How do you use discriminant analysis and logistic regression to classify and predict outcomes?

In discriminant analysis, the goal is to find a linear combination of the input variables that provides the best discrimination between the groups. The input variables are also called predictors or independent variables, while the groups to be classified are referred to as the dependent variable or the categorical variable. However, it works best when the groups can be separated by straight lines or planes. There are a few things to keep in mind when using discriminant analysis. It doesn't work well if the relationship between the characteristics and the groups is not a straight line or if there are extreme values that can distort the results. If the groups have different sizes, it can affect the accuracy of the classification.

Logistic regression and linear discriminant analyses are multivariate statistical methods that can be used for the evaluation of the associations between various covariates and a categorical outcome. Both methodologies have been extensively applied in research, Logistic regression is a form of regression which is used when the dependent variable is dichotomous, discrete, or categorical, and the explanatory variables are of any kind. The discriminant analysis focuses on the association between multiple independent variables and a categorical dependent variable by forming a composite of the independent variables.

Discriminant analysis is particularly valuable in situations where you need to classify observations into distinct groups based on a set of predictor variables. Its utility shines in fields such as finance, where it can be used to assess credit risk by classifying borrowers into different risk categories, or in healthcare, where it might differentiate between various patient outcomes based on clinical data. One of the strengths of discriminant analysis lies in its ability to create a model that not only predicts category membership but also provides insight into which variables contribute most to the classification.

Logistic regression is a workhorse in the economist's toolbox. Unlike some methods that impose rigid assumptions on data distribution, logistic regression thrives in messy real-world scenarios. It can accommodate diverse covariance structures within outcome categories, reflecting the inherent complexity of economic phenomena. This adaptability empowers researchers to model the probability of events like loan defaults or stock market fluctuations with greater accuracy. The resulting logistic regression equation goes beyond mere prediction – it unveils odds ratios, quantifying the directional and proportional influence of each explanatory variable on the outcome category.

In my experience, this method can be better explained by a simple and concise distinction: logistic regression is a supervised machine learning algorithm used for binary classification tasks. It can be used to predict the probability of an event belonging to one of two classes based on input variables: a success or a failure. For example, likelihood of a loan default, likelihood of a patient developing a specific medical condition, predict voting preferences, etc. Logistic regression is a valuable and widely used due to its simplicity, efficiency, interpretability, and probability estimation capabilities. However, it is essential to consider its assumptions, limitations, and potential challenges when applying it in practice.

Selecting the champion between discriminant analysis and logistic regression hinges on the specifics of your economic inquiry. Discriminant analysis shines when dealing with continuous, normally distributed explanatory variables, particularly in large datasets. However, logistic regression thrives in the face of messy real-world data. It adeptly handles categorical, ordinal, or skewed explanatory variables, often encountered in economic analysis. This adaptability makes logistic regression a strong choice for exploring consumer spending patterns, where income levels might be categorical and spending habits might exhibit non-normal distributions. Ultimately, the victor hinges on your data characteristics and research goals.

It is crucial to carefully assess the characteristics of the classification task and the dataset at hand before deciding on the appropriate method. Exploring alternative algorithms and considering the specific limitations and assumptions can help in choosing the most suitable approach for the given context. For example, a common challenge in classification problems happens when dealing with high-dimensional datasets where the number of predictors is much larger than the number of observations. In this case, both discriminant analysis and logistic regression will face challenges. This situation may lead to overfitting. Dimensionality reduction techniques designed for high-dimensional data should be explored instead.

When choosing between discriminant analysis and logistic regression for classification and prediction, consider the nature of your data and the assumptions of each method. Discriminant analysis is ideal for normally distributed data with equal covariance matrices, while logistic regression is robust for non-normal data and doesn't require assumptions about covariance. Assess your data's distribution and covariance structure to make an informed choice.

R's glm function in R is designed for logistic regression and provides detailed statistical summaries. Python, through libraries like scikit-learn and statsmodels, also excels in logistic regression. These libraries offer comprehensive tools for data manipulation, model fitting, and evaluation. Python integrates well with other steps in a data pipeline, such as web scraping or database communication. SQL can also be used for logistic regression, although it's less conventional. Logistic regression in SQL involves creating a binary variable for the outcome and using mathematical functions to calculate odds ratios. This method is more labor-intensive and less precise compared to R or Python.