Regression analysis is a central topic in statistical modeling, aiming to estimate the relationships between a dependent variable, commonly referred to as the response variable, and one or more independent variables, i.e., explanatory variables. Linear regression is by far the most popular method for performing this task in several fields of research, such as prediction, forecasting, or causal inference. Beyond various classical methods to solve linear regression problems, such as Ordinary Least Squares, Ridge, or Lasso regressions - which are often the foundation for more advanced machine learning (ML) techniques - the latter have been successfully applied in this scenario without a formal definition of statistical significance. At most, permutation or classical analyses based on empirical measures (e.g., residuals or accuracy) have been conducted to reflect the greater ability of ML estimations for detection. In this paper, we introduce a method, named Statistical Agnostic Regression (SAR), for evaluating the statistical significance of an ML-based linear regression based on concentration inequalities of the actual risk using the analysis of the worst case. To achieve this goal, similar to the classification problem, we define a threshold to establish that there is sufficient evidence with a probability of at least 1-eta to conclude that there is a linear relationship in the population between the explanatory (feature) and the response (label) variables. Simulations in only two dimensions demonstrate the ability of the proposed agnostic test to provide a similar analysis of variance given by the classical $F$ test for the slope parameter.

Transfer learning is beneficial by allowing the expressive features of models pretrained on large-scale datasets to be finetuned for the target task of smaller, more domain-specific datasets. However, there is a concern that these pretrained models may come with their own biases which would propagate into the finetuned model. In this work, we investigate bias when conceptualized as both spurious correlations between the target task and a sensitive attribute as well as underrepresentation of a particular group in the dataset. Under both notions of bias, we find that (1) models finetuned on top of pretrained models can indeed inherit their biases, but (2) this bias can be corrected for through relatively minor interventions to the finetuning dataset, and often with a negligible impact to performance. Our findings imply that careful curation of the finetuning dataset is important for reducing biases on a downstream task, and doing so can even compensate for bias in the pretrained model.

This paper compares the performances of three supervised machine learning algorithms in terms of predictive ability and model interpretation on structured or tabular data. The algorithms considered were scikit-learn implementations of extreme gradient boosting machines (XGB) and random forests (RFs), and feedforward neural networks (FFNNs) from TensorFlow. The paper is organized in a findings-based manner, with each section providing general conclusions supported by empirical results from simulation studies that cover a wide range of model complexity and correlation structures among predictors. We considered both continuous and binary responses of different sample sizes. Overall, XGB and FFNNs were competitive, with FFNNs showing better performance in smooth models and tree-based boosting algorithms performing better in non-smooth models. This conclusion held generally for predictive performance, identification of important variables, and determining correct input-output relationships as measured by partial dependence plots (PDPs). FFNNs generally had less over-fitting, as measured by the difference in performance between training and testing datasets. However, the difference with XGB was often small. RFs did not perform well in general, confirming the findings in the literature. All models exhibited different degrees of bias seen in PDPs, but the bias was especially problematic for RFs. The extent of the biases varied with correlation among predictors, response type, and data set sample size. In general, tree-based models tended to over-regularize the fitted model in the tails of predictor distributions. Finally, as to be expected, performances were better for continuous responses compared to binary data and with larger samples.