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Testing the sum of regression coefficients

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Testing the sum of regression coefficients

Hector Maletta
My son has to test the hypothesis that the sum of the coefficients in a
linear regression equation equals 1. That would be the null hypothesis, I
suppose. The equation has two predictors, so in fact he wants to test the
null hypothesis that B1+B2=1. How you do this? I am extremely busy at the
moment so I am transmitting the query to this forum. Thanks in advance.



Hector

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Re: Testing the sum of regression coefficients

Marta García-Granero-2
Hi Hector:

> My son has to test the hypothesis that the sum of the coefficients in a
> linear regression equation equals 1. That would be the null hypothesis, I
> suppose. The equation has two predictors, so in fact he wants to test the
> null hypothesis that B1+B2=1. How you do this? I am extremely busy at the
> moment so I am transmitting the query to this forum.
(sorry, I'm cooking right now because I have people invited for dinner
tonight, I can't extend a lot on the topic).

Using the same logic I use when I want to test particular hypotheses
concerning beta coefficients in logistic regression, I guess the idea is
to compute a generalised t test in the form:

    (b1+b2+...+bk)-1
t=----------------------
      SE(sum(bi))

The SE is the tricky part. I think it must be something like:

SE=SQRT(SE^2(b1)+SE^2(b2)+...+SE^2(bk)-R12SE(b1)*SE(b2)-R12SE(b1)*SE(b3)....)

To get the correlation among coefficients, you need the following syntax
(notice the /OUTFILE=COVB(Coefficients) ):

DATASET DECLARE Coefficients.
REGRESSION
  /MISSING LISTWISE
  /STATISTICS COEFF OUTS R ANOVA ZPP
  /CRITERIA=PIN(.05) POUT(.10)
  /NOORIGIN
  /DEPENDENT salario
  /METHOD=ENTER salini tiempemp expprev
  /OUTFILE=COVB(Coefficients) .

You will get a SD&covariance matrix you need to turn into correlations
using MCONVERT.

I promise that tomorrow morning (CET) I will extend my skimpy
explanation a bit more. Also, I will take a look at my regression books.

Now, a mussaka awaits me.

Best regards,
Marta García-Granero

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Re: Testing the sum of regression coefficients

Mike Speed
Use GLM with the x's as the covariates ( there are no factors)
Setup and then hit paste.
Add the following:

\lmatrix = "b1 + b2" all 0 1 1.
\kmatrix = 1.


F. Michael Speed
Professor
Director of Online Learning
Department of Statistics
Associate Dean for Technology Mediated Instruction
College of Science
TAMU
 979-845-3182



-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of Marta García-Granero
Sent: Tuesday, November 27, 2007 12:22 PM
To: [hidden email]
Subject: Re: Testing the sum of regression coefficients

Hi Hector:

> My son has to test the hypothesis that the sum of the coefficients in a
> linear regression equation equals 1. That would be the null hypothesis, I
> suppose. The equation has two predictors, so in fact he wants to test the
> null hypothesis that B1+B2=1. How you do this? I am extremely busy at the
> moment so I am transmitting the query to this forum.
(sorry, I'm cooking right now because I have people invited for dinner
tonight, I can't extend a lot on the topic).

Using the same logic I use when I want to test particular hypotheses
concerning beta coefficients in logistic regression, I guess the idea is
to compute a generalised t test in the form:

    (b1+b2+...+bk)-1
t=----------------------
      SE(sum(bi))

The SE is the tricky part. I think it must be something like:

SE=SQRT(SE^2(b1)+SE^2(b2)+...+SE^2(bk)-R12SE(b1)*SE(b2)-R12SE(b1)*SE(b3)....)

To get the correlation among coefficients, you need the following syntax
(notice the /OUTFILE=COVB(Coefficients) ):

DATASET DECLARE Coefficients.
REGRESSION
  /MISSING LISTWISE
  /STATISTICS COEFF OUTS R ANOVA ZPP
  /CRITERIA=PIN(.05) POUT(.10)
  /NOORIGIN
  /DEPENDENT salario
  /METHOD=ENTER salini tiempemp expprev
  /OUTFILE=COVB(Coefficients) .

You will get a SD&covariance matrix you need to turn into correlations
using MCONVERT.

I promise that tomorrow morning (CET) I will extend my skimpy
explanation a bit more. Also, I will take a look at my regression books.

Now, a mussaka awaits me.

Best regards,
Marta García-Granero

=====================
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[hidden email] (not to SPSSX-L), with no body text except the
command. To leave the list, send the command
SIGNOFF SPSSX-L
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INFO REFCARD

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Re: Testing the sum of regression coefficients

Chen Wei-2
Hi Mike:

I have been sending you email (to your two email addresses) over the last
two weeks, but all failed to reach you. Can you please reply when you see
this email? Both your email address and mine are correct, but when I send
you email, I always get a mail delivery failure message.

