       Re: statistics questions

• To: mathgroup at smc.vnet.net
• Subject: [mg61891] Re: statistics questions
• Date: Thu, 3 Nov 2005 04:59:09 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```On 11/2/05 at 4:09 AM, chris.chiasson at gmail.com (Chris Chiasson)
wrote:

<pseudo code snipped>

>I would assume there is a NonlinearRegress option to directly
>obtain the confidence interval/margin of error for the mean
>response. Does anyone know what that is?

It is unclear to me what you are defining as mean response in this context. Normally, I would take mean response to be the mean of the observed response variable (y). But this is computed as simply Mean[y] and the confidence interval is estimated by knowing somethimg about the distribution of y. Of course, since this is really independent of NonlinearRegress this must not be what you have in mind.

>Why do the Mean and Single PredictionCITable results (use
>RegressionReport -> {MeanPredictionCITable,
>SinglePredictionCITable, SummaryReport} ) have variable height
>confidence intervals (easily seen by subtracting lower from upper)?

Because they are two different things with different uncertainties associated with each.

With MeanPredictedCITable you are computing the confidence interval about the average response for a given x value. With SinglePredictCITable you are computing the confidence interval for the observed response. This has an additional source of variabiliy since the observed response varies from the mean. That is in units where the variance is unity, the variance for the observed response will be 1 + the variance for the mean response at a given set of conditions (specific values of the independent parameters).

Notice that when

fit = NonlinearRegress[data, ... ] or
fit = Regress[data, ... ]

doing

First[SinglePredictionCITable /. fit][[All,3]]^2 -
First[MeanPredictionCITable /. fit][[All,3]]^2

results in a list whose elements are all equal to the esitmated variance.

And back to your earlier question. If you meant the mean response for a given set of conditions then the confidence interval you want is given by MeanPredictCITable.

>Why is the input syntax inconsistent from NonlinearRegress to
>Regress? I have also noticed that NonlinearRegress takes the exact
>same parameters as FindFit, but they take their last two arguments
>in reverse order.

I've no idea.

>What is the point of NonlinearFit if Mathematica has FindFit?

None now. But remember the package with NonlinearFit existed before FindFit was added to the kernel. So, this is simply a holdover from earlier versions which likely will disappear in some future version of the package.

>I have noticed that FindFit and NonlinearRegress both return fits
>with significance arithmetic intact, while Regress does not.

I am not really sure why this difference exists. I can hazard a guess. For the linear regression case, variance in the response variable will certainly dominate any lack of precision due to arithmetic. However, for the general non-linear case I don't see how the same can be guaranteed.
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