average), and their freshmen college grades
(Math 11. Statistics 11, Biology, Chemistry, Physics, English 11). The results
revealed that grades in high school English and Statistics 11 had the highest
discriminating power in classifying the students into below average and above
average.
Bodong (2001), applying the discriminant analysis, was able to classify a
qualified from non-qualified freshmen applicants in Benguet State University. The
variables utilized were the fourth year weighted average, high school Math, Science
and English grades, and IQ scores. Results showed that the IQ score and English
grade in high school gave clear separation between admitted and not admitted
freshmen applicants.
Antiyag and Tognaon (2006) used discriminant analysis to show the
performance indicators of Benguet State University student achievers and non-
achievers to select the ‘best’ from among possible discriminating variables that will
give clear separation between achiever and non-achiever, and estimate the chance
of misclassification tools. The study was therefore concluded that the Physics and
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
10
English grades are the most important variables to consider in the selection of
achievers and non-achievers. The results also indicated that classification of the
subject into group membership is important.
Benito and Cabasoy (2006) applied stepwise discriminant analysis on the
body measurement of graduating student, in determining the set of body
measurement of young male and female adults that allows for the best
discrimination between sizes (small, medium, large) and the accuracy of predicting
cases classified into correct classification. They concluded that the neck, hip, waists
and length of arms are the variables with high discriminating power in the
separation between sizes.
Domiles and Tamayo (2005) also used discriminant analysis in classifying
admitted and not admitted high school freshmen applicants at Benguet State
University-Secondary Laboratory School, and in classifying the admitted high
school freshmen applicants with regard to what section (Science, Vo-Ag and HE)
they will be placed. They concluded that IQ scores and aptitude scores are the most
important variables to be considered in the selection of high school freshmen
applicants. Also, general point average and IQ scores are the most important
variables to be considered in the classification of section.
Inciong (2001) used discriminant analysis to classify the nutritional status
(overweight, normal, underweight) of pre-school age at La Trinidad, Benguet. The
variables considered were the weight, sex, age, date first seen (months), number of
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
11
brothers and sisters, barangay where they belong, parents’ educational attainment
and occupation. It was concluded that the weight of the child has the greatest
contribution in determining the nutritional status of the selected pre-school age
children. The researcher relates this study to find similarities especially in the tool
applied through the variables considered are different.
Castillo (2003), as cited by Cornelio and Dacus (2007), employed
discriminant analysis in classifying food security in selected indicators in certain
regions in the Philippines. A total of 65 variables were considered describing the
1,200 households from the National Capital Region, Region IV and Region VII
with regard to their food consumption, energy and nutrient intake, and other socio-
economic characteristics. These three regions were selected to represent high,
middle and low income regions. Results showed that the three regions with three
different economic conditions have different food security indicators. Household,
thus, be classified according to level of food security in ways unique to each
region.
The business environment of Korean housing industry has changed recently
from a supplier’s market to a buyer’s. Kim, J. (2000) attempted to offer a
characteristics profile and a forecasting model classifying the housing purchase
consumers into three groups: a single-family housing purchase group, an apartment
housing purchase group, and a non-purchase group. These groups were classified
by employing discriminant analysis and were predicted using the discriminant
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
12
function: a linear combination of demographic, socio-economic and residential
characteristics.
Another application of discriminant analysis is the study of Coleman and
Taylor (1996) entitled, “Determination of a Discriminant Function as a Prediction
Model for Effectiveness of Speed Zoning in Urban Areas”. Speed zoning is the
application of a different speed limit to a section of roadway than is applicable to
adjoining highway segments. Speed zoning traditionally has been based on one of
two similar procedures, one relying primarily on the 85th percentile speed and
engineering judgment and the second, which includes 85th percentile speed, with
some form of quantification of environmental and geometric variables to reduce the
speed limit below the 85th percentile speed. Three problems exist with the current
practice of establishing speed zones. First is that traffic engineers have no way of
predicting if their speed zoning actions will result in better compliance with the
speed limit. Second it is unclear whether the section will make the driving
environment safer resulting in fewer accidents. The third problem is that where
states have procedures which quantify and use environmental and geometric
variables, the empirical basis for their exclusion or inclusion has not been
validated. Coleman and Taylor utilized discriminant analysis to determine if a
quantifiable relationship between accident parameters, speed parameters, roadside
friction, and environmental/geometric variables can be used to predict the
effectiveness of proposed speed zoning actions. The findings are that the most
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
13
significant variables identified by the discriminant functions for effective zones
were: skewness index (negatively skewed), signalization, driveway frequency, and
85th percentile speed.
As cited by Bellovary (2000), one of the most well-known bankruptcy
prediction models was developed by Altman using discriminant analysis. Thus,
Bellovary summarized and analyzed existing research on bankruptcy prediction
studies in order to facilitate more productive future researches in this area.
Moreover, analysis of accuracy of the models suggests that multivariate
discriminant analysis and neural networks are the most promising methods for
bankruptcy prediction models.
Discriminant analysis applied to SAR studies using topological descriptors
allowed Galvez (1996) to plot frequency distribution diagrams: a function of the
number of drugs within an interval of values of discriminant function against these
values. Galvez used these representations, pharmacological distribution diagrams
(PDDs), in structurally heterogeneous groups where generally they adopt skewed
Gaussian shapes or present several maxima. The maxima afford intervals of
discriminant function in which existed a good expectancy to find new active drugs.
A set of beta-blockers with contrasted activity had been selected to test the ability
of PDDs as a visualizing technique, for the identification of new beta-blocker
active compounds.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
14
Alcohol dependence often cannot be diagnosed based on self-report alone.
Various biochemical and haematological parameters to screen alcohol use
disorders. Hence, Vaswani and Rao (2005) a study to develop discriminant
equations based on lipid and liver measures independently for identifying alcohol
dependent and non-dependent subjects. One hundred subjects fulfilling the criteria
of alcohol dependence and seventy healthy controls were included. The socio-
demographic details, caloric intake, height, weight and blood pressure were
recorded, and samples were analyzed for various lipid measures as well as liver
function using diagnostic values such as sensitivity, specificity, positive predictive
value (PV+), negative predictive value (PV-), and discriminant analysis. Two
equations were constructed based on liver and lipid measures independently. 84.7%
of the subjects on the basis of total cholesterol (TC), apolipoprotein B (ApoB) and
low density lipoprotein-cholesterol (LDL?HDL-c) and 89.1% on the basis of
aspartate amino transferase (AST) and gamma glutamyl transferase (GGT) were
correctly classified into their respective groups.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
THEORETICAL FRAMEWORK
Discriminant Analysis
Discriminant analysis is the appropriate statistical technique when the
dependent variable is categorical (nominal or non-metric) and the independent
variables are metric. It is capable of handling either two or multiple groups. When
three or more classifications are involved, the technique is referred to as multiple
discriminant analysis (MDA).
Discriminant analysis involves deriving the linear combination of the two or
more independent variables that will discriminate best between the prior defined
groups. This is achieved by the statistical decision rule of maximizing the between-
group variance relative to the within-group variance; this relationship is expressed
as the ratio of between-group to within-group variance.
Discriminant analysis is appropriate for testing the hypothesis that the group
means of the two or more groups are equal. It multiplies each independent variable
by its corresponding weight and adds these products together.
The discriminant analysis procedure starts from the assignment of the
individual observation to its prior grouping by letting C1 denotes the first
population, C2 the second population, and X = (x1, x2, …, xp) be column vector of
observations in a full set of p measurement that has a multi-normal distribution
with mean μc in the Cth group and common positive covariance matrix Σ.
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The linear discriminant function based on q of the p variables for
discriminating the groups is given by:
DF = a1X1 + a2X2 + … + apXp = a’X
(1)
where: DF = discriminant
score
ai = discriminant
weight
for independent variable
X
=
independent
variable
Such that, the vector of coefficients maximizes a’Σa, where Σ is the
common covariance matrix of the groups and is a vector coefficient which indicates
the contribution of the variables to differentiation along the function. The signs of
Σ are important because it indicates whether the variables are making positive or
negative contribution.
