prior specification in MCMCglmm

There's a mistake in my email below.
The generalization of an inverse-gamma prior for multi-response models
should, I think, just be:
list(V=diag(x), nu=1.002)

Sorry about that,
Ned




-----Original Message-----
From: Ned Dochtermann [mailto:ned.dochtermann at gmail.com]
Sent: Tuesday, April 26, 2011 1:12 PM
To: 'r-sig-mixed-models at r-project.org'; 'bonamy at horus.ens.fr'
Subject: Re: [R-sig-ME] prior specification in MCMCglmm

Celine and Pierre,

I too am still very unclear on priors however a month ago Jarrod replied to
some questions of mine regarding multi-response models with this:

	"From experience I find

	list(V=diag(2), nu=2, alpha.mu=c(0,0), alpha.V=diag(2)*a))

	where a is something large (e.g. 1000, depending on the scale of the
	data) works well for the two standard deviations and the
correlation, in terms of informativeness.

	You can't use parameter expanded priors for the residual term yet,
so you will have to stick with the 	standard inverse-Wishart (or use
another program). Generally, data are highly informative for the residual
part so often the posterior is not very sensitive to the prior
specification. Nevertheless, you should 	check alternatives:

	V=diag(2), nu=1.002 gives the inverse-gamma prior for the variances
with shape=scale=0.001 V=diag(2)*1e-6, 	nu=3 is flat for the correlation
from -1 to 1"

I think that this would generalize to G1=(list(V=diag(x), nu=1.00x)) for x
response variables for an inverse-gamma prior for G but I'm not entirely
positive that such is the case. I'm also not positive how to generalize the
flat prior (mainly would nu stay at three?), the first one I assume
generalizes to (V=diag(x), nu=x, alpha.mu=c(0,0...0x),alpha.V=diag(x)*a).

The whole thread for this discussion starts here:
https://stat.ethz.ch/pipermail/r-sig-mixed-models/2011q1/005694.html

The whole prior issue is still a mystery to me so I continue to be
uncomfortable with those approaches where they're necessary; nonetheless for
some data I'm analyzing they're basically necessary (e.g. multi-response
generalized mixed models). By and large the univariate analyses I've
conducted produce highly concordant results regardless of whether fitting
via REML, ML, Laplace or MCMC and regardless of priors for the latter. This
gives me a bit of confidence in the quantitative results and quite a bit of
confidence in the inferences. Unfortunately such comparisons aren't as
feasible (for me) with multi-response models.

Ned

--
Ned Dochtermann
Department of Biology
University of Nevada, Reno

ned.dochtermann at gmail.com
http://wolfweb.unr.edu/homepage/mpeacock/Ned.Dochtermann/
http://www.researcherid.com/rid/A-7146-2010
--

Message: 5
Date: Tue, 26 Apr 2011 09:41:58 +0200
From: "Pierre B. de Villemereuil" <bonamy at horus.ens.fr>
To: Celine Teplitsky <teplitsky at mnhn.fr>
Cc: r-sig-mixed-models at r-project.org
Subject: Re: [R-sig-ME] prior specification in MCMCglmm
Message-ID: <4DB67746.4000607 at horus.ens.fr>
Content-Type: text/plain

Dear Ciline,

I'm not very comfortable with covariance priors, but my guess is that,
is this case, you've got to really specify a inverse-Wishart as a prior.

You should check into Hadfield's article introducing MCMCglmm, they use
something like V=diag(dimV)/4 and n=dimV. Why ? I have no idea. If your
data are not standardized, I don't think you should divide by 4 (but
then, you specify your prior as if your guess for variance components is
that they all equal 1), but for the rest...

Sorry I can not help further. Maybe somebody else would be able to help
on this subject ?

Pierre.


Le 26/04/2011 09:17, Celine Teplitsky a icrit :

Dear Pierre,

thanks a lot! It does help, but I will need time to fully understand
the paper.

Just one further question if I may, what prior would you use for a
covariance then?

Many thanks again,

All the best

Celine

Dear Ciline !

One usual "non informative" prior on variance component is V=1, and
nu=0.002, which correspond to a inverse-Gamma(0.001,0.001). This is
usual, but that is not to say that it is really non informative.
Indeed, inverse-Gamma(e,e) is weakly informative, since the posterior
can depend on the choice of e.

