# Mixed model vs. n-way ANOVA for hierarchical data and proportions

I have a experimental setup (plant germination). I don't have concrete data yet. I am looking at how many plants germinate in given conditions, broken down as follows:

1) Two different temperatures.

2) For each of the two temperatures, I have 4 salinity concentrations.

3) For each salinity concentrations I have 2 levels of habitat from which the seeds were collected.

4) For each habitat type I have specific locations from which the seeds were collected.

The measure of interest (response) is the proportion of seeds that germinated at various time intervals. I know some people approached this via n-way ANOVA (judging from the literature). But it seems to be that a mixed model may be (more) appropriate.

QUESTIONS:

1) What would be a better approach and why?

2) If one used a mixed model, which of the variables (location, habitat type, salinity) would you include as a random effect?

3) Given that these are proportion or percentage data, what would be the best approach?

• – Scortchi - Reinstate Monica Jan 13 '17 at 15:56
• Thanks for that, although those two posts (and related ones) did not really give me the answers to my questions. – Tilen Jan 13 '17 at 18:17
• No - I just thought they're closely related - enough to be worth linking to & perhaps helpful to you &/or potential answerers.. – Scortchi - Reinstate Monica Jan 13 '17 at 18:39
• Indeed. I did read them though, so your effort wasn't wasted. :) Thanks again! – Tilen Jan 13 '17 at 22:04
• Thank you for asking for statistical advice before carrying out the experiments. That is too frequently not the case. The answer from @Stefan provides many helpful links for further study. – EdM Feb 6 '17 at 17:11

1) What would be a better approach and why?

(General) linear mixed models are much more flexible in modeling your data. For example, as you already mentioned you have to identify random terms. Random terms are variables for which you want variances to be estimated, i.e. you are not interested in mean values but more interested in capturing and accounting for the variation between those groups in your analysis. Furthermore, you would also add those variables in the random statement on which you performed multiple measurements, for example Subjects in a repeated-measures design. Here's more to read for you regarding this question:

2) If one used a mixed model, which of the variables (location, habitat type, salinity) would you include as a random effect?

Since there are multiple levels of nesting in your design (hierarchical design), this should go in the random statement. However, identifying the exact nature of the nesting structure depends on the statistical software you use and how your variables are set up. There are many posts here on CrossValidated on how to do this. For example: Mixed Effects Model with Nesting

3) Given that these are proportion or percentage data, what would be the best approach?

If you are using germination success (yes/no) as an outcome and want to have random effects calculated, I would go with a general linear mixed effects model and specify the binary nature of your outcome in the model. For example see here:

And in case you are using R, have a look here too: http://glmm.wikidot.com/faq#toc27

• Many thanks @Stefan! Indeed, I will be using R for this. I did consider a binary response GLMM, but I guess an actual binomial one (number of germinations out of n possible seeds) would likely be an even better alternative...? – Tilen Feb 7 '17 at 11:35
• @Tilen Yes, both works (see linked examples and example therein). Here some more information on worked glmer() 's: ms.mcmaster.ca/~bolker/R/misc/foxchapter/bolker_chap.html – Stefan Feb 7 '17 at 14:25
• No problem @Tilen ! If it answered your question you could consider accepting it. You could also provide a snippet of your data structure (update your question) and I can help you figuring out the random statement. Thanks! – Stefan Feb 12 '17 at 19:27
• @ Stefan: Done! I was away last week, so it took me a while to get through all the linked examples you provided. Many thanks again, they were really useful. I do not have data yet, so there's nothing I can currently provide. Also, the experimental design explained is something my co-workers are working on, I was merely interested in how one would go about analysing such a setup, to help them. :) – Tilen Feb 13 '17 at 21:28
• @Tilen Thanks! I am glad it helped you. Oops, yes I forgot that you were in the planning phase of the experiment :) Good luck! – Stefan Feb 14 '17 at 1:33