# Fixed effect vs random effect when all possibilities are included in a mixed effects model

In a mixed effects model the recommendation is to use a fixed effect to estimate a parameter if all possible levels are included (e.g., both males and females). It is further recommended to use a random effect to account for a variable if the levels included are just a random sample from a population (enrolled patients from the universe of possible patients) and you want to estimate the population mean and variance instead of the means of the individual factor levels.

I am wondering if you are logically obliged to always use a fixed effect in this manner. Consider a study on how foot / shoe size changes through development and is related to, say, height, weight and age. ${\rm Side}$ clearly must be included in the model somehow to account for the fact that the measurements over the years are nested within a given foot and are not independent. Moreover, right and left are all the possibilities that can exist. In addition, it can be very true that for a given participant their right foot is larger (or smaller) than their left. However, although foot size does differ somewhat between the feet for all people, there is no reason to believe that right feet will on average be larger than left feet. If they are in your sample, this is presumably due to something about the genetics of the people in your sample, rather than something intrinsic to right-foot-ness. Finally, ${\rm side}$ seems like a nuisance parameter, not something you really care about.

Let me note that I made this example up. It may not be any good; it is just to get the idea across. For all I know, having a large right foot and a small left foot was necessary for survival in the paleolithic.

In a case like this, would it make (more / less / any) sense to incorporate ${\rm side}$ in the model as a random effect? What would be the pros and cons of using a fixed vs. random effect here?

• why would you want to treat side as a random factor given that there are two and only two levels of the factor? Where does the randomness come from in your problem setting? – Aaron Zeng Oct 21 '14 at 21:18
• @AaronZeng, setting aside the quality of my example, that is my question. Is there ever any reason to represent levels with random effects if you have all possible levels. What if the factor in question had >2 levels? – gung Oct 21 '14 at 21:21
• @gung I just came back to this thread - did any of the answers help? If not - what more are you interested to learn? Maybe you have your own answer (if yes, I'd be interested to learn more about this issue!)? – Tim Nov 24 '15 at 9:28
• It's been a long time since I've been back here, @Tim. I appreciate both answers (I upvoted them), but they aren't quite what I was looking for (probably due to an insufficiently clear question statement). I have thought about compiling an answer from some stuff Ben Bolker has posted in various places, but it would be a bit of work & I've never actually gotten it done. It is still something I should do, though. Thanks for the nudge. – gung Nov 24 '15 at 18:06

The general problem with "fixed" and "random" effects is that they are not defined in a consistent way. Andrew Gelman quotes several of them:

(1) Fixed effects are constant across individuals, and random effects vary. For example, in a growth study, a model with random intercepts $a_i$ and fixed slope $b$ corresponds to parallel lines for different individuals $i$, or the model $y_{it} = a_i + b_t$. Kreft and De Leeuw (1998) thus distinguish between fixed and random coefficients.

(2) Effects are fixed if they are interesting in themselves or random if there is interest in the underlying population. Searle, Casella, and McCulloch (1992, Section 1.4) explore this distinction in depth.

(3) “When a sample exhausts the population, the corresponding variable is fixed; when the sample is a small (i.e., negligible) part of the population the corresponding variable is random.” (Green and Tukey, 1960)

(4) “If an effect is assumed to be a realized value of a random variable, it is called a random effect.” (LaMotte, 1983)

(5) Fixed effects are estimated using least squares (or, more generally, maximum likelihood) and random effects are estimated with shrinkage (“linear unbiased prediction” in the terminology of Robinson, 1991). This definition is standard in the multilevel modeling literature (see, for example, Snijders and Bosker, 1999, Section 4.2) and in econometrics.

and notices that they are not consistent. In his book Data Analysis Using Regression and Multilevel/Hierarchical Models he generally avoids using those terms and in their work he focuses on fixed or varying between groups intercepts and slopes because

Fixed effects can be viewed as special cases of random effects, in which the higher-level variance (in model (1.1), this would be $\sigma^2_\alpha$ ) is set to $0$ or $\infty$. Hence, in our framework, all regression parameters are “random,” and the term “multilevel” is all-encompassing.

This is especially true with Bayesian framework - commonly used for mixed models - where all the effects are random per se. If you are thinking Bayesian, you are not really concerned with "fixed" effects and point estimates and have no problem with treating all the effects as random.

