BIBLIOGRAPHY EDWARD ANTIPUESTO, EFRAIM DICKSON...
BIBLIOGRAPHY
EDWARD ANTIPUESTO, EFRAIM DICKSON DULAWEN and BOBBY F.
ROARING. March 2009. Multivariate Linear Regression Model for Academic
Performance of the Freshmen Achiever at Benguet State University. Benguet State
University, La Trinidad , Benguet.
Adviser: DR. SALVACION Z. BELIGAN
ABSTRACT
The study was conducted to determine predictors of academic performance
among freshmen achievers using multivariate linear regression. The specific objectives of
the study were to1) Describe the relationships existing between the different factors and
the academic performance; 2) Find a suitable regression model that would predict the
best factor affecting academic performance among freshmen achievers; 3) Identify the
factors that may predict the student’s academic performance. High school average,
gender, course, age, and type of high they have attended were investigated as a possible
predictor of academic performance. Multivariate linear regression analysis was used to
perform to build a predictive model that could determine whether the following variables
could predict the academic performance of the freshman academic achievers.
The test for the significance of the regression coefficients, the F-test did not
support the rejection of the hypothesis that the different predictor variables have no
significant effect on the student’s performance in Math 11. In addition, none of the
regressors are contributory to the performance of the students in English and Soc Sci
grade.
Generally, fitting univariate regression model on the three dependent variables,
the test of significance for the beta coefficient was found not significant.
Fitting the multivariate regression model, the result showed significant result.
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TABLE OF CONTENT
Page
BIBLIOGRAPHY………………………………………………..…….………..…i
ABSTRACT……..…………………………………………………………….........i
TABLE OF CONTENT……..…………………….…………….…………………iii
INTRODUCTION
Background of the Study…………….…………………………………..…1
Statement of the Problem……….…………...………………………...……2
Objectives of the Study…….……………..…………………………...……3
Importance of the Study…..……………….………………………..……...4
Scope and Delimitation…….………………………...………………….….5
REVIEW OF RELATED LITERATURE…………...……….……………………6
THEORETICAL FRAMEWORK…………………………………………...……10
Definitions of terms ……………………………………………………….23
METHODOLOGY…………………………………..……………………….……25
RESULTS AND DISCUSSION………………………………………………..…27
SUMMARY, CONCLUSIONS,
RECOMMENDATION…….………………………………….…….……...….…34
LITERATURE CITED……………………………………….…………...………37
APPENDICES……………………………………………….………...………….39
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1
INTRODUCTION
Background of the study
The very heart of a learner’s education Mission is to have stable and
progressing academic achievements. The indicators of the learner’s academic
achievements are the marks and grades written on a learner’s official academic
records.
Educators have long been looking for a predictor of academic performance
for students as they enter college because low academic performance of students
means waste not only personal and societal time, as well as resources of the
educational institution. Students who can be identified as “at risk” for failure early
in their academic careers can be targeted for interventions that will increase the
likelihood of success. Because student academic performance are vital to students’
college life, understanding factors that diminish students’ satisfaction and
perseverance is necessary if these problems are to be addressed and overcome.
It is therefore important that school administrators should study the factors
contributory to the academic performance of every students.
A number of factors affecting academic performance have been studied by
several authors.
One of the studies reviewed is on the relationship between high school
grade point average (GPA) and college performance (Benford, 2006).
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Results show that high school performance of students graduated from
either private of public high school had a significant effect on their college
performance on the same subject.
Using multivariate regression analysis, Beligan (2004) found that students’
performance in Math - based subjects was affected by Age, Gender and the type of
high school they had attended.
Foreign studies also showed that the academic performance of a student is
multidetermined by number of contributing influences, and factors including
academic factors, personality variables, family characteristics, and environmental
factors (Benguiristain, 2003).
Student’s academic performance is, however, affected by multiple factors
and should not also be confined to one. To reach a valid generalization on students’
academic performance, we need to consider different factors. This study, therefore,
is expected to fill the gaps.
Statement of the problem
This study aimed to evaluate the factors of academic performance among
BSU freshmen academic achievers.
Specifically the study sought to answer the following questions:
Multivariate Linear Regression Model for
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1. What are the relationships existing between the different factors
and to the academic performance?
2. What is the suitable regression model that would predict the best
factor affecting academic performance among freshmen achievers?
3. What are the factors that may predict the student’s academic
performance?
Objectives of the study
The purpose of this study is to determine the predictors of academic
performance among freshmen achievers using multivariate linear regression. The
specific objectives of the study were to:
1. Describe the relationships existing between the different factors and to
the academic performance.
2. Find a suitable regression model that would predict the best factor
affecting academic performance among freshmen achievers.
3. Identify the factors that may predict the student’s academic performance.
Multivariate Linear Regression Model for
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Significance of the study
This study would shed the light to further strengthen the existing foundation
by determining the predictors of academic performance among the freshmen
achievers.
For the students, this study would help them to acquire strong determination
in carrying their studies and become aware of their responsibilities and for them to
be academically productive.
The study would also help school administrators as well as the parents and
guardians in formulating and knowing the best predictors of academic success.
In addition, work in this area has the potential to provide important
suggestions to improve standards and quality of education and also the performance
of the students.
To assist faculty in helping students achieve academic success, variables
should be identified to help predict whether or not students might have difficulties
in achieving academic success, and therefore whether or not those students will
have difficulties in completing a degree.
The results of this study would help identify students possibly risk for
failure so that interventional measures can be developed and implemented to
promote educational productivity and reduce attrition.
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Scope and Delimitation
Several limitations to this study existed. A population consisted of freshmen
academic achievers enrolled at Benguet State University during school year 2008-
2009. They were chosen because they have a minor subject which was used as their
output variable. In addition, independent variables were also limited.
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REVIEW OF RELATED LITERATURE
What makes a successful person? Vince Lombardi observed, “The
difference between a successful person and others is not a lack of strength, not a
lack of knowledge, but rather in a lack of will” (The Official Site of Vince
Lambordi, n.d.). He recognized, as do so many of us, that motivation may be the
most important factor in determining success. In the field of research, however,
motivation and biographical factors conceptually related to success can not be
directly measured and are difficult to manipulate as variables. Therefore, a limited
number of studies have successfully linked such factors to success in specific areas.
In educational environment, a student motivation, success orientation, and network
of support can surpass the importance of writing and reading skills.
Analyzing multivariate context data using a series of univariate methods
compromises the very essence and richness of a multifaceted phenomenon, such as
education production.
The measurements often involves the use of linear regression, correlation,
or multivariate-discriminant analysis to indicate the nature of the relationship
(Goldhaber & Brewer, 1997a; Monk, 1992; Montmarquette & Majseredjian, 1989;
Montimore, Simmons, Stoll, Lewis, & Ecob, 1988; Murnane, 1975;Smith &
Tomlinson 1989; Walberg, 1982; Walberg & Fowler, 1989; Walberg & Weintien,
1982; Wenglinsky, 1997). Approximately 400 studies have been reported including
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those that have been evaluated and summarized using meta-analysis (Greenwald, et
al., 1996; Hanushek, 1997; Hedges, Laine & Greenwald , 1994).
Most of these academic performance studies examined the relationship
between two or more variables and one educational output measure (usually student
achievement) using univariate analysis (Hanushek, 1997).
In education, there is often more than one output measure for a given set of
inputs. Academic performance studies must examine the relationship between sets
of variables and sets of outputs measures as multivariate composites (Pedhazur,
1997).
Heinbuch and Samuels (1995) and Tatsuoka (1988) have noted that very
little concerted effort has been made to establish the nature of the relationship
between multiple educational inputs and multiple measures of student achievement
using multivariate methods. Pedhazur also noted that simply calculating zero-order
correlations for all possible pairs of variables using univariate tests on data with
multivariate constructs “ . . . affects the prescribed α level” (p.895) thus increasing
the chance for a Type 1 error. Moreover, analyzing multivariate context data using
a series of univariate methods compromises the very essence and richness of a
multifaceted phenomenon, such as education production.
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In a multivariate context, education production examines the relationship
between a set of two or more educational input variables and a set of two or more
educational output variables (Pedhazur 1997; Tatsuoka, 1973; Tatsuoka, 1988).
However, colleges and universities do not currently acknowledge such
factors in their selection process, which is intended to choose student most likely to
succeed at their school. Most colleges and universities limit their admission
decisions to the traditional cognitive predictors of academic performance, usually
high school general weighted average (GWA), scores in standardized tests such as
the SAT, and high school rank (Aiken, 1964).
Another study conducted by Reiter (1964) tried to determine if there was a
relationship between non-academic factors and academic performance of college
freshmen and sophomore students. Participants were 76 randomly selected male
and female college students from an introductory psychology course. His study
shows that high school achievements were a superior predictor of academic success
(Reiter, 1964). Also, Reiter (1964) stated that non-intellectual factors, such as
anxiety, academic discipline, and SAT composite scores, were not proven to be true
predictors of academic success.
A number of studies are shown that several variables other than academic
characteristics are associated with success in college. Larose, Robertson, Roy, and
Legault (1998) argue that the frequency and quality of the interactions between
students and their faculty are related to higher levels of college success. Other
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variables related to college success are the ability to adapt to the college
environment, personal motivation, and students’ relationship with their peers
(Larose et al., 1998). In addition to these findings, Aiken (1964) suggests that
students who withdraw from classes are less motivated to achieve than students
who do not withdraw.
A study conducted buy Naumann, Bendalos, and Gutkin (2003) at
Midwestern University implies that first-generation college students rely more
heavily on motivational factors in order to succeed than do second-generation
students. These results were not surprising given “first-generation students did not
typically have the same source of support throughout careers as did second-
generation students” (Naumann, Bendalos, and Gutkin, 2003, p.8).
Gregorc (1979) described a person’s learning style as consisting of distinct
behaviors which serve as stable indicators of how a person learns and adapts to
his/her learning environment. The most extensively researched and applied learning
style construct has been the field-dependence/independence dimension (Guild and
Garger, 1985).
Multivariate Linear Regression Model for
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THEORETICAL FRAMEWORK
Multivariate Linear Regression
Multivariate linear models are, in one sense, like repeated measures models
– in both cases, it may be helpful to think of the analysis as involving multiple
dependent measures. In repeated measures analysis, the multiple dependent
variables are the same measure, repeated (usually over time). In multivariate
models, the multiple dependent variables are measures of multiple outcomes,
usually measured with different metrics, and usually measured at the same point in
time.
The coefficient of the linear regression model is estimated under the
assumption that the random term assumes normal distribution with zero mean and
constant variance. The values of the random term are also assumed to be
independent.
The general multivariate model may be expressed as:
a1Y1 + a2Y2 + a3Y3 +…..+akYk = b0 + b1X1 + b2X2 + b3X3 + ……. bmXm
Where a1, a2, a3,….., ak are the coefficients associated with Y1,Y2,Y3,…,Yk
the linear composite of educational output measures : b0 is the intercept of the
composite regression line on line linear composite f output measures, and
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b1,b2,b3,….,bm are the regression coefficients associated with X1,X2,X3,…,Xm the
linear composite of educational input variables.
This study used multivariate methods to examine the relationships between
a set of three educational input variables and two educational output measures.
Based on the multivariate model explained above, the overall production function
model for this study is:
a1Y1 + a2Y2 + a3Y3= b0 + b1X1 + b2X2 + b3X3 + ……. b5X5
Where X1, X2, X3, X4, X4, X5, are thirteen educational input variables;
Y1,Y2,Y3 are the four output measures a1,a2,a3 are the coefficients or weights of the
linear composite of outputs in the regression model ; b1, b2, b3,….,b5 are the
coefficients or weights of the linear composite of inputs in the regression model.
Let us assume that there are p response variates Y’=Y1, Y2,….Yp and that
these variables and linearly related with r-fixed variates X1, X2,….,Xr. The
theoretical model for response variables is
E(Yk/X)= βok + β1kX1+…+βrkXr.
In matrix notation, the above model can be expressed as
Yj’= Xj’β+ ej’,
j=1,2,…,N
Y=Xβ + Є
Multivariate Linear Regression Model for
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Where Y11 Y12……..Y1p
Y21 Y22……..Y2p
Y=
:
:
:
YN1 YN2……..YNp N x p
1
X11………X1r
1
X21………X2r
X=
:
:
:
1 XN1……..XNr N x r + 1
β= is the unknown parameter
βo
=
β1
:
βrk
Є = is the matrix of random error
Where e11
e12……..e1p
e21
e22………e2p
Є=
:
:
:
eN1 eN2……..eNp
N x p
Multivariate Linear Regression Model for
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e1’
=
e2’
:
eN’
Each vector ej’ will be assumed to have the following properties:
(1) E(ej)= 0 for all j
(2) E(ej ej’)= ∑ for all j and j’
Where ∑ is the variance-covariance matrix of the random vector ej defined
as
σ11
σ12…….. σ1p
σ21
σ22……… σ2p
∑ =
:
:
:
σN1 σN2…….. σNp
(3) ej ~ Np(0 ∑) for all j
(4) For any I1 the N x 1 random vector ej will be assume to satisfy E(ej)= 0 and
E(e
2
j ej’)= σi (IN)
Multivariate Linear Regression Model for
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Estimation of the parameters β
Similar to the multiple linear regression, the method of least squares is need
to estimate the matrix β.
