Highlights of Chapter 7

1. Highlights of Chapter 7

2. Mathematical expectation • If f(x) is the probability function of the random variable X, then ∑u(x) f(x) =∑u(x) f(x) is the mathematical expectation or the expected value of the function u(X) E[u(X) ] =∑u(xi) f(xi) • E[u(x) ] is a weighted mean of u(x) , where the weights are the discrete probabilities f(xi) = P(x)| x=xi). Let X be the random variable defined by the outcome of the cast of the die. Thus f(xi) = 1/6 for all possible outcomes (x=1,2,3,4,5,6). Assume that one attaches a utility |1,x=1,2,3 u(x) = |10,x=4 |40,x=6 E[u(x)]= ∑u(x) f(x) =1(1/6) +1(1/6) +1(1/6 ) +10(1/6 ) +10(1/6 ) +60(1/6) ) =1(3/6 ) +10(2/6 ) +40(1/6 ) = 10.5

3. Contingencies • Straight-line contingencies • At high levels of rainfall storage, net benefits fall to 0 • At no rainfall, the net benefits reach maximum, reflecting the maximum net return to agriculture – this is the loss of agricultural output due to drought. • One would assign a probability to two or more outcomes and use simple mathematical expectation to estimate expected values • Non-linear contingencies • The relationship between value and rainfall need not be linear • If B were the correct relationship, the net benefits of the storage facility is much lower. • Estimates of the probabilities for B come from weather records, or expert opinion. • One can assign probabilities along B (P values)

4. Markov models Applied to Cancer Screening • Markov processes are a particularly useful form for modelling contingencies. • A decision tree analysis offers a convenient representation of how an individual might transition among these states. • Figure 1 illustrates a simple decision tree associated with a screening test such as the FOBT (Fecal Occult Blood Test). • The first decision is whether to administer/participate in the test—this is governed by a simple binary probability of P1 and 1 – P1. • The outcome in either case is the same, but with possibly different probabilities (P values). • P values reflect genetic predispositions, policy, environmental hazards and behaviour. FOBT is a common screen for those over 50. It detects traces of blood in stool samples. A positive test can mean cancer or some other bowel condition and a colonoscopy is the usual follow-up test. A negative test means nothing will be done (assuming the patient is asymptomatic) until the next screen. At some age, some physicians will recommend going to the colonoscopy directly. There are few risks associated with FOBT, but a colonoscopy does carry some risk and a cost to the patient (possible injury/disease and time.

5. Assumptions of the Model • Estimating the probabilities is essential for a decision model-based health policy analysis. • These probabilities come from the expert opinion, the literature that comprises evidence from many jurisdictions, and estimates based on empirical analysis of data relevant to the population in question. • Time is absent from the model. • In reality, those who test positive with the FOBT would proceed to further confirmative testing such as colonoscopy. Those who are not tested and who have cancer would likely not discover this until some years later, when symptoms had emerged. • The standard of care for the FOBT is an annual FOBT, which means that tests are repeated on the same population and a constant fraction are found to have disease in each testing cycle • No provision has been made for false positives (which would likely be verified in the follow-up colonoscopy) and false negatives (triggering unneeded tests) • Another limitation to a decision tree model alone is that it maps the outcome for a single individual. • Finally, the model includes no treatment profile or what may be termed the clinical pathway. • Those diagnosed using the FOBT should be placed into a clinical process that is less costly than those who remain untreated and then enter a more complex treatment process. This concept represents the core assumption of primary care and needs to be empirically verified using real world data

6. Modelling transitions over time The full Markov model comprises a series of states and transitions among states over time. This is an illustration of a disease progression—adding a diagnostic (screening) step (at T – 1) represents an enhancement to the model. Monte Carlo simulation applies the Markov model repeatedly using a distribution (or several distributions) on the P values, representing behaviours and attributes of the population and shown in the square boxes

7. Modelling transitions over time Each branch has a P value

8. The Full FOBT Model There are four outcomes at each cycle (typically annual): screen now, screen next year, never screen, and die.

9. Probabilities • The Markov model relies on three main sets of probabilities to direct cases through an appropriate sequence of interventions and costs: • The proportion of the population that has an FOBT: • The goal of this exercise is to test the idea that the PIN initiative does indeed reduced net costs as a result of identifying and managing disease at early stages. • The proportion of FOBTs that result in positive versus negative results: • The chance of obtaining a positive FOBT result that later turns out to be false (no cancer is actually present) is relatively high, the model does not account for these. The relatively high number of false positives mainly results from lack of compliance with pre-test instructions. In such cases the individual is asked to retake the test. • The proportion of those who undergo colonoscopy that are diagnosed with each stage of colorectal cancer: • In reality, the course and treatment of the cancers, once diagnosed, will be unique to each individual. By focusing on the detection of the cancer and not the management once diagnosed, the model looks at the population in aggregate

10. Colorectal cancer stages • Colorectal cancer – the five stages: • Stage 0: the lining of the colon/rectum is affected. This stage is usually treated with a polypectomy during the colonoscopy itself. • Stage 1: the cancer has spread to the middle layers of the colon/rectum wall. Resection surgery will normally be performed, followed by monitoring. • Stage 2: the cancer affects nearby tissue. Treatment will involve resection surgery and chemotherapy. • Stage 3: nearby lymph nodes are affected. Again, treatment will include resection surgery, chemotherapy, and radiation (if the cancer is found in the rectum). • Stage 4: the cancer has spread to other organs in the body. Treatment will involve resection surgery, chemotherapy, and radiation. • Once classified with a stage of cancer, the individual is assigned an average lifetime cost based on the stage of cancer, and is then “absorbed” by the model (i.e., they have made their contribution to the model and are removed from the active process). • The model does not try to account for every unique course of action to deal with the diagnosed cancer (or the recurrence of cancer), but looks at the population in aggregate. • Those individuals not scheduled to undergo an FOBT in the current year are assumed to continue with their normal activities, only being exposed to the possibility of death due to other causes within that one year before returning the following years for an FOBT.

11. Colorectal cancer treatment costs • A fixed cost for the FOBT and its processing is assigned at the screening stage each time the individual undergoes the test. • An average lifetime cost associated with the stage of cancer is assigned once the individual has been diagnosed. and reflects costs associated with initial diagnosis, treatment, and follow-up of the cancer, as well as costs associated with recurrence and terminal care.. • It is possible that an individual with stage 4 cancer may incur similar or even lower costs than someone with stage 3 cancer if they decline treatment due to the inevitable fact of non-recovery, or may require services for a shorter period of time due to earlier death. • The costs are associated with each stage of colorectal cancer and do not differ depending on whether an individual is in the screening or non-screening population. • The stage 3 and 4 lifetime costs are the same for both populations; as with the probabilities, the cost savings that come from increased screening and early detection are manifested in the fact that the screened population is likely to require only the less complex (and less costly) treatments.

12. Colorectal cancer treatment costs

13. The value of the FOBT Caution: This is a feasibility study based on assumed data. The parameters have not been validated using case-level information and therefore, the estimates are subject to error and should not be taken as final.

14. Markov modelling applied to vaccination This is a two cycle (year) vaccination model Ca : administrative costs Cs : adverse side effects Ce|v: cost if epidemic occurs given vaccination Ce|nv : cost if epidemic occurs and no vaccination program P1: epidemic occurs in year 1 1 – P1: epidemic does not occur in year 1 P2: epidemic occurs in year 2 1-P2: epidemic does not occur in year 2