Best case scenario argument

Say you have unlimited amounts of data and unlimited computational power. In this case you could imagine estimating every effect as fixed, since fixed effects give you more flexibility (enable us to compare the individual effects). However, even in this case, most of us would be reluctant to use fixed effects for everything.

For example, imagine that you want to model exam results of schools in some region and you have data on all the 100 schools in the region. In this case you could threat schools as fixed - since you have data on all the levels - but in practice you probably would rather think of them as random. Why is that?

1. One reason is that generally in this kind of cases you are not interested in effects of individual schools (and it is hard to compare all of them), but rather a general variability between schools.

2. Another argument in here is model parsimony. Generally you are not interested in "every possible influence" model, so in your model you include few fixed effects that you want to test and control for the other possible sources of variability. This makes mixed effects models fit the general way of thinking about statistical modeling where you estimate something and control for other things. With complicated (multilevel or hierarchical) data you have many effects to include, so you threat some as "fixed" and some as "random" so to control for them.

3. In this scenario you also wouldn't think of the schools as each having its own, unique, influence on the results, but rather as about schools having some influence in general. So this argument would be that we believe that is is not really possible to estimate the unique effects of individual schools and so we threat them as random sample of possible schools effects.

Mixed effects models are somewhere in between "everything fixed" and "everything random" scenarios. The data we encounter makes us to lower our expectations about estimate everything as fixed effects, so we decide what effects we want to compare and what effects we want to control, or have general feeling about their influence. It is not only about what the data is, but also how we think of the data while modeling it.

• Lots of good points here, @Tim. I am wondering what your take is on the gung's example in the OP; there was a long discussion in the comments under my answer but I think by now it is finally more or less resolved. Would be good to know if you agree or perhaps disagree with what I wrote. – amoeba Oct 2 '16 at 12:41
• @amoeba it's an interesting answer (I already +1'd) and I agree with your point. I think that essentially gung is right (the same as Gelman - who's always right :) ) that there is no single answer. There is a huge literature and multiple ways to employ mixed effect models and no clear-cut distinction. Moreover, there are people who by default always use fixed effects for everything and there are ones who use random effects whenever they can, even in cases that we would generally rather consider as fixed effects... It also depends on what exactly you want to model. – Tim Oct 2 '16 at 17:39

## Executive summary

It is indeed often said that if all possible factor levels are included in a mixed model, then this factor should be treated as a fixed effect. This is not necessarily true FOR TWO DISTINCT REASONS:

(1) If the number of levels is large, then it can make sense to treat the [crossed] factor as random.

I agree with both @Tim and @RobertLong here: if a factor has a large number of levels that are all included in the model (such as e.g. all countries in the world; or all schools in a country; or maybe the entire population of subjects is surveyed, etc.), then there is nothing wrong with treating it as random --- this could be more parsimonious, could provide some shrinkage, etc.

lmer(size ~ age + subjectID)                     # fixed effect
lmer(size ~ age + (1|subjectID))                 # random effect


(2) If the factor is nested within another random effect, then it has to be treated as random, independent of its number of levels.

There was a huge confusion in this thread (see comments) because other answers are about case #1 above, but the example you gave is an example of a different situation, namely this case #2. Here there are only two levels (i.e. not at all "a large number"!) and they do exhaust all possibilities, but they are nested inside another random effect, yielding a nested random effect.

lmer(size ~ age + (1|subject) + (1|subject:side)  # side HAS to be random


## Detailed discussion of your example

Sides and subjects in your imaginary experiment are related like classes and schools in the standard hierarchical model example. Perhaps each school (#1, #2, #3, etc.) has class A and class B, and these two classes are supposed to be roughly the same. You will not model classes A and B as a fixed effect with two levels; this would be a mistake. But you will not model classes A and B as a "separate" (i.e. crossed) random effect with two levels either; this would be a mistake too. Instead, you will model classes as a nested random effect inside schools.

