Statistical inference of linear regression models for small sample skewed data

For small sample experimental observation data, the severe multicollinearity between indicator variables and the asymmetry of model error distribution can lead to the inability to construct a suitable accurate statistical model. In order to overcome the impact of inaccurate estimates of the shape parameters and skewness parameters of the error distribution on the statistical inference of linear regression models, this paper proposes a method using sensitivity analysis based on small sample data, which can more accurately estimate the shape parameters and skewness parameters of the model error distribution assuming that the error follows a skewed normal distribution. After obtaining parameter estimates of the error distribution, effective statistical inference can be made for linear regression models with severe multicollinearity. Firstly, Bayesian regression combined with the Markov Chain Monte Carlo (MCMC) method was used to estimate model coefficients, and then a posterior interval estimation was used to screen indicator variables. The simulation results showed that the method can effectively establish the final model. By constructing skewed simulation data with small samples, it was verified that our method provides a valuable alternative for statistical inference of linear regression models with multicollinearity for skewed data with small samples. Finally, a quantitative structure-activity relationship model between the quantitative parameters of crown ether molecules and the copper isotope fractionation coefficient Δ⁶⁵Cu (%) was constructed using the method proposed in this paper.