Writing lme mixed model as equation I have a linear mixed model I have constructed in lme as follows with random intercept and slope.
summary(lme(X ~x1*x2 + x3*x4 + x5 + random=  ~1+ site|ID, data=data))

Site are different locations, ID is participants IDs.
Previous questions Asked here are similar but have slightly different model.
How do I write this in the form of a model equation? I think this might be useful to have as a comparison for others constructing models in with lme.
Edit. X1-4 are categorical and 5 is continuous. (In the real example this is a RCT, dummy variables for time points in interaction with trial-arm then a covariate X5).
 A: There is a very interesting package called equatiomatic which can produce equations for a large range of model types. lme is not supported at present but it is trivial to re-write your model using lmer:
m0 <- lmer(X ~ x1*x2 + x3*x4 + x5 + site + (1+site|ID), data=data)
extract_eq(m0)

which produces the following:
$$
\begin{aligned}
  \operatorname{X}_{i}  &\sim N \left(\mu, \sigma^2 \right) \\
    \mu &=\alpha_{j[i]} + \beta_{1}(\operatorname{x1}) + \beta_{2}(\operatorname{x2}) + \beta_{3}(\operatorname{x3}) + \beta_{4}(\operatorname{x4}) + \beta_{5}(\operatorname{x5}) + \beta_{6j[i]}(\operatorname{site}) + \beta_{7}(\operatorname{x1} \times \operatorname{x2}) + \beta_{8}(\operatorname{x3} \times \operatorname{x4}) \\    
\left(
  \begin{array}{c} 
    \begin{aligned}
      &\alpha_{j} \\
      &\beta_{6j}
    \end{aligned}
  \end{array}
\right)
  &\sim N \left(
\left(
  \begin{array}{c} 
    \begin{aligned}
      &\mu_{\alpha_{j}} \\
      &\mu_{\beta_{6j}}
    \end{aligned}
  \end{array}
\right)
, 
\left(
  \begin{array}{cc}
     \sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{6j}} \\ 
     \rho_{\beta_{6j}\alpha_{j}} & \sigma^2_{\beta_{6j}}
  \end{array}
\right)
 \right)
    \text{, for ID j = 1,} \dots \text{,J}
\end{aligned}
$$