Using the model
Yj’= X’jβ + ej’
for j=1, 2,…, N
The matrix to be minimized is (e’ e) Є Є’ which is the matrix of error sums
of squares and sums of products.
Since
Є = Y – Xβ and Є’ = Y’ – β’X’
Then
Є’Є = (Y’ – β’X’)(Y – Xβ)
=Y’Y - 2β’X’Y + β’X’X
Taking the partial derivation of Є’Є with respect to β and equates the
results to 0, we would get
Є’Є= -2Y’Y + 2X’Xβ = 0
The resulting normal equations are:
Multivariate Linear Regression Model for
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X’Xβ= X’Y
Where,
X’X is the (r+1) x (r+1) matrix of unconnected sum of squares and sums of
products
N
∑Xj1…………∑X jr
∑Xj1 ∑Xj12……… ∑Xj1Xjr
X’X =
:
:
:
∑Xjr ∑XjrXjl……. ∑Xjr2
β =(βim) is the (r+1) x p matrix of estimated intercepts and partial regression
coefficients
β01
β02…….. β0p
β11
β12……… β1p
β =
:
:
:
βr1 βr2……… βrp
β =
β1
β2…….. βp
And
X’Y is the (r+1) x p matrix of uncorrected sums of products of X and Y:
Multivariate Linear Regression Model for
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1
1 …….. 1
Y11 Y
12…….. Y1p
β11
β12……… β1p
Y 21 Y22……… Y 2p
β =
:
:
:
:
:
:
βr1 βr2……… βrp
YN1 YN2……… YNp
∑Yj1
∑Yj2…………… ∑Y jp
∑Xj1Yj1
∑Xj1Yj2……… ∑Xj1Yjp
X’X = :
: :
∑XjnYj1 ∑XjrYj2………. ∑XjrYjp
g
01
g02…….. g0p
g11
g12……… g1p
= :
:
:
g
r1 gr2……… grp
It will be assumed that the matrix X’X is nonsingular so that the solution
to the equation (X’X)β=X’Y is
β = (X’X)-1X’Y
= WG
Multivariate Linear Regression Model for
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Where
w
00
w01…….. w0r
W = (X’X)-1= w10
w11……...w1r
:
:
:
wr0 wr1……. wrn
After multiplying W x G, we would get these particular solutions:
Intercepts:β0
βoi=∑(Wok)(gki),
i=1,2,…p
Partial regression coefficients
βi=βim=∑(Wik)(gkm), i.m=2,…….
Finally, the estimated p regression equations are:
Ŷ1 = β01 + β 11X1 + β r1Xr
Ŷ2 = β02 + β 12X1 + β r2Xr
Ŷp = β0p + β1pX1 + βrpXr
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Estimation of ∑
The estimate of the variance-covariance matrix ∑ will be derived from the
estimate residual matrix Є = Y – Xβ.
Є’Є = (Y –Xβ)’(Y-Xβ)
= (Y’- β’X’)(Y –Xβ)
= Y’Y – βX’Y – Y’Xβ + β’X’Xβ
From the normal equation (X’X)β= X’Y
Є’Є = Y’Y – β’X’Y
The elements of Є’Є are defined:
eil = Yi’ Yl - ∑
eim= Yi’Ym - ∑
From the previous section, the residual matrix Є’Є can be written as
Є’Є = E’ [I – XWX’]E
And
E(e’e)=
E{
Є’[ I – XWX’]Є}
Multivariate Linear Regression Model for
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= ∑ tr [ I –XWX’]
= ∑ [ tr (IN) – tr (XWX’)]
= ∑ ( N – (r+1)) = ∑ (N-r-1)
Therefore, the unbiased estimator of ∑ is ∑’
∑’ = S =
The elements of S are:
Sample variances= S 2
i/x = Yi’ Yi – βigi
= ∑
2 ∑
Sample covariance: Sim
Sim = Yi’Ym – βi’gm
= ∑YjiYjm – ∑
Estimate of the variance of β
By
definition,
V(β) = E[β’ – β][β’ – β]’
But
β’=WX’Y = WX’(Xβ+Є)
Multivariate Linear Regression Model for
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=WX’Xβ + WX’Є
Therefore,
β’ – β = WX’Є and
(β’- β) (β’- β) = (WX’Є) (WX’Є)’
=WX’(ЄЄ’)XW’
E(β’- β) (β’- β)’ = E[WX’(ЄЄ’) XW’
= WX’ E(ЄЄ’) XW’
= WX’ (∑(IN)XW’
=∑(W)
Hence,
V(β)= ∑(W)
Where
V(β) = Var (β1,β2,…..βp)
Var(β1) Cov(β1β2) ….. Cov(β1βp)
= Cov(β1β2) Var(β2) …..
Cov(β2βp)
:
:
:
Cov(βpβ1) Cov(βpβ2) Var(βp)
Multivariate Linear Regression Model for
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S 2
1/x
S12/x
S1p/x
∑(W) =
S2
2
1/x
S2/x
S2p/x W
: : :
S
2
p1/x
Sp2/x
Sp/x
Therefore,
Var(βi) = (Si/x)(W) for
i=1,2,……p
Cov(βiβm)= (Sim/x)(W) i m=1,2,…..p
Test of hypothesis
Ho: CB=K
Where:
C is the nh x (r+1) of rank nh
K is the nh x p matrix of constants = 0
A. The Union-Intersection Test
1. Compute the matrix of sum of squares due to the hypothesis
Ho :
H= (Cβ’-K)’ [CWC’] (Cβ’-K)
2. Compute the matrix of sums of squares and sums of products,E.
E=Y’Y -β’X’X
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3. Obtain the largest characteristics root of HE-1. Let C=char(max) HE-1. The
numbers of characteristics roots of HE-1 . Since rank of H is min(nh,p),
hence rank HE-1 is min(nh,p). Therefore the degree of the characteristics
polynomial is nh.
4. Compute the statistics θ,
Θ=
5. Compare the value of θ with the tabular value θα(s,m,n) where
s= min(VH,p) VH= nh
m=|
|
n=
Reject Ho: CB=K if θ θα(s,m,n)
The equivalent F-statistics is
Fc=
c(n+1)/(m+1)
which follows an F-distribution with 2m+2 and 2n+2 degrees of freedom
B. The Likelihood Ratio Test or Wilk’s Lambda
After computing H and E as given above obtain the statistic U
U= | | U
|
|
α(p,VH,VE)
The table below gives the transform of U to provide upper tail tests using F-
distribution.
Multivariate Linear Regression Model for
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Table 1. Transformation of U with upper tail tests using F-distribution.
Parameters
Statistics having F-distribution
Degrees of
Freedom
VH = 1 for any p
1
p, V
E+ VH -p
VH = 2 for any p
1
√
1
2p,2(VE+ VH -p)
√
p=1 any VH
1
V
H, VE
p=2 any VH
1
√
2VH, 2VE
√
Definition of terms
Academic Performance - refers to the performance that is characterized by a
narrow concern for book learning and formal rules, without knowledge or
experience of practical matters.
Community Ability - refers to the performance of students in Basic English
subjects.
Dummy variables-These variables may be thought of as additional variables
for which statistical adjustment is desired.
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Mathematical Ability - refers to the performance of the students in
mathematics.
Multivariate Analysis - comprises a set of techniques dedicated to the
analysis of data sets with more than one variable.
Predictor Variables - refers to the variables from which projections are
made in a prediction study.
Regression Analysis - refers to the estimation of the linear relationship
between a dependent variable and one or more independent variables or covariates.
Study Habits - giving proper time and energy to your studies and various
“tricks” to help you study more effectively.
Predictor Variable - refer to the variable that are use to predict the values of
another variable in a mode.
Multivariate Linear Regression Model for
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METHODOLOGY
Location of the study
The study was conducted at Benguet State University during the 2
semester of the school year 2008-2009 at La Trinidad, Benguet.
Respondents of the Study
The respondents of the study were all the forty eight freshmen academic
achievers of School Year 2008-2009.
Data gathering procedure
The grades of the selected achievers were obtained from the Office of the
University Registrar and Office of the Students’ Affairs of Benguet State
University instrumentation.
A questionnaire was prepared and floated to obtain the age, course, school
attended, and high school average and gender data.
Statistical analysis of data
The gathered and tabulated data were analyzed using of descriptive and
inferential methods. The descriptive method includes the analysis of the data using
frequency, percentage, mean, and rank.
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The inferential method utilized in determining the influence of different
factors on the grades of the student’s achievers. The dependent variables are the
grades in the subjects namely: Math, English and Social Science. The independent
variables include the High School Grade, Age of the respondents, Gender, Type of
High School attended and the course. For qualitative independent variables,
dummies were defined and entered into the model.
Fitting the multiple regression and multivariate regression models, the use
of the PROC REG procedure of the SAS software was compelled to facilitate the
computations.
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RESULTS AND DISCUSSION
Descriptive Statistics
Table 2 shows the computed means and standard deviation of the grades on
the three dependent variables (Math 11, English 11, and Social Science 11) and two
independent variables, (High school grade and Age). The respondents’ earned
grades in the three subjects as the dependent variables ranged from 1.26 to 1.60
with social science having the least grades of 1.26 and math having the highest
average of 1.68. Their high school mean grade and age of the respondents were
90.2% and 16.98% respectively.
Table 2. Mean and standard deviation of the variables.
VARIABLE N MEAN
STANDARD
MAX MIN
DEVIATION
Math 11
49
1.68
0.35
1.00
2.75
English 11
49
1.60
0.24
1.25
2.25
Soc sci 11
49
1.26
0.25
1.00
1.75
HS GRADE(X1)
49
90.28
2.43
84.33
95.59
AGE(X2) 49
16.98
1.49
16.00
24
Multivariate Linear Regression Model for
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28
Table 3 presents the distribution of achievers by course. About 35 percent
of the achievers are enrolled in the nursing course and about 29 and 14 percent are
taking up BSE and BEE respectively. The remaining 22 percent of the achievers are
taking up other degree programs such as: DVM, AET, BSF, BSAS, BLIS, BSES,
and BSA. It can be seen in the table that most of the achiever are dominated by
Nursing and Education students
On the distribution of achievers according to type of High School attended,
majority of the respondents are graduates of the National High School (67.35%).
Achievers from private schools comprise the 20 percent of the respondents. The
remaining 10 percent of the achiever respondents are from Barangay/Regional High
School.
Table 3. Distribution of respondents by course, type of high school graduated, and
gender
COURSE COURSE PERCENT CUMULATIVE
CUMULATIVE
PERCENT
FREQUENCY
BSN 17
0.35
0.35
17
BEE&BSE 21
0.43
0.78 38
OTHERDEGREE 11
0.22
100.00
49
REGIONAL HS
5
0.10
0.12
6
NATIONAL HS
33
0.67
0.80
39
PRIVATE HS
10
0.20
100.00
49
FEMALE 36
0.73
0.73
39
MALE 13
0.27
100.00
49
Multivariate Linear Regression Model for
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29
As to gender, female achievers outnumbered male achievers. About 73
percent of the academic achievers of the Benguet State University are females and
the remaining are males.
Univariate Multiple Regression Model
Before fitting a multivariate regression model, a univariate multiple linear
model was first fitted for each dependent variable. Taking Math as the first
dependent variable, the univariate multiple linear regression model gave the
following results in Table 4.
Table 4. Beta coefficients for Math grade
DEPENDENT
β0 INTERCEPT
HS AVERAGE
AGE (β2)
VARIABLE
(β1)
YC1T1 0.612
0.010279
-0.00607
YC1T2 0.603
0.010279
-0.00607
YC1T3 0.472
0.010279
-0.00607
YC2T1 0.948
0.010279
-0.00607
YC2T2 0.940
0.010279
-0.00607
YC2T3 0.808
0.010279
-0.00607
YC3T1 0.887
0.010279
-0.00607
YC3T2 0.878
0.010279
-0.00607
YC3T3 0.747
0.010279
-0.00607
Where: C1= CN
T1=REGIONAL HIGH SCHOOL
C2=CTE
T2=NATIONAL HIGH SCHOOL
C3=OTHER DEGREE
T3=PRIVATE HIGH SCHOOL
Multivariate Linear Regression Model for
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As to the goodness – of – fit of the fitted regression model given below,
Math = HGWA + Age + Gender (Dummy) + Course (Dummy) +
Type of HS (Dummy)
only 11 percent of the variability in Math grades can be explained by the
independent variables in the model.
Moreover, the test for the significance of the regression coefficients, the F-
test did not support the rejection of the hypothesis that the different predictor
variables have no significant effect on the student’s performance in Math 11.
Regressing the same predictors on the student’s performance in English as
the dependent variable, the independent variables in the fitted model,
English = HGWA + Age + Gender (Dummy) + Course (Dummy) +
Type of HS (Dummy)
they explained only 18.06 percent of the variability in the English grades obtained
by the respondents.
The significance of the effects of the different regressors was found not
significant. This means that none of the regressors are contributory to the
performance of the students in English. The models derived for the combination of
the different dummy variables are shown in Table 5.
