Examination of distance based regression methods for different data structures

In order to the parameter estimations of the regression model to be obtained as a result of simple and multiple linear regression analysis to be reliable, some assumptions about the model must be provided. One of the few models developed as a solution for situations where assumptions cannot be provide in parameter estimation methods is Distance Based Regression methods. The purpose of these methods is to properly address problems with measure value estimators, including categorical or a mix of real-valued and categorical explanatory variables. Distance-based regression is an alternative method for parameter estimation in linear regression models when mixed-type explanatory variables are used. Distance-based regression is similar to classical linear regression, except that explanatory variables are measured by distance measures rather than raw values. In this study, datasets with sample sizes of 10, 25, 50, 100, 250 and 500 produced for Binomial, Normal, t, Chi-square and Poisson distributions of Euclidean, Gower and Manhattan distance measures and real data with discrete and continuous distribution. It was aimed to determine the effect on the data sets (10, 50 and 100 sample sizes) by comparing the results obtained from the Linear Regression method. R packages "dbstats", "cluster" and "tidyverse" were used to perform the analysis. As a result, it has been determined that the use of Manhattan distance in data with Poisson distribution may produce unsuccessful results, especially in small sample sizes (n<50). Although there is no significant difference between Gower and Euclidean distances in different distributions according to sample sizes, it has been determined that the use of Euclidean distance measure in some distributions produces results that cause fluctuation. However, it has been understood that the Gower distance can be recommended as a more suitable choice since it has a more stable structure. For the applicability of the Least Square Estimation method, it may be recommended to use Distance Based Regression methods in cases where the necessary assumptions mentioned in this study cannot be met.