Analysis of competing risks data is often complicated by the incomplete or selectively missing information on the cause of failure. Standard approaches typically assume that the cause of failure is missing at random (MAR), an assumption that is generally untestable and frequently implausible in observational studies. We propose a novel sensitivity analysis framework for the proportional cause-specific hazards model that accommodates missing-not-at-random (MNAR) scenarios. A sensitivity parameter is used to quantify the association between missingness and the unobserved cause of failure. Regression coefficients are estimated as functions of this parameter, and a simultaneous confidence band is constructed via a wild bootstrap procedure. This allows identification of a range of MNAR scenarios for which effects remain statistically significant; we refer to this range as a robustness interval. The validity of the proposed approach is justified both theoretically, via empirical process theory, and empirically, through simulation studies. We apply the method to the analysis of data from an HIV cohort study in sub-Saharan Africa, where a substantial proportion of causes of failure are missing and the MAR assumption is implausible. The analysis shows that key findings regarding risk factors for care interruption and mortality are robust across a broad spectrum of MNAR scenarios, underscoring the method's utility in situations with MNAR causes of failure.

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