Multivariate Linear Regression Model for
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31
Table 5. Beta coefficients for English grade
DEPENDENT
β0 INTERCEPT
HS AVERAGE
AGE (β2)
VARIABLE
(β1)
YC1T1
2.995 -0.02024
0.015401
YC1T2
2.957 -0.02024
0.015401
YC1T3
3.088 -0.02024
0.015401
YC2T1
3.097 -0.02024
0.015401
YC2T2
3.059 -0.02024
0.015401
YC2T3
3.190 -0.02024
0.015401
YC3T1
3.152 -0.02024
0.015401
YC3T2
3.114 -0.02024
0.015401
YC3T3
3.245 -0.02024
0.015401
Where: C1= CN
T1=REGIONAL HIGH SCHOOL
C2=CTE
T2=NATIONAL HIGH SCHOOL
C3=OTHER DEGREE
T3=PRIVATE HIGH SCHOOL
Moreover, using the third dependent variable, the Social Science grade,
with the same predictor variables, about 53% in the variability in Social Science
grade can be attributed to HGWA, Age, Gender, Course and Type of High School
attended.
English = HGWA + Age + Gender (Dummy) + Course (Dummy) +
Type of HS (Dummy)
Multivariate Linear Regression Model for
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Table 6. Beta coefficients for Social Science grade
DEPENDENT
β0 INTERCEPT
HS AVERAGE
AGE (β2)
VARIABLE
(β1)
YC1T1
2.466 -0.009 -0.022
YC1T2
2.340 -0.009 -0.022
YC1T3
2.331 -0.009 -0.022
YC2T1
2.764 -0.009 -0.022
YC2T2
2.638 -0.009 -0.022
YC2T3
2.629 -0.009 -0.022
YC3T1
2.908 -0.009 -0.022
YC3T2
3.114 -0.009 -0.022
YC3T3
3.245 -0.009 -0.022
Where: C1= CN
T1=REGIONAL HIGH SCHOOL
C2=CTE
T2=NATIONAL HIGH SCHOOL
C3=OTHER DEGREE
T3=PRIVATE HIGH SCHOOL
However, the test of significance for the beta coefficient is not significant.
Taking Social Science as the third dependent variable, the univariate multiple linear
model gave the following results in Table 6.
Multivariate Regression Model
Fitting a multivariate regression model with the grades in the three subjects
as dependent variables using the same predictor variables, the MANOVA test gave
significant result as shown in table 7
Multivariate Linear Regression Model for
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33
Generally, fitting univariate regression model on the three dependent
variables, the test of significance for the beta coefficient was found not significant.
Fitting the multivariate regression model, the result showed significant
result.
Table 7. Multivariate test for the overall model.
Statistic Value
F
Pr>F
Wilks' Lambda
0.31987
2.2229
0.0028
Pillai's Trace
0.85308
1.9867
0.0083
Hotelling-Lawley Trace
1.61989
2.4748
0.0008
Roy's Greatest Root
1.26915
6.3457
0.0001
Eigenvalues Math:
0.56
English : 0.19
SocSci : 0.10
Multivariate Linear Regression Model for
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SUMMARY, RECCOMENDATION AND CONCLUSION
This study was conducted to determine the predictors of academic
performance among academic achievers using Multivariate Linear Regression.
Specifically the study aim to 1) describes the relationship existing between the
different factors and to the academic performance. 2) Find a suitable regression
model that would predict the best factors affecting academic performance of
achievers. 3) Identify the factors that may predict the student’s academic
performance.
The variables collected are information about to those freshmen achievers
on there grade in Math 11, English 11, Soc Sci 11, High School grade, Gender,
Age, Course and Type of High School. These variables are gathered using a
questionable while the number of student was gathered from the Registrar’s office
and the Office of the Student’s Affairs. Out of the thousand students of the
university, we just selected the 49 freshmen academic achievers.
The statistical tool used in this study was Multivariate Regression Analysis.
This statistical was used to accommodate the multivariate nature of educational
input and output measures. Prior to analysis, all variables in the data set were
examined through SAS program for accuracy of data entry. A Multivariate
Regression Analysis was performed in order to determine the predictive
Multivariate Linear Regression Model for
Academic Performance of the Freshmen
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35
capabilities of the independent variables. The Multivariate Regression Analysis
indicated that the composite of predictor had a highly significant linear relationship
with the output measures (Math 11, English 11 and Soc Sci. In addition, there are
more given effect of the independent variable that contributes to the model.
Based on the conclusion, it is recommended that the result of this study may
be out of great help to the college. It maybe of help in counseling and guidance
among the university on some factors that might predict academic performance to
the student. Knowledge of the student’s predictor’s will guide in strengthening their
stat. Strategies to suit the need of student’s.
Based on the results, the researchers recommend the following:
1. The results of the study could help in fostering student’s awareness on
the factors that could affect their academic performance.
2. The results of the study may also be of great help to parents and
guardians in motivating their children as well as in knowing the factors that affect
their children’s’ achievement.
3. Further investigation of this study is also recommended.
In conclusion, if a goal of the university is to be efficient and effective at
educating its students, and subsequently have them graduate, then further research
is needed to identify if other variables exist that can predict students' ability to
Multivariate Linear Regression Model for
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36
improve their academic performance. Future studies should continue to explore
additional variables as well as investigate student performance beyond the first year
in order to confirm the trend for effectiveness of these predictors.
Multivariate Linear Regression Model for
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37
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BENFORD,R, GESS-NEWSOME,J.(2006). Factors affecting Student Academic
Success in Gateway at Northern Arizona University. Rusell Benford Julie
Gess-Newsome Center for Science Teaching and Learning Northern
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Alexandria,VA: Association for Supervision and Curriculum Development.
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HANUSHEK, E. A. (1997) Assessing the effect of school resources and student
performance: An update. Education Evaluation and Policy Analysis, 19,
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Performance of College Students. Morehouse College
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teacher allocation in Boston public Schools. Education Evaluation and
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NURKOWSKI, L. C(1995). Transfer Student Persistence and Academic Success
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TATSUOKA, M. M. (1973). Multivariate analysis in education research. In F. N.
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http:// www.jstor.org/view/10448039/ap060023/06a00030/0
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39
APPENDIX
Appendix A
COLLEGE OF ARTS AND SCIENCES
MATH-PHYSICS-STATISTICS DEPARTMENT
APPLICATION FOR MANUSCRIPT/PROJECT DEFENSE
Date: ________
Group Members:
Edward Antipuesto
Efraim
Dickson
Dulawen
Major
Field:
Statistics
Bobby F. Roaring
Minor Field: Info. Tech.
Degree: Bachelor of Science in Applied Statistics
Title of Thesis:
“Multivariate Linear Regression Model for Academic Performance of the Freshmen Achiever
at Benguet State University”
Endorsed: Dr. Salvacion Z. Beligan
(Adviser)
Date and Time of Defense: March 19, 2009, 11:00 AM
Place of Defense: CAS An 210
Noted: Maria Azucena B. Lubrica
Department Chairman
Report of result of manuscript/project defense
Name and Signature
Remarks*
_____________________ __________________
Adviser
_____________________ __________________
Member
_____________________ __________________
Member
_____________________ ___________________
MARIA AZUCENA B. LUBRICA
Department chairman
Multivariate Linear Regression Model for
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Appendix B
Benguet State University
College of Arts and Sciences
Math-Physics-Statistics Department
La Trinidad, Benguet
February 9, 2009
DR. EDNA CHUA
Office of the Student Affairs
Benguet State University
MADAME:
We, the undersigned fourth year students taking up Bachelor of Sciences in Applied Statistics at Benguet State University,
are conducting a research entitled “MULTIVARIATE REGRESSION ANALYSIS OF ACADEMIC PERFORMANCE OF
THE FRESHMEN ACHIEVERS AT BENGUET STATE UNIVERSITY”.
In view hereof, we would like to request permission from your good office to gather the following:
a) High school average
b) IQ
c) Entrance Exam Results of the Freshmen Academic Achievers.
Attached to this letter are the names and course of the freshmen academic achiever.
Thank you very much for your favorable consideration.
Respectfully yours,
EDWARD ANTIPUESTO
EFRAIM DICKSON L. DULAWEN
BOBBY F. ROARING
Researchers
Noted:
DR. SALVACION Z. BELIGAN
Adviser
Approved:
DR. EDNA CHUA
Office of the Student Affairs
Multivariate Linear Regression Model for
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Appendix C
Benguet State University
College of Arts and Sciences
Math-Physics-Statistics Department
La Trinidad, Benguet
February 9, 2009
DR. MARLENE B. ATINYAO
Office of the University Registrar
Benguet State University
MADAME:
We, the undersigned fourth year students taking up Bachelor of Sciences in Applied Statistics at Benguet State
University, are conducting a research entitled “MULTIVARIATE REGRESSION ANALYSIS OF ACADEMIC
PERFORMANCE OF THE FRESHMEN ACHIEVERS AT BENGUET STATE UNIVERSITY”.
In view hereof, we would like to request permission from your good office to gather the following:
a) High school average
b) IQ
c) Entrance Exam Results of the Freshmen Academic Achievers.
Attached to this letter are the names and course of the freshmen academic achiever.
Thank you very much for your favorable consideration.
Respectfully yours,
EDWARD ANTIPUESTO
EFRAIM DICKSON L. DULAWEN
BOBBY F. ROARING
Researchers
Noted:
DR. SALVACION Z. BELIGAN
Adviser
Approved:
DR. MARLENE B. ATINYAO
Director, OUR
Multivariate Linear Regression Model for
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Appendix D
Benguet State University
College of Arts and Sciences
Math-Physics-Statistics Department
La Trinidad, Benguet
Dear Respondents,
The undersigned Bachelor of Science in Applied Statistics students are conducting a research entitled, “MULTIVARIATE
REGRESSION ANALYSIS OF ACADEMIC PERFORMANCE OF THE FRESHMEN ACHIEVERS AT BENGUET
STATE UNIVERSITY”. In this connection, may we request that you provide us with the information needed?
Your truthful and complete answers will help the success of this study. Rest assured that answers would be treated
confidentially.
Heartfelt thanks for your anticipated cooperation.
Reverently yours,
_____________________
EDWARD ANTIPUESTO
_____________________________
Noted:
EFRAIM DICKSON L. DULAWEN
___________________
__________________________
BOBBY F. ROARING
DR. SALVACION Z. BELIGAN
Researchers
Adviser
SURVEY QUESTIONNAIRE
INSTRUCTION: Please fill in the blank or put a check (
) mark on the given space with the correct entries. All entries will
be regarded as strictly confidential.
RESPONDENTS PROFILE
1. Name: ______________________
2. Age: _______________________
3. Gender: [ ]-Male
[ ]-Female
4. Course: _____________________
5. Type of high school graduated:
[ ]-Barangay High School
[ ]-City/Regional High School
[ ]-National High School
[ ]-Private High School
6. What was your High School average (Pls. specify)? ________
Multivariate Linear Regression Model for Academic Performance of the Freshmen
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43
Appendix E
RAW DATA
MATH
ENGLISH SOC SCI
HS
AGE GENDER
COURSE TYPE
OF
11
11
11
GRADE
HS
1.75 1.50 1.00
91.36
17 0
1
3
1.75 1.25 1.00
93.13
17 0
1
3
1.75 1.25 1.00
94.00
16 0
1
4
1.50 1.50 1.00
89.24
16 1
1
3
1.50 1.50 1.00
91.94
16 0
1
4
1.50 1.50 1.00
90.80
16 0
1
3
1.50 1.50 1.00
92.83
17 0
1
3
1.50 1.25 1.00
91.26
16 0
1
3
1.50 1.50 1.50
92.51
16 0
1
2
1.50 1.50 1.25
89.69
17 0
1
3
1.50 1.50 1.25
91.72
17 0
1
3
1.50 1.50 1.25
89.55
16 1
1
3
1.50 2.00 1.00
90.08
17 0
1
3
1.50 1.50 1.00 8.54 17 1
1
4
1.50 1.50 1.25
90.77
16 0
1
3
1.50 1.75 1.00
91.62
17 0
1
4
2.00 1.50 1.00
95.59
17 0
1
4
1.50 1.25 1.00
87.75
17 1
2
3
1.25 1.25 1.25
88.00
16 0
2
2
1.50 1.75 1.25
93.33
16 0
2
4
1.50 1.75 1.25
91.79
22 1
2
3
2.25 1.50 1.25
89.65
19 0
2
3
1.75 1.25 1.50
89.59
18 1
2
3
2.75 2.00 1.25
91.07
17 0
2
2
1.50 2.00 1.25
86.00
18 0
2
3
2.00 1.75 1.50
90.03
16 0
2
3
1.75 1.50 1.50
91.38
16 0
2
3
Multivariate Linear Regression Model for
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2.25 1.50 1.75
88.75
16 0
2
3
2.00 1.75 1.50
93.40
17 0
2
4
2.25 1.50 1.50
89.93
18 0
2
3
2.00 2.00 1.75
86.74
16 1
2
4
1.00 1.50 1.75
91.20
18 1
3
4
1.25 2.00 1.25
85.95
18 1
3
4
1.50 1.75 1.75
88.38
17 0
3
2
2.00 1.50 1.50
85.20
16 1
3
3
1.75 1.75 1.50
88.26
16 1
3
3
2.50 1.75 1.50
90.10
17 0
3
3
2.00 1.75 1.50
84.33
24 0
3
3
1.25 1.50 1.00
88.75
16 0
4
3
1.75 1.50 1.00
92.38
17 0
4
3
1.25 1.25 1.00
92.67
17 1
4
3
1.50 1.50 1.25
87.84
16 0
5
3
1.50 1.50 1.00
91.15
18 0
5
3
2.25 1.75 1.50
88.35
16 0
6
2
1.25 2.00 1.25
90.53
17 0
7
3
1.75 1.75 1.00
94.82
16 0
8
3
1.50 1.75 1.25
88.59
18 1
9
3
1.75 2.25 1.00
92.00
16 0
10
3
1.75 1.25 1.50
91.00
16 0
11
3
Multivariate Linear Regression Model for
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Appendix F
SAMPLE RESULT OF MULTIVARIATE REGRESSION ANALYSIS
Model: MODEL1
Dependent Variable: MATH
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 8 1.49237 0.18655 1.714 0.1251
Error 40 4.35457 0.10886
C Total 48 5.84694
Multivariate Linear Reg
ression Model for Academic Performance of the Freshmen
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46
Root MSE 0.32995 R‐square 0.2552
Dep Mean 1.68367 Adj R‐sq 0.1063
C.V. 19.59678
NOTE: Model is not full rank. Least‐squares solutions for the parameters are not unique. Some
statistics will be misleading. A reported DF of 0 or B means that the estimate is biased.
