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This panel provides supports for data entry and interpretation of results only. Detailed discussions on Discriminant Analysis is provided in the Discriminant Analysis Explained Page
Data EntryThe data is a table, each row being a case and each column values.
OptionsThe program provides two buttons for 2 options in how the results are presented
ResultsIn order to conserve space for result output, the outcome groups are labeled as group 1, 2, 3.... Table 1 links group number to group names in the input data.Table 2 shows the mean and Standard Deviations for all the independent variables The values of the independent variables are normalized to z (z=(value-mean)/Standard Deviation). z values are not displayed, but are used for all subsequent calculations The normalized values (z) are used to estimate the coefficients of the Linear Discriminant Functions, and test their statistical significance using the Chi Squares test. Table 3 shows the Chi Squares and Type I error for statistical significance. The criteria p<0.05 is used to determine statistical significance. Using the default reference data, the total number of functions (nf) is one less than the number of outcome groups (ng), nf=ng-1 Table 4 displays the coefficient values. The columns are functions and the rows are independent variable. These are used to estimate function scores for each case Table 5 the centroid values, the center for each function in each ourcome group. The columns are functions, and the rows are outcome groups. The remaining calculations will now depend on the option the user chose.
Table 6 matches the original designated group against the allocated group. This allows the user to understand the precision of the Discriminant analysis. With the default reference data, a perfect match was produced. This is not surprising, as the data was artificially created to achieve this. With real data however, there would be overlaps, and some of the cases would be allocated to the wrong outcome group. Table 7 displays the function scores and Maximum Likelihood values for each case. PlotsIf there is more than 1 Discriminant functions used, the program proceeds to plot all combinations of functions. Each plot is a 2 dimensional representation of two factors. For each outcome group, a line is drawn from the factor scores in each case to the centrod values in the group the case came from. The plot allows the user to examine the relationship between the function scores in different groups.Parameters for calculation Bayesian ProbabilitiesThe program presents all the data obstained from the reference data that are necessary to estimate Bayesian probabilities of belonging to each outcome group from each case. These includes the number of functions (nf), number of outome group (hg) and their names, the mean and Standard Deviation of the independent variables, the function coefficients and centroid values. These can be copied and pasted to DiscriminantTemplate.html and used to estimate Bayesian probability on additional data. As the html file provides all the necessary explanations, they will not be elaborated here.ReferencesThe algorithm on this page was developed using php, after studying the formulae provided in the following text booksOverall JE and Klett CJ (1972) Applied Multivariate Analysis. McGraw Hill Series in Psychology. McGraw Hill Book Company New York. Library of Congress No. 73-14716407-047935-6
Press WH, Flannery VP, Teukolsky SA, Vetterling WT (1989). Numerical Recipes in Pascal. Cambridge University Press IBSN 0-521-37516-9 p.395-396 and p.402-404. Jacobi method for finding Eigen values and Eigen vectors Norusis MJ (1979) SPSS Statistical Algorithms Release 8. SPSS Inc Chicago Chapterr 23 : Discriminant p. 69-83. Extensive algoritm provided by SPSS. George D and Mallery P (1999) SPSS for Windows Step by Step. A Simple Guide and Reference. Allyn and Bacon, Sydney. ISBN 0-205-28395-0 Chapter 26. The Discriminant Procedure p.313-328. |