XLSTAT: Discriminant Analysis in Excel

 

Dataset for Example

The attached data are from [Fisher M. (1936). The Use of Multiple Measurements in Taxonomic Problems. Annals of Eugenics, 7, 179 -188] and correspond to 150 Iris flowers, described by four variables (sepal length, sepal width, petal length, petal width) and their species. Three different species have been included in this study: setosa, versicolor and virginica.

Our goal is to test if the four variables allow to discriminate the species, and to visualize the observations on a 2-dimensional map that shows as well as possible how separated the groups are. Iris setosa, versicolor and virginica.

Setting Up

• After opening XLSTAT, select the XLSTAT / Analyzing data / Discriminant analysis command.

• Once you've clicked on the button, the Discriminant analysis dialog box appears.

• The qualitative dependent variable corresponds here to the "Species" variable.

• The quantitative Explanatory variables are the four descriptive variables.

• In the Options tab, we uncheck the Equality of covariance matrices  option (assuming that the covariance matrices of the three species are equal would be incorrect)

         

• To keep the plots clear, the Labels option was unchecked in the Charts tab.

  Computations start after clicking OK, and results will then be displayed.

Interpreting the Results

The first results displayed are the various matrices used for the computations. The two Box tests confirm that we need to reject the hypothesis that the covariance matrices are equal between the groups.

The Wilks' Lambda test allows us to test if the vector of the means for the various groups are equal or not (you can understand it as a multidimensional version of the Fisher's LSD or the Tukey's HSD tests). We see that the difference between the means vectors of the groups is significant.

The next table shows the eigenvalues and the corresponding % of variance.

99% of the variance is represented with the first factor. There are only two factors: the maximum number of factors is equal to k-1, when n>p>k, where n is the number of observations, p is the number of explanatory variables, and k is the number of groups.

The following chart shows how the initial variables are correlated with the two factors (this chart corresponds to the factor loadings table). Factor F1 is correlated with Sepal length, Petal length, and Petal width and that F2 is correlated with Sepal width.

The table shows the discriminant functions. Assuming equal covariance matrices yields linear functions; otherwise, as here, they are quadratic. Observations are assigned to the group with the highest function value. These functions can also predict group membership for new data.

The table shows each observation’s factor scores, group membership probabilities, and squared Mahalanobis distances to group centroids. Observations are assigned to the group with the highest posterior probability, which incorporates prior probabilities via Bayes’ theorem.

Three observations (5, 9, 12) were reclassified. This may indicate measurement errors, unusual flower growth, imprecise species identification, or missing discriminative information.

The following chart represents the observations on the factor axes. It allows to confirm that the species are very well discriminated on the factor axes extracted from the original explanatory variables.

The confusion matrix summarizes reclassification results, showing that 98% of observations were correctly classified.

As the corresponding option has been activated in the "Outputs" tab of the dialog box, the predictions for the cross-validation are computed. Cross-validation allows to see what would be the prediction for a given observation if it is left out of the estimation sample. We can see here that only one more observation (Obs8) is miss-classified.

The confusion matrix of the cross-validation is displayed below.

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