The following parameters have been set to 0, since the variables are a linear combination
of
other variables as shown.
T3 = +1.0000 * INTERCEP ‐1.0000 * T1 ‐1.0000 * T2
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP B 0.694579 2.43541495 0.285 0.7770
Multivariate Linear Reg
ression Model for Academic Performance of the Freshmen
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HGWA 1 0.010279 0.02512236 0.409 0.6846
AGE 1 ‐0.006073 0.03506443 ‐0.173 0.8634
GENDER 1 ‐0.227511 0.12190480 ‐1.866 0.0693
C1 1 0.004804 0.13286341 0.036 0.9713
C2 1 0.341181 0.14171705 2.407 0.0208
C3 1 0.279586 0.19160862 1.459 0.1523
T1 B 0.140017 0.19766260 0.708 0.4828
T2 B 0.102657 0.13157800 0.780 0.4399
T3 0 0 . . .
Dependent Variable: ENG
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Multivariate Linear Reg
ression Model for Academic Performance of the Freshmen
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Model 8 0.51518 0.06440 1.102 0.3821
Error 40 2.33686 0.05842
C Total 48 2.85204
Root MSE 0.24171 R‐square 0.1806
Dep Mean 1.59694 Adj R‐sq 0.0168
C.V. 15.13556
NOTE: Model is not full rank. Least‐squares solutions for the parameters are not unique. Some
statistics will be misleading. A reported DF of 0 or B means that the estimate is biased.
The following parameters have been set to 0, since the variables are a linear combination
of
other variables as shown.
T3 = +1.0000 * INTERCEP ‐1.0000 * T1 ‐1.0000 * T2
Multivariate Linear Reg
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Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP B 3.384601 1.78409146 1.897 0.0651
HGWA 1 ‐0.020236 0.01840368 ‐1.100 0.2781
AGE 1 0.015401 0.02568685 0.600 0.5522
GENDER 1 ‐0.111806 0.08930277 ‐1.252 0.2178
C1 1 ‐0.164187 0.09733063 ‐1.687 0.0994
C2 1 ‐0.062799 0.10381647 ‐0.605 0.5487
C3 1 ‐0.007576 0.14036511 ‐0.054 0.9572
T1 B ‐0.113289 0.14480003 ‐0.782 0.4386
T2 B ‐0.151477 0.09638899 ‐1.572 0.1239
T3 0 0 . . .
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Dependent Variable: SOC
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 8 1.55410 0.19426 5.622 0.0001
Error 40 1.38212 0.03455
C Total 48 2.93622
Root MSE 0.18588 R‐square 0.5293
Dep Mean 1.25510 Adj R‐sq 0.4351
C.V. 14.81031
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NOTE: Model is not full rank. Least‐squares solutions for the parameters are not unique. Some
statistics will be misleading. A reported DF of 0 or B means that the estimate is biased.
The following parameters have been set to 0, since the variables are a linear combination
of
other variables as shown.
T3 = +1.0000 * INTERCEP ‐1.0000 * T1 ‐1.0000 * T2
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP B 2.382049 1.37206180 1.736 0.0902
HGWA 1 ‐0.009449 0.01415341 ‐0.668 0.5082
AGE 1 ‐0.022418 0.01975457 ‐1.135 0.2632
GENDER 1 ‐0.021456 0.06867861 ‐0.312 0.7564
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C1 1 ‐0.062635 0.07485246 ‐0.837 0.4077
C2 1 0.235317 0.07984042 2.947 0.0053
C3 1 0.379675 0.10794828 3.517 0.0011
T1 B 0.125302 0.11135897 1.125 0.2672
T2 B ‐0.000142 0.07412829 ‐0.002 0.9985
T3 0 0 . . .
Multivariate Test:
L Ginv(X'X) L' LB‐cj
0.0057974335 0.0005044996 0.0086155014 ‐0.001313812 0.0058847404 0.0163624548
0.0005044996 0.0112940136 0.0006179345 ‐0.000140828 ‐0.008479562 ‐0.015947632
0.0086155014 0.0006179345 0.1365074533 0.0039390604 ‐0.003122265 ‐0.021829978
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‐0.001313812 ‐0.000140828 0.0039390604 0.1621532104 0.0978031711 0.0967026652
0.0058847404 ‐0.008479562 ‐0.003122265 0.0978031711 0.1844841043 0.1359956205
0.0163624548 ‐0.015947632 ‐0.021829978 0.0967026652 0.1359956205 0.3372444604
0.011761556 0.0063620468 0.0698618214 0.0383235684 0.0154849678 0.0259728851
0.0089674611 ‐0.004563661 0.0269253009 0.0391040946 0.0439800302 0.0704872815
0.011761556 0.0089674611 0.010279155 ‐0.020236126 ‐0.009449088
0.0063620468 ‐0.004563661 ‐0.00607292 0.0154006991 ‐0.022418391
0.0698618214 0.0269253009 ‐0.227511242 ‐0.111806137 ‐0.021455628
0.0383235684 0.0391040946 0.0048038604 ‐0.164186815 ‐0.062635445
0.0154849678 0.0439800302 0.3411810716 ‐0.062798934 0.2353170282
0.0259728851 0.0704872815 0.279585682 ‐0.007575886 0.3796746199
0.3588920024 0.1303478278 0.1400173804 ‐0.113288808 0.125302437
0.1303478278 0.1590308249 0.1026566118 ‐0.151476506 ‐0.000141511
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Inv(L Ginv(X'X) L') Inv()(LB‐cj)
283.35573878 ‐43.27367347 ‐18.5222449 19.93244898 ‐6.458571429 ‐18.51428571
‐43.27367347 106.97959184 4.2653061224 ‐7.653061224 4.2857142857 7.1428571429
‐18.5222449 4.2653061224 9.5510204082 ‐1.510204082 0.2857142857 2.1428571429
19.93244898 ‐7.653061224 ‐1.510204082 11.102040816 ‐4.857142857 ‐2.428571429
‐6.458571429 4.2857142857 0.2857142857 ‐4.857142857 10 ‐2
‐18.51428571 7.1428571429 2.1428571429 ‐2.428571429 ‐2 6
‐3.071632653 ‐2.897959184 ‐1.326530612 ‐0.734693878 0.5714285714 0.2857142857
‐6.475102041 4.693877551 ‐0.020408163 ‐0.795918367 ‐0.714285714 ‐0.857142857
‐3.071632653 ‐6.475102041 ‐0.989438776 ‐5.72755102 ‐11.49158163
‐2.897959184 4.693877551 1.4336734694 2.5969387755 1.7551020408
‐1.326530612 ‐0.020408163 ‐1.887755102 ‐0.260204082 0.6836734694
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‐0.734693878 ‐0.795918367 ‐1.87244898 ‐1.647959184 ‐2.836734694
0.5714285714 ‐0.714285714 2.6785714286 0.3928571429 1.9285714286
0.2857142857 ‐0.857142857 0.2142857143 0.8214285714 1.9642857143
4.4897959184 ‐3.469387755 0.8316326531 0.2653061224 0.9744897959
‐3.469387755 10.408163265 0.0051020408 ‐0.795918367 ‐0.9234693885
T, the H + E Matrix
5.8469387755 0.7525510204 1.3290816327
0.7525510204 2.8520408163 0.4757653061
1.3290816327 0.4757653061 2.9362244898
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Eigenvectors
0.0517543588 0.1270359725 0.520020392
0.4131660836 ‐0.289104225 ‐0.158534347
0.1453177832 0.5176258271 ‐0.298329281
Eigenvalues
0.5593056922
0.1913680346
0.1024034267
Multivariate Statistics and F Approximations
S=3 M=2 N=18
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Statistic Value F Num DF Den DF Pr > F
Wilks' Lambda 0.31986707 2.2229 24 110.8128 0.0028
Pillai's Trace 0.85307715 1.9867 24 120 0.0083
Hotelling‐Lawley Trace 1.61988940 2.4748 24 110 0.0008
Roy's Greatest Root 1.26914662 6.3457 8 40 0.0001
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
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Appendix G
TABLE RESULTS
Table 8. Univariate ANOVA with Math grade (Y1) as dependent
Source DF Sum
of
Mean
F Value
Prob>F
Square
Square
Model 8
1.49237
0.18655
1.714
0.1251
Error 48
4.35457
0.10886
C Total
48
5.84694
Root MSE
0.32995
Dep Mean
1.68367
R-square 0.2552
Adj R-sq
0.1063
C.V. 19.59678
Table 9. Parameter estimates for Math grade
Parameter
Parameter
Standard
T for HO:
Prob > |T|
Variable
Estimate
Error
Parameter = 0
Intercept 0.69458
2.43541
0.285
0.7770
HGWA 0.01028
0.02512
0.409
0.6846
AGE 0.00607
0.03506
-0.173
0.8634
GENDER
-0.22750
0.12190
-1.866
0.0693
C1 0.00480
0.13286
0.036
0.9713
C2 0.34118
0.14172
2.407
0.0208
C3 0.27959
0.19161
1.459
0.1523
T1 0.14002
0.19766
0.708
0.4828
T2 0.10266
0.13158
0.780
0.4399
T3 0
-
-
-
Where: C1= CN
T1=REGIONAL HIGH SCHOOL
C2=CTE
T2=NATIONAL HIGH SCHOOL
C3=OTHER DEGREE
T3=PRIVATE HIGH SCHOOL
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Table 10. Univariate ANOVA with English grade (Y2) as dependent
Source
DF
Sum of
Mean
F Value
Prob>F
Square
Square
Model 8
0.51518
0.0644
1.102
0.3821
Error 48
2.33686
0.0584
C Total
48
2.85204
Root
MSE
0.24171
Dep
Mean
1.59694
R-square
0.18060
Adj
R-sq
0.01680
C.V.
15.13556
Table 11. Parameter estimates for English grade
Parameter
Parameter
Standard Error
T for HO:
Prob > |T|
Variable
Estimate
Parameter = 0
Intercept 3.384601
1.7840915
1.897
0.0651
HGWA -0.020236
0.0184037
-1.100
0.2781
AGE 0.015401
0.0256869
0.600
0.5522
GENDER
-0.111806
0.0893028
-1.252
0.2178
C1 -0.164187
0.0973306
-1.687
0.0994
C2 -0.062799
0.1417171
2.407
0.5487
C3 -0.007576
0.1403651
-0.054
0.9572
T1 -0.113289
0.1448000
-0.782
0.4386
T2 -0.151477
0.096389
-1.572
0.1239
T3 0
-
-
-
Where: C1= CN
T1=REGIONAL HIGH SCHOOL
C2=CTE
T2=NATIONAL HIGH SCHOOL
C3=OTHER DEGREE
T3=PRIVATE HIGH SCHOOL
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Table 12. Univariate ANOVA with Social Science grade (Y3) as dependent
Source DF Sum
of
Mean
F Value
Prob>F
Square
Square
Model 3
1.55410
0.19426
5.622
0.0001
Error 45
1.38212
0.03455
C Total
48
2.93622
Root MSE
0.1858
Dep Mean
1.2551
R-square 0.5293
Adj R-sq
0.4351
C.V. 14.8103
Table 13. Parameter estimates for Social Science grade
Parameter
Parameter
Standard
T for HO:
Prob > |T|
Variable
Estimate
Error
Parameter = 0
Intercept 2.3820
1.37206
1.736
0.0902
HGWA 0.0094
0.01415
-0.668
0.5082
AGE -0.0224
0.01975
-1.135
0.2632
GENDER
-0.0215
0.06868
-0.312
0.7564
C1 -0.0626
0.07485
-0.837
0.4077
C2 0.2353
0.07984
2.947
0.0053
C3 0.3796
0.10795
3.517
0.0011
T1 0.1253
0.11136
1.125
0.2672
T2 -0.0001
0.07413
-0.002
0.9985
T3 0
-
-
-
Where: C1= CN
T1=REGIONAL HIGH SCHOOL
C2=CTE
T2=NATIONAL HIGH SCHOOL
C3=OTHER DEGREE
T3=PRIVATE HIGH SCHOOL
Multivariate Linear Regression Model for
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EDWARD ANTIPUESTO, EFRAIM DICKSON DULAWEN and BOBBY F.
ROARING. March 2009. Multivariate Linear Regression Model for Academic
Performance of the Freshmen Achiever at Benguet State University. Benguet State
University, La Trinidad , Benguet.
