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Gompertz law, New Deale
Malin Breast Cancer Paper:8, t
Key PointsMDEALE 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.
FNZ,jP p!Survival and Life Expectancy (LE)
*Life Expectancy (LE) with continuous death
Life Tables for males, US 2004Captions for Life Tables&Cohort Life Expectancy after treatmentv:DEALE: If death rate is constant d, Life Expectancy = 1/doDerivation of DEALE equation
Discounting is like deathDiscounting 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 1r
So these years = S0(1d)(1r) = S0(1dr+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 S0(1d0r)(1d1r) disc. years etc.
So discounted LE = "(1dr)n if death rate is constant
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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?THZ\GW/B ,wMortality and ageBut, 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 healthjHZP*Z\PHY(*\,s,&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 d40 +b.
This ignores the fact that d40 includes some b.
b may change predictably in years from incidence. m03:kf3 b"S
k4Using 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?>NL`1Effect of added hazard in Fixed Lifetime LE model21Assume 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+s2 +& sL1 = (1sL)/(1s) = 16.4
halfcycle correction 16.4(1d/2) = 15.96
Or new Life expect. = +" 0 to L : exp(.0513t)dt
= [1exp(.0513L)]/.0513. If L = 33.3, this is 15.96
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$$*((h,F,00t7#3!~+A better formula for LE with dread diseaseWeighted 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 +(1p)/(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) 307311,1992.; i;BB0QG t8G/MMalin paper ContextIn 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.0fj+,
FramingUse costeffectiveness 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)bnS`+nS`"S A
Treatments7Diagnostic evaluation always given
Therapy for DCIS always given
Surgery: mastectomy or BCS
postop radiation
Adjuvant therapy
tamoxifen for ER+ and chemo
reconstruction after mastectomy
BMT: expensive risky last chance procedure
Followup: regular always given
intensive shown to have no benefit in trials
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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)
0/YngH:_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 costeffective.
+ZFPZ2P+ZLP!ZBP+/2+8 !$$B((P*O*d!Calculating Life expectancy gainsStudied 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.
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(1(Life tables for women with breast cancerStart 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 bB,:K$+JPresenting Results0For all recommendations, we talked about number of lives saved, not QALYs gained.
Put together a minimum package of very costeffective treatments
Costed some more expensive treatments: more radiation, breast reconstruction
Compared this to giving the minimum package to more uninsured women with cancer.01MPPDI]/`Ksx,, g(HH(dh ~` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dd? ?
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Notes: Number dying = Death rate x number living at start of interval
Years lived in interval includes partial years for those dying
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Years lived in last age bin (here 100+) = number at start x LE (100)
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Not how long a baby born in 2004 will live.
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For proof see next slide. Note L=1/d <> d= 1/L[[TB
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Explain survival curve KaplanMeier : Imagine cohort of 1,000 people. Vertical S(t) axis= number of people alive at start of each subsequent year, marked on X axis= time from birth or start of treatment. What is S(t) S(t+1)? Death, also S(t+1) = s(t) S(t). These curves come from models or data on survival e.g. of 10s of 1000s of women with various adjuvant therapies in Breast Cancer.
life expectancy = average number of years they live = total number of years cohort lives/1,000. As shown, each vertical slice = another year. Can add them 1 year at at time to get the area under the survival curve as total years before they all die. In the graph, everyone dies on Dec 31. It is more realistic to subtract 1/2 year from each vertical slice  Why? area of little triangle on top those dying might die on average halfway through the year) This is called the half cycle correction in Treeage.
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Cohort LE after treatment comes from longitudinal studies and can be transformed to probability of survival. So if there are say 1000 in treatment cohort we followed, we divide those surviving to each period by 1000, we start with S(0) = 1, the 900 alive after a year becomes .9, and S(t) is the probability a patient is still alive at time t, and LE is the area under the survival curve/1.
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YAssuming the death rate is constant is a good approximation for people with some severe diseases such as lung cancer, on dialysis for kidney failure or advanced CHF. This is the Markov assumption:
With constant death rate, we get a simple formula for LE, as shown.
The proof depends on formula for the sum of a geometric series.
We have drawn it as continuous but its true for staircase survival also. DRAW ONE ON BOARD FOR LATER USE of DEALE STAIRCASE.
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sOn average, the hazard of dying in the whole population is not constant, and death rates increase geometrically with age, doubling every 8 years , with men almost double women. Heart disease goes up at about the same rate.
Crude death rates are heavily influenced by the age of the population  young countries will necessarily grow a lot, and crude death rates will rise. In my committee to measure national health we recommend LE as a measure of current health, not the crude death rate.