In your imaginary foot-size study, subject and side are random effects and side is nested inside subject. This essentially means that a combined variable is formed, e.g. John-Left, John-Right, Mary-Left, Mary-Right, etc., and there are two crossed random effects: subjects and subjects-sides. So for subject $i=1\ldots n$ and for side $j=1,2$ we would have:

$$\text{Size}_{ijk} = \mu+\alpha\cdot\text{Height}_{ijk}+\beta\cdot\text{Weight}_{ijk}+\gamma\cdot\text{Age}_{ijk}+\epsilon_i + \color{red}{\epsilon_{ij}} + \epsilon_{ijk}$$ $$\epsilon_i\sim\mathcal N(0,\sigma^2_\mathrm{subjects}),\quad\quad\text{Random intercept for each subject}$$ $$\color{red}{\epsilon_{ij}}\sim\mathcal N(0,\sigma^2_\text{subject-side}),\quad\quad\text{Random int. for side nested in subject}$$ $$\epsilon_{ijk}\sim\mathcal N(0,\sigma^2_\text{noise}),\quad\quad\text{Error term}$$

As you wrote yourself, "there is no reason to believe that right feet will on average be larger than left feet". So there should be no "global" effect (neither fixed nor random crossed) of right or left foot at all; instead, each subject can be thought of having "one" foot and "another" foot, and this variability we should include into the model. These "one" and "another" feet are nested within subjects, hence nested random effects.

More details in response to the comments. [Sep 26]

My model above includes Side as a nested random effect within Subjects. Here is an alternative model, suggested by @Robert, where Side is a fixed effect:

$$\text{Size}_{ijk} = \mu+\alpha\cdot\text{Height}_{ijk}+\beta\cdot\text{Weight}_{ijk}+\gamma\cdot\text{Age}_{ijk} + \color{red}{\delta\cdot\text{Side}_j}+\epsilon_i + \epsilon_{ijk}$$

I challenge @RobertLong or @gung to explain how this model can take care of the dependencies existing for consecutive measurements of the same Side of the same Subject, i.e. of the dependencies for data points with the same $ij$ combination.

It cannot.

The same is true for @gung's hypothetical model with Side as a crossed random effect:

$$\text{Size}_{ijk} = \mu+\alpha\cdot\text{Height}_{ijk}+\beta\cdot\text{Weight}_{ijk}+\gamma\cdot\text{Age}_{ijk} +\epsilon_i + \color{red}{\epsilon_j} + \epsilon_{ijk}$$

It fails to account for dependencies as well.

Demonstration via a simulation [Oct 2]

Here is a direct demonstration in R.

I generate a toy dataset with five subjects measured on both feet for five consecutive years. The effect of age is linear. Each subject has a random intercept. And each subject has one of the feet (either the left or the right) larger than another one.

set.seed(17)

demo = data.frame(expand.grid(age = 1:5,
side=c("Left", "Right"),
subject=c("Subject A", "Subject B", "Subject C", "Subject D", "Subject E")))
demo$size = 10 + demo$age + rnorm(nrow(demo))/3

for (s in unique(demo$subject)){ # adding a random intercept for each subject demo[demo$subject==s,]$size = demo[demo$subject==s,]$size + rnorm(1)*10 # making the two feet of each subject different for (l in unique(demo$side)){
demo[demo$subject==s & demo$side==l,]$size = demo[demo$subject==s & demo$side==l,]$size + rnorm(1)*7
}
}

plot(1:50, demo$size)  Apologies for my awful R skills. Here is how the data look like (each consecutive five dots is one feet of one person measured over the years; each consecutive ten dots are two feet of the same person): Now we can fit a bunch of models: require(lme4) summary(lmer(size ~ age + side + (1|subject), demo)) summary(lmer(size ~ age + (1|side) + (1|subject), demo)) summary(lmer(size ~ age + (1|subject/side), demo))  All models include a fixed effect of age and a random effect of subject, but treat side differently. 1. Model 1: fixed effect of side. This is @Robert's model. Result: age comes out not significant ($t=1.8$), residual variance is huge (29.81). 2. Model 2: crossed random effect of side. This is @gung's "hypothetical" model from OP. Result: age comes out not significant ($t=1.4$), residual variance is huge (29.81). 3. Model 3: nested random effect of side. This is my model. Result: age is very significant ($t=37$, yes, thirty-seven), residual variance is tiny (0.07). This clearly shows that side should be treated as a nested random effect. Finally, in the comments @Robert suggested to include the global effect of side as a control variable. We can do it, while keeping the nested random effect: summary(lmer(size ~ age + side + (1|subject/side), demo)) summary(lmer(size ~ age + (1|side) + (1|subject/side), demo))  These two models do not differe much from #3. Model 4 yields a tiny and insignificant fixed effect of side ($t=0.5\$). Model 5 yields an estimate of side variance equal to exactly zero.

