On average, we die a decade earlier than expected

“Doctors say he’s got a 50/50 chance of living… though there’s only a 10% chance of that.”

I’ve lately had occasion to contemplate my own mortality. How long should I expect to live? The most recent life table published by the Centers for Disease Control (see the reference at the end of this post) indicates an expected lifespan of 76.5 years for a male. This is based on a model of age at death as a random variable X with the probability density shown in the following figure.

Probability distributions of age at death based on the United States period life table for 2014.

The expected lifespan of 76.5 years is E[X] (using the red curve for males). In other words, if we observed a large number of hypothetical (male) infants born in the reference period 2014– and they continued to experience 2014 mortality rates throughout their lifetimes– then their ages at death would follow the above distribution, with an average of 76.5 years.

However, I have more information now: I have already survived roughly four decades of life. So it makes sense to ask, what is my conditional expected age at death, given that I have already survived to, say, age 40? In other words, what is E[X | X \geq 40]?

This value is 78.8 years; I can expect to live to a greater age now than I thought I would when I was first born. The following figure shows this conditional expected age at death E[X | X \geq x], as well as the corresponding expected additional lifespan E[X-x | X \geq x], as a function of current age x.

Conditional expected age at death and expected additional lifespan, vs. current age.

For another example, suppose that I survive to age 70. Instead of expecting just another 6.5 years, my expected additional lifespan has jumped to 14.5 years.

Which brings us to the interesting observation motivating this post: suppose instead that I die at age 70. I will have missed out on an additional 14.5 years of life on average, compared to the rest of the septuagenarians around me. Put another way, at the moment of my death, I perceive that I am dying 14.5 years earlier than expected.

But this perceived “loss” always occurs, no matter when we die! (In terms of the above figure, the expected value E[X-x | X \geq x] is always positive.) We can average this effect over the entire population, and find that on average males die 12.2 years earlier than expected, and females die 10.8 years earlier than expected.

Reference:

  1. Arias, E., United States Life Tables 2014, National Vital Statistics Reports, 66(4) August 2017 [PDF]

Following are the probabilities P(\left\lfloor{X}\right\rfloor = x) for the United States 2014 period life table used in this post, derived from the NVSR data in the above reference, extended to maximum age 120 using the methodology described in the technical notes.

