Analysis of Science Students’ SSCE Result using Canonical Correlation Analysis

Analysis of Science Students’ SSCE Result using Canonical Correlation Analysis

Abstract

This research intends to establish whether student’s performance in other subjects has any impact on Mathematical science subjects. This research work adopts the method of Canonical Correlation Analysis to investigate the relationship between the Sets of the subjects. The data used for the analysis is students’ WAEC results for the year 2011, Federal Science College, Sokoto. Out of fourteen subjects areas which the students sat for, nine subjects were used for the study. Two sets were formed. Set-A consists of Mathematics, Chemistry and Physics, while Set-B consists of Biology, Hausa-Language, Agricultural Science, Economics, English Language, and Geography. Three Canonical roots were obtained and only one is statistically significant showing a slight correlation between the two Sets of data.     

However, there should be equal concentration on other subjects to improve the students’ performance in the Mathematical science subjects.

Introduction

Multivariate Statistics is a useful set of methods for analyzing a large amount of information in an integrated frame, focusing on the simplicity (Simon, 1969) and latent order (Wheatley 1994) in seemingly complex array of variables. Benefits to using multivariate statistics include: Expanding sense of knowing, allowing rich and realistic research designs and having flexibility built on similar univariate methods. 

Canonical correlation analysis is the most generalized member of the family of multivariate statistical techniques. It is directly related to several dependence methods, similar to regression; canonical correlation’s goal is to quantify the strength of the relationship, in this case between the two sets of variables. It also resembles discriminant analysis in its ability to determine independent dimension for each variable set, in this situation with the objective of producing the maximum correlation between the dimensions. Thus, canonical correlation identifies the optimum structure or the dimensionality of each variable set that maximizes the relationship between dependent and independent variable sets. In doing so, it develops a number independent canonical function that maximize the correlation between the linear composites, also known as canonical variates, which are sets of dependent and independent variables. Each canonical function is actually based on the correlation between two canonical variates, one variate for the dependent variables and one for the independent variables. Among unique feature of canonical correlation is that the variates are derived to maximize their correlation. Moreover, canonical correlation does not stop with the derivation of a single relationship between the sets of variables, instead a number of canonical functions.

Data used for the Analysis

The study is concerned at analyzing SSCE examination results of some students of Federal Science College, Sokoto. Out of Fourteen Subjects areas which the students sat for, nine subjects are used. The selection was based on the number of students that sat for the interested subjects of study. Linear Canonical Correlation Analysis were used for the study. Wilk’s Lambda test is used for the test of significant of the canonical variate. The data consist of scores of 100 students for 2011 WEST AFRICA EXAMINATION COUNCIL (WAEC) Exams in nine subjects which were divided into Set-A = {Mathematics, Chemistry and Physics} and Set-B = {Biology, Hausa Language, Agricultural Science, Economics, English Language and Geography} 

Methodology

Anderson (1958) gives a detailed Mathematical concept of canonical correlation analysis. Let X be a q-dimensional random vector and Y be a p-dimensional random vector. Suppose that X and Y have means and  respectively and that

                  

          

          

Let us now consider the linear combinations

g=           

and

f=           

The correlation between g and f is defined as show below 

          

Wilk’s Lambda test

The Wilk’s Lambda test of hypothesis is given as:

, i.e. there is no relationship between the canonical variates.

, i.e. there is relationship between the canonical variates.

Test statistic:

        

Where:

is the correlation between 

is the correlation between   

is the correlation between   

Significance Level:

Decision rule:

Reject if   and otherwise accept. Rencher (2002) 

Analysis and Discussion

An initial step in canonical correlation analysis is an inspection of the correlation matrix of the given data.

Let S denote the data such that

S = {set-A, set-B}

Where:

Set – A = {Mathematics, Chemistry, Physics}

Set – B = {Biology, Hausa Language, Agric. Science, Economics, English Language,  

                 Geography}                             

Proper analysis begins with a simple examination of the correlation significance Dunn et, al, (1977)

Table 1: Canonical correlation coefficient of Set – A and Set – B 

Canonical

Functions

Canonical Correlation

Eigen values

% of Variance Explained

1

2

3

0.4310

0.3891

0.1447

0.1858

0.1514

0.0209

51.9

42.3

5.8

Table 1 shows the Canonical correlation of the three canonical variates and their corresponding Eigen values. The Eigen values of the canonical variates can be tested by employing Wilk’s Lambda criterion to test for the significant by using Wilk’s Lambda test, Rencher (1998)

Hypothesis:

 

Reject   , we have the following table:

Table 2: To test that the canonical correlations are zero

S/NO

N

P

Q

Df

p-value

1

100

6

3

18

0.0057

0.05

2

100

5

2

10

0.0657

0.05

3

100

4

1

4

0.7378

0.05

Since one of the canonical correlation tested is significant i.e. at the first canonical correlation, p – value, it implies that the null hypothesis is rejected, which indicates that one out of the three canonical correlation is significantly different from zero. Where P is the number of variables considered in a certain canonical variate, while Q is the number of variables considered in the opposite canonical variate and df is the degree of freedom used at each level of canonical function.    

