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 the correct error bars. 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 *high 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).

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 every and all 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 --say-- criterion performance or any *a priori*-determined
value. As shown in Vignettes 2 and 3, 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. We begin with data
format.

The format of the data in `superb`

follows the general convention "1 subject =
1 line", that is, all the information regarding a given participant must stand
on a single line. This format is called in R the **wide format**; in SPSS, SAS
and other statistics software, this is the standard data organization. Further, the data
must be contained within a `data.frame`

with named columns.

Note that it is easy to go from **wide format** to **long format**, with, e.g.,
Navarro's `wideToLong`

excellent function [@n15].

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.

#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 then 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:

superbPlot(dta, WSFactors = "Moment(3)", variables = c("Week 1", "Week 2", "Week 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=16) + scale_x_discrete(labels=c("1" = "Week 1", "2" = "Week 2", "3"="Week 3"))

Note that missing data are not handled by `superb`

. The cells with NA must be
removed or imputed prior to perform 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.

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

superbPlot(dta, WSFactors = "Moment(3)", variables = c("Week 1", "Week 2", "Week 3"), adjustments = list(purpose = "difference"), plotStyle="line" )

superbPlot(dta, WSFactors = "Moment(3)", variables = c("Week 1", "Week 2", "Week 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=16) + scale_x_discrete(labels=c("1" = "Week 1", "2" = "Week 2", "3"="Week 3"))

The first argument is the data.frame, in wide format (here `dta`

). The second argument
describe the experimental design. 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 third argument indicates the columns in the data.frame
containing the measurements. Three columns are involved. The name must match the column
names in the data.frame. If unsure, check with

```
head(dta)
```

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 graphing
directives can be added. Figure 2 was actually obtained with these commands:

superbPlot(dta, WSFactors = "Moment(3)", variables = c("Week 1", "Week 2", "Week 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=16) + scale_x_discrete(labels=c("1" = "Week 1", "2" = "Week 2", "3"="Week 3"))

The directives in `errorbarParams`

are injected inside the `geom_errorbar`

in charge of
drawing the error bars whereas the remaining directives are applied to the whole plot.
The reader is referred to the package `ggplot2`

for more on these graphing directives.

Is is known that within-subject designs are 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 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:

superbPlot(dta, WSFactors = "Moment(3)", variables = c("Week 1", "Week 2", "Week 3"), statistic = "mean", errorbar = "CI", adjustments = list(purpose = "difference", decorrelation = "CM"), 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=16) + 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.

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

options(superb.feedback = 'none')

# add (ficticious) cluster membership for each participant in the column "cluster" dta$cluster <- sort(rep(1:5, 3)) superbPlot(dta, WSFactors = "Moment(3)", variables = c("Week 1", "Week 2", "Week 3"), adjustments = list(purpose = "difference", decorrelation = "CM", samplingDesign = "CRS", popSize = 100), 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=16) + 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 `adjusments`

list. They are cumulative, i.e., more than one adjustment can be used, depending on the
situation.

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