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?
 A: 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.

A: 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.
