BIBLIOGRAPHY DAS-ILEN, ELVIRA D. and PADILAN,...
BIBLIOGRAPHY
DAS-ILEN, ELVIRA D. and PADILAN, DIXY E. April 2008. Efficiency of
CATANOVA in Measuring Association between Socio-Economic Variables and
Motivations and Aspirations of Working Students. Benguet State University, La
Trinidad, Benguet.
ADVISER: SALVACION Z. BELIGAN, Ph.D.
ABSTRACT
The study was conducted to determine the association between socio-economic
variables and motivations and aspirations of working students at Benguet State
University using the Analysis of Variance for Categorical data (CATANOVA) as an
alternative tool for Pearson’s Chi-square test. Specifically, this study aimed to determine
the relationship of working student’s motivation and aspiration with their socio-economic
variables and to determine the proportion of variation of the response variable
(motivations/aspirations) attributed to the independent variables (socio-economic
variables).
A survey questionnaire was floated and distributed to the respondents through
random sampling in order to obtain the needed data for this study.
Both chi-square and C statistics were computed to determine the significance of
the association between the response and the independent variables.
Based on the results of the analysis employing both CATANOVA and Pearson’s
Chi-square students motivations was associated to the father’s occupation, household size
and sibling’s position. Aspiration of working students was associated to gender and
family income. With the use of CATANOVA, the percent contribution of the socio-
economic variables on the variability of the working student’s motivations and
aspirations were determined.
CATANOVA statistics is at par with the Pearson’s Chi-square statistics in
determining associations but more efficient than Pearson’s Chi-square in terms of their
variability.
ii
TABLE OF CONTENTS
Page
Bibliography …………………………………………………………….………i
Abstract …………………………………………………………………………i
Table of Contents ……………………………………………………………...iii
INTRODUCTION
Background of the Study…………………………………….…………1
Objectives of the Study ………………………………………………...3
Importance of the Study ………………………………………………. 3
Scope and Delimitation of the Study ……………….………………….4
REVIEW OF RELATED LITERATURE
Studies Using CATANOVA ………………………………….….….... 5
Studies on the Association of Educational
Aspirations and Socio-economic Variables ……………..….……..….. 8
THEORETICAL FRAMEWORK
Analysis of Variance for Categorical Data ……………………….…...11
Pearson’s Chi-square Test ………….………………………………….23
Distribution
of
Pearson’s Chi-Square …….……………………………24
Efficiency of Pearson’s Chi-square
and CATANOVA ………………………………….……….………….24
Definition of Terms ……………………………….…………………...25
METHODOLOGY
Locale of the Study ……………………………….……………………27
Respondents of the Study ……………………….………….…………..27
iii
Data Collection Instrument …………………….…………….……...…27
Analysis of Data ………………………………….………….…………28
RESULTS AND DISCUSSION
Associations Between Educational Motivations and
Socio-Demographic Profile of the Working Students ………..…………29
Associations Between Educational Aspirations and
Socio-Demographic Profile of the Working Students ……….…………31
Efficiency of CATANOVA and the Pearson’s
Chi-square Test in Measuring Associations ...…………………...……..32
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
Summary
…………………………………………….….………………34
Conclusions
………………………………………….…….……………34
Recommendations
………………………………….……….…………..35
LITERATURE CITED ………………………………………….….…………..36
APPENDICES ……………………………………………………….…………37
iv
INTRODUCTION
Background of the Study
Researchers usually grope on what statistical tool to be used in
determining relationships of categorical variables. Several techniques for
analyzing and testing contingency data are available in many literatures. The Chi-
Square test is one of the most commonly used for determining the association or
dependence of two variables. The Chi-Square test is most appropriate when the
sample has been selected from an infinite or large bivariate multinomial
population and the sample size n is reasonable large (Neter, et al., 1988).
However, according to Light and Margolin (1974), Chi-square test is no longer
appropriate when sample size n is small, that is, when more than 20% of the
observed cell frequencies are lower than 5. Thus, a statistical tool known as
Categorical Analysis of Variance (CATANOVA) derived earlier by Light and
Margolin (1971) is suggested for use for small sample comparisons.
CATANOVA is a variant of the standard One-Way ANOVA since it also
partition the variation of nominal data into additive components. The procedure
deals with the frequency counts and row and column marginal totals in a two
dimensional contingency table. Similar to the parametric One-Way ANOVA, the
general approach to categorical data is to compute the total variation in the data
and then partition this into specific components. Aside from the measurement of
association statistic, CATANOVA can also determine the proportion of variation
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
2
attributable to the independent variable which other test can not do, like the chi-
square test.
For the application of CATANOVA, the researchers of this study sought
to utilize the empirical data on the educational aspirations of working students.
One of the most important concerns of any College or University is the
full development of students’ potential through meaningful and relevant programs
that respond to their varied backgrounds, orientations, personal and professional
needs. To meet their needs, the provision of a well-defined student assistance
program requires much attention, as it contributes to the total development of
prospective professionals through varied learning experience in a work setting
within the school. As provided for in the National Compensation Circular No. 64
(1990), students assistance maybe hired to render emergency or temporary
services for the following reasons: 1) to provide practicum training for students,
2) to provide extra income for students and 3) to emphasize dignity in labor.
Though these objectives of the NCC are for the benefits of the students, many
students found working and studying at the same time a difficult task because it
does not only consume much the time of students but also prevent them from
joining extra curricular activities that will promote their personal growth.
Hence, the purpose of this study is two-fold, first, to find the relationship
between students’ motivation to work and their demographic profile and the
relationship between educational aspirations and socio-economic background of
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
3
the respondents, and second, to assess the applicability of CATANOVA in
contingency tables with different dimensions.
Objectives of the Study
The study aimed to determine the efficiency of CATANOVA in
measuring the degree of association between two categorical variables observed
from working students at Benguet State University. Specifically, the study aimed:
1. To determine the association between educational motivation and socio-
economic background of working students.
2. To determine the association between educational aspirations and socio-
economic profile of the working students; and
3. To determine the efficiency of CATANOVA and Pearson Chi-square
test in measuring the degree of association between two categorical variables.
Importance of the Study
Results of this study maybe utilized as an input to school administrators in
policy formulation regarding student assistance programs.
The study is also important to researchers who use statistics in
determining the degree of association between two variables measured in nominal
scale and to any researcher who is dealing with quantitative variables.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
4
Moreover, the result of the study will serve as a guide for researchers in
determining the efficiency of CATANOVA over the Pearson’s Chi-square test
using the data on motivations and aspirations of working students.
Scope and Delimitation of the Study
This study was conducted at Benguet State University, La Trinidad,
Benguet and it focused on the aspirations and motivations of working students in
relation to their socio-economic profile using the CATANOVA.
The respondents were students from Benguet State University who are
working inside the school as students’ assistants, and those who are working
outside BSU as service crews in McDonalds and Jollibee for a minimum of five
(5) months.
The researchers gathered the needed information from the different
colleges of Benguet State University through the administration of survey
questionnaires.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
5
REVIEW OF RELATED LITERATURE
Studies Using CATANOVA
Anestesiol (1995) evaluated the incidence of colonization and infection by
methicillin-resistant in PICU. They studied two-hundred patients with duration of
hospitalization longer than 24 hours out of the 255 patients who were hospitalized
during the same period. The patients were divided in two groups according to the
presence or the absences of MRS. The difference of the two populations were
compared using the t-test and the CATANOVA. Wilcoxon's test was used to
analyze the relation between the two values. The results were significant when p =
0.05 and Ct = 3.81. They concluded that the clinical impact of MRS in terms of
morbidity and mortality in this PICU is modest. The prevention and limitation of
the spread of MRS could be obtained by simple but essential measures of control.
Anderson et. al. (1980) presented an extension of the analysis of variance
for categorical data (CATANOVA) procedure to multidimensional contingency
tables involving several factors and a response variable measured on a nominal
scale. Using an appropriate measure of total variation for multinomial data, partial
and multiple association measures are developed as R2 quantities which parallel
the analogous statistics in multiple linear regression for quantitative data. In
addition, test statistics are derived in terms of these R2 criteria. Finally, this
CATANOVA approach is illustrated within the context of 2 three-way
contingency table from a multicenter clinical trial.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
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Singh B. (2004) examined the CATANOVA method for its suitability for
analysis of nominal data from repeated measures design. The modified tests are
developed for single and multi-group repeated measures designs, separately. The
computed results reveal that in single group repeated measures designs the actual
size of existing CATANOVA test is more for negative correlation and less for
positive correlation than the stated size. This implies that the existing test may
yield the non-existing effects as significant for negative correlation and may not
even detect the real effects for positive correlation among repeated observations.
Similar results are obtained for repeated factor and interaction effects and just
reverse result for group effect in multi-group repeated measures design. These
results show that the existing CATANOVA tests are not valid for repeated
measures designs and hence the modified tests should be used for analysis of
nominal data from such designs.
Souza et.al. (2001) studied the genetic divergence among organisms;
generally the analysis is done directly from the DNA molecule. Therefore, a
possible outcome is categorical being one out of four categories (looking at the
nucleotide level). Light and Margolin (1971) developed an analysis of variance
for categorical data (CATANOVA) and Pinheiro et al. (2000) employed a
similar measure of variation and extended the CATANOVA procedure taking into
account several positions in the sequence for balanced designs. Here we consider
variable number of sequences in each group, where, the samples are unbalanced.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
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In order to test the null hypothesis of homogeneity among groups, the asymptotic
distribution of the test statistic was found and its power is evaluated. An
application of the test to real data is illustrated using resampling methods such as
the bootstrap to generate the empirical distribution of the test statistics.
Light’s et.al. (1979) revealed that the CATANOVA method for analysis of
two ways classified nominal data was developed analogous to the least squares
method of fitting constants for two way cross-classified quantitative data with
disproportionate cell frequencies. This method has been compared with simple
chi-square method through a numerical example. Exact small sample behavior in
two-way contingency tables is investigated for Pearson's chi-square statistic 2
χ ,
Light and Margolin's C statistic and its related 2
R measure of association,
^
Kullback's minimum discrimination information statistic f (2) , and Goodman and
Kruskal's Lambda. 2
R was shown to be identical to Goodman and Kruskal’s λ ,
leading to a test for independence based on λ . In small samples from a product
of multinomial model, the null distribution of C is better approximated by a chi-
square distribution than is the null distribution of 2
χ ; both are considerably better
^
approximated by a 2
χ distribution than is the null distribution of f (2) . It is
proved for tables with two columns and any number of rows that if the column
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
8
^
totals are equal, then 2
χ ; thus, 2
χ is more conservative than f(2) . Hence, use of
^
f ( )
2 should be avoided in testing independence in tables with small samples.
Light’s et. al. (1971) discussed a measure of variation for categorical data.
He developed an analysis of variance for a one-way table, where the response
variable is categorical. The data can be viewed alternatively as falling in a two-
dimensional contingency table with one margin fixed. Components of variation
are derived, and their properties are investigated under a common multinomial
model. Using these components, we propose a measure of the variation in the
response variable explained by the grouping variable. A test statistic is
constructed on the basis of these properties, and its asymptotic behavior under the
null hypothesis of independence is studied. Empirical sampling results confirming
the asymptotic behavior and investigating power are included.
Lesser’s et. al. (1980) studied the educational aspirations of 617
adolescent students in Denmark and was classified into five categories.
CATANOVA and Chi-square test was used to determine the relationship of
gender and educational aspirations of the students. They found out that gender is
statistically associated to educational aspirations of the students although, only 2.1
percent of the variation was explained by knowledge of a respondents’ gender.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
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Studies about Aspirations and Socio-
Economic Characteristics
Aspirations are strong desires to reach something high or great. Young
people's aspirations guide what students learn in school, how they prepare for
adult life, and what they eventually do (Walberg, 1989). This Digest reports on
educational aspirations of rural youth compared with students living elsewhere,
and suggests ways communities can work together to raise the sights of their
young people.
Kayser (1973) tested the predictability of the aspirational changes using
the simple, two-state, and discrete-time Markov model and to test for differences
in selected background characteristics for students with different aspirational
patterns. Major findings that there was a differential turnover between college and
non-college plans, that students with a given level but different histories of
aspirations were similar in selected background characteristics such as family
status, significant others' support, and income aspirations, and that the two-state,
discrete-time Markov model predicted the changes in educational aspirations for
the students sampled. A major conclusion was that the assumption of aspirational
stability was supported and the process appears to be Markovian in nature with
one general process in operation over all the high school years.
Chiale et.al. (1965) investigated associations of overweight status and
changes in overweight status over time with life satisfaction and future aspirations
among a community sample of young women. Analyses were conducted using the
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
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SPSS version 11.0 statistical software package (SPSS, Inc., Chicago, IL).
2
χ analysis was used to investigate differences in aspirations and life satisfaction
among women in the four BMI categories (cross-sectional analysis) and across
women in the four overweight status change categories (longitudinal analysis).
Young women’s aspirations were cross-sectional related to BMI category, such
that obese women were less likely to aspire to further education, although this
relationship maybe explained largely by current occupation. Even after adjusting
for current occupation, young women who were obese were more dissatisfied with
work/career/study, family relationships, partner relationships, and social activities.
Weight status was also longitudinally associated with aspirations and life
satisfaction. Women who were overweight or obese at both surveys were more
likely than other women to aspire to "other" types of employment (including self-
employed and unpaid work in the home) as opposed to full-time employment.
They were also less likely to be satisfied with study or partner relationships.
Women who resolved their overweight/obesity status were more likely to aspire to
being childless than other women.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
11
THEORETICAL FRAMEWORK
Analysis of Variance for Categorical
Data (CATANOVA)
In this study, a technique derived by Light and Margolin (1974) was
utilized as measure of association between categorical variables. The analysis
begins with the partition of the total variation in the data into an explainable
component and an unexplained component or noise. The general approach of this
technique is to compute the total variation in the data and then partition this
variation into specific components. The distributions of the various components
are derived under an assumed model.
CATANOVA is a variant of the standard one-way analysis of variance. It
is an analysis of variance of a one-way table with replication, where the variable
being observed is nominal.
Since the data is nominal, this is one obstacle in defining measure of
variation for categorical data because there is a tendency to think of variation as a
measure of departure of a set of individual observations from their mean.
Moreover, with categorical data, the mean is an undefined concept. However,
there is an alternative way of looking at variation. Gini noted that the sum of
squares of deviations from their mean for n quantitative measurements can be
expressed solely as a function of the squares of the pair wise differences for all
(n pairs. Specifically, if X ,..., X denote the measurements, then
2 )
1
n
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
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n
n
2
1
SS = ∑ (X − X
X
X
i
) = ∑ ( −
i
j )
1
2 n
i =
j =1
(1)
n
n
= 1 ∑ ∑ d 2ij
2 n i=1 j=1
where,
n
X
X = ∑ i
i=1
n
d = X − X .
ij
i
j
Assuming there are G unordered experimental groups and unordered I
response categories. Each response is in one of the I categories. Denote the
number of responses in one category i for group j,i = ,...,
1
I and j − ,...,
1
G by n .
ij
The number of responses, or group size, for group j is n =
G n . The
+ j
∑j= ij
1
number of responses in the ith category is n = ∑G n .. Thus, the total number of
j+
ij
1
responses in the study is;
n = ∑G n = I n = I
G n
(2)
j=
+ j
1
∑i= i+
1
∑i=1∑j= ij
1
An alternative way of viewing this data is via an I and G contingency
table where n is the count in the (i, j) cell. For this two-dimensional
ij
contingency table, or the equivalent one-way analysis of variance with categorical
responses, the total variation in the response variable or “total sum of squares” is
equal to;
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
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n
1
TSS =
−
∑I n2
(3)
i=
i+
2
2n
1
In one-way analysis of variance with quantitative data one would compute
the within-group component of variation and the between-group component of
variation. We do this by applying the formula of variation for categorical
responses X ,..., X which is;
1
n
1 n n
n
n
2
1
∑∑d =
d
(4)
ij
∑∑ ,
2
ij
n j 1= i 1=
2n j 1= i 1=
where,
d = 1 if x and x name different categories
ij
i
j
= 0 if x and x name the same category.
i
j
To the ( ijn pairs of responses within group j, j = ,...,
1
.
