time-varying-covariates.Rmd

---
title: "Time-varying covariates"
author: "Daniel Lüdecke"
date: "26 3 2019"
output: html_document
---

```{r setup, include=FALSE,echo=FALSE}
library(knitr)
knitr::opts_chunk$set(
  echo = TRUE,
  collapse = TRUE, 
  warning = FALSE,
  comment = "#>", 
  dev = "png"
)
```

Time-varying covariates are most flexibly modelled with splines. Here are some examples, from simple longitudinal models to models with time-varying covariates.

  * `y` = outcome (continuous / Gaussian)
  * `t` = time-points
  * `tv` = time-varying covariate
  * `tc` = time-constant covariate
  * `id` = subject-ID

```{r eval=FALSE}
# Model 1 - constant change in time
lmer(y ~ t + tv + tc + (1 + t | id), data)

# Model 2 - constant change in time, different slopes depending on covariate
# (interaction)
lmer(y ~ t * tv + tc + (1 + t | id), data)

# Model 3 - non-linear change in time
lmer(y ~ t + I(t^2) + tv + tc + (1 + t | id), data)

# Model 4 - non-linear change in time, different slopes depending on covariate
# (interaction)
lmer(y ~ t * tv + I(t^2) * tv + tc + (1 + t | id), data)

# Model 5 - non-linear change in time, time-varying covariate
# (interaction with non-linear covariate)
lmer(y ~ y ~ t * tv + I(t^2) * tv + t * I(tv^2) + I(t^2) * I(tv^2) + tc + (1 + t | id), data)

# Model 6 - non-linear change in time, time-varying covariate
# (cubic instead of quadratic interaction)
model6 <- lmer(
  y ~ t * tv + I(t^2) * tv + I(t^3) * tv + t * I(tv^2) + t * I(tv^3) +
    I(t^2) * I(tv^2) + I(t^2) * I(tv^3) + I(t^3) * I(tv^2) +
    I(t^3) * I(tv^3) + tc + (1 + t | id),
  data
)
```

This final model with splines is almost identical to model 6, when comparing marginal effects. The time-varying covariate needs to be specified in the `by`-argument from the spline-term. Time `t` is non-linear (spline), `tv` varies over time.

```{r eval=FALSE}
model7 <- lmer(y ~ s(t) + s(t, by = tv) + tc + (1 + t | id), data)
```

Note that for time-varying covariates, these may correlate with the group-level predictors. In such cases, a "complex random effects within-between" model is recommended, see https://strengejacke.github.io/mixed-models-snippets/random-effects-within-between-effects-model.html.