Discriminant Analysis
Also known as: LDA, Fisher discriminant analysis, discriminant function analysis, canonical discriminant analysis
Discriminant analysis finds linear combinations of predictor variables that best separate two or more known groups. It is used both to understand which predictors distinguish the groups and to classify new observations into those groups with minimum error.
Key highlights
- Simultaneously identifies which predictors drive group separation and produces a classification rule, combining explanation and prediction.
- Produces easily interpretable discriminant function coefficients and structure coefficients (correlations of predictors with the function).
- For well-separated multivariate normal groups it is the optimal linear classifier in the sense of minimising expected misclassification.
- Handles more than two groups by extracting multiple orthogonal discriminant functions.
- Computationally efficient and widely implemented in standard statistical software.
Intuition
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How it works
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When to use it
Use discriminant analysis when you have a categorical dependent variable (group membership) and multiple continuous predictor variables, and your goals are either to describe which predictors distinguish the groups or to build a classification rule. It is most appropriate when the predictors are approximately multivariate normal within each group and the within-group covariance matrices are equal (homoscedasticity). Do not use it when predictors are predominantly categorical (prefer logistic regression or log-linear models), when group sizes are very unequal without correction, when the normality and homoscedasticity assumptions are severely violated, or when the number of predictors approaches or exceeds the sample size.
Strengths & limitations
- Simultaneously identifies which predictors drive group separation and produces a classification rule, combining explanation and prediction.
- Produces easily interpretable discriminant function coefficients and structure coefficients (correlations of predictors with the function).
- For well-separated multivariate normal groups it is the optimal linear classifier in the sense of minimising expected misclassification.
- Handles more than two groups by extracting multiple orthogonal discriminant functions.
- Computationally efficient and widely implemented in standard statistical software.
- Assumes multivariate normality of predictors within groups and equal within-group covariance matrices; violations reduce classification accuracy and distort significance tests.
- Sensitive to outliers, which can pull group centroids and inflate within-group scatter.
- Requires that the number of cases per group substantially exceeds the number of predictors; with many predictors relative to sample size, the within-group scatter matrix becomes singular.
- Only finds linear boundaries between groups; non-linear group structures are missed unless quadratic discriminant analysis or kernel methods are used.
- Classification accuracy evaluated on the training sample is optimistically biased without cross-validation or a holdout set.
Common pitfalls
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Applications
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Frequently asked
What is the difference between discriminant analysis and logistic regression?
Both can classify observations into groups using multiple predictors, but they differ in assumptions and approach. Discriminant analysis assumes multivariate normality within groups and equal covariance matrices; when these hold it is slightly more powerful. Logistic regression makes no distributional assumptions about the predictors and handles categorical predictors more naturally; it is generally preferred when normality or homoscedasticity cannot be justified.
How many discriminant functions are extracted?
The number of discriminant functions is the minimum of (number of groups minus one) and (number of predictors). For two groups there is exactly one function; for k groups there are at most k-1. Each successive function captures residual group separation not explained by earlier ones.
How is classification accuracy evaluated?
The primary summary is the classification table (confusion matrix) showing how many observations were correctly assigned to each group. Hit rates on the training sample are inflated; unbiased estimates come from leave-one-out cross-validation or from classifying a held-out test sample. Chance-corrected accuracy (kappa or Press's Q statistic) adjusts for group size imbalance.
What is the difference between standardised coefficients and structure coefficients?
Standardised discriminant coefficients (weights applied to standardised predictors) indicate each predictor's unique contribution after partialling out others — they are analogous to beta weights in regression. Structure coefficients are simple correlations between each original predictor and the discriminant score; they indicate how closely a predictor relates to the function as a whole, including shared variance. Both are needed for complete interpretation.
When should I use quadratic rather than linear discriminant analysis?
When the within-group covariance matrices differ substantially across groups (a significant Box M test combined with a meaningful practical difference), quadratic discriminant analysis (QDA) should be considered. QDA estimates separate covariance matrices per group and produces curved rather than linear boundaries, often improving classification accuracy in that situation, though it requires larger samples.
Sources
- 1.Fisher, R. A. (1936). The use of multiple measurements in taxonomic problems. Annals of Eugenics, 7(2), 179–188.
- 2.Hair, J. F., Black, W. C., Babin, B. J. & Anderson, R. E. (2019). Multivariate Data Analysis (8th ed.). Cengage Learning.ISBN 978-1473756540
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Cite this page
ScholarGate. (2026, June 3). Discriminant Analysis. ScholarGate. https://scholargate.app/statistics/discriminant-analysis