Discuss the similarities and differences between discriminant analysis and logistic regression. How might a researcher decide between using one or the other? Would it be possible to run both procedures on the same data? Explain your answer.

Discriminant analysis and logistic regression are both statistical methods used for predictive modeling and classification purposes. While they share similarities in terms of their objectives, they differ in their assumptions, measurement levels, output, and interpretability. In this discussion, I will compare and contrast these two methods and discuss how a researcher might decide between using one or the other.

Discriminant analysis is a statistical technique used to predict the group membership of observations based on a set of predictor variables. It assumes that the predictors are normally distributed within each group and have equal covariance matrices across groups. Discriminant analysis focuses on finding a linear combination of predictors that maximally discriminates between groups. The objective is to minimize within-group variance and maximize between-group variance. The output of discriminant analysis includes discriminant function coefficients, which represent the weights assigned to each predictor in the linear combination, and the discriminant scores, which indicate the relative distance of each observation from the group centroids.

On the other hand, logistic regression is a statistical method used to model the relationship between a binary dependent variable and one or more predictor variables. It assumes a logistic or sigmoid-shaped relationship between the predictor variables and the log odds of success, and it estimates the coefficients of the predictors that maximize the likelihood of the observed data. The output of logistic regression includes regression coefficients, which represent the change in the log odds of success associated with a unit change in the corresponding predictor, and odds ratios, which indicate the multiplicative effect of each predictor on the odds of success.

One key difference between discriminant analysis and logistic regression is the type of dependent variable they can handle. Discriminant analysis is suitable for predicting group membership or categorical outcomes with more than two categories, while logistic regression is specifically designed for binary outcomes. Moreover, logistic regression can be extended to handle multiple categories through techniques such as multinomial logistic regression or ordered logistic regression.

Another difference lies in the assumptions made by these two methods. Discriminant analysis assumes that the predictors are normally distributed within each group, while logistic regression makes no such assumptions about the distribution of predictors. Additionally, logistic regression assumes that the relationship between the predictors and the log odds of success is linear on the logit scale. If this assumption is violated, transformations or non-linear extensions of logistic regression, such as polynomial terms or splines, can be used.

In terms of interpretability, logistic regression provides more straightforward interpretation of the estimated coefficients. The regression coefficients represent the change in the log odds of success associated with a unit change in the corresponding predictor. In contrast, discriminant analysis is less interpretable, as the discriminant function coefficients do not have a straightforward probabilistic interpretation. Furthermore, discriminant analysis does not provide odds ratios or probability estimates for individual observations.

When deciding between discriminant analysis and logistic regression, researchers should consider several factors. First, they need to assess the type of outcome variable they are working with; if the outcome is binary, logistic regression is the appropriate choice, while discriminant analysis is more suitable for multiclass classification problems. Second, researchers need to evaluate the assumptions and data requirements of each method. If the assumptions are violated, alternative methods or data transformations should be considered. Lastly, researchers should consider the interpretability of the results and the specific research question they are addressing. Logistic regression is often preferred when the focus is on estimating the effects of predictors on the odds of success, while discriminant analysis may be more appropriate when the goal is to classify observations into groups.

It is possible to run both discriminant analysis and logistic regression on the same data. However, it is important to note that the two methods have different objectives and assumptions. Using both methods can provide complementary insights into the relationships between the predictors and the outcome, but the interpretation and results may differ. Researchers should carefully compare and evaluate the results from each method before drawing conclusions. Additionally, running both procedures can help assess the robustness of the findings and provide an opportunity for sensitivity analyses.