# 1 © 2008 Emmett Keeler RAND More realistic Life Exp. calculations Deale Gompertz law, New Deale Malin Breast Cancer Paper.

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1 © 2008 Emmett Keeler RAND More realistic Life Exp. calculations Deale Gompertz law, New Deale Malin Breast Cancer Paper

2 © 2008 Emmett Keeler RAND Key Points DEALE is easy to use, but not very accurate Life tables are the gold standard for calculating the impact of varied additive hazards on discounted LE, But Mixed DEALEs work well In Malin, CEA is used to devise a low cost breast cancer package for uninsured women-- shows tradeoffs between covering more people and generosity of care.

Survival and Life Expectancy (LE) Let S(t) = number alive at start of year t, s(t) = survival rate from year t to t+1, so S(t+1) = s(t)S(t). What is S(t) - S(t+1)? Life Expectancy = Average years to be lived = (∑ S(t)) / S(0). “age” at death for those dying S(t) S(t+1) Number surviving, S(t) Years from start Start t t+1 S(0) another year for survivors 1000? area under survival curve

Life Expectancy (LE) with continuous death Let S(t) = number alive at time t, and dS/dt = -h(t)S(t). h(t) is called the hazard of dying. S(t) = S(0) exp (∫-h(x)dx) Life Expectancy = Average years lived = (∫S(t)) / S(0). If h(t) = h, LE = 1/h S(t) Number surviving, S(t) Years from start Start t S(0) more years for survivors

Life Tables for males, US 2004 Age death rate in interval number living at start deaths in interval years lived in interval years lived in this and all future intervals life expect at start 0-1.0071000799475,17575.2 1-2.001 9931992.574,17174.7 100+1.0012.5 30 2.4 ??

6 © 2008 Emmett Keeler RAND Captions for Life Table Excerpt from http://www.cdc.gov/nchs/nvss.htm http://www.cdc.gov/nchs/nvss.htm Notes: Number dying = Death rate x number living at start of interval Years lived in interval includes partial years for those dying 74,171/993 = 74.7 gives LE at age 1. Years lived in last age bin (here 100+) = number at start x LE (100) Period LE is a synthetic measure of current population health based on 2004 death rates  Not how long a baby born in 2004 will live. Cohort LE is a measure of how long a certain cohort of people lived. Cancer trials collect this.

Prob of Survival 1.0 Time from Beginning of intervention 0 1 2 3 Area under curve = Life Expectancy after intervention If we divide S(t) by the number treated then S(0) = 1, S(t) is a probability, and LE = area under survival curve/1. Cohort Life Expectancy after treatment

DEALE: If death rate is constant d, Life Expectancy = 1/d Here death rate =. 25, so LE = 4 years. For proof see next slide. Note L=1/d d= 1/L

9 © 2008 Emmett Keeler RAND Derivation of DEALE equation Calculus version: survival S(t) = exp(-dt), for hazard of dying d LE = area under survival curve = ∫ exp(-dt)dt = -1/d * exp(-dt) | when t = ∞ - when t = 0 = -1/d [0 - 1] = 1/d LE = 1 + s +s 2 + s 3 + …s n sLE = s +s 2 + s 3 + …s n + s n+1 LE - sLE = 1 - s n+1 ~1 for large n LE = ∑s i = 1/(1-s) = 1/d for large n

10 © 2008 Emmett Keeler RAND Discounting is like death Discounting future years at rate r% is formally like assuming r% additional deaths each year. At the start of the second year, we have a proportion d who have died. When we add in years from year 2 in total years lived, each year has value 1-r So these years = S 0 (1-d)(1-r) = S 0 (1-d-r+dr)  dr is the product of two small #s and so negligible  If we divide year into smaller time periods, dr disappears.  in the third year we have S 0 (1-d 0 -r)(1-d 1 -r) disc. years etc. So discounted LE = ∑(1-d-r) n if death rate is constant

11 © 2008 Emmett Keeler RAND Example: Using the DEALE to calculate discounted life expectancy Assume a 50 year old white woman will have the average 2005 US LE of 33.3 years after cure. Assume a discount rate of 5%. What is her discounted life expectancy? Death rate = 1/33.3 =.03. So d+r =.08, so discounted life expectancy = 1/.08 = 12.5. What if we assume she will never die, I.e. L = ∞. What is her discounted LE?

12 © 2008 Emmett Keeler RAND Mortality and age But, overall death rates increases steadily with age (Gompertz,1826).  Death rate doubles every 8.5 years 30 - 85;  women’s rate = 60% that of men the same age  similarly for incidence of heart disease So, crude death rate is smaller than 1/L  Especially in countries with young populations.  crude rate a poor measure of current health

13 © 2008 Emmett Keeler RAND What is impact of added hazards on LE? Need some model to fit data, and then to do calculations. We often assume a baseline or “normal” death rate and model death from additional risks as added to that.  So if the death rate from disease B is b, then the death rate for a 40 year old with B is modeled as d 40 +b. This ignores the fact that d 40 includes some b.  b may change predictably in years from incidence.

