Maybe, it helps to first understand the difference between fixed effects and random effects in multi-level models. First of all, both models you suggest are multi-level, more specifically two-level as your population of fishes is grouped in tanks. In both models, the predicted values can be expressed as the sum of the population average, the two main effects and the effect of Tank.

mu_i = beta_0 + x_Temp * beta_Temp + x_Food * beta_Food + x_Tank * beta_Tank

(in reality, b_Tank is a dummy variable)

The really only difference between fixed and random effects is that random effects are drawn from a Normal distribution and the variance of this distribution is estimated alongside. (Formally, a fixed effects estimate is drawn from a uniform distribution.)

beta_Tank,i ~ Normal(0, var_Tanks)

In my book, I explain random effects by the term population (link below, see chapters 6.3 and 6.7). A population is a set of entities that are similar and therefore clump together at the center to some extent, but vary around this center to some (other) extent.

For example: humans are a population with respect to tallness. Tallness varies, but most people are clumped together in a range, say 160 - 190. As a counter-example: the set of vehicles, bikes, motor cycles, cars, trains, aircrafts are not a population with respect to weight, because they form several clusters in very different places along the weight scale (i.e., not centered).

Now to your tanks. As this is a carefully designed experimental setup, I would assume that they are similar enough to satisfy the population criterion. Unless, and that seems to be your concern, complex dynamics within each Tank can make the set of Tanks "fall apart". A drastic example would be, that due to some subtle differences in starting conditions, either all Fish in Tank will die, or they will all thrive. If that is a possibility, I would rather not use random effects.

https://www.researchgate.net/project/Book-New-statistics-for-the-design-researcher

In a mixed multilevel model each tank is a level 2 unit and each fish is a level 1 unit. The degree of similarity within a tank (the degree of dependence ) is estimated by an Intra Cluster Correlation. This is defined as the ratio of the level 2 variance divide by the (sum of the level 2 and level 1 variance). If the tank makes no difference, then there will be no level 2 variance and ICC = 0. If however there are differences between tanks but all fish in each tank are the same, then the ICC will equal 1. In the mixed model the tank level random effect applies to all the fish in the tank , they have the same tank effect. The advantage of random effects over fixed effects (a dummy variable for each tank) is you get this measure of similarity within (ICC), you can include level 2 variables (attribute of the tank) and level 1 variables (attribute of a fish)and the standard errors of these (fixed) regression parameters for variables measured at level 1 and 2 are corrected for estimated degree of dependence.

Article Fixed and Random effects models: making an informed choice

Article Do multilevel models ever give different results?.

Article Explaining Fixed Effects: Random Effects Modeling of Time-Se...

There's a short paragraph on "controlling for non-independence in data-points" here: https://peerj.com/articles/4794/

The advantage of mixed effects models is that you can also account for non-independence among "slopes". As you said, you may assume more similarity from fish within tanks, but - e.g. - over time, fish within a tank may vary, too. In this case, or in general for longitudinal data, a random-slope-intercept-model is a good choice.

Furthermore, mixed models also have "shrinkage" on your estimates, reducing Type I/II error probabilities and giving more precise estimates.

Finally, if you have longitudinal data with time-varying predictors (fixed effects) that may correlate with your group (random) effects, this paper might be of interest for you: Bell, Andrew, Malcolm Fairbrother, and Kelvyn Jones. 2018. “Fixed and Random Effects Models: Making an Informed Choice.” Quality & Quantity. https://doi.org/10.1007/s11135-018-0802-x.

I tried to write down the formulas from the above paper in R, which you will find here:

https://strengejacke.github.io/mixed-models-snippets/random-effects-within-between-effects-model.html

Best

Daniel

Hmmm, interesting. So basically you are saying that with these models the replicates do not need to be indpendent of one another.

For example, if I am interested in the abundance of snails on an intertidal, I can measure densities in adjacent abutting 1x1m quadrats (i.e., non-random) and that still allows me to effectively estimate density at that site. This is despite the fact that a predator with a home range broader than 1x1m could live in one or more of the quadrats and influence what is in the adjacent quadrats. Does that mean it effectively accounts for the spatial autocorrelation in this example?

You have changed the question!

The two level model can be extended to include additional spatial autocorrelation at a number of different scales.

The final chapter of this shows how a cross-classified multiple membership model can include spatial weights:

Book Developing multilevel models for analysing contextuality, he...

The model deals with counts and repeated measures over time.

And this paper considers varying the scale to see at what scale a process comes into focus:

Article Modelling residential segregation as unevenness and clusteri...

