Adetunji,A.A et. al. /International Journal of Modern Sciences and Engineering Technology (IJMSET) ISSN 2349-3755; Available at https://www.ijmset.com Volume 1, Issue 4, 2014, pp.55-59
Test for Ordered Alternative in Location Tests, an Application to Students Admission Ige, S. O. 2
Adetunji, A. A. 1
Department of Mathematics and Statistics, Federal Polytechnic, Ado-Ekiti, Ekiti State, Nigeria
[email protected]
Department of Mathematics and Statistics, Federal Polytechnic, AdoEkiti, Ekiti State, Nigeria
[email protected]
Abstract
Education is an instrument par excellence for effecting national development but the rate of admission denial in higher institution of learning in Nigeria is getting beyond reach. Infrastructure expansion is on the downward trend with associated higher demand for admission into various institution. This research paper observes the rate of admission in Nigeria with particular reference to the Federal Polytechnic, Ado-Ekiti, Ekiti State, Nigeria over a period of ten years. Jonckheere-Terpstra Nonparametric ordered alternative test was used and the result shows a significant increase in the number of admitted students over the sampled period. The results indicate an important need for the government to see to the establishment of more institutions of higher learning and ensure proper funding of the various established higher institutions to enable them their infrastructures.
Keywords: Nonparametric, Admission, Students, and Jonckheere-Terpstra 1. INTRODUCTION: In statistical modelling, a parametric statistic distribution of model involves several unknown constants called parameters. However, non-parametric statistics (also called “distribution free statistics”) means the statistics do not assume data or population have any parameters (such as mean and variance) or characteristic structure (such as normal distribution) [1]. The methods are often the only method available for data that simply specify order or counts of numbers of events or of individuals in various categories since those populations do not have normal distribution. Also, the method can be of great advantage because of its easy usage and when the parametric equivalent is not available or the available one is highly cumbersome to utilise. Most of the parametric techniques have their non-parametric equivalent. In the parametric test, there is one-sample t-test (Wilcoxon Signed-Rank test), two-sample t-test (Wilcoxon Rank-Sum test), and one way ANOVA (Kruskal-Wallis test). The Kruskal-Wallis test is an omnibus test and is used to compare population location parameters among two or more groups based on independent samples. If the expectation is on the nature of the data is that there is trend in the data (mostly increasing), the researcher may want to test hypotheses about mean or median (θi) for H0: all θi are equal against HA: θ1 ≤ θ2 ≤ θ3 ≤ … ≤ θk with at least one of the inequalities is strict. In this case, a special case of Kruskal-Wallis test which is called Jonckheere-Terpstra is used. Nonparametric Statistics Nonparametric statistics included descriptive and inferential statistics not based on paramatrized families of probability distributions. The ty mm pical parameters are the mean, variance, etc. Unlike parametric statistics, nonparametric statistics make no assumptions about the probability distributions of the variables being assessed [2]. Definitions In statistics, the term "non-parametric statistics" has at least two different meanings: 1. The first meaning of non-parametric covers techniques that do not rely on data belonging to any particular distribution. These include, among others: · distribution free methods, which do not rely on assumptions that the data are drawn from a given probability distribution. As such it is the opposite of parametric statistics. It includes nonparametric descriptive statistics, statistical models, inference and statistical tests. © IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved
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Adetunji,A.A et. al. /International Journal of Modern Sciences and Engineering Technology (IJMSET) ISSN 2349-3755; Available at https://www.ijmset.com Volume 1, Issue 4, 2014, pp.55-59
· non-parametric statistics (in the sense of a statistic over data, which is defined to be a function on a sample that has no dependency on a parameter), whose interpretation does not depend on the population fitting any parameterised distributions. Order statistics, which are based on the ranks of observations, are one example of such statistics and these play a central role in many nonparametric approaches. 2. The second meaning of non-parametric covers techniques that do not assume that the structure of a model is fixed. Typically, the model grows in size to accommodate the complexity of the data. In these techniques, individual variables are typically assumed to belong to parametric distributions, and assumptions about the types of connections among variables are also made. These techniques include, among others: · non-parametric regression, which refers to modeling where the structure of the relationship between variables is treated non-parametrically, but where nevertheless there may be parametric assumptions about the distribution of model residuals. · non-parametric hierarchical Bayesian models, such as models based on the Dirichlet process, which allow the number of latent variables to grow as necessary to fit the data, but where individual variables still follow parametric distributions and even the process controlling the rate of growth of latent variables follows a parametric distribution [3]. Applications and Purpose Non-parametric