ANALYSIS OF SSCE EXAMINATION RESULT USING CANONICAL CORRELATION AND PRINCIPAL COMPONENT ANALYSES

1.0 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.

There are several disbelieves on multivariate method these include: needing large samples, a belief that multivariate methods are challenging to learn, and results that may seem more complex to interpret than those from univariate methods. Most multivariate methods can embrace multiple theories and hypothesis; most can analyze several independent variables, and some allow several dependent variables, for example Canonical correlation, factor analysis, principal component and multivariate analysis of variance (MANOVA). Some allow the examination of several groups or samples (e.g. logistic regression; analysis of covariance and discriminant function analysis) (Anastasi and Urbina, 1997; Harlow 2005)

Canonical correlation analysis is the most generalized member of the family of multivariate statistical techniques. It is the 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 dependent variable sets.

Canonical correlation analysis deals with the association between composites sets of multiple dependent and independent variables. 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.

Canonical correlation analysis reduces each of these patterns to derived variables, the canonical U and V variables. This means, for example that each such relation can be visually inspected using a familiar biavariate scatter diagram. The largest canonical correlation corresponds to the strongest relation between independent and dependent variables. Sub-sequent canonical correlations correspond to relation of decreasing strength. The significant of this feature for human factor research is that we often find different response patterns under different environmental conditions. For example, different patterns of flight mode selection under different phases of flight Canonical correlation analysis allows these patterns to be character objectively and allows their relative strengths to be measured. Anderson (1958).

Principal component analysis is one of the methods that can be used to analyse multivariate dataset. It can reduce the dimensionality of large data set which consists of a number of interrelated variables to smaller components. Hussain et. al. (2011)

In principal component Analysis, we seek to maximize the variance of a linear combination of the variables. For example, we might want to rank students on the basis of their scores on achievement test in English, Mathematics, Reading, and so on. An average score would provide a single scale on which to compare the students, but with unequal weights we can spread the students out further on the scale and obtain a better ranking.

Essentially, Principal component is a one-sample technique applied to data with no groupings among the observations and no partitioning of the variables into subset y and x.

Principal components are concerned only with the core structure of a single sample of observations on p variables. None of the variables is designated as dependent, and no grouping of observations is assumed. Rencher (2002)

The first principal component is the linear combination with maximal variance; we are essentially searching for a dimension along which the observations are maximally separated or spread out. The second principal component is the linear combination with maximal variance in a direction orthogonal to the first principal component, and so on.

In general, the principal components define different dimensions from those define by discriminant function or canonical variates.

Principal components are used to reduce the number of dimensions. Another useful dimension reduction device is to evaluate the first two principal components for each observation vector and construct a scatter plot to check for multivariate normality, outliers, and so on. Rencher (2002).

1.1 Significant of the study

Various researches had been carried out on Canonical Correlation that focuses majorly on socio economic and management, such as; to analyze Asset Liability Management (ALM), the relationship between information Technology and Investment and analyzing education production in Malaysia. Also; Principal Components has been used to identify common driving in votality term structure and in understanding meteorological characteristic in predicting weather condition.

Infact; it has not been established; whether students performance in non-mathematical subjects has any impact on their performance in mathematical subjects and the prediction of the subjects contributing to their inconsistency.

1.2 Aim and Objectives

The aim of this research work is to study the relationship between mathematical subjects and non-mathematical subjects and the view of verifying the variability in the students’ performance. The above aim is achieved through the following objectives:

To analyze the Canonical Correlation between the mathematical and non-mathematical subjects.

To test the significance of canonical variate using Wilk’s Lambda Test.

To test the homogeneity of variance among the variables using Bartlett’s Test.

To analyze the principal components among the variables.

To verify the variables that contributes significantly to the percentage of variance in the components.

1.3 Scopes and Limitation of the study

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 and Principal Component Analysis were used for the study. Bartlett’s test is used to test for the homogeneity of variance of the variables.

2.0 LITERATURE REVIEW

The nature of Canonical Correlation analysis is also closely related to principal components analysis. The primary purpose of which is to define the underlying structure among the variables in the analysis. Canonical correlation analysis corresponds to principal component analysis in the creation of the optimum structure or dimensionality of each variable set that maximizes the relationship between independent and dependent variables sets. Whereas principal components analysis attempt to explain the linear relationship among a set of observed variables and an unknown number of factors/variates. However, Canonical correlation focuses more on the linear relationship between two variates.

