The Discriminant Analysis Report

The Discriminant Analysis report provides discriminant results based on your selected Discriminant Method. The Discriminant Method and the Classification variable are shown at the top of the report. If you selected the Regularized method, its associated parameters are also shown.

You can change Discriminant Method by selecting the option from the red triangle menu. The results in the report update to reflect the selected method.

Example of a Discriminant Analysis Report

The default Discriminant Analysis report contains the following sections:

•	When you select the Wide Linear discriminant method, a Principal Components report appears. See Principal Components.

•	The Canonical Plot shows the points and multivariate means in the two dimensions that best separate the groups. See Canonical Plot.

•	The Discriminant Scores report provides details about how each observation is classified. See Discriminant Scores.

•	The Score Summaries report provides an overview of how well observations are classified. See Discriminant Scores.

Principal Components

This report only appears for the Wide Linear method. Consider the following notation:

•	Denote the n by p matrix of covariates by X, where n is the number of observations and p is the number of covariates.

•	For each observation in X, subtract the covariate mean and divide the difference by the pooled standard deviation for the covariate. Denote the resulting matrix by Xs.

The report gives the following:

Number

The number of eigenvalues extracted. Eigenvalues are extracted until Cum Percent is at least 99.99%, indicating that 99.99% of the variation has been explained.

Eigenvalue

The eigenvalues of the covariance matrix for Xs, namely (Xs’Xs)/(n - p), arranged in decreasing order.

Cum Percent

The cumulative sum of the eigenvalues as a percentage of the sum of all eigenvalues. The eigenvalues sum to the rank of Xs’Xs.

Singular Value

The singular values of Xs arranged in decreasing order.

Canonical Plot

The Canonical Plot is a biplot. Canonical Plot for Iris.jmp shows the Canonical Plot for a linear discriminant analysis of the data table Iris.jmp. The points have been colored by Species.

Canonical Plot for Iris.jmp

The biplot axes are the first two canonical variables. These define the two dimensions that provide maximum separation among the groups. The biplot shows how each observation is represented in terms of canonical variables and how each covariate contributes to the canonical variables.

•	The observations and the multivariate means of each group are represented as points on the biplot. They are expressed in terms of the first two canonical variables.

•	The set of rays that appears in the plot represents the covariates. The rays show how each covariate loads onto the first two standardized canonical variables. The direction of a ray indicates the degree of association of that covariate with the first two canonical variables.

Notice these additional details about the Canonical Plot and its association with other parts of the Discriminant report:

•	The point corresponding to each multivariate mean is denoted by a plus (“+”) marker.

•	A 95% confidence level ellipse is plotted for each mean. If two groups differ significantly, the confidence ellipses tend not to intersect.

•	Show or hide the 95% confidence ellipses by selecting Canonical Options > Show Means CL Ellipses from the red triangle menu.

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The labeled rays show the directions of the covariates in the canonical space. These rays emanate from the point (0,0), which represents the grand mean of the data in terms of the canonical variables. They represent the degree of association, or loading, of each covariate on each canonical variable.

•	Obtain the values of the loadings by selecting Canonical Options > Show Canonical Details from the red triangle menu. At the bottom of the Canonical Details report, click Standardized Scoring Coefficients. See Standardized Scoring Coefficients for details.

•	Show or hide the rays by selecting Canonical Options > Show Biplot Rays from the red triangle menu.

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Drag the center of the biplot rays to other places in the graph. Specify their position and scaling by selecting Canonical Options > Biplot Ray Position from the red triangle menu. The default Radius Scaling shown in the Canonical Plot is 1.5, unless an adjustment is needed to make the rays visible.

•	An ellipse denoting a 50% contour is plotted for each group. This depicts a region in the space of the first two canonical variables that contains approximately 50% of the observations, assuming normality.

•	Show or hide the 50% contours by selecting Canonical Options > Show Normal 50% Contours from the red triangle menu.

•	Color code the points to match the ellipses by selecting Canonical Options > Color Points from the red triangle menu.

Classification into Three or More Categories

For the Iris.jmp data, there are three Species, so only two canonical variables. The plot in Canonical Plot for Iris.jmp shows good separation of the three groups using the two canonical variables.

The rays in the plot indicate the following:

•	Petal length is positively associated with Canonical1 and negatively associated with Canonical2. It loads more heavily on Canonical1 than on Canonical2.

•	Petal width is positively associated with both Canonical1 and Canonical2. It loads more heavily on Canonical2 than on Canonical1.

•	Sepal width is negatively associated with Canonical1 and positively associated with Canonical2. It loads more heavily on Canonical2 than on Canonical1.

•	Sepal length is weakly associated with Canonical1 and very weakly associated with Canonical2.

Classification into Two Categories

When the classification variable has only two levels, the points are plotted against the single canonical variable, denoted by Canonical1 in the plot. The covariates load on Canonical1 only. The rays are shown with a vertical component only in order to separate them. Project the rays onto the Canonical1 axis to compare their association with the single canonical variable.