G
2 )
The variation in the response variable within the th
j group is then
I
n
1
ij −
∑n2
(5)
ij
2
2n+ j i=1
Hence the total within-group variation or within –group sum of squares
(WSS) is found by assuming (5) over j ;
G
I
n
1
WSS = ∑⎛
⎞
+
⎜ j
2
⎜
−
∑n ⎟
ij ⎟
1
2
2n
j= ⎝
+ j i=1
⎠
(6)
G
I
= n − 1 ∑ 1 ∑n2ij
2
2 j=1 n+ j i=1
For balance design situations where n = n , then
+ j
+
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
14
1 ⎛
G
I
⎞
2
2
WSS =
Gn −
+
∑∑n .
2 ⎜⎜
ij ⎟⎟
n ⎝
j 1
= i 1
=
⎠
The between-group variation or sum of squares (BSS) is defined as the
differences between TSS and WSS. Hence:
1 ⎛ G
I
I
1
1
BSS = ⎜
2
2
⎜∑
∑ ⎞
n ⎟ =
i
n
1
.
(7)
ij ⎟
∑ = i+
2 j=
⎝ 1 n
1
2n
+ j i=
⎠
For balance design situations where n = n , this reduces to
+ j
+
1 ⎡ ⎛ G I
⎞
I
⎤
BSS =
⎢G ∑∑n2 − n2 .
ij
∑ i+⎥
2n G
+
⎢ ⎜⎜
⎟
⎟
⎣ ⎝ j=1 i=1
⎠ i=1
⎥
⎦
We now turn to the problem posed in the introduction on measures of
association for categorical. The three components of variation we have just
derived enable us to define a measure of association between the grouping and
response variables which maybe given a “proportion of variation” explained
interpretation. We define this measure as:
G
I
I
⎛
1
⎞
⎜∑
∑
1
n2 ⎟ − ∑n2
ij
⎜
⎟
+
j=1 n
2
⎝
+ j i=1
n
i
i
⎠
=1
BSS
R =
=
(8)
I
1
2
TSS
n − ∑n
n
i+
i=1
n
It has the property that 2
R = 0 if ij
= f ; i = 1,…,I; j = 1,….,G, i.e.,
i
n+ j
if for each j, j = 1 there exists a I such that n = n , i.e., if there is perfect
ij
+ j
predictability. Otherwise, 0
2
< R < 1.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
15
BSS, TSS and a Proposed Test Statistics
The previously proposed method of partitioning categorical variation will
be referred to as a categorical analysis of variation, or CATANOVA. We
developed the following C statistics to test the null hypothesis p = p .
ij
i
(n − )1(I − )1BSS
C =
(9)
TSS
This test statistic, referred to as the CATANOVA C statistic, is asymptotically
approximated as 2
χ(
under H .
I 1
− )(G 1
− )
0
Recalling the measure of association defined in (8), we note that
2
C = (n − )(
1 I − )
1 R . Thus, to test the significance of the measure of association,
we need to multiply it by (n − )(
1 I − )
1 and approximate its distribution under H
0
by 2
χ(
.
I 1
− )(G 1
− )
The derivation of the test statistic proceeds along asymptotic lines. This
enables us to invoke further results for quadratic forms in normal variates,
because V is asymptotically multivariate normal,
V ≈ N (μ, Ζ)
(10)
In investigating BSS, we first prove that under H , the asymptotic
0
distribution of BSS does not depend on μ . BSS can be written as;
2
I
G ⎡
1
n
⎤
BSS = V ′BV = ∑∑
ij
ni+
⎢
−
⎥ .
(11)
2 i=1 j=1 ⎢ n
n
j
⎥
+
⎣
⎦
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
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G
Let ).
θ = n − n p = n − E (n Then θ = ∑θ = n − np . Moreover,
ij
ij
+ j
i
ij
o
ij
ij
ij
i+
i
j 1
=
under H ,
0
θ ′ = (θ ,...,θ ,...,θ ,...,θ )
11
I1
1G
IG
is asymptotically
θ ′ ≈ N( ,
0 Z )
(12)
where,
Z is as in Z = n Z and Z = M ⊗ Z where,
j
+ j
o
o
⎡ p 1
( − p )
− p p
− p p
1
1
1
2
L
1
I
⎤
⎢
⎥
p 1
( − p )
− p p
.
⎢
2
2
L
2
I
⎥
Z =
o
⎢
⎥
M
⎢
⎥
⎣
p 1
( − p )
I
I ⎦
Now,
BSS = V BV
′
⎡
2
1
θ + n p
⎤
=
ij
ij
i
θ
np
i+
i
=
⎢
∑ ∑
−
n ⎥
i
j
2
⎢
n
n
+ j
+
⎥
j
⎣
⎦
2
I
G ⎡
1
θ
⎤
ij
θi
=
⎢
∑∑
− + n ⎥
+ j
2 i=1 j=1 ⎢ n
n
+
⎥
j
⎣
⎦
= θ ′ θ
B
Hence, under H in studying BSS , we may treat μ as 0 . From this, it follows
0
that under H , BSS is asymptotically distributed as,
0
IG
BSS ≈ ∑
2
λ χ
(13)
i
1
i=1
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
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where,
{λ ,i = ,...,
1
is the set of characteristic roots of BZ :
i
}
IG
1 ⎡⎛
−1
1
⎞
⎤
BZ =
⎜ M − μ ⎟ ⊗ I M ⊗ Z
⎢
G
1 ⎥[
o ]
2 ⎣⎝
n
⎠
⎦
(14)
1 ⎡
1
⎤
=
I − μ M ⊗ Z
⎢ G
G
⎥
o
2 ⎣
n
⎦
where,
1 ⎡⎛
−1
1
⎞
⎤
B = T −W = ⎢⎜M − μ
and
G ⎟ ⊗ I ⎥
2 ⎣⎝
n
⎠
⎦
Z = M ⊗ Z
o
where,
Z is defined in (12).
o
The characteristic roots of BZ under H are then one half the products of the
0
characteristic roots of
I
1
−
M
(15)
G
n U G
With the characteristic roots of Z , the characteristic roots of (15) are the
o
solutions to the determinantal equation:
1
I (I − λ) −
M = 0
(16)
G
U
n G
The left-hand side of (2.7) is:
G 1
.
L H.S. = λ(I λ) −
−
Efficiency of CATANOVA in Measuring Association between Socio-Economic
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18
Hence, λ = 0 is a root withy multiplicity one and λ = 1 is a root with
multiplicity G −1 .
Thus, (2.4) is really
I
1
BSS ≈ ∑
2
λ χ
(17)
i
G−1
2 i=1
where,
{λ ,i = ,...,
1
I is the set of characteristic roots of Z .
i
}
o
Recall that:
⎡ p1(1− p1 ) − p p
− p p
1
2
1
I
⎤
⎢
⎥
Z =
.
(18)
o
⎢
M
⎥
⎢
⎣
p1(1− pI )⎥⎦
i.e., Z , is a multinomial covariance matrix. Roy et. Al., studied the determination
o
of the characteristic roots of the general multinomial covariance matrix without
presenting the roots. The characteristic equation they obtain is:
I
2
⎪
⎧
⎛ p
⎞
i
⎪
⎫ I
1
⎨ − ∑
⎬∏(p − λ
(19)
i
) = 0
⎪
⎜
⎜
⎟
⎟
λ
i=
p −
1 ⎝
i
⎠⎪ i 1
⎩
⎭ =
Certainly λ = 0 is a root. This also follows because a multinomial
covariance matrix is singular. The determination of the other roots in general
appears to be an unsolved problem. We can make some further observations,
however, by noting that (19) is equivalent to:
∏I (
I
p = λ
p
p
λ
(20)
i
)
2
−
i=
∑ i ∏ ( −
j
) = 0
1
j≠i
i 1
=
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19
Then;
a. Unless there is a j ≠ i such that p = p ,λ = p is not a root.
i
j
i
b. If no two probabilities p , p i ≠ j , are equal, then between each two
i
j
successively ordered probabilities lies a characteristics roots, i.e., if we order the
probabilities and the characteristic roots, we have:
p < λ < p < λ < ... < p
< λ
< p
1
i
1
i
i2
i2
i(I 1
− )
i(I 1
− )
iI
To see this, note that the left-hand side of (20) as a function of λ
changes continuously from positive to negative or vise versa as λ changes from
p
to p
.
ij
i( j 1
+ )
c. If k of the probabilities p = i = ,...,
1
I, are equal, i.e.,
i
p = p
... = p
= p , then in (20), (
) 1−
− k
p λ
is a factor and λ = p is then a
ir
(ir+1)
(ir+k −1)
characteristic root with multiplicity k = 1 .
Unfortunately the set of{λ ,i = ,...,
1
I − . Suppose, however, that under H we
i
}
1
o
I 1
1
approximate the distribution of BSS ~ ∑−λ
,
2
iχ G−1
2 i=1
By (suggested also by Thompson):
1 I
S ~ ∑−1 2
λ x (I−1)(G−1) ,
(21)
2 i=1
where,
Efficiency of CATANOVA in Measuring Association between Socio-Economic
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I 1
∑−
I
λ
1
2
−
i
∑ p
traceZ
i
i 1
=
o
i 1
λ =
=
=
=
.
I −1
I −1
I −1
From (c) above, if p =
= p
1
...
= then
1
λ = is a root multiplicity I −1 and
i
i
I
I
1
2
BSS ~
X (
.
I 1
− )(G 1
− )
2I
Thus, under H , at the center of the simplex defined by
o
({
I
p ,..., p :
≥ ,
0 = ,...,
1
and ∑ p = 1
(22)
i
i )
p
i
I
i
}
i
i=1
The approximation of (17) by (21) is exact. Moreover, for I = 2 and all G , the
approximation of (17) by (21) is exact.
In general, over the simplex of (22),
1
I 1
1
E BSS =
∑ − λ G − = λ I − G −
0
( i 1= i)( )1 ( )1( )1
2
2
⎛ 1
⎞
=
2
E⎜ λx (I 1−)(G 1−) ⎟ = E(S ).
⎝ 2
⎠
Thus, the approximation in (17) has the correct first moment under H over the
0
entire simplex. Further, under H :
0
Var (BSS)
(G − )
1
I
1
=
∑ −1 2λ = G trace Z
i
( − )1
( 20)
i=1
2
2
1
= (G − )
1 [∑I [p 1 p 2 2
p2 p2
(23)
i ( −
i ) ]+
I
I
i=1
∑i=1∑j=i+ i j
1
]
2
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21
1
2
= (G − )
1 ⎡
I
2
∑ p − 2 I 3
I
2
⎤
⎢
p
p
i=
i
∑
+
i=
i
∑
1
1
( i 1= i ) .
2
⎥
⎣
⎦
The variance of the approximation is:
1
Var (S )= (I − )
1 (G − ) 2
1 λ
2
1 ⎛ G −1
2
⎞
= ⎜
⎟(1
I
2
− ∑ p
(24)
i 1
=
i ) .
2 ⎝ I −1 ⎠
For I =2 and all G, (23) and (24) are exactly equal over the entire simplex defined
by (22) because the approximation is exact. Moreover, for all I and G, (23) and
(24) are exactly equal at the center of the simplex defined (22), for the same
reason. Numerical comparisons between (23) and (24) for various values of I and
(p ,..., p indicate moderate to good agreement “near” the center of the simplex.
i
n )
For example, with I = 4 and p = 0. ,
2 p = 0. ,
2 and p = 0. ,
4 then (23)
1
2
4
1
= (G − )
1 ( 1824
.
). Hence, we will approximate the asymptotic distribution of BSS
2
as:
⎛
I
⎞
2
⎜1− ∑ p ⎟
⎜
i
i 1
=
⎟ 2
BSS ~ ⎜
χ
.
(25)
2(I − )
(I 1
− )(G− )
1
1 ⎟
⎜
⎟
⎝
⎠
1 ⎛
I
2 ⎞
This still leaves the problem that ⎜1− ∑ p ⎟ is unknown. The minimum
2 ⎝
i
i=1
⎠
1 ⎛
I
1 I
2 ⎞
variance unbiased estimator under H of ⎜1− ∑ p ⎟ =
∑ p − can be
i (1
p1 )
o
2 ⎝
i
i=1
⎠
2 i 1=
shown to be:
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1 ⎛ n
I
⎞ ⎛ n + ⎛
⎞
n + ⎞
i
⎜
⎟∑⎜
⎜
⎟ I
i
−
⎟
2n ⎝ n −1⎠ i=1 ⎝ n ⎝
⎠
n ⎠
⎡
1
∑ n2 ⎤
=
⎢∑n
i
i
−
+
+
⎥
(
2 n
i
− )
1 ⎢
(26)
i
n
⎣
⎥
⎦
I
1
⎡
1
2 ⎤
=
n − ∑n
(
2 n
⎢
− )
1 ⎣
n
i+ ⎥
i=1
⎦
= TMS
1 ⎛
I
2 ⎞
We already knew that I TMS
(
) = ⎜1−
p
o
∑ ⎟
2 ⎝
i
i=1
⎠
n
Since n i+ coverage’s in probability to p under H , the asymptotic variance of
n
i
o
TMS is zero.
1 ⎛
I
2 ⎞
Thus,
under H , we view TMS as a constant equal to ⎜1− ∑ p ⎟ then,
o
2 ⎝
i
i=1
⎠
(n − )1(I − )1BSS
C=
TSS
⎡⎛ G
I
1
⎞
I
1
⎤
⎢ ∑
∑n2 −
n2
ij
∑ i+ ⎥
⎜
⎜
⎟
⎟
j=1 n + j
1
n
=
(n-1)
(I-1)
⎢⎝
i=
⎠
i=1
⎥
⎢
(27)
I
⎥
1
⎢
n − ∑n2i+
⎥
⎢
n i=1
⎥
⎣
⎦
Maybe approximated as 2
χ(
.
I 1
− )(G 1
− )
Pearson’s Chi-square Test
Pearson's chi-square test (χ2) is one of a variety of chi-square tests –
statistical procedures whose results are evaluated by reference to the chi-square
distribution. It tests a null hypothesis that the relative frequencies of occurrence of
Efficiency of CATANOVA in Measuring Association between Socio-Economic
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observed events follow a specified frequency distribution. The events are assumed
to be independent and have the same distribution, and the outcomes of each event
must be mutually exclusive. A simple example is the hypothesis that an ordinary
six-sided die is "fair", i.e., all six outcomes occur equally often. Pearson's chi-
square is the original and most widely-used chi-square test.
Chi-square is calculated by finding the difference between each observed
and theoretical frequency for each possible outcome, squaring them, dividing each
by the theoretical frequency, and taking the sum of the results. The number of
degrees of freedom is equal to the number of possible outcomes, minus 1:
n (O − E 2
i
i )
2
χ = ∑
(28)
i=1
Ei
where,
O = an observed frequency;
i
E = an expected (theoretical) frequency, asserted by the null
i
hypothesis;
n = the number of possible outcomes of each event.
