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The annual percent change (APC) has been adopted as a useful measure for analyzing the changing trends of cancer mortality and incidence rates by the NCI SEER program. Difficulties, however, arise when comparing the sample APCs between two overlapping regions because of induced dependence (e.g., comparing the cancer mortality change rate of California with that of the national level). This paper deals with a new perspective for understanding the sample distribution of the test-statistics for comparing the APCs between overlapping regions. Our proposal allows for computational readiness and easy interpretability. We further propose a more general family of estimators, namely, the so-called minimum power divergence estimators, including the maximum likelihood estimators as a special case. Our simulation experiments support the superiority of the proposed estimator to the conventional maximum likelihood estimator. The proposed method is illustrated by the analysis of the SEER cancer mortality rates observed from 1991 to 2006.