Cause-deleted lifetables can only result in an equal gap (before and after) if the deleted causes are *not* proportional

cause_deletion_gap_test.R

library(HMDHFDplus)
library(tidyverse)

mlt <- readHMDweb("ESP","mltper_1x1",username = Sys.getenv("us"),password = Sys.getenv("pw"))



# 1950 Spanish males
mx1 <- c(0.08382, 0.01361, 0.00764, 0.00464, 0.0032, 0.00245, 0.00222, 
         0.00177, 0.00156, 0.00141, 0.00149, 0.00137, 0.0012, 0.00137, 
         0.00155, 0.00184, 0.00176, 0.00202, 0.00291, 0.00292, 0.00324, 
         0.003, 0.00339, 0.00369, 0.00398, 0.00407, 0.00417, 0.00395, 
         0.00403, 0.00374, 0.00438, 0.004, 0.00421, 0.00425, 0.00426, 
         0.00459, 0.00436, 0.00435, 0.00507, 0.00452, 0.00561, 0.00492, 
         0.00629, 0.00622, 0.0064, 0.00756, 0.00716, 0.00801, 0.00897, 
         0.00846, 0.01257, 0.00921, 0.01205, 0.01244, 0.01418, 0.01493, 
         0.01645, 0.0172, 0.02093, 0.0186, 0.02561, 0.02031, 0.02754, 
         0.02943, 0.02963, 0.03569, 0.03683, 0.04137, 0.04593, 0.04097, 
         0.06245, 0.05064, 0.07113, 0.07384, 0.07546, 0.08916, 0.09797, 
         0.10053, 0.13222, 0.09951, 0.15648, 0.10813, 0.1549, 0.15988, 
         0.19702, 0.20494, 0.22352, 0.22641, 0.26305, 0.257, 0.30614, 
         0.20722, 0.33845, 0.27717, 0.27676, 0.3711, 0.39205, 0.41341, 
         0.4351, 0.45704, 0.47914, 0.50133, 0.52352, 0.54561, 0.56752, 
         0.58918, 0.61049, 0.63139, 0.6518, 0.67168, 0.69095)
# 2000 Spanish males
mx2 <- c(0.00472, 0.00045, 0.00029, 0.00022, 0.00017, 0.00019, 0.00019, 
         0.00016, 0.00015, 0.00015, 0.00015, 0.00013, 0.00015, 0.00019, 
         0.00032, 0.00035, 0.00052, 7e-04, 7e-04, 0.00073, 0.00086, 0.00085, 
         8e-04, 0.00083, 0.00091, 9e-04, 0.00098, 9e-04, 0.00106, 0.00101, 
         0.00105, 0.00122, 0.00127, 0.00135, 0.00138, 0.00164, 0.00167, 
         0.00174, 0.00191, 0.00191, 0.00221, 0.00217, 0.00244, 0.00266, 
         0.00292, 0.00309, 0.00315, 0.00378, 0.00383, 0.00419, 0.0049, 
         0.00492, 0.00544, 0.00592, 0.00637, 0.00678, 0.00774, 0.00815, 
         0.00879, 0.00969, 0.01044, 0.0119, 0.01274, 0.01409, 0.01554, 
         0.01716, 0.01797, 0.0205, 0.02162, 0.02436, 0.02678, 0.02972, 
         0.0328, 0.03627, 0.03961, 0.04446, 0.04931, 0.0549, 0.06143, 
         0.06833, 0.07615, 0.08362, 0.09272, 0.10069, 0.11235, 0.12252, 
         0.1417, 0.15362, 0.17027, 0.18321, 0.20229, 0.23436, 0.24819, 
         0.27569, 0.29697, 0.31857, 0.34459, 0.37157, 0.39938, 0.42785, 
         0.45681, 0.48606, 0.51541, 0.54466, 0.57359, 0.60204, 0.62981, 
         0.65675, 0.68271, 0.70759, 0.73128)
mx2ex <- function(mx){
  sum(exp(-cumsum(mx))) + .5
}

# sizeable gap
(gap <- mx2ex(mx2) - mx2ex(mx1)) 

# proportional reduction = gap shrinks (entropy)
(gap <- mx2ex(mx2*.9) - mx2ex(mx1*.9))