Estimation Of Effect Size And Relative Importance Of Predictors Under The Collinearity Condition Of Linear Model

Objective: In the face of natural complexities and collinearity,variable importance estimation using multiple regression may be ambiguous and risky.The shortcomings of Dominance analysis,Relative Weights,Product Measure have been extensively described in statistical literature that these tools fail to address collinearity.We introduce two methods which will be helpful in interpreting the typical multiple regression analyses conducted on medical data: regression Commonality analysis(CA)and Bayesian Dominance Hierarchies approach based on a statistical model of paired comparisons.Methods: CA decomposes the variance of R~2 into unique and common(or shared)variance(or effects)of predictors,and hence,it can significantly improve exploratory capabilities in studies where multiple regressions are widely used,particularly when predictors are correlated.CA can explicitly identify the magnitude and location of collinearity and suppression in a regression model.Besides,we have improved CA method and compared with Dominance analysis for determining variables importance.Bayesian Dominance Hierarchies approach based on Bradley-Terry pairwise comparison model,first computing additional contribution in R~2 within each model size at different levels by Dominance analysis,through the pairwise comparison of win/loss table was constructed at all levels,build the advantage index of likelihood function and hierarchical prior distribution to infer the posterior distribution advantage index.Using the Gibbs sampling method of Markov Chain Monte Carlo(MCMC),the dominance index of each variable and its 95% confidence interval are obtained by repeated iterations.The empirical analysis of the Commonality analysis and the Bayesian Dominance Hierarchies method are all realized through the R statistical software.Results: In this paper,using two medical empirical datasets to explore variables structural relationship and variables importance.The data of 47 regions based on fertility variables,occupation,examination ability,education level,belief and infant mortality rate were analyzed.Analysis results show that the occupation,examination ability,level of education,there are linear,these four variables between the beliefs of occupation,examination ability by inhibiting the variance by the level of education,belief and independent interpretation of fertility,enhance the level of education,religion,occupation of procreation ability ability to explain,to help understand the key role the level of education,belief changes on fertility in people.The improved CA was consistent with Dominance analysis for determining variables importance,rank row relative importance of each variable on fertility was leavel of education > examination ability > the infant mortality rate > belief > occupation.The unfolded protein energy as the dependent variable,the lipophilic constant(PIE),the lipophilic constant(PIF),a chain of amino acids from protein internal water free energy(DGR),surface water contact area of amino acid(SAC),molecular index(MR),polar parameter(Lam)the molecular area,amino acid(Vol)as the sample data of 19 independent variables to analyze the Bayesian Dominance Hierarchies.The Gibbs sampling method of MCMC iterated 10000 times to get the point estimation and 95% confidence interval of the dominant index of each variable.The dominance index of PIF is the largest,followed by DGR,PIE,Lam,Vol,MR and SAC.So the relative importance of the independent variable is PIF>DGR> PIE >Lam >Vol >MR >SAC.Conclusion: CA can significantly improve the ability of model exploration in multiple regression research,particularly when predictors are correlated,CA can explicitly identify the magnitude and location of collinearity and suppression in a regression model.The effect of the improved CA and Dominance analysis on the estimation of relative importance of independent variables is consistent.When the independent variable is highly correlated,can use Bayesian Dominance Hierarchies method to estimate the relative importance,this method provides more comprehensive inference for related variables group relative advantage ability,may also be a useful supplement to analyze other than multiple regressions.