14 © 2008 Emmett Keeler RAND Using the DEALE to calculate impact of dread disease Consider a 50 year old white woman with the average US 2005 LE of 33.33 years. She gets breast cancer and after treatment is assumed to have a 7% chance of dying from it each year. Assume hazard of “normal” and breast cancer deaths add. “Normal” death hazard = 1/33.33 =.03. Combined hazard =.07+.03 = 0.1, so new life expectancy is 1/.01 = 10 years. What is her Discounted LE, with a 5% discount rate?

15 © 2008 Emmett Keeler RAND Effect of added hazard in Fixed Lifetime LE model Assume without disease, death rate is 0 for L years, and then the person dies.  what does the survival curve look like? What if there is an added hazard of 5% per year? Staircase: new LE = 1+s+s 2 +…s L-1 = (1-s L )/(1-s) = 16.4  half-cycle correction 16.4(1-d/2) = 15.96 Or new Life expect. = ∫ 0 to L : exp(-.0513t)dt = [1-exp(-.0513L)]/.0513. If L = 33.3, this is 15.96

16 © 2008 Emmett Keeler RAND A better formula for LE with dread disease Weighted average of DEALE and Fixed Lifetime model of LE Suppose in someone over 50, normal LE = L, hazard from dread disease = d LE = p(1- exp(-dL))/d +(1-p)/(d + 1/L), with p =.5 to.75 works well over a wide range of d and L.  Easier to implement in EXCEL than a life table.  for the 50 year old woman, with p =.75 L =33.3, d =.05, we have LE =.75 * 16 +.25 * 12.5 = 15.1 Keeler E, Bell R. New Deales: other approximations of Life Expectancy, Med Dec Making (12) 307-311,1992.

17 © 2008 Emmett Keeler RAND Malin paper Context In 1999, California paid for mammographic screening of uninsured women, but not subsequent treatment. California was considering giving up to \$15 million for their treatment. Wanted advice on what to cover. I helped Jennifer Malin with quick project.

18 © 2008 Emmett Keeler RAND Framing Use cost-effectiveness analysis to rate different treatments -- costs from CA government perspective = direct medical, but health benefits to women. For budget, need estimates of “incidence.”  actual: cases expected from current screening levels  potential: if screens = uninsured x incident cancer <65 Early (curable) breast cancer in women under 65. Studied 8 representative women.  45 or 60  ER+ (can use tamoxifen) or ER-  lymph node involved (40%) or not (20% 10 year survival)

19 © 2008 Emmett Keeler RAND Treatments Diagnostic evaluation always given Therapy for DCIS always given Surgery: mastectomy or BCS  post-op radiation  Adjuvant therapy tamoxifen for ER+ and chemo  reconstruction after mastectomy  BMT: expensive risky last chance procedure Follow-up: regular always given  intensive shown to have no benefit in trials

20 © 2008 Emmett Keeler RAND Data on benefits and costs EBCTCG had great data on treatment of early breast cancer: surgery, radiation and chemo. We also looked at reconstruction and BMT where data was not as good. Had expert opinion from earlier papers. Costs from Medicare allowed charges for services, AWP for drugs (with PHS discounts in sens. analysis)

21 © 2008 Emmett Keeler RAND Issues Utility of life during and after treatment  disutility of treatment x length of treatment?  later life disutility? Evidence vs. Standard of care  radiation after surgery does not improve survival BCS dominates mastectomy + reconstruction.  if it is possible.  but law mandates private insurers cover reconstruction No evidence on BMT effectiveness  we calculated how good it would have to be to be cost- effective.

22 © 2008 Emmett Keeler RAND Calculating Life expectancy gains Studied 8 types of women  age 45, 60  node - and + = 20, 40% 10 year survival,  ER+ (can use tamoxifen), ER- breast cancer. Used reported odds from EBCTCG for survival to 10 years, then constant added BC risk over normal women for rest of life.  constant added hazard fit well for first 10 years Lots of calculations to get discounted LE, so used mixed Deale, EXCEL  Validated results against Life Tables for one treatment always within.03 years.

23 © 2008 Emmett Keeler RAND Life tables for women with breast cancer Start with standard life table for population of interest Add in another column for hazard of dying if you have breast cancer by age  this column depends on age of onset  type of disease and treatment Use combined hazard = sum of these columns

24 © 2008 Emmett Keeler RAND Presenting Results For all recommendations, we talked about number of lives saved, not QALYs gained. Put together a minimum package of very cost- effective treatments Costed some more expensive treatments: more radiation, breast reconstruction Compared this to giving the minimum package to more uninsured women with cancer.

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