A chi-square probability of 0.05 or less is commonly interpreted by
applied workers as justification for rejecting the null hypothesis that the row
variable is unrelated (that is, only randomly related) to the column variable. The
alternate hypothesis is not rejected when the variables have an associated
relationship.
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Variables and Motivations and Aspirations of Working Students
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24
Distribution of Pearson’s Chi-Square
The null distribution of the Pearson statistic with j rows and k columns is
approximated by the chi-square distribution with (k − 1) (j − 1) degrees of
freedom. [28] This approximation arises as the true distribution, under the null
hypothesis, if the expected value is given by a multinomial distribution. For large
sample sizes, the central limit theorem says this distribution tends toward a certain
multivariate normal distribution.
Efficiency of Pearson’s Chi-square and C
The efficiency of both the 2
χ and the C statistics maybe obtained using
2k
the formula given below:
E =
V (C)
(29)
2k
E =
(30)
V ( 2
χ )
where k is the degrees of freedom of the r x c contingency table computed as
(r – 1)(c – 1).
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Definition of Terms
Aspiration. A strong desire to achieve a goal.
CATANOVA. It is a technique designed to identify the variation between
treatments of interest to the researcher. A measure of variation for categorical
data.
Categorical. An unordered and discrete variable. It is data that can only be
put into unordered groups.
Chi-square Test. A statistics used in the analysis of enumeration data. It
reflects discrepancies between the observed and expected or theoretical
frequencies of individuals, objects, or events falling in various categories.
Economic Profile. Refers to the respondent’s age, sex, civil status, parent’s
occupation, parent’s annual income and etc.
Household size. Refers to the number of family members of the
respondents.
Motivation. A factor that encourages a person to pursue or achieve
something.
Nominal data. These are categorical data where the order of the categories
is arbitrary.
One-way analysis of variance. A procedure for comparing the mean scores
of two or more groups based on one categorical independent variable.
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Pearson’s’ Chi-square. It is used to test the hypothesis of no association of
columns and rows in tabular data. Also known as the chi-square goodness-of-fit
or chi-square test of independence.
Predictor Variables. These are variables from which projections are made
in a prediction study. These include the socio-economic profile of the
respondents.
Proportion of variation ( 2
R ). Used to measure association.
Response variable. A variable on which information is collected and
which there is an interest because of its direct policy relevance.
Test of goodness of fit. It is used to test if an observed distribution
conforms to any other distribution. Establishes whether or not an observed
frequency distribution differs from a theoretical distribution.
Test of independence. Assess whether paired observations on two
variables expressed in a contingency table are independent of each other.
Working students. These are youths working their way through college.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
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27
METHODOLOGY
Locale of the study
The study was conducted at Benguet State University, La Trinidad,
Benguet during the school year 2007-2008.
Respondents of the Study
A sample of 100 students was chosen at random from the College of
Agriculture, College Nursing, College of Arts and Sciences, College of
Engineering, College of Home Technology and Economics, College of Teacher
Education, College of Forestry, and College of Veterinary Medicine.
This study included students working as student assistants within Benguet
State University as the sampling units.
Data Collection Instrument
A questionnaire consisting of several questions on the respondents’ socio-
economic background and questions on their motivations and aspirations for
working was developed and pre-tested.
The dependent variables included student’s motivations and aspirations
and the predictor variables were the socio-economic background of the
respondents which included gender, father and mother’s occupation, mother and
fathers’ educational attainment, household size, number of siblings, sibling
position and parents’ annual income.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
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Analysis of Data
The responses of the respondents on the different questions were tabulated
in an r x c contingency table. The Chi-square test and the CATANOVA were both
computed for each contingency table.
Significance of results of the 2
χ and C values were compared to the
tabular value at 0.01 and 0.05 level of significance to determine the association of
the respondents’ socio-economic variables with their motivations and aspirations.
The
R2 which is a measure of the variation in the dependent variable due
to the independent variable was likewise determined only in CATANOVA. The
efficiency of both the 2
χ and CATANOVA were computed using equation (29)
and (30).
Efficiency of CATANOVA in Measuring Association between Socio-Economic
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/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
29
RESULTS AND DISCUSSION
Associations Between Educational
Motivations and Socio-Demographic
Profile of the Working Students
For purposes of illustrating the computation of CATANOVA, the
responses of 100 working students were classified into educational motivations by
father’s educational attainment, by household size, and by sibling’s position. As
revealed in Table 1, both CATANOVA C and the 2
χ test did not show
significant association between educational motivations and father’s educational
attainment. This result indicates that student’s motivation to work while studying
Table 1. Association between student’s motivations to work as student’s assistants
and father’s occupation and father’s education.
EDUCATIONAL MOTIVATION
DEMOGRAPHIC
Earn Extra
Develop
Spend Free
PROFILE
Money
Interpersonal
Time Wisely
for Allowance
Skills
Father’s
Occupation
Government
3 8 0
Employee
Self-Employed
5
3
4
Farming
19
15
13
Laboring
16
8
6
df=6 2
χ =10.78, P=.09, C=10.39 P=.10; R2(%)=5.2
Father’s
Education
Elementary 11
11 8
High School
20
9
13
College 12
14
2
df=4 2
χ =9.00,
P=.06, C=8.03
P=.09; R2(%)=4.05
Efficiency of CATANOVA in Measuring Association between Socio-Economic
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/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
30
is weakly motivated by their father’s educational attainment. However, the
variation in the student’s motivation to work attributed to their father’s
educational attainment was about 4.05 percent. The father’s occupation was
likewise weakly associated to their children’s motivation to work while studying.
As revealed by both tests, no significant association between father’s educational
attainment and father’s occupation and student’s motivations to work.
As shown in Table 2, chi-square test did not show significant association
between educational motivations by household size. However, in the C test, there
is a significant association between the 2 variables. This result indicates that
student’s motivation to work was motivated by household size. Although
statistically associated to student’s educational motivation, household size
contributed only 2.30% of the variation in the student’s educational motivations.
For the association between sibling’s position and students’ educational
motivation, both 2
χ and C tests revealed significant results. These findings
indicate that student’s motivation to work is strongly motivated by sibling’s
position. The variation in the students’ motivation to work is attributed to their
sibling’s position by about 5.50 percent.
Table 2, shows significant association between students’ motivation and
household size and between student’s motivation and sibling’s position using the
C test.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
31
Table 2. Association between student’s motivations to work as student’s assistants
and household size and sibling’s position.
EDUCATIONAL MOTIVATION
DEMOGRAPHIC
Earn Extra
Develop
Spend Free
PROFILE
Money
Interpersonal
Time Wisely
for Allowance
Skills
Household
Size
1-5 9
6
5
6-9 18
21
13
10 or more
16
7
5
df=4 2
χ =4.00ns, P=.40, C=9.23* P=.05; R2(%)=2.30
Sibling’s
Position
Eldest 19
10
3
Middle 11
17
14
Youngest 13
7
6
df=4 2
χ =10.56*, P=.03, C=10.86* P=.02; R2(%)=5.50
* = significant ns = not significant
These results suggest that both household size and sibling position had
significant contribution on the student’s to work as student assistants 2.30 % of
the variation in the student’s educational motivations.
Association Between Educational
Aspirations and Socio-Demographic
Profile of the Working Students
Table 3 shows both C and the 2
χ test showed strong evidence to declare
significant association between student’s educational aspirations and gender and
between student’s educational aspiration and parent’s annual income.
The results indicate that student’s aspiration to work while studying are strongly
attributed to their gender and parent’s annual income. Gender explained
Efficiency of CATANOVA in Measuring Association between Socio-Economic
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/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
32
Table 3. Association between student’s aspirations to work as student’s assistants
and gender and family income.
DEMOGRAPHIC
PROFILE
EDUCATIONAL ASPIRATION
Gender
Finish a Degree
Preparation for Future Job
Male
14
21
Female 41
24
df=1 2
χ =4.90*, p=.03 C=4.85*, p=.03 R2(%)=4.9
Family Income
30,000 and below
6
12
31,000 to 50,000
28
10
51,000 to 80,000
10
10
81,000 to 100,000
7
6
101,000 or more
4
7
df=4 2
χ =10.53*, p=.03 C=10.42*, p=.03 R2(%)=10.53
* - significant
4.9 % of the variation in student’s aspiration while parent’s income explained
10.53 % variation in the student’s aspiration to work.
Efficiency of CATANOVA
and the Pearson’s Chi-square Test
in Measuring Associations
Table 4 presents the efficiency of the Pearson’s chi-square and
CATANOVA C association tests for two variables cross-tabulated in contingency
tables with different dimensions. For a 4 x 3 contingency table, the variability of
the Pearson’s chi-square statistic and the C statistics considered higher than the
expected variance of 2k for the theoretical 2
χ distribution.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
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/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
33
Table 4. Efficiency of CATANOVA and Chi-square Test with 100 Sample Size
Efficiency of
Efficiency of
Pearson’s Chi-
CATANOVA
Motivation Associated to
Dimension
square
2k
2k
V (C)
V ( 2
χ )
Father’s Occupation
4 x 3
0.99
0.98
Father’s Educ’l Attainment
3 x 3
1.01
1.00
Household Size
3 x 3
1.01
1.00
Sibling’s Position
3 x 3
1.01
1.00
Aspiration Associated to
Gender
2 x 2
1.06
1.06
Parents Annual Income
5 x 2
1.04
1.04
The efficiency of Pearson’s Chi-square in a 3 x 3 contingency table was
found higher than the efficiency of the C statistic. This means that the Pearson’s
chi-square statistics is more variable than the theoretical 2
χ distribution.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
34
SUMMARY, CONCLUSION AND RECOMMENDATION
Summary
Both chi-square and C statistics were computed to determine the
significance of association between the response and independent variables.
Based on the results of the analysis employing both CATANOVA and
Pearson’s Chi-square, students’ motivations was associated to the father’s
occupation, house hold size and sibling’s position. Aspiration of working students
was associated to gender and family income. With the use of CATANOVA, the
percent contribution of the socio-economic variables on the variability of the
working students’ motivations and aspirations were determined.
CATANOVA
statistics
is
at par with the Pearson’s Chi-square statistics in
determining associations but more efficient than Pearson’s Chi-square in terms of
their variability.
Conclusion
Based on the above results, the following conclusions were drawn:
The father’s occupation, educational attainment, household size and
sibling position had significant bearing on the respondent’s motivation to work.
The respondent’s aspirations to earn a degree were explained by their
gender and their family income.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
35
CATANOVA as a tool for determining association is at par with the Chi-
square test but considered more efficient than the Pearson’s Chi-square.
Recommendations
In dealing with qualitative data where the response variable is categorized
with no order CATANOVA C is recommended for used because it does not only
measure the relationship between the response and predictor variables it also
measures the percent contribution explained by the predictor variables.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
36
LITERATURE CITED
BALATERO, R.E. 2003. Factors Affecting Academic Performance of Working
Students of Benguet State University, La Trinidad Benguet.
COBB, R. A., MCINTIRE, W.G., and PRATT, P. A.1989. Vocational and
educational aspirations of high school students: A problem for rural
America. In R. Quaglia (Ed.), Research in Rural Education
COCHRAN, W.G. (1950). “The Comparison of Percentages in Matched
Samples,” Biometrika.
D’AMBRA, ANTONELLO and PASQUALE SARNACCHIARO, 2007.
Explorative Data Analysis and CATANOVA for Ordinal Variables: An
Integrated Approach
HOEFFDING, W.1965. “Asymptotically Optimal tests for Multinomial
Distributions,” Annals of Mathematical Statistics.
LIGHT, R.J. MARGOLIN, H. B. 1969. “Analysis of Variance for Categorical
Data, with Applications to Agreement and Association,” Unpublished
Ph.D. dissertation, Department of Statistics, Harvard University.
MACLI-ING and GUIMPAYAN. 2001. Factors Affecting Academic
Performance of Working Students in McDonalds Center mall.
Undergraduate Thesis. Benguet State University, La Trinidad Benguet.
PEARSON, K.1900. “On the Criterion that a Given System of Deviation from the
probable in the Case of the Correlated System of Variables is Such that it
can Be Reasonably Supposed to have Arisen from Random Sampling,”
Philosophical Magazine.
RJ ANDERSON, JR LANDIS, 1980. Communications in Statistics-Theory and
Methods, informaworld.com
WALBERG, H. J. 1989. Student aspirations: National and international
perspectives. In R. Quaglia (Ed.), Research in Rural Education.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
37
APPENDICES
Appendix A: Letter of Communication
Benguet State University
College of Arts and Sciences
La Trinidad, Benguet
MR. ROMEO BABARAN
Manager
Jollibee La Trinidad
La Trinidad, Benguet
We the undersigned are students from Benguet State University
taking up Bachelor of Science in Applied Statistics. We are conducting our
research entitled “Efficiency of CATANOVA in Measuring Association Between
Socio-economic Variables and Motivations and Aspirations of Working
students.” This is to comply with the requirements of the course.
In this connection, may we request for permission that we be allowed to
administer our questionnaire to the working students of Benguet State University
who are having part time job in your fast food establishment.
Your favorable action for this request is highly appreciated.
Thank you.
Researchers,
ELVIRA D. DAS-ILEN
DIXY E. PADILAN
Noted by:
SALVACION Z. BELIGAN
Adviser
Recommending Approval:
MARIA AZUCENA B. LUBRICA
Chairman-MPS Department
AUREA MARIE M. SANDOVAL
CAS-Dean
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
38
Benguet State University
College of Arts and Sciences
La Trinidad, Benguet
The Manager
McDonald La Trinidad
La Trinidad, Benguet
Greetings!
We the undersigned are students from Benguet State University taking up
Bachelor of Science in Applied Statistics. We are conducting our research entitled
“Efficiency of CATANOVA in Measuring Association Between Socio-economic
Variables and Motivations and Aspirations of Working students.” This is to
comply with the requirements of the course.
In this connection, may we request for permission that we be allowed to
administer our questionnaire to the working students of Benguet State University
who are having part time job in your fast food establishment.
Your favorable action for this request is highly appreciated.
Thank you.
Researchers,
ELVIRA D. DAS-ILEN
DIXY E. PADILAN
Noted by:
SALVACION Z. BELIGAN
Adviser
Recommending Approval:
MARIA AZUCENA B. LUBRICA
Chairman-MPS Department
AUREA MARIE M. SANDOVAL
CAS-Dean
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
39
Appendix B: Sample Survey Questionnaire
Benguet State University
College of Arts and Sciences
La Trinidad, Benguet
Dear Respondents:
Greetings!
We the undersigned are students from Benguet State University taking up
Bachelor of Science in Applied Statistics. We are conducting our research entitled
“Efficiency of CATANOVA in Measuring Association Between Socio-economic
Variables and Motivations and Aspirations of Working students.” This is to
comply with the requirements of the course.
In this connection, may we solicit your valued cooperation in answering
the following questions honestly. Rest assured, your answers will be kept
confidential.
Thank you and more power.
Researchers,
ELVIRA D. DAS-ILEN
DIXY E. PADILAN
Noted by:
SALVACION Z. BELIGAN
Adviser
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
40
DIRECTION: Please check or write on the blank that corresponds to your answer
Personal Data:
Name
(optional):
_______________
Gender:
_____
Age:
_____
Civil
Status:
___
Degree/Course:
____________________
Year
Level:
_____
Number of Units: ________
A. Socio-economic Profile:
Direction. Please write your answer on the blank provided for the needed
information.