2.1 Review of Related Literatures

Anderson (1958) used Canonical correlation to study interrelations of a data with n=25, p=q=2. There were two correlations, with the first one as 0.789 which was higher than any of the individual correlations of a variable of their first set with a variable of the other. The second canonical correlation is very close to zero. This means that to study the relation between two head dimensions of the first sons and second sons we can confine our attention to the first canonical variate. Since the second canonical variates correlates slightly.

Noor and Ang (2005): used Canonical Correlation Analysis to analyse educational production by investigating the effects of school inputs, environmental inputs and gender influence in the production of a joint educational production function in mathematics and science subjects for eight grade students in Malaysia.

Antonio et. al. (2010):used principal component to identify the common driving changes in the volatility term structure, which has become the factor in the valuation of securities such as caps, swaptions, bonds with embedded options, interest rate derivatives etc.

Hussain et. al. (2011): seeks to use Principal component analysis in understanding the meteorological characteristics helps in predicting the Weather conditions, as found that the first two principal component can be described as the variation of geo-potential values at all levels and relative humidity with temperature values of 850 and 1000 level respectively.

3.0 Materials and Methods

3.1 Data used for the study

The data used for the study were collected from Federal Science College, Sokoto. 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}.

3.2 Canonical Correlation Analysis

Anderson (1958) gave 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

(3.1)

(3.2)

(3.3)

Let us now consider the linear combinations

g= (3.4)

and

f= (3.5)

The correlation between g and f is defined as show below

(3.6)

3.2 Principal component Analysis

Hussain et. al. (2011): The steps involved in the analysis of principal component analysis include the method of getting the data, standardizing the data, calculating the covariance matrix and visualising the results. Algebraically, principal components are particular linear combinations of the p random variables.

Geometrically, these linear combinations represent the selection of new coordinate system obtained by rotating the original system with their development does not require a multivariate normal assumption. On the other hand, principal components derived for multivariate normal populations have useful interpretations in terms of the constant ellipsoids.

Step 1: get the data

Consider the linear combinations:

(3.7)

Step 2: standardise the data

Sometimes it makes sense to compute principal component for raw data. This is appropriate when all the variables are in the same units. Standardizing the data is often preferable when the variables are in different units or when the variance of different columns is substantial. This can be done by subtracting the means of each column and dividing by its standard deviation namely:

In matrix notation, it is given by:

(3.8)

Where is the diagonal standard deviation matrix. From this, we obtain mean of Z equal zero, E(Z) = 0.

Step 3: calculate the covariance matrix.

Further, the covariance matrix of Z is calculated using the formula below

(3.9)

Where also known as correlation.

Step 4: calculate the eigenvectors and Eigen values of the covariance matrix

The principal components of Z may be obtained from eigenvectors of the correlation matrix of X, refer to equation 3.39.

The principal component of the standardised variables with Cov(Z) = is given by

, (3.10)

The eigenvectors of correlation matrix are also known as principal components coefficients or principal component loadings

Moreover,

(3.11)

and

, (3.12)

In this case, are the Eigen values-eigenvectors pairs with .

As seen from equation 3.47, the total (standard variables) population variance is simply q, the sum of the diagonal elements of the matrix,then the proportion of the total variance explained by the principal component of Z is :

, (3.13)

Where theare the Eigen values of .

In short, principal component analysis consists of finding linear transformations of the original variables, that have the property of being uncorrelated.

The Y variables are chosen in such a way that has maximum variance, has maximum variance to being uncorrelated with, and so on.

3.5 Tests for Significance

3.5.1 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:

(3.14)

Where:

is the correlation between

Significance Level:

Decision rule:

Reject if and otherwise accept. Rencher (2002)

3.5.2 Bartlett’s Test

The Bartlett’s test of hypothesis is given as:

Test Statistic:

(3.15)

In the above, is the variance of the group, N is the total sample size, is the sample size of the group, k is the number of groups, and is the pooled variance. The pooled variance is weighted average of the group variance and is defined as:

. (3.16)

Significance Level:

Critical Region: The variances are judged to be unequal if,

Where is the upper critical value of the chi-square distribution with k-1 degree of freedom and a significance level of .

3.6 Deciding How Many Component to Retain

In every application, a decision must be made on how many principal components should be retained in order to effectively summarize the data. The following guide lines have been proposed:

1. Retain sufficient components to account for a specified percentage of the total variance, say 80%.

2. Retain the components whose Eigen values are greater than the average of the eigen values, . For a correlation matrix, this average is 1.