Adviser: DR. SALVACION Z. BELIGAN
ABSTRACT
The study was conducted to determine predictors of academic performance
among freshmen achievers using multivariate linear regression. The specific objectives of
the study were to1) Describe the relationships existing between the different factors and
the academic performance; 2) Find a suitable regression model that would predict the
best factor affecting academic performance among freshmen achievers; 3) Identify the
factors that may predict the student’s academic performance. High school average,
gender, course, age, and type of high they have attended were investigated as a possible
predictor of academic performance. Multivariate linear regression analysis was used to
perform to build a predictive model that could determine whether the following variables
could predict the academic performance of the freshman academic achievers.
The test for the significance of the regression coefficients, the F-test did not
support the rejection of the hypothesis that the different predictor variables have no
significant effect on the student’s performance in Math 11. In addition, none of the
regressors are contributory to the performance of the students in English and Soc Sci
grade.
Generally, fitting univariate regression model on the three dependent variables,
the test of significance for the beta coefficient was found not significant.
Fitting the multivariate regression model, the result showed significant result.
ii
TABLE OF CONTENT
Page
BIBLIOGRAPHY………………………………………………..…….………..…i
ABSTRACT……..…………………………………………………………….........i
TABLE OF CONTENT……..…………………….…………….…………………iii
INTRODUCTION
Background of the Study…………….…………………………………..…1
Statement of the Problem……….…………...………………………...……2
Objectives of the Study…….……………..…………………………...……3
Importance of the Study…..……………….………………………..……...4
Scope and Delimitation…….………………………...………………….….5
REVIEW OF RELATED LITERATURE…………...……….……………………6
THEORETICAL FRAMEWORK…………………………………………...……10
Definitions of terms ……………………………………………………….23
METHODOLOGY…………………………………..……………………….……25
RESULTS AND DISCUSSION………………………………………………..…27
SUMMARY, CONCLUSIONS,
RECOMMENDATION…….………………………………….…….……...….…34
LITERATURE CITED……………………………………….…………...………37
APPENDICES……………………………………………….………...………….39
iii
1
INTRODUCTION
Background of the study
The very heart of a learner’s education Mission is to have stable and
progressing academic achievements. The indicators of the learner’s academic
achievements are the marks and grades written on a learner’s official academic
records.
Educators have long been looking for a predictor of academic performance
for students as they enter college because low academic performance of students
means waste not only personal and societal time, as well as resources of the
educational institution. Students who can be identified as “at risk” for failure early
in their academic careers can be targeted for interventions that will increase the
likelihood of success. Because student academic performance are vital to students’
college life, understanding factors that diminish students’ satisfaction and
perseverance is necessary if these problems are to be addressed and overcome.
It is therefore important that school administrators should study the factors
contributory to the academic performance of every students.
A number of factors affecting academic performance have been studied by
several authors.
One of the studies reviewed is on the relationship between high school
grade point average (GPA) and college performance (Benford, 2006).
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Results show that high school performance of students graduated from
either private of public high school had a significant effect on their college
performance on the same subject.
Using multivariate regression analysis, Beligan (2004) found that students’
performance in Math - based subjects was affected by Age, Gender and the type of
high school they had attended.
Foreign studies also showed that the academic performance of a student is
multidetermined by number of contributing influences, and factors including
academic factors, personality variables, family characteristics, and environmental
factors (Benguiristain, 2003).
Student’s academic performance is, however, affected by multiple factors
and should not also be confined to one. To reach a valid generalization on students’
academic performance, we need to consider different factors. This study, therefore,
is expected to fill the gaps.
Statement of the problem
This study aimed to evaluate the factors of academic performance among
BSU freshmen academic achievers.
Specifically the study sought to answer the following questions:
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1. What are the relationships existing between the different factors
and to the academic performance?
2. What is the suitable regression model that would predict the best
factor affecting academic performance among freshmen achievers?
3. What are the factors that may predict the student’s academic
performance?
Objectives of the study
The purpose of this study is to determine the predictors of academic
performance among freshmen achievers using multivariate linear regression. The
specific objectives of the study were to:
1. Describe the relationships existing between the different factors and to
the academic performance.
2. Find a suitable regression model that would predict the best factor
affecting academic performance among freshmen achievers.
3. Identify the factors that may predict the student’s academic performance.
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Significance of the study
This study would shed the light to further strengthen the existing foundation
by determining the predictors of academic performance among the freshmen
achievers.
For the students, this study would help them to acquire strong determination
in carrying their studies and become aware of their responsibilities and for them to
be academically productive.
The study would also help school administrators as well as the parents and
guardians in formulating and knowing the best predictors of academic success.
In addition, work in this area has the potential to provide important
suggestions to improve standards and quality of education and also the performance
of the students.
To assist faculty in helping students achieve academic success, variables
should be identified to help predict whether or not students might have difficulties
in achieving academic success, and therefore whether or not those students will
have difficulties in completing a degree.
The results of this study would help identify students possibly risk for
failure so that interventional measures can be developed and implemented to
promote educational productivity and reduce attrition.
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Scope and Delimitation
Several limitations to this study existed. A population consisted of freshmen
academic achievers enrolled at Benguet State University during school year 2008-
2009. They were chosen because they have a minor subject which was used as their
output variable. In addition, independent variables were also limited.
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REVIEW OF RELATED LITERATURE
What makes a successful person? Vince Lombardi observed, “The
difference between a successful person and others is not a lack of strength, not a
lack of knowledge, but rather in a lack of will” (The Official Site of Vince
Lambordi, n.d.). He recognized, as do so many of us, that motivation may be the
most important factor in determining success. In the field of research, however,
motivation and biographical factors conceptually related to success can not be
directly measured and are difficult to manipulate as variables. Therefore, a limited
number of studies have successfully linked such factors to success in specific areas.
In educational environment, a student motivation, success orientation, and network
of support can surpass the importance of writing and reading skills.
Analyzing multivariate context data using a series of univariate methods
compromises the very essence and richness of a multifaceted phenomenon, such as
education production.
The measurements often involves the use of linear regression, correlation,
or multivariate-discriminant analysis to indicate the nature of the relationship
(Goldhaber & Brewer, 1997a; Monk, 1992; Montmarquette & Majseredjian, 1989;
Montimore, Simmons, Stoll, Lewis, & Ecob, 1988; Murnane, 1975;Smith &
Tomlinson 1989; Walberg, 1982; Walberg & Fowler, 1989; Walberg & Weintien,
1982; Wenglinsky, 1997). Approximately 400 studies have been reported including
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those that have been evaluated and summarized using meta-analysis (Greenwald, et
al., 1996; Hanushek, 1997; Hedges, Laine & Greenwald , 1994).
Most of these academic performance studies examined the relationship
between two or more variables and one educational output measure (usually student
achievement) using univariate analysis (Hanushek, 1997).
In education, there is often more than one output measure for a given set of
inputs. Academic performance studies must examine the relationship between sets
of variables and sets of outputs measures as multivariate composites (Pedhazur,
1997).
Heinbuch and Samuels (1995) and Tatsuoka (1988) have noted that very
little concerted effort has been made to establish the nature of the relationship
between multiple educational inputs and multiple measures of student achievement
using multivariate methods. Pedhazur also noted that simply calculating zero-order
correlations for all possible pairs of variables using univariate tests on data with
multivariate constructs “ . . . affects the prescribed α level” (p.895) thus increasing
the chance for a Type 1 error. Moreover, analyzing multivariate context data using
a series of univariate methods compromises the very essence and richness of a
multifaceted phenomenon, such as education production.
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In a multivariate context, education production examines the relationship
between a set of two or more educational input variables and a set of two or more
educational output variables (Pedhazur 1997; Tatsuoka, 1973; Tatsuoka, 1988).
However, colleges and universities do not currently acknowledge such
factors in their selection process, which is intended to choose student most likely to
succeed at their school. Most colleges and universities limit their admission
decisions to the traditional cognitive predictors of academic performance, usually
high school general weighted average (GWA), scores in standardized tests such as
the SAT, and high school rank (Aiken, 1964).
Another study conducted by Reiter (1964) tried to determine if there was a
relationship between non-academic factors and academic performance of college
freshmen and sophomore students. Participants were 76 randomly selected male
and female college students from an introductory psychology course. His study
shows that high school achievements were a superior predictor of academic success
(Reiter, 1964). Also, Reiter (1964) stated that non-intellectual factors, such as
anxiety, academic discipline, and SAT composite scores, were not proven to be true
predictors of academic success.
A number of studies are shown that several variables other than academic
characteristics are associated with success in college. Larose, Robertson, Roy, and
Legault (1998) argue that the frequency and quality of the interactions between
students and their faculty are related to higher levels of college success. Other
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variables related to college success are the ability to adapt to the college
environment, personal motivation, and students’ relationship with their peers
(Larose et al., 1998). In addition to these findings, Aiken (1964) suggests that
students who withdraw from classes are less motivated to achieve than students
who do not withdraw.
A study conducted buy Naumann, Bendalos, and Gutkin (2003) at
Midwestern University implies that first-generation college students rely more
heavily on motivational factors in order to succeed than do second-generation
students. These results were not surprising given “first-generation students did not
typically have the same source of support throughout careers as did second-
generation students” (Naumann, Bendalos, and Gutkin, 2003, p.8).
Gregorc (1979) described a person’s learning style as consisting of distinct
behaviors which serve as stable indicators of how a person learns and adapts to
his/her learning environment. The most extensively researched and applied learning
style construct has been the field-dependence/independence dimension (Guild and
Garger, 1985).
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THEORETICAL FRAMEWORK
Multivariate Linear Regression
Multivariate linear models are, in one sense, like repeated measures models
– in both cases, it may be helpful to think of the analysis as involving multiple
dependent measures. In repeated measures analysis, the multiple dependent
variables are the same measure, repeated (usually over time). In multivariate
models, the multiple dependent variables are measures of multiple outcomes,
usually measured with different metrics, and usually measured at the same point in
time.
The coefficient of the linear regression model is estimated under the
assumption that the random term assumes normal distribution with zero mean and
constant variance. The values of the random term are also assumed to be
independent.
The general multivariate model may be expressed as:
a1Y1 + a2Y2 + a3Y3 +…..+akYk = b0 + b1X1 + b2X2 + b3X3 + ……. bmXm
Where a1, a2, a3,….., ak are the coefficients associated with Y1,Y2,Y3,…,Yk
the linear composite of educational output measures : b0 is the intercept of the
composite regression line on line linear composite f output measures, and
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b1,b2,b3,….,bm are the regression coefficients associated with X1,X2,X3,…,Xm the
linear composite of educational input variables.
This study used multivariate methods to examine the relationships between
a set of three educational input variables and two educational output measures.
Based on the multivariate model explained above, the overall production function
model for this study is:
a1Y1 + a2Y2 + a3Y3= b0 + b1X1 + b2X2 + b3X3 + ……. b5X5
Where X1, X2, X3, X4, X4, X5, are thirteen educational input variables;
Y1,Y2,Y3 are the four output measures a1,a2,a3 are the coefficients or weights of the
linear composite of outputs in the regression model ; b1, b2, b3,….,b5 are the
coefficients or weights of the linear composite of inputs in the regression model.
Let us assume that there are p response variates Y’=Y1, Y2,….Yp and that
these variables and linearly related with r-fixed variates X1, X2,….,Xr. The
theoretical model for response variables is
E(Yk/X)= βok + β1kX1+…+βrkXr.
In matrix notation, the above model can be expressed as
Yj’= Xj’β+ ej’,
j=1,2,…,N
Y=Xβ + Є
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Where Y11 Y12……..Y1p
Y21 Y22……..Y2p
Y=
:
:
:
YN1 YN2……..YNp N x p
1
X11………X1r
1
X21………X2r
X=
:
:
:
1 XN1……..XNr N x r + 1
β= is the unknown parameter
βo
=
β1
:
βrk
Є = is the matrix of random error
Where e11
e12……..e1p
e21
e22………e2p
Є=
:
:
:
eN1 eN2……..eNp
N x p
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e1’
=
e2’
:
eN’
Each vector ej’ will be assumed to have the following properties:
(1) E(ej)= 0 for all j
(2) E(ej ej’)= ∑ for all j and j’
Where ∑ is the variance-covariance matrix of the random vector ej defined
as
σ11
σ12…….. σ1p
σ21
σ22……… σ2p
∑ =
:
:
:
σN1 σN2…….. σNp
(3) ej ~ Np(0 ∑) for all j
(4) For any I1 the N x 1 random vector ej will be assume to satisfy E(ej)= 0 and
E(e
2
j ej’)= σi (IN)
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Estimation of the parameters β
Similar to the multiple linear regression, the method of least squares is need
to estimate the matrix β.
Using the model
Yj’= X’jβ + ej’
for j=1, 2,…, N
The matrix to be minimized is (e’ e) Є Є’ which is the matrix of error sums
of squares and sums of products.