1. Parents occupation. __________ Mother
__________ Father
2. Parents highest educational attainment. __________ Mother ________ Father
3. Household size (# of family members). _____
4. Number of siblings in the family (including you): _____
5. Your sibling position. _____
6. Annual income of parents (just encircle your answer on the choices given
below).
a. below 30,000
b. 31,000 to 50,000
c. 51,000 to 80,000
d. 81,000 to 100,000
e. 101,000 and above
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
41
B. On Motivation and Aspiration:
Direction. Please encircle only your main reason why you work while studying.
Motivations:
a. Augment the limited allowance from parents.
b. To develop interpersonal skills.
c. To spend free time wisely.
Aspirations:
a. To finish a degree.
b. For job preparation.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
42
Appendix C: Variables Raw Data
t
n
me
in
l Atta
na
s
a
tio
ze
e
Of Parents
s
c
ational Attainment
uc
on
ccupation
ccupation
Ed
ndent
s
O
er
e
r
's
er of Sibling
a
tion
spo
th
s
e
hold Si
nnual Incom
spiration
Re
Gend
Father's O
Mother'
Father's Edu
Mo
Hou
Numb
Sibling Positi
A
Motiv
A
1
1
3
5
2
2
3
3
3
2
1
2
2
1
3
3
2
3
3
3
3
5
1
2
3
2
4
1
1
3
1
1
3
1
3
2
4
1
4
2
2
3
3
2
3
5
1
1
5
2
4
5
1
3
2
3
3
5
2
2
6
2
3
3
2
3
2
2
3
2
3
1
7
1
3
5
1
2
2
2
1
1
1
1
8
1
3
4
2
2
2
2
2
3
3
2
9
2
3
5
1
3
1
1
1
1
2
1
10
2
3
2
1
2
3
3
2
2
1
1
11
2
3
3
3
3
2
2
2
5
3
2
12
2
3
5
1
2
2
1
1
2
1
1
13
2
1
5
3
3
2
2
3
4
2
1
14
1
4
5
2
2
3
3
1
1
2
2
15
2
3
4
1
3
2
1
1
3
2
1
16
1
3
3
2
3
2
2
2
1
1
1
17
2
3
5
3
3
2
2
2
3
3
2
18
2
4
5
3
3
1
1
1
2
1
1
19
2
4
2
2
3
2
1
2
2
2
2
20
2
1
5
3
2
2
1
2
4
2
2
21
2
2
4
3
3
2
2
2
4
2
2
22
2
4
5
2
3
1
1
1
2
2
1
23
2
3
3
2
3
1
1
3
1
1
1
24
1
3
3
3
3
2
1
2
1
2
2
25
1
4
5
1
3
3
3
3
2
1
2
26
2
1
5
3
3
1
1
1
2
1
1
27
2
3
5
2
3
3
3
1
1
2
2
28
2
3
4
2
3
2
1
2
3
3
1
29
2
4
5
3
3
2
2
2
2
1
1
30
2
3
5
3
2
2
2
2
4
2
1
31
2
4
5
2
3
3
2
1
2
1
1
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
43
32
2
3
5
1
2
2
2
2
2
1
1
33
2
4
5
3
2
3
3
2
3
1
1
34
2
4
2
1
1
1
2
1
1
3
2
35
1
4
5
2
3
3
2
1
1
1
2
36
1
2
4
3
3
3
3
2
5
1
2
37
2
1
1
3
3
1
1
2
3
2
1
38
1
2
5
2
3
2
2
2
1
3
2
39
1
3
3
1
2
2
1
3
1
2
1
40
1
1
1
3
3
2
1
1
2
2
1
41
2
2
2
2
2
3
1
1
2
2
1
42
2
4
5
2
2
3
3
2
2
1
1
43
1
3
4
1
2
3
3
2
2
3
2
44
2
4
5
3
3
2
2
2
3
2
2
45
1
3
1
3
3
2
2
2
5
2
2
46
2
3
3
1
3
2
1
2
3
3
2
47
2
2
3
1
1
2
2
3
4
3
1
48
2
2
1
3
3
3
3
1
5
1
1
49
1
3
5
1
2
2
3
3
4
2
2
50
1
1
5
3
3
3
2
1
5
2
2
51
1
3
4
2
2
2
2
1
4
1
1
52
2
3
5
2
3
1
1
2
3
3
2
53
2
2
4
2
3
2
2
2
2
1
1
54
2
3
3
1
1
3
3
3
4
1
1
55
2
3
2
2
3
1
1
2
3
1
2
56
1
4
5
1
2
2
1
3
2
2
2
57
1
3
3
1
2
1
1
2
3
2
2
58
2
2
1
3
3
3
2
1
4
1
2
59
1
1
2
3
3
3
3
2
5
2
1
60
1
3
5
2
3
2
1
2
5
1
2
61
1
2
5
3
3
2
2
1
3
1
1
62
1
3
4
2
2
2
2
1
2
3
1
63
2
3
3
2
3
3
3
2
1
2
2
64
2
4
3
1
3
3
2
3
2
3
2
65
2
3
5
1
3
1
1
3
2
1
2
66
2
4
5
3
1
2
1
2
3
2
1
67
2
2
2
2
2
3
3
2
4
3
1
68
2
3
2
2
2
2
2
1
4
1
2
69
1
4
4
1
3
3
2
2
2
3
1
70
2
4
5
3
3
2
1
1
2
1
1
71
2
4
5
2
2
2
2
2
2
2
1
72
1
3
3
1
3
2
3
2
1
2
2
73
2
3
5
2
3
2
2
2
2
3
1
74
2
3
3
1
2
2
2
1
1
1
2
75
2
4
4
2
3
2
2
1
2
1
1
76
2
3
5
2
3
1
1
1
2
1
1
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
44
77
1
4
5
2
3
2
1
1
2
1
1
78
2
4
5
2
2
1
2
3
2
1
1
79
2
4
1
2
3
1
1
2
3
1
1
80
2
3
4
1
3
3
3
3
4
1
1
81
2
3
5
1
2
1
1
2
2
2
1
82
1
3
5
2
3
2
1
2
2
3
1
83
2
2
5
2
3
2
1
1
2
3
2
84
2
2
2
3
3
3
3
2
2
2
2
85
2
1
1
3
3
2
2
3
3
1
1
86
2
3
5
2
3
1
1
3
5
1
1
87
2
3
3
1
2
1
1
2
2
3
1
88
2
3
3
1
2
1
1
3
2
2
1
89
2
1
5
3
3
2
1
1
4
2
2
90
1
3
5
2
3
1
1
3
2
3
2
91
1
5
3
1
2
3
2
3
3
1
2
92
1
1
2
2
2
2
2
1
3
1
1
93
1
1
2
2
2
2
2
1
3
2
2
94
2
3
5
2
3
2
3
2
1
2
1
95
2
4
5
3
3
2
1
1
2
1
1
96
1
4
4
2
3
2
1
3
1
3
2
97
1
3
3
1
2
2
1
3
2
2
1
98
2
4
5
2
3
3
2
2
3
3
1
99
2
3
1
3
3
2
2
3
3
1
2
100
2
4
4
1
3
3
1
1
1
1
2
Legend:
Gender:
1
=
male
2 = female
Parent’s occupation:
1 = government employee
2 = self-employed
3 = farming
4 = laboring
5 = housekeeping
Parent’s educational attainment:
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
45
1 = elementary
2 = high school
31,000 to 50,000 = 2
3 = college
51,000 to 80,000 = 3
Household size:
81,000 to 100,000 = 4
1 to 5 members = 1
101,000 and above = 5
6 to 9 members = 2
Motivations:
10 and above members = 3
1= to augment the limited
Number of Siblings:
from parents
1 to 4 siblings = 1
2= to develop interpersonal
5 to 8 siblings = 2
skills
9 and above siblings = 3
3=to spend time productively
Siblings Position:
Aspirations:
Eldest = 1
1= to finish a degree
Middle
=
2
2= preparation for future job
Youngest
=
3
Parent’s Annual income:
30,000 and below = 1
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
46
Appendix D: Sample Computations
Table 1. Gender-Aspirations Cross Tabulation
Gender
Aspirations Male
Female
Total
1
14
41
55
2
21
24
45
Total
35
65
100
I
n
1
TSS= −
∑n2
i+
2
2n i=1
n 1 G 1
WSS=
2
− ∑
n
2
2
ij
j 1
= n+ j
BSS= TSS – WSS
R2= BSS/TSS
C= (n-1)(I-1) R2
where,
n= total number of observation.
nij=the count in the ijth cell
ni+= the total count in the ith cell
n+j= the total in the jth cell.
100
1
TSS=
−
([ 2
2
55 + 45 )]
2
)
100
(
2
=50-25.25
=24.75
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
47
100 1 ⎡⎛142 + 212 ⎞ ⎛ 412 + 242 ⎞⎤
WSS=
− ⎢
+
⎜
⎜
⎟
⎟ ⎜⎜
⎟
⎟⎥
2
2 ⎣⎝
35
⎠ ⎝
65
⎠⎦
= 50 -0.05(18.2 + 34.723077)
= 50 – 26.4615385
= 23.5384615
BSS= 24.75 – 23.5384615
= 1.211538
.
1 211538
R 2 =
75
.
24
= 0. 048951
C= (100-1)(2-1)(0.048951)
= 4.846154
The efficiency is between gender and motivation.
ed
Approximat
Eff( 2
χ )=
Observed( 2
χ )
.
4 000
=
.
3 758
=1.06
ed
Approximat
Eff(C)=
Observed(C)
.
4 000
=
=
1.06
.
3 766
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
48
Appendix E: EMPIRICAL SAMPLING UNDER Ho
ANOVA for Categorical Data
a = Approximated b = Observed CATANOVA c = Observed CHI-SQUARE
PERCENTILE
Simulation
yi
G Group
Mean
Variance
Reference Empirical
number
sample
percentile critical
size
value
1
⎛ 1 1 1 ⎞
2 100 all
2.000a
4.000a
.99
.890
4.440
⎜ , , ⎟
⎝ 3 3 3 ⎠
1.973b
3.766b
.90
.944
5.763
1.982c
3.758c
.95
.940
9.238
2
⎛ 1 1 1 ⎞
2 50 all
2.000a
4.000a
.90
.890
4.459
⎜ , , ⎟
⎝ 3 3 3 ⎠
1.994b
3.889b
.99
.951
6.034
2.027c
3.918c
.99
.989
9.092
3
(.70,.15,.13)
2 30 all
2.000a
4.000a
.90
.903
4.720
2.027b
4.160b
.95
.955
6.211
2.077c
3.826 c
.99
.991
9.463
4
(.70,.15,.15)
2 60 all
2.000 a
4.000a
.90
.890
4.433
1.951 b
3.940b
.95
.947
3.870
1.963 c
3.380c
.99
.992
9.854
5
(.70,.15,.15)
2 100all 2.000a
4.000a
.90
.900
4.606
1.951b
3.940b
.95
.955
6.149
1.963c
3.380c
.99
.990
9.310
6
⎛ 1 1 1 ⎞
2 15all 2.000a
4.000a
.90
.895
4.528
⎜ , , ⎟
⎝ 2 4 4 ⎠
2.024b
4.224b
.95
.956
6.265
2.078c
3.911c
.99
.992
9.770
7
(.90,.03,.03)
2 80all 2.000a
4.000a
.90
.904
4.710
1.986b
3.989b
.95
.955
6.166
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
49
2.013c 3.033c .99
.990
9.126
8
(.48,.48,.04)
2 80all 2.000a
4.000a
.90
.916
4.973
2.020b
5.498b
.95
.968
6.970
2.101c
3.399c
.99
.995
10.619
9
(.30,.30,.40)
2 50all 2.000a
4.000a
.90
.895
4.496
1.889b
3.562b
.95
.942
3.671
1.929c
3.752c
.99
.982
8.085
10
⎛ 1 1 1 ⎞
2 20,40 2.000a
4.000a
.90
.896
4.569
⎜ , , ⎟
⎝ 3 3 3 ⎠
1.954b
3.822b
.95
.952
6.117
1.989c
3.973c
.99
.988
9.019
11
⎛ 1 1 1 ⎞
2 30,100 2.000a
4.000a
.90
.890
4.422
⎜ , , ⎟
⎝ 3 3 3 ⎠
1.940b
3.602b
.95
.938
3.447
1.957c
3.587c
.99
.986
8.688
12
(.70,.15,.15)
2 20,40 2.000a
4.000a
.90
.900
4.618
2.029b
4.061b
.95
.945
5.805
2.051c
3.446c
.99
.990
9.306
13
(.90,.05,.05)
2 20,40 2.000a
4.000a
.90
.902
4.642
2.014b
4.024b
.95
.954
6.182
2.092c
3.867c
.99
.970
9.344
14
⎛ 1 1 1 2 ⎞
2 50all 3.000a
6.000a
.90
.901
6.298
⎜ , , , ⎟
⎝ 5 5 5 5 ⎠
2.919b
6.095b
.95
.947
7.684
2.963c
5.725c
.99
.989
11.180
15
⎛ 1 1 1 1 1 ⎞ 2 50all 4.000a
8.000a
.90
.895
7.619
⎜ , , , , ⎟
⎝ 5 5 5 5 5 ⎠
3.999b
7.959b
.95
.947
9.260
4.062c
7.926c
.99
.987
12.813
16
(.30,.30,.40)
3 50all 4.000a
8.000a
.90
.897
7.678
4.012b
8.005b
.95
.950
9.460
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
50
4.038c 7.954c .99
.988
13.095
17
⎛ 1 1 1 2 ⎞
3 80all 6.000a
12.000a
.90
.903
10.868
⎜ , , , ⎟
⎝ 5 5 5 5 ⎠
6.116b
12.190b
.95
.949
12.544
6.155c
12.113c
.99
.984
15.949
18
(.16,.16,.16,.16,.
2 100all 3.000a
10.000a
.90
.900
9.266
16,.20)
3.035b
9.967b
.95
.954
11.391
5.060c
9.854c
.99
.988
14.688
19
(.20,.20,.20,.40)
4 80all 9.000a
18.000a
.90
.895
14.283
8.908b
17.054b
.95
.945
16.621
8.971c
16.336c
.99
.986
21.064
20
⎛ 1 1 1 ⎞
4 30all 6.000a
12.000a
.90
.926
11.770
⎜ , , ⎟
⎝ 3 3 3 ⎠
6.280b
12.860b
.95
.961
13.357
6.347c
13.018c
.99
.986
16.128
21
⎛ 1 1 1 1 ⎞
3 100all 8.000a
16.000a
.90
.890
12.986
⎜ , , , ⎟
⎝ 5 5 5 5 ⎠
7.967b
15.887b
.95
.940
15.047
7.992c
16.028c
.99
.987
19.580
22
(.30,.30,.40)
3 30all 8.000a
16.000a
.90
.890
12.993
7.794b
15.389b
.95
.940
15.027
7.815c
15.438c
.99
.981
18.554
23
⎛ 1 1 1 1 1 ⎞ 4 100all 12.000a 24.000a
.90
.908
18.873
⎜ , , , , ⎟
⎝ 5 5 5 5 5 ⎠
12.200b
25.296b
.95
.956
21.569
12.198c
24.856c
.99
.993
27.498
24
(.10,.10,.10,.10,.
6 100all 35.000a 70.000a
.90
.918
47.617
10,.10,.10,.30)
35.035b
80.488b
.95
.958
51.247
34.940c
69.204c
.99
.991
59.326
Source: Journal of the American Statistical Association, September 1971
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
DAS-ILEN, ELVIRA D. and PADILAN, DIXY E. April 2008. Efficiency of
CATANOVA in Measuring Association between Socio-Economic Variables and
Motivations and Aspirations of Working Students. Benguet State University, La
Trinidad, Benguet.
