Highlights of Chapter 7

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(x i) = 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

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 valuesNon-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)

Markov modelsApplied to Cancer Screening

FOBT

No Test

Test No Disease

Disease

Disease

No Disease

P1

1-P1

P2

1- P2

P3

1- P3

• 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.

• 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

Assumptions of the Model

Well1000

Disease0

Dead0

Well750

Disease150

Dead 150

Well525

105 145

.7

.7

.7

.7

.7

.15

.15

.15

.15

.3

.3

1000 ENTER THE MODEL

T

T + 1

T + 2

1

1

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

Modelling transitions over time

Negative(return to FOBT in 2 years)

FOBT

Positive

No FOBT

Stage 0

Stage 1

Stage 2

Stage 3

Stage 4

Stage 3

Stage 4

Return to“No FOBT”

Less complex (costly) treatment

More complex (costly) treatment

Very complex (and costly) treatment

More complex (costly) treatment

Very complex (and costly) treatment

Modelling transitions over time

Each branch has a P value

Dead (due to other causes)

Never screen

Continue with normal activities

Symptoms: Stage 3 cancer

Symptoms: Stage 4 cancer

Re-route to Never screen

Incur Stage 3 costs

Incur Stage 4 costs

Dead

Screeners: no FOBT in current year – continue with normal activities

Re-route to Screeners: no FOBT in current year

Screeners: undergo FOBT in current year

Negative test results

Positive test results(follow-up with colonoscopy)

Diagnosed: Stage 0 cancerIncur Stage 0 costs

Diagnosed: Stage 1 cancer Incur Stage 1 costs

Diagnosed: Stage 2 cancerIncur Stage 2 costs

Diagnosed: Stage 3 cancerIncur Stage 3 costs

Diagnosed: Stage 4 cancerIncur Stage 4 costs

The Full FOBT Model

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

The Markov model relies on three main sets of probabilities to direct cases through an appropriate sequence of interventions and costs:

1.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.

2.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.

3.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

Probabilities

• 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.

Colorectal cancer stages

• 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.

Colorectal cancer treatment costs

Table C1: Estimated intervention costs for the colorectal cancer model Intervention Costs Notes

FOBT & processing* $54.23 Includes costs associated with visits to physician, the FOBT kit, and processing. These include only the estimated billing by the physician, the cost of the kit, the actual lab test, and subsequent reporting. They do not include the opportunity cost of the time spent by the patient at the office, completion of the test protocol at home, or return of the completed test to the lab.

Stage 0 diagnosis* $620.70 Includes costs associated with positive FOBT consultation, colonoscopy, and polypectomy.

Stage 1 diagnosis** $29,459.58

These represent average lifetime costs per patient, for both colon and rectal cancer. Includes costs associated with initial treatment, well-patient follow-up, local recurrence treatment & active care, metastases treatment & active care, and terminal care. Costs were initially in 1998 Cdn $, and have been adjusted to 2009 values.

Stage 2 diagnosis** $35,889.39

Stage 3 diagnosis** $42,697.42

Stage 4 diagnosis** $44,832.48

Note: *Source: Flanagan, W. (2002). Modelling colorectal cancer screening in POHEM. Technical Report for the National Committee on Colorectal Cancer Screening.

**Source: Maroun, J. (2003). Lifetime costs of colon and rectal cancer management in Canada. Chronic Diseases in Canada, 24(4).

Colorectal cancer treatment costs

Table 1: Varying the intervention probabilities for the screened and non screened populations

Non-

screened Scenario 1 Scenario 2 Scenario 3

Regular two-year screening 42% 50% 67% 80% Undergo FOBT screening in current year

21% 25% 33.5% 40%

Undergo FOBT screening next year 21% 25% 33.5% 40% Never screen 58% 50% 33% 20%

Expected cost Incremental savings from FOBT Non-screened $21,000.54 Screened Scenario 1 (50/50) $20,709.73 $290.81 Screened Scenario 2 (67/33) $20,091.76 $908.78 Screened Scenario 3 (80/20) $19,619.20 $1,381.34 Note: The expected cost and savings are calculated over a five-year period. Costs are approximately +/- $84.

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.

The value of the FOBT

Markov modelling applied to vaccination

This is a two cycle (year) vaccination model

Ca : administrative costsCs : adverse side effectsCe|v : cost if epidemic occurs given vaccinationCe|nv : cost if epidemic occurs and no vaccination programP1 : epidemic occurs in year 11 – P1 : epidemic does not occur in year 1P2 : epidemic occurs in year 21-P2: epidemic does not occur in year 2