A graphic for Random intercepts via distributions

Good afternoon,

I came across a picture (https://github.com/hkil/m/blob/master/mlm.PNG)
that tries to show the concept of random-intercept models using
distributions.

I think, however, the picture erroneously mixes regression concepts (e.g.,
error terms) with distributional properties of those regression concepts.

I appreciate confirmation from the expert members?

Thanks,
Simon

	[[alternative HTML version deleted]]

     Can you clarify your concern?

I can see things to quibble about here (the scales of the level-2 and
level-1 diagrams are different; I don't know why they're using e_{ij}
for the residual error of individual i in school j but U_{0j} for the
deviation of school j around the grand mean; it's a little confusing to
have "level 1" above "level 2" in the text but level 2 above level 1 in
the picture; it's potentially confusing for the arrow showing the
deviation from the baseline to intersect with the population density
curve [technically, the deviation doesn't have a "level", so could be
drawn instead as an arrow between two vertical lines rather than from a
line to a particular point ...

... but nothing that seems actively misleading.

   Others may have other opinions or see something I'm missing.

On 7/8/20 6:27 PM, Simon Harmel wrote:

Good afternoon,

I came across a picture (https://github.com/hkil/m/blob/master/mlm.PNG)
that tries to show the concept of random-intercept models using
distributions.

I think, however, the picture erroneously mixes regression concepts

(e.g.,

error terms) with distributional properties of those regression concepts.

I appreciate confirmation from the expert members?

Thanks,
Simon

      [[alternative HTML version deleted]]

Thanks Ben. The notations e_{ij} for the residual error of individual i in school j and U_{0j} for the deviation of school j's mean from the grand mean is just how educational methodologists denote these concepts.

 But specifically, I thought regression concepts like e_{ij} and U_{0j} all should be correctly shown on a scatter plot like this: https://github.com/hkil/m/blob/master/mlm2.PNG.

So, with your suggestions is this a better picture?: https://github.com/hkil/m/blob/master/mlm3.PNG

Is there a relationship between the scale of the fist-level distributions, and the second-level distribution that the picture should observe?

Thanks,
Simon


On Wed, Jul 8, 2020 at 5:51 PM Ben Bolker <bbolker at gmail.com> wrote:

     Can you clarify your concern?

I can see things to quibble about here (the scales of the level-2 and
level-1 diagrams are different; I don't know why they're using e_{ij}
for the residual error of individual i in school j but U_{0j} for the
deviation of school j around the grand mean; it's a little confusing to
have "level 1" above "level 2" in the text but level 2 above level 1 in
the picture; it's potentially confusing for the arrow showing the
deviation from the baseline to intersect with the population density
curve [technically, the deviation doesn't have a "level", so could be
drawn instead as an arrow between two vertical lines rather than from a
line to a particular point ...

... but nothing that seems actively misleading.

   Others may have other opinions or see something I'm missing.

On 7/8/20 6:27 PM, Simon Harmel wrote:

Good afternoon,

I came across a picture (https://github.com/hkil/m/blob/master/mlm.PNG)
that tries to show the concept of random-intercept models using
distributions.

I think, however, the picture erroneously mixes regression concepts (e.g.,
error terms) with distributional properties of those regression concepts.

I appreciate confirmation from the expert members?

Thanks,
Simon

      [[alternative HTML version deleted]]

   I agree that the second version you link to might be slightly more
technically correct, but I don't think there's anything about harmful.
   The most important correction (IMO) would be to make the red
(level-2) distribution much wider, so that it actually matched the
scale of the level-1 distribution.  (The other problem with the
picture is that for prettiness, the beta_{0x} values we can see appear
evenly spaced, which is unrealistic ...)

On Wed, Jul 8, 2020 at 7:25 PM Simon Harmel <sim.harmel at gmail.com> wrote:

Thanks Ben. The notations e_{ij} for the residual error of individual i

in school j and U_{0j} for the deviation of school j's mean from the grand
mean is just how educational methodologists denote these concepts.

 But specifically, I thought regression concepts like e_{ij} and U_{0j}

all should be correctly shown on a scatter plot like this:
https://github.com/hkil/m/blob/master/mlm2.PNG.

So, with your suggestions is this a better picture?:

https://github.com/hkil/m/blob/master/mlm3.PNG

Is there a relationship between the scale of the fist-level

distributions, and the second-level distribution that the picture should
observe?

Thanks,
Simon


On Wed, Jul 8, 2020 at 5:51 PM Ben Bolker <bbolker at gmail.com> wrote:

     Can you clarify your concern?

I can see things to quibble about here (the scales of the level-2 and
level-1 diagrams are different; I don't know why they're using e_{ij}
for the residual error of individual i in school j but U_{0j} for the
deviation of school j around the grand mean; it's a little confusing to
have "level 1" above "level 2" in the text but level 2 above level 1 in
the picture; it's potentially confusing for the arrow showing the
deviation from the baseline to intersect with the population density
curve [technically, the deviation doesn't have a "level", so could be
drawn instead as an arrow between two vertical lines rather than from a
line to a particular point ...

... but nothing that seems actively misleading.

   Others may have other opinions or see something I'm missing.

On 7/8/20 6:27 PM, Simon Harmel wrote:

Good afternoon,

I came across a picture (

https://github.com/hkil/m/blob/master/mlm.PNG)

that tries to show the concept of random-intercept models using
distributions.

I think, however, the picture erroneously mixes regression concepts

(e.g.,

error terms) with distributional properties of those regression

concepts.

I appreciate confirmation from the expert members?

Thanks,
Simon

      [[alternative HTML version deleted]]

Thanks so much, will do all that!