ADVISER: SALVACION Z. BELIGAN, Ph.D.
ABSTRACT
The study was conducted to determine the association between socio-economic
variables and motivations and aspirations of working students at Benguet State
University using the Analysis of Variance for Categorical data (CATANOVA) as an
alternative tool for Pearson’s Chi-square test. Specifically, this study aimed to determine
the relationship of working student’s motivation and aspiration with their socio-economic
variables and to determine the proportion of variation of the response variable
(motivations/aspirations) attributed to the independent variables (socio-economic
variables).
A survey questionnaire was floated and distributed to the respondents through
random sampling in order to obtain the needed data for this study.
Both chi-square and C statistics were computed to determine the significance of
the association between the response and the independent variables.
Based on the results of the analysis employing both CATANOVA and Pearson’s
Chi-square students motivations was associated to the father’s occupation, household size
and sibling’s position. Aspiration of working students was associated to gender and
family income. With the use of CATANOVA, the percent contribution of the socio-
economic variables on the variability of the working student’s motivations and
aspirations were determined.
CATANOVA statistics is at par with the Pearson’s Chi-square statistics in
determining associations but more efficient than Pearson’s Chi-square in terms of their
variability.
ii
TABLE OF CONTENTS
Page
Bibliography …………………………………………………………….………i
Abstract …………………………………………………………………………i
Table of Contents ……………………………………………………………...iii
INTRODUCTION
Background of the Study…………………………………….…………1
Objectives of the Study ………………………………………………...3
Importance of the Study ………………………………………………. 3
Scope and Delimitation of the Study ……………….………………….4
REVIEW OF RELATED LITERATURE
Studies Using CATANOVA ………………………………….….….... 5
Studies on the Association of Educational
Aspirations and Socio-economic Variables ……………..….……..….. 8
THEORETICAL FRAMEWORK
Analysis of Variance for Categorical Data ……………………….…...11
Pearson’s Chi-square Test ………….………………………………….23
Distribution
of
Pearson’s Chi-Square …….……………………………24
Efficiency of Pearson’s Chi-square
and CATANOVA ………………………………….……….………….24
Definition of Terms ……………………………….…………………...25
METHODOLOGY
Locale of the Study ……………………………….……………………27
Respondents of the Study ……………………….………….…………..27
iii
Data Collection Instrument …………………….…………….……...…27
Analysis of Data ………………………………….………….…………28
RESULTS AND DISCUSSION
Associations Between Educational Motivations and
Socio-Demographic Profile of the Working Students ………..…………29
Associations Between Educational Aspirations and
Socio-Demographic Profile of the Working Students ……….…………31
Efficiency of CATANOVA and the Pearson’s
Chi-square Test in Measuring Associations ...…………………...……..32
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
Summary
…………………………………………….….………………34
Conclusions
………………………………………….…….……………34
Recommendations
………………………………….……….…………..35
LITERATURE CITED ………………………………………….….…………..36
APPENDICES ……………………………………………………….…………37
iv
INTRODUCTION
Background of the Study
Researchers usually grope on what statistical tool to be used in
determining relationships of categorical variables. Several techniques for
analyzing and testing contingency data are available in many literatures. The Chi-
Square test is one of the most commonly used for determining the association or
dependence of two variables. The Chi-Square test is most appropriate when the
sample has been selected from an infinite or large bivariate multinomial
population and the sample size n is reasonable large (Neter, et al., 1988).
However, according to Light and Margolin (1974), Chi-square test is no longer
appropriate when sample size n is small, that is, when more than 20% of the
observed cell frequencies are lower than 5. Thus, a statistical tool known as
Categorical Analysis of Variance (CATANOVA) derived earlier by Light and
Margolin (1971) is suggested for use for small sample comparisons.
CATANOVA is a variant of the standard One-Way ANOVA since it also
partition the variation of nominal data into additive components. The procedure
deals with the frequency counts and row and column marginal totals in a two
dimensional contingency table. Similar to the parametric One-Way ANOVA, the
general approach to categorical data is to compute the total variation in the data
and then partition this into specific components. Aside from the measurement of
association statistic, CATANOVA can also determine the proportion of variation
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
2
attributable to the independent variable which other test can not do, like the chi-
square test.
For the application of CATANOVA, the researchers of this study sought
to utilize the empirical data on the educational aspirations of working students.
One of the most important concerns of any College or University is the
full development of students’ potential through meaningful and relevant programs
that respond to their varied backgrounds, orientations, personal and professional
needs. To meet their needs, the provision of a well-defined student assistance
program requires much attention, as it contributes to the total development of
prospective professionals through varied learning experience in a work setting
within the school. As provided for in the National Compensation Circular No. 64
(1990), students assistance maybe hired to render emergency or temporary
services for the following reasons: 1) to provide practicum training for students,
2) to provide extra income for students and 3) to emphasize dignity in labor.
Though these objectives of the NCC are for the benefits of the students, many
students found working and studying at the same time a difficult task because it
does not only consume much the time of students but also prevent them from
joining extra curricular activities that will promote their personal growth.
Hence, the purpose of this study is two-fold, first, to find the relationship
between students’ motivation to work and their demographic profile and the
relationship between educational aspirations and socio-economic background of
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
3
the respondents, and second, to assess the applicability of CATANOVA in
contingency tables with different dimensions.
Objectives of the Study
The study aimed to determine the efficiency of CATANOVA in
measuring the degree of association between two categorical variables observed
from working students at Benguet State University. Specifically, the study aimed:
1. To determine the association between educational motivation and socio-
economic background of working students.
2. To determine the association between educational aspirations and socio-
economic profile of the working students; and
3. To determine the efficiency of CATANOVA and Pearson Chi-square
test in measuring the degree of association between two categorical variables.
Importance of the Study
Results of this study maybe utilized as an input to school administrators in
policy formulation regarding student assistance programs.
The study is also important to researchers who use statistics in
determining the degree of association between two variables measured in nominal
scale and to any researcher who is dealing with quantitative variables.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
4
Moreover, the result of the study will serve as a guide for researchers in
determining the efficiency of CATANOVA over the Pearson’s Chi-square test
using the data on motivations and aspirations of working students.
Scope and Delimitation of the Study
This study was conducted at Benguet State University, La Trinidad,
Benguet and it focused on the aspirations and motivations of working students in
relation to their socio-economic profile using the CATANOVA.
The respondents were students from Benguet State University who are
working inside the school as students’ assistants, and those who are working
outside BSU as service crews in McDonalds and Jollibee for a minimum of five
(5) months.
The researchers gathered the needed information from the different
colleges of Benguet State University through the administration of survey
questionnaires.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
5
REVIEW OF RELATED LITERATURE
Studies Using CATANOVA
Anestesiol (1995) evaluated the incidence of colonization and infection by
methicillin-resistant in PICU. They studied two-hundred patients with duration of
hospitalization longer than 24 hours out of the 255 patients who were hospitalized
during the same period. The patients were divided in two groups according to the
presence or the absences of MRS. The difference of the two populations were
compared using the t-test and the CATANOVA. Wilcoxon's test was used to
analyze the relation between the two values. The results were significant when p =
0.05 and Ct = 3.81. They concluded that the clinical impact of MRS in terms of
morbidity and mortality in this PICU is modest. The prevention and limitation of
the spread of MRS could be obtained by simple but essential measures of control.
Anderson et. al. (1980) presented an extension of the analysis of variance
for categorical data (CATANOVA) procedure to multidimensional contingency
tables involving several factors and a response variable measured on a nominal
scale. Using an appropriate measure of total variation for multinomial data, partial
and multiple association measures are developed as R2 quantities which parallel
the analogous statistics in multiple linear regression for quantitative data. In
addition, test statistics are derived in terms of these R2 criteria. Finally, this
CATANOVA approach is illustrated within the context of 2 three-way
contingency table from a multicenter clinical trial.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
6
Singh B. (2004) examined the CATANOVA method for its suitability for
analysis of nominal data from repeated measures design. The modified tests are
developed for single and multi-group repeated measures designs, separately. The
computed results reveal that in single group repeated measures designs the actual
size of existing CATANOVA test is more for negative correlation and less for
positive correlation than the stated size. This implies that the existing test may
yield the non-existing effects as significant for negative correlation and may not
even detect the real effects for positive correlation among repeated observations.
Similar results are obtained for repeated factor and interaction effects and just
reverse result for group effect in multi-group repeated measures design. These
results show that the existing CATANOVA tests are not valid for repeated
measures designs and hence the modified tests should be used for analysis of
nominal data from such designs.
Souza et.al. (2001) studied the genetic divergence among organisms;
generally the analysis is done directly from the DNA molecule. Therefore, a
possible outcome is categorical being one out of four categories (looking at the
nucleotide level). Light and Margolin (1971) developed an analysis of variance
for categorical data (CATANOVA) and Pinheiro et al. (2000) employed a
similar measure of variation and extended the CATANOVA procedure taking into
account several positions in the sequence for balanced designs. Here we consider
variable number of sequences in each group, where, the samples are unbalanced.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
7
In order to test the null hypothesis of homogeneity among groups, the asymptotic
distribution of the test statistic was found and its power is evaluated. An
application of the test to real data is illustrated using resampling methods such as
the bootstrap to generate the empirical distribution of the test statistics.
Light’s et.al. (1979) revealed that the CATANOVA method for analysis of
two ways classified nominal data was developed analogous to the least squares
method of fitting constants for two way cross-classified quantitative data with
disproportionate cell frequencies. This method has been compared with simple
chi-square method through a numerical example. Exact small sample behavior in
two-way contingency tables is investigated for Pearson's chi-square statistic 2
χ ,
Light and Margolin's C statistic and its related 2
R measure of association,
^
Kullback's minimum discrimination information statistic f (2) , and Goodman and
Kruskal's Lambda. 2
R was shown to be identical to Goodman and Kruskal’s λ ,
leading to a test for independence based on λ . In small samples from a product
of multinomial model, the null distribution of C is better approximated by a chi-
square distribution than is the null distribution of 2
χ ; both are considerably better
^
approximated by a 2
χ distribution than is the null distribution of f (2) . It is
proved for tables with two columns and any number of rows that if the column
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
8
^
totals are equal, then 2
χ ; thus, 2
χ is more conservative than f(2) . Hence, use of
^
f ( )
2 should be avoided in testing independence in tables with small samples.
Light’s et. al. (1971) discussed a measure of variation for categorical data.
He developed an analysis of variance for a one-way table, where the response
variable is categorical. The data can be viewed alternatively as falling in a two-
dimensional contingency table with one margin fixed. Components of variation
are derived, and their properties are investigated under a common multinomial
model. Using these components, we propose a measure of the variation in the
response variable explained by the grouping variable. A test statistic is
constructed on the basis of these properties, and its asymptotic behavior under the
null hypothesis of independence is studied. Empirical sampling results confirming
the asymptotic behavior and investigating power are included.
Lesser’s et. al. (1980) studied the educational aspirations of 617
adolescent students in Denmark and was classified into five categories.
CATANOVA and Chi-square test was used to determine the relationship of
gender and educational aspirations of the students. They found out that gender is
statistically associated to educational aspirations of the students although, only 2.1
percent of the variation was explained by knowledge of a respondents’ gender.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
9
Studies about Aspirations and Socio-
Economic Characteristics
Aspirations are strong desires to reach something high or great. Young
people's aspirations guide what students learn in school, how they prepare for
adult life, and what they eventually do (Walberg, 1989). This Digest reports on
educational aspirations of rural youth compared with students living elsewhere,
and suggests ways communities can work together to raise the sights of their
young people.
Kayser (1973) tested the predictability of the aspirational changes using
the simple, two-state, and discrete-time Markov model and to test for differences
in selected background characteristics for students with different aspirational
patterns. Major findings that there was a differential turnover between college and
non-college plans, that students with a given level but different histories of
aspirations were similar in selected background characteristics such as family
status, significant others' support, and income aspirations, and that the two-state,
discrete-time Markov model predicted the changes in educational aspirations for
the students sampled. A major conclusion was that the assumption of aspirational
stability was supported and the process appears to be Markovian in nature with
one general process in operation over all the high school years.
Chiale et.al. (1965) investigated associations of overweight status and
changes in overweight status over time with life satisfaction and future aspirations
among a community sample of young women. Analyses were conducted using the
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
10
SPSS version 11.0 statistical software package (SPSS, Inc., Chicago, IL).
2
χ analysis was used to investigate differences in aspirations and life satisfaction
among women in the four BMI categories (cross-sectional analysis) and across
women in the four overweight status change categories (longitudinal analysis).
Young women’s aspirations were cross-sectional related to BMI category, such
that obese women were less likely to aspire to further education, although this
relationship maybe explained largely by current occupation. Even after adjusting
for current occupation, young women who were obese were more dissatisfied with
work/career/study, family relationships, partner relationships, and social activities.
Weight status was also longitudinally associated with aspirations and life
satisfaction. Women who were overweight or obese at both surveys were more
likely than other women to aspire to "other" types of employment (including self-
employed and unpaid work in the home) as opposed to full-time employment.
They were also less likely to be satisfied with study or partner relationships.
Women who resolved their overweight/obesity status were more likely to aspire to
being childless than other women.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
11
THEORETICAL FRAMEWORK
Analysis of Variance for Categorical
Data (CATANOVA)
In this study, a technique derived by Light and Margolin (1974) was
utilized as measure of association between categorical variables. The analysis
begins with the partition of the total variation in the data into an explainable
component and an unexplained component or noise. The general approach of this
technique is to compute the total variation in the data and then partition this
variation into specific components. The distributions of the various components
are derived under an assumed model.
CATANOVA is a variant of the standard one-way analysis of variance. It
is an analysis of variance of a one-way table with replication, where the variable
being observed is nominal.
Since the data is nominal, this is one obstacle in defining measure of
variation for categorical data because there is a tendency to think of variation as a
measure of departure of a set of individual observations from their mean.
Moreover, with categorical data, the mean is an undefined concept. However,
there is an alternative way of looking at variation. Gini noted that the sum of
squares of deviations from their mean for n quantitative measurements can be
expressed solely as a function of the squares of the pair wise differences for all
(n pairs. Specifically, if X ,..., X denote the measurements, then
2 )
1
n
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
12
n
n
2
1
SS = ∑ (X − X
X
X
i
) = ∑ ( −
i
j )
1
2 n
i =
j =1
(1)
n
n
= 1 ∑ ∑ d 2ij
2 n i=1 j=1
where,
n
X
X = ∑ i
i=1
n
d = X − X .
ij
i
j
Assuming there are G unordered experimental groups and unordered I
response categories. Each response is in one of the I categories. Denote the
number of responses in one category i for group j,i = ,...,
1
I and j − ,...,
1
G by n .
ij
The number of responses, or group size, for group j is n =
G n . The
+ j
∑j= ij
1
number of responses in the ith category is n = ∑G n .. Thus, the total number of
j+
ij
1
responses in the study is;
n = ∑G n = I n = I
G n
(2)
j=
+ j
1
∑i= i+
1
∑i=1∑j= ij
1
An alternative way of viewing this data is via an I and G contingency
table where n is the count in the (i, j) cell. For this two-dimensional
ij
contingency table, or the equivalent one-way analysis of variance with categorical
responses, the total variation in the response variable or “total sum of squares” is
equal to;
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
13
n
1
TSS =
−
∑I n2
(3)
i=
i+
2
2n
1
In one-way analysis of variance with quantitative data one would compute
the within-group component of variation and the between-group component of
variation. We do this by applying the formula of variation for categorical
responses X ,..., X which is;
1
n
1 n n
n
n
2
1
∑∑d =
d
(4)
ij
∑∑ ,
2
ij
n j 1= i 1=
2n j 1= i 1=
where,
d = 1 if x and x name different categories
ij
i
j
= 0 if x and x name the same category.
i
j
To the ( ijn pairs of responses within group j, j = ,...,
1
.