methods are widely used for studying populations that take on a ranked order (such as movie reviews receiving one to four stars). The use of non-parametric methods may be necessary when data have a ranking but no clear numerical interpretation, such as when assessing preferences. In terms of levels of measurement, non-parametric methods result in "ordinal" data. As non-parametric methods make fewer assumptions, their applicability is much wider than the corresponding parametric methods. In particular, they may be applied in situations where less is known about the application in question. Also, due to the reliance on fewer assumptions, nonparametric methods are more robust. Another justification for the use of non-parametric methods is simplicity. In certain cases, even when the use of parametric methods is justified, non-parametric methods may be easier to use. Due both to this simplicity and to their greater robustness, non-parametric methods are seen by some statisticians as leaving less room for improper use and misunderstanding. The wider applicability and increased robustness of non-parametric tests comes at a cost: in cases where a parametric test would be appropriate, non-parametric tests have less power. In other words, a larger sample size can be required to draw conclusions with the same degree of confidence[4]. Education is an important instrument for the development of the individual and the society as it is a weapon against poverty, illiteracy, disease, and other socio-economical vices. It is a deliberate, planful, conscious and directed process where by people can learn their culture and participate in it effectively [5]. It is associated with a number of values-literacy, knowledge, good moral upbringing or behaviour and good citizenship [6]. The importance of education in Nigeria in the development of the individual and the nation is highly recognized The Federal Government of Nigeria, as shown in the National Policy on Education[7] , "Education is an instrument par excellence for effecting national development". Consequently parents are extremely desirous of giving qualitative education to their children by sending them to the university, the highest educational institution in the world. Hence, the lust for admission in various institution of higher in the country has been on the increase over the years even when the resources to cater for this increment in various institution is on the reverse trend. Based on the assumption, this research hypotheses that there is increment on the number of admitted students in Nigeria higher institutions with particular reference to the Federal Polytechnic, Ado-Ekiti, Ekiti State, Nigeria over a ten year period. Some factors militate against thousands of qualified Nigerians getting admission into Nigerian higher institutions. These factors include: shortage of adequate manpower; federal government policy on admission, lack of adequate facilities lack of adequate facilities, shortage of adequate manpower and the effect of catchments area and quota system. 2. MATERIALS AND METHODS: The data used for this work was obtained from the Academic Unit of the Federal Polytechnic, Ado-Ekiti, Ekiti State, Nigeria. It covers total number of admitted student in the four faculties of the school for a ten year period © IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved
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Adetunji,A.A et. al. /International Journal of Modern Sciences and Engineering Technology (IJMSET) ISSN 2349-3755; Available at https://www.ijmset.com Volume 1, Issue 4, 2014, pp.55-59
Jonckheere's Trend Test In statistics, the Jonckheere trend test[8] (sometimes called the Jonckheere–Terpstra[9] test) is a test for an ordered alternative hypothesis within an independent samples (between-participants) design. It is similar to the Kruskal–Wallis test in that the null hypothesis is that several independent samples are from the same population. However, with the Kruskal–Wallis test there is no a priori ordering of the populations from which the samples are drawn. When there is an a priori ordering, the Jonckheere test has more statistical power than the Kruskal–Wallis test. The null and alternative hypotheses can be conveniently expressed in terms of population medians for k populations (where k > 2). Letting θi be the population median for the ith population, the null hypothesis is: H0: θ1 = θ2 = … = θk The alternative hypothesis is that the population medians have an a priori ordering e.g.: HA: θ1 ≤ θ2 ≤ … ≤ θk with at least one strict inequality. Procedure The test can be seen as a special case of Maurice Kendall’s more general method of rank correlation[7] and makes use of the Kendall’s S statistic. This can be computed in one of two ways: The ‘direct counting’ method · Arrange the samples in the predicted order · For each score in turn, count how many scores in the samples to the right are larger than the score in question. This is P. · For each score in turn, count how many scores in the samples to the right are smaller than the score in question. This is Q. · S=P–Q The ‘nautical’ method · Cast the data into an ordered contingency table, with the levels of the independent variable increasing from left to right, and values of the dependent variable increasing from top to bottom. · For each entry in the table, count all other entries that lie to the ‘South East’ of the particular entry. This is P. · For each entry in the table, count all other entries that lie to the ‘South West’ of the particular entry. This is Q. · S=P–Q Note that there will always be ties in the independent variable (individuals are ‘tied’ in the sense that they are in the same group) but there may or may not be ties in the dependent variable. If there are no ties – or the ties occur within a particular sample (which does not effect the value of the test statistic) Normal approximation to S The standard normal distribution can be used to approximate the distribution of S under the null hypothesis for cases in which exact tables are not available. The mean of the distribution of S will always be zero, and assuming that there are no ties scores between the values in two (or more) different samples the variance is given by: ∑