Measure of the Discriminatory Power
of Variables and Functions
The discriminatory power of a discriminant function refers to the distance
of the group centroids relative to the amount of the dispersion within the groups
based on the given variables. If the magnitude is too small, then it is meaningless to
use the variable to discriminate between groups. Wilk’s lambda (λ) statistic is a
generalized likelihood criterion used in determining the variable that contributes
most to the discriminatory power of a model which is a linear combination of p
variables. The formula is given by:
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
17
q
λ
=
∏
1
(2)
i = k+1 (1 + еi)
where: е i
=
eigenvalue of the ith model
k
=
number of derived functions
If
λ approaches 0, this indicates that the two groups are well separated and
if λ is close to 1, this indicates that no group differences exist. This criterion is
based on the overall multivariate F-ratio for the test of differences between the
group centroids. The variable which maximizes the F-ratio also minimizes λ, a
measure of group discrimination. The significance of λ is tested using the formula:
1
–
λ½
=
ms – p (k-1) / 2 + 1
(3)
λ½
p (k-1)
where:
m
=
N – 1 (p + k) / 2
s
=
√ p2 (k-1)2 – 4
√ p2 + (k-1)2 – 5
This is found to be distributed as F with p (k-1) numerator degrees of
freedom and ms – p (k+1) / 2 + 1 denominator degrees of freedom.
Another statistic that can be used to judge the substantive utility of a
discriminant function is the canonical coefficient ri * defined as:
ri * = √ еi / (1 + еi) (4)
where: еi
=
eigenvalue of the ith model
Discriminant Analysis on the Performance of Academic Scholars and
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This coefficient is a measure of association which summarizes the degrees
of relatedness between the group and the discriminant function. A value of zero
denotes no relationship at all, and a value close to 1 represents a high degree of
association. The square of the canonical coefficient is the average squared
canonical coefficient (ASCC). This ASCC is the proportion of variation in the
discriminant function explained by the variable.
Case Classification and Misclassification Probabilities
Classification can be achieved through a series of classification functions,
one for each group. The equation for one group would be of the form:
Ci = Ci0 + Σ CijXj = Ci0 + Ci0 + Ci1X1 + Ci2X2 + … + CipXp
(5)
where:
Ci is the classification score for the group I,
Cij’s are the classification coefficients
Xj’s are the raw scores of the discriminating variables.
Problem of Classification
For two groups situation, let Xt = (x1, x2…Xp) denotes the vector of
measurements which are the basis for classifying an individual into one of two
groups C1 and C2. According to Fisher (1936), the p-multivariate variables need to
be transformed into a multivariate variable by finding the linear combination of the
X’s which maximally discriminates the group.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
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Let
X1 and X2 denotes the vectors of means for the subjects on the p
variables in group 1 and 2. The location of group 1 on the discriminant function is
then given by Y1 = at x1 and the location of group 2 by y2 = at x2. The midpoint
between the two groups on the discriminant function is the given by:
M
=
½
(y1 + y2) = (½)
(x1 – x2) t S-1 (x1 – x2) (6)
If we let Zi denote the score for the ith subject on the discriminant function,
then the decision rule is as follows:
If
Zi ≥ m, then classify subject in group 1;
If
Zi ≤ m, then classify subject in group 2;
On the classification probability, let f1(x) and f2(x) be the probability
distribution function associated with the p x 1 random vector X for the populations
C1 and C2, respectively. Let Ω be the sample space for x. Let R1 be the set of X
values for which we classify objects as G1 in R2 = Ω – R1 be the remaining values
for which we classify objects to G2. The conditional probability, p (2/1), of
classifying an object in G2 when in fact, it is from G1 is
P (2/1)
=
p (X ε R2 G1) = ∫ R2 = Ω - R1 f1(x) dx
P (1/2)
=
p (X ε R1 G2) = ∫ R2 f1(x) dx
Let
P1 be the prior probability of C1 and P2 be the prior probability of C2;
thus, P1 + P2 = 1. The following are the classification probabilities for X:
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
20
P (correct classifying as C1) = P
(C1/C1) * P1
P (misclassifying as C1) =
P
(C1/C2) * P2
P (correct classifying as C2) = P
(C2/C2) * P2
P (misclassifying as C2) =
P
(C2/C1) * P1
The Huberty one sided Z –test was used to determine the hit-ratio or how
good the discriminant function to correct new cases in the group. The formula is
given as:
Z
=
( O – E ) √ N
(7)
√[ E ( N – E ) ]
where:
O
=
actual frequency of the individuals belonging to the
correct group
E
=
expected frequency of the individuals that should
belong to the group
N = sample
size
Discriminant Analysis on the Performance of Academic Scholars and
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Definition of Terms
Academic Performance. It is the grade point average or the general
weighted average (GWA) of the students at the end of the semester or school year
involving the grades in all the subjects enrolled in.
Centroid. This is the mean value of the discriminant Z-scores of a particular
category or group.
Criterion Variable. This is the dependent variable, also called the grouping
variable. It is the object of classification efforts.
Discriminant Analysis. This is a technique for distinguishing or classifying
observations into groups. It provides sorting procedures into previously chosen
variables, and reveals which combinations of variables discriminate among the
groups.
Discriminant Function. The model of equation formed using discriminant
analysis usually in the form of Y = C1X1 + C2X2 + … + CpXp, where: Y represents
the dependent categorical variable, the C’s are the discriminant weight, and the X’s
are the independent variables.
Discriminant Score. It is the value resulting from applying a discriminant
function formula to the data for a given case.
Discriminating Variables. These are the independent variables, also called
predictors.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
22
Eigenvalue. The characteristic root of each discriminant function reflects
the ratio of importance of the dimensions which classify cases of the dependent
variable.
General Weighted Average. This is the grade point average or the weighted
mean grade of all grade points a student earned by enrolment.
Hit Rate. It is the percentage of group cases correctly classified.
Misclassification Probability. This is the probability of given individual to
be classified in the incorrect group.
Non-Scholars. These are students who are unable to acquire financial aid or
scholarship.
Scholars. These are individuals or students who are awarded scholarship or
financial aid to further their studies.
Scholarship. It is an award usually based on academic achievement,
community involvement, or similar factors. It may be awarded regardless of
financial need. Some scholarships must be applied for, while others are awarded
automatically.
Test. This refers to a systematic procedure for observing a behavior or
describing it with the aid of a numerical scale or category system.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
METHODOLOGY
Respondents of the Study
The respondents of the study consisted of a total of 287 freshmen students
of Benguet State University. Out of the 287 students, 87 are academic scholars and
200 are non-academic scholars.
Sampling Method
The selection of the 87 academic scholars was done by total sampling while
the selection of the 200 non-academic scholars was made using simple random
sampling. This means that each member of the target population has an equal
chance of being included in the sample.
Data Collection
The grades of the non-academic scholars on the following subjects:
English11, Filipino11, Social Science11 and Mathematics11, were obtained from
the College of Arts and Sciences Dean’s Office while the grades of the academic
scholars on the aforementioned general education subjects were obtained at the
Students Scholarship Grant Unit-Office of Student Affairs, Benguet State
University.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
24
Statistical Analysis
The respondents were first classified as academic scholars and non-
academic scholars. The gathered data on the four general education courses were
then subjected for discriminant analysis.
Discriminant analysis usually identifies which subject or variable
contributes most to the discrimination between the scholars and non-scholars, and
also to determine the chances of all the elements to be in the correct classification.
Discriminant analysis works by creating a new variable, which is a
combination of the original variables. This is known as the Discriminant Function
with the formula given as:
DF = W1X1 + W2X2 + … + WpXp.
(DF – score, W – weight, X – independent variable)
The means and standard deviation of the variables under each group were
first obtained. To generate the discriminant function and to estimate the chance of
correct classification, discriminant analysis using the SAS program was employed.
Statistical Packages for Social Sciences or SPSS was also utilized in
obtaining the Covariance Matrices and the Canonical Discriminant Functions.
