Predictive model in the presence of missing data: the centroid criterion for variable selection

Abstract

IntroductionIn many studies, covariates are not always fully observed because of missing data process. Usually, subjects with missing data are excluded from the analysis but the number of covariates can be greater than the size of the sample when the number of removed subjects is high. Subjective selection or imputation procedures are used but this leads to biased or powerless models.\n\nThe aim of our study was to develop a method based on the selection of the nearest covariate to the centroid of a homogeneous cluster of covariates. We applied this method to a forensic medicine data set to estimate the age of aborted fetuses.\n\nAnalysisO_ST_ABSMethodsC_ST_ABSWe measured 46 biometric covariates on 50 aborted fetuses. But the covariates were complete for only 18 fetuses.\n\nFirst, to obtain homogeneous clusters of covariates we used a hierarchical cluster analysis.\n\nSecond, for each obtained cluster we selected the nearest covariate to the centroid of the cluster, maximizing the sum of correlations [Formula] (the centroid criterion).\n\nThird, with the covariate selected this way, the sample size was sufficient to compute a classical linear regression model.\n\nWe have shown the almost sure convergence of the centroid criterion and simulations were performed to build its empirical distribution.\n\nWe compared our method to a subjective deletion method, two simple imputation methods and to the multiple imputation method.\n\nResultsThe hierarchical cluster analysis built 2 clusters of covariates and 6 remaining covariates. After the selection of the nearest covariate to the centroid of each cluster, we computed a stepwise linear regression model. The model was adequate (R2=90.02%) and the cross-validation showed low prediction errors (2.23 10-3).\n\nThe empirical distribution of the criterion provided empirical mean (31.91) and median (32.07) close to the theoretical value (32.03).\n\nThe comparisons showed that deletion and simple imputation methods provided models of inferior quality than the multiple imputation method and the centroid method.\n\nConclusionWhen the number of continuous covariates is greater than the sample size because of missing process, the usual procedures are biased. Our selection procedure based on the centroid criterion is a valid alternative to compose a set of predictors.