---
title: "Three steps to make your plot"
bibliography: "../inst/REFERENCES.bib"
csl: "../inst/apa-6th.csl"
output: 
  rmarkdown::html_vignette
description: >
  This vignette reviews the steps needed to realize your summary plot with correct
  error bars.
vignette: >
  %\VignetteIndexEntry{Three steps to make your plot}
  %\VignetteEngine{knitr::rmarkdown}
  \usepackage[utf8]{inputenc}
---


```{r, echo = FALSE, warning=FALSE, message = FALSE, results = 'hide'}
cat("this will be hidden; use for general initializations.\n")
library(superb)
library(ggplot2)
```

The package ``superb`` facilitates the production of 
summary statistic plots having adjusted error bars [@cgh21]. Error bars are
measures of precision; they are meant to convey some indications of
the trust we can place in a given result. Those results that come
will large error bars (wide intervals) should be considered less
reliable that those with short error bars. 

Error bars, which are drawn from measures of precision, comes in many flavors. There is
the simple *standard error* (SE), the more famous *confidence interval* (CI), and 
new proposals such as the *highest density interval* (HDI). The package ``superb``
has build-in functions for both the SE and CI, but any other measure of precision
can be added (as shown in [Vignette 4](https://dcousin3.github.io/superb/articles/Vignette4.html)).

All measures of precision are tailored for a given statistic. Hence, we have the
*standard error of the mean*, the *standard error of the median*, the *confidence
interval of the skew*, etc. For each and every statistic, there is a corresponding
measure of precision.  In what follows, we discuss the confidence interval, but
everything that follows apply equally well to any measure of precision.

The basic confidence interval can be termed a "stand-alone" confidence interval 
because it indicates the precision of a statistic in isolation. It is useful to
compare this statistics to a criterion performance or any *a priori*-determined
value. As shown in [Vignette 2](https://dcousin3.github.io/superb/articles/Vignette2.html) 
and [Vignette 3](https://dcousin3.github.io/superb/articles/Vignette3.html), 
such "stand-alone" measures are inadequate
when an observed result is to be compared to other observed results [also see 
@c05; @c17].

In order to use ``superb``, four choices need to be made, described next. Prior to 
that, it is necessary to check the data format. Herein we illustrate the means with 
95% confidence intervals; these can be changed, as shown in [Vignette 4](https://dcousin3.github.io/superb/articles/Vignette4.html).

## Data format

The data from which to make a plot must be contained within a ``data.frame`` with named columns.
(I can also be in a tibble or related format as long as they convert to ``data.frame()``)
The format of the data can be of two forms, **long format** and **wide format**.

In the **wide format**, the general convention is "1 subject = 
1 line", that is, all the information regarding a given participant must stand 
on a single line. In SPSS, SAS
and other statistics software, this is the standard data organization.  You can
convert from **wide format** to **long format**, with ``superbToWide()``
or Navarro's ``wideToLong``'s excellent function [@n15]. When there are multiple
measures in your data, they can be used using the ``cbind()`` or ``crange()``
notations: With ``cbind()``, you must name all the variables whereas with 
``crange()`` you name a range of column with the first and last only column
name of the range.

In the **long format**, each line contains a single measure. if there are
repated measures, they are on consecutive lines, but can be assigned to a 
participant because there is an ID column. Then the formula must include a
``|`` to indicate that the data are nested with participants' ID.

As an illustration, we use the following ficticious data set showing the 
performance of 15 participants on their motivation scores on Week 1, Week 2 
and Week 3 for a program to stop smoking (**long format**).

```{r}
#Motivation data for 15 participants over three weeks in wide format:
dta <- matrix( c(
    45, 50,  59,
    47, 58,  64,
    53, 63,  72,
    57, 64,  81,
    58, 67,  86,
    61, 70,  98,
    61, 75, 104,
    63, 79, 100,
    63, 79,  84,
    71, 81,  96,
    72, 83,  82,
    74, 84,  82,
    76, 86,  93,
    84, 90,  85,
    90, 96,  89
), ncol=3, byrow=T)

# put column names and convert to data.frame:
colnames(dta) <- c("Week 1", "Week 2", "Week 3")
dta           <- as.data.frame(dta)
```

The mean scores per week is illustrated below:

```{r, message=FALSE, echo=FALSE, fig.height=3, fig.width=4, fig.cap="**Figure 1**. Mean scores along with 95% confidence interval per week for a program to stop smoking."}
superb(
    cbind(`Week 1`, `Week 2`, `Week 3`) ~ ., 
    dta, 
    WSFactors = "Moment(3)",
    statistic = "mean", errorbar = "CI",
    adjustments = list(purpose = "single", decorrelation = "none"),
    plotStyle="line",
    errorbarParams = list(width = .2)
) + 
coord_cartesian( ylim = c(50,100) ) +
ylab("Mean +- 95% CI") +
labs(title="(stand-alone)\n95% confidence interval")+
theme_gray(base_size=10) +
scale_x_discrete(labels=c("1" = "Week 1", "2" = "Week 2", "3"="Week 3"))
``` 

Instead of ``cbind(`Week 1`, `Week 2`, `Week 3`)``, we could have used 
``crange(`Week 1`, `Week 3`)`` to refer to all the columns from `Week 1` to ``Week3`.