The flow of the analysis is shown in Figure 1.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
25
1. THE PROBLEM
2. THE DESIGN
- classification of observations
- identification of dependent and
into scholars and non-scholars
independent variables
- identification of sample size
4. THE DISCRIMINANT FUNCTION
3. THE ASSUMPTIONS
- identification of discriminant variable
- normality of the variables
- implication of the discriminant
- linearity of relationships
functions
- equal dispersion of matrices
5. EVALUATION OF FUNCTIONS
- discriminant weights
6. VALIDATION OF RESULT
- Wilk’s Lambda
- Posterior Probability of
- canonical correlations
Classification Cases
- graphical display of the centroids
8. PREDICTIVE ACCURACY
7. CLASSIFICATION MATRICES
- assessing of the significance
- criterion for hit ratio
of accuracy
Figure 1. Flow chart for discriminant analysis
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
RESULTS AND DISCUSSION
Summary Statistics
Table 1 shows the grade point average of the scholars and non-scholars in
the four subject areas under test. Mathematics had the highest mean of 2.50 with a
standard deviation of 0.53 while Filipino had the lowest mean and standard
deviation of 2.02 and 0.51 respectively. For the scholars, Filipino had the highest
mean of 1.80 with standard deviation of 0.42 and Mathematics had the lowest mean
of 1.39 with 0.31 standard deviation.
Generally, Mathematics had the highest mean of 1.95 while Filipino with
the mean of 1.91 was the lowest. For the overall mean, non-scholars had higher
mean of 2.25 while the scholars had lower mean of 1.60.
Table 1. Grade point average of scholars and non-scholars
NON-SCHOLAR SCHOLAR
TOTAL
SUBJECT
Standard
Standard
Mean
Mean
Mean
Deviation
Deviation
SS 2.16
0.67
1.68
0.32
1.92
ENG 2.32
0.90
1.52
0.23
1.92
MATH 2.50
0.53
1.39
0.31
1.95
FIL 2.02
0.51
1.80
0.42
1.91
Overall Mean
2.25 1.60 1.93
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
27
The Two Group Discriminant Function
The derived discriminant functions for the non-scholars and academic
scholars in Benguet State University are shown as equation 1 and equation 2,
respectively.
Y1
=
-22.36 – 2.91SS + 7.29Eng + 9.43Math + 5.39Fil
(1)
Y2
=
-10.51 – 0.06SS + 3.99Eng + 3.34Math + 5.68Fil
(2)
The constant values of -22.36 and -10.51 in equation 1 and equation 2, in
that order, corresponds to the y-intercept of the multiple regression model. The
coefficients for X1, X2, X3 and X4 correspond to the regression coefficients in
Multiple Regression. The computed values of the coefficients for Social Science,
English, Mathematics and Filipino in equation 1 are as follows: -2.90, 7.29, 9.43
and 5.39, respectively.
Moreover, the computed coefficient values for Social Science, English,
Mathematics and Filipino in equation 2 are as follows: -0.059, 3.99, 3.34 and 5.68,
respectively.
In function 1 and 2, the academic subjects English, Math and Filipino were
found to have large weights or large discriminatory power. This result suggests that
English, Math and Filipino grades are the three subjects that contribute highly in
differentiating non-scholars and scholars.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
28
Discriminatory Power of Academic Subjects in
the Derived Discriminant Function
Based on computed Wilk’s Lambda (λ), a criterion used in choosing the
variable contributes most to the discriminatory power of the model, Mathematics
grade had the lowest value of 0.458 and Filipino grade had the highest value of
0.961. Based on the significance of the variables’ discriminatory power in
discriminating academic scholars and non-academic scholars, a computed Wilk’s
lambda approaching zero indicates high separation between the groups. These
findings are supported by the Multivariate F-tests which are all significant at 1%
level of significance (Table 2). From the above results, Math and English grades of
the students are the best indicators for separating academic scholars and non-
academic scholars.
Table 2. Discriminatory power of the different academic subjects in classifying
individuals into group membership
Wilk’s
ACADEMIC SUBJECT
F - value
Significance
Lambda
Social Science 11
0.879
39.379
0.000
English 11
0.630
167.675
0.000
Mathematics 11
0.458
337.296
0.000
Filipino 11
0.961
11.713
0.001
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
29
40
1 1
1
1
30
1
1 1
1 1
1
1 1
1
Misclassified
1
1 1
1
1
1 1
1
1
1
1 11
11 1
1
1
1 1 1
1
1
1
11 11
1 1 111
1 1
1 1
1
2
1 21
1 1 11 1
1
1 1
1
1 1 1 1
11 1
20
2 2
1 111
1
1
1 1 1
1
11
1
2
1
1 1 1 1
11 1
1
1 1 11
1 11
1 1
1 1 1
11
2
1
1
1
1
1 111 1
Non-Scholars
2 2
1 1
1 1
1 1
1
1 1
1
1 1
2
2
1
11
11
1 1 1 1
222
2 21
1
11
22
2 22
21
1 1
11 1
1 1
1
1
2 22
2 21 21
1 1 1
2 22 2
2
2
2
1 1
1
10
2 2
2
2
2 22
2
2
2
2 2 1 11
2
2
22 2
2
2
2
12
2 2
2
2
2
2 2
2
2
2
2
Scholars
DF2
0
-10
0
10
20
30
40
50
DF1
Figure 2. Scatter plot of the derived discriminant functions
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
30
Classification into Group Membership
In the analysis of discriminant functions, 287 samples (200 non-scholars
and 87 academic scholars) were utilized. The number of individuals and percents
classified into groups for the original samples, including the computed value of Z
are presented in Table 3.
Table 3. Classification matrix for the discriminant function
NON-
FROM GROUP
SCHOLARS TOTAL
SCHOLARS
Non-Scholars
173* 27 200
Percent
86.5 13.5 100
Scholars
2 85* 87
Percent
2.3 97.7 100
* Correctly classified cases
Hit Rate:
89.90 %
Z: 13.46
Significance:
0.000
The table shows the result of classification computed from derived
discriminant functions. Out of 287, 89.90 percent of the sample size was correctly
classified and 10.10 percent of the samples were misclassified.
Among the non-scholars, 173 or 86.50 percent of the individuals were
classified correctly, while 27 or 13.50 percent of the original 200 samples were
incorrectly classified as non-scholars. For the group of academic scholars, 85 or
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
31
97.70 percent of the students were in the correct classification and two or 2.30
percent of the individuals originally classified as scholars were assigned to non-
academic scholars.
A test on the 95 percent hit-rate ratio obtained a Z-value of -4.11 (p<.01)
suggests that the hit-rate ratio of 90% is significantly lower than the assumed hit-
rate of 95%. This result indicates that the derived function can classify the
individuals into its correct classification by less than 95% at a time.
Presented in Table 4 is the list of misclassified non-academic scholars to
academic scholars with corresponding probabilities of misclassification. In this
study, 27 out of 200 non-academic scholars could probably be academic scholars
by discriminant analysis.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
32
Table 4. Posterior probability of membership in each group
POSTERIOR PROBABILITY
MISCLASSIFIED
OF MEMBERSHIP
OBSERVATION
Classified Non-Academic
Academic
From Group into Group
Scholar
Scholar
14
NS AS 0.2402 0.7598
25
NS AS 0.4602 0.5398
36
NS AS 0.2558 0.7442
38
NS AS 0.4088 0.2912
51
NS AS 0.2722 0.7278
63
NS AS 0.0670 0.9330
65
NS AS 0.2402 0.7598
71
NS AS 0.0123 0.9877
76
NS AS 0.0143 0.9857
85
NS AS 0.2639 0.7361
88
NS AS 0.3791 0.6209
92
NS AS 0.1240 0.8760
98
NS AS 0.4189 0.5811
115
NS AS 0.4946 0.5054
131
NS AS 0.4395 0.5605
139
NS AS 0.3912 0.6088
143
NS AS 0.2499 0.7501
144
NS AS 0.3134 0.6866
170
NS AS 0.4114 0.5886
174
NS AS 0.1275 0.8725
176
NS AS 0.0670 0.0167
179
NS AS 0.2700 0.0167
180
NS AS 0.4166 0.0167
183
NS AS 0.0619 0.0167
185
NS AS 0.0724 0.0167
193
NS AS 0.2442 0.0167
194
NS AS 0.4088 0.0167
AS – Academic Scholars
NS – Non-Academic Scholars
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
SUMMARY, CONCLUSION AND RECOMMENDATIONS
Summary
The variables observed from the 200 non-scholars and 87 scholars were
final grades in Social Science 11, English 11, Math 11 and Filipino 11. The data on
the aforementioned variables were obtained from the Dean’s Office of College of
Arts and Sciences at Benguet State University and analyzed using the SAS
Discriminant Analysis Procedure.