Concerning the WAMwiki suggestion to use the phenotypic variance to
set the prior, this quite not orthodox since no information coming
from your current dataset should be used to define the prior (but you
could use previous data to parametrize your prior).

I would suggest you to refit your model considering V=1 and nu=0.002
as a (so-called) non informative prior. Other solutions exist like
using the parameter expansion and a chi2 distribution by setting
V=1,nu=1000,alpha.mu=0 and alpha.V=1, which is also weakly
informative (it has more weight in variance values less than 10).

For more information about priors on variance component, and
parameter expansion, it would suggest you to read :

   1. A. Gelman, + Prior distributions for variance parameters in
      hierarchical models ;, /Bayesian analysis/ 1, n^o . 3 (2006):
      515--533.

In the hope I'm helping.

Pierre de Villemereuil.


Le 25/04/2011 13:26, Celine Teplitsky a icrit :

Dear all,

I realise that Jarrod is doing field work, but I'm really hoping
someone can answer my question while he's not around.

I am running animal models estimating covariances between life
history traits, and I'm having trouble knowing which prior to use.

Thing is, if I use a prior as described on the Wam wiki site with
V=PhenotypicVar/4 (as I have 3 random effects + residual), I have
very nice results, with some significant genetic correlations
between some life history traits.

However, one reviewer asked about prior sensitivity because CI were
pretty large, so I went back to MCMCglmm course notes and saw that
non informative prior were supposed to be V=diag(nbDimV)*0 and
n=nbDimV-3. This led to an error message about G being ill
conditioned, so I tried with diag(nbDimV)*0.001 and
diag(nbDimV)*0.01 instead of diag(nbDimV)*0, and diag(nbDimV)*0.01
worked... But then I have the posterior of additive genetic variance
collapsing on 0 for some trait. So my guess would be that I should
use those latest priors, and believe my nice results did not exist.
But as Hadfield et al paper and the Wam wiki website do not
recommend those priors, I am a bit confused. Could someone help me
figure out what would be the right thing to do?

All my apologies if this is a silly question, but I'm feeling a bit
lost here

Thanks a lot in advance

Celine

--

Celine Teplitsky
Dipartement Ecologie et Gestion de la Biodiversiti UMR 7204
Uniti Conservation des Esphces, Restauration et Suivi des Populations
Case Postale 51
55 rue Buffon 75005 Paris

Webpage :http://www2.mnhn.fr/cersp/spip.php?rubrique96
Fax : (33-1)-4079-3835
Phone: (33-1)-4079-3443

	[[alternative HTML version deleted]]



------------------------------

Message: 6
Date: Tue, 26 Apr 2011 11:53:02 +0200
From: peter dalgaard <PDalgd at gmail.com>
To: Junqian Gordon Xu <xjqian at gmail.com>
Cc: r-sig-mixed-models at r-project.org
Subject: Re: [R-sig-ME] interaction term in null hypothesis
Message-ID: <E1153437-757B-42E9-9A72-89EF9A368101 at gmail.com>
Content-Type: text/plain; charset="us-ascii"


On Apr 26, 2011, at 00:05 , Junqian Gordon Xu wrote:

I have a quick question for a simple model as below:

Fix + (1 | Rand) + (1 | Rand : Fix)

Which one is the null hypothesis:

1 + (1 | Rand)

or

1 + (1 | Rand) + (1 | Rand : Fix)

To me the interaction term (1 | Rand : Fix) does not make much sense if
no fixed effect term is present in the model, but I'm not sure.

It does make sense, at least sometimes. For one thing, such interactions are
often aliased to aspects of the experimental design; e.g., if you have
randomized different treatment to left side and the right side of test
subjects, then the random interaction is equivalent to the (random)
difference between the two sides within the same subject. Also, you could
conceivably have a kill-or-cure drug with positive effects for some and
negative effects for others, with the question being whether the total
effect is positive or negative.