3. Use the scree graph, a plot of (= Eigen values) versus (i = no of components), and look for a natural break between the ‘‘Large’’ Eigen values and the ‘‘Small’’

Eigen values. Rencher(2002)

3.8 Statistical package used for the study

The statistical package used on the data is NCSS 2007; it was used for the study of canonical correlation and principal component analysis.

4.0 Introduction

This chapter deals with the Analysis and Discussion of the studied data considered in the research work. The Analysis of Canonical Correlation and principal Components were adopted.

4.1 Canonical correlation analysis

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 4.1.1: Canonical correlation coefficient of Set – A and Set – B

Canonical

Functions

Canonical Correlation

Eigen values

% of Variance Explained

0.4310

0.3891

0.1447

0.1858

0.1514

0.0209

51.9

42.3

5.8

Table 4.1.1 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 4.1.2: To test that the canonical correlations are zero

S/NO

p-value

100

0.0057

0.05

100

0.0657

0.05

100

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 4.1.3: 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 4.1.3 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 4.1.4: 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 4.1.4 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.

4.2 Bartlett’s test:

The Bartlett’s test requires measuring the Homogeneity of variance across variables, i.e. the subjects.

Hypothesis:

For at least one pair ( i , j)

Reject

From the Bartlett’s Test, Approximation chi-square equals 113.04 with degree of freedom of 36, probability level of 0.000, at , we therefore reject and conclude that the variances across the variables are not equal.

In this regard, this calls for the use of Principal Component Analysis; to see the variables i.e. the subjects that posse’s high variability contribution to the set of components considered.

Therefore, table 4.2.1 shows the Eigen values in column two, which are the proportions of total variance in all the variables, which are accounted for by the components. From the output component one gives the highest variance explained followed by component two which gives the second highest variance explained and so on. The second component is formed from the variance remaining after those associated with the first component has been extracted, thus this account for the second largest amount of variance. It is worth while to note that the principal component coefficient which gives the variance explained for each component gives the value of less than 30% of the variance explained. Therefore more than one component is needed to describe the variability of the data. In other to obtain a meaningful interpretation of the principal component analysis, we need to reduce to fewer than nine (9) components. In this study, we use the common decision in which we retain only the component with about 80% of variance explained. Therefore, from column 3 i.e. extraction Eigen Values for the retained components, we observed that six components are retained together with their percentage of variance explained by each component. The cumulative variance give as well, shows that the first six components account for about 82.37% of the total variance in the data. Rencher (2002)

A component’s Eigen value may be computed as the sum of its squared component loadings for the entire variable.

A component’s Eigen value divided by the number of variables (which equals the sum of variances because the variance of a standardized variance equals to 1.0) gives the percentage of variance in all the variables which it explains. The ratio of Eigen values is the ratio of explanatory importance of the component with respect to the variable. And, so if a component has a low Eigen value less than the standardized variance i.e. 1, then it is contributing little to the explanatory importance of variance in the variable and may be ignored as redundant with more important components.

Table 4.2.1: Total Variance Explained by each Component

Components

Initial Eigen Values

Extraction Eigen Values for the retained components

S/NO

Total

% of variance

Cumulative

Total

% of Variance

Cumulative %

2.4726

1.3276

1.0777

0.8765

0.8360

0.8231

0.6276

0.5257

0.4330

27.47

14.75

11.97

9.74

9.29

9.15

6.97

5.84

4.81

27.47

42.22

54.20

63.94

73.23

82.37

89.35

95.19

100.00

2.4726

1.3276

1.0777

0.8765

0.8360

0.8231

27.47

14.75

11.97

9.74

9.29

9.15

27.47

42.22

54.20

63.94

73.23

82.37

The Cattell’s scree Test plots the components as the X-axis and the corresponding Eigen values as the Y-axis. The Eigen values are plotted in the sequence of the principal factors. The number of factors is chosen where the plot levels off to a linear decreasing pattern, or as one moves to the right, towards later factors, the Eigen values drop. When the drop ceases and the curve makes an elbow towards less step decline, Cattel’s Scree Test (1966): says that to drop the entire further factor after the one starting the elbow.

