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- Re: Execute t test for Extremum of Parabola

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08-24-2017 08:22 AM

Hallow,

I run a quadric regression:

Y = β1 X + β2〖〖X 〗^2 + ε

The Extremum of Parabola is

(–β1 )/2β2

Both

β1 and β2

are significance, but as I understood it's not enough to determined that the Extremum of Parabolaֲ which equal toֲ

β1 and β2

is significant.

Is there aֲ SAS procedure that execute t test for Extremum of Parabolaֲ ?

Thank you,

Lior

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Solution

08-24-2017
01:57 PM

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08-24-2017 12:56 PM

Reparameterize your equation to make the extremum a parameter:

Y = B2*X*(X-2*Xe)

where Xe is the position of the extremum. Use proc nlin to estimate B2 and Xe. Proc nlin will give you a confidence interval for Xe.

PG

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08-24-2017 09:59 AM

but as I understood it's not enough to determined that the Extremum of Parabola

Why do you say this? Have you read this somewhere? Can you provide us with a reference?

Is there something about this specific problem that requires you to compute statistically significant differences between the extremum? Because the "usual" statistical modeling doesn't look at the extremum, it only tests to see if the and slopes are significant. (And by the way, your model doesn't have an intercept, why is that?)

I have thought about this and if you really want to test if the extremum is not some value, you'd probably have to work out the math yourself, or do something like a randomization test.

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08-24-2017 11:29 AM

Hi Paige Miller,

Thank you for your quick response.

I don't have reference for that requirement. i've been told to do so by the Professor who guide me in my PhD.

In my specific problem, the extremum of the parabola is very important (it is constitute a peak which I set in the second stage in other regression as a dummy variable).

if for example I receive that the peak of performance is in year 2 , I want to verify that it not in year 1 or 3.

the reason I don't have intercept is that I imply Fixed Effect (the "absorb" and "class" procedures).

Regarding the randomization test you have mention could it be done by SAS?

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08-24-2017 11:40 AM

So let me make sure I am 100% understanding this requirement.

You want the X value (not the Y value) at the extremum, and you want to test to see if this X value is not some other number (for example, it is not 3). Is that correct?

The hypothesis tests I am familiar all deal with testing some aspect of the fitted model in the Y direction. I am not even sure it is kosher to test a hypothesis in the X direction, but assuming this is okay, a randomization test would work. I think you'd have to find someone who has written a macro to do perform randomization tests in general, and then adapt it for the case of this extremum.

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08-24-2017 10:35 AM

Since a TTest compares a sample of values to a suspected mean or two samples for the same mean I am not sure where an extremum, which is a single point would be a candidate.

Perhaps you are looking to compare if a single value y is within confidence limits of the regression at the same x value??

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08-24-2017 11:16 AM - edited 08-24-2017 11:20 AM

ballardw wrote:

Perhaps you are looking to compare if a single value y is within confidence limits of the regression at the same x value??

Maybe he wants a test for H0: Extremum = k and then the confidence intervals might work; if the confidence interval around the regression at the extremum contains k then you accept H0; otherwise you reject.

But I still am not on board with the idea that it's not enough to just test intercept and slopes ... I'd like to hear the reasoning why a test of the extremum is needed.

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08-24-2017 11:58 AM

Hi,

I think you formulate it right:

H0: Extremum = k and then the confidence intervals might work; if the confidence interval around the regression at the extremumֲ contains k then you accept H0; otherwise you reject.

so I understand from you that it is doable but needed a macro programing?

I will consult again with my professor regarding the reasoning for that test.

Thanks.

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08-24-2017 11:42 AM

Hi Ballardw,

Actually its not have to be a t test , it can be any other statistical test which can confirm that the extremum of the parabula is significant.

I am not interested to compare if a single value yֲ is within confidence limits of the regression at the same x value but just to verfy that in this x value , the value of y is the highest.

thanks,

Lior

Solution

08-24-2017
01:57 PM

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08-24-2017 12:56 PM

Reparameterize your equation to make the extremum a parameter:

Y = B2*X*(X-2*Xe)

where Xe is the position of the extremum. Use proc nlin to estimate B2 and Xe. Proc nlin will give you a confidence interval for Xe.

PG

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## Re: Execute t test for Extremum of Parabola

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08-24-2017 01:55 PM

Thank you very much.

I will try that.

Regards,

Lior