Estimating Parameters

First we describe how we estimate each of the parameters of the model using public health data for colorectal cancer.

• We assume a time unit of weeks.

• The current guidelines endorsed by the CDC and the USPSTF recommend routine screen- ings beginning at age 50 with a FOBT test every year, sigmoidoscopy every 5 years, and a colonoscopy every 10 years (CDC 2013b). We assume that patients enter the system at age 50 and wait a random time at home before their rst screen. We assume that the time at home between age 50 and the rst screen follows the same distribution as the time spent at home between any subsequent screens. Our base estimate for the average time (at home) until the next screen is scheduled is 520 weeks (10 years). While colonoscopy screening is recommended to begin at age 50, not all patients will immediately request a screen once turning age 50. We could adjust this parameter, by assuming that patients enter the model at age 40 or 45 so that a higher number have requested a screen by age 50.

• The death rate due to other causes, α = ₁₆₂₃1_.5284. From data collected from the Social Security Administration (SSA) 2009 period Actuarial Life Table we observe the remaining life expectancy at age 50 to be 29.35 years for males and 33.02 years for females (SSA - Oce of the Cheif Actuary 2013). Using US Census data, we calculate the population distribution between sexes to at age 50 to be approximately 51% female and 49% male (U. S. Census Bureau 2013). We use a weighted average on the two expected remaining

average remaining lifetime when entering the system is given by an exponential distribution with mean 31.2217 years. We solve for alpha by taking the inverse of this mean converted to weeks. Since the death rate due to colorectal cancer is relatively small, we do not adjust the life expectancy to be only deaths due to other causes; therefore we slightly over-estimate the death rate due to other causes.

• The rate of people who reach screening age, λ = 10.662patients per week. We observe data from the US Census Bureau on population distribution by age and we estimate a current rate of approximately 12,500 people in the US turn 50 every day (U. S. Census Bureau 2013). This amounts to 87,500 people per week. We divide this by an estimate of the number of clinics administering colonoscopies, 8207, to get the average number of patients per clinic who turn 50 per week (See et al. 2004).

• The cancer incidence rate, β = 0.00001924. From data collected from the Center for Disease Control and Prevention we observe that 52.7 men per 100,000 men and 39.7 per 100,000 women are diagnosed with colorectal cancer each year (CDC 2013a). We average the incidence rates to get the incidence rate for all people assuming the population is 51% female and 49% male to get an incidence rate of 46.070 per 100,000 people (U. S. Census Bureau 2013). Using the data on number of people who turn 50 each week we solve the following equation for β using the estimates below and the mortality rate calculated

above. Note that using this equation to estimateβ assumes a stable population size. An

alternative estimate from the National Cancer Institute of 102,480 new cases of colorectal cancer in 2013 or the estimate from the rst paragraph of the introduction could also be used on the right hand side of the equation and will provide a slightly higher estimate (currently the right hand side is estimated to be 138,210 new cases per year). Our estimates

forαandβ, imply that about3%of all people will develop colorectal cancer at some point in their lifetime (though it may go undetected in some who die from other causes before detection).

(T urn50P er Y ear)P(Cancer Af ter50) = (Incidence P er100000)(₁₀₀₀₀₀ U SP opulation) (T urn50P er Y ear) = (12500) (365) = 4,562,500

P(Cancer Af ter50) = _α+β_β (Incidence P er100000) = 47.070 (U SP opulation) = 300,000,000

Substituting and solving forβ we get the above parameter estimate.

(4,562,500) ₁ β

1623.5284 +β

!

= (47.070) (3,000,000) 100000

• The rate until cancer is detected via other methods θ = ₂₆₀1 . We found this parameter dicult to estimate, so we assume in the base case that the average time until cancer is detected via other methods once the person has cancer to be half of the established screening guideline.

• The cost of colonoscopy capacity, c= 3.081. We observe an estimate for the average cost of a colonoscopy to be $3,081 which is the average cost of a colonoscopy estimated by BCBSNC for an uninsured patient ((CostHelper.com 2013)). This cost does not include the cost associated with lost utility or lost productivity when the patient actually attends a screen. We write the costs in terms of thousands of dollars per week. Rosenthal (2013) states the average price in the US as $1,185, but this appears to be an out of pocket price,

• The cost of delay when cancer is detected via screening b1 = 0.3851. This parameter is

also dicult to estimate due to the slow growing nature of colorectal cancer and lack of available data. Luo et al. (2010) provide estimates of 1-year costs attributable to colon cancer by stage of diagnosis. They nd that patients diagnosed with cancer in local stage had the lowest costs ($27,551), followed by patients with distant stage ($29,933), and patients with regional cancer had the highest cost ($30,748). From these ndings we see that there is a high xed cost for treating cancer from any stage (even local) and the increase in costs by stage above does not account for other costs such as increased chance of death from cancer or disutility for the patient. We assume that 8 weeks of delay for a patient with cancer will cost as much in treatment as one colonoscopy.

• The cost of delay when cancer is detected other methods b2 = 0.7703. This parameter is

also dicult to estimate so we assume it to be twice the delay cost when cancer is detected via screening.

• The base capacity isµ= 33.327. We observe from See et al. (2004) an estimate of 14.2 million colonoscopies for 8,207 practices in 2002. This translates to an average screening rate of 33.327 colonoscopies per week per clinic. Assuming that utilization for colono- scopies is very high (as observed by high waiting time) we use this estimate as the estimate for capacity. However, the authors also nd that all physicians combined report that they could increase to 22.4 million colonoscopies within 1 year. This translates to a capacity of µ = 52.488. We can use the rst estimate as a low estimate for average weekly clinic capacity and the second estimate as a high estimate for average weekly clinic capacity.