title: Synthesis of clinical prediction models under different sets of covariates with one individual patient data.
abstract: BACKGROUND
  Recently, increased development of clinical prediction models has been reported in the medical literature. However, evidence synthesis methodologies for these prediction models have not been sufficiently studied, especially for practical situations such as a meta-analyses where only aggregated summaries of important predictors are available. Also, in general, the covariate sets involved in the prediction models are not common across studies. As in ordinary model misspecification problems, dropping relevant covariates would raise potentially serious biases to the prediction models, and consequently to the synthesized results.
  
  
  METHODS
  We developed synthesizing methods for logistic clinical prediction models with possibly different sets of covariates. In order to aggregate the regression coefficient estimates from different prediction models, we adopted a generalized least squares approach with non-linear terms (a sort of generalization of multivariate meta-analysis). Firstly, we evaluated omitted variable biases in this approach. Then, under an assumption of homogeneity of studies, we developed bias-corrected estimating procedures for regression coefficients of the synthesized prediction models.
  
  
  RESULTS
  Numerical evaluations with simulations showed that our approach resulted in smaller biases and more precise estimates compared with conventional methods, which use only studies with common covariates or which utilize a mean imputation method for omitted coefficients. These methods were also applied to a series of Japanese epidemiologic studies on the incidence of a stroke.
  
  
  CONCLUSIONS
  Our proposed methods adequately correct the biases due to different sets of covariates between studies, and would provide precise estimates compared with the conventional approach. If the assumption of homogeneity within studies is plausible, this methodology would be useful for incorporating prior published information into the construction of new prediction models.
authors.affiliation: Department of Statistical Science, School of Multidisciplinary Sciences, SOKENDAI (The Graduate University for Advanced Studies), Tokyo, Japan. yoneoka.daisuke@ism.ac.jp.
authors.forename: Daisuke
authors.initials: D
authors.lastname: Yoneoka
authors.affiliation: Department of Data Science, The Institute of Statistical Mathematics, Tokyo, Japan. henmi@ism.ac.jp.
authors.forename: Masayuki
authors.initials: M
authors.lastname: Henmi
authors.affiliation: Research Center for Cancer Prevention and Screening, National Cancer Center, Tokyo, Japan. nsawada@ncc.go.jp.
authors.forename: Norie
authors.initials: N
authors.lastname: Sawada
authors.affiliation: Department of Global Health Policy, Graduate School of Medicine, The University of Tokyo, Tokyo, Japan. mnminoue@m.u-tokyo.ac.jp.
authors.forename: Manami
authors.initials: M
authors.lastname: Inoue
mesh_terms: D000328:Adult; D000368:Aged; D000465:Algorithms; D003627:Data Interpretation, Statistical; D006801:Humans; D015201:Meta-Analysis as Topic; D008875:Middle Aged; D015233:Models, Statistical; D015999:Multivariate Analysis; D017711:Nonlinear Dynamics; D011446:Prospective Studies; D012372:ROC Curve; D018570:Risk Assessment; D020521:Stroke / Q000175:diagnosis*