Now consider the first canonical variate pair  and  with canonical correlation Coefficient, so that the proportion of variance common to the first canonical variate pair is  showing about 18.58% of the proportion of variance captured by the first canonical variate.

Similarly  is the canonical correlation coefficient between the second canonical variate pair and so  which indicates about 15.14% of the proportion of variance captured,    shows the canonical correlation coefficient between the third canonical variate pair and so  indicating 2.1% of the proportion of variance captured.

Although; canonical correlation analysis has many tables for interpretation. Further interpretations of the canonical correlation coefficients will be done as suggested by Dunn et, al, (1997), using Canonical Loadings and Canonical Cross Loadings.

Table 3: Canonical loading for Set –A and Set – B

         Sets

    Subjects

    

    

    

  

      Set – 1

Mathematics

Chemistry

Physics

-0.2908

-0.7016

-0.2828

 -0.2846

 -0.6093

 1.0876

-0.9885

0.6484

-0.0140

     

        Set – 2

Biology

Hausa Language

Agric. Science

Economics

English Language

Geography

-0.3753

-0.1636

-0.1861

-0.1489

-0.0809

-0.6385

-0.0152

-0.7342

-0.3976

-0.0167

0.1179

0.5338

-0.2374

0.5464

-0.5660

-0.0530

-0.6729

0.5121

The canonical loadings in Table 3 provided information about the relative contribution of variables to each independent canonical relationship, the first pair of canonical variates can be written as follows:

=-0.3753Biology-0.1636HausaLanguage-0.1861AgriculturalScience-

-0.1489Economics-0.0809English Language-0.6385Geography.

The correlation  between   is called the first canonical correlation coefficient.

Of the individual variable physics loading heaviest with the value (-0.2828) followed by mathematics (-0.2908) and chemistry (-0.7016) loading for the ordering for the criterion variables.

The values attached to each subjects are their partial correlation to their corresponding canonical variables and indicating the individual contribution to the canonical pair.

Table 4: Canonical cross loading for Set-A and Set-B 

      Sets

  Subjects

      

      

       

     

      Set-1

Mathematics

Chemistry

Physics

-0.2540

-0.3934

-0.2869      

-0.0636

-0.0907

0.2903

-0.1145

0.0486

-0.0028

      

      Set-2

Biology

Hausa Language

Agric. Science

Economics

English Language

Geography

-0.2738

-0.1831

-0.1781

-0.1455

-0.1651

-0.3604

-0.0218

-0.3087

-0.2017

-0.0030

0.0368

0.1453

-0.0281

0.0421

-0.0559

-0.0090

-0.0956

0.0363

Table 4 shows the Canonical Cross loading of the two canonical functions. In the first canonical function, we see that both Mathematics and Physics slightly have high correlations with independent canonical variate -0.2540 and -0.2869 respectively. While the Weakest correlation came from Set-B, i.e. Geography with -0.3604 followed by Biology with -0.2738.  

However, the canonical correlation which examines the linear relationship between Set – A and Set – B variables is by creating the combinations. The first canonical correlation explains the maximum relationship between the canonical variates and each successive canonical correlation is estimated so as to be orthogonal yet still explain the maximum relationship not accounted for by the previous canonical correlation. This reflects the high variance among these variables. By squaring the terms in the canonical loading, we find percentage of the variance for each of the variable explained by function 1.

Conclusion

It can be seen that set-A and set-B are slightly correlated as sought for. Canonical correlation analysis measured the strength of relationship of the canonical pair and the subjects that strongly contributed. The first pair with a measure of correlation of 0.431 with the proportion of variability of about 52%, the second pair with a measure of correlation 0.3891 with the proportion of variability of about 42% and the third canonical pair with a measure of correlation 0.1447 having a proportion of variability of about 6%. From the output of the analysis carried out on the entire data, it is clearly seen that the correlation between the mathematical science subjects and others is moderate at the first canonical variate due to the much emphasis and concentration given to the Mathematical science subjects compare to the Other subjects, it is recommended that, a similar emphasis and concentration should be given to other subjects by the Authority of the school, because the better understanding of the Other subjects  will  enhance  understanding of the Mathematical science subjects as it curtails all its expressions and explanations in English Language.

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