Since
Є = Y – Xβ and Є’ = Y’ – β’X’
Then
Є’Є = (Y’ – β’X’)(Y – Xβ)
=Y’Y - 2β’X’Y + β’X’X
Taking the partial derivation of Є’Є with respect to β and equates the
results to 0, we would get
Є’Є= -2Y’Y + 2X’Xβ = 0
The resulting normal equations are:
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X’Xβ= X’Y
Where,
X’X is the (r+1) x (r+1) matrix of unconnected sum of squares and sums of
products
N
∑Xj1…………∑X jr
∑Xj1 ∑Xj12……… ∑Xj1Xjr
X’X =
:
:
:
∑Xjr ∑XjrXjl……. ∑Xjr2
β =(βim) is the (r+1) x p matrix of estimated intercepts and partial regression
coefficients
β01
β02…….. β0p
β11
β12……… β1p
β =
:
:
:
βr1 βr2……… βrp
β =
β1
β2…….. βp
And
X’Y is the (r+1) x p matrix of uncorrected sums of products of X and Y:
Multivariate Linear Regression Model for
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1
1 …….. 1
Y11 Y
12…….. Y1p
β11
β12……… β1p
Y 21 Y22……… Y 2p
β =
:
:
:
:
:
:
βr1 βr2……… βrp
YN1 YN2……… YNp
∑Yj1
∑Yj2…………… ∑Y jp
∑Xj1Yj1
∑Xj1Yj2……… ∑Xj1Yjp
X’X = :
: :
∑XjnYj1 ∑XjrYj2………. ∑XjrYjp
g
01
g02…….. g0p
g11
g12……… g1p
= :
:
:
g
r1 gr2……… grp
It will be assumed that the matrix X’X is nonsingular so that the solution
to the equation (X’X)β=X’Y is
β = (X’X)-1X’Y
= WG
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Where
w
00
w01…….. w0r
W = (X’X)-1= w10
w11……...w1r
:
:
:
wr0 wr1……. wrn
After multiplying W x G, we would get these particular solutions:
Intercepts:β0
βoi=∑(Wok)(gki),
i=1,2,…p
Partial regression coefficients
βi=βim=∑(Wik)(gkm), i.m=2,…….
Finally, the estimated p regression equations are:
Ŷ1 = β01 + β 11X1 + β r1Xr
Ŷ2 = β02 + β 12X1 + β r2Xr
Ŷp = β0p + β1pX1 + βrpXr
Multivariate Linear Regression Model for
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Estimation of ∑
The estimate of the variance-covariance matrix ∑ will be derived from the
estimate residual matrix Є = Y – Xβ.
Є’Є = (Y –Xβ)’(Y-Xβ)
= (Y’- β’X’)(Y –Xβ)
= Y’Y – βX’Y – Y’Xβ + β’X’Xβ
From the normal equation (X’X)β= X’Y
Є’Є = Y’Y – β’X’Y
The elements of Є’Є are defined:
eil = Yi’ Yl - ∑
eim= Yi’Ym - ∑
From the previous section, the residual matrix Є’Є can be written as
Є’Є = E’ [I – XWX’]E
And
E(e’e)=
E{
Є’[ I – XWX’]Є}
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= ∑ tr [ I –XWX’]
= ∑ [ tr (IN) – tr (XWX’)]
= ∑ ( N – (r+1)) = ∑ (N-r-1)
Therefore, the unbiased estimator of ∑ is ∑’
∑’ = S =
The elements of S are:
Sample variances= S 2
i/x = Yi’ Yi – βigi
= ∑
2 ∑
Sample covariance: Sim
Sim = Yi’Ym – βi’gm
= ∑YjiYjm – ∑
Estimate of the variance of β
By
definition,
V(β) = E[β’ – β][β’ – β]’
But
β’=WX’Y = WX’(Xβ+Є)
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=WX’Xβ + WX’Є
Therefore,
β’ – β = WX’Є and
(β’- β) (β’- β) = (WX’Є) (WX’Є)’
=WX’(ЄЄ’)XW’
E(β’- β) (β’- β)’ = E[WX’(ЄЄ’) XW’
= WX’ E(ЄЄ’) XW’
= WX’ (∑(IN)XW’
=∑(W)
Hence,
V(β)= ∑(W)
Where
V(β) = Var (β1,β2,…..βp)
Var(β1) Cov(β1β2) ….. Cov(β1βp)
= Cov(β1β2) Var(β2) …..
Cov(β2βp)
:
:
:
Cov(βpβ1) Cov(βpβ2) Var(βp)
Multivariate Linear Regression Model for
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S 2
1/x
S12/x
S1p/x
∑(W) =
S2
2
1/x
S2/x
S2p/x W
: : :
S
2
p1/x
Sp2/x
Sp/x
Therefore,
Var(βi) = (Si/x)(W) for
i=1,2,……p
Cov(βiβm)= (Sim/x)(W) i m=1,2,…..p
Test of hypothesis
Ho: CB=K
Where:
C is the nh x (r+1) of rank nh
K is the nh x p matrix of constants = 0
A. The Union-Intersection Test
1. Compute the matrix of sum of squares due to the hypothesis
Ho :
H= (Cβ’-K)’ [CWC’] (Cβ’-K)
2. Compute the matrix of sums of squares and sums of products,E.
E=Y’Y -β’X’X
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3. Obtain the largest characteristics root of HE-1. Let C=char(max) HE-1. The
numbers of characteristics roots of HE-1 . Since rank of H is min(nh,p),
hence rank HE-1 is min(nh,p). Therefore the degree of the characteristics
polynomial is nh.
4. Compute the statistics θ,
Θ=
5. Compare the value of θ with the tabular value θα(s,m,n) where
s= min(VH,p) VH= nh
m=|
|
n=
Reject Ho: CB=K if θ θα(s,m,n)
The equivalent F-statistics is
Fc=
c(n+1)/(m+1)
which follows an F-distribution with 2m+2 and 2n+2 degrees of freedom
B. The Likelihood Ratio Test or Wilk’s Lambda
After computing H and E as given above obtain the statistic U
U= | | U
|
|
α(p,VH,VE)
The table below gives the transform of U to provide upper tail tests using F-
distribution.
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Table 1. Transformation of U with upper tail tests using F-distribution.
Parameters
Statistics having F-distribution
Degrees of
Freedom
VH = 1 for any p
1
p, V
E+ VH -p
VH = 2 for any p
1
√
1
2p,2(VE+ VH -p)
√
p=1 any VH
1
V
H, VE
p=2 any VH
1
√
2VH, 2VE
√
Definition of terms
Academic Performance - refers to the performance that is characterized by a
narrow concern for book learning and formal rules, without knowledge or
experience of practical matters.
Community Ability - refers to the performance of students in Basic English
subjects.
Dummy variables-These variables may be thought of as additional variables
for which statistical adjustment is desired.
Multivariate Linear Regression Model for
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Mathematical Ability - refers to the performance of the students in
mathematics.
Multivariate Analysis - comprises a set of techniques dedicated to the
analysis of data sets with more than one variable.
Predictor Variables - refers to the variables from which projections are
made in a prediction study.
Regression Analysis - refers to the estimation of the linear relationship
between a dependent variable and one or more independent variables or covariates.
Study Habits - giving proper time and energy to your studies and various
“tricks” to help you study more effectively.
Predictor Variable - refer to the variable that are use to predict the values of
another variable in a mode.
Multivariate Linear Regression Model for
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METHODOLOGY
Location of the study
The study was conducted at Benguet State University during the 2
semester of the school year 2008-2009 at La Trinidad, Benguet.
Respondents of the Study
The respondents of the study were all the forty eight freshmen academic
achievers of School Year 2008-2009.
Data gathering procedure
The grades of the selected achievers were obtained from the Office of the
University Registrar and Office of the Students’ Affairs of Benguet State
University instrumentation.
A questionnaire was prepared and floated to obtain the age, course, school
attended, and high school average and gender data.
Statistical analysis of data
The gathered and tabulated data were analyzed using of descriptive and
inferential methods. The descriptive method includes the analysis of the data using
frequency, percentage, mean, and rank.
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The inferential method utilized in determining the influence of different
factors on the grades of the student’s achievers. The dependent variables are the
grades in the subjects namely: Math, English and Social Science. The independent
variables include the High School Grade, Age of the respondents, Gender, Type of
High School attended and the course. For qualitative independent variables,
dummies were defined and entered into the model.
Fitting the multiple regression and multivariate regression models, the use
of the PROC REG procedure of the SAS software was compelled to facilitate the
computations.
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RESULTS AND DISCUSSION
Descriptive Statistics
Table 2 shows the computed means and standard deviation of the grades on
the three dependent variables (Math 11, English 11, and Social Science 11) and two
independent variables, (High school grade and Age). The respondents’ earned
grades in the three subjects as the dependent variables ranged from 1.26 to 1.60
with social science having the least grades of 1.26 and math having the highest
average of 1.68. Their high school mean grade and age of the respondents were
90.2% and 16.98% respectively.
Table 2. Mean and standard deviation of the variables.
VARIABLE N MEAN
STANDARD
MAX MIN
DEVIATION
Math 11
49
1.68
0.35
1.00
2.75
English 11
49
1.60
0.24
1.25
2.25
Soc sci 11
49
1.26
0.25
1.00
1.75
HS GRADE(X1)
49
90.28
2.43
84.33
95.59
AGE(X2) 49
16.98
1.49
16.00
24
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Table 3 presents the distribution of achievers by course. About 35 percent
of the achievers are enrolled in the nursing course and about 29 and 14 percent are
taking up BSE and BEE respectively. The remaining 22 percent of the achievers are
taking up other degree programs such as: DVM, AET, BSF, BSAS, BLIS, BSES,
and BSA. It can be seen in the table that most of the achiever are dominated by
Nursing and Education students
On the distribution of achievers according to type of High School attended,
majority of the respondents are graduates of the National High School (67.35%).
Achievers from private schools comprise the 20 percent of the respondents. The
remaining 10 percent of the achiever respondents are from Barangay/Regional High
School.
Table 3. Distribution of respondents by course, type of high school graduated, and
gender
COURSE COURSE PERCENT CUMULATIVE
CUMULATIVE
PERCENT
FREQUENCY
BSN 17
0.35
0.35
17
BEE&BSE 21
0.43
0.78 38
OTHERDEGREE 11
0.22
100.00
49
REGIONAL HS
5
0.10
0.12
6
NATIONAL HS
33
0.67
0.80
39
PRIVATE HS
10
0.20
100.00
49
FEMALE 36
0.73
0.73
39
MALE 13
0.27
100.00
49
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As to gender, female achievers outnumbered male achievers. About 73
percent of the academic achievers of the Benguet State University are females and
the remaining are males.
Univariate Multiple Regression Model
Before fitting a multivariate regression model, a univariate multiple linear
model was first fitted for each dependent variable. Taking Math as the first
dependent variable, the univariate multiple linear regression model gave the
following results in Table 4.
Table 4. Beta coefficients for Math grade
DEPENDENT
β0 INTERCEPT
HS AVERAGE
AGE (β2)
VARIABLE
(β1)
YC1T1 0.612
0.010279
-0.00607
YC1T2 0.603
0.010279
-0.00607
YC1T3 0.472
0.010279
-0.00607
YC2T1 0.948
0.010279
-0.00607
YC2T2 0.940
0.010279
-0.00607
YC2T3 0.808
0.010279
-0.00607
YC3T1 0.887
0.010279
-0.00607
YC3T2 0.878
0.010279
-0.00607
YC3T3 0.747
0.010279
-0.00607
Where: C1= CN
T1=REGIONAL HIGH SCHOOL
C2=CTE
T2=NATIONAL HIGH SCHOOL
C3=OTHER DEGREE
T3=PRIVATE HIGH SCHOOL
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As to the goodness – of – fit of the fitted regression model given below,
Math = HGWA + Age + Gender (Dummy) + Course (Dummy) +
Type of HS (Dummy)
only 11 percent of the variability in Math grades can be explained by the
independent variables in the model.
Moreover, the test for the significance of the regression coefficients, the F-
test did not support the rejection of the hypothesis that the different predictor
variables have no significant effect on the student’s performance in Math 11.
Regressing the same predictors on the student’s performance in English as
the dependent variable, the independent variables in the fitted model,
English = HGWA + Age + Gender (Dummy) + Course (Dummy) +
Type of HS (Dummy)
they explained only 18.06 percent of the variability in the English grades obtained
by the respondents.
The significance of the effects of the different regressors was found not
significant. This means that none of the regressors are contributory to the
performance of the students in English. The models derived for the combination of
the different dummy variables are shown in Table 5.
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Table 5. Beta coefficients for English grade
DEPENDENT
β0 INTERCEPT
HS AVERAGE
AGE (β2)
VARIABLE
(β1)
YC1T1
2.995 -0.02024
0.015401
YC1T2
2.957 -0.02024
0.015401
YC1T3
3.088 -0.02024
0.015401
YC2T1
3.097 -0.02024
0.015401
YC2T2
3.059 -0.02024
0.015401
YC2T3
3.190 -0.02024
0.015401
YC3T1
3.152 -0.02024
0.015401
YC3T2
3.114 -0.02024
0.015401
YC3T3
3.245 -0.02024
0.015401
Where: C1= CN
T1=REGIONAL HIGH SCHOOL
C2=CTE
T2=NATIONAL HIGH SCHOOL
C3=OTHER DEGREE
T3=PRIVATE HIGH SCHOOL
Moreover, using the third dependent variable, the Social Science grade,
with the same predictor variables, about 53% in the variability in Social Science
grade can be attributed to HGWA, Age, Gender, Course and Type of High School
attended.