G
2 )
The variation in the response variable within the th
j group is then
I
n
1
ij −
∑n2
(5)
ij
2
2n+ j i=1
Hence the total within-group variation or within –group sum of squares
(WSS) is found by assuming (5) over j ;
G
I
n
1
WSS = ∑⎛
⎞
+
⎜ j
2
⎜
−
∑n ⎟
ij ⎟
1
2
2n
j= ⎝
+ j i=1
⎠
(6)
G
I
= n − 1 ∑ 1 ∑n2ij
2
2 j=1 n+ j i=1
For balance design situations where n = n , then
+ j
+
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
14
1 ⎛
G
I
⎞
2
2
WSS =
Gn −
+
∑∑n .
2 ⎜⎜
ij ⎟⎟
n ⎝
j 1
= i 1
=
⎠
The between-group variation or sum of squares (BSS) is defined as the
differences between TSS and WSS. Hence:
1 ⎛ G
I
I
1
1
BSS = ⎜
2
2
⎜∑
∑ ⎞
n ⎟ =
i
n
1
.
(7)
ij ⎟
∑ = i+
2 j=
⎝ 1 n
1
2n
+ j i=
⎠
For balance design situations where n = n , this reduces to
+ j
+
1 ⎡ ⎛ G I
⎞
I
⎤
BSS =
⎢G ∑∑n2 − n2 .
ij
∑ i+⎥
2n G
+
⎢ ⎜⎜
⎟
⎟
⎣ ⎝ j=1 i=1
⎠ i=1
⎥
⎦
We now turn to the problem posed in the introduction on measures of
association for categorical. The three components of variation we have just
derived enable us to define a measure of association between the grouping and
response variables which maybe given a “proportion of variation” explained
interpretation. We define this measure as:
G
I
I
⎛
1
⎞
⎜∑
∑
1
n2 ⎟ − ∑n2
ij
⎜
⎟
+
j=1 n
2
⎝
+ j i=1
n
i
i
⎠
=1
BSS
R =
=
(8)
I
1
2
TSS
n − ∑n
n
i+
i=1
n
It has the property that 2
R = 0 if ij
= f ; i = 1,…,I; j = 1,….,G, i.e.,
i
n+ j
if for each j, j = 1 there exists a I such that n = n , i.e., if there is perfect
ij
+ j
predictability. Otherwise, 0
2
< R < 1.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
15
BSS, TSS and a Proposed Test Statistics
The previously proposed method of partitioning categorical variation will
be referred to as a categorical analysis of variation, or CATANOVA. We
developed the following C statistics to test the null hypothesis p = p .
ij
i
(n − )1(I − )1BSS
C =
(9)
TSS
This test statistic, referred to as the CATANOVA C statistic, is asymptotically
approximated as 2
χ(
under H .
I 1
− )(G 1
− )
0
Recalling the measure of association defined in (8), we note that
2
C = (n − )(
1 I − )
1 R . Thus, to test the significance of the measure of association,
we need to multiply it by (n − )(
1 I − )
1 and approximate its distribution under H
0
by 2
χ(
.
I 1
− )(G 1
− )
The derivation of the test statistic proceeds along asymptotic lines. This
enables us to invoke further results for quadratic forms in normal variates,
because V is asymptotically multivariate normal,
V ≈ N (μ, Ζ)
(10)
In investigating BSS, we first prove that under H , the asymptotic
0
distribution of BSS does not depend on μ . BSS can be written as;
2
I
G ⎡
1
n
⎤
BSS = V ′BV = ∑∑
ij
ni+
⎢
−
⎥ .
(11)
2 i=1 j=1 ⎢ n
n
j
⎥
+
⎣
⎦
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16
G
Let ).
θ = n − n p = n − E (n Then θ = ∑θ = n − np . Moreover,
ij
ij
+ j
i
ij
o
ij
ij
ij
i+
i
j 1
=
under H ,
0
θ ′ = (θ ,...,θ ,...,θ ,...,θ )
11
I1
1G
IG
is asymptotically
θ ′ ≈ N( ,
0 Z )
(12)
where,
Z is as in Z = n Z and Z = M ⊗ Z where,
j
+ j
o
o
⎡ p 1
( − p )
− p p
− p p
1
1
1
2
L
1
I
⎤
⎢
⎥
p 1
( − p )
− p p
.
⎢
2
2
L
2
I
⎥
Z =
o
⎢
⎥
M
⎢
⎥
⎣
p 1
( − p )
I
I ⎦
Now,
BSS = V BV
′
⎡
2
1
θ + n p
⎤
=
ij
ij
i
θ
np
i+
i
=
⎢
∑ ∑
−
n ⎥
i
j
2
⎢
n
n
+ j
+
⎥
j
⎣
⎦
2
I
G ⎡
1
θ
⎤
ij
θi
=
⎢
∑∑
− + n ⎥
+ j
2 i=1 j=1 ⎢ n
n
+
⎥
j
⎣
⎦
= θ ′ θ
B
Hence, under H in studying BSS , we may treat μ as 0 . From this, it follows
0
that under H , BSS is asymptotically distributed as,
0
IG
BSS ≈ ∑
2
λ χ
(13)
i
1
i=1
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17
where,
{λ ,i = ,...,
1
is the set of characteristic roots of BZ :
i
}
IG
1 ⎡⎛
−1
1
⎞
⎤
BZ =
⎜ M − μ ⎟ ⊗ I M ⊗ Z
⎢
G
1 ⎥[
o ]
2 ⎣⎝
n
⎠
⎦
(14)
1 ⎡
1
⎤
=
I − μ M ⊗ Z
⎢ G
G
⎥
o
2 ⎣
n
⎦
where,
1 ⎡⎛
−1
1
⎞
⎤
B = T −W = ⎢⎜M − μ
and
G ⎟ ⊗ I ⎥
2 ⎣⎝
n
⎠
⎦
Z = M ⊗ Z
o
where,
Z is defined in (12).
o
The characteristic roots of BZ under H are then one half the products of the
0
characteristic roots of
I
1
−
M
(15)
G
n U G
With the characteristic roots of Z , the characteristic roots of (15) are the
o
solutions to the determinantal equation:
1
I (I − λ) −
M = 0
(16)
G
U
n G
The left-hand side of (2.7) is:
G 1
.
L H.S. = λ(I λ) −
−
Efficiency of CATANOVA in Measuring Association between Socio-Economic
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Hence, λ = 0 is a root withy multiplicity one and λ = 1 is a root with
multiplicity G −1 .
Thus, (2.4) is really
I
1
BSS ≈ ∑
2
λ χ
(17)
i
G−1
2 i=1
where,
{λ ,i = ,...,
1
I is the set of characteristic roots of Z .
i
}
o
Recall that:
⎡ p1(1− p1 ) − p p
− p p
1
2
1
I
⎤
⎢
⎥
Z =
.
(18)
o
⎢
M
⎥
⎢
⎣
p1(1− pI )⎥⎦
i.e., Z , is a multinomial covariance matrix. Roy et. Al., studied the determination
o
of the characteristic roots of the general multinomial covariance matrix without
presenting the roots. The characteristic equation they obtain is:
I
2
⎪
⎧
⎛ p
⎞
i
⎪
⎫ I
1
⎨ − ∑
⎬∏(p − λ
(19)
i
) = 0
⎪
⎜
⎜
⎟
⎟
λ
i=
p −
1 ⎝
i
⎠⎪ i 1
⎩
⎭ =
Certainly λ = 0 is a root. This also follows because a multinomial
covariance matrix is singular. The determination of the other roots in general
appears to be an unsolved problem. We can make some further observations,
however, by noting that (19) is equivalent to:
∏I (
I
p = λ
p
p
λ
(20)
i
)
2
−
i=
∑ i ∏ ( −
j
) = 0
1
j≠i
i 1
=
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19
Then;
a. Unless there is a j ≠ i such that p = p ,λ = p is not a root.
i
j
i
b. If no two probabilities p , p i ≠ j , are equal, then between each two
i
j
successively ordered probabilities lies a characteristics roots, i.e., if we order the
probabilities and the characteristic roots, we have:
p < λ < p < λ < ... < p
< λ
< p
1
i
1
i
i2
i2
i(I 1
− )
i(I 1
− )
iI
To see this, note that the left-hand side of (20) as a function of λ
changes continuously from positive to negative or vise versa as λ changes from
p
to p
.
ij
i( j 1
+ )
c. If k of the probabilities p = i = ,...,
1
I, are equal, i.e.,
i
p = p
... = p
= p , then in (20), (
) 1−
− k
p λ
is a factor and λ = p is then a
ir
(ir+1)
(ir+k −1)
characteristic root with multiplicity k = 1 .
Unfortunately the set of{λ ,i = ,...,
1
I − . Suppose, however, that under H we
i
}
1
o
I 1
1
approximate the distribution of BSS ~ ∑−λ
,
2
iχ G−1
2 i=1
By (suggested also by Thompson):
1 I
S ~ ∑−1 2
λ x (I−1)(G−1) ,
(21)
2 i=1
where,
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I 1
∑−
I
λ
1
2
−
i
∑ p
traceZ
i
i 1
=
o
i 1
λ =
=
=
=
.
I −1
I −1
I −1
From (c) above, if p =
= p
1
...
= then
1
λ = is a root multiplicity I −1 and
i
i
I
I
1
2
BSS ~
X (
.
I 1
− )(G 1
− )
2I
Thus, under H , at the center of the simplex defined by
o
({
I
p ,..., p :
≥ ,
0 = ,...,
1
and ∑ p = 1
(22)
i
i )
p
i
I
i
}
i
i=1
The approximation of (17) by (21) is exact. Moreover, for I = 2 and all G , the
approximation of (17) by (21) is exact.
In general, over the simplex of (22),
1
I 1
1
E BSS =
∑ − λ G − = λ I − G −
0
( i 1= i)( )1 ( )1( )1
2
2
⎛ 1
⎞
=
2
E⎜ λx (I 1−)(G 1−) ⎟ = E(S ).
⎝ 2
⎠
Thus, the approximation in (17) has the correct first moment under H over the
0
entire simplex. Further, under H :
0
Var (BSS)
(G − )
1
I
1
=
∑ −1 2λ = G trace Z
i
( − )1
( 20)
i=1
2
2
1
= (G − )
1 [∑I [p 1 p 2 2
p2 p2
(23)
i ( −
i ) ]+
I
I
i=1
∑i=1∑j=i+ i j
1
]
2
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21
1
2
= (G − )
1 ⎡
I
2
∑ p − 2 I 3
I
2
⎤
⎢
p
p
i=
i
∑
+
i=
i
∑
1
1
( i 1= i ) .
2
⎥
⎣
⎦
The variance of the approximation is:
1
Var (S )= (I − )
1 (G − ) 2
1 λ
2
1 ⎛ G −1
2
⎞
= ⎜
⎟(1
I
2
− ∑ p
(24)
i 1
=
i ) .
2 ⎝ I −1 ⎠
For I =2 and all G, (23) and (24) are exactly equal over the entire simplex defined
by (22) because the approximation is exact. Moreover, for all I and G, (23) and
(24) are exactly equal at the center of the simplex defined (22), for the same
reason. Numerical comparisons between (23) and (24) for various values of I and
(p ,..., p indicate moderate to good agreement “near” the center of the simplex.
i
n )
For example, with I = 4 and p = 0. ,
2 p = 0. ,
2 and p = 0. ,
4 then (23)
1
2
4
1
= (G − )
1 ( 1824
.
). Hence, we will approximate the asymptotic distribution of BSS
2
as:
⎛
I
⎞
2
⎜1− ∑ p ⎟
⎜
i
i 1
=
⎟ 2
BSS ~ ⎜
χ
.
(25)
2(I − )
(I 1
− )(G− )
1
1 ⎟
⎜
⎟
⎝
⎠
1 ⎛
I
2 ⎞
This still leaves the problem that ⎜1− ∑ p ⎟ is unknown. The minimum
2 ⎝
i
i=1
⎠
1 ⎛
I
1 I
2 ⎞
variance unbiased estimator under H of ⎜1− ∑ p ⎟ =
∑ p − can be
i (1
p1 )
o
2 ⎝
i
i=1
⎠
2 i 1=
shown to be:
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1 ⎛ n
I
⎞ ⎛ n + ⎛
⎞
n + ⎞
i
⎜
⎟∑⎜
⎜
⎟ I
i
−
⎟
2n ⎝ n −1⎠ i=1 ⎝ n ⎝
⎠
n ⎠
⎡
1
∑ n2 ⎤
=
⎢∑n
i
i
−
+
+
⎥
(
2 n
i
− )
1 ⎢
(26)
i
n
⎣
⎥
⎦
I
1
⎡
1
2 ⎤
=
n − ∑n
(
2 n
⎢
− )
1 ⎣
n
i+ ⎥
i=1
⎦
= TMS
1 ⎛
I
2 ⎞
We already knew that I TMS
(
) = ⎜1−
p
o
∑ ⎟
2 ⎝
i
i=1
⎠
n
Since n i+ coverage’s in probability to p under H , the asymptotic variance of
n
i
o
TMS is zero.
1 ⎛
I
2 ⎞
Thus,
under H , we view TMS as a constant equal to ⎜1− ∑ p ⎟ then,
o
2 ⎝
i
i=1
⎠
(n − )1(I − )1BSS
C=
TSS
⎡⎛ G
I
1
⎞
I
1
⎤
⎢ ∑
∑n2 −
n2
ij
∑ i+ ⎥
⎜
⎜
⎟
⎟
j=1 n + j
1
n
=
(n-1)
(I-1)
⎢⎝
i=
⎠
i=1
⎥
⎢
(27)
I
⎥
1
⎢
n − ∑n2i+
⎥
⎢
n i=1
⎥
⎣
⎦
Maybe approximated as 2
χ(
.
I 1
− )(G 1
− )
Pearson’s Chi-square Test
Pearson's chi-square test (χ2) is one of a variety of chi-square tests –
statistical procedures whose results are evaluated by reference to the chi-square
distribution. It tests a null hypothesis that the relative frequencies of occurrence of
Efficiency of CATANOVA in Measuring Association between Socio-Economic
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observed events follow a specified frequency distribution. The events are assumed
to be independent and have the same distribution, and the outcomes of each event
must be mutually exclusive. A simple example is the hypothesis that an ordinary
six-sided die is "fair", i.e., all six outcomes occur equally often. Pearson's chi-
square is the original and most widely-used chi-square test.
Chi-square is calculated by finding the difference between each observed
and theoretical frequency for each possible outcome, squaring them, dividing each
by the theoretical frequency, and taking the sum of the results. The number of
degrees of freedom is equal to the number of possible outcomes, minus 1:
n (O − E 2
i
i )
2
χ = ∑
(28)
i=1
Ei
where,
O = an observed frequency;
i
E = an expected (theoretical) frequency, asserted by the null
i
hypothesis;
n = the number of possible outcomes of each event.
A chi-square probability of 0.05 or less is commonly interpreted by
applied workers as justification for rejecting the null hypothesis that the row
variable is unrelated (that is, only randomly related) to the column variable. The
alternate hypothesis is not rejected when the variables have an associated
relationship.
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Distribution of Pearson’s Chi-Square
The null distribution of the Pearson statistic with j rows and k columns is
approximated by the chi-square distribution with (k − 1) (j − 1) degrees of
freedom. [28] This approximation arises as the true distribution, under the null
hypothesis, if the expected value is given by a multinomial distribution. For large
sample sizes, the central limit theorem says this distribution tends toward a certain
multivariate normal distribution.