(
∑
)
( )= (i) Where n is the total number of scores, and ti is the number of scores in the ith sample. The approximation to the standard normal distribution can be improved by the use of a continuity correction: Sc = |S| – 1. Thus 1 is subtracted from a positive S value and 1 is added to a negative S value. The z-score equivalent is then given by: = (ii) ( )
Ties If scores are tied between the values in two (or more) different samples there are no exact table for the S distribution and an approximation to the normal distribution has to be used. In this case no continuity correction is applied to the value of S and the variance is given by:
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Adetunji,A.A et. al. /International Journal of Modern Sciences and Engineering Technology (IJMSET) ISSN 2349-3755; Available at https://www.ijmset.com Volume 1, Issue 4, 2014, pp.55-59
∑
( )=
∑
∑
∑
+
∑
∑
(
∑
)(
)
∑
+
∑
(
∑
)
(iii)
where ti is a row marginal total and ui a column marginal total in the contingency table. The z-score equivalent is then given by: = (iv) ( )
3. RESULTS AND DISCUSSION: Hypothesis Statement Let θi be the population median (the median of total number of admitted students for the ith year), the null hypothesis is: H0: θ1 = θ2 = … = θk The alternative hypothesis is that the population medians have an a priori ordering, assuming there is yearly increment in the number of students admitted (Median of the admitted in the year 2000 < Median of the admitted in the year 2001 < … < Median of the admitted in the year 2019) HA: θ1 ≤ θ2 ≤ … ≤ θk with at least one strict inequality. Using SPSS version 19, the table below shows the result of Jonckheere-Terpstra Test Table 1: Jonckheere-Terpstra Test Number admitted Number of Levels in Year
10
N
42029
Observed J-T Statistic
4.076E8
Mean J-T Statistic
3.963E8
Std. Deviation of J-T Statistic
1428223.198
Std. J-T Statistic
7.843
Asymp. Sig. (2-tailed)
.000
Decision Rule: Reject H0 if Sig. value < α (0.05) Conclusion: Since Sig. value (0.000) < α (0.05), we reject conclude that the number of admitted student at the Federal polytechnic, Ado-Ekiti is in increasing order under the years reviewed. 4. POLICY IMPLICATION The findings above revealed an upward trend in the number of yearly admitted student over the course of period being reviewed; hence, the following suggestion may assist the policy maker in providing solution to the challenge: · the establishment of more institutions of higher learning by the government to cope with the increasing number of candidates; · proper funding of the various established higher institutions to enable them their infrastructures; · the recruitment of more capable personnel into various units, sections and departments of various institution in order to cope with the demand of the expansion currently being witnessing;
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Adetunji,A.A et. al. /International Journal of Modern Sciences and Engineering Technology (IJMSET) ISSN 2349-3755; Available at https://www.ijmset.com Volume 1, Issue 4, 2014, pp.55-59
5. REFERENCES [1] http://en.wikipedia.org/wiki/Non-parametric_statistics [2] Bagdonavicius, V., Kruopis, J., Nikulin, M.S. (2011). "Non-parametric tests for complete data", ISTE & WILEY: London & Hoboken. ISBN 978-1-84821-269-5. [3] Corder, G.W. & Foreman, D.I. (2009). Nonparametric Statistics for Non-Statisticians: A Step-byStep Approach, Wiley. ISBN 978-0-470-45461-9. [4] Gibbons, Jean Dickinson and Chakraborti, Subhabrata (2003). Nonparametric Statistical Inference, 4th Ed. CRC. ISBN 0-8247-4052-1. [5] Butts, R.F. (1955). A cultural history of Western Education. London: McGraw Hill Books Co. [6] Amadi, L.E. (1998). "Education: the Nations Base for Progress". Nigerian Journal of Curriculum and Instruction, 7 (1): 1-4 [7] Federal Government of Nigeria (1981 and 1998). National Policy on Education. Lagos: Federal Ministry of Education. [8] Jonckheere, A. R. (1954). “A distribution-free k-sample test against ordered alternatives”. Biometrika, 41: 133–145. [9] Terpstra, T. J. (1952). “The asymptotic normality and consistency of Kendall's test against trend, when ties are present in one ranking”. Indagationes Mathematicae, 14: 327–333.
AUTHORS’ BRIEF BIOGRAPHY:
ADETUNJI, Ademola Abiodun: He is an Instructor in the Department of Mathematics and Statistics, Federal Polytechnic, Ado-Ekiti, Ekiti State, Nigeria. He is a young, vibrant, dynamic, and ambitious researcher with a number of scholarship and publications to his short academic experience. He has attended local and international conferences where he presented papers. He holds Masters of Science (MSc) in Statistics from University of Ilorin, Nigeria
IGE, Samuel O.:He a lecturer in the Deaprtment of Mathematics and Statistics, Federal Polytechnic, Ado-Ekiti, Ekiti State, Nigeria. He holds Masters of Science degree and Baschelor of Science degree both at the University of Ilorin, Nigeria. He has been lecturing since 1993 and has supervised over 50 students for Higher National Diploma in Statistics in various fields.
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