Applying Discriminant Function Analysis in Research

When a researcher needs to sort observations into distinct, naturally occurring groups – say, identifying which farms are likely to be profitable, which loan applicants are likely to default, or which customers belong to a high-value segment – they need a method that goes beyond simple comparison. Discriminant Function Analysis (DFA) is exactly that method. It is a multivariate statistical technique that classifies cases into predefined groups by identifying the combination of predictor variables that best separates those groups. Originally developed by Sir Ronald Fisher in 1936, DFA remains one of the foundational tools in classification and predictive modeling across business, finance, social sciences, and agribusiness research.

What is discriminant function analysis? 🔗

Discriminant Function Analysis is used when the dependent variable (the outcome) is categorical and the independent variables (the predictors) are continuous. The central goal is to identify a linear combination of predictor variables – called a discriminant function – that maximally separates two or more groups. When applied to a new observation, this function produces a discriminant score, which is then used to assign that observation to one of the groups.

Put simply: DFA answers the question, “Given a set of measured characteristics about an individual or unit, which group does it most likely belong to?”

According to Statistics Solutions, DFA examines whether significant differences exist among groups in terms of predictor variables, and it also evaluates the accuracy of that classification. The number of discriminant functions that can be derived is always one less than the number of groups – so if you have two groups, you get one function; three groups yield two functions, and so on.

The mechanics: how DFA works 🔗

According to UCLA’s statistical computing resources, DFA performs a multivariate test of differences between groups and also determines the minimum number of dimensions needed to describe those differences. The procedure works by solving for coefficients (weights) that, when multiplied by the predictor variables and summed, produce the maximum discrimination between groups.

The resulting discriminant function takes the general form:

D = b₁X₁ + b₂X₂ + … + bₙXₙ

Where D is the discriminant score, b values are the coefficients, and X values are the predictor variables. A new observation is classified into the group for which its discriminant score is closest to that group’s mean (called the centroid).

Researchers use statistical tests like Wilks’ Lambda and the associated chi-square statistic to assess whether the discriminant function significantly separates the groups. The classification matrix – which shows the number of correctly and incorrectly classified cases – is the primary tool for evaluating predictive accuracy.

Two types: two-group vs. multiple discriminant analysis 🔗

DFA comes in two primary forms based on the number of outcome categories:

Two-group discriminant analysis 🔗

Used when the dependent variable has exactly two categories – for example, defaulted vs. non-defaulted borrowers, or high-yield vs. low-yield crop fields. In this case, only one discriminant function is derived, and it provides a single score to separate the two groups.

Multiple discriminant analysis (MDA) 🔗

Used when there are three or more groups – such as customer segments classified as low, medium, and high value. Multiple Discriminant Analysis identifies the combinations of variables that best differentiate the categories, helping create predictive models across more complex classification scenarios. Its utility extends across marketing, environmental science, and finance, facilitating a deeper understanding of the patterns underlying multi-group data.

Key assumptions of DFA 🔗

For DFA to produce reliable results, several statistical conditions need to be met. Research published in PMC identifies the following core assumptions:

• Multivariate normality: The predictor variables must follow a multivariate normal distribution. This generally limits predictors to continuous or ratio-scale variables.
• Homogeneity of variance-covariance matrices: The variance-covariance structure must be equal across groups. This is tested using Box’s M test.
• Independence of observations: Each case in the dataset must be independent of the others.
• Sample size: A general rule is a minimum of 20 cases per predictor variable to ensure stable estimates.

DFA is also sensitive to outliers – cases with extreme values can distort the discriminant function and reduce classification accuracy. That said, research has shown that DFA can be relatively robust to slight violations of the normality assumption, particularly with larger and balanced sample sizes.

Descriptive vs. predictive discriminant analysis 🔗

A useful distinction in applied research is between two modes of DFA:

Descriptive discriminant analysis (DDA) 🔗

Here, the focus is on understanding which variables best differentiate pre-existing groups. The goal is explanation – for example, understanding what financial characteristics separate profitable agribusiness firms from loss-making ones. The discriminant function helps identify the variables with the greatest contribution to group separation.

Predictive discriminant analysis (PDA) 🔗

Here, the goal is classification – assigning new, unknown cases to one of the groups. Once the discriminant function is derived from a training dataset, it is applied to new observations to predict group membership. Predictive DA produces scores for individuals on the predictor variables to predict the category to which each individual belongs, making it the more widely used form in applied business and agribusiness research.

Step-by-step: conducting a DFA 🔗

The process of conducting a DFA follows a structured sequence. Statistics Solutions outlines the major steps as follows:

• Step 1 – Define the groups: Identify the naturally occurring, mutually exclusive groups you want to classify.
• Step 2 – Select predictor variables: Choose the continuous variables you believe will help discriminate between the groups. These directly influence classification accuracy.
• Step 3 – Estimate the discriminant function: Use statistical software (SPSS, SAS, R, or Python) to derive the function coefficients. The direct method assesses all predictors simultaneously; the stepwise method enters them sequentially, retaining only significant ones.
• Step 4 – Test significance: Use Wilks’ Lambda or the F-statistic to confirm whether the function significantly separates the groups.
• Step 5 – Evaluate classification accuracy: Examine the classification matrix. High accuracy on new (holdout) cases – not just the training data – indicates a robust model.
• Step 6 – Classify new observations: Apply the derived function to classify new cases based on their predictor variable scores.