Regards
Gordon

--
Peter Dalgaard
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com



------------------------------

There's a mistake in my email below.
The generalization of an inverse-gamma prior for multi-response models
should, I think, just be:
list(V=diag(x), nu=1.002)

Sorry about that,
Ned




-----Original Message-----
From: Ned Dochtermann [mailto:ned.dochtermann at gmail.com]
Sent: Tuesday, April 26, 2011 1:12 PM
To: 'r-sig-mixed-models at r-project.org'; 'bonamy at horus.ens.fr'
Subject: Re: [R-sig-ME] prior specification in MCMCglmm

Celine and Pierre,

I too am still very unclear on priors however a month ago Jarrod replied

some questions of mine regarding multi-response models with this:

	"From experience I find

	list(V=diag(2), nu=2, alpha.mu=c(0,0), alpha.V=diag(2)*a))

	where a is something large (e.g. 1000, depending on the scale of the
	data) works well for the two standard deviations and the
correlation, in terms of informativeness.

	You can't use parameter expanded priors for the residual term yet,
so you will have to stick with the 	standard inverse-Wishart (or use
another program). Generally, data are highly informative for the residual
part so often the posterior is not very sensitive to the prior
specification. Nevertheless, you should 	check alternatives:

	V=diag(2), nu=1.002 gives the inverse-gamma prior for the variances
with shape=scale=0.001 V=diag(2)*1e-6, 	nu=3 is flat for the

from -1 to 1"

I think that this would generalize to G1=(list(V=diag(x), nu=1.00x)) for x
response variables for an inverse-gamma prior for G but I'm not entirely
positive that such is the case. I'm also not positive how to generalize

flat prior (mainly would nu stay at three?), the first one I assume
generalizes to (V=diag(x), nu=x, alpha.mu=c(0,0...0x),alpha.V=diag(x)*a).

The whole thread for this discussion starts here:
https://stat.ethz.ch/pipermail/r-sig-mixed-models/2011q1/005694.html

The whole prior issue is still a mystery to me so I continue to be
uncomfortable with those approaches where they're necessary; nonetheless

some data I'm analyzing they're basically necessary (e.g. multi-response
generalized mixed models). By and large the univariate analyses I've
conducted produce highly concordant results regardless of whether fitting
via REML, ML, Laplace or MCMC and regardless of priors for the latter.

gives me a bit of confidence in the quantitative results and quite a bit

confidence in the inferences. Unfortunately such comparisons aren't as
feasible (for me) with multi-response models.

Ned

--
Ned Dochtermann
Department of Biology
University of Nevada, Reno

ned.dochtermann at gmail.com
http://wolfweb.unr.edu/homepage/mpeacock/Ned.Dochtermann/
http://www.researcherid.com/rid/A-7146-2010
--

Message: 5
Date: Tue, 26 Apr 2011 09:41:58 +0200
From: "Pierre B. de Villemereuil" <bonamy at horus.ens.fr>
To: Celine Teplitsky <teplitsky at mnhn.fr>
Cc: r-sig-mixed-models at r-project.org
Subject: Re: [R-sig-ME] prior specification in MCMCglmm
Message-ID: <4DB67746.4000607 at horus.ens.fr>
Content-Type: text/plain

Dear Ciline,

I'm not very comfortable with covariance priors, but my guess is that,
is this case, you've got to really specify a inverse-Wishart as a prior.

You should check into Hadfield's article introducing MCMCglmm, they use
something like V=diag(dimV)/4 and n=dimV. Why ? I have no idea. If your
data are not standardized, I don't think you should divide by 4 (but
then, you specify your prior as if your guess for variance components is
that they all equal 1), but for the rest...

Sorry I can not help further. Maybe somebody else would be able to help
on this subject ?

Pierre.


Le 26/04/2011 09:17, Celine Teplitsky a icrit :

Dear Pierre,

thanks a lot! It does help, but I will need time to fully understand
the paper.

Just one further question if I may, what prior would you use for a
covariance then?

Many thanks again,

All the best

Celine

Dear Ciline !

One usual "non informative" prior on variance component is V=1, and
nu=0.002, which correspond to a inverse-Gamma(0.001,0.001). This is
usual, but that is not to say that it is really non informative.
Indeed, inverse-Gamma(e,e) is weakly informative, since the posterior
can depend on the choice of e.

Concerning the WAMwiki suggestion to use the phenotypic variance to
set the prior, this quite not orthodox since no information coming
from your current dataset should be used to define the prior (but you
could use previous data to parametrize your prior).

I would suggest you to refit your model considering V=1 and nu=0.002
as a (so-called) non informative prior. Other solutions exist like
using the parameter expansion and a chi2 distribution by setting
V=1,nu=1000,alpha.mu=0 and alpha.V=1, which is also weakly
informative (it has more weight in variance values less than 10).

