# Re: [R] comparing two regression models with different dependent variable

From: Joris Meys <jorismeys_at_gmail.com>
Date: Thu, 10 Jun 2010 11:33:30 +0200

This is only valid in case your X matrix is exactly the same, thus when you have an experiment with multiple response variables (i.e. paired response data). When the data for both models come from a different experiment, it ends here.

You also assume that y1 and y2 are measured in the same scale, and can be substracted. If you take two models, one with response Y in meters and one with response Y in centimeters, all others equal, your method will find the models "significantly different" whereas they are exactly the same except for a scaling parameter. If we're talking two different responses, the substraction of both responses doesn't even make sense.

The hypothesis you test is whether there is a significant relation between your predictors and the difference of the "reward" response and the "punishment" response. If that is the hypothesis of interest, the difference can be interpreted in a sensible way, AND both the reward learning curve and the punishment learning curve are measured simultaneously for every participant in the study, you can intrinsically compare both models by modelling the difference of the response variable.

As this is not the case (learning curves from punishment and reward can never be made up simultaneously), your approach is invalid.

Cheers
Joris

On Thu, Jun 10, 2010 at 9:00 AM, Gabor Grothendieck <ggrothendieck_at_gmail.com> wrote:
> We need to define what it means for these models to be the same or
> different.  With the usual lm assumptions suppose for i=1, 2 (the two
> models) that:
>
> y1 = a1 + X b1 + error1
> y2 = a2 + X b2 + error2
>
> which implies the following which also satisfies the usual lm assumptions:
>
> y1-y2 = (a1-a2) + X(b1-b2) + error
>
> Here X is a matrix, a1 and a2 are scalars and all other elements are
> vectors.  We say the models are the "same" if b1=b2 (but allow the
> intercepts to differ even if the models are the "same").
>
> If y1 and y2 are as in the built in anscombe data frame and x3 and x4
> are the x variables, i.e. columns of X, then:
>
>> fm1 <- lm(y1 - y2 ~ x3 + x4, anscombe)
>> # this model reduces to the following if b1 = b2
>> fm0 <- lm(y1 - y2 ~ 1, anscombe)
>> anova(fm0, fm1)
> Analysis of Variance Table
>
> Model 1: y1 - y2 ~ 1
> Model 2: y1 - y2 ~ x3 + x4
>  Res.Df    RSS Df Sum of Sq      F Pr(>F)
> 1     10 20.637
> 2      8 18.662  2    1.9751 0.4233 0.6687
>
> so we cannot reject the hypothesis that the models are the "same".
>
>
> On Wed, Jun 9, 2010 at 11:19 AM, Or Duek <orduek_at_gmail.com> wrote:
>> Hi,
>> I would like to compare to regression models - each model has a different
>> dependent variable.
>> The first model uses a number that represents the learning curve for reward.
>> The second model uses a number that represents the learning curve from
>> punishment stimuli.
>> The first model is significant and the second isn't.
>> I want to compare those two models and show that they are significantly
>> different.
>> How can I do that?
>> Thank you.
>>
>>        [[alternative HTML version deleted]]
>>
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>>
>
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```--
Joris Meys
Statistical consultant

Ghent University
Faculty of Bioscience Engineering
Department of Applied mathematics, biometrics and process control

tel : +32 9 264 59 87
Joris.Meys_at_Ugent.be
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