Applications of DFA in research 🔗

IBM’s documentation on Linear Discriminant Analysis highlights that DFA is applied broadly across industries wherever classification problems arise. Below are some of the most well-documented applications.

Credit scoring and financial risk 🔗

One of the most prominent real-world uses of DFA is in credit risk assessment. Lenders use discriminant analysis to separate creditworthy borrowers from those likely to default, using financial ratios like earnings before depreciation and interest to sales (EBDIT/Sales) and cash flow to sales as discriminating factors. The model assigns a composite score to each applicant; scores above a cut-off threshold indicate creditworthy customers, and those below indicate higher risk.

A landmark application is Edward Altman’s Z-score model (1968), which used multiple discriminant analysis to predict corporate bankruptcy. Despite known limitations in meeting distributional assumptions, Altman’s model – built on five financial ratios – remains one of the most widely used bankruptcy prediction tools in practice. Firms scoring above 2.675 were classified as financially sound, while those below were flagged as bankruptcy-prone.

Marketing and customer segmentation 🔗

In marketing, DFA helps classify customers into distinct behavioral or demographic groups. Businesses use discriminant analysis to predict customer trends and tailor strategies to each segment. For instance, a company might classify customers as brand-loyal versus likely to churn, using purchase history, product preferences, and engagement metrics as predictors. The outcome informs targeted retention and acquisition strategies, improving both engagement and return on investment.

Agribusiness and agricultural research 🔗

In agribusiness, DFA is used to classify farms, producers, or production systems based on performance variables. Researchers can classify farm operations into high-profit and low-profit groups using predictors like farm size, input costs, crop yield, and market access. DFA is also applied to predict crop yield outcomes – classifying fields into high-yield and low-yield categories based on historical data on weather, soil quality, and farming practices – allowing for more targeted resource allocation decisions.

Similarly, market segmentation in the agri-food sector benefits from DFA. A dairy processor, for example, can classify consumers as high-consumption or low-consumption groups based on age, income, and dietary habits. Identifying which variables drive the segmentation enables more precise, data-backed marketing campaigns.

Medicine and social sciences 🔗

In medicine, DFA is used to assess disease severity and prognosis. Patients are divided into groups based on clinical outcomes, and the discriminant function identifies which laboratory or clinical variables most strongly distinguish between those groups, supporting more accurate future diagnoses. In social sciences, DFA has been applied to voter behavior analysis, educational outcome classification, and psychological profiling.

DFA vs. logistic regression: choosing the right tool 🔗

DFA is often compared with logistic regression, as both methods address classification problems with a categorical outcome. The key difference is that DFA requires the predictor variables to be continuous and multivariate normally distributed, while logistic regression places no distributional requirement on the predictors and can handle categorical independent variables as well. When DFA’s assumptions are fully met, it is statistically more powerful than logistic regression. However, because those assumptions are often violated in real-world datasets, logistic regression has become the more commonly preferred method in practice.

Research by Press and Wilson (RAND) confirms that when at least one predictor variable is qualitative – which is common – logistic regression with maximum likelihood estimation is the preferred approach. DFA retains an advantage when working with smaller, balanced samples where variance-covariance homogeneity can be established.

Validating and interpreting DFA results 🔗

A critical – and often overlooked – step in DFA is validation. As StatSoft’s overview of discriminant analysis notes, classifying the same cases used to build the discriminant function will always produce inflated accuracy estimates. True predictive validity can only be assessed by classifying new cases not used in model development – a process called cross-validation. A common approach is to split the data into an estimation sample (used to derive the function) and a holdout validation sample (used to test it).

The standardized discriminant function coefficients are used to interpret the relative contribution of each predictor to group separation – similar to standardized regression coefficients in ordinary regression. Larger absolute values indicate stronger discriminating power. The structure matrix (correlations between predictors and the discriminant function) provides an additional interpretive layer, revealing which variables are most associated with the discriminant dimensions.

What do you think? If you were designing a DFA study to classify agribusiness firms into viable and at-risk categories, which predictor variables would you prioritize – financial ratios, operational metrics, or market access indicators? And given the strict assumptions DFA requires, in what research contexts do you think it still holds a clear edge over logistic regression?

References

1. https://en.wikipedia.org/wiki/Linear_discriminant_analysis
2. https://pmc.ncbi.nlm.nih.gov/articles/PMC9750738/
3. https://www.statisticssolutions.com/discriminant-analysis/
4. https://stats.oarc.ucla.edu/spss/dae/discriminant-function-analysis/
5. https://www.vaia.com/en-us/explanations/math/statistics/discriminant-analysis/
6. https://pmc.ncbi.nlm.nih.gov/articles/PMC2798100/
7. https://www.sciencedirect.com/topics/neuroscience/discriminant-function-analysis
8. https://www.ibm.com/think/topics/linear-discriminant-analysis
9. https://www.tutorialspoint.com/what-is-the-use-of-discriminant-analysis-in-credit-score-model
10. https://www.researchoptimus.com/article/what-is-descriminant-analysis.php
11. https://www.rand.org/pubs/papers/P6277.html
12. https://www.uaq.mx/statsoft/stdiscan.html