English = HGWA + Age + Gender (Dummy) + Course (Dummy) +
Type of HS (Dummy)
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Table 6. Beta coefficients for Social Science grade
DEPENDENT
β0 INTERCEPT
HS AVERAGE
AGE (β2)
VARIABLE
(β1)
YC1T1
2.466 -0.009 -0.022
YC1T2
2.340 -0.009 -0.022
YC1T3
2.331 -0.009 -0.022
YC2T1
2.764 -0.009 -0.022
YC2T2
2.638 -0.009 -0.022
YC2T3
2.629 -0.009 -0.022
YC3T1
2.908 -0.009 -0.022
YC3T2
3.114 -0.009 -0.022
YC3T3
3.245 -0.009 -0.022
Where: C1= CN
T1=REGIONAL HIGH SCHOOL
C2=CTE
T2=NATIONAL HIGH SCHOOL
C3=OTHER DEGREE
T3=PRIVATE HIGH SCHOOL
However, the test of significance for the beta coefficient is not significant.
Taking Social Science as the third dependent variable, the univariate multiple linear
model gave the following results in Table 6.
Multivariate Regression Model
Fitting a multivariate regression model with the grades in the three subjects
as dependent variables using the same predictor variables, the MANOVA test gave
significant result as shown in table 7
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Generally, fitting univariate regression model on the three dependent
variables, the test of significance for the beta coefficient was found not significant.
Fitting the multivariate regression model, the result showed significant
result.
Table 7. Multivariate test for the overall model.
Statistic Value
F
Pr>F
Wilks' Lambda
0.31987
2.2229
0.0028
Pillai's Trace
0.85308
1.9867
0.0083
Hotelling-Lawley Trace
1.61989
2.4748
0.0008
Roy's Greatest Root
1.26915
6.3457
0.0001
Eigenvalues Math:
0.56
English : 0.19
SocSci : 0.10
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SUMMARY, RECCOMENDATION AND CONCLUSION
This study was conducted to determine the predictors of academic
performance among academic achievers using Multivariate Linear Regression.
Specifically the study aim to 1) describes the relationship existing between the
different factors and to the academic performance. 2) Find a suitable regression
model that would predict the best factors affecting academic performance of
achievers. 3) Identify the factors that may predict the student’s academic
performance.
The variables collected are information about to those freshmen achievers
on there grade in Math 11, English 11, Soc Sci 11, High School grade, Gender,
Age, Course and Type of High School. These variables are gathered using a
questionable while the number of student was gathered from the Registrar’s office
and the Office of the Student’s Affairs. Out of the thousand students of the
university, we just selected the 49 freshmen academic achievers.
The statistical tool used in this study was Multivariate Regression Analysis.
This statistical was used to accommodate the multivariate nature of educational
input and output measures. Prior to analysis, all variables in the data set were
examined through SAS program for accuracy of data entry. A Multivariate
Regression Analysis was performed in order to determine the predictive
Multivariate Linear Regression Model for
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35
capabilities of the independent variables. The Multivariate Regression Analysis
indicated that the composite of predictor had a highly significant linear relationship
with the output measures (Math 11, English 11 and Soc Sci. In addition, there are
more given effect of the independent variable that contributes to the model.
Based on the conclusion, it is recommended that the result of this study may
be out of great help to the college. It maybe of help in counseling and guidance
among the university on some factors that might predict academic performance to
the student. Knowledge of the student’s predictor’s will guide in strengthening their
stat. Strategies to suit the need of student’s.
Based on the results, the researchers recommend the following:
1. The results of the study could help in fostering student’s awareness on
the factors that could affect their academic performance.
2. The results of the study may also be of great help to parents and
guardians in motivating their children as well as in knowing the factors that affect
their children’s’ achievement.
3. Further investigation of this study is also recommended.
In conclusion, if a goal of the university is to be efficient and effective at
educating its students, and subsequently have them graduate, then further research
is needed to identify if other variables exist that can predict students' ability to
Multivariate Linear Regression Model for
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improve their academic performance. Future studies should continue to explore
additional variables as well as investigate student performance beyond the first year
in order to confirm the trend for effectiveness of these predictors.
Multivariate Linear Regression Model for
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HANUSHEK, E. A. (1997) Assessing the effect of school resources and student
performance: An update. Education Evaluation and Policy Analysis, 19,
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HEINBUCH, S. E., & SAMUELS, J.A. (1995). Getting more bang for the
education buck.Public Productivity & Management Review, 18(3), 233-
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JOHNSON, R. K (2003). Intrinsic and Extrinsic Motivations Predict Academic
Performance of College Students. Morehouse College
MONK D. H. (1992). Education resources for improving schools: A case study of
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NURKOWSKI, L. C(1995). Transfer Student Persistence and Academic Success
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Multivariate Linear Regression Model for
Academic Performance of the Freshmen
Achiever at Benguet State University / Antipuesto, Edward; et al. 2009
39
APPENDIX
Appendix A
COLLEGE OF ARTS AND SCIENCES
MATH-PHYSICS-STATISTICS DEPARTMENT
APPLICATION FOR MANUSCRIPT/PROJECT DEFENSE
Date: ________
Group Members:
Edward Antipuesto
Efraim
Dickson
Dulawen
Major
Field:
Statistics
Bobby F. Roaring
Minor Field: Info. Tech.
Degree: Bachelor of Science in Applied Statistics
Title of Thesis:
“Multivariate Linear Regression Model for Academic Performance of the Freshmen Achiever
at Benguet State University”
Endorsed: Dr. Salvacion Z. Beligan
(Adviser)
Date and Time of Defense: March 19, 2009, 11:00 AM
Place of Defense: CAS An 210
Noted: Maria Azucena B. Lubrica
Department Chairman
Report of result of manuscript/project defense
Name and Signature
Remarks*
_____________________ __________________
Adviser
_____________________ __________________
Member
_____________________ __________________
Member
_____________________ ___________________
MARIA AZUCENA B. LUBRICA
Department chairman
Multivariate Linear Regression Model for
Academic Performance of the Freshmen
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40
Appendix B
Benguet State University
College of Arts and Sciences
Math-Physics-Statistics Department
La Trinidad, Benguet
February 9, 2009
DR. EDNA CHUA
Office of the Student Affairs
Benguet State University
MADAME:
We, the undersigned fourth year students taking up Bachelor of Sciences in Applied Statistics at Benguet State University,
are conducting a research entitled “MULTIVARIATE REGRESSION ANALYSIS OF ACADEMIC PERFORMANCE OF
THE FRESHMEN ACHIEVERS AT BENGUET STATE UNIVERSITY”.
In view hereof, we would like to request permission from your good office to gather the following:
a) High school average
b) IQ
c) Entrance Exam Results of the Freshmen Academic Achievers.
Attached to this letter are the names and course of the freshmen academic achiever.
Thank you very much for your favorable consideration.
Respectfully yours,
EDWARD ANTIPUESTO
EFRAIM DICKSON L. DULAWEN
BOBBY F. ROARING
Researchers
Noted:
DR. SALVACION Z. BELIGAN
Adviser
Approved:
DR. EDNA CHUA
Office of the Student Affairs
Multivariate Linear Regression Model for
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41
Appendix C
Benguet State University
College of Arts and Sciences
Math-Physics-Statistics Department
La Trinidad, Benguet
February 9, 2009
DR. MARLENE B. ATINYAO
Office of the University Registrar
Benguet State University
MADAME:
We, the undersigned fourth year students taking up Bachelor of Sciences in Applied Statistics at Benguet State
University, are conducting a research entitled “MULTIVARIATE REGRESSION ANALYSIS OF ACADEMIC
PERFORMANCE OF THE FRESHMEN ACHIEVERS AT BENGUET STATE UNIVERSITY”.
In view hereof, we would like to request permission from your good office to gather the following:
a) High school average
b) IQ
c) Entrance Exam Results of the Freshmen Academic Achievers.
Attached to this letter are the names and course of the freshmen academic achiever.
Thank you very much for your favorable consideration.
Respectfully yours,
EDWARD ANTIPUESTO
EFRAIM DICKSON L. DULAWEN
BOBBY F. ROARING
Researchers
Noted:
DR. SALVACION Z. BELIGAN
Adviser
Approved:
DR. MARLENE B. ATINYAO
Director, OUR
Multivariate Linear Regression Model for
Academic Performance of the Freshmen
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42
Appendix D
Benguet State University
College of Arts and Sciences
Math-Physics-Statistics Department
La Trinidad, Benguet
Dear Respondents,
The undersigned Bachelor of Science in Applied Statistics students are conducting a research entitled, “MULTIVARIATE
REGRESSION ANALYSIS OF ACADEMIC PERFORMANCE OF THE FRESHMEN ACHIEVERS AT BENGUET
STATE UNIVERSITY”. In this connection, may we request that you provide us with the information needed?
Your truthful and complete answers will help the success of this study. Rest assured that answers would be treated
confidentially.
Heartfelt thanks for your anticipated cooperation.
Reverently yours,
_____________________
EDWARD ANTIPUESTO
_____________________________
Noted:
EFRAIM DICKSON L. DULAWEN
___________________
__________________________
BOBBY F. ROARING
DR. SALVACION Z. BELIGAN
Researchers
Adviser
SURVEY QUESTIONNAIRE
INSTRUCTION: Please fill in the blank or put a check (
) mark on the given space with the correct entries. All entries will
be regarded as strictly confidential.
RESPONDENTS PROFILE
1. Name: ______________________
2. Age: _______________________
3. Gender: [ ]-Male
[ ]-Female
4. Course: _____________________
5. Type of high school graduated:
[ ]-Barangay High School
[ ]-City/Regional High School
[ ]-National High School
[ ]-Private High School
6. What was your High School average (Pls. specify)? ________
Multivariate Linear Regression Model for Academic Performance of the Freshmen
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43
Appendix E
RAW DATA
MATH
ENGLISH SOC SCI
HS
AGE GENDER
COURSE TYPE
OF
11
11
11
GRADE
HS
1.75 1.50 1.00
91.36
17 0
1
3
1.75 1.25 1.00
93.13
17 0
1
3
1.75 1.25 1.00
94.00
16 0
1
4
1.50 1.50 1.00
89.24
16 1
1
3
1.50 1.50 1.00
91.94
16 0
1
4
1.50 1.50 1.00
90.80
16 0
1
3
1.50 1.50 1.00
92.83
17 0
1
3
1.50 1.25 1.00
91.26
16 0
1
3
1.50 1.50 1.50
92.51
16 0
1
2
1.50 1.50 1.25
89.69
17 0
1
3
1.50 1.50 1.25
91.72
17 0
1
3
1.50 1.50 1.25
89.55
16 1
1
3
1.50 2.00 1.00
90.08
17 0
1
3
1.50 1.50 1.00 8.54 17 1
1
4
1.50 1.50 1.25
90.77
16 0
1
3
1.50 1.75 1.00
91.62
17 0
1
4
2.00 1.50 1.00
95.59
17 0
1
4
1.50 1.25 1.00
87.75
17 1
2
3
1.25 1.25 1.25
88.00
16 0
2
2
1.50 1.75 1.25
93.33
16 0
2
4
1.50 1.75 1.25
91.79
22 1
2
3
2.25 1.50 1.25
89.65
19 0
2
3
1.75 1.25 1.50
89.59
18 1
2
3
2.75 2.00 1.25
91.07
17 0
2
2
1.50 2.00 1.25
86.00
18 0
2
3
2.00 1.75 1.50
90.03
16 0
2
3
1.75 1.50 1.50
91.38
16 0
2
3
Multivariate Linear Regression Model for
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2.25 1.50 1.75
88.75
16 0
2
3
2.00 1.75 1.50
93.40
17 0
2
4
2.25 1.50 1.50
89.93
18 0
2
3
2.00 2.00 1.75
86.74
16 1
2
4
1.00 1.50 1.75
91.20
18 1
3
4
1.25 2.00 1.25
85.95
18 1
3
4
1.50 1.75 1.75
88.38
17 0
3
2
2.00 1.50 1.50
85.20
16 1
3
3
1.75 1.75 1.50
88.26
16 1
3
3
2.50 1.75 1.50
90.10
17 0
3
3
2.00 1.75 1.50
84.33
24 0
3
3
1.25 1.50 1.00
88.75
16 0
4
3
1.75 1.50 1.00
92.38
17 0
4
3
1.25 1.25 1.00
92.67
17 1
4
3
1.50 1.50 1.25
87.84
16 0
5
3
1.50 1.50 1.00
91.15
18 0
5
3
2.25 1.75 1.50
88.35
16 0
6
2
1.25 2.00 1.25
90.53
17 0
7
3
1.75 1.75 1.00
94.82
16 0
8
3
1.50 1.75 1.25
88.59
18 1
9
3
1.75 2.25 1.00
92.00
16 0
10
3
1.75 1.25 1.50
91.00
16 0
11
3
Multivariate Linear Regression Model for
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Appendix F
SAMPLE RESULT OF MULTIVARIATE REGRESSION ANALYSIS
Model: MODEL1
Dependent Variable: MATH
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 8 1.49237 0.18655 1.714 0.1251
Error 40 4.35457 0.10886
C Total 48 5.84694
Multivariate Linear Reg
ression Model for Academic Performance of the Freshmen
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Root MSE 0.32995 R‐square 0.2552
Dep Mean 1.68367 Adj R‐sq 0.1063
C.V. 19.59678
NOTE: Model is not full rank. Least‐squares solutions for the parameters are not unique. Some
statistics will be misleading. A reported DF of 0 or B means that the estimate is biased.
The following parameters have been set to 0, since the variables are a linear combination
of
other variables as shown.
