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ParameterCITable vs. ParameterConfidenceRegion: rectangular vs.

  • To: mathgroup at smc.vnet.net
  • Subject: [mg89897] ParameterCITable vs. ParameterConfidenceRegion: rectangular vs.
  • From: andreas.kohlmajer at gmx.de
  • Date: Tue, 24 Jun 2008 03:26:53 -0400 (EDT)

Hi!

I have some difficulties interpretating the difference between
ParameterCITable and ParameterConfidenceRegion. This is my data:

(* Mathematica 6.0 *)
Needs["LinearRegression`"];
f = {#, 0.1 # + 0.0005 #^2 + RandomReal[NormalDistribution[0, 5]]} &;
(* parabola *)
SeedRandom[123];
data = Table[f[n], {n, 0, 500, 10}]; (* equally distributed from 0 to
500 *)

Regress gives me the CI table and the confidence region:

fit = Regress[data, {x, x^2}, x, IncludeConstant -> False,
   RegressionReport -> {SummaryReport, ParameterConfidenceRegion,
     ParameterCITable}];

I can plot the rectangular confidence interval and the ellipsoid
together:

Show[Graphics[{Red, Opacity[0.25],
   Rectangle[##] & @@ ((ParameterCITable /. fit)[[1, All, 3]] //
      Transpose)}], Graphics[{ParameterConfidenceRegion /. fit}],
 AspectRatio -> 1]

For further calculations, I need the ratio of the x^2 to x-
coefficient. If I use the rectangular region together with interval
arithmetic, I get a huge range for the ratio:

(Last[#]/First[#] &)@(Interval[#] & /@ (ParameterCITable /. fit)[[1,
     All, 3]])

If I use the ellipsoid instead, I get a different result:

({Min[#],
    Max[#]} &)@((Last[#]/
      First[#] &) /@ (Graphics[{ParameterConfidenceRegion /. fit}][[1,
       1, 1, 3, 2, 1]])) (* get min-max range from graphics object *)

Which confidence region is correct? Are both regions statistically
correct? Which region is statistically more precise? Which region
should be used for further computation?

(keywords: ParameterCITable, ParameterConfidenceRegion, Interval
Arithmetic)



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