The results showed that Math, English and Filipino 11 grades have high
discriminating powers in differentiating academic scholars and non-scholars. From
the derived discriminant functions, 86.5 percent of the non-scholars and 97.7
percent of the academic scholars were correctly classified. Thus, there was 13.5
percent and 2.3 percent misclassification in the grouping of non-scholars and
academic scholars respectively into its correct classification.
Out of 200 non-academic scholars, 27 could become academic scholars by
discriminant analysis.
Conclusion
From the above findings, it can be concluded that Mathematics, English and
Filipino are the three subjects that can be used to differentiate an academic scholars
from non-academic scholars.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
34
The derived discriminant function was able to classify individuals into its
correct grouping by only 90%.
Out of the 200 non-academic scholars, 27 of them could be classified as
academic scholars.
Recommendation
Based on the above results, the researchers have come up with the following
recommendations:
1. Include other performance related variables to come up with a good
discriminant function with 100% hit-rate.
2. The derived function may be utilized by the registrar’s office in
determining academic scholars.
3. Revisions of the guidelines on scholarship in the Benguet State
University code is suggested.
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
35
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PEDHAZUR, E. J. (1997). Multiple Regression in Behavioral Research:
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http://safa.uwaterloo.ca/scholarships/scholarship.html
www.arts.yorku.ca/advising/handbook2002/glossary.html
en.wikipedia.org/wiki/Scholarship
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
38
APPENDIX A.
DISCRIMINANT ANALYSIS OF SCHOLARS AND NON-SCHOLARS
CLASSIFICATION OF TEST DATA
Discriminant Analysis Classification Results for Test Data: WORK.TEST
Classification Results using Linear Discriminant Function
Posterior Probability of Membership in GRP:
Obs. Classified into GRP 1 2
1 1 0.9377 0.0623
2 1 0.9696 0.0304
3 1 0.8993 0.1007
4 1 0.9943 0.0057
5 1 0.9934 0.0066
6 1 0.9997 0.0003
7 1 0.9866 0.0134
8 1 0.8647 0.1353
9 1 0.9994 0.0006
10 1 0.9875 0.0125
11 1 0.9899 0.0101
12 1 0.9987 0.0013
13 1 0.9884 0.0116
14 2 0.2402 0.7598
15 1 0.7897 0.2103
16 1 0.6121 0.3879
17 1 0.9997 0.0003
18 1 0.9973 0.0027
19 1 0.9970 0.0030
20 1 0.9977 0.0023
21 1 0.9971 0.0029
22 1 0.9853 0.0147
23 1 0.8925 0.1075
24 1 0.9962 0.0038
25 2 0.4602 0.5398
26 1 0.9973 0.0027
27 1 0.9874 0.0126
28 1 0.9931 0.0069
29 1 0.9952 0.0048
30 1 0.9940 0.0060
31 1 0.7586 0.2414
32 1 0.5999 0.4001
33 1 0.9987 0.0013
34 1 0.9871 0.0129
35 1 0.9714 0.0286
36 2 0.2558 0.7442
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
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Classification Results using Linear Discriminant Function
Posterior Probability of Membership in GRP:
Obs. Classified into GRP 1 2
37 1 0.9988 0.0012
38 2 0.4088 0.5912
39 1 0.9851 0.0149
40 1 0.9867 0.0133
41 1 0.9447 0.0553
42 1 0.9838 0.0162
43 1 0.9997 0.0003
44 1 0.9986 0.0014
45 1 0.9750 0.0250
46 1 0.9974 0.0026
47 1 0.9977 0.0023
48 1 0.9785 0.0215
49 1 0.9395 0.0605
50 1 0.9988 0.0012
51 2 0.2722 0.7278
52 1 0.7683 0.2317
53 1 0.8787 0.1213
54 1 0.9988 0.0012
55 1 0.7772 0.2228
56 1 0.7791 0.2209
57 1 0.9988 0.0012
58 1 0.9755 0.0245
59 1 0.6486 0.3514
60 1 0.9971 0.0029
61 1 0.9869 0.0131
62 1 0.9771 0.0229
63 2 0.0670 0.9330
64 1 0.9860 0.0140
65 2 0.2402 0.7598
66 1 0.9974 0.0026
67 1 0.9892 0.0108
68 1 0.9943 0.0057
69 1 0.9984 0.0016
70 1 0.9999 0.0001
71 2 0.0123 0.9877
72 1 0.9513 0.0487
73 1 0.9941 0.0059
74 1 0.9677 0.0323
75 1 0.6121 0.3879
76 2 0.0143 0.9857
77 1 0.9941 0.0059
78 1 0.9918 0.0082
79 1 0.9677 0.0323
80 1 0.6440 0.3560
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
40
Classification Results using Linear Discriminant Function
Posterior Probability of Membership in GRP:
Obs. Classified into GRP 1 2
81 1 0.6440 0.3560
82 1 0.9972 0.0028
83 1 0.9962 0.0038
84 1 0.9867 0.0133
85 2 0.2639 0.7361
86 1 0.9657 0.0343
87 1 0.9997 0.0003
88 2 0.3791 0.6209
89 1 0.9989 0.0011
90 1 0.9991 0.0009
91 1 0.9999 0.0001
92 2 0.1240 0.8760
93 1 0.9974 0.0026
94 1 0.9995 0.0005
95 1 0.9649 0.0351
96 1 0.9995 0.0005
97 1 0.9967 0.0033
98 2 0.4189 0.5811
99 1 0.9886 0.0114
100 1 0.9999 0.0001
101 1 0.8741 0.1259
102 1 0.5670 0.4330
103 1 0.9489 0.0511
104 1 0.9998 0.0002
105 1 0.9992 0.0008
106 1 0.9938 0.0062
107 1 1.0000 0.0000
108 1 1.0000 0.0000
109 1 0.9974 0.0026
110 1 0.9869 0.0131
111 1 0.9937 0.0063
112 1 0.9504 0.0496
113 1 0.9991 0.0009
114 1 0.9972 0.0028
115 2 0.4946 0.5054
116 1 0.9220 0.0780
117 1 0.7754 0.2246
118 1 0.9988 0.0012
119 1 0.9997 0.0003
120 1 0.9952 0.0048
121 1 0.9873 0.0127
122 1 0.9473 0.0527
123 1 0.9776 0.0224
124 1 0.9853 0.0147
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
41
Classification Results using Linear Discriminant Function
Posterior Probability of Membership in GRP:
Obs. Classified into GRP 1 2
125 1 1.0000 0.0000
126 1 0.9987 0.0013
127 1 0.9970 0.0030
128 1 0.9345 0.0655
129 1 0.7472 0.2528
130 1 0.9994 0.0006
131 2 0.4395 0.5605
132 1 0.9673 0.0327
133 1 0.9938 0.0062
134 1 0.9997 0.0003
135 1 0.9975 0.0025
136 1 0.9986 0.0014
137 1 0.8694 0.1306
138 1 0.9990 0.0010
139 2 0.3912 0.6088
140 1 0.9968 0.0032
141 1 0.9967 0.0033
142 1 0.9551 0.0449
