# Is a tilde or an equals sign correct in linear mixed model formulas?

I know the formula for a linear mixed model (LMM) is often (always?) written with a tilde, rather than an equals sign, between the LHS and RHS. For example, one would write

outcome ~ 1 + var1 + (1|var2)

to denote that the outcome variable is modeled by an intercept plus var1 plus var2 with random effects (random intercept model).

I am doing proofs of a paper with LMM equations in the methods, and the journal for some reason has trouble rendering the tilde symbol properly (appears instead as a dash). So my questions is, is it also acceptable to write the formula with an equals sign instead of a tilde? Is there a difference, or would this be the incorrect way of writing a formula for a LMM?

• I would like to revolt against the weird journal not being able to render a tilde ~, but I have to say that it is no problem to use an equal sign. See for instance the use on Wikipedia which is very standard. The tilde sign is a notation used in R to write regression equations but elsewhere it has a different meaning as mentioned in Alexia's answer. Commented Nov 23, 2021 at 18:13
• More about the tilde see: stats.stackexchange.com/questions/531826/… Commented Nov 23, 2021 at 18:14
• If this is actual programming code, then you may need the need the tilde for the code to run Commented Nov 23, 2021 at 18:22
• There's certainly a difference. "$=$" means that the two things on either side are identical (e.g. if you're talking about numbers, they have identical values), which is not the case here. What you could do is (a) simply report the RHS of the model and write your discussion in a way that explains that its a model for $y$, or (b) give up and use "$=$" - or perhaps some other symbol like "$\approx$" - but on first use specify that you don't mean mathematical equality (or whatever the symbol you use conventionally means) and then explain what you're using it to mean instead. Commented Nov 23, 2021 at 22:21
• The formula you included in the OP is R code used for supplying a LMM to the lme4::lmer() function; it is not universal notation (e.g., it wouldn't work for nlme::lme()) and therefore should not be how you denote your model in a paper. Use the model in @AdamO's answer, which is formal statistical notation.
– Noah
Commented Nov 24, 2021 at 19:27

If the journal is typesetting in LaTeX, the formula needs to be in math mode and use the \sim expression $$\sim$$ which is different from ~ in text mode - you can see the difference, no? But that's not your job, and I'm sure the journal has a steep publication fee because why?

Anyway, the correct way to express a LMM is not using the $$\sim$$ notation because it means "is distributed as". If I were a reviewer or editor, I would insist to define variables: so say "var1" is obesity/overweight. You have also defined a random intercept in var2. Is this actually a covariate or a subject identifier? You can already see how this is become confusing for the audience. Assuming var2 is a subject identifier, this is a simple example of a random intercepts model, or repeated measures ANOVA. Lastly suppose Y is creatinine level, define them as $$\text{Obese}, \text{Subject}$$, and $$\text{Creat}$$ respectively.

My preferred way to express your mixed model is WAY more formal, such as the below

$$\text{Creat}_{ij} = \beta_0 + \beta_1 \text{Obese} + \epsilon_j+ \epsilon_{i,j}$$

Then interpret the model by saying,

The repeated measures ANOVA models creatinine level for the $$i$$-th subject at time $$j$$ with a fixed effect of obesity at baseline, a random intercept for subject, and $$\epsilon_{i,j}$$ a random error term.

• I am not seeing a mixed model here: e.g., there's no $j$-level random term like $\mu_{0j}$? Commented Nov 23, 2021 at 20:39
• @Alexis that's the $\beta_{2,j}$. This is similar to the Diggle, Heagerty, Liang, Zeger, notation. Commented Nov 23, 2021 at 21:01
• Got it... I am not accustomed to seeing mixed models written sans explicit notation for random effects (random intercepts, and random slopes), but know that there are a bunch of different ways of presenting these models. Commented Nov 23, 2021 at 21:06
• I would probably say $\beta_0 + \beta_1 \textrm{Obese} + \epsilon_{i,T} T_i + \epsilon_{r,ij}; \epsilon_T \sim N(0,\sigma^2_T); \epsilon_{r} \sim N(0, \sigma^2_\epsilon)$ to complete the specification, and note that $T_i$ is an indicator variable. Commented Nov 23, 2021 at 22:48
• (1|var2) explicitly denotes variation in intercepts by levels of a (categorical) grouping variable (your "subject identifier") var2. I agree your notation above (although I would generally augment it with the distributional information about $\epsilon_j$ and $\epsilon_{i,j}$ - you could argue that the distributional information is relatively unimportant if asymptotics etc., but I would rather be complete). (Also, I agree that the OP shouldn't use R syntax in a formal setting, but this is fairly widely used: bbolker.github.io/mixedmodels-misc/… ) Commented Nov 29, 2021 at 17:43

The tilde as a relational operator is frequently used in a statistical context to indicate "is distributed as," and read "the term(s) on the left are distributed as indicated by the terms on the right." You will frequently see this indicating a simple distribution model, such as the "$$\varepsilon \sim \mathcal{N}(0,\sigma^{2})$$" in a regression context, indicating the errors are modeled with a normal distribution centered on zero and with variance $$\sigma^{2}$$. In a regression context you may also see an expectation of the dependent variable conditional on independent variables, or complex error structures expressed this way. (Aside: What you have written looks more like R code than mathematical notation in a textual context, and @AdamO makes a good point about deficiencies in this notation.)

By contrast the equals sign as a definitional operator is used to indicate just that: mathematical equality in a definitional sense; for example, in a regression context, you may define the way every observed value of a dependent variable is a function of both a fixed, or deterministic model, and a random, or stochastic model (the latter of which is often subsequently expressed with tilde notation to indicate distribution model, as in the snippet in the above paragraph). Sometimes a link function is left implicit in the mathematical formalism (e.g., where the text indicates something like "where the term to the left of the equals sign is the anti-link function of the dependent variable"), and sometimes it will be explicitly articulated in the mathematical formalism on either the left or right side of the equals sign.

• +1 it's important to note that R's formula language is endemic to R. It's idiosyncratic and has some deficiencies. The R formula expression does not suffice to explain a regression model in a scientific journal. For instance, y ~ x*w implies that $E[Y|X,W] = \gamma X W$ meaning there's no intercept or the 1st order effects of $X$ or $W$, but R adds those in for you. Commented Nov 23, 2021 at 18:28
• You mean that the distribution has SD $\sigma$, I guess -- although in my experience the notation $N(\mu, \sigma^2)$ is as or more common than $N(\mu, \sigma)$. Commented Nov 23, 2021 at 18:38
• @NickCox Changed. Commented Nov 23, 2021 at 20:40
• @AdamO the notation is not unique to R (so I wouldn't call it endemic). It's based on Wilkinson and Rogers notation and is used by programs like GLIM and Genstat; R makes some implementation changes for various reasons (e.g. it has to replace "$.$" with something else, so it uses "$:$") and because it broadens the use of the notation, there's a few oddities that creep in to avoid clashes with other bits of R. Commented Nov 23, 2021 at 22:29
• @Glen_b good point. However, I daresay we both agree: computer code does not suffice to express a linear model any more than unformatted computer output suffices to report the results of a regression. Commented Nov 24, 2021 at 16:41