Thanks.

Chen Wei

----- Original Message -----
From: "Mike Speed" <[hidden email]>
To: <[hidden email]>
Sent: Wednesday, November 28, 2007 2:30 AM
Subject: (瑞星提示-此邮件可能是垃圾邮件)Re: Testing the sum of regression
coefficients


> Use GLM with the x's as the covariates ( there are no factors)
> Setup and then hit paste.
> Add the following:
>
> \lmatrix = "b1 + b2" all 0 1 1.
> \kmatrix = 1.
>
>
> F. Michael Speed
> Professor
> Director of Online Learning
> Department of Statistics
> Associate Dean for Technology Mediated Instruction
> College of Science
> TAMU
> 979-845-3182
>
>
>
> -----Original Message-----
> From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of
> Marta García-Granero
> Sent: Tuesday, November 27, 2007 12:22 PM
> To: [hidden email]
> Subject: Re: Testing the sum of regression coefficients
>
> Hi Hector:
>
>> My son has to test the hypothesis that the sum of the coefficients in a
>> linear regression equation equals 1. That would be the null hypothesis, I
>> suppose. The equation has two predictors, so in fact he wants to test the
>> null hypothesis that B1+B2=1. How you do this? I am extremely busy at the
>> moment so I am transmitting the query to this forum.
> (sorry, I'm cooking right now because I have people invited for dinner
> tonight, I can't extend a lot on the topic).
>
> Using the same logic I use when I want to test particular hypotheses
> concerning beta coefficients in logistic regression, I guess the idea is
> to compute a generalised t test in the form:
>
>    (b1+b2+...+bk)-1
> t=----------------------
>      SE(sum(bi))
>
> The SE is the tricky part. I think it must be something like:
>
> SE=SQRT(SE^2(b1)+SE^2(b2)+...+SE^2(bk)-R12SE(b1)*SE(b2)-R12SE(b1)*SE(b3)....)
>
> To get the correlation among coefficients, you need the following syntax
> (notice the /OUTFILE=COVB(Coefficients) ):
>
> DATASET DECLARE Coefficients.
> REGRESSION
>  /MISSING LISTWISE
>  /STATISTICS COEFF OUTS R ANOVA ZPP
>  /CRITERIA=PIN(.05) POUT(.10)
>  /NOORIGIN
>  /DEPENDENT salario
>  /METHOD=ENTER salini tiempemp expprev
>  /OUTFILE=COVB(Coefficients) .
>
> You will get a SD&covariance matrix you need to turn into correlations
> using MCONVERT.
>
> I promise that tomorrow morning (CET) I will extend my skimpy
> explanation a bit more. Also, I will take a look at my regression books.
>
> Now, a mussaka awaits me.
>
> Best regards,
> Marta García-Granero
>
> =====================
> To manage your subscription to SPSSX-L, send a message to
> [hidden email] (not to SPSSX-L), with no body text except the
> command. To leave the list, send the command
> SIGNOFF SPSSX-L
> For a list of commands to manage subscriptions, send the command
> INFO REFCARD
>
> =====================
> To manage your subscription to SPSSX-L, send a message to
> [hidden email] (not to SPSSX-L), with no body text except the
> command. To leave the list, send the command
> SIGNOFF SPSSX-L
> For a list of commands to manage subscriptions, send the command
> INFO REFCARD

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Re: Testing the sum of regression coefficients

Richard Ristow
In reply to this post by Hector Maletta
At 11:42 AM 11/27/2007, Hector Maletta wrote:

>My son has to test the hypothesis that the sum of the coefficients
>in a linear regression equation equals 1: the equation has two
>predictors, so he wants to test the null hypothesis that B1+B2=1.

You've had a couple of solutions, so take this as an exercise on my
part: solving the problem with REGRESSION, by re-framing it.

The model is

Y = B1*X1 + B2*X2 + e.

Let
D = 1 - (B1 + B2)
and you want to test the null hypothesis D=0. So, make D a
coefficient of the model:

Transform the above equation, and get
B2 = D - 1 + B1.
which gives the re-framed model
Y = B1*X1 + (D - 1 + B1)*X2 + e.
or
y = B1*X1 + D*X2 -1*X2 + B1*X2 + e.
or
(Y-X2) = B1*(X1+X2) + D*X2 + e.

So, transforming
Yd = Y-X2
Xs = X+X2
and estimating the model

Yd = B1*Xs + D*x2 + e.

should do it.

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