1.2 45 3.16 72.7 SR 1.3 67 3.38 102.4 SR 1.1 48 3.61 33.7 SR 1.6 36 3.51 58.2 SR 1.5 47 3.20 44.2 DR 1.5 74 3.21 91.8 DR 1.7 47 3.39 53.1 DR 1.6 56 3.36 88.5 DR 1.1 27 3.30 36.3 SW 1.0 53 3.55 74.7 SW 0.9 37 3.23 94.2 SW 1.2 23 3.07 53.8 SW 1.4 44 3.34 20.7 DW 1.3 34 3.24 9.5 DW 1.1 37 3.24 17.8 DW 1.4 55 3.35 35.9 DW

Data Input for Discriminant Analysis Using Reference Data     The data is for a single analysis     It is a table of multiple columns     Each row contains data from a case or a record     All columns except the last are predictors and must be numerical     The last column (on the right) is outcome group name, single character or text with no gaps

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 Entry

The data is a table, each row being a case and each column values.

• All rows must have the same number of columns
• Columns are separated by spaces or tabs
• All columns, except the last one on the right, are values of the independent vaiables, and must be numerical
• The last column on the right is the group labels, and is treated as text. It is a single character or word with no gaps

The default example data are artificially created to demonstrate the algorithm, and does not reflect reality. It purports to be a study to discriminate the type of wine (in alphabetical order, DR for dry red, DW for dry white, SR for sweet red, and SW for sweet white). It uses 4 measurements to do so, col 1 is measurement of tannin, col 2 color, col 3 acidity, and col 4 sugar

Options

The program provides two buttons for 2 options in how the results are presented

• Calculate Discriminant Functions from Data (Use All Functions) presents all of the discriminant functions produced (number of outcome groups -1). This is the standard output in most algorithms for Discriminant Analysis. With the default example where there are 4 outcome groups, all 3 functions are used
• Calculate Discriminant Functions from Data (Use Only Significant Functions) uses only those discriminant functions tested as statistically significant by the Chi Squares Test (p<0.05). With the default example with 4 outcome groups, all 3 functions are initially produced. However, as the third function is not statistically significant, only the first 2 functions are used to produce the results for presentation

The pros and cons of the options are discussed in Discriminant Analysis Explained Page

Results

In 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.

• If the user chose to use all functions, then all the functions (nf) are used for the remaining calculations. With the default reference data, there are 4 outcome groups, so 3 functions, and 3 functions will be used.
• If the user chose to use only significant functions then the non-significant ones will be discarded. With the default reference data, the third function is not statistically significant, so only the first 2 are used for the remaining calculations

The programs now use the z values and the coefficients to calculate the Maximum Likelihood (probability assuming the same apriori probability for all outcomes regardless of sample sizes in the input data). Each case is then allocated to the group with the highest probability.

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.

Plots

If 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 Probabilities

The 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.

References

The algorithm on this page was developed using php, after studying the formulae provided in the following text books

Overall 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

• Chapter 2 p.24-56 : Matrix math. Particularly the Square Root method of matrix inversion. Also calculations for between and within group Sum Product and Covariance matrices
• Chapter 10 p.280-306 : Multiple Discriminant Analysis, particularly the algorithm.
• Chapter 13 p. 345-371 Normal Probability Density Model for classification.
• Chapter 14 p. 373-383 Use of Canonical Correlates for Classification. Chapters 13 and 14 provided the algorithms for calculating Maximum Likelihood and the Bayesian Probabilities

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.

Wikipedia on Discriminant Analysis.