## Execute t test for Extremum of Parabola

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# Execute t test for Extremum of Parabola

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
Posts: 5,129

## Re: Execute t test for Extremum of Parabola

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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Posts: 2,173

## Re: Execute t test for Extremum of Parabola

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.

--
Paige Miller
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Posts: 14

## Re: Execute t test for Extremum of Parabola

Posted in reply to PaigeMiller

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?

Posts: 2,173

## Re: Execute t test for Extremum of Parabola

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.

--
Paige Miller
Super User
Posts: 12,148

## Re: Execute t test for Extremum of Parabola

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??

Posts: 2,173

## Re: Execute t test for Extremum of Parabola

[ Edited ]

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.

--
Paige Miller
Occasional Contributor
Posts: 14

## Re: Execute t test for Extremum of Parabola

Posted in reply to PaigeMiller

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.

Occasional Contributor
Posts: 14

## Re: Execute t test for Extremum of Parabola

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
Posts: 5,129

## Re: Execute t test for Extremum of Parabola

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
Highlighted
Occasional Contributor
Posts: 14

## Re: Execute t test for Extremum of Parabola

Thank you very much.

I will try that.

Regards,

Lior

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