[6] ICE Willi-Warp

See **O'Brien BJ, Gertsen K,
Willan AR, Faulkner LA.**** [Is there a kink in consumers' threshold value for
cost-effectiveness in health care? Health Econ 2002; 11(2): 175-180]**
for background information and a discussion of

WTP < WTA

This key observation states that consumers' **Willingness-to-Pay** (more
to get more benefit)** **is always less than their** Willingness-to-Accept** (less
benefit to save money.) The earlier work on "Net Benefit" by
**Stinnett AA, and Mullahy J. [Net health
benefits: a new framework for the analysis of uncertainty in cost-effectiveness
analysis. Med Dec Making,
Special Issue on Pharmacoeconomics 1998; 18: S68-S80]
**is thus clearly rather unrealistic. In fact, Net Benefit uses the

See equation (5)
of **Obenchain RL. [ICE Preference Maps: Nonlinear Generalizations of Net Benefit and
Acceptability. Health Services and Outcomes Research Methodology
2008; 8: 31-56]** for
Bob O's

**Launch Willi-Warp Animation: **Watch WTP and
WTA change with Lambda = One held fixed.

This LINK generalizes all earlier work in the sense that it allows for cases where WTP < WTA or WTP = WTA and even (unrealistic) cases where WTP > WTA. In fact, equation (4) of Obenchain(2008) states that:

where *w*(x, y) is a "standardized" willingness of
**WTP / Lambda**
within the NE quadrant or
**WTA / Lambda**
within the SW quadrant, where** s = y/x**
is the standardized ICE ratio and where
**Eta =
Gamma / Beta**
is the map “power-parameter ratio.” This relationship assumes that
__
w(x, y) is the derivative (SLOPE) of
the curved, constant ICE preference contour @ (x, y) that passes through this point__
and, as a result, can be represented graphically as follows:

**ICE Diagonal Symmetry**

Note that the absolute ICE angle, +theta, above is 45 degrees + ArcTan(s). Thus, as s increases from 0 to infinity

(an infinite range) within the North-East ICE quadrant, theta increases only from +45 degrees to +135 degrees.

**The above graphic displays a pair of standardized
“dual rays” that [i] contain the same
distributions of ICE preferences (varying as a function of ICE radius), [ii] correspond to equal absolute ICE polar
angles relative to the upper-left to lower-right (x =
-y)
diagonal, and [iii] have both standardized slopes (s =
y/x) and standardized willingnesses (w =
WTP / Lambda
or WTA / Lambda) at identical radii that are numerical
reciprocals. The case depicted here corresponds to 0 < s < w < 1 < 1/w
< 1/s because the "power parameter ratio" (Eta = Gamma / Beta) of the
corresponding ICE preference map is greater than one (realistic.)**

Equation (1) of Obenchain(2008) introduces the corresponding 2-parameter family of potentially
**"nonlinear" ICE preference maps** as
being of the general form:

where the positive Beta "ICE radius" power determines **
Returns-to-Scale** and the positive Gamma "signed ICE angle" power determines
map "directionality." Specifically, when **Gamma < Beta**, the map tends to be somewhat "roundish"
and quite unrealistic in the sense that **WTA** < **WTP** below the **
x = y (lower-left to upper-right) diagonal (ICE angle theta = +90 or -90
degrees)**:

On the other hand, when **Beta < Gamma**, the resulting map tends to be "directional" and realistic in the sense that
**WTP** < **WTA **below the **x = y (lower-left to upper-right)
diagonal**:

The following is an extreme map where (within the South-East ICE
quadrant; ICE angle theta between +45 and -45 degrees) **WTP** can be as low as zero and
**WTA** can be as high as infinity!
This is a so-called **"ICE Omega" map** where the power parameter ratio (Eta =
Gamma
/ Beta) assumes the maximum possible value (5.828...) consistent with the **ICE monotonicity axiom**.

In summary then,
Bob O's **"LINK"** function, which
states that **Lambda** __must be the geometric
mean of all well-matched WTP and WTA pairings__, shows how (in
the

In turn, the concept that
**Lambda** can be __meaningfully held fixed__ is the KEY to
separating **exogenous and contradictory economic
uncertainty** about **Lambda** from the
**endogenous statistical uncertainty** in
patient-level data about a specific treatment comparison (**T** vs. **S**.)
In other words,
Bob O's **"LINK"** function provides
the theoretical basis for OBJECTIVELY determining **
Lambda** for a given disease state or medical condition by __eliciting
well-matched pairs of WTP and WTA numerical values from patients
(and their trained care givers) ...using any relevant disease-specific
units of effectiveness__. (The awkward and vague concepts of a QALY and
of how to re-express effectiveness outcomes in QALYs then become outmoded!)