Scaling Coefficients for Linear Case

Determine which level of the ith attribute in

(x10 ,..., xi-10 , xi , xi+10 ,..., xn0 )

is equivalent to x{ k }

therefore

liv(xi) = lk

where v(x{ k }) < v(x{ i })

Solve with n simultaneous equations for n unkknowns

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Notes:

Note that this does not require a specific health state for assessment.

Done in the familiar terms of the patient.

Grants opportunity to demonstrate proper understanding of the terminology.

Tests the attribute scales’ operational quality early in the process, both individually and in combination.

Need to give an example (illustrated with patient again)

To illustrate the method, imagine the oversimplification where a patient measures health by only three two level attribute scales: exercise (some or none), dietary restrictions (none or any), and presence of any disease (true or false). The patient is asked to imagine they will have a life expectancy of 10 years as an individual who’s state is best described as S1={renal disease, protein restricted diet, and no exercise }. Further they are asked if they would trade that state of health for one with 5 years life expectancy described as S2={no disease, protein restricted diet, and exercise }. For illustration, assume it is determined that the patient is ambivalent about the choice between 10 years in the first (S1) and 7 years in the second (S2) state of health. The patient is also asked similarly for the number of years they would trade for in a third state S3={no disease, no dietary restrictions, and no exercise}; let’s say it is 9 years. With the values known for the levels of each attribute (1.0 or 0.0 in this case) only the scaling coefficients of the linear model are unknown. Using the method of simultaneous equations we can solve for each scaling coefficient knowing that the sum of the coefficients themselves must sum to one.

Utility(S1) = 1Attribute1 + l2Attribute2 + l3Attribute3 = l1(0.0) + l2(0.0) + l3(1.0)

Utility(S2) = l1Attribute1 + l2Attribute2 + l3Attribute3 = l1(1.0) + l2(0.0) + l3(1.0)

Utility(S3) = l1Attribute1 + l2Attribute2 + l3Attribute3 = l1(0.0) + l2(1.0) + l3(0.0)

and

Utility(S1) 10yr = Utility(S2) 7yr = Utility(S3) 9yr

therefore

[l1(0.0) + l2(0.0) + l3(1.0)]10yr = [l1(1.0) + l2(0.0) + l3(1.0)]7yr = [l1(0.0) + l2(1.0) + l3(0.0)]9yr

means

l1= 0.587, l2= 0.217, and l3= 0.196.

This means that the expected utility for this patient for any given health state is given by the formula:

Utility (given state of health) = 0.587 Disease Attribute Level + 0.217 Dietary Attribute Level + 0.196 Exercise Attribute Level