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In this project entitled statistical analysis on education trust fund allocation to tertiary institutions in six geo-political zones of Nigeria, the average allocation to zones, method of distributions, extraction of principal components, classification of the components into factors and to test if there is any significant difference in the allocation among the zones was carried out using principal components analysis, factor analysis, normality test just to mention but a few. The average allocation to all the zones within the period under review was #14,605,429,76. The allocation to zones was normally distributed indicating unbiasedness in the allocations. University allocation is the principal factor component in the ETF allocation among the institutions revealing high contribution of university with 0.201 in the first component, followed by monotechnics, polytechnics and colleges of education. With little difference in the allocations among polytechnics, monotechnics and colleges of education, they were grouped into one factor and university in another factor. Based on the results obtained; no zone is more favored and their distribution is unbiased
1.1 BACKGROUND OF STUDY
In Principal Components Analysis (PCA) and Factor Analysis (FA) one wishes to extract from a set of P variables a reduced set of M components or factors that accounts for most of the variance in a P variables in other words, we wish to reduce a set of P variables to a set of M underlying super ordinate dimensions.
These underlying factors are inferred from the correlations among the P variables. Each factor is estimated as a weighted sum of the P variables. The factor is thus;
F1 = W1X1 + Wi2X2 + W1pXp+ K.
One may also express each of the P variables as a linear combination of the M factors,
Xj = Aij F1 + A2j F2 + Amj Fm + k+ Uj
Where Uj is the variance that is unique to variable j, variance
that cannot be explained by any of the common factors. Principal component analysis is a variable reduction
procedure which provides guidelines regarding the necessary sample size and number of items per component. It also
shows how to determine the number of components to retain, interpret the rotated solution, create factor scores and summarize the results.
It is appropriate when you have obtained measures on a number of observed variables and wish to develop a smaller number of artificial variables called Principal Components that will account for most of the variance in the observed variables. The principal components may then be used as predictor variables in subsequent analysis.
Principal component is defined as a linear combination of optimally weighted observed variables. The “linear combination” here refers to the fact that scores on a component are created by adding together scores on the observed variables being analyzed and “optimally weighted” refers to the fact that the observed variables are weighted in such a way that the resulting components account for a maximal amount of variance in the data set.
Factor analysis is a mathematical tool which can be used to examine a wide range of data sets. It is the most familiar multivariate procedure used in the behavioral sciences; it includes both component analysis and common
factor analysis. In factor analysis, you need only the correlation or covariance matrix not the actual scores. The purpose of factor is to discover simple patterns in the patterns of relationship among the variables. In particular, it seeks to discover if the observed variable can be explained largely or entirely in terms of a much smaller number of variable called factors.
Onyeagu (2003) explained the difference between factor analysis and principal component analysis. Factor analysis is covariance (or correlation) oriented. In principal component analysis, all components are needed to produce an inter-correlation (covariance) exactly. In factor analysis, a few factors will reproduce the inter-correlations (covariance) exactly.
Wang (2007) differentiate the principal component analysis and factor analysis as in principal component analysis the major objective is to select a number of component that will express as much of the total variance in the data as possible.
However, the factors formed in the factor analysis are generated to identify the latent variables that are
contributing to the common variance in the data. A factor analysis attempts to exclude unique variance from the analysis; whereas a principal component analysis does not differentiate common and unique variance. PCA analyzes variance while FA analyses covariance.
The PCA and FA have some similarities such as their measurement scale is interval or ratio level, linear relationship between observed variables, normal distribution for each observed variables. Each pair of observed variables has a bivariate normal distribution and lastly PCA and FA are both variable reduction techniques. If communalities are large, close to 1.00, results could be similar.
1.2 SOME FACTS ABOUT NIGERIA EDUCATION
The literacy and educational characteristic of population aged 6 years and above were enumerated in 1991 population census. The literacy was 60% for males and 40% for females. The literacy level in the country appears to have improved over years, while the sex differential on literacy among persons in the age group 35-39 was almost twice as high for male (68.3%) and female (35.8%). In contrast, the
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