For more information about priors on variance component, and
parameter expansion, it would suggest you to read :

   1. A. Gelman, + Prior distributions for variance parameters in
      hierarchical models ;, /Bayesian analysis/ 1, n^o . 3 (2006):
      515--533.

In the hope I'm helping.

Pierre de Villemereuil.


Le 25/04/2011 13:26, Celine Teplitsky a icrit :

Dear all,

I realise that Jarrod is doing field work, but I'm really hoping
someone can answer my question while he's not around.

I am running animal models estimating covariances between life
history traits, and I'm having trouble knowing which prior to use.

Thing is, if I use a prior as described on the Wam wiki site with
V=PhenotypicVar/4 (as I have 3 random effects + residual), I have
very nice results, with some significant genetic correlations
between some life history traits.

However, one reviewer asked about prior sensitivity because CI were
pretty large, so I went back to MCMCglmm course notes and saw that
non informative prior were supposed to be V=diag(nbDimV)*0 and
n=nbDimV-3. This led to an error message about G being ill
conditioned, so I tried with diag(nbDimV)*0.001 and
diag(nbDimV)*0.01 instead of diag(nbDimV)*0, and diag(nbDimV)*0.01
worked... But then I have the posterior of additive genetic variance
collapsing on 0 for some trait. So my guess would be that I should
use those latest priors, and believe my nice results did not exist.
But as Hadfield et al paper and the Wam wiki website do not
recommend those priors, I am a bit confused. Could someone help me
figure out what would be the right thing to do?

All my apologies if this is a silly question, but I'm feeling a bit
lost here

Thanks a lot in advance

Celine

--

Celine Teplitsky
Dipartement Ecologie et Gestion de la Biodiversiti UMR 7204
Uniti Conservation des Esphces, Restauration et Suivi des Populations
Case Postale 51
55 rue Buffon 75005 Paris

Webpage :http://www2.mnhn.fr/cersp/spip.php?rubrique96
Fax : (33-1)-4079-3835
Phone: (33-1)-4079-3443

	[[alternative HTML version deleted]]



------------------------------

Message: 6
Date: Tue, 26 Apr 2011 11:53:02 +0200
From: peter dalgaard <PDalgd at gmail.com>
To: Junqian Gordon Xu <xjqian at gmail.com>
Cc: r-sig-mixed-models at r-project.org
Subject: Re: [R-sig-ME] interaction term in null hypothesis
Message-ID: <E1153437-757B-42E9-9A72-89EF9A368101 at gmail.com>
Content-Type: text/plain; charset="us-ascii"


On Apr 26, 2011, at 00:05 , Junqian Gordon Xu wrote:

I have a quick question for a simple model as below:

Fix + (1 | Rand) + (1 | Rand : Fix)

Which one is the null hypothesis:

1 + (1 | Rand)

or

1 + (1 | Rand) + (1 | Rand : Fix)

To me the interaction term (1 | Rand : Fix) does not make much sense if
no fixed effect term is present in the model, but I'm not sure.

It does make sense, at least sometimes. For one thing, such interactions

often aliased to aspects of the experimental design; e.g., if you have
randomized different treatment to left side and the right side of test
subjects, then the random interaction is equivalent to the (random)
difference between the two sides within the same subject. Also, you could
conceivably have a kill-or-cure drug with positive effects for some and
negative effects for others, with the question being whether the total
effect is positive or negative.

Regards
Gordon

--
Peter Dalgaard
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com



------------------------------

That's my understanding as well. I think that the shape and
"informativeness" of the prior are essentially conflated. For example, a
flat prior would not be informing the posterior much so its shape and the
amount of information imparted are related.

Of course it is equally likely that I'm completely wrong on this.

Ned

--
Ned Dochtermann
Department of Biology
University of Nevada, Reno

ned.dochtermann at gmail.com
http://wolfweb.unr.edu/homepage/mpeacock/Ned.Dochtermann/
http://www.researcherid.com/rid/A-7146-2010
--


-----Original Message-----
From: Celine Teplitsky [mailto:teplitsky at mnhn.fr]
Sent: Wednesday, April 27, 2011 7:37 AM
To: Ned Dochtermann
Cc: r-sig-mixed-models at r-project.org; bonamy at horus.ens.fr
Subject: Re: [R-sig-ME] prior specification in MCMCglmm

Ned,

thanks a lot, I'll try that and compare results with the different priors.

