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I am aiming to see if there is any difference of root production (as new roots) between forests of different ages (site 1 =10 years, site2 = 22 years, site3 =58 years).

As experimental design, I have 7 tubes per sites. In each of the tubes, I have for 4 depths. In each of the depths, I measure the number of root at 80 and 95 degrees direction in the soil. And I repeated those measurements 3 times.

My design is hence: measurement time (3) x site (3) x tubes (15) x depth (4) x angles (2). In total I have 504 datapoints.

Therefore, I have the angle nested into depth, nested into tubes, nested into site, and in addition I have a repeated measurement.

I would like to know if my root number is different across sites, and also I would like to know if the root number is different at depth for each site.

I am hesitating between the option below on how to write my linear mixed model in R. In addition, can be a fixed effect also a nested random effect- if not why? do you have any litterature on that?*

Model1 = lmer(Root number  ~ Site * Depth + (1|Tube)+ (1|MeasurementTime)+ (1|Angle), data = DataprodF)

--> Model 1 return a singular fit, why? is it because I don't have enough data?

Model2 = lmer(Root number  ~ Site * Depth + (1|Tube/Angle)+ (1|MeasurementTime), 
               data = DataprodF)

--> Model 2 return a singular fit

Model3 = lmer(Root number ~ Site * Depth + (1|Site/Depth/Tube/Angle)+ (1|MeasurementTime),  data = DataprodF)

--> Model 3 return a singular fit

A sample of my data : All except root number are as factor. App.sess is measurement time (after 4, 6 and 8 months). Angle is A= 80 degrees and B =90 degrees.

enter image description here

Thanks a lot for your help!

Novice_stat
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  • This does not seem like nested factors (at least not to the extent you suggest). Have you read [this](https://stats.stackexchange.com/questions/228800/crossed-vs-nested-random-effects-how-do-they-differ-and-how-are-they-specified/228814#228814) ? With very few levels of some of your factors it may not makes sense for them to be random. Why do you think they are nested? To he nested, a lower- evel factor level has to belong to one and only one upper-level factor level. – Robert Long Jul 30 '20 at 03:29
  • thanks for the link robert. I have read it. I still think I have nested factors. Here, the tubes are the school, and the depths are the class, and the angle are "working group" within the class. Is that not correct? – Novice_stat Jul 30 '20 at 08:40
  • Well, for example, Angle `A` does not "belong" to tube 5, other tubes also have Angle `A`. Is there anything different about angle A in tube 5 and angle A in tube 4 ? In a school setting, we could have class 1 in school 1 and class 1 in school 2, but these 2 classes are completely different. – Robert Long Jul 30 '20 at 08:47
  • The only difference between the angle in tube 4 and 5, is there spatial location. One is within tube 5 and the other one is within tube 4. For the school setting, I have got what you said, I think I should have different tubes numbers for each sites, because they are differents tubes. It is like school 1, tube 1, 2,3; school 2 tubes 4,5,6 ect.. That make my nesting explicit. and I suppose the angle will need to have an unique identifier too. I am affraid that all with overfitt my model. – Novice_stat Jul 30 '20 at 09:00
  • I agree regarding tube, but I still don't see angle as nested. To me it seems similar to a treatment variable where different subjects get one of several treatments. We wouldn't consider treatment to be nested in subject - all the subjects who got a particular treatment got the same treatment. In your case all the tubes were placed at different angles but the tubaes that were placed at a particular angle were placed at the *same* angle, right ?? – Robert Long Jul 30 '20 at 09:51
  • Thanks a lot for clarifying and your help Robert- It is much appreciated and it clearer for me know now. I think, for the angles, I have miss explain the design. The tubes are not inserted with an angle, I took the image at the same depth, but one on the right direction and one of the left. It is like being in a car, and looking throught the right and left windows. – Novice_stat Jul 30 '20 at 10:38
  • Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/111219/discussion-between-robert-long-and-novice-stat). – Robert Long Jul 30 '20 at 10:51

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