Note that, by default, missing data are not handled by ``superb``. Ideally,
the cells with NA must be removed or imputed prior to perform the plot. Alternatively, 
you can look into the summary statistic ``meanNArm()``. We made it on purpose to be
difficult to make you plot with missing data because you should have handled these
prior to making your summary plot.


## Step 1: Decide of the purpose of the plot

In Figure 1 above, did you look at the result of, say, Week 1, in isolation? or did you compare 
it to the results obtained in the other weeks? The second perspective is actually to look at
the __difference__ between the results. if such is the case, the error bar shown on this plot
are actually misleading because they are too short. The reason for that is further explored in
[Vignette 2](https://dcousin3.github.io/superb/articles/Vignette2.html).

In most experiments, one condition is **compared** to other conditions. 
In that case, we are interested in **pair-wise** differences between means, not
by single results in isolation. Yet, the regular confidence intervals are valid only for results
in isolation. Whenever you wish to compare a result to other results, to examine differences
between conditions (which is, most of the time), you need to adjust the confidence interval 
lengths so that they remain adequate inference tools.

In ``superb``, you obtain an adjustment to error bar length with an option 
for the purpose of the plot. The default ``purpose = "single"`` returns stand-alone error 
bars (as in Figure 1); ``purpose = "difference"`` returns error bars valid for pair-wise 
comparisons. The minimum specification for the data frame above would therefore be

```{r, message=FALSE, echo=TRUE, results='hide', fig.show='hide'}
superb(
    cbind(`Week 1`, `Week 2`, `Week 3`) ~ .,
    dta, 
    WSFactors = "Moment(3)",
    adjustments = list(purpose = "difference"),
    plotStyle="line"
    )
```

```{r, message=FALSE, echo=FALSE, fig.height=3, fig.width=4, fig.cap="**Figure 2**. Mean scores along with difference-adjusted 95% confidence interval per week for a program to stop smoking."}
superb(
    cbind(`Week 1`, `Week 2`, `Week 3`) ~ .,
    dta, 
    WSFactors = "Moment(3)",
    statistic = "mean", errorbar = "CI",
    adjustments = list(purpose = "difference"),
    plotStyle="line",
    errorbarParams = list(width = .2)
) + 
coord_cartesian( ylim = c(50,100) ) +
ylab("Mean +- 95% CI") +
labs(title="Difference-adjusted\n95% confidence interval")+
theme_gray(base_size=10) +
scale_x_discrete(labels=c("1" = "Week 1", "2" = "Week 2", "3"="Week 3"))
``` 

The first argument is the formula describing on the left-hand
side the measurements, and on the right-hand side, the betwee-
subject conditions. The second argument is a
data.frame, here in wide format. The data frame is ``dta``. 
Because we have multiple dependent variables (collected
with ``cbind()``, the third argument
describe the within-subject experimental factor(s). 
Here, there is a single within-subject factor (``WSFactors``), 
 called ``Moment``. In within-subject factors, it is necessary to indicate how many
level the factor has (here 3).

The column named in the formula must match the column
names in the data.frame. If unsure, check with

```{r}
head(dta)
```

Note that the measurements could be named using `crange()` 
instead of `cbind()`. The specification `crange()` is used
to refer to a range of variables from the first given to
the last given. Here, the formula could have been given as
``crange(`Week 1`, `Week 3`) ~ .``

The argument in which the adjustments will be listed is called ``adjustments`` which is
a list with ---for the moment--- an adjustment for the purpose of the plot: ``purpose =
 "difference"``. The standalone CI is the default (it can be obtained explicitly with 
``purpose= "single"``). These two expressions, *single* and *difference*, are from @b12. 

Note that the plot obtained is a ``ggplot`` object to which additional graphic
directives can be added. Figure 2 was actually obtained with these commands:

```{r, message=FALSE, echo=TRUE, fig.height=3, fig.width=4, results='hide', fig.show='hide'}
superb(
    cbind(`Week 1`, `Week 2`, `Week 3`) ~ .,
    dta, 
    WSFactors = "Moment(3)",
    statistic = "mean", errorbar = "CI",
    adjustments = list(purpose = "difference"),
    plotStyle="line"
) + 
coord_cartesian( ylim = c(50,100) ) +
ylab("Mean +- 95% CI") +
labs(title="Difference-adjusted\n95% confidence interval")+
theme_gray(base_size=10) +
scale_x_discrete(labels=c("1" = "Week 1", "2" = "Week 2", "3"="Week 3"))
``` 

The graphic directives are applied to the whole plot.
The reader is referred to the package ``ggplot2`` for more on these graphic directives.


