---
title: "Data Standardization"
output: 
  rmarkdown::html_vignette:
    toc: true
    fig_width: 10.08
    fig_height: 6
vignette: >
  \usepackage[utf8]{inputenc}
  %\VignetteIndexEntry{Data Standardization}
  %\VignetteEngine{knitr::rmarkdown}
---

```{r message=FALSE, warning=FALSE, include=FALSE}
library(knitr)
options(knitr.kable.NA = "")
knitr::opts_chunk$set(
  comment = ">",
  message = FALSE,
  warning = FALSE,
  dpi = 300
)

pkgs <- c(
  "datawizard",
  "poorman",
  "see",
  "ggplot2",
  "parameters",
  "lme4"
)

if (!all(sapply(pkgs, requireNamespace, quietly = TRUE))) {
  knitr::opts_chunk$set(eval = FALSE)
}
```

# Introduction

To make sense of their data and effects, scientists might want to standardize
(Z-score) their variables. This makes the data unitless, expressed only in terms
of deviation from an index of centrality (e.g., the mean or the median).
However, aside from some benefits, standardization also comes with challenges
and issues, that the scientist should be aware of.

## Methods of Standardization

The `datawizard` package offers two methods of standardization via the
`standardize()` function:

- **Normal standardization**: center around the *mean*, with *SD* units
  (default).

- **Robust standardization**: center around the *median*, with *MAD* (median
  absolute deviation) units (`robust = TRUE`).

Let's look at the following example:

```{r}
library(datawizard)
library(effectsize) # for data

# let's have a look at what the data look like
data("hardlyworking", package = "effectsize")
head(hardlyworking)

# let's use both methods of standardization
hardlyworking$xtra_hours_z <- standardize(hardlyworking$xtra_hours)
hardlyworking$xtra_hours_zr <- standardize(hardlyworking$xtra_hours, robust = TRUE)
```

We can see that different methods give different central and variation values:

```{r, eval=FALSE}
library(dplyr)
library(tidyr)

hardlyworking %>%
  select(starts_with("xtra_hours")) %>%
  pivot_longer(everything()) %>%
  group_by(name) %>%
  summarise(
    mean = mean(value),
    sd = sd(value),
    median = median(value),
    mad = mad(value)
  )
```

```{r, echo=FALSE}
library(poorman)

hardlyworking %>%
  select(starts_with("xtra_hours")) %>%
  reshape_longer(colnames_to = "name", values_to = "value") %>%
  group_by(name) %>%
  summarise(
    mean = mean(value),
    sd = sd(value),
    median = median(value),
    mad = mad(value)
  ) %>%
  knitr::kable(digits = 4)
```

`standardize()` can also be used to standardize a full data frame - where each
numeric variable is standardized separately:

```{r}
hardlyworking_z <- standardize(hardlyworking)
```

```{r, eval=FALSE}
hardlyworking_z %>%
  select(-xtra_hours_z, -xtra_hours_zr) %>%
  pivot_longer(everything()) %>%
  group_by(name) %>%
  summarise(
    mean = mean(value),
    sd = sd(value),
    median = median(value),
    mad = mad(value)
  )
```

```{r, echo=FALSE}
hardlyworking_z %>%
  select(-xtra_hours_z, -xtra_hours_zr) %>%
  reshape_longer(colnames_to = "name", values_to = "value") %>%
  group_by(name) %>%
  summarise(
    mean = mean(value),
    sd = sd(value),
    median = median(value),
    mad = mad(value)
  ) %>%
  knitr::kable(digits = 4)
```

Weighted standardization is also supported via the `weights` argument, and
factors can also be standardized (if you're into that kind of thing) by setting
`force = TRUE`, which converts factors to treatment-coded dummy variables before
standardizing.

## Variable-wise *vs.* Participant-wise 

Standardization is an important step and extra caution is required in
**repeated-measures designs**, in which there are three ways of standardizing
data:

- **Variable-wise**: The most common method. A simple scaling of each column.

- **Participant-wise**: Variables are standardized "within" each participant,
  *i.e.*, for each participant, by the participant's mean and SD.

- **Full**: Participant-wise first and then re-standardizing variable-wise.

Unfortunately, the method used is often not explicitly stated. This is an issue
as these methods can generate important discrepancies (that can in turn
contribute to the reproducibility crisis). Let's investigate these 3 methods.