143 2 0.2499 0.7501
144 2 0.3134 0.6866
145 1 0.9723 0.0277
146 1 0.9702 0.0298
147 1 0.9833 0.0167
148 1 0.9943 0.0057
149 1 0.9436 0.0564
150 1 0.9970 0.0030
151 1 0.9031 0.0969
152 1 0.9995 0.0005
153 1 0.9997 0.0003
154 1 0.9686 0.0314
155 1 0.9973 0.0027
156 1 0.8694 0.1306
157 1 0.9636 0.0364
158 1 0.9985 0.0015
159 1 0.9871 0.0129
160 1 0.9986 0.0014
161 1 0.8873 0.1127
162 1 0.9999 0.0001
163 1 0.9988 0.0012
164 1 0.9985 0.0015
165 1 0.5946 0.4054
166 1 0.9970 0.0030
167 1 0.9975 0.0025
168 1 0.9699 0.0301
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
42
Classification Results using Linear Discriminant Function
Posterior Probability of Membership in GRP:
Obs. Classified into GRP 1 2
169 1 0.8478 0.1522
170 2 0.4114 0.5886
171 1 0.9925 0.0075
172 1 0.6046 0.3954
173 1 0.7624 0.2376
174 2 0.1275 0.8725
175 1 0.8925 0.1075
176 2 0.0670 0.9330
177 1 0.9499 0.0501
178 1 0.9988 0.0012
179 2 0.2700 0.7300
180 2 0.4166 0.5834
181 1 0.8993 0.1007
182 1 0.9736 0.0264
183 2 0.0619 0.9381
184 1 0.8694 0.1306
185 2 0.0724 0.9276
186 1 0.9988 0.0012
187 1 0.8787 0.1213
188 1 0.9986 0.0014
189 1 0.9948 0.0052
190 1 0.6676 0.3324
191 1 0.9974 0.0026
192 1 0.9941 0.0059
193 2 0.2442 0.7558
194 2 0.4088 0.5912
195 1 0.8532 0.1468
196 1 0.8895 0.1105
197 1 0.9457 0.0543
198 1 0.9997 0.0003
199 1 0.9931 0.0069
200 1 0.9968 0.0032
201 2 0.0299 0.9701
202 2 0.0035 0.9965
203 2 0.0074 0.9926
204 2 0.4292 0.5708
205 2 0.0992 0.9008
206 1 0.5512 0.4488
207 2 0.0128 0.9872
208 2 0.0034 0.9966
209 2 0.0055 0.9945
210 2 0.0328 0.9672
211 2 0.1357 0.8643
212 2 0.0139 0.9861
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
43
Classification Results using Linear Discriminant Function
Posterior Probability of Membership in GRP:
Obs. Classified into GRP 1 2
213 2 0.0035 0.9965
214 2 0.0545 0.9455
215 2 0.0139 0.9861
216 2 0.0128 0.9872
217 2 0.0154 0.9846
218 2 0.0626 0.9374
219 2 0.0034 0.9966
220 2 0.0265 0.9735
221 2 0.0034 0.9966
222 2 0.0032 0.9968
223 2 0.3765 0.6235
224 2 0.0074 0.9926
225 2 0.2271 0.7729
226 2 0.0078 0.9922
227 2 0.0139 0.9861
228 2 0.0066 0.9934
229 2 0.0143 0.9857
230 2 0.0167 0.9833
231 2 0.0066 0.9934
232 2 0.0146 0.9854
233 2 0.0034 0.9966
234 2 0.1206 0.8794
235 2 0.0072 0.9928
236 2 0.0008 0.9992
237 2 0.0034 0.9966
238 2 0.0083 0.9917
239 2 0.0499 0.9501
240 2 0.2761 0.7239
241 2 0.0312 0.9688
242 2 0.0341 0.9659
243 2 0.0030 0.9970
244 2 0.0154 0.9846
245 2 0.0007 0.9993
246 2 0.0065 0.9935
247 2 0.0008 0.9992
248 2 0.1499 0.8501
249 2 0.0167 0.9833
250 2 0.0154 0.9846
251 2 0.0124 0.9876
252 2 0.0626 0.9374
253 2 0.0060 0.9940
254 2 0.0345 0.9655
255 2 0.0578 0.9422
256 2 0.0071 0.9929
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
44
Classification Results using Linear Discriminant Function
Posterior Probability of Membership in GRP:
Obs. Classified into GRP 1 2
257 2 0.0312 0.9688
258 2 0.0013 0.9987
259 2 0.0076 0.9924
260 2 0.0118 0.9882
261 2 0.0129 0.9871
262 2 0.4039 0.5961
263 2 0.2458 0.7542
264 2 0.0632 0.9368
265 2 0.0059 0.9941
266 2 0.0007 0.9993
267 2 0.0299 0.9701
268 1 0.9833 0.0167
269 2 0.0017 0.9983
270 2 0.0016 0.9984
271 2 0.0535 0.9465
272 2 0.4472 0.5528
273 2 0.0234 0.9766
274 2 0.0137 0.9863
275 2 0.0070 0.9930
276 2 0.0632 0.9368
277 2 0.0026 0.9974
278 2 0.4395 0.5605
279 2 0.0014 0.9986
280 2 0.0318 0.9682
281 2 0.0029 0.9971
282 2 0.0545 0.9455
283 2 0.0055 0.9945
284 2 0.0658 0.9342
285 2 0.1228 0.8772
286 2 0.0149 0.9851
287 2 0.1395 0.8605
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
45
APPENDIX B.
DISCRIMINANT ANALYSIS OF SCHOLARS AND NON-SCHOLARS
OUTPUT CLASSIFICATION RESULTS OF TEST DATA
OBS SS ENG MATH FIL 1 2 INTO
1 2.75 2.00 2.75 2.25 0.93767 0.06233 1
2 2.75 2.25 2.75 2.50 0.96962 0.03038 1
3 2.25 2.25 2.25 1.50 0.89933 0.10067 1
4 1.50 2.50 2.25 1.75 0.99432 0.00568 1
5 2.00 2.00 2.75 2.00 0.99335 0.00665 1
6 2.00 2.50 3.00 2.25 0.99970 0.00030 1
7 1.75 2.00 2.50 1.75 0.98658 0.01342 1
8 2.50 2.00 2.50 2.50 0.86472 0.13528 1
9 2.25 3.00 2.75 2.50 0.99940 0.00060 1
10 2.50 1.75 3.00 1.50 0.98751 0.01249 1
11 2.75 2.50 2.75 1.50 0.98985 0.01015 1
12 2.50 2.50 3.00 2.25 0.99868 0.00132 1
13 2.25 3.00 2.25 2.25 0.98841 0.01159 1
14 1.00 1.50 1.50 1.50 0.24015 0.75985 2
15 1.75 2.00 2.00 1.50 0.78967 0.21033 1
16 1.25 1.75 1.75 1.50 0.61208 0.38792 1
17 1.75 2.75 2.75 2.75 0.99967 0.00033 1
18 2.25 2.50 2.75 2.00 0.99731 0.00269 1
19 2.50 2.25 3.00 2.25 0.99698 0.00302 1
20 2.00 2.25 2.75 1.25 0.99765 0.00235 1
21 2.25 3.00 2.50 2.75 0.99705 0.00295 1
22 2.50 2.25 2.75 2.50 0.98528 0.01472 1
23 2.25 2.25 2.25 1.75 0.89252 0.10748 1
24 2.00 1.75 3.00 2.50 0.99616 0.00384 1
25 3.00 1.75 2.50 1.50 0.46017 0.53983 2
26 2.50 2.75 2.75 2.25 0.99734 0.00266 1
27 2.00 2.75 2.25 2.25 0.98740 0.01260 1
28 1.50 2.00 2.50 2.00 0.99307 0.00693 1
29 2.75 2.75 2.75 1.75 0.99519 0.00481 1
30 2.75 2.75 2.75 2.50 0.99402 0.00598 1
31 3.00 1.75 2.75 2.25 0.75856 0.24144 1
32 1.75 2.25 1.75 2.25 0.59988 0.40012 1
33 2.00 3.00 2.50 2.50 0.99869 0.00131 1
34 2.25 2.00 2.75 1.75 0.98712 0.01288 1
35 2.50 2.00 2.75 2.00 0.97139 0.02861 1
36 1.25 1.75 1.50 1.50 0.25583 0.74417 2
37 1.75 2.75 2.50 2.00 0.99877 0.00123 1
38 1.25 1.50 1.75 1.50 0.40877 0.59123 2
39 2.25 2.00 2.75 2.25 0.98512 0.01488 1
40 2.00 2.25 2.50 2.00 0.98672 0.01328 1
41 2.75 2.50 2.50 2.25 0.94466 0.05534 1
42 2.00 1.75 2.75 2.25 0.98384 0.01616 1
43 2.50 3.00 3.00 2.50 0.99973 0.00027 1
44 2.25 2.25 3.00 2.25 0.99856 0.00144 1
45 2.75 2.75 2.50 2.25 0.97497 0.02503 1
