[R-sig-ME] random intercept and random slope

[André Barbosa]

Dear list members,
I have read several threads on this list about the use of random variables and its interpretation. I seem to have learned a lot about model fitting, from plotting the raw data and checking the slopes and intercepts, to getting rid of the p-value mindset in which I had had my basic statistics courses at university.
However, since my statistical courses never covered Mixed Effect Models, I am still unsure if I am doing the right thing or not. Having said that, would you please take a look at the following data and see if my rational is correct? The model is quite simple, I believe.
I have 8 different plant species, which were mixed two-by-two in all possible combinations. So, each species has 7 pairs + it being alone (monoculture). My response variable is a ratio (observed /expected productivity values, where observed is the productivity of species “a” achieved when mixed with another species, and expected is its value when in monoculture).
Each species had its own nutrient content analyzed. As I had three nutrient variables measured from each plant, I calculated indices of dissimilarity for each of those pairs.
My starting model (without specifying random or fixed effects) would be:
ratio ~ dissimilarity | species
I expected, based on previous studies, that the relationship between ratio and dissimilarity would yield different slopes for each species, from negative to positive – expectation confirmed by potting a xyplot function of my data. Thus, species should be random. Looking at the xyplot of my data, I also see that the intercepts are somehow variable, ranging between 0.5 and 1.5 (response data points do not extend much further from this range, either). For this reason, I thought on including intercepts as random, as well, which leave me without fixed variables.
So, I decided to test:
lmer(ratio ~ 1 + (dissimilarity|species))
Here follows a subset of my data:
species            pair            ratio            dissimilarity
a            a+b            1.090935            1.870297012
a            a+c            1.182509            0.691033781
a            a+d            1.505538            1.441237522
a            a+e            1.547295            0.953060747
a            a+f            1.463782            1.306913498
a            a+g            1.197587            1.331087471
a            a+h            1.113263            1.097840225
b            b+a            0.899969            1.870297012
b            b+c            1.102478            1.548604304
b            b+d            1.218110            1.669409077
b            b+e            1.095748            1.536191709
b            b+f            1.306822            1.579788658
b            b+g            1.299480            1.084382658
b            b+h            1.219945            1.137927922
c            c+a            1.092199            1.441237522
c            c+b            1.486702            1.669409077
c            c+d            0.847517            1.688612802
c            c+e            0.210150            0.651183878
c            c+f            1.459219            1.064428069
c            c+g            0.87810            0.590191888
c            c+h            0.91223            1.37455314
d            d+a            1.32486            0.953060747
d            d+b            1.37737            1.536191709
d            d+c            1.23869            1.310607287
d            d+e            1.15714            0.651183878
d            d+f            0.97390            1.22540051
d            d+g            0.92355            0.640371057
d            d+h            0.79097            1.224534999

The output from my model, using the whole data set is:
> summary(random.model)
Linear mixed model fit by REML
Formula: ratio~ 1 + (dissimilarity | species)
   Data: k
   AIC   BIC  logLik deviance REMLdev
 11.29 21.42 -0.6465   -3.273   1.293
Random effects:
 Groups   Name        Variance Std.Dev. Corr
 species  (Intercept) 0.092206 0.30365
          dissimilarity        0.030348 0.17421  -1.000
 Residual             0.048348 0.21988
Number of obs: 56, groups: species, 8

Fixed effects:
            Estimate Std. Error t value
(Intercept)  1.18024    0.04087   28.88

----

My questions:

1. Is my approach correct in plotting the independent variable “dissimilarity” as random intercept, as in ~ 1 + (dissimilarity|species)?
2. On a publication, can I report the variance component of the random terms (in percentage) as the main result of the test?
3. Would it be possible to run McMC on a model that does not have Fixed Effects?
4. Would there be any other metrics that I should report as well?

I am sure that my questions are pretty basic, but I would strongly appreciate any input from you. Thank you!
Andre