R help - Jan 2012 - Calculating & plotting a linear regression between two correlated variables

Hi,
I have a Community (COM) composed of 6 species: A, B, C, D, E & F.
The density of my Community is thus (Eq.1): dCOM = dA + dB + dC + dE + dF

I would like to calculate and plot a linear regression between the density
of each of my species and the density of the whole community (illustrating
how the density of each species varies with variations of the whole
community).
For example, I would like to plot dA = a * dCOM + b, with a and b the slope
and intercept of the regression.

The problem is that dA and dCOM are correlated because dA contributes to
values of dCOM (see Eq.1 above), and thus I'm probably not allowed to use a
"simple" linear regression (because parametric statistics do not allow
for
correlated observations).
>From what I red (eg, www.ats.ucla.edu/stat/r/faq/spatial_regression.htm),
Linear Mixed Models allow for correlated observations by adding a correction
to the values. The webpage also says that  we can use the correlation option
in the lme function (nlme package) to find the type of correction to be
used, but I can?t figure out how to do this for my dataset.  

Can someone help me please ?
You will find an example of my dataset below (density of species A [dA] and
density of the whole community [dCOM]) and my R script.


Example from my dataset:
dA	dCOM
0.611	0.73
0.474	0.669
0.203	0.388
0.011	0.213
0.407	0.722
0.148	0.437
0.084	0.281
0	0.054
0.402	0.93
0.044	0.285
0.011	0.147
0	0.091
0.547	0.767
0.559	0.699
0.321	0.441
0.084	0.262
0.428	0.761
0.234	0.398
0.019	0.191
0	0.053
0.302	0.509
0.06	0.213
0.045	0.184
0.033	0.118
0.374	0.585
0.256	0.5
0.22	0.401
0.136	0.275
0.503	0.684
0.48	0.617
0.257	0.387
0.038	0.159
0.382	0.611
0.192	0.393
0.047	0.243
0.036	0.188

R script for this example:

dummy <- rep(1,36)  #  grouping variable in our data
model1 <- lme(fixed = dA ~ dCOM, data = bb1, random = ~ 1 | dCOM/dummy,
correlation= ?,method = "ML")

Thank you very much.

Julie.


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Hi Julie,

Mixed effects models are typically used to allow for correlations
between observations of the outcome variable---the fact that you are
trying to model dA ~ dCOM rather assumes you expect some sort of
association between the two.  It is not exactly clear what you want to
deal with using a correlation structure, but it seems to me your model
is this:

dA = a * (dA + dB + dC + dE + dF) + b

which is a little circular because dA is explaining itself.  Without
knowing more, I would think about something like:

dA = a1 * dB + a2 * dC + a3 * dE + a4 * dF + b

if you want a single slope for all a1 - a4 effects, you could move to
a structural equation modelling framework and impose the constraint
that:

a1 = a2 = a3 = a4

in the parameter estimation.

Cheers,

Josh

On Sun, Jan 22, 2012 at 12:31 PM, JulieV <sharkette002 at hotmail.com>
wrote:
> Hi,
> I have a Community (COM) composed of 6 species: A, B, C, D, E & F.
> The density of my Community is thus (Eq.1): dCOM = dA + dB + dC + dE + dF
>
> I would like to calculate and plot a linear regression between the density
> of each of my species and the density of the whole community (illustrating
> how the density of each species varies with variations of the whole
> community).
> For example, I would like to plot dA = a * dCOM + b, with a and b the slope
> and intercept of the regression.
>
> The problem is that dA and dCOM are correlated because dA contributes to
> values of dCOM (see Eq.1 above), and thus I'm probably not allowed to
use a
> "simple" linear regression (because parametric statistics do not
allow for
> correlated observations).
>
> >From what I red (eg,
www.ats.ucla.edu/stat/r/faq/spatial_regression.htm),
> Linear Mixed Models allow for correlated observations by adding a
correction
> to the values. The webpage also says that ?we can use the correlation
option
> in the lme function (nlme package) to find the type of correction to be
> used, but I can?t figure out how to do this for my dataset.
>
> Can someone help me please ?
> You will find an example of my dataset below (density of species A [dA] and
> density of the whole community [dCOM]) and my R script.
>
>
> Example from my dataset:
> dA ? ? ?dCOM
> 0.611 ? 0.73
> 0.474 ? 0.669
> 0.203 ? 0.388
> 0.011 ? 0.213
> 0.407 ? 0.722
> 0.148 ? 0.437
> 0.084 ? 0.281
> 0 ? ? ? 0.054
> 0.402 ? 0.93
> 0.044 ? 0.285
> 0.011 ? 0.147
> 0 ? ? ? 0.091
> 0.547 ? 0.767
> 0.559 ? 0.699
> 0.321 ? 0.441
> 0.084 ? 0.262
> 0.428 ? 0.761
> 0.234 ? 0.398
> 0.019 ? 0.191
> 0 ? ? ? 0.053
> 0.302 ? 0.509
> 0.06 ? ?0.213
> 0.045 ? 0.184
> 0.033 ? 0.118
> 0.374 ? 0.585
> 0.256 ? 0.5
> 0.22 ? ?0.401
> 0.136 ? 0.275
> 0.503 ? 0.684
> 0.48 ? ?0.617
> 0.257 ? 0.387
> 0.038 ? 0.159
> 0.382 ? 0.611
> 0.192 ? 0.393
> 0.047 ? 0.243
> 0.036 ? 0.188
>
> R script for this example:
>
> dummy <- rep(1,36) ?# ?grouping variable in our data
> model1 <- lme(fixed = dA ~ dCOM, data = bb1, random = ~ 1 | dCOM/dummy,
> correlation= ?,method = "ML")
>
> Thank you very much.
>
> Julie.
>
>
> --
> View this message in context:
http://r.789695.n4.nabble.com/Calculating-plotting-a-linear-regression-between-two-correlated-variables-tp4319051p4319051.html
> Sent from the R help mailing list archive at Nabble.com.
>
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.


-- 
Joshua Wiley
Ph.D. Student, Health Psychology
Programmer Analyst II, Statistical Consulting Group
University of California, Los Angeles
https://joshuawiley.com/

Hi Josh,

Thanks for your response ! 

Actually, I already tried to plot it with a "classical" regression and
I
know the relation is linear:

dA = 0.765 * dCOM - 0.089 
p(slope) < 0.0001 
p(intercept) = 0.0003

The fact is that I can not use these results as my variables dA and dCOM are
correlated (as mentioned above, Eq.1). What I need to find out is which
correction I should do on my data, and how, to be able to calculte the
regression p-values correctly with Linear Mixed Models.

I am interested in this because I know that my species decline at different
rates when my community is declining. 
For example, with decreasing values of dCOM, dA reaches 0 before dB.


Julie

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