**[3] ICE Angle Serendipity**

**The QUESTION addressed here:
Technically, the definition of the Bootstrap Confidence
Wedge seems to use only ICE Angle Order Statistics and, thus, appears to totally
ignore ICE Radius.
Doesn't that mean the observed ICE Radius
is rather "unimportant?" **

**Figure One: Visualization using ICE Polar Coordinates**

**Here we see a Boostrap Distribution of ICE Uncertainty
and the ICE Radius and ICE
Angle of its "center point" ...i.e. the point representing the observed pair of
Treatment Differences (New minus Standard) in
Effectiveness (horizontal) and
Cost (vertical.)**

**Figure Two: Definition of the 95% Confidence Wedge.**

**Now, we superimpose the pair of ICE Rays defined
by "Counting Outwards" from the observed ICE Ratio (Slope) by a specified number
of ICE Angle Order Statistics (e.g. 11,875 out of
25,000.)**

**Figure Three: What Would Happen if the
ICE Radius were to Suddenly Increase (with
Everything Else Held Fixed)?**

**Figure Four: And the Answer is that the 95%
ICE Angle Confidence
Wedge would then Automatically Narrow as shown
here!**

###
**In other words, t**he ICE
Radius of the
observed ICE Outcome Difference Pair (New
minus Standard) is primarily a measure of overall
ICE PRECISION (Inverse
Uncertainty.)

**Basing your Confidence Wedge solely upon ICE Angle Order
Statistics really Gets you
Something for Nothing!
While the calculations may appear to ignore ICE Radius
they are actually quite sensitive to it ...in the
sense that they would automatically adjust for any changes in it.**

**PS.
Another key insight into the relative importance of ICE
Angles and ICE Radii is provided by the
ICE Preference Mapping work of
Obenchain(2008.) What do you see in the "highly
realistic"
NONLINEAR
map with
Beta < Gamma displayed below? This map tends to be**** "directional"
with the highly desirable property that WTP < WTA below the x = y (lower-left to upper-right)
diagonal where New is preferred over the Standard Treatment...**

### What I see here is that**
ICE Radius is clearly much
more important than ICE Angle within the
South-East and North-West ICE Quadrants. See **
Obenchain and Sacristan
(1997) for more on this point.

**Note that the "4-step
graphical argument"
given earlier on this page used a North-East Quadrant example ...where
ICE Radius is actually somewhat less important than
ICE Angle!**

**ICE Radius
is totally unimportant only in the (unrealistic) limit as the Beta "power
parameter" decreases to ZERO. There are then NO
Returns-to-Scale corresponding to changes in ICE
Radius, the ICE Cartesian Monotonicity Axiom is violated, and Preferences
vary only with ICE Angle ...as in the
**
Laupacis, Feeny, Detsky and Tugwell (1992) "Pie-Chart" Map!