Canonical Plot for Fitness.jmp shows a Canonical Plot for the sample data table Fitness.jmp. The seven continuous variates are used to classify an individual into the categories M (male) or F (female). Since the classification variable has only two categories, there is only one canonical variable.

Canonical Plot for Fitness.jmp

The points in Canonical Plot for Fitness.jmp have been colored by Sex. Note that the two groups are well separated by their values on Canonical1.

Although the rays corresponding to the seven covariates have a vertical component, in this case you must interpret the rays only in terms of their projection onto the Canonical1 axis. You note the following:

•	MaxPulse, Runtime, and RunPulse have little association with Canonical1.

•	Weight, RstPulse, and Age are positively associated with Canonical1. Weight has the highest degree of association. The covariates RstPulse and Age have a similar, but smaller, degree of association.

•	Oxy is negatively associated with Canonical1.

Discriminant Scores

The Discriminant Scores report provides the predicted classification of each observation and supporting information.

Row

Row of the observation in the data table.

Actual

Classification of the observation as given in the data table.

SqDist(Actual)

Value of the saved formula SqDist[<level>] for the classification of the observation given in the data table. For details, see Score Options.

Prob(Actual)

Estimated probability of the observation’s actual classification.

-Log(Prob)

Negative of the log of Prob(Actual). Large values of this negative log-likelihood identify observations that are poorly predicted in terms of membership in their actual categories.

A plot of -Log(Prob) appears to the right of the -Log(Prob) values. A large bar indicates a poor prediction. An asterisk(*) indicates observations that are misclassified.

If you are using a validation or a test set, observations in the validation set are marked with a “v” and those in the test set are marked with a “t”.

Predicted

Predicted classification of the observation. The predicted classification is the category with the highest predicted probability of membership.

Prob(Pred)

Estimated probability of the observation’s predicted classification.

Others

Lists other categories, if they exist, that have a predicted probability that exceeds 0.1.

Show Interesting Rows Only shows the Discriminant Scores report for the Iris.jmp sample data table using the Linear discriminant method. The option Score Options > Show Interesting Rows Only option is selected, showing only misclassified rows or rows with predicted probabilities between 0.05 and 0.95.

Show Interesting Rows Only

Score Summaries

The Score Summaries report provides an overview of the discriminant scores. The table in Score Summaries for Iris.jmp shows Actual and Predicted classifications. If all observations are correctly classified, the off-diagonal counts are zero.

Score Summaries for Iris.jmp

The Score Summaries report provides the following information:

Columns

If you used Stepwise Variable Selection to construct the model, the columns entered into the model are listed. See Score Summaries Report Showing Selected Covariates.

Source

If no validation is used, all observations comprise the Training set. If validation is used, a row is shown for the Training and Validation sets, or for the Training, Validation, and Test sets.

Number Misclassified

Provides the number of observations in the specified set that are incorrectly classified.

Percent Misclassified

Provides the percent of observations in the specified set that are incorrectly classified.

Entropy RSquare

A measure of fit. Larger values indicate better fit. For details, see Entropy RSquare.

Note: It is possible for Entropy RSquare to be negative.

-2LogLikelihood

Twice the negative log-likelihood of the observations in the training set, based on the model. Larger values indicate better fit. Provided for the training set only. For more details, see Fitting Linear Models.

Confusion Matrices

Shows matrices of actual by predicted counts for each level of the categorical X. If you are using JMP Pro with validation, a matrix is given for each set of observations. If you are using JMP with excluded rows, the excluded rows are considered the validation set and a separate Validation matrix is given. For more information, see Validation in JMP and JMP Pro.

Entropy RSquare

The Entropy RSquare is a measure of fit. It is computed for the training set and for the validation and test sets if validation is used.

Entropy RSquare for the Training Set

For the training set, Entropy RSquare is computed as follows:

•	A discriminant model is fit using the training set.

•	Predicted probabilities based on the model are obtained.

•	Using these predicted probabilities, the likelihood is computed for observations in the training set. Call this Likelihood_FullTraining.

•	The reduced model (no predictors) is fit using the training set.

•	The predicted probabilities for the levels of X from the reduced model are used to compute the likelihood for observations in the training set. Call this quantity Likelihood_ReducedTraining.

•	The Entropy RSquare for the training set is:

Entropy RSquare for Validation and Test Sets

For the validation set, Entropy RSquare is computed as follows:

•	A discriminant model is fit using only the training set.

•	Predicted probabilities based on the training set model are obtained for all observations.

•	Using these predicted probabilities, the likelihood is computed for observations in the validation set. Call this Likelihood_FullValidation.

•	The reduced model (no predictors) is fit using only the training set.

•	The predicted probabilities for the levels of X from the reduced model are used to compute the likelihood for observations in the validation set. Call this quantity Likelihood_ReducedValidation.

•	The Validation Entropy RSquare is:

The Entropy RSquare for the test set is computed in a manner analogous to the Entropy RSquare for the Validation set.