What I just begin to understand is that nu is not really a degree of
belief you can play with but a more specific parameter of a
distribution, right?

All the best

Celine

There's a mistake in my email below.
The generalization of an inverse-gamma prior for multi-response models
should, I think, just be:
list(V=diag(x), nu=1.002)

Sorry about that,
Ned




-----Original Message-----
From: Ned Dochtermann [mailto:ned.dochtermann at gmail.com]
Sent: Tuesday, April 26, 2011 1:12 PM
To: 'r-sig-mixed-models at r-project.org'; 'bonamy at horus.ens.fr'
Subject: Re: [R-sig-ME] prior specification in MCMCglmm

Celine and Pierre,

I too am still very unclear on priors however a month ago Jarrod replied

to

some questions of mine regarding multi-response models with this:

	"From experience I find

	list(V=diag(2), nu=2, alpha.mu=c(0,0), alpha.V=diag(2)*a))

	where a is something large (e.g. 1000, depending on the scale of the
	data) works well for the two standard deviations and the
correlation, in terms of informativeness.

	You can't use parameter expanded priors for the residual term yet,
so you will have to stick with the 	standard inverse-Wishart (or use
another program). Generally, data are highly informative for the residual
part so often the posterior is not very sensitive to the prior
specification. Nevertheless, you should 	check alternatives:

	V=diag(2), nu=1.002 gives the inverse-gamma prior for the variances
with shape=scale=0.001 V=diag(2)*1e-6, 	nu=3 is flat for the

correlation

from -1 to 1"

I think that this would generalize to G1=(list(V=diag(x), nu=1.00x)) for x
response variables for an inverse-gamma prior for G but I'm not entirely
positive that such is the case. I'm also not positive how to generalize

the

flat prior (mainly would nu stay at three?), the first one I assume
generalizes to (V=diag(x), nu=x, alpha.mu=c(0,0...0x),alpha.V=diag(x)*a).

The whole thread for this discussion starts here:
https://stat.ethz.ch/pipermail/r-sig-mixed-models/2011q1/005694.html

The whole prior issue is still a mystery to me so I continue to be
uncomfortable with those approaches where they're necessary; nonetheless

for

some data I'm analyzing they're basically necessary (e.g. multi-response
generalized mixed models). By and large the univariate analyses I've
conducted produce highly concordant results regardless of whether fitting
via REML, ML, Laplace or MCMC and regardless of priors for the latter.

This

gives me a bit of confidence in the quantitative results and quite a bit

of

confidence in the inferences. Unfortunately such comparisons aren't as
feasible (for me) with multi-response models.

Ned

--
Ned Dochtermann
Department of Biology
University of Nevada, Reno

ned.dochtermann at gmail.com
http://wolfweb.unr.edu/homepage/mpeacock/Ned.Dochtermann/
http://www.researcherid.com/rid/A-7146-2010
--

Message: 5
Date: Tue, 26 Apr 2011 09:41:58 +0200
From: "Pierre B. de Villemereuil"<bonamy at horus.ens.fr>
To: Celine Teplitsky<teplitsky at mnhn.fr>
Cc: r-sig-mixed-models at r-project.org
Subject: Re: [R-sig-ME] prior specification in MCMCglmm
Message-ID:<4DB67746.4000607 at horus.ens.fr>
Content-Type: text/plain

Dear Ciline,

I'm not very comfortable with covariance priors, but my guess is that,
is this case, you've got to really specify a inverse-Wishart as a prior.

You should check into Hadfield's article introducing MCMCglmm, they use
something like V=diag(dimV)/4 and n=dimV. Why ? I have no idea. If your
data are not standardized, I don't think you should divide by 4 (but
then, you specify your prior as if your guess for variance components is
that they all equal 1), but for the rest...

Sorry I can not help further. Maybe somebody else would be able to help
on this subject ?

Pierre.


Le 26/04/2011 09:17, Celine Teplitsky a icrit :

Dear Pierre,

thanks a lot! It does help, but I will need time to fully understand
the paper.

Just one further question if I may, what prior would you use for a
covariance then?

