19-linear_mixed_effects_models3.Rmd

# Linear mixed effects models 3

## Learning goals

- Pitfalls in fitting `lmers()`s (and what to do about it). 
- Understanding `lmer()` syntax even better.
- ANOVA vs. lmer 

## Load packages and set plotting theme

```{r, message=FALSE}
library("knitr")       # for knitting RMarkdown 
library("kableExtra")  # for making nice tables
library("janitor")     # for cleaning column names
library("broom.mixed") # for tidying up linear mixed effects models 
library("patchwork")   # for making figure panels
library("lme4")        # for linear mixed effects models
library("afex")        # for ANOVAs
library("car")         # for ANOVAs
library("datarium")    # for ANOVA dataset
library("modelr")      # for bootstrapping
library("boot")        # also for bootstrapping
library("ggeffects")   # for plotting marginal effects
library("emmeans")     # for marginal effects
library("tidyverse")   # for wrangling, plotting, etc. 
```

```{r}
theme_set(theme_classic() + #set the theme 
            theme(text = element_text(size = 20))) #set the default text size

# knitr display options 
opts_chunk$set(comment = "",
               fig.show = "hold")

# # set contrasts to using sum contrasts
# options(contrasts = c("contr.sum", "contr.poly"))

# suppress grouping warning messages
options(dplyr.summarise.inform = F)
```

## Load data sets

### Reasoning data

```{r}
df.reasoning = sk2011.1
```

## Understanding the lmer() syntax

Here is an overview of how to specify different kinds of linear mixed effects models.

```{r, echo=F}
tibble(formula = c("`dv ~ x1 + (1 | g)`",
                   "`dv ~ x1 + (0 + x1 | g)`",
                   "`dv ~ x1 + (x1 | g)`",
                   "`dv ~ x1 + (x1 || g)`",
                   "`dv ~ x1 + (1 | school) + (1 | teacher)`",
                   "`dv ~ x1 + (1 | school) + (1 | school:teacher)`"),
       description = c("Random intercept for each level of `g`",
                       "Random slope for each level of `g`",
                       "Correlated random slope and intercept for each level of `g`",
                       "Uncorrelated random slope and intercept for each level of `g`",
                       "Random intercept for each level of `school` and for each level of `teacher` (crossed)",
                       "Random intercept for each level of `school` and for each level of `teacher` in `school` (nested)")) %>% 
  kable()
```

Note that this `(1 | school/teacher)` is equivalent to `(1 | school) + (1 | teacher:school)` (see [here](https://stats.stackexchange.com/questions/228800/crossed-vs-nested-random-effects-how-do-they-differ-and-how-are-they-specified)). 

## ANOVA vs. lmer

### Between subjects ANOVA

Let's start with a between subjects ANOVA (which means we are in `lm()` world). We'll take a look whether what type of `instruction` participants received made a difference to their `response`. 

First, we use the `aov_ez()` function from the "afex" package to do so. 

```{r}
aov_ez(id = "id",
       dv = "response",
       between = "instruction",
       data = df.reasoning)
```

Looks like there was no main effect of `instruction` on participants' responses. 

An alternative route for getting at the same test, would be via combining `lm()` with `joint_tests()` (as we've done before in class). 

```{r}
lm(formula = response ~ instruction,
   data = df.reasoning %>% 
     group_by(id, instruction) %>% 
     summarize(response = mean(response)) %>% 
     ungroup()) %>% 
  joint_tests()
```

The two routes yield the same result. Notice that for the `lm()` approach, I calculated the means for each participant in each condition first (using `group_by()` and `summarize()`). 

### Repeated-measures ANOVA

Now let's take a look whether `validity` and `plausibility` affected participants' responses in the reasoning task. These two factors were varied within participants. Again, we'll use the `aov_ez()` function like so: 

```{r}
aov_ez(id = "id",
       dv = "response",
       within = c("validity", "plausibility"),
       data = df.reasoning %>% 
         filter(instruction == "probabilistic"))
```

For the linear model route, given that we have repeated observations from the same participants, we need to use `lmer()`. The repeated measures ANOVA has the random effect structure as shown below: 

```{r}
lmer(formula = response ~ 1 + validity * plausibility + (1 | id) + 
       (1 | id:validity) + (1 | id:plausibility),
     data = df.reasoning %>% 
        filter(instruction == "probabilistic") %>%
        group_by(id, validity, plausibility) %>%
        summarize(response = mean(response))) %>% 
  joint_tests()
```

Again, we get a similar result using the `joint_tests()` function. 

Note though that the results of the ANOVA route and the `lmer()` route weren't identical here (although they were very close). For more information as to why this happens, see [this post](https://stats.stackexchange.com/questions/117660/what-is-the-lme4lmer-equivalent-of-a-three-way-repeated-measures-anova).

### Mixed ANOVA

Now let's take a look at both between- as well as within-subjects factors. Let's compare the `aov_ez()` route

```{r}
aov_ez(id = "id",
       dv = "response",
       between = "instruction",
       within = c("validity", "plausibility"),
       data = df.reasoning)
```

with the `lmer()` route: 

```{r}
lmer(formula = response ~ instruction * validity * plausibility + (1 | id) + 
       (1 | id:validity) + (1 | id:plausibility),
      data = df.reasoning %>%
        group_by(id, validity, plausibility, instruction) %>%
        summarize(response = mean(response))) %>% 
  joint_tests()
```
Here, both routes yield the same results. 

## Additional resources

### Readings

- [Nested and crossed random effects in lme4](https://www.muscardinus.be/statistics/nested.html)

## Session info

Information about this R session including which version of R was used, and what packages were loaded. 

```{r}
sessionInfo()
```