46 1.75 2.50 2.50 1.75 0.99739 0.00261 1
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
46
OUTPUT CLASSIFICATION RESULTS OF TEST DATA
OBS SS ENG MATH FIL 1 2 INTO
47 2.75 3.00 2.75 2.00 0.99773 0.00227 1
48 3.00 3.00 2.50 2.00 0.97854 0.02146 1
49 2.25 2.00 2.50 2.00 0.93948 0.06052 1
50 2.25 2.75 2.75 2.00 0.99882 0.00118 1
51 1.50 2.00 1.50 1.50 0.27217 0.72783 2
52 2.25 2.50 2.00 2.50 0.76830 0.23170 1
53 1.25 1.75 2.00 1.50 0.87867 0.12133 1
54 1.75 2.75 2.50 2.00 0.99877 0.00123 1
55 2.50 1.75 2.50 1.75 0.77717 0.22283 1
56 2.75 2.00 2.50 2.00 0.77907 0.22093 1
57 2.00 2.50 2.75 1.75 0.99881 0.00119 1
58 2.00 2.50 2.25 1.75 0.97547 0.02453 1
59 2.25 1.75 2.25 1.25 0.64855 0.35145 1
60 2.00 1.75 3.00 1.50 0.99713 0.00287 1
61 3.00 2.25 3.00 2.25 0.98686 0.01314 1
62 2.75 2.25 2.75 1.50 0.97714 0.02286 1
63 1.50 1.50 1.50 1.50 0.06699 0.93301 2
64 1.75 3.00 2.00 2.75 0.98605 0.01395 1
65 1.00 1.50 1.50 1.50 0.24015 0.75985 2
66 2.75 3.00 2.75 2.50 0.99737 0.00263 1
67 3.00 2.75 2.75 2.00 0.98921 0.01079 1
68 1.50 2.50 2.25 1.75 0.99432 0.00568 1
69 1.50 2.00 2.75 2.25 0.99837 0.00163 1
70 1.75 3.00 2.75 1.50 0.99990 0.00010 1
71 2.50 2.00 1.50 3.00 0.01229 0.98771 2
72 1.75 2.50 2.00 1.50 0.95135 0.04865 1
73 2.00 2.50 2.50 2.00 0.99414 0.00586 1
74 1.75 1.75 2.50 2.00 0.96768 0.03232 1
75 1.25 1.75 1.75 1.50 0.61208 0.38792 1
76 2.25 2.00 1.50 5.00 0.01434 0.98566 2
77 2.50 2.00 3.00 1.75 0.99407 0.00593 1
78 1.00 1.50 2.50 2.00 0.99181 0.00819 1
79 1.00 2.00 2.00 2.00 0.96770 0.03230 1
80 2.25 2.25 2.00 1.75 0.64404 0.35596 1
81 2.25 2.25 2.00 1.75 0.64404 0.35596 1
82 2.25 2.00 3.00 1.75 0.99717 0.00283 1
83 2.00 1.75 3.00 2.50 0.99616 0.00384 1
84 2.00 2.25 2.50 2.00 0.98672 0.01328 1
85 1.75 1.75 1.75 1.50 0.26387 0.73613 2
86 2.00 2.00 2.50 2.50 0.96568 0.03432 1
87 2.50 3.00 3.00 2.75 0.99971 0.00029 1
88 1.75 2.00 1.75 2.50 0.37912 0.62088 2
89 3.00 3.00 3.00 2.25 0.99888 0.00112 1
90 3.00 3.00 3.00 1.50 0.99910 0.00090 1
91 2.25 3.00 3.00 2.25 0.99988 0.00012 1
92 1.50 1.75 1.50 2.00 0.12401 0.87599 2
93 3.00 2.75 3.00 2.25 0.99745 0.00255 1
94 2.00 2.75 2.75 1.75 0.99948 0.00052 1
95 1.50 1.50 2.50 2.00 0.96494 0.03506 1
96 2.50 2.75 3.00 1.75 0.99950 0.00050 1
97 2.00 1.75 3.00 2.00 0.99668 0.00332 1
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
47
OUTPUT CLASSIFICATION RESULTS OF TEST DATA
OBS SS ENG MATH FIL 1 2 INTO
98 1.75 1.50 2.00 1.50 0.41890 0.58110 2
99 3.00 2.25 3.00 1.75 0.98863 0.01137 1
100 2.00 3.00 3.00 2.25 0.99994 0.00006 1
101 1.50 1.50 2.25 1.50 0.87408 0.12592 1
102 2.00 2.50 1.75 3.00 0.56703 0.43297 1
103 3.00 2.75 2.50 2.25 0.94890 0.05110 1
104 2.25 2.75 3.00 1.50 0.99978 0.00022 1
105 1.00 1.75 2.75 2.00 0.99921 0.00079 1
106 1.00 2.00 2.25 1.50 0.99376 0.00624 1
107 5.00 3.00 5.00 2.00 1.00000 0.00000 1
108 5.00 3.00 5.00 2.50 1.00000 0.00000 1
109 1.75 2.50 2.50 1.75 0.99739 0.00261 1
110 2.25 2.50 2.50 2.25 0.98687 0.01313 1
111 2.75 2.25 3.00 2.25 0.99369 0.00631 1
112 2.50 2.75 2.25 2.00 0.95040 0.04960 1
113 3.00 3.00 3.00 1.50 0.99910 0.00090 1
114 2.25 2.00 3.00 1.75 0.99717 0.00283 1
115 2.50 2.75 1.75 1.75 0.49459 0.50541 2
116 2.25 1.50 2.75 2.50 0.92198 0.07802 1
117 1.50 1.75 2.00 1.50 0.77536 0.22464 1
118 2.75 2.75 3.00 2.25 0.99878 0.00122 1
119 2.00 3.00 2.75 2.25 0.99973 0.00027 1
120 2.75 2.75 2.75 1.75 0.99519 0.00481 1
121 2.50 2.25 2.75 2.00 0.98726 0.01274 1
122 2.25 2.00 2.50 1.50 0.94727 0.05273 1
123 2.50 3.00 2.25 2.00 0.97764 0.02236 1
124 2.50 2.25 2.75 2.50 0.98528 0.01472 1
125 1.75 3.00 3.00 2.00 0.99997 0.00003 1
126 2.00 2.50 2.75 2.00 0.99872 0.00128 1
127 2.50 2.25 3.00 2.25 0.99698 0.00302 1
128 2.00 1.75 2.50 2.00 0.93451 0.06549 1
129 1.75 2.50 1.75 2.75 0.74722 0.25278 1
130 1.50 2.25 2.75 1.50 0.99942 0.00058 1
131 2.00 1.75 2.00 1.50 0.43950 0.56050 2
132 1.50 1.50 2.50 1.75 0.96733 0.03267 1
133 2.25 2.75 2.50 2.50 0.99377 0.00623 1
134 2.25 2.75 3.00 2.00 0.99974 0.00026 1
135 2.50 2.75 2.75 2.00 0.99753 0.00247 1
136 2.25 2.25 3.00 2.25 0.99856 0.00144 1
137 1.00 1.50 2.00 1.50 0.86941 0.13059 1
138 2.00 2.50 2.75 1.25 0.99897 0.00103 1
139 1.25 1.50 1.75 1.75 0.39122 0.60878 2
140 2.25 3.00 2.50 3.00 0.99683 0.00317 1
141 5.00 5.00 2.75 3.00 0.99674 0.00326 1
142 2.75 2.50 2.50 1.50 0.95507 0.04493 1
143 1.75 1.75 1.75 1.75 0.24991 0.75009 2
144 5.00 2.50 3.00 2.25 0.31343 0.68657 2
145 2.00 2.00 2.50 1.75 0.97225 0.02775 1
146 2.00 2.00 2.50 2.00 0.97021 0.02979 1
147 2.50 1.75 3.00 2.50 0.98333 0.01667 1
148 2.25 2.25 2.75 1.75 0.99432 0.00568 1
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OUTPUT CLASSIFICATION RESULTS OF TEST DATA
OBS SS ENG MATH FIL 1 2 INTO
149 2.25 2.75 2.25 5.00 0.94357 0.05643 1
150 2.25 2.00 3.00 2.00 0.99695 0.00305 1
151 2.75 2.25 2.50 1.50 0.90305 0.09695 1
152 2.75 3.00 3.00 2.00 0.99950 0.00050 1
153 2.50 3.00 3.00 2.50 0.99973 0.00027 1
154 1.25 1.75 2.25 1.75 0.96865 0.03135 1
155 2.25 3.00 2.50 2.50 0.99726 0.00274 1
156 1.00 1.50 2.00 1.50 0.86941 0.13059 1
157 2.25 2.25 2.50 3.00 0.96356 0.03644 1
158 2.00 2.00 3.00 2.00 0.99854 0.00146 1
159 1.50 2.25 2.25 1.75 0.98713 0.01287 1
160 2.00 2.00 3.00 1.75 0.99865 0.00135 1
161 2.25 1.75 2.50 1.50 0.88729 0.11271 1
162 1.75 2.50 3.00 2.00 0.99987 0.00013 1
163 2.50 2.50 3.00 2.00 0.99877 0.00123 1
164 1.75 2.25 2.75 2.25 0.99850 0.00150 1
165 1.25 1.75 1.75 1.75 0.59458 0.40542 1
166 2.50 2.25 3.00 2.25 0.99698 0.00302 1
167 2.50 2.75 2.75 2.00 0.99753 0.00247 1
168 3.00 2.50 2.75 2.75 0.96994 0.03006 1
