# When nature passes through (0;0): what are the consequences for a linear mixed model?

I'm analyzing the data for my master thesis; it's about photosynthetic efficiency for 10 different genotypes of a certain plant species (genotype=Accession in my dataset).

For each Accession, I've been measuring PPFD (light intensity) that is the continuous independent variable, and rETR (photosynthetic efficiency) that is the continuous dependent variable in the linear model I would like to run.

For the light intensities I've been measuring and the species I'm using, we can in fact consider the relation between PPFD and rETR approximately linear.

What I'm interested in is to see if the slopes given by the interaction between PPFD and Accession differ. In particular these Accessions come from Asia and Africa, so another thing would be to check if the "Africa-slopes" are different from the "Asia-slopes".

It's a given fact, that the origin of my lines should be in (0;0), since without light there is no electron transport in the plant and it cannot be negative. Therefore I'm after a model with no intercept.

I had a complete randomized block design, with 5 Blocks (for a total of 5x10 plants). In a block, one of the Accessions died out, so I'm temporarily excluding the whole block from the analysis (I measured the Blocks one by one, but always changing order).

I measured three leaves per plant, with all the Blocks measured twice per day (Mo and Af) and I repeated the measurements for 6 days. After a quick analysis I decided it was ok to average rETR for the 3 leaves within one plant, since it didn't look like having a significant effect.

Therefore I remain with 3 blocking factors nested within each other: Date (6 levels) MoAf (2 levels) Block (4 levels) I thought I could use these to control for environmental noise.

What I did first was to run a linear model with lm(), without considering my blocking factors, so to have a general idea of what was going on:

    LM3E = lm(rETR ~ 0 + PPFD:Accession, data=avpsN, na.action=na.omit)


Then I tried to fit a mixed effect model with lmer(), to take into account my blocking factors and compare it with the previous one. I was only able to ask for a random intercept, but it is not really what I want:

    lER3C = lmer(rETR ~ 0 + PPFD:Accession + (1|Date:MoAf:Block), data=avps,   na.action=na.omit)


And then, out of curiosity, I tried to fit something similar to LM3E, but using lmer to see if they were going to give the same outcome. Since lmer() does not work without random effects, I had to specify as random what I would like to keep as fixed:

    lER3 = lmer(rETR ~ 0 + (PPFD-1|Accession), data=avps, na.action=na.omit)


My concerns are:

1. What are the differences between lER3 and lER3E? The slope coefficients given by the two models are exactly the same. Is it better to specify Accession as a random effect (so the opposite of what I would like to do?) or in this case it does not matter, but it would if I also included my blocking factors?

2. If I'm trying to specify a linear model with origin in (0;0), lER3C is somehow pushing me away from what should be "reality" (no rETR in darkness), but I think it also better corrects for variation between days etc. (Maybe not really a question for stackexchange?)

This is my first question posted here, so I'm sorry if it's far too long and verbose :)