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Draw some charts on board?
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You can do the same thing assuming people die continually. Then instead of descending stairs, we have a smooth downslope, and instead of a sum, we have an integral.
I wont go through the calculus versions here, but if you know calculus you can check them out yourself.
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EBCTCG trialists collaborative group = trials with 10s of thousands of women randomized to all these treatments. JM also searched recent abstracts etc. for effects of new technologies.
BMT was a bad example of technological diffusion  a lot of people were winging this expensive, risky therapy without being required to be in trials, and without any evidence it worked.
Problem of charges  what would this government program for poor women pay? We assumed standard allowed charges and looked at some discounts in sensitivity analysis.8>
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bER+ women can use Tamoxifen a cheap treatment that works well on them.
Odds of survival to 10 years for various treatments  see paper. Why odds? that s what the trial reports because they analyze with logistic or cox.
These 8 cases x treatment packages x various discount rates & lots of calculations.82GS,
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bBriefed the foundation that was trying to get a sensible measure passed.
They liked lives rather than discounted QALYs, as who would not. Women were of comparable ages, so lives made sense, particularly in this within disease context.
We generally like the efficient care for more, but once you touch someone it is hard to deny them any standard care.
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pn Additive hazard model is used for both estimating b, and then in using it in other models.
If you don t have or believe cause of death, reference disease population is young, may assume all cause mortality is essentially disease mortality.
The Framingham model we looked at earlier did not have this form  the risk factors multiplied. But the form we have here is more common for the impact of independent diseases or of a dread disease on LE. It probably doesn t work for little kids in Africa either, as any of the diseases takes out the same kids.
may or may not want to take out that risk from regular death rates. Often it is negligible.
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Death rate without disease is assumed to be 0 in early stage of fixed lifetime model.
Draw fixed life time and also additional hazard.
This model works surprisingly well, as you can see in my paper.:V1?H
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NScreening bills of uninsured were paid out of tobacco settlement.
Some private foundations provided some treatment, but state was considering doing more.
How could they stretch their limited funds for supporting treatment?
They asked Jennifer Malin, a selfconfident young doctor, and she asked me to help.
How would you have done it?:OBXEo,WH
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4dBudget goes to medical costs, want maximum health benefit for that. good setup for CEA.
This incidence is the incidence of breast cancer found in positive mammographies, not of breast cancer per se.
So again we have issue of current people getting screened, vs what would happen if treatment were also covered. We did both.
Curable because that is way more efficient and luckily incurable are eligible for Medicaid, under 65 because most women get insurance at that age.
Outcomes and best treatment depends on pateint characteristics, we picked 8 types to study  reasonable data on breakdown to these types8eItCW
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Look at drawing, and assume no one dies but you are counting discounted years of life left, I.e. draw in top horizontal line.
With people dying at rate d, take bigger chunk of each year.
And what is that sum at bottom = 1/(d+r) because X in this case is d+rh~<F>#GH
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If L=infinity, d = 1/L = 0 so discounted LE = 1/(0+ .05) = 20 years. That is the most anyone can live at 5% discount rate.8,4^H
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normal + b + r = .3 +.7 +.5 = .15
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lBC treatment very onerous  people suffer now so they will live longer. some studies and opinions on how low QOL is during therapy  we multiplied time in therapy x (1QOL) to get the health cost of treatment. A few weeks that were subtracted from much larger LE gains. after treatment, global QOL of survivors was reported to be very high.  no change for BCS vs mastectomy, but some other satisfaction measures were higher with BCS.
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oMild digression: period LE at birth is a measure used often in population health comparisons. It is a fictional summary statistic based on a survival curve where each drop comes from last years death rates, not from a cohort. It is a reasonable measure of population health, and my committee to measure national health recommends it.
Cohort LE  see next slide
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rUS life tables were used in many CEAs I have helped with. Here is an excerpt from Vital Statistics website: represents deaths in US for men in 2004 by a Life Table. Rows represent age intervals, I ll go through columns: note number died = AxB, years lived is between 1000 that started and 993 that finished. Cumulative survival = area under survival curve from this age on. It is done in Excel by adding from the bottom. So at age 0, it = cum survival at age 1 + 994. Life expectancy is then cum survival/ the number alive at the start, as shown. So this table gives the life expectancy at every age.
At some point, here age 100, they stop. Take a look at the handout or the NCHS website if you are interested.
This is a very simple spreadsheet program to write. They should do it for you, but I am not sure they do.::H!H
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