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Stepwise Procedures In Discriminant Analysis
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ESTIMATING MISCLASSIFICATION RATES
Probability of Misclassification:
This is a measure of the tendency that an individual or objects are wrongly classified. It involves misclassifications errors like;
Classification of an object into population, ï°i, given that it is actually from population, ï°j.
Classification of an object into population, ï°j, given that it is actually from population, ï°i.
It can be explained further with the help of the confusion matrix for classification below;
n1, n2 and n3 are sample sizes from population
N is total sample sizes = n1 + n2 + n3 ;
Cij = number of objects correctly classified (i=j) ; Wij = number of objects wrongly classified (i = j). Therefore, the probabilities of misclassification are
Actual Error Rate (AER) is the probability that a classification function based on the present sample will misclassify a future observation. Apparent Error Rate (A PER) is the estimate of actual error rate, which is the proportion of misclassification resulting from resubstitution.We can define the following:
nij = Correctly classified (i=j) nij = wrongly classified (i =j)
IMPROVED ESTIMATES OF ERROR RATES
Estimates of error rates can be improved with the following techniques:
Holdout Method : This is the leaving-one-out or the cross-validation method which ensures that all but one observation is used in the computation of the classification rule and the omitted observation is then classified with the rule. This procedure which increases the computation load is repeated for each observation in a sample of size N = Σini. Each observation is then classified by a function based on N
– 1 observations.
Partitioning the sample : In order to avoid bias, the sample is split into two parts:
A training sample which is used to construct the classification rule and
A validation sample which is used to evaluate the classification rule.
Since the observations in the validation sample which is used to evaluate the classification rule are not used in constructing the classification rule, the resulting error rate is unbiased.
Two disadvantages exist with this method:
Large samples that may not be available are required.
The classification function to be used in practice is not evaluated.
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ABSRACT - [ Total Page(s): 1 ]
Abstract
Several multivariate measurements require variables
selection and ordering. Stepwise procedures ensure a step by step method
through which these variables are selected and ordered usually for
discrimination and classification purposes. Stepwise procedures in discriminant
analysis show that only important variables are selected, while redundant
variables (variables that contribute less in the presence of other variables) are
discarded. The use of stepwise procedures ... Continue reading---
APPENDIX A - [ Total Page(s): 1 ] ... Continue reading---
APPENDIX B - [ Total Page(s): 1 ] APPENDIX II BACKWARD ELIMINATION METHOD The procedure for the backward elimination of variables starts with all the x’s included in the model and deletes one at a time using a partial  or F. At the first step, the partial  for each xi isThe variable with the smallest F or the largest  is deleted. At the second step of backward elimination of variables, a partial  or F is calculated for each q-1 remaining variables and again, the variable which is th ... Continue reading---
TABLE OF CONTENTS - [ Total Page(s): 1 ]TABLE OF CONTENTSPageTitle PageApproval pageDedicationAcknowledgementAbstractTable of ContentsCHAPTER 1: INTRODUCTION1.1 Discriminant Analysis1.2 Stepwise Discriminant analysis1.3 Steps Involved in discriminant Analysis1.4 Goals for Discriminant Analysis1.5 Examples of Discriminant analysis problems1.6 Aims and Obj ectives1.7 Definition of Terms1.7.1 Discriminant function1.7.2 The eigenvalue1.7.3 Discriminant Score1.7.4 Cut off1.7 ... Continue reading---
CHAPTER ONE - [ Total Page(s): 2 ] DEFINITION OF TERMS Discriminant Function This is a latent variable which is created as a linear combination of discriminating variables, such that Y = L1X1 + L2X2 + + Lp Xp where the L’s are the discriminant coefficients, the x’s are the discriminating variables. The eigenvalue: This is the ratio of importance of the dimensions which classifies cases of the dependent variables. There is one eigenvalue for each discriminant functio ... Continue reading---
CHAPTER TWO - [ Total Page(s): 3 ] 5 is called the mahalanobis (squared) distance for known parameters. For unknown parameters, the Mahalanobis (squared) distance is obtained by estimating p1, p2 and S by X1, X2 and S, respectively. Following the same technique the Mahalanobis (Squared) distance, D , for the unknown parameters is D2 = (X- X)+S-1 (X1- X2) . The distribution of D can be used to test if there are significant differences between the two groups.2.4 WELCH’S CRITERION Welch (1939) suggest ... Continue reading---
CHAPTER FOUR - [ Total Page(s): 3 ]CHAPTER FOUR DATA ANALYSISMETHOD OF DATA COLLECTIONThe data employed in this work are as collected by G.R. Bryce andR.M. Barker of Brigham Young University as part of a preliminary study of a possible link between football helmet design and neck injuries.Five head measurements were made on each subject, about 30 subjects per group:Group 1 = High School Football players Group 2 = Non-football playersThe five variables areWDIM = X1 = head width at wi ... Continue reading---
CHAPTER FIVE - [ Total Page(s): 1 ]CHAPTER FIVERESULTS, CONCLUSION AND RECOMMENDATIONRESULTSAs can be observed from the results of the analysis, when discriminant analysis was employed, the variable CIRCUM(X2) has the highest Wilks’ lambda of 0.999 followed by FBEYE (X2) (0.959). The variable EYEHD (X4) has the least Wilks’ lambda of 0.517 followed by EARHD (X5) (0.705). Also the least F-value was recorded with the variable CIRCUM (X2) (0.074) followed by the variable FBEYE (X2) (2.474), while the variable EYEHD (X4 ... Continue reading---
REFRENCES - [ Total Page(s): 1 ] REFERENCES Anderson, T.W. (1958). An introduction to multivariate statistical Analysis. John Wiley & Sons Inc., New York. Cohen, J. (1968). Multiple regression as a general data-analytic system. Psychological Bulletin 70, 426-443. Cooley W.W. and Lohnes P.R. (1962). Multivariate procedures for the Behavioural Sciences, New York John Wiley and Sons Inc. Efroymson, M.A. (1960). Multiple regression analysis. In A. Raston & H.S. Wilfs (Eds.) Mathematical methods for ... Continue reading---
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ABSRACT - [ Total Page(s): 1 ]
Abstract
Several multivariate measurements require variables
selection and ordering. Stepwise procedures ensure a step by step method
through which these variables are selected and ordered usually for
discrimination and classification purposes. Stepwise procedures in discriminant
analysis show that only important variables are selected, while redundant
variables (variables that contribute less in the presence of other variables) are
discarded. The use of stepwise procedures ... Continue reading---