In this article, we are interested in capturing heterogeneity in clustered or longitudinal data. Traditionally such heterogeneity is modeled by either fixed effects (FE) or random effects (RE). In FE models, the degree of freedom for the heterogeneity equals the number of clusters/subjects minus 1, which could result in less efficiency. In RE models, the heterogeneity across different clusters/subjects is described by, for example, a random intercept with 1 parameter (for the variance of the random intercept), which could lead to oversimplification and biases (for the estimates of subject-specific effects). Our “fused effects” model stands in between these two approaches: we assume that there are unknown number of distinct levels of heterogeneity, and use the fusion penalty approach for estimation and inference. We evaluate and compare the performance of our method to the FE and RE models by simulation studies. We apply our method to the Ocular Hypertension Treatment Study to capture the heterogeneity in the progression rate of primary open-angle glaucoma of left and right eyes of different subjects.

Original languageEnglish
Pages (from-to)1901-1916
Number of pages16
JournalStatistics in medicine
Issue number8
StatePublished - Apr 15 2021


  • fusion penalty
  • high-dimensional data
  • precision medicine
  • variable selection


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