Re: ParameterCITable vs. ParameterConfidenceRegion:

• To: mathgroup at smc.vnet.net
• Subject: [mg89959] Re: [mg89897] ParameterCITable vs. ParameterConfidenceRegion:
• From: DrMajorBob <drmajorbob at att.net>
• Date: Wed, 25 Jun 2008 06:27:11 -0400 (EDT)
• References: <868851.1214299401133.JavaMail.root@m08>
• Reply-to: drmajorbob at longhorns.com

```> 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]])

Interval[{0.0046359, 0.00870396}]

or equivalently

1/Divide @@ Interval /@ (ParameterCITable /. fit)[[1, All, 3]]

Interval[{0.0046359, 0.00870396}]

I'm not sure that's a "huge" range, but anyway... here's the other result:

({Min[#],
Max[#]} &)@((Last[#]/
First[#] &) /@ (Graphics[{ParameterConfidenceRegion /. fit}][[1,
1, 1, 3, 2, 1]]))

{0.00432882, 0.00955912}

or equivalently

Through[{Min, Max}@
Cases[Cases[Graphics[{ParameterConfidenceRegion /. fit}],
Line[pts_] :> pts, Infinity], {x_Real, y_Real} :> y/x, Infinity]]

{0.00432882, 0.00955912}

The two results don't look all that different to me, but that difference
is, explicitly, the difference between viewing the CR as an ellipse (first
method) vs viewing it as a rectangle (second method). The first takes
account of a relationship between the two parameters, while the second
treats them as independent, and don't forget, the elliptical version uses
a SAMPLE of points adaptively chosen by the Graphics function for viewing
purposes.

(I'm frankly shocked that the two results were so nearly identical.)

Neither set of ratios, IMHO, has any simple statistical interpretation,
but the elliptical CR is obviously more descriptive than the rectangular
one.

Bobby

On Tue, 24 Jun 2008 02:26:53 -0500, <andreas.kohlmajer at gmx.de> wrote:

> 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)
>
>
>

--
DrMajorBob at longhorns.com

```

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