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We propose a novel combinatorial inference framework to conduct general uncertainty quantification in ranking problems. We consider the widely adopted Bradley-Terry-Luce (BTL) model, where each item is assigned a positive preference score that determines the Bernoulli distributions of pairwise comparisons’ outcomes. Our proposed method aims to infer general ranking properties of the BTL model. The general ranking properties include the “local” properties such as if an item is preferred over another and the “global” properties such as if an item is among the top K-ranked items. We further generalize our inferential framework to multiple testing problems where we control the false discovery rate (FDR) and apply the method to infer the top-K ranked items. We also derive the information-theoretic lower bound to justify the minimax optimality of the proposed method. We conduct extensive numerical studies using both synthetic and real data sets to back up our theory. 

Abstract Background Interstitial lung abnormalities (ILA) can be detected on computed tomography (CT) in lung cancer patients and have an association with mortality in advanced non-small cell lung cancer (NSCLC) patients. The aim of this study is to demonstrate the significance of ILA for mortality in patients with stage I NSCLC using Boston Lung Cancer Study cohort. Methods Two hundred and thirty-one patients with stage I NSCLC from 2000 to 2011 were investigated in this retrospective study (median age, 69 years; 93 males, 138 females). ILA was scored on baseline CT scans prior to treatment using a 3-point scale (0 = no evidence of ILA, 1 = equivocal for ILA, 2 = ILA) by a sequential reading method. ILA score 2 was considered the presence of ILA. The difference of overall survival (OS) for patients with different ILA scores were tested via log-rank test and multivariate Cox proportional hazards models were used to estimate hazard ratios (HRs) including ILA score, age, sex, smoking status, and treatment as the confounding variables. Results ILA was present in 22 out of 231 patients (9.5%) with stage I NSCLC. The presence of ILA was associated with shorter OS (patients with ILA score 2, median 3.85 years [95% confidence interval (CI): 3.36 – not reached (NR)]; patients with ILA score 0 or 1, median 10.16 years [95%CI: 8.65 - NR]; P  <  0.0001). In a Cox proportional hazards model, the presence of ILA remained significant for increased risk for death (HR = 2.88, P  = 0.005) after adjusting for age, sex, smoking and treatment. Conclusions ILA was detected on CT in 9.5% of patients with stage I NSCLC. The presence of ILA was significantly associated with a shorter OS and could be an imaging marker of shorter survival in stage I NSCLC.