Age   P(all)              P(male)             P(female)
===========================================================
  0 0.005831            0.006325            0.005313
  1 0.000367843         0.000391508         0.000343167
  2 0.000246463         0.000276133         0.000216767
  3 0.000182814         0.000206546         0.000157072
  4 0.000156953         0.000183668         0.000129216
  5 0.000141037         0.000160804         0.000120255
  6 0.000125127         0.000142914         0.000106328
  7 0.000112203         0.000128008         0.0000963806
  8 0.000100276         0.000112117         0.0000884231
  9 0.0000913317        0.0000992073        0.0000834481
 10 0.0000883454        0.0000932456        0.0000824477
 11 0.0000952854        0.000103156         0.0000874072
 12 0.000119095         0.000137857         0.000101304
 13 0.000164729         0.000203286         0.000124134
 14 0.000227209         0.000294457         0.000155893
 15 0.000293617         0.000391501         0.000190617
 16 0.000362946         0.000491412         0.000227306
 17 0.000442117         0.000609009         0.000265957
 18 0.000529115         0.000743227         0.000302594
 19 0.000616971         0.000881116         0.000338207
 20 0.00070566          0.00101964          0.000373784
 21 0.000786255         0.00114394          0.000408332
 22 0.000847887         0.0012343           0.000438875
 23 0.000886654         0.00128494          0.000465417
 24 0.000909534         0.00130883          0.000490933
 25 0.000929392         0.0013228           0.0005174
 26 0.000952147         0.00134161          0.000545804
 27 0.000977786         0.00136426          0.00057515
 28 0.0010063           0.00139366          0.000605432
 29 0.00103864          0.00142878          0.00063566
 30 0.00107285          0.0014657           0.000668788
 31 0.00110792          0.00150147          0.000705793
 32 0.00114483          0.0015361           0.000745674
 33 0.00118454          0.00156959          0.000792356
 34 0.00122898          0.00160582          0.000845811
 35 0.00128495          0.00165443          0.000908956
 36 0.00135141          0.00171632          0.000981746
 37 0.00142538          0.00178654          0.00106021
 38 0.00150387          0.0018631           0.00114136
 39 0.00158781          0.00194786          0.00122516
 40 0.00168489          0.00204935          0.00131843
 41 0.00179886          0.00217314          0.00142304
 42 0.00192761          0.00231898          0.00153401
 43 0.00207581          0.00249327          0.00165615
 44 0.00224899          0.00270038          0.00179419
 45 0.00243725          0.00292846          0.00194116
 46 0.00265083          0.0031884           0.00210855
 47 0.0029103           0.00350311          0.00231349
 48 0.00321588          0.00387243          0.0025555
 49 0.0035468           0.00427362          0.00281771
 50 0.00387592          0.00467212          0.00307957
 51 0.00420287          0.00507013          0.00333606
 52 0.00454693          0.00549737          0.00359937
 53 0.00492128          0.00597165          0.0038758
 54 0.00532664          0.00649003          0.00417147
 55 0.00575619          0.00703581          0.00448667
 56 0.00619215          0.00758275          0.00481225
 57 0.00662626          0.00813029          0.00513737
 58 0.00705499          0.00866988          0.00545972
 59 0.00748745          0.00921066          0.00578812
 60 0.00794918          0.00978879          0.0061402
 61 0.0084469           0.010399            0.00653116
 62 0.0089597           0.010994            0.00696556
 63 0.00947691          0.0115436           0.00744933
 64 0.0100035           0.0120603           0.00797895
 65 0.0105466           0.0125691           0.00854801
 66 0.0111425           0.0131347           0.00916566
 67 0.0118165           0.0137895           0.00985298
 68 0.0126025           0.0145881           0.010625
 69 0.0135386           0.0155721           0.0115183
 70 0.014622            0.016711            0.0125556
 71 0.0157853           0.0179169           0.0136863
 72 0.0169733           0.0191484           0.0148415
 73 0.0181664           0.0203416           0.0160437
 74 0.0193544           0.0214907           0.0172763
 75 0.0205581           0.0226235           0.0185559
 76 0.0219039           0.023887            0.0199909
 77 0.0233782           0.0252875           0.0215559
 78 0.0249405           0.0266573           0.0233399
 79 0.0266659           0.0281283           0.0253501
 80 0.0283006           0.0295587           0.0272207
 81 0.0298041           0.0307938           0.0290306
 82 0.0311707           0.0318902           0.0307088
 83 0.0326118           0.0329808           0.0325375
 84 0.0338734           0.0336728           0.0344093
 85 0.0348103           0.0342521           0.0357896
 86 0.0356915           0.0345144           0.0373244
 87 0.036144            0.0342741           0.0384714
 88 0.0361034           0.0334914           0.0391388
 89 0.0355212           0.0321521           0.0392438
 90 0.0343716           0.0302738           0.0387212
 91 0.0326583           0.0279093           0.0375332
 92 0.0304192           0.0251463           0.0356786
 93 0.0277276           0.0221028           0.0331989
 94 0.02469             0.0189177           0.030181
 95 0.0214386           0.0157381           0.0267542
 96 0.0181203           0.0127037           0.0230805
 97 0.0148823           0.00993297          0.0193396
 98 0.011857            0.00751144          0.0157101
 99 0.00914934          0.00548606          0.0123497
100 0.00682791          0.0038652           0.00937893
101 0.0049216           0.00262443          0.0068711
102 0.00342273          0.00171608          0.00484973
103 0.00229461          0.00108015          0.00329442
104 0.00148199          0.000654343         0.00215221
105 0.000921789         0.000381551         0.00135156
106 0.000552114         0.000214241         0.000815772
107 0.00031851          0.000115918         0.00047333
108 0.000177059         0.0000604924        0.000264139
109 0.0000949139        0.0000304833        0.000141878
110 0.0000491106        0.0000148533        0.0000734294
111 0.0000245565        0.00000700891       0.0000366663
112 0.0000118822        0.00000320819       0.0000176914
113 0.00000557214       0.00000142697       0.00000826206
114 0.00000253659       0.000000617876      0.00000374136
115 0.00000112287       0.000000260924      0.00000164594
116 0.0000004842        0.00000010766       0.00000070483
117 0.000000203758      0.0000000434809     0.000000294373
118 0.0000000838243     0.0000000172192     0.000000120143
119 0.0000000337717     0.0000000066978     0.0000000480077
120 0.0000000216956     0.00000000409582    0.0000000304297
This entry was posted in Uncategorized. Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

w

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.