Efficiency of Pearson’s Chi-square and C
The efficiency of both the 2
χ and the C statistics maybe obtained using
2k
the formula given below:
E =
V (C)
(29)
2k
E =
(30)
V ( 2
χ )
where k is the degrees of freedom of the r x c contingency table computed as
(r – 1)(c – 1).
Efficiency of CATANOVA in Measuring Association between Socio-Economic
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Definition of Terms
Aspiration. A strong desire to achieve a goal.
CATANOVA. It is a technique designed to identify the variation between
treatments of interest to the researcher. A measure of variation for categorical
data.
Categorical. An unordered and discrete variable. It is data that can only be
put into unordered groups.
Chi-square Test. A statistics used in the analysis of enumeration data. It
reflects discrepancies between the observed and expected or theoretical
frequencies of individuals, objects, or events falling in various categories.
Economic Profile. Refers to the respondent’s age, sex, civil status, parent’s
occupation, parent’s annual income and etc.
Household size. Refers to the number of family members of the
respondents.
Motivation. A factor that encourages a person to pursue or achieve
something.
Nominal data. These are categorical data where the order of the categories
is arbitrary.
One-way analysis of variance. A procedure for comparing the mean scores
of two or more groups based on one categorical independent variable.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
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26
Pearson’s’ Chi-square. It is used to test the hypothesis of no association of
columns and rows in tabular data. Also known as the chi-square goodness-of-fit
or chi-square test of independence.
Predictor Variables. These are variables from which projections are made
in a prediction study. These include the socio-economic profile of the
respondents.
Proportion of variation ( 2
R ). Used to measure association.
Response variable. A variable on which information is collected and
which there is an interest because of its direct policy relevance.
Test of goodness of fit. It is used to test if an observed distribution
conforms to any other distribution. Establishes whether or not an observed
frequency distribution differs from a theoretical distribution.
Test of independence. Assess whether paired observations on two
variables expressed in a contingency table are independent of each other.
Working students. These are youths working their way through college.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
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/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
27
METHODOLOGY
Locale of the study
The study was conducted at Benguet State University, La Trinidad,
Benguet during the school year 2007-2008.
Respondents of the Study
A sample of 100 students was chosen at random from the College of
Agriculture, College Nursing, College of Arts and Sciences, College of
Engineering, College of Home Technology and Economics, College of Teacher
Education, College of Forestry, and College of Veterinary Medicine.
This study included students working as student assistants within Benguet
State University as the sampling units.
Data Collection Instrument
A questionnaire consisting of several questions on the respondents’ socio-
economic background and questions on their motivations and aspirations for
working was developed and pre-tested.
The dependent variables included student’s motivations and aspirations
and the predictor variables were the socio-economic background of the
respondents which included gender, father and mother’s occupation, mother and
fathers’ educational attainment, household size, number of siblings, sibling
position and parents’ annual income.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
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28
Analysis of Data
The responses of the respondents on the different questions were tabulated
in an r x c contingency table. The Chi-square test and the CATANOVA were both
computed for each contingency table.
Significance of results of the 2
χ and C values were compared to the
tabular value at 0.01 and 0.05 level of significance to determine the association of
the respondents’ socio-economic variables with their motivations and aspirations.
The
R2 which is a measure of the variation in the dependent variable due
to the independent variable was likewise determined only in CATANOVA. The
efficiency of both the 2
χ and CATANOVA were computed using equation (29)
and (30).
Efficiency of CATANOVA in Measuring Association between Socio-Economic
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/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
29
RESULTS AND DISCUSSION
Associations Between Educational
Motivations and Socio-Demographic
Profile of the Working Students
For purposes of illustrating the computation of CATANOVA, the
responses of 100 working students were classified into educational motivations by
father’s educational attainment, by household size, and by sibling’s position. As
revealed in Table 1, both CATANOVA C and the 2
χ test did not show
significant association between educational motivations and father’s educational
attainment. This result indicates that student’s motivation to work while studying
Table 1. Association between student’s motivations to work as student’s assistants
and father’s occupation and father’s education.
EDUCATIONAL MOTIVATION
DEMOGRAPHIC
Earn Extra
Develop
Spend Free
PROFILE
Money
Interpersonal
Time Wisely
for Allowance
Skills
Father’s
Occupation
Government
3 8 0
Employee
Self-Employed
5
3
4
Farming
19
15
13
Laboring
16
8
6
df=6 2
χ =10.78, P=.09, C=10.39 P=.10; R2(%)=5.2
Father’s
Education
Elementary 11
11 8
High School
20
9
13
College 12
14
2
df=4 2
χ =9.00,
P=.06, C=8.03
P=.09; R2(%)=4.05
Efficiency of CATANOVA in Measuring Association between Socio-Economic
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30
is weakly motivated by their father’s educational attainment. However, the
variation in the student’s motivation to work attributed to their father’s
educational attainment was about 4.05 percent. The father’s occupation was
likewise weakly associated to their children’s motivation to work while studying.
As revealed by both tests, no significant association between father’s educational
attainment and father’s occupation and student’s motivations to work.
As shown in Table 2, chi-square test did not show significant association
between educational motivations by household size. However, in the C test, there
is a significant association between the 2 variables. This result indicates that
student’s motivation to work was motivated by household size. Although
statistically associated to student’s educational motivation, household size
contributed only 2.30% of the variation in the student’s educational motivations.
For the association between sibling’s position and students’ educational
motivation, both 2
χ and C tests revealed significant results. These findings
indicate that student’s motivation to work is strongly motivated by sibling’s
position. The variation in the students’ motivation to work is attributed to their
sibling’s position by about 5.50 percent.
Table 2, shows significant association between students’ motivation and
household size and between student’s motivation and sibling’s position using the
C test.
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31
Table 2. Association between student’s motivations to work as student’s assistants
and household size and sibling’s position.
EDUCATIONAL MOTIVATION
DEMOGRAPHIC
Earn Extra
Develop
Spend Free
PROFILE
Money
Interpersonal
Time Wisely
for Allowance
Skills
Household
Size
1-5 9
6
5
6-9 18
21
13
10 or more
16
7
5
df=4 2
χ =4.00ns, P=.40, C=9.23* P=.05; R2(%)=2.30
Sibling’s
Position
Eldest 19
10
3
Middle 11
17
14
Youngest 13
7
6
df=4 2
χ =10.56*, P=.03, C=10.86* P=.02; R2(%)=5.50
* = significant ns = not significant
These results suggest that both household size and sibling position had
significant contribution on the student’s to work as student assistants 2.30 % of
the variation in the student’s educational motivations.
Association Between Educational
Aspirations and Socio-Demographic
Profile of the Working Students
Table 3 shows both C and the 2
χ test showed strong evidence to declare
significant association between student’s educational aspirations and gender and
between student’s educational aspiration and parent’s annual income.
The results indicate that student’s aspiration to work while studying are strongly
attributed to their gender and parent’s annual income. Gender explained
Efficiency of CATANOVA in Measuring Association between Socio-Economic
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Table 3. Association between student’s aspirations to work as student’s assistants
and gender and family income.
DEMOGRAPHIC
PROFILE
EDUCATIONAL ASPIRATION
Gender
Finish a Degree
Preparation for Future Job
Male
14
21
Female 41
24
df=1 2
χ =4.90*, p=.03 C=4.85*, p=.03 R2(%)=4.9
Family Income
30,000 and below
6
12
31,000 to 50,000
28
10
51,000 to 80,000
10
10
81,000 to 100,000
7
6
101,000 or more
4
7
df=4 2
χ =10.53*, p=.03 C=10.42*, p=.03 R2(%)=10.53
* - significant
4.9 % of the variation in student’s aspiration while parent’s income explained
10.53 % variation in the student’s aspiration to work.
Efficiency of CATANOVA
and the Pearson’s Chi-square Test
in Measuring Associations
Table 4 presents the efficiency of the Pearson’s chi-square and
CATANOVA C association tests for two variables cross-tabulated in contingency
tables with different dimensions. For a 4 x 3 contingency table, the variability of
the Pearson’s chi-square statistic and the C statistics considered higher than the
expected variance of 2k for the theoretical 2
χ distribution.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
33
Table 4. Efficiency of CATANOVA and Chi-square Test with 100 Sample Size
Efficiency of
Efficiency of
Pearson’s Chi-
CATANOVA
Motivation Associated to
Dimension
square
2k
2k
V (C)
V ( 2
χ )
Father’s Occupation
4 x 3
0.99
0.98
Father’s Educ’l Attainment
3 x 3
1.01
1.00
Household Size
3 x 3
1.01
1.00
Sibling’s Position
3 x 3
1.01
1.00
Aspiration Associated to
Gender
2 x 2
1.06
1.06
Parents Annual Income
5 x 2
1.04
1.04
The efficiency of Pearson’s Chi-square in a 3 x 3 contingency table was
found higher than the efficiency of the C statistic. This means that the Pearson’s
chi-square statistics is more variable than the theoretical 2
χ distribution.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
34
SUMMARY, CONCLUSION AND RECOMMENDATION
Summary
Both chi-square and C statistics were computed to determine the
significance of association between the response and independent variables.
Based on the results of the analysis employing both CATANOVA and
Pearson’s Chi-square, students’ motivations was associated to the father’s
occupation, house hold size and sibling’s position. Aspiration of working students
was associated to gender and family income. With the use of CATANOVA, the
percent contribution of the socio-economic variables on the variability of the
working students’ motivations and aspirations were determined.
CATANOVA
statistics
is
at par with the Pearson’s Chi-square statistics in
determining associations but more efficient than Pearson’s Chi-square in terms of
their variability.
Conclusion
Based on the above results, the following conclusions were drawn:
The father’s occupation, educational attainment, household size and
sibling position had significant bearing on the respondent’s motivation to work.
The respondent’s aspirations to earn a degree were explained by their
gender and their family income.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
35
CATANOVA as a tool for determining association is at par with the Chi-
square test but considered more efficient than the Pearson’s Chi-square.
Recommendations
In dealing with qualitative data where the response variable is categorized
with no order CATANOVA C is recommended for used because it does not only
measure the relationship between the response and predictor variables it also
measures the percent contribution explained by the predictor variables.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
36
LITERATURE CITED
BALATERO, R.E. 2003. Factors Affecting Academic Performance of Working
Students of Benguet State University, La Trinidad Benguet.
COBB, R. A., MCINTIRE, W.G., and PRATT, P. A.1989. Vocational and
educational aspirations of high school students: A problem for rural
America. In R. Quaglia (Ed.), Research in Rural Education
COCHRAN, W.G. (1950). “The Comparison of Percentages in Matched
Samples,” Biometrika.
D’AMBRA, ANTONELLO and PASQUALE SARNACCHIARO, 2007.
Explorative Data Analysis and CATANOVA for Ordinal Variables: An
Integrated Approach
HOEFFDING, W.1965. “Asymptotically Optimal tests for Multinomial
Distributions,” Annals of Mathematical Statistics.
LIGHT, R.J. MARGOLIN, H. B. 1969. “Analysis of Variance for Categorical
Data, with Applications to Agreement and Association,” Unpublished
Ph.D. dissertation, Department of Statistics, Harvard University.
MACLI-ING and GUIMPAYAN. 2001. Factors Affecting Academic
Performance of Working Students in McDonalds Center mall.
Undergraduate Thesis. Benguet State University, La Trinidad Benguet.
PEARSON, K.1900. “On the Criterion that a Given System of Deviation from the
probable in the Case of the Correlated System of Variables is Such that it
can Be Reasonably Supposed to have Arisen from Random Sampling,”
Philosophical Magazine.
RJ ANDERSON, JR LANDIS, 1980. Communications in Statistics-Theory and
Methods, informaworld.com
WALBERG, H. J. 1989. Student aspirations: National and international
perspectives. In R. Quaglia (Ed.), Research in Rural Education.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
37
APPENDICES
Appendix A: Letter of Communication
Benguet State University
College of Arts and Sciences
La Trinidad, Benguet
MR. ROMEO BABARAN
Manager
Jollibee La Trinidad
La Trinidad, Benguet
We the undersigned are students from Benguet State University
taking up Bachelor of Science in Applied Statistics. We are conducting our
research entitled “Efficiency of CATANOVA in Measuring Association Between
Socio-economic Variables and Motivations and Aspirations of Working
students.” This is to comply with the requirements of the course.
In this connection, may we request for permission that we be allowed to
administer our questionnaire to the working students of Benguet State University
who are having part time job in your fast food establishment.
Your favorable action for this request is highly appreciated.
Thank you.
Researchers,
ELVIRA D. DAS-ILEN
DIXY E. PADILAN
Noted by:
SALVACION Z. BELIGAN
Adviser
Recommending Approval:
MARIA AZUCENA B. LUBRICA
Chairman-MPS Department
AUREA MARIE M. SANDOVAL
CAS-Dean
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
38
Benguet State University
College of Arts and Sciences
La Trinidad, Benguet
The Manager
McDonald La Trinidad
La Trinidad, Benguet
Greetings!
We the undersigned are students from Benguet State University taking up
Bachelor of Science in Applied Statistics. We are conducting our research entitled
“Efficiency of CATANOVA in Measuring Association Between Socio-economic
Variables and Motivations and Aspirations of Working students.” This is to
comply with the requirements of the course.
In this connection, may we request for permission that we be allowed to
administer our questionnaire to the working students of Benguet State University
who are having part time job in your fast food establishment.
Your favorable action for this request is highly appreciated.
Thank you.
Researchers,
ELVIRA D. DAS-ILEN
DIXY E. PADILAN
Noted by:
SALVACION Z. BELIGAN
Adviser
Recommending Approval:
MARIA AZUCENA B. LUBRICA
Chairman-MPS Department
AUREA MARIE M. SANDOVAL
CAS-Dean
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
39
Appendix B: Sample Survey Questionnaire
Benguet State University
College of Arts and Sciences
La Trinidad, Benguet
Dear Respondents:
Greetings!
We the undersigned are students from Benguet State University taking up
Bachelor of Science in Applied Statistics. We are conducting our research entitled
“Efficiency of CATANOVA in Measuring Association Between Socio-economic
Variables and Motivations and Aspirations of Working students.” This is to
comply with the requirements of the course.
In this connection, may we solicit your valued cooperation in answering
the following questions honestly. Rest assured, your answers will be kept
confidential.
Thank you and more power.
Researchers,
ELVIRA D. DAS-ILEN
DIXY E. PADILAN
Noted by:
SALVACION Z. BELIGAN
Adviser
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
40
DIRECTION: Please check or write on the blank that corresponds to your answer
Personal Data:
Name
(optional):
_______________
Gender:
_____
Age:
_____
Civil
Status:
___
Degree/Course:
____________________
Year
Level:
_____
Number of Units: ________
A. Socio-economic Profile:
Direction. Please write your answer on the blank provided for the needed
information.
1. Parents occupation. __________ Mother
__________ Father
2. Parents highest educational attainment. __________ Mother ________ Father
3. Household size (# of family members). _____
4. Number of siblings in the family (including you): _____
5. Your sibling position. _____
6. Annual income of parents (just encircle your answer on the choices given
below).
a. below 30,000
b. 31,000 to 50,000
c. 51,000 to 80,000
d. 81,000 to 100,000
e. 101,000 and above
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
41
B. On Motivation and Aspiration:
Direction. Please encircle only your main reason why you work while studying.
Motivations:
a. Augment the limited allowance from parents.
b. To develop interpersonal skills.
c. To spend free time wisely.