• I don't really think that, in this example, side meets any of the usual definitions/guidelines of when a factor should be treated as random vs fixed. In particular, making inferences beyond the sampled levels of the factor is meaningless. Moreover, with only 2 levels of the factor, treating it as fixed seems an unambigious and straightforward way to approach the modelling. – Robert Long Sep 26 '16 at 19:28
• Robert, thanks for the reply. Either I am completely confused or I failed to explain properly what I mean. Treating side as a fixed effect means assuming that one of the sides (e.g. Right) is always bigger than the other (Left), by a certain amount. This amount is the same for all people. This is explicitly not what the OP had in mind. He wrote that in some people Right might be larger and in some other people Left. However, we need to account for the side because of correlated errors. Why can't we treat as a nested random effect then? It's exactly like classes within schools. – amoeba Sep 26 '16 at 19:44
• I don't know that it necessarily implies that. What it does say is that, in this sample, there may be a systematic difference between sides (which may or may not be an artifact due to sampling variation). I prefer to think about including it as a fixed effect as "controlling" for non-independence and nothing more - in the same way that we would add a confounder to a model and not even dream to try to interpret it's coefficient. – Robert Long Sep 26 '16 at 20:09
• I'm upvoting your answer after further reflection. You raise some really interesting points. I don't have time at the moment to delve into the maths of this. I'd like to find a toy dataset to play with if possible (if you know of one, please let me know) – Robert Long Sep 27 '16 at 15:22
• +1, on further reflection, you do seem to be right about the peculiarities of this study. Is the larger point that there isn't a single answer to the fixed vs random effect when all possibilities are included, & each case must be assessed individually, I wonder? – gung Oct 1 '16 at 23:45

I don't think you are logically obliged to always use a fixed effect in the manner described in the OP. Even when the usual definitions/guidelines for when to treat a factor as random are not met, I might be inclined to still model it as random when there are a large number of levels, so that treating the factor as fixed would consume many degrees of freedom and result in a cumbersome and less parsimonious model.

• This seems like a reasonable point, & I appreciate that you weren't blinded by my example. I gather from this, & your comment to @amoeba's answer, that "when there are a large number of levels" (vs "with only 2 levels of the factor") seems to be key. – gung Sep 26 '16 at 19:42
• +1 because I agree with this point, bit it drives me nuts that I failed to explain my point and that neither you nor @gung see what I meant. Treating the side either as fixed or as a crossed random effect necessarily means assuming that one of the sides (e.g. Right) is always bigger than the other (Left), for all subjects. This is explicitly not what gung wrote in his OP, stating that "there is no reason to believe that right feet will on average be larger than left feet". I still see gung's example as a clear case for nested random effect, in full analogy with classes within schools. – amoeba Sep 26 '16 at 19:48
• @amoeba interesting point but I don't agree. I'll comment in the comments to your answer... – Robert Long Sep 26 '16 at 20:07

If you're talking about the situation where you know all possible levels of a factor of interest, and also have data to estimate the effects, then definitely you don't need to represent levels with random effects.

The reason that you want to set random effect to a factor is because you wish to make inference on the effects of all levels of that factor, which are typically unknown. To make that kind of inference, you impose the assumption that the effects of all levels form a normal distribution in general. But given your problem setting, you can estimates the effects of all levels. Then there is certainly no need to set random effects and impose additional assumption.

It's like the situation that you are able to get all the values of the population (thus you know the true mean), but you are trying to take a large sample from the population and use central limit theorem to approximate the sampling distribution, and then make inference on the true mean.

• One comment: sometimes you have all levels but still use random effect for them. E.g. you conduct nation-vide study on education and have data on all the schools, but still you'll use random effect for schools rather then using dummies for each school. – Tim Oct 2 '16 at 13:33