## Step 2: Decide how to handle the within-subject measures

Is is known that within-subject designs are generally more powerful at detecting differences.
The implication of this is that they afford more statistical power and consequently
the error bars should be shorter. The stand-alone CI are oblivious to this fact; it
is however possible to inform them that you used a within-subject design.

The method to handle within-subject data comes from the observation that repeated 
measures tend to be correlated [e.g., @gc19]. Informing the CI of this
correlation is a process called **decorrelation**. To this day, there exists 
three methods for decorrelation:

* ``CM``: this method, called from the two authors Cousineau and Morey [@c05;@m08], will
decorrelate the data but each measurement may have different adjustments;
* ``LM``: this method, the first developed by Loftus and Masson [-@lm94], will 
make all the error bars have the same length;
* ``CA``: this method, called *correlation-based adjustment* was proposed in @c19. 
As per CM, the error bars can be different in length.

The three methods were compared in @c19 and shown to be mathematically based on the
same concepts and estimating the same precision. It is therefore a matter of 
personal preference which one you use.

Figure 2 above will become Figure 3 if you add a decorrelation adjustment:

```{r, message=TRUE, echo=TRUE, fig.height=3, fig.width=4}
superb(
    cbind(`Week 1`, `Week 2`, `Week 3`) ~ .,
    dta, 
    WSFactors = "Moment(3)",
    statistic = "mean", errorbar = "CI",
    adjustments = list(purpose = "difference", decorrelation = "CM"), #new!
    plotStyle="line",
    errorbarParams = list(width = .2)
) + 
coord_cartesian( ylim = c(50,100) ) +
ylab("Mean +- 95% CI") +
labs(title="Correlation- and Difference-adjusted\n95% confidence interval")+
theme_gray(base_size=10) +
scale_x_discrete(labels=c("1" = "Week 1", "2" = "Week 2", "3"="Week 3"))
``` 

Unless you change the options to ``options(superb.feedback = 'none')``, the command will issue some additional
information. In the present data set, $\varepsilon$ is 0.54 which is low. A Mauchly test
of sphericity indicates rejection of sphericity, so interpret the error bars with caution. 
All the messages issued beginning with ``"FYI"`` are just information. Hereafter, the warnings
are inhibited.


## Step 3: Specify the sampling procedure and the population size

The stand-alone confidence intervals are appropriate when your sample was obtained
randomly, a method formally called *Simple Randomize Sampling* (SRS). However, this
is not the only sampling method possible. Another commonly employed sampling procedure 
is cluster sampling (formally *Cluster Randomized Sampling*, CRS). The CRS is the
only one (beyond SRS) where the exact adjustment is known [@cl16] and thus SRS (no 
adjustment) and CRS (adjustments that tend to widen the error bars) are the
only two sampling adjustments currently implemented in ``superb``. 

Other sampling methods includes *Stratified Sampling*, *Snowball Sampling*, *Convenience
Sampling*, etc., none of which have a known impact on the precision of the measures.

Also, determine if the population size is finite or infinite.
When the population under scrutiny is finite, you may have a sizeable proportion of 
the population in your sample, which improves precision. In this case, the error bars will
be shortened.

These adjustments are implemented in ``superb`` with additional adjustments, such as

```{r, echo = FALSE}
options(superb.feedback = 'none')
```

```{r, message=FALSE, echo=TRUE , results='hide', fig.height=3, fig.width=4}
# add (ficticious) cluster membership for each participant in the column "cluster"
dta$cluster <- sort(rep(1:5, 3))

superb(
    cbind(`Week 1`, `Week 2`, `Week 3`) ~ .,
    dta, 
    WSFactors = "Moment(3)",
    adjustments = list(purpose = "difference", decorrelation = "CM",
                       samplingDesign = "CRS", popSize = 100),        #new!
    plotStyle = "line",
    clusterColumn = "cluster",  # identify the column containing cluster membership
    errorbarParams = list(width = .2)
) + 
coord_cartesian( ylim = c(50,100) ) +
ylab("Mean +- 95% CI") +
labs(title="Cluster- Correlation, and Difference-adjusted\n95% confidence interval")+
theme_gray(base_size=10) +
scale_x_discrete(labels=c("1" = "Week 1", "2" = "Week 2", "3"="Week 3"))
``` 

As you can see, the adjustments can be obtained with a single option inside the ``adjustments``
list. They are cumulative, i.e., more than one adjustment can be used, depending on the 
situation.


# References