Aspirations:
a. To finish a degree.
b. For job preparation.
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
42
Appendix C: Variables Raw Data
t
n
me
in
l Atta
na
s
a
tio
ze
e
Of Parents
s
c
ational Attainment
uc
on
ccupation
ccupation
Ed
ndent
s
O
er
e
r
's
er of Sibling
a
tion
spo
th
s
e
hold Si
nnual Incom
spiration
Re
Gend
Father's O
Mother'
Father's Edu
Mo
Hou
Numb
Sibling Positi
A
Motiv
A
1
1
3
5
2
2
3
3
3
2
1
2
2
1
3
3
2
3
3
3
3
5
1
2
3
2
4
1
1
3
1
1
3
1
3
2
4
1
4
2
2
3
3
2
3
5
1
1
5
2
4
5
1
3
2
3
3
5
2
2
6
2
3
3
2
3
2
2
3
2
3
1
7
1
3
5
1
2
2
2
1
1
1
1
8
1
3
4
2
2
2
2
2
3
3
2
9
2
3
5
1
3
1
1
1
1
2
1
10
2
3
2
1
2
3
3
2
2
1
1
11
2
3
3
3
3
2
2
2
5
3
2
12
2
3
5
1
2
2
1
1
2
1
1
13
2
1
5
3
3
2
2
3
4
2
1
14
1
4
5
2
2
3
3
1
1
2
2
15
2
3
4
1
3
2
1
1
3
2
1
16
1
3
3
2
3
2
2
2
1
1
1
17
2
3
5
3
3
2
2
2
3
3
2
18
2
4
5
3
3
1
1
1
2
1
1
19
2
4
2
2
3
2
1
2
2
2
2
20
2
1
5
3
2
2
1
2
4
2
2
21
2
2
4
3
3
2
2
2
4
2
2
22
2
4
5
2
3
1
1
1
2
2
1
23
2
3
3
2
3
1
1
3
1
1
1
24
1
3
3
3
3
2
1
2
1
2
2
25
1
4
5
1
3
3
3
3
2
1
2
26
2
1
5
3
3
1
1
1
2
1
1
27
2
3
5
2
3
3
3
1
1
2
2
28
2
3
4
2
3
2
1
2
3
3
1
29
2
4
5
3
3
2
2
2
2
1
1
30
2
3
5
3
2
2
2
2
4
2
1
31
2
4
5
2
3
3
2
1
2
1
1
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
43
32
2
3
5
1
2
2
2
2
2
1
1
33
2
4
5
3
2
3
3
2
3
1
1
34
2
4
2
1
1
1
2
1
1
3
2
35
1
4
5
2
3
3
2
1
1
1
2
36
1
2
4
3
3
3
3
2
5
1
2
37
2
1
1
3
3
1
1
2
3
2
1
38
1
2
5
2
3
2
2
2
1
3
2
39
1
3
3
1
2
2
1
3
1
2
1
40
1
1
1
3
3
2
1
1
2
2
1
41
2
2
2
2
2
3
1
1
2
2
1
42
2
4
5
2
2
3
3
2
2
1
1
43
1
3
4
1
2
3
3
2
2
3
2
44
2
4
5
3
3
2
2
2
3
2
2
45
1
3
1
3
3
2
2
2
5
2
2
46
2
3
3
1
3
2
1
2
3
3
2
47
2
2
3
1
1
2
2
3
4
3
1
48
2
2
1
3
3
3
3
1
5
1
1
49
1
3
5
1
2
2
3
3
4
2
2
50
1
1
5
3
3
3
2
1
5
2
2
51
1
3
4
2
2
2
2
1
4
1
1
52
2
3
5
2
3
1
1
2
3
3
2
53
2
2
4
2
3
2
2
2
2
1
1
54
2
3
3
1
1
3
3
3
4
1
1
55
2
3
2
2
3
1
1
2
3
1
2
56
1
4
5
1
2
2
1
3
2
2
2
57
1
3
3
1
2
1
1
2
3
2
2
58
2
2
1
3
3
3
2
1
4
1
2
59
1
1
2
3
3
3
3
2
5
2
1
60
1
3
5
2
3
2
1
2
5
1
2
61
1
2
5
3
3
2
2
1
3
1
1
62
1
3
4
2
2
2
2
1
2
3
1
63
2
3
3
2
3
3
3
2
1
2
2
64
2
4
3
1
3
3
2
3
2
3
2
65
2
3
5
1
3
1
1
3
2
1
2
66
2
4
5
3
1
2
1
2
3
2
1
67
2
2
2
2
2
3
3
2
4
3
1
68
2
3
2
2
2
2
2
1
4
1
2
69
1
4
4
1
3
3
2
2
2
3
1
70
2
4
5
3
3
2
1
1
2
1
1
71
2
4
5
2
2
2
2
2
2
2
1
72
1
3
3
1
3
2
3
2
1
2
2
73
2
3
5
2
3
2
2
2
2
3
1
74
2
3
3
1
2
2
2
1
1
1
2
75
2
4
4
2
3
2
2
1
2
1
1
76
2
3
5
2
3
1
1
1
2
1
1
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
44
77
1
4
5
2
3
2
1
1
2
1
1
78
2
4
5
2
2
1
2
3
2
1
1
79
2
4
1
2
3
1
1
2
3
1
1
80
2
3
4
1
3
3
3
3
4
1
1
81
2
3
5
1
2
1
1
2
2
2
1
82
1
3
5
2
3
2
1
2
2
3
1
83
2
2
5
2
3
2
1
1
2
3
2
84
2
2
2
3
3
3
3
2
2
2
2
85
2
1
1
3
3
2
2
3
3
1
1
86
2
3
5
2
3
1
1
3
5
1
1
87
2
3
3
1
2
1
1
2
2
3
1
88
2
3
3
1
2
1
1
3
2
2
1
89
2
1
5
3
3
2
1
1
4
2
2
90
1
3
5
2
3
1
1
3
2
3
2
91
1
5
3
1
2
3
2
3
3
1
2
92
1
1
2
2
2
2
2
1
3
1
1
93
1
1
2
2
2
2
2
1
3
2
2
94
2
3
5
2
3
2
3
2
1
2
1
95
2
4
5
3
3
2
1
1
2
1
1
96
1
4
4
2
3
2
1
3
1
3
2
97
1
3
3
1
2
2
1
3
2
2
1
98
2
4
5
2
3
3
2
2
3
3
1
99
2
3
1
3
3
2
2
3
3
1
2
100
2
4
4
1
3
3
1
1
1
1
2
Legend:
Gender:
1
=
male
2 = female
Parent’s occupation:
1 = government employee
2 = self-employed
3 = farming
4 = laboring
5 = housekeeping
Parent’s educational attainment:
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
45
1 = elementary
2 = high school
31,000 to 50,000 = 2
3 = college
51,000 to 80,000 = 3
Household size:
81,000 to 100,000 = 4
1 to 5 members = 1
101,000 and above = 5
6 to 9 members = 2
Motivations:
10 and above members = 3
1= to augment the limited
Number of Siblings:
from parents
1 to 4 siblings = 1
2= to develop interpersonal
5 to 8 siblings = 2
skills
9 and above siblings = 3
3=to spend time productively
Siblings Position:
Aspirations:
Eldest = 1
1= to finish a degree
Middle
=
2
2= preparation for future job
Youngest
=
3
Parent’s Annual income:
30,000 and below = 1
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
46
Appendix D: Sample Computations
Table 1. Gender-Aspirations Cross Tabulation
Gender
Aspirations Male
Female
Total
1
14
41
55
2
21
24
45
Total
35
65
100
I
n
1
TSS= −
∑n2
i+
2
2n i=1
n 1 G 1
WSS=
2
− ∑
n
2
2
ij
j 1
= n+ j
BSS= TSS – WSS
R2= BSS/TSS
C= (n-1)(I-1) R2
where,
n= total number of observation.
nij=the count in the ijth cell
ni+= the total count in the ith cell
n+j= the total in the jth cell.
100
1
TSS=
−
([ 2
2
55 + 45 )]
2
)
100
(
2
=50-25.25
=24.75
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
47
100 1 ⎡⎛142 + 212 ⎞ ⎛ 412 + 242 ⎞⎤
WSS=
− ⎢
+
⎜
⎜
⎟
⎟ ⎜⎜
⎟
⎟⎥
2
2 ⎣⎝
35
⎠ ⎝
65
⎠⎦
= 50 -0.05(18.2 + 34.723077)
= 50 – 26.4615385
= 23.5384615
BSS= 24.75 – 23.5384615
= 1.211538
.
1 211538
R 2 =
75
.
24
= 0. 048951
C= (100-1)(2-1)(0.048951)
= 4.846154
The efficiency is between gender and motivation.
ed
Approximat
Eff( 2
χ )=
Observed( 2
χ )
.
4 000
=
.
3 758
=1.06
ed
Approximat
Eff(C)=
Observed(C)
.
4 000
=
=
1.06
.
3 766
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
48
Appendix E: EMPIRICAL SAMPLING UNDER Ho
ANOVA for Categorical Data
a = Approximated b = Observed CATANOVA c = Observed CHI-SQUARE
PERCENTILE
Simulation
yi
G Group
Mean
Variance
Reference Empirical
number
sample
percentile critical
size
value
1
⎛ 1 1 1 ⎞
2 100 all
2.000a
4.000a
.99
.890
4.440
⎜ , , ⎟
⎝ 3 3 3 ⎠
1.973b
3.766b
.90
.944
5.763
1.982c
3.758c
.95
.940
9.238
2
⎛ 1 1 1 ⎞
2 50 all
2.000a
4.000a
.90
.890
4.459
⎜ , , ⎟
⎝ 3 3 3 ⎠
1.994b
3.889b
.99
.951
6.034
2.027c
3.918c
.99
.989
9.092
3
(.70,.15,.13)
2 30 all
2.000a
4.000a
.90
.903
4.720
2.027b
4.160b
.95
.955
6.211
2.077c
3.826 c
.99
.991
9.463
4
(.70,.15,.15)
2 60 all
2.000 a
4.000a
.90
.890
4.433
1.951 b
3.940b
.95
.947
3.870
1.963 c
3.380c
.99
.992
9.854
5
(.70,.15,.15)
2 100all 2.000a
4.000a
.90
.900
4.606
1.951b
3.940b
.95
.955
6.149
1.963c
3.380c
.99
.990
9.310
6
⎛ 1 1 1 ⎞
2 15all 2.000a
4.000a
.90
.895
4.528
⎜ , , ⎟
⎝ 2 4 4 ⎠
2.024b
4.224b
.95
.956
6.265
2.078c
3.911c
.99
.992
9.770
7
(.90,.03,.03)
2 80all 2.000a
4.000a
.90
.904
4.710
1.986b
3.989b
.95
.955
6.166
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
49
2.013c 3.033c .99
.990
9.126
8
(.48,.48,.04)
2 80all 2.000a
4.000a
.90
.916
4.973
2.020b
5.498b
.95
.968
6.970
2.101c
3.399c
.99
.995
10.619
9
(.30,.30,.40)
2 50all 2.000a
4.000a
.90
.895
4.496
1.889b
3.562b
.95
.942
3.671
1.929c
3.752c
.99
.982
8.085
10
⎛ 1 1 1 ⎞
2 20,40 2.000a
4.000a
.90
.896
4.569
⎜ , , ⎟
⎝ 3 3 3 ⎠
1.954b
3.822b
.95
.952
6.117
1.989c
3.973c
.99
.988
9.019
11
⎛ 1 1 1 ⎞
2 30,100 2.000a
4.000a
.90
.890
4.422
⎜ , , ⎟
⎝ 3 3 3 ⎠
1.940b
3.602b
.95
.938
3.447
1.957c
3.587c
.99
.986
8.688
12
(.70,.15,.15)
2 20,40 2.000a
4.000a
.90
.900
4.618
2.029b
4.061b
.95
.945
5.805
2.051c
3.446c
.99
.990
9.306
13
(.90,.05,.05)
2 20,40 2.000a
4.000a
.90
.902
4.642
2.014b
4.024b
.95
.954
6.182
2.092c
3.867c
.99
.970
9.344
14
⎛ 1 1 1 2 ⎞
2 50all 3.000a
6.000a
.90
.901
6.298
⎜ , , , ⎟
⎝ 5 5 5 5 ⎠
2.919b
6.095b
.95
.947
7.684
2.963c
5.725c
.99
.989
11.180
15
⎛ 1 1 1 1 1 ⎞ 2 50all 4.000a
8.000a
.90
.895
7.619
⎜ , , , , ⎟
⎝ 5 5 5 5 5 ⎠
3.999b
7.959b
.95
.947
9.260
4.062c
7.926c
.99
.987
12.813
16
(.30,.30,.40)
3 50all 4.000a
8.000a
.90
.897
7.678
4.012b
8.005b
.95
.950
9.460
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
50
4.038c 7.954c .99
.988
13.095
17
⎛ 1 1 1 2 ⎞
3 80all 6.000a
12.000a
.90
.903
10.868
⎜ , , , ⎟
⎝ 5 5 5 5 ⎠
6.116b
12.190b
.95
.949
12.544
6.155c
12.113c
.99
.984
15.949
18
(.16,.16,.16,.16,.
2 100all 3.000a
10.000a
.90
.900
9.266
16,.20)
3.035b
9.967b
.95
.954
11.391
5.060c
9.854c
.99
.988
14.688
19
(.20,.20,.20,.40)
4 80all 9.000a
18.000a
.90
.895
14.283
8.908b
17.054b
.95
.945
16.621
8.971c
16.336c
.99
.986
21.064
20
⎛ 1 1 1 ⎞
4 30all 6.000a
12.000a
.90
.926
11.770
⎜ , , ⎟
⎝ 3 3 3 ⎠
6.280b
12.860b
.95
.961
13.357
6.347c
13.018c
.99
.986
16.128
21
⎛ 1 1 1 1 ⎞
3 100all 8.000a
16.000a
.90
.890
12.986
⎜ , , , ⎟
⎝ 5 5 5 5 ⎠
7.967b
15.887b
.95
.940
15.047
7.992c
16.028c
.99
.987
19.580
22
(.30,.30,.40)
3 30all 8.000a
16.000a
.90
.890
12.993
7.794b
15.389b
.95
.940
15.027
7.815c
15.438c
.99
.981
18.554
23
⎛ 1 1 1 1 1 ⎞ 4 100all 12.000a 24.000a
.90
.908
18.873
⎜ , , , , ⎟
⎝ 5 5 5 5 5 ⎠
12.200b
25.296b
.95
.956
21.569
12.198c
24.856c
.99
.993
27.498
24
(.10,.10,.10,.10,.
6 100all 35.000a 70.000a
.90
.918
47.617
10,.10,.10,.30)
35.035b
80.488b
.95
.958
51.247
34.940c
69.204c
.99
.991
59.326
Source: Journal of the American Statistical Association, September 1971
Efficiency of CATANOVA in Measuring Association between Socio-Economic
Variables and Motivations and Aspirations of Working Students
/ Elvira D. Das-Ilen & Dixy E. Padilan. 2008
Document Outline
- Efficiency of
CATANOVA in Measuring Association between Socio-Economic Variables and
Motivations and Aspirations of Working Students.
- BIBLIOGRAPHY
- ABSTRACT
- TABLE OF CONTENTS
- INTRODUCTION
- REVIEW OF RELATED LITERATURE
- THEORETICAL FRAMEWORK
- METHODOLOGY
- RESULTS AND DISCUSSION
- SUMMARY, CONCLUSION AND RECOMMENDATION
- LITERATURE CITED
- APPENDICES