### The Data

We will take the `emotion` dataset in which participants were exposed to
negative pictures and had to rate their emotions (**valence**) and the amount of
memories associated with the picture (**autobiographical link**). One could make
the hypothesis that for young participants with no context of war or violence,
the most negative pictures (mutilations) are less related to memories than less
negative pictures (involving for example car crashes or sick people). In other
words, **we expect a positive relationship between valence** (with high values
corresponding to less negativity) **and autobiographical link**.

Let's have a look at the data, averaged by participants:

```{r, eval=FALSE}
# Download the 'emotion' dataset
load(url("https://raw.github.com/neuropsychology/psycho.R/master/data/emotion.rda"))

# Discard neutral pictures (keep only negative)
emotion <- emotion %>% filter(Emotion_Condition == "Negative")

# Summary
emotion %>%
  drop_na(Subjective_Valence, Autobiographical_Link) %>%
  group_by(Participant_ID) %>%
  summarise(
    n_Trials = n(),
    Valence_Mean = mean(Subjective_Valence),
    Valence_SD = sd(Subjective_Valence)
  )
```

```{r, echo=FALSE}
load(url("https://raw.github.com/neuropsychology/psycho.R/master/data/emotion.rda"))

# Discard neutral pictures (keep only negative)
emotion <- emotion %>% filter(Emotion_Condition == "Negative")

# Summary
emotion %>%
  subset(!(is.na(Subjective_Valence) | is.na(Autobiographical_Link))) %>% 
  group_by(Participant_ID) %>%
  summarise(
    n_Trials = n(),
    Valence_Mean = mean(Subjective_Valence),
    Valence_SD = sd(Subjective_Valence)
  )
```

As we can see from the means and SDs, there is a lot of variability **between**
participants both in their means and their individual *within*-participant SD.

### Effect of Standardization 

We will create three data frames standardized with each of the three
techniques.

```{r}
Z_VariableWise <- emotion %>%
  standardize()

Z_ParticipantWise <- emotion %>%
  group_by(Participant_ID) %>%
  standardize()

Z_Full <- emotion %>%
  group_by(Participant_ID) %>%
  standardize() %>%
  ungroup() %>%
  standardize()
```

Let's see how these three standardization techniques affected the **Valence**
variable.

### Across Participants

We can calculate the mean and SD of *Valence* across all participants:

```{r, eval=FALSE}
# Create a convenient function to print
summarise_Subjective_Valence <- function(data) {
  df_name <- deparse(substitute(data))
  data %>%
    ungroup() %>%
    summarise(
      DF = df_name,
      Mean = mean(Subjective_Valence),
      SD = sd(Subjective_Valence)
    )
}
# Check the results
rbind(
  summarise_Subjective_Valence(Z_VariableWise),
  summarise_Subjective_Valence(Z_ParticipantWise),
  summarise_Subjective_Valence(Z_Full)
)
```

```{r, echo=FALSE}
# Create a convenient function to print
summarise_Subjective_Valence <- function(data) {
  df_name <- deparse(substitute(data))
  data <- data %>%
    ungroup() %>%
    summarise(
      Mean = mean(Subjective_Valence),
      SD = sd(Subjective_Valence)
    )
  cbind(DF = df_name, data)
}
# Check the results
rbind(
  summarise_Subjective_Valence(Z_VariableWise),
  summarise_Subjective_Valence(Z_ParticipantWise),
  summarise_Subjective_Valence(Z_Full)
) %>%
  knitr::kable(digits = 2)
```


The **means** and the **SD** appear as fairly similar (0 and 1)...

```{r, fig.width=7, fig.height=4.5, results='markup', fig.align='center'}
library(see)
library(ggplot2)

ggplot() +
  geom_density(aes(Z_VariableWise$Subjective_Valence,
    color = "Z_VariableWise"
  ), size = 1) +
  geom_density(aes(Z_ParticipantWise$Subjective_Valence,
    color = "Z_ParticipantWise"
  ), size = 1) +
  geom_density(aes(Z_Full$Subjective_Valence,
    color = "Z_Full"
  ), size = 1) +
  see::theme_modern() +
  labs(color = "")
```

and so do the marginal distributions...

