A confidence sequence is a sequence of confidence intervals that is uniformly valid over an unbounded time horizon. Our work develops confidence sequences whose widths go to zero, with nonasymptotic coverage guarantees under nonparametric conditions. We draw connections between the Cramér-Chernoff method for exponential concentration, the law of the iterated logarithm (LIL) and the sequential probability ratio test-our confidence sequences are time-uniform extensions of the first; provide tight, nonasymptotic characterizations of the second; and generalize the third to nonparametric settings, including sub-Gaussian and Bernstein conditions, self-normalized processes and matrix martingales. We illustrate the generality of our proof techniques by deriving an empirical-Bernstein bound growing at a LIL rate, as well as a novel upper LIL for the maximum eigenvalue of a sum of random matrices. Finally, we apply our methods to covariance matrix estimation and to estimation of sample average treatment effect under the Neyman-Rubin potential outcomes model.

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Background and objectives Mortality and CKD risk have not been described in military casualties with posttraumatic AKI requiring RRT suffered in the Iraq and Afghanistan wars.Design, setting, participants, & measurements This is a retrospective case series of post-traumatic AKI requiring RRT in 51 military health care beneficiaries (October 7, 2001-December 1, 2013, evacuated to the National Capital Region, documenting in-hospital mortality and subsequent CKD. Participants were identified using electronic medical and procedure records.Results Age at injury was 2666 years; of the participants, 50 were men, 16% were black, 67% were white, and 88% of injuries were caused by blast or projectiles. Presumed AKI cause was acute tubular necrosis in 98%, with rhabdomyolysis in 72%. Sixty-day all-cause mortality was 22% (95% confidence interval [95% CI], 12% to 35%), significantly less than the 50% predicted historical mortality (P,0.001). The VA/NIH Acute Renal Failure Trial Network AKI integer score predicted 60-day mortality risk was 33% (range, 6%-96%) (n=49). Of these, nine died (mortality, 18%; 95% CI, 10% to 32%), with predicted risks significantly miscalibrated (P,0.001). The area under the receiver operator characteristic curve for the AKI integer score was 0.72 (95% CI, 0.56 to 0.88), not significantly different than the AKI integer score model cohort (P=0.27). Of the 40 survivors, one had ESRD caused by cortical necrosis. Of the remaining 39, median time to last follow-up serum creatinine was 1158 days (range, 99-3316 days), serum creatinine was 0.8560.24 mg/dl, and eGFR was 118623 ml/min per 1.73 m 2 . No eGFR was ,60 ml/min per 1.73 m 2 , but it may be overestimated because of large/medium amputations in 54%. Twenty-five percent (n=36) had proteinuria; one was diagnosed with CKD stage 2.Conclusions Despite severe injuries, participants had better in-hospital survival than predicted historically and by AKI integer score. No patient who recovered renal function had an eGFR,60 ml/min per 1.73 m 2 at last follow-up, but 23% had proteinuria, suggesting CKD burden.

A sensitivity analysis in an observational study tests whether the qualitative conclusions of an analysis would change if we were to allow for the possibility of limited bias due to confounding. The design sensitivity of a hypothesis test quantifies the asymptotic performance of the test in a sensitivity analysis against a particular alternative. We propose a new, nonasymptotic, distribution-free test, the uniform general signed rank test, for observational studies with paired data, and examine its performance under Rosenbaum’s sensitivity analysis model. Our test can be viewed as adaptively choosing from among a large underlying family of signed rank tests, and we show that the uniform test achieves design sensitivity equal to the maximum design sensitivity over the underlying family of signed rank tests. Our test thus achieves superior design sensitivity, indicating it will perform well in sensitivity analyses on large samples. We support this conclusion with simulations and a data example, showing that the advantages of our test extend to moderate sample sizes as well.

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