On Wed, Jul 8, 2020 at 6:45 PM Ben Bolker <bbolker at gmail.com> wrote:

   I agree that the second version you link to might be slightly more
technically correct, but I don't think there's anything about harmful.
   The most important correction (IMO) would be to make the red
(level-2) distribution much wider, so that it actually matched the
scale of the level-1 distribution.  (The other problem with the
picture is that for prettiness, the beta_{0x} values we can see appear
evenly spaced, which is unrealistic ...)

On Wed, Jul 8, 2020 at 7:25 PM Simon Harmel <sim.harmel at gmail.com>

wrote:

Thanks Ben. The notations e_{ij} for the residual error of individual i

in school j and U_{0j} for the deviation of school j's mean from the

grand

mean is just how educational methodologists denote these concepts.

 But specifically, I thought regression concepts like e_{ij} and U_{0j}

all should be correctly shown on a scatter plot like this:
https://github.com/hkil/m/blob/master/mlm2.PNG.

So, with your suggestions is this a better picture?:

https://github.com/hkil/m/blob/master/mlm3.PNG

Is there a relationship between the scale of the fist-level

distributions, and the second-level distribution that the picture should
observe?

Thanks,
Simon


On Wed, Jul 8, 2020 at 5:51 PM Ben Bolker <bbolker at gmail.com> wrote:

     Can you clarify your concern?

I can see things to quibble about here (the scales of the level-2 and
level-1 diagrams are different; I don't know why they're using e_{ij}
for the residual error of individual i in school j but U_{0j} for the
deviation of school j around the grand mean; it's a little confusing

to

have "level 1" above "level 2" in the text but level 2 above level 1

in

the picture; it's potentially confusing for the arrow showing the
deviation from the baseline to intersect with the population density
curve [technically, the deviation doesn't have a "level", so could be
drawn instead as an arrow between two vertical lines rather than from

a

line to a particular point ...

... but nothing that seems actively misleading.

   Others may have other opinions or see something I'm missing.

On 7/8/20 6:27 PM, Simon Harmel wrote:

Good afternoon,

I came across a picture (

https://github.com/hkil/m/blob/master/mlm.PNG)

that tries to show the concept of random-intercept models using
distributions.

I think, however, the picture erroneously mixes regression concepts

(e.g.,

error terms) with distributional properties of those regression

concepts.

I appreciate confirmation from the expert members?

Thanks,
Simon

      [[alternative HTML version deleted]]

        [[alternative HTML version deleted]]

Simon, will you have those pictures available somewhere for educational
purposes? My field is ecology, where mixed models are more often presented
otherwise, not stressing that hierarchy among levels so much as in
education studies. However, I find the pictures could be useful to my
students,  along with the caveats you and Ben Bolker have made.





El jue., 9 jul. 2020 a las 2:11, Simon Harmel (<sim.harmel at gmail.com>)
escribi?:

Thanks so much, will do all that!

On Wed, Jul 8, 2020 at 6:45 PM Ben Bolker <bbolker at gmail.com> wrote:

   I agree that the second version you link to might be slightly more
technically correct, but I don't think there's anything about harmful.
   The most important correction (IMO) would be to make the red
(level-2) distribution much wider, so that it actually matched the
scale of the level-1 distribution.  (The other problem with the
picture is that for prettiness, the beta_{0x} values we can see appear
evenly spaced, which is unrealistic ...)

On Wed, Jul 8, 2020 at 7:25 PM Simon Harmel <sim.harmel at gmail.com>

wrote:

Thanks Ben. The notations e_{ij} for the residual error of individual

i

in school j and U_{0j} for the deviation of school j's mean from the

grand

mean is just how educational methodologists denote these concepts.

 But specifically, I thought regression concepts like e_{ij} and

U_{0j}

all should be correctly shown on a scatter plot like this:
https://github.com/hkil/m/blob/master/mlm2.PNG.

So, with your suggestions is this a better picture?:

https://github.com/hkil/m/blob/master/mlm3.PNG

Is there a relationship between the scale of the fist-level

distributions, and the second-level distribution that the picture should
observe?

Thanks,
Simon


On Wed, Jul 8, 2020 at 5:51 PM Ben Bolker <bbolker at gmail.com> wrote:

     Can you clarify your concern?

I can see things to quibble about here (the scales of the level-2 and
level-1 diagrams are different; I don't know why they're using e_{ij}
for the residual error of individual i in school j but U_{0j} for the
deviation of school j around the grand mean; it's a little confusing

to

have "level 1" above "level 2" in the text but level 2 above level 1

in

the picture; it's potentially confusing for the arrow showing the
deviation from the baseline to intersect with the population density
curve [technically, the deviation doesn't have a "level", so could be
drawn instead as an arrow between two vertical lines rather than

from a

line to a particular point ...

... but nothing that seems actively misleading.

   Others may have other opinions or see something I'm missing.

On 7/8/20 6:27 PM, Simon Harmel wrote:

Good afternoon,

I came across a picture (

https://github.com/hkil/m/blob/master/mlm.PNG)

that tries to show the concept of random-intercept models using
distributions.

I think, however, the picture erroneously mixes regression concepts

(e.g.,

error terms) with distributional properties of those regression

concepts.

I appreciate confirmation from the expert members?

Thanks,
Simon

      [[alternative HTML version deleted]]

        [[alternative HTML version deleted]]

--

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Javier Seoane Pinilla
Profesor Titular
Centro de Investigaci?n en Biodiversidad y Cambio Global (CIBC-UAM)
Departamento de Ecologia
Universidad Autonoma de Madrid
Edificio de Biolog?a
Darwin, 2
E-28049 Madrid
SPAIN

e-mail: javier.seoane at uam.es
webpage: http://teguam.es/miembros/javier-seoane/
Tlf: +34 91 497 3639
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