### At the Participant Level

However, we can also look at what happens in the participant level. Let's look at
the first 5 participants:

```{r, eval=FALSE}
# Create convenient function
print_participants <- function(data) {
  df_name <- deparse(substitute(data))
  data %>%
    group_by(Participant_ID) %>%
    summarise(
      DF = df_name,
      Mean = mean(Subjective_Valence),
      SD = sd(Subjective_Valence)
    ) %>%
    head(5) %>%
    select(DF, everything())
}

# Check the results
rbind(
  print_participants(Z_VariableWise),
  print_participants(Z_ParticipantWise),
  print_participants(Z_Full)
)
```


```{r, echo=FALSE}
# Create convenient function
print_participants <- function(data) {
  df_name <- deparse(substitute(data))
  data %>%
    group_by(Participant_ID) %>%
    summarise(
      Mean = mean(Subjective_Valence),
      SD = sd(Subjective_Valence)
    ) %>%
    cbind(DF = df_name, .) %>% 
    head(5) %>%
    select(DF, everything())
}

# Check the results
rbind(
  print_participants(Z_VariableWise),
  print_participants(Z_ParticipantWise),
  print_participants(Z_Full)
) %>%
  knitr::kable(digits = 2)
```

Seems like *full* and *participant-wise* standardization give similar results,
but different ones than *variable-wise* standardization.

### Compare

Let's do a **correlation** between the **variable-wise and participant-wise
methods**.

```{r, fig.width=7, fig.height=4.5, results='markup', fig.align='center'}
r <- cor.test(
  Z_VariableWise$Subjective_Valence,
  Z_ParticipantWise$Subjective_Valence
)

data.frame(
  Original = emotion$Subjective_Valence,
  VariableWise = Z_VariableWise$Subjective_Valence,
  ParticipantWise = Z_ParticipantWise$Subjective_Valence
) %>%
  ggplot(aes(x = VariableWise, y = ParticipantWise, colour = Original)) +
  geom_point(alpha = 0.75, shape = 16) +
  geom_smooth(method = "lm", color = "black") +
  scale_color_distiller(palette = 1) +
  ggtitle(paste0("r = ", round(r$estimate, 2))) +
  see::theme_modern()
```

While the three standardization methods roughly present the same characteristics
at a general level (mean 0 and SD 1) and a similar distribution, their values
are not exactly the same!

Let's now answer the original question by investigating the **linear relationship between valence and autobiographical link**. We can do this by
running a mixed-effects model with participants entered as random effects.

```{r}
library(lme4)
m_raw <- lmer(
  formula = Subjective_Valence ~ Autobiographical_Link + (1 | Participant_ID),
  data = emotion
)
m_VariableWise <- update(m_raw, data = Z_VariableWise)
m_ParticipantWise <- update(m_raw, data = Z_ParticipantWise)
m_Full <- update(m_raw, data = Z_Full)
```

We can extract the parameters of interest from each model, and find:

```{r}
# Convenient function
get_par <- function(model) {
  mod_name <- deparse(substitute(model))
  parameters::model_parameters(model) %>%
    mutate(Model = mod_name) %>%
    select(-Parameter) %>%
    select(Model, everything()) %>%
    .[-1, ]
}

# Run the model on all datasets
rbind(
  get_par(m_raw),
  get_par(m_VariableWise),
  get_par(m_ParticipantWise),
  get_par(m_Full)
)
```

As we can see, **variable-wise** standardization only affects **the coefficient** (which is expected, as it changes the unit), but not the test
statistic or statistical significance. However, using **participant-wise**
standardization *does* affect the coefficient **and** the significance. 

**No method is better or more justified, and the choice depends on the specific case, context, data and goal.**

### Conclusion

1. **Standardization can be useful in *some* cases and should be justified**.

2. **Variable and Participant-wise standardization methods *appear* to produce similar data**.

3. **Variable and Participant-wise standardization can lead to different results**.

4. **The chosen method can strongly influence the results and should therefore be explicitly stated and justified to enhance reproducibility of results**.

We showed here yet another way of **sneakily tweaking the data** that can change
the results. To prevent its use as a bad practice, we can only highlight the
importance of open data, open analysis/scripts, and preregistration.

# See also 

- `datawizard::demean()`: <https://easystats.github.io/datawizard/reference/demean.html>
- `standardize_parameters(method = "pseudo")` for mixed-effects models
<https://easystats.github.io/effectsize/reference/standardize_parameters.html>

# References
