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)