And this considers some of the structures that a multilevel model can handle

http://www.bristol.ac.uk/cmm/software/mlwin/mlwin-resources.html#mlmbkgrd

and all of it is about the substantive benefits of understanding the nature of the dependency; and the technical benefit of better estimated standard errors come as a concomitant.

Thanks Kelvyn,

It's interesting to think of the problem as just another exercise in dependencies. I can see how there are similarities to spatial autocorrelation and repeated measures. I will have to sit down and spend so time to mull over the extent to which the dependencies of fish within a tank equate to spatial or temporal examples. Both the spatial and tempral examples seem to have a more predictable structure than behavioural interrelationships, unless the intensity of the behavioural interactions are quantified and maybe the social hierarchies is known. Indeed it is this predictable spatial and temporal structure that spatial autocorrelation and repeated measures exploit. Hmmm...

I am enjoying this exchange.

When I teach mixed multilevel models I particularly enjoy three things

1 discussing the implicit correlation structure that is imposed by a particular model. This is easiest seen in repeated measures. The simplest model with occasions nested in individuals with a random effect at each level generates a compound symmetry assumption with the correlation having a block diagonal structure (if occasions are sorted within individuals) so that there is just one correlation estimate for all individuals and all occasions, so however far apart the occasions are they are equally correlated. A very unrealistic assumption but maybe complex enough to capture the essence of what is going on. I am reminded of the George Box quote - all models are wrong but some are useful. You can make things a bit more realistic by adding more terms - a Toeplitz structure allows the dependence/autocorrelation to depend on the time lag between occasions , while the unstructured model is the most general of all so that each pair of occasions is allowed to have its own correlation estimate. But even that is for all individuals!

The same things apply to other situations in which these models are used. So if we have pupils nested in classes there is an implicit block diagonal correlation structure, the ICC gives both an estimate of how much of the variation lies at the class level and also how correlated two randomly chosen pupils in the same class are. You are hoping (it cannot be anything else) that you are getting some handle on this dependency but it maybe too unrealistic and that there are in reality overlapping networks of learners in the class. This could be modeled if you knew about them, but you usually do not. In the spatial case the standard two level model with people in places allows areas to be potentially important but does not include information about which areas are next to each other; and you maybe missing spillover effects (thereby breaking the models' SUTVA assumption). You can put additional spatial auto-correlation into the model but that requires specification of some spatial weighting scheme to define the joins. What impresses me about my colleague Peter Hagget's work is that he evaluates a number of alternative spatial structures finding for example that measles epidemics spreading by hierarchical diffusion at the start of an epidemic and then subsequently by contagion/ distance. The models are capable of greater sophistication but it is rather rare to do this. I am certainly guilty of this practice.

2 Modelling heterogeneity: the mixed model allows the estimation of complex within and between heterogeneity as a function of a measured variable. This is well known as random slopes modelling but I think is better seen as estimating a variance function. For example, the unexplained variation between classes in progress (conditioning on a base score) is structured by that score so that the class they are in really matters for low ability on entry pupils. You can also do this within class between pupils , that is explicit modelling of heteroscedasticity. I could see all of this being important for fish in a tank ( I know nothing about this!) so that the between tank variation and the within tank variation is dependent on the size of the fish ( a measured attribute). This of course implies that the degree of contexuality - the difference tanks make - depends on the nature of the measured attributes of the fish.

3 Fixed effects as opposed to random effects; the former (the gold standard in economics) is to put a dummy in for each higher level unit - 100 tanks needs 100 dummies. Random effects assumes the differences between tanks comes from a distribution and it is teh variance of that distribution that is estimated. The fixed effects approach gets rid of the tank effect as a nuisance - you estimate them but ignore them , you focus on the other measured level 1 variables and getting correct standard errors for those; contextual effects are not see as of importance. In contrast with random effects you are substantively interested in the between and within variation and you want to include variables at both level to structure this variance (random slopes etc) and to account for it. In RE models but not in FE models you can include level 2 variables to do this and you can have cross-level interactions to try and account for the random-slope variation.

Sorry for going on at such length!

On 1 this considers correlation structures in repeated measures and spatial models:

Book Developing multilevel models for analysing contextuality, he...

on 2 , this focuses on variance functions at each level

Article Modelling Complexity: Analysing Between-Individual and Betwe...

and on 3 , these papers debate fixed versus random

Article Explaining Fixed Effects: Random Effects Modeling of Time-Se...

Article Fixed and Random effects models: making an informed choice

Article Understanding and misunderstanding Group mean centering: a c...