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I had previously read some on this topic and my understanding was that a fixed-effect model is when you are sampling an entire population, for example, a school district, a school, or a village. And a random-effects model would be used when taking from any of the populations a sample where the goal of the random-effects model would be to estimate the population. I am reading Successful hunting increases testosterone and cortisol in a subsistence population and the authors describe:

Individuals were modelled as random effects to control for non-independence of multiple specimens collected from the same participant.

I am not understanding this that well and am asking for further explanation, please. While on the one hand, it seems because the entire population (viz. the village) was not sampled we would use the random-effects model.

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    $\begingroup$ Is there some reason you aren't asking this on Cross Validated? If not, I recommend you move the non-biology questions to that site and delete them from this one; NB: please do not crosspost. ——— Please also post each question separately — this helps you get answers to each question and makes those answers much easier to find for future users. Please see the tour and consult the help center starting with How to Ask for details. $\endgroup$
    – tyersome
    Jun 29 at 4:24
  • $\begingroup$ @Tyersome IMO this is a good fit for the site. It's contextualising a statistical question in terms of biology - the biology is key to the question. $\endgroup$
    – user438383
    Jun 29 at 8:27
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    $\begingroup$ There are too many questions here; I've removed the additional questions, but you can ask them separately. $\endgroup$
    – Bryan Krause
    Jun 30 at 17:17
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    $\begingroup$ For the quoted text, the mixed vs. fixed has nothing to do with sampling the entire population vs. a subset but rather with multiple samples per individual. This makes the samples non-independent, so a mixed model is necessary. The question is better answered here: stats.stackexchange.com/questions/4700/… $\endgroup$
    – kmm
    Jun 30 at 18:18

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While I am a biologist by trade, not a statistician, I still do a fair bit of statistical modeling. It seems like you were taught a flawed, or at least incomplete understanding of mixed effects modeling. While your experimental design is what determines whether an effect is fixed or random (at least it should be), the description you gave doesn't really match with how they are used (at least in my experience with modeling). I'll try to break that statement down a bit just based on what you wrote in the question.

... my understanding was that a fixed-effect model is when you are sampling an entire population, for example, a school district, a school, or a village. And a random-effects model would be used when taking from any of the populations a sample where the goal of the random-effects model would be to estimate the population.

First, you don't need to sample the entire population to measure a fixed effect. If you surveyed 10% of the people in a village at random, you still model your response variable as a fixed effect for that village. But, say instead you sampled 10% of the household units in a village, sampling each person within each household once. There are likely to be various environmental or genetic factors that are shared within a household that vary between different households. Because of that non-independence, each set of household samples can be thought of as it's own sub-population of samples that aren't truly independent from one another. So you would include a random effect term for households in your model, to account for those variations. But the actual response variable (or variables) of interest would still be modeled as fixed effects.

The same goes for the example in the paper. Even if the entire population had been sampled, each individual has their own unique set of genes and metabolic processes that can vary from person to person, so those repeated measures are not independent from one another. Modeling individuals as random effects allows you to see how your variables of interest are related at a population level without being confounded by those little difference between people. This is one reason that cross-sectional study designs, where each subject is sampled only once, can be limited when it comes to generating generalizable inferences about a larger population. Without those longitudinal samples, it's not possible to account for any of that inter-individual variation in your models.

Mixed models can get pretty complex pretty easily with more comprehensive sampling efforts and study designs. Say you have longitudinal samples collected from multiple households across multiple villages. Now you have to consider how all of those confounding variables (village, household, individual) are related to one another and design statistical models in which each of those terms are appropriately nested to properly account for random effects.

For some more reading, I think this chapter from Applied Statistics for Experimental Biology by Jeffrey Walker does a pretty good job of explaining mixed how models work with some good examples of how they are constructed for different scenarios.

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    $\begingroup$ I think the confusion OP has about "sampling the whole population" should be about sampling the levels of factor, rather than the population. Generally, you use fixed effects when your data include all the levels of a factor (for example: Smokers and Non-Smokers cover everyone; smoking/non is a fixed effect), and random effects when they don't (for example, you sampled students in Paris, London, and Berlin; city is a random effect). However, that's not a comprehensive explanation of the difference, just a superficial guideline. It's certainly important for OP to learn more before using this. $\endgroup$
    – Bryan Krause
    Jun 30 at 16:12
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    $\begingroup$ (...cont) If you are "sampling each person within each household once" you don't need any random effects for the household - your error by household is wrapped up in your overall error term. However, if you sampled multiple people in each household, you may want a random effect of household to account for the part of the error that is shared among people in a household. It's only necessary when you have multiple observations there. $\endgroup$
    – Bryan Krause
    Jun 30 at 17:08
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    $\begingroup$ (...cont) Same thing at the village level: if you're sampling multiple villages, you'd probably have village as a random effect, not a fixed effect, if those villages are somewhat arbitrarily selected from all the possible villages in the world or in a country or whatever. If, however, you want to compare Springfield to Shelbyville as a research question, you'd model village as a fixed effect: you haven't chosen those villages randomly in that case, you've selected them specifically as part of your research question. $\endgroup$
    – Bryan Krause
    Jun 30 at 17:09

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