Many thanks again,

All the best

Celine

Dear Ciline !

One usual "non informative" prior on variance component is V=1, and
nu=0.002, which correspond to a inverse-Gamma(0.001,0.001). This is
usual, but that is not to say that it is really non informative.
Indeed, inverse-Gamma(e,e) is weakly informative, since the posterior
can depend on the choice of e.

Concerning the WAMwiki suggestion to use the phenotypic variance to
set the prior, this quite not orthodox since no information coming
from your current dataset should be used to define the prior (but you
could use previous data to parametrize your prior).

I would suggest you to refit your model considering V=1 and nu=0.002
as a (so-called) non informative prior. Other solutions exist like
using the parameter expansion and a chi2 distribution by setting
V=1,nu=1000,alpha.mu=0 and alpha.V=1, which is also weakly
informative (it has more weight in variance values less than 10).

For more information about priors on variance component, and
parameter expansion, it would suggest you to read :

    1. A. Gelman, + Prior distributions for variance parameters in
       hierarchical models ;, /Bayesian analysis/ 1, n^o . 3 (2006):
       515--533.

In the hope I'm helping.

Pierre de Villemereuil.


Le 25/04/2011 13:26, Celine Teplitsky a icrit :

Dear all,

I realise that Jarrod is doing field work, but I'm really hoping
someone can answer my question while he's not around.

I am running animal models estimating covariances between life
history traits, and I'm having trouble knowing which prior to use.

Thing is, if I use a prior as described on the Wam wiki site with
V=PhenotypicVar/4 (as I have 3 random effects + residual), I have
very nice results, with some significant genetic correlations
between some life history traits.

However, one reviewer asked about prior sensitivity because CI were
pretty large, so I went back to MCMCglmm course notes and saw that
non informative prior were supposed to be V=diag(nbDimV)*0 and
n=nbDimV-3. This led to an error message about G being ill
conditioned, so I tried with diag(nbDimV)*0.001 and
diag(nbDimV)*0.01 instead of diag(nbDimV)*0, and diag(nbDimV)*0.01
worked... But then I have the posterior of additive genetic variance
collapsing on 0 for some trait. So my guess would be that I should
use those latest priors, and believe my nice results did not exist.
But as Hadfield et al paper and the Wam wiki website do not
recommend those priors, I am a bit confused. Could someone help me
figure out what would be the right thing to do?

All my apologies if this is a silly question, but I'm feeling a bit
lost here

Thanks a lot in advance

Celine

--

Celine Teplitsky
Dipartement Ecologie et Gestion de la Biodiversiti UMR 7204
Uniti Conservation des Esphces, Restauration et Suivi des Populations
Case Postale 51
55 rue Buffon 75005 Paris

Webpage :http://www2.mnhn.fr/cersp/spip.php?rubrique96
Fax : (33-1)-4079-3835
Phone: (33-1)-4079-3443

	[[alternative HTML version deleted]]



------------------------------

Message: 6
Date: Tue, 26 Apr 2011 11:53:02 +0200
From: peter dalgaard<PDalgd at gmail.com>
To: Junqian Gordon Xu<xjqian at gmail.com>
Cc: r-sig-mixed-models at r-project.org
Subject: Re: [R-sig-ME] interaction term in null hypothesis
Message-ID:<E1153437-757B-42E9-9A72-89EF9A368101 at gmail.com>
Content-Type: text/plain; charset="us-ascii"


On Apr 26, 2011, at 00:05 , Junqian Gordon Xu wrote:

I have a quick question for a simple model as below:

Fix + (1 | Rand) + (1 | Rand : Fix)

Which one is the null hypothesis:

1 + (1 | Rand)

or

1 + (1 | Rand) + (1 | Rand : Fix)

To me the interaction term (1 | Rand : Fix) does not make much sense if
no fixed effect term is present in the model, but I'm not sure.

It does make sense, at least sometimes. For one thing, such interactions

are

often aliased to aspects of the experimental design; e.g., if you have
randomized different treatment to left side and the right side of test
subjects, then the random interaction is equivalent to the (random)
difference between the two sides within the same subject. Also, you could
conceivably have a kill-or-cure drug with positive effects for some and
negative effects for others, with the question being whether the total
effect is positive or negative.

Regards
Gordon

--
Peter Dalgaard
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com



------------------------------