Figure 4.1 : Cattell Scree Test graph

Table 4.2.2 shows the communalities which measures the percent of variance in a given row explained by all the components. That is, the communality is the squared multiple correlation for the variable using the components as predictors. Communality for a variable is the sum of squared components loadings for that variable (row) and is the percent of variance due to the variable explained by all the components. For full orthogonal Principal Component Analysis, the communality will be 1.0 and all of the variance in the variables will be explained by all the components. Which their number equals that of the variables and is written under initial. The extracted communalities, is the percent of variance in a given variable explained by the components are the extracted, which are normally fewer in number than the original variables which led the coefficient to be less than 1.0.

Table 4.2.2: Communalities Extracted By each Variable

Variables

Initial

Extraction

Economics

Geography

English Language

Hausa Language

Mathematics

Agric. Science

Biology

Chemistry

Physics

1.0000

0.9769

0.7832

0.8567

0.7856

0.8570

0.8222

0.8650

0.6963

0.7706

Table 4.2.3: Component Loadings, showing Correlation between Component and variables

Variables

Comp. 1

Comp. 2

Comp. 3

Comp. 4

Comp. 5

Comp. 6

Economics

Geography

Eng. Language

Hau. Language

Mathematics

Agric. Science

Biology

Chemistry

Physics

-0.3435

-0.6309

-0.4650

-0.4235

-0.5520

-0.4050

-0.5623

-0.6901

-0.5470

-0.0590

-0.1332

0.0448

0.6985

-0.9970

0.6056

0.0694

-0.1269

-0.6472

0.6317

0.0061

0.4910

-0.0289

-0.2723

-0.3188

0.3291

-0.3273

-0.2130

0.4752

-0.1475

-0.6077

0.3117

-0.0619

-0.2570

0.0222

0.3036

-0.0065

0.1008

-0.5694

0.1571

-0.0172

0.6793

-0.0845

-0.0251

-0.0354

-0.0791

-0.4694

-0.1466

-0.0590

0.1415

-0.0549

-0.3417

0.6592

0.0508

-0.0289

Table 4.2.3 is going to be used for interpretation. Interpretation for component

loadings in principal components is similar to interpretation of coefficients for factor analysis and coefficients in multiple regressions as well as canonical loadings in Canonical correlation Analysis. We want to have some criterion, which helps us determine which of these are large and which of these are considered to be negligible.

1. In Component 1, the percentage amount of variability explained, contributed by the coefficient of each variable. Economics has the highest coefficient with (-0.3435) followed by Agricultural Science (-0.4050) and Hausa Language (-0.4650), and to a lesser extent Chemistry (-0.6901).

2. Component 2, Hausa Language with (0.6985) has the highest contribution to the variability, followed by Agricultural Science (0.6056) and Biology (0.0694) and English Language (0.0448).

3. Component 3 is primarily related to Economics (0.6317), English Language (0.4910), and Biology (0.3291). As Economics increases, the other two subjects increase as well.

4. Component 4 is primarily related to Economics (0.6793), Hausa Language (0.3117) and Chemistry (0.3036). As Economics increases, the two other subjects increase as well, almost all the other subjects decrease.

5. Component 5 is primarily related to Mathematics (0.6592), and English Language (0.1517).

6. Component 6 is primarily related to Biology (0.6592) only. As From the Table 4.6 0f communalities, it can be seen that all the causes are well represented, we can think of the value as multiple values for regression model predicting the variable of interest. The communality for a given variable can be interpreted as the proportion of variance in that variable explained by the 6 factors. In other word, if multiple regressions is performed on Biology against the 6 factors, therefore = 0.865 which is about 86.5% of the variable due to variation in Biology is explained by factors model. The results suggest that principal component analysis does the best job of explaining variation in Biology.

5.3 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%.

Principal component analysis was also applied and showed six groups of closely inter-related subjects based on the fact that six components were used. It is also shown in Table 4.7 that values that close zero correlating a variable and a component can be dropped which indicates variable reduction. The strongest inter-related subjects are found in the beginning column of table 4.7 and decrease through the last column. The 27% of the variability captured by the inter-related variables is due to the contribution of all the subjects but Economics, Agricultural Science and Hausa Language contribute significantly.

Mathematics, Chemistry and Physics are directly related, where as the other subjects are also related to each other. However, it can be seen that Mathematics, Chemistry and Physics are inversely related to the other subjects. That is, an increase in the performance in Mathematics, Chemistry and Physics will lower the performance of the other subjects and this might due to the much emphasis and concentration give to the mathematical subjects.

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ANALYSIS OF SSCE EXAMINATION RESULT USING CANONICAL CORRELATION AND PRINCIPAL COMPONENT ANALYSES

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