169 2.25 1.25 2.75 2.25 0.84782 0.15218 1
170 1.50 1.75 1.75 1.75 0.41143 0.58857 2
171 2.25 1.75 3.00 2.25 0.99255 0.00745 1
172 1.75 1.75 2.00 1.75 0.60461 0.39539 1
173 1.50 1.75 2.00 1.75 0.76237 0.23763 1
174 1.00 1.75 1.25 1.75 0.12746 0.87254 2
175 2.25 2.25 2.25 1.75 0.89252 0.10748 1
176 1.50 1.50 1.50 1.50 0.06699 0.93301 2
177 2.25 2.50 2.25 1.75 0.94988 0.05012 1
178 2.50 3.00 2.75 2.25 0.99883 0.00117 1
179 1.25 1.75 1.50 1.25 0.27000 0.73000 2
180 2.75 2.00 2.25 2.25 0.41662 0.58338 2
181 2.25 2.25 2.25 1.50 0.89933 0.10067 1
182 2.75 2.25 2.75 2.00 0.97364 0.02636 1
183 1.25 1.25 1.50 1.50 0.06192 0.93808 2
184 1.00 1.50 2.00 1.50 0.86941 0.13059 1
185 1.75 1.75 1.50 1.50 0.07244 0.92756 2
186 1.50 2.50 2.50 1.75 0.99876 0.00124 1
187 1.25 1.75 2.00 1.50 0.87867 0.12133 1
188 2.00 3.00 2.50 2.75 0.99859 0.00141 1
189 2.50 2.50 2.75 1.75 0.99477 0.00523 1
190 1.75 2.25 1.75 1.25 0.66761 0.33239 1
191 3.00 2.75 3.00 2.25 0.99745 0.00255 1
192 2.00 2.50 2.50 2.00 0.99414 0.00586 1
193 1.50 2.00 1.50 2.00 0.24419 0.75581 2
194 1.25 1.50 1.75 1.50 0.40877 0.59123 2
195 2.00 1.50 2.50 2.25 0.85320 0.14680 1
196 2.00 2.50 2.00 2.00 0.88953 0.11047 1
197 2.75 2.00 2.75 1.75 0.94569 0.05431 1
198 2.25 2.75 3.00 3.00 0.99966 0.00034 1
199 2.50 2.00 3.00 2.25 0.99314 0.00686 1
Discriminant Analysis on the Performance of Academic Scholars and
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OUTPUT CLASSIFICATION RESULTS OF TEST DATA
OBS SS ENG MATH FIL 1 2 INTO
200 3.00 2.75 3.00 3.00 0.99683 0.00317 1
201 1.50 1.75 1.25 2.00 0.02992 0.97008 2
202 2.00 1.50 1.25 1.50 0.00354 0.99646 2
203 1.75 1.50 1.25 1.50 0.00740 0.99260 2
204 1.50 1.75 1.75 1.50 0.42924 0.57076 2
205 2.75 1.50 2.25 3.00 0.09919 0.90081 2
206 1.75 1.75 2.00 2.50 0.55117 0.44883 1
207 1.75 1.25 1.50 2.00 0.01279 0.98721 2
208 1.50 1.50 1.00 1.50 0.00340 0.99660 2
209 1.50 1.25 1.25 2.25 0.00547 0.99453 2
210 2.75 2.00 1.75 2.25 0.03279 0.96721 2
211 1.75 1.50 1.75 1.50 0.13574 0.86426 2
212 2.00 1.50 1.50 2.00 0.01390 0.98610 2
213 2.00 1.50 1.25 1.50 0.00354 0.99646 2
214 1.50 1.50 1.50 2.25 0.05452 0.94548 2
215 2.00 1.50 1.50 2.00 0.01390 0.98610 2
216 1.75 1.25 1.50 2.00 0.01279 0.98721 2
217 1.50 1.50 1.25 1.50 0.01540 0.98460 2
218 1.50 1.50 1.50 1.75 0.06256 0.93744 2
219 1.50 1.50 1.00 1.50 0.00340 0.99660 2
220 1.50 1.25 1.50 2.00 0.02646 0.97354 2
221 1.50 1.50 1.00 1.50 0.00340 0.99660 2
222 1.50 1.50 1.00 1.75 0.00316 0.99684 2
223 1.50 1.75 1.75 2.25 0.37654 0.62346 2
224 1.75 1.50 1.25 1.50 0.00740 0.99260 2
225 1.00 1.50 1.50 1.75 0.22707 0.77293 2
226 1.75 2.00 1.00 1.75 0.00780 0.99220 2
227 2.00 1.50 1.50 2.00 0.01390 0.98610 2
228 1.25 1.50 1.00 1.75 0.00660 0.99340 2
229 1.50 1.50 1.25 1.75 0.01433 0.98567 2
230 1.75 1.75 1.25 1.50 0.01673 0.98327 2
231 1.25 1.50 1.00 1.75 0.00660 0.99340 2
232 1.50 1.00 1.50 1.25 0.01461 0.98539 2
233 1.50 1.50 1.00 1.50 0.00340 0.99660 2
234 2.00 1.75 1.75 2.25 0.12065 0.87935 2
235 1.50 1.75 1.00 1.75 0.00718 0.99282 2
236 2.25 1.75 1.00 1.75 0.00078 0.99922 2
237 1.50 1.50 1.00 1.50 0.00340 0.99660 2
238 1.50 1.75 1.00 1.25 0.00830 0.99170 2
239 2.25 1.75 1.75 3.00 0.04990 0.95010 2
240 1.50 1.50 1.75 1.00 0.27610 0.72390 2
241 2.00 1.75 1.50 2.00 0.03116 0.96884 2
242 1.25 1.50 1.25 1.25 0.03411 0.96589 2
243 1.50 1.00 1.25 1.50 0.00299 0.99701 2
244 1.50 1.50 1.25 1.50 0.01540 0.98460 2
245 1.75 1.25 1.00 1.75 0.00066 0.99934 2
246 1.75 1.00 1.50 1.50 0.00653 0.99347 2
247 2.00 1.50 1.00 1.50 0.00077 0.99923 2
248 1.50 1.75 1.50 1.25 0.14987 0.85013 2
249 1.75 1.75 1.25 1.50 0.01673 0.98327 2
250 1.50 1.50 1.25 1.50 0.01540 0.98460 2
Discriminant Analysis on the Performance of Academic Scholars and
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OUTPUT CLASSIFICATION RESULTS OF TEST DATA
OBS SS ENG MATH FIL 1 2 INTO
251 1.50 1.50 1.25 2.25 0.01241 0.98759 2
252 1.50 1.50 1.50 1.75 0.06256 0.93744 2
253 1.75 1.50 1.25 2.25 0.00595 0.99405 2
254 1.50 1.75 1.25 1.50 0.03447 0.96553 2
255 1.25 1.25 1.50 1.75 0.05781 0.94219 2
256 1.25 1.50 1.00 1.50 0.00710 0.99290 2
257 2.00 1.75 1.50 2.00 0.03116 0.96884 2
258 1.75 1.00 1.25 1.75 0.00133 0.99867 2
259 1.25 1.50 1.00 1.25 0.00764 0.99236 2
260 1.50 1.00 1.50 2.00 0.01177 0.98823 2
261 2.00 1.50 1.50 2.25 0.01293 0.98707 2
262 2.00 1.75 2.00 2.00 0.40386 0.59614 2
263 1.25 1.25 1.75 1.25 0.24581 0.75419 2
264 1.75 1.75 1.50 2.00 0.06321 0.93679 2
265 1.50 1.25 1.25 2.00 0.00589 0.99411 2
266 1.75 1.25 1.00 1.50 0.00071 0.99929 2
267 2.25 1.50 1.75 2.00 0.02991 0.97009 2
268 1.75 2.00 2.50 2.50 0.98334 0.01666 1
269 2.00 1.25 1.25 1.25 0.00167 0.99833 2
270 1.75 1.50 1.00 1.50 0.00162 0.99838 2
271 2.25 1.75 1.75 2.75 0.05348 0.94652 2
272 1.50 1.75 1.75 1.25 0.44723 0.55277 2
273 2.00 1.75 1.50 3.00 0.02344 0.97656 2
274 1.75 1.25 1.50 1.75 0.01375 0.98625 2
275 2.00 1.75 1.25 2.00 0.00696 0.99304 2
276 1.75 1.75 1.50 2.00 0.06321 0.93679 2
277 2.00 1.50 1.25 2.50 0.00265 0.99735 2
278 2.00 1.75 2.00 1.50 0.43950 0.56050 2
279 1.75 1.00 1.25 1.50 0.00143 0.99857 2
280 1.25 1.50 1.25 1.50 0.03178 0.96822 2
281 1.50 1.50 1.00 2.00 0.00294 0.99706 2
282 1.50 1.50 1.50 2.25 0.05452 0.94548 2
283 1.50 1.25 1.25 2.25 0.00547 0.99453 2
284 1.50 2.00 1.25 2.00 0.06577 0.93423 2
285 1.25 1.50 1.50 1.75 0.12282 0.87718 2
286 2.00 1.50 1.50 1.75 0.01493 0.98507 2
287 1.25 1.50 1.50 1.25 0.13946 0.86054 2
Discriminant Analysis on the Performance of Academic Scholars and
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APPENDIX C.
