# Application of discriminant analysis to predict the class of degree for graduating students in a university system

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3 08 Int. J. Phys. Sci. So that: (c) Mean Vectors, d The deviation of mean vectors of Group from Group gives: d G G G G U W d Thus, substituting these values of the discriminant weights, U in equation (), we get: Z 0.047( GPA) ( STA0) () Multivariate test of significance Problems arising in multivariate populations are direct generalization from the Univariate case. Thus, we decided to test for equality of Group Means and equality of Variance-Covariance matrices. Equality of group means The hypothesis of interest is: H µ µ 0 : () () Vs H : µ () µ () Using F-transformation of Hotelling s T,as our test statistic. Where T N + N P F. T ( N + N ) P N N D M and N + N D M H 0 is rejected if CAL > P, N+ N P ; F F α Mahalanobis distance. At 5% level of significance, we rejected the hypothesis of equality of group means. This implies that there exist significant differences between the group means. Equality of covariance matrices The hypothesis of interest is: H V V 0 : Vs H : V V As our test statistic, we use Box s M test g M ( N g) Log / S / Vi Log / Si / i A reasonable approximation, when each N i > 0 and N i is large relative to P < 6 and g < 6, is obtained by using the Chi-Square approximation. Where: B H 0 is rejected if ( c) M g P + 3P C 6( P + )( g ) i Vi N g g M ( N g) Log / S / Vi Log / Si / i B ( c) M > α ( v) ; where P( P + )( g ) V At 5% level of significance, we accepted the hypothesis of equality of Variance-Covariance matrices. This analysis shows that the equality of variance assumption required when using Hotelling s T statistics is tenable. Classification rule We define the cut off as: C Z + Z, Z Z We first of all compute Z and Z which denote the functions at Group Centriods, by substituting the means of GPA and STA 0 for each of the two groups. In the linear discriminant function, we obtain Z and Z on calculation. Thus, the discriminating procedure is as follows: Assign an individual to group if Z DS > 0.0 and group if Z 0.0. DS RESULTS The data were processed on a microcomputer using SPSS statistical software package and confusion matrices for the analysis sample and hold-out sample are shown in Tables -3 below. In Table, the rows totals are the observed categories for the class of degree and the columns totals are the

4 Erimafa et al. 09 Table. Confusion matrix for actual and predicted categories of class of degree. Class of Degree Predicted Class of Degree Total BCD PCD rd Pass Total Table. Confusion matrix for actual and predicated class of degree with percentage. Group Predicted Group Membership Total Original Count % Total Hit Ratio 87.5%. Probability of Misclassification Table 3. Confusion matrix for validated data. Group Predicted Group Membership Total Original Count % Total Hit Ratio 88.%. Probability of Misclassification predicted categories for the class of degree. It was observed that 50 out of 55 individuals predicted to graduate with Second Class Upper ( ) or Second Class Lower division ( ) did so. This represents Hit Ratio of 90.9%. Also, of 65 individuals predicted to graduate with Third Class or Pass, some 55 did so. This also represents a Hit Ratio of 84.6%. In Table, success in identifying students that will graduate with better classes of degree (BCD) was 83.3%, essentially similar to that of the hold-out sample (Table 3). Also, in Table, success in identifying students that will graduate with Poor Class of Degree (PCD) was 9.7%, again essentially the same as the hold-out sample (Table 3). Looking at the Total Hit-Ratio for both the historical sample (Table ) and the hold-out sample (Table 3), the results are essentially similar. Hence this shows that, the classification result of the historical sample (analysis sample) was not biased upward. Conclusion The consistent high hit rates for both the analysis sample and hold-out sample, i.e., the overall percentage of correct classifications which is 87.5 and 88.%, as seen in the confusion matrices (Tables and 3), for this study, which is a measure of predictive ability shows that discriminant analysis can be used to predict students graduating class of degree from knowledge of variable(s) that have relationship with performance. This study tends to illustrate the logicality and wisdom in examining related statistical technique useful for the purpose of prediction. The use of discriminant analysis in this manner that is, conducting discriminant analysis for predictive purpose enables us to identify the students who might be termed at risk; these are students that will graduate with Poor Class of Degree, PCD. It also identifies STA 0 as having a booster effect on final graduating Cumulative Gra-

5 00 Int. J. Phys. Sci. APPENDI A 0 HISTORICAL DATA VALUES OF GPA AND STA 0 FOR TWO GROUPS GROUP ( N 60) GROUP ( N 60) GROUP ( VALIDATED DATA VALUES OF GPA AND STA 0 FOR TWO GROUPS 0 N ) GROUP ( N ) de Point Average (CGPA), as well as brought to light the difficulty in understanding its concept. Therefore there is need for an instructional intervention. In conclusion, this study shows that discriminant analysis provides results that are both more interpretable and statistically sound, in addition to being a statistically correct procedure for prediction purpose than traditional measures. ACKNOWLEDGEMENT I thank the Editor and the reviewers for their insightful comments that enhanced the paper. REFERENCES Adebayo SB, Jolayemi ET (998b). On the Effect of Rare Outcome on some Agreement/Concordance Indices. Nig. J. Pure and Appl. Sci. 3: Adebayo SB, Jolayemi ET (999). Effect of Rare Outcome on the Measure of Agreement Index. J. Nig. Stat. Assoc. 3:-0. Anderson TW (958). An Introduction to Multivariate Statistical Analysis. New York: John Wiley Charles BK, June EH (970). Predicting Graduation Withdrawal and Failure in College by Multiple Discriminant Analysis. J. Edu. Measur. 7(): Huberty CJ (989). Problems with stepwise methods: Better alternatives. In: B Thompson (ED.), Advances in social science methodology. Greenwich, CT: JAI Press. Vol. : Cook RD, Weisberg S (98). Residuals and Influence in Regression, London: Chapman and Hall. Fisher RA (936). The use of Multiple Measurements in Taxonomic

6 Erimafa et al. 0 problems. Ann. Eug. 7: Gilbert ES (968). On Discrimination using Qualitative Variable. J. Amr. Stat. Soc. pp Huberty CJ (994). Applied Discriminant Analysis. New York: Wiley and sons. Huberty CJ, Barton RM (989). An Introduction to Discriminant Analysis. Measurement and Evaluation in Counseling and Development, : Jolayemi ET (999a). On the Measure of Agreement between two raters, Biomed. J. 3: Moore DH (973). Evaluation of five Discrimination procedures for Binary Variables. J. Amer. Stat. Soc. 3: Stevens J (996). Applied Multivariate Statistics for the Social Sciences.In: Mahwah NJ, (3 rd Ed). Lawrence Erlbaum Associate, Inc. Thompson B (995). Stepwise regression and stepwise discriminant analysis need not apply here: A guidelines editorial. Edu. Psychol. Measur. 55(4): Usoro AE (006). Discriminant Analysis and its Application to the Classification of Students on the Basis of their Academic Performances. J. Res. Phy. Sci. (3):

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