T3 = +1.0000 * INTERCEP ‐1.0000 * T1 ‐1.0000 * T2
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP B 0.694579 2.43541495 0.285 0.7770
Multivariate Linear Reg
ression Model for Academic Performance of the Freshmen
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HGWA 1 0.010279 0.02512236 0.409 0.6846
AGE 1 ‐0.006073 0.03506443 ‐0.173 0.8634
GENDER 1 ‐0.227511 0.12190480 ‐1.866 0.0693
C1 1 0.004804 0.13286341 0.036 0.9713
C2 1 0.341181 0.14171705 2.407 0.0208
C3 1 0.279586 0.19160862 1.459 0.1523
T1 B 0.140017 0.19766260 0.708 0.4828
T2 B 0.102657 0.13157800 0.780 0.4399
T3 0 0 . . .
Dependent Variable: ENG
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Multivariate Linear Reg
ression Model for Academic Performance of the Freshmen
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Model 8 0.51518 0.06440 1.102 0.3821
Error 40 2.33686 0.05842
C Total 48 2.85204
Root MSE 0.24171 R‐square 0.1806
Dep Mean 1.59694 Adj R‐sq 0.0168
C.V. 15.13556
NOTE: Model is not full rank. Least‐squares solutions for the parameters are not unique. Some
statistics will be misleading. A reported DF of 0 or B means that the estimate is biased.
The following parameters have been set to 0, since the variables are a linear combination
of
other variables as shown.
T3 = +1.0000 * INTERCEP ‐1.0000 * T1 ‐1.0000 * T2
Multivariate Linear Reg
ression Model for Academic Performance of the Freshmen
Achiever at Benguet State University / Antipuesto, Edward; et al. 2009
49
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP B 3.384601 1.78409146 1.897 0.0651
HGWA 1 ‐0.020236 0.01840368 ‐1.100 0.2781
AGE 1 0.015401 0.02568685 0.600 0.5522
GENDER 1 ‐0.111806 0.08930277 ‐1.252 0.2178
C1 1 ‐0.164187 0.09733063 ‐1.687 0.0994
C2 1 ‐0.062799 0.10381647 ‐0.605 0.5487
C3 1 ‐0.007576 0.14036511 ‐0.054 0.9572
T1 B ‐0.113289 0.14480003 ‐0.782 0.4386
T2 B ‐0.151477 0.09638899 ‐1.572 0.1239
T3 0 0 . . .
Multivariate Linear Reg
ression Model for Academic Performance of the Freshmen
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Dependent Variable: SOC
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 8 1.55410 0.19426 5.622 0.0001
Error 40 1.38212 0.03455
C Total 48 2.93622
Root MSE 0.18588 R‐square 0.5293
Dep Mean 1.25510 Adj R‐sq 0.4351
C.V. 14.81031
Multivariate Linear Reg
ression Model for Academic Performance of the Freshmen
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51
NOTE: Model is not full rank. Least‐squares solutions for the parameters are not unique. Some
statistics will be misleading. A reported DF of 0 or B means that the estimate is biased.
The following parameters have been set to 0, since the variables are a linear combination
of
other variables as shown.
T3 = +1.0000 * INTERCEP ‐1.0000 * T1 ‐1.0000 * T2
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP B 2.382049 1.37206180 1.736 0.0902
HGWA 1 ‐0.009449 0.01415341 ‐0.668 0.5082
AGE 1 ‐0.022418 0.01975457 ‐1.135 0.2632
GENDER 1 ‐0.021456 0.06867861 ‐0.312 0.7564
Multivariate Linear Reg
ression Model for Academic Performance of the Freshmen
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C1 1 ‐0.062635 0.07485246 ‐0.837 0.4077
C2 1 0.235317 0.07984042 2.947 0.0053
C3 1 0.379675 0.10794828 3.517 0.0011
T1 B 0.125302 0.11135897 1.125 0.2672
T2 B ‐0.000142 0.07412829 ‐0.002 0.9985
T3 0 0 . . .
Multivariate Test:
L Ginv(X'X) L' LB‐cj
0.0057974335 0.0005044996 0.0086155014 ‐0.001313812 0.0058847404 0.0163624548
0.0005044996 0.0112940136 0.0006179345 ‐0.000140828 ‐0.008479562 ‐0.015947632
0.0086155014 0.0006179345 0.1365074533 0.0039390604 ‐0.003122265 ‐0.021829978
Multivariate Linear Reg
ression Model for Academic Performance of the Freshmen
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‐0.001313812 ‐0.000140828 0.0039390604 0.1621532104 0.0978031711 0.0967026652
0.0058847404 ‐0.008479562 ‐0.003122265 0.0978031711 0.1844841043 0.1359956205
0.0163624548 ‐0.015947632 ‐0.021829978 0.0967026652 0.1359956205 0.3372444604
0.011761556 0.0063620468 0.0698618214 0.0383235684 0.0154849678 0.0259728851
0.0089674611 ‐0.004563661 0.0269253009 0.0391040946 0.0439800302 0.0704872815
0.011761556 0.0089674611 0.010279155 ‐0.020236126 ‐0.009449088
0.0063620468 ‐0.004563661 ‐0.00607292 0.0154006991 ‐0.022418391
0.0698618214 0.0269253009 ‐0.227511242 ‐0.111806137 ‐0.021455628
0.0383235684 0.0391040946 0.0048038604 ‐0.164186815 ‐0.062635445
0.0154849678 0.0439800302 0.3411810716 ‐0.062798934 0.2353170282
0.0259728851 0.0704872815 0.279585682 ‐0.007575886 0.3796746199
0.3588920024 0.1303478278 0.1400173804 ‐0.113288808 0.125302437
0.1303478278 0.1590308249 0.1026566118 ‐0.151476506 ‐0.000141511
Multivariate Linear Reg
ression Model for Academic Performance of the Freshmen
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Inv(L Ginv(X'X) L') Inv()(LB‐cj)
283.35573878 ‐43.27367347 ‐18.5222449 19.93244898 ‐6.458571429 ‐18.51428571
‐43.27367347 106.97959184 4.2653061224 ‐7.653061224 4.2857142857 7.1428571429
‐18.5222449 4.2653061224 9.5510204082 ‐1.510204082 0.2857142857 2.1428571429
19.93244898 ‐7.653061224 ‐1.510204082 11.102040816 ‐4.857142857 ‐2.428571429
‐6.458571429 4.2857142857 0.2857142857 ‐4.857142857 10 ‐2
‐18.51428571 7.1428571429 2.1428571429 ‐2.428571429 ‐2 6
‐3.071632653 ‐2.897959184 ‐1.326530612 ‐0.734693878 0.5714285714 0.2857142857
‐6.475102041 4.693877551 ‐0.020408163 ‐0.795918367 ‐0.714285714 ‐0.857142857
‐3.071632653 ‐6.475102041 ‐0.989438776 ‐5.72755102 ‐11.49158163
‐2.897959184 4.693877551 1.4336734694 2.5969387755 1.7551020408
‐1.326530612 ‐0.020408163 ‐1.887755102 ‐0.260204082 0.6836734694
Multivariate Linear Reg
ression Model for Academic Performance of the Freshmen
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55
‐0.734693878 ‐0.795918367 ‐1.87244898 ‐1.647959184 ‐2.836734694
0.5714285714 ‐0.714285714 2.6785714286 0.3928571429 1.9285714286
0.2857142857 ‐0.857142857 0.2142857143 0.8214285714 1.9642857143
4.4897959184 ‐3.469387755 0.8316326531 0.2653061224 0.9744897959
‐3.469387755 10.408163265 0.0051020408 ‐0.795918367 ‐0.9234693885
T, the H + E Matrix
5.8469387755 0.7525510204 1.3290816327
0.7525510204 2.8520408163 0.4757653061
1.3290816327 0.4757653061 2.9362244898
Multivariate Linear Reg
ression Model for Academic Performance of the Freshmen
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56
Eigenvectors
0.0517543588 0.1270359725 0.520020392
0.4131660836 ‐0.289104225 ‐0.158534347
0.1453177832 0.5176258271 ‐0.298329281
Eigenvalues
0.5593056922
0.1913680346
0.1024034267
Multivariate Statistics and F Approximations
S=3 M=2 N=18
Multivariate Linear Reg
ression Model for Academic Performance of the Freshmen
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Statistic Value F Num DF Den DF Pr > F
Wilks' Lambda 0.31986707 2.2229 24 110.8128 0.0028
Pillai's Trace 0.85307715 1.9867 24 120 0.0083
Hotelling‐Lawley Trace 1.61988940 2.4748 24 110 0.0008
Roy's Greatest Root 1.26914662 6.3457 8 40 0.0001
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
Multivariate Linear Reg
ression Model for Academic Performance of the Freshmen
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Appendix G
TABLE RESULTS
Table 8. Univariate ANOVA with Math grade (Y1) as dependent
Source DF Sum
of
Mean
F Value
Prob>F
Square
Square
Model 8
1.49237
0.18655
1.714
0.1251
Error 48
4.35457
0.10886
C Total
48
5.84694
Root MSE
0.32995
Dep Mean
1.68367
R-square 0.2552
Adj R-sq
0.1063
C.V. 19.59678
Table 9. Parameter estimates for Math grade
Parameter
Parameter
Standard
T for HO:
Prob > |T|
Variable
Estimate
Error
Parameter = 0
Intercept 0.69458
2.43541
0.285
0.7770
HGWA 0.01028
0.02512
0.409
0.6846
AGE 0.00607
0.03506
-0.173
0.8634
GENDER
-0.22750
0.12190
-1.866
0.0693
C1 0.00480
0.13286
0.036
0.9713
C2 0.34118
0.14172
2.407
0.0208
C3 0.27959
0.19161
1.459
0.1523
T1 0.14002
0.19766
0.708
0.4828
T2 0.10266
0.13158
0.780
0.4399
T3 0
-
-
-
Where: C1= CN
T1=REGIONAL HIGH SCHOOL
C2=CTE
T2=NATIONAL HIGH SCHOOL
C3=OTHER DEGREE
T3=PRIVATE HIGH SCHOOL
Multivariate Linear Regression Model for
Academic Performance of the Freshmen
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59
Table 10. Univariate ANOVA with English grade (Y2) as dependent
Source
DF
Sum of
Mean
F Value
Prob>F
Square
Square
Model 8
0.51518
0.0644
1.102
0.3821
Error 48
2.33686
0.0584
C Total
48
2.85204
Root
MSE
0.24171
Dep
Mean
1.59694
R-square
0.18060
Adj
R-sq
0.01680
C.V.
15.13556
Table 11. Parameter estimates for English grade
Parameter
Parameter
Standard Error
T for HO:
Prob > |T|
Variable
Estimate
Parameter = 0
Intercept 3.384601
1.7840915
1.897
0.0651
HGWA -0.020236
0.0184037
-1.100
0.2781
AGE 0.015401
0.0256869
0.600
0.5522
GENDER
-0.111806
0.0893028
-1.252
0.2178
C1 -0.164187
0.0973306
-1.687
0.0994
C2 -0.062799
0.1417171
2.407
0.5487
C3 -0.007576
0.1403651
-0.054
0.9572
T1 -0.113289
0.1448000
-0.782
0.4386
T2 -0.151477
0.096389
-1.572
0.1239
T3 0
-
-
-
Where: C1= CN
T1=REGIONAL HIGH SCHOOL
C2=CTE
T2=NATIONAL HIGH SCHOOL
C3=OTHER DEGREE
T3=PRIVATE HIGH SCHOOL
Multivariate Linear Regression Model for
Academic Performance of the Freshmen
Achiever at Benguet State University / Antipuesto, Edward; et al. 2009
60
Table 12. Univariate ANOVA with Social Science grade (Y3) as dependent
Source DF Sum
of
Mean
F Value
Prob>F
Square
Square
Model 3
1.55410
0.19426
5.622
0.0001
Error 45
1.38212
0.03455
C Total
48
2.93622
Root MSE
0.1858
Dep Mean
1.2551
R-square 0.5293
Adj R-sq
0.4351
C.V. 14.8103
Table 13. Parameter estimates for Social Science grade
Parameter
Parameter
Standard
T for HO:
Prob > |T|
Variable
Estimate
Error
Parameter = 0
Intercept 2.3820
1.37206
1.736
0.0902
HGWA 0.0094
0.01415
-0.668
0.5082
AGE -0.0224
0.01975
-1.135
0.2632
GENDER
-0.0215
0.06868
-0.312
0.7564
C1 -0.0626
0.07485
-0.837
0.4077
C2 0.2353
0.07984
2.947
0.0053
C3 0.3796
0.10795
3.517
0.0011
T1 0.1253
0.11136
1.125
0.2672
T2 -0.0001
0.07413
-0.002
0.9985
T3 0
-
-
-
Where: C1= CN
T1=REGIONAL HIGH SCHOOL
C2=CTE
T2=NATIONAL HIGH SCHOOL
C3=OTHER DEGREE
T3=PRIVATE HIGH SCHOOL
Multivariate Linear Regression Model for
Academic Performance of the Freshmen
Achiever at Benguet State University / Antipuesto, Edward; et al. 2009
Document Outline
- Multivariate Linear Regression Model for Academic
Performance of the Freshmen Achiever at Benguet State University
- BIBLIOGRAPHY
- ABSTRACT
- TABLE OF CONTENT
- INTRODUCTION
- REVIEW OF RELATED LITERATURE
- THEORETICAL FRAMEWORK
- METHODOLOGY
- RESULTS AND DISCUSSION
- LITERATURE CITED
- APPENDIX