Discriminant Analysis of Scholars and Non-Scholars
Classification of Test Data
Discriminant Analysis
287 Observations 286 DF Total
4 Variables 285 DF Within Classes
2 Classes 1 DF Between Classes
Class Level Information
Output Prior
GRP SAS Name Frequency Weight Proportion Probability
1 _1 200 200.0000 0.696864 0.500000
2 _2 87 87.0000 0.303136 0.500000
Discriminant Analysis Pooled Covariance Matrix Information
Covariance Natural Log of the Determinant
Matrix Rank of the Covariance Matrix
4 -6.6095354
Discriminant Analysis Pairwise Generalized Squared Distances Between
Groups
2 _ _ -1 _ _
D (i|j) = (X - X )' COV (X - X )
i j i j
Generalized Squared Distance to GRP
From GRP 1 2
1 0 7.80453
2 7.80453 0
Discriminant Analysis Linear Discriminant Function
_ -1 _ -1 _
Constant = -.5 X' COV X Coefficient Vector = COV X
j j j
GRP
1 2
CONSTANT -22.35626 -10.51363
SS -2.90547 0.05859
ENG 7.28604 3.98555
MATH 9.43072 3.33555
FIL 5.38761 5.68003
Discriminant Analysis on the Performance of Academic Scholars and
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Discriminant Analysis Classification Summary for
Calibration Data: WORK.GRADE
Resubstitution Summary using Linear Discriminant Function
Generalized Squared Distance Posterior Probability of Membership in each
Function:
GRP:
2 _ -1 _ 2 2
D (X) = (X-X )' COV (X-X ) Pr(j|X) = exp(-.5 D (X)) / SUM exp(-.5 D (X))
j j j j k k
Number of Observations and Percent Classified into GRP:
From GRP 1 2 Total
1 173 27 200
86.50 13.50 100.00
2 2 85 87
2.30 97.70 100.00
Total 175 112 287
Percent 60.98 39.02 100.00
Priors 0.5000 0.5000
Error Count Estimates for GRP:
1 2 Total
Rate 0.1350 0.0230 0.0790
Priors 0.5000 0.5000
Discriminant Analysis on the Performance of Academic Scholars and
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APPENDIX D.
TWO-GROUP DISCRIMINANT ANALYSIS
Group Statistics
Valid N (listwise)
GROUP
Mean
Std. Deviation
Unweighted
Weighted
1.00
SS
2.1550
.6671
200
200.000
ENG
2.2700
.5175
200
200.000
MATH
2.4987
.5272
200
200.000
FIL
2.0175
.5096
200
200.000
2.00
SS
1.6839
.3200
87
87.000
ENG
1.5201
.2326
87
87.000
MATH
1.3851
.3093
87
87.000
FIL
1.8046
.4201
87
87.000
Total
SS
2.0122
.6225
287
287.000
ENG
2.0427
.5673
287
287.000
MATH
2.1611
.6965
287
287.000
FIL
1.9530
.4934
287
287.000
Tests of Equality of Group Means
Wilks'
Lambda
F
df1
df2
Sig.
SS
.879
39.379
1
285
.000
ENG
.630
167.675
1
285
.000
MATH
.458
337.296
1
285
.000
FIL
.961
11.713
1
285
.001
Discriminant Analysis on the Performance of Academic Scholars and
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Analysis 1
Box's Test of Equality of Covariance Matrices
Log Determinants
Log
GROUP
Rank
Determinant
1.00
4
-5.862
2.00
4
-9.877
Pooled within-groups
4
-6.610
The ranks and natural logarithms of determinants
printed are those of the group covariance matrices.
Test Results
Box's M
132.136
F
Approx.
12.964
df1
10
df2
134083.8
Sig.
.000
Tests null hypothesis of equal population covariance matrices.
Discriminant Analysis on the Performance of Academic Scholars and
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Summary of Canonical Discriminant Functions
Eigenvalues
Canonical
Function Eigenvalue % of Variance Cumulative % Correlation
1
1.660a
100.0
100.0
.790
First 1 canonical discriminant functions were used in the
a.
analysis.
Wilks' Lambda
Wilks'
Test of Function(s) Lambda
Chi-square
df
Sig.
1
.376
276.891
4
.000
Standardized Canonical Discriminant Function Coefficients
Function
1
SS
-.620
ENG
.533
MATH
1.030
FIL
-.051
Discriminant Analysis on the Performance of Academic Scholars and
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Structure Matrix
Function
1
MATH
.844
ENG
.595
SS
.288
FIL
.157
Pooled within-groups correlations between discriminating
variables and standardized canonical discriminant functions
Variables ordered by absolute size of correlation within function.
Functions at Group Centroids
Function
GROUP
1
1.00
.847
2.00
-1.947
Unstandardized canonical discriminant
functions evaluated at group means
Discriminant Analysis on the Performance of Academic Scholars and
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Classification Statistics
Classification Processing Summary
Processed
287
Excluded
Missing or out-of-range
0
group codes
At least one missing
0
discriminating variable
Used in Output
287
Prior Probabilities for Groups
Cases Used in Analysis
GROUP
Prior
Unweighted
Weighted
1.00
.500
200
200.000
2.00
.500
87
87.000
Total
1.000
287
287.000
Classification Function Coefficients
GROUP
1.00
2.00
SS
-2.905
5.859E-02
ENG
7.286
3.986
MATH
9.431
3.336
FIL
5.388
5.680
(Constant)
-23.049
-11.207
Fisher's linear discriminant functions
Discriminant Analysis on the Performance of Academic Scholars and
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58
APPENDIX E.
COMPUTATIONS
Z- test:
Z
= (258-273)√287
√273 (287-273)
= 254.1
61.82
= -4.11
Discriminant Analysis on the Performance of Academic Scholars and
Non-Scholars in Benguet State University / Liezyl T. Astodillo; et al. 2009
Document Outline
- Discriminant Analysis on the Performance of
Academic Scholars and Non-Scholars in Benguet State University
- BIBLIOGRAPHY
- ABSTRACT
- TABLE OF CONTENTS
- INTRODUCTION
- REVIEW OF LITERATURE
- THEORETICAL FRAMEWORK
- METHODOLOGY
- RESULTS AND DISCUSSION
- SUMMARY, CONCLUSION AND RECOMMENDATIONS
- LITERATURE CITED
- APPENDIX