Deaths related to a specific cause: ratio of two rates

Dear Colleagues, I would like to discuss with you an analytical problem I have encountered.

I am interested in estimating the fraction of deaths attributable to a specific cause. I have two rates over time for all-cause and cancer-related deaths (as for example in the graph below, with solid lines indicating 95%CI)

These rates have been estimated in a cohort of people: I have used a spline for calendar time (time scale) and then predicted rates for very tiny intervals of calendar time. For each time point, I have therefore the estimate and the standard error separately for all-cause and cancer death.

I would need to estimate the ratio of these two rates (cancer/all-cause), which corresponds to the fraction of death due to cancer over time. If I had two exposures like men and women (instead of two outcomes all-cause and cancer death), I would estimate the ratio of the two rates and obtain the standard error as the square root of the sum of the two variances (men and women) because they are independent. But in this case all-cause deaths include cancer deaths.

Do you have any suggestion or reference(s) on how to estimate the standard error of the ratio in this case?

Thank you all very much for your feedback.

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