# Using a custom statistic with its error bar" In superb: Get Precision of Means Under Various Designs and Sampling Schemes

```cat("this will be hidden; use for general initializations.")
library(superb)
library(ggplot2)
```

By default, `superbPlot` generates mean plots along with 95% confidence intervals of the mean. However, these choices can be changed.

• To change the summary statistics, use the argument `statistic =`;

• To change the interval function, use the argument `errorbar =`. The abbreviation `CI` stands for confidence interval; `SE` stands for standard error.

• With `CI`, to change the confidence level, use the argument `gamma =`;

The defaults are `statistic = "mean", errorbar = "CI", gamma = 0.95`. For error bar functions that accept a gamma parameter (e.g., `CI`, the gamma parameter is automatically transfer to the function). For other functions that do not accept a gamma parameter (e.g., `SE`), the gamma parameter is unused.

## Changing the summary function to be plotted

In what follow, we use `GRD()` to generate a random dataset with an interaction then make plots varying the statistics displayed.

```# shut down 'warnings', 'design' and 'summary' messages
options(superb.feedback = 'none')

# Generate a random dataset from a (3 x 2) design, entirely within subject.
# The sample size is very small (n=5) and the correlation between scores is high (rho = .8)
dta <- GRD(
WSFactors  = "Moment(3): Dose(2)",
Effects    = list("Dose*Moment"=custom(0,0,0,1,1,3)),
SubjectsPerGroup = 50,
Population = list( mean=10, stddev = 5, rho = .80)
)

# a quick function to call superbPlot
makeplot <- function(statfct, errorbarfct, gam, rg, subttl) {
superbPlot(dta,
WSFactors  = c("Moment(3)","Dose(2)"),
variables = c("DV.1.1","DV.2.1","DV.3.1","DV.1.2","DV.2.2","DV.3.2"),
statistic = statfct,
errorbar  = errorbarfct,
gamma     = gam,
plotStyle = "line",
) + ylab(subttl) + coord_cartesian( ylim = rg )
}

p1 <- makeplot("mean",      "CI", .95, c(6,14), "Mean +- 95% CI of the mean")
p2 <- makeplot("mean",      "SE", .00, c(6,14), "Mean +- SE of the mean")
p3 <- makeplot("median",    "CI", .95, c(6,14), "Median +- 95% CI of the median")
p4 <- makeplot("fisherskew","CI", .95, c(-2,+2), "Fisher skew +- 95% CI")

library(gridExtra)
p <- grid.arrange(p1,p2,p3,p4, ncol=2)
```

## Determining that a summary function is valid to use in superbPlot

Any summary function can be accepted by superbPlot, as long as it is given within double-quote. The built-in statistics functions such as `mean` and `median` can be given. Actually, any descriptive statistics, not just central tendency, can be provided. That includes `IQR`, `mad`, etc. To be valid, the function must return a number when given a vector of numbers.

In doubt, you can test if the function is valid for `superbPlot` with (note the triple colon):

```superb:::is.stat.function("mean")
```

Any error bar function can be accepted by superbPlot. These functions must be named `"interval function"."descriptive statistic"`. For example, the following `CI.mean` is the confidence interval of the mean. Other functions are `SE.mean`, `SE.median`, `CI.fisherskew`, etc. The `superb` library provides some 20+ such functions. @htc14 and @htc15 reviewed some of these functions.

The error bar functions can be of three types:

• a function that returns a width. Standard error functions are example of this type of function. With width function, the error bar extend plus and minus that width around the descriptive statistics.

• an interval function. Such functions returns the actual lower and upper limits of the interval and therefore are used as is to draw the bar (i.e., they are not relative to the descriptive statistics). Confidence interval functions are of this type (as they are not necessarily symmetrical about the descriptive statistics).

• `"none"`. This keyword produces an error bar of null width.

The interval function can be tested to see if it exists:

```superb:::is.errorbar.function("SE.mean")
```

To see if a gamma is accepted for a certain interval function, you can try

```superb:::is.gamma.required("SE.mean")
```

## Creating a custom-made descriptive statistic function to be used in superbPlot

As an example, we create from scratch a descriptive statistic function that will be fed to `superbPlot`. Following @gc18 , we implement the single-group design Cohen's d (\$d_1\$). This descriptive statistic is computed relative to an hypothetical population mean. Herein, we use the data from `dataFigure1` and assume that the population mean is 100.

```    # create a descriptive statistics
d1 <- function(X) {
mean(X-100)/sd(X)
}

# we can test it with the data from group 1...
grp1 <- dataFigure1\$score[dataFigure1\$grp==1]
grp2 <- dataFigure1\$score[dataFigure1\$grp==2]
d1(grp1)

# or check that it is a valid statistic function
superb:::is.stat.function("d1")
```

Once we have the function, we can ask for a plot of this function with

```    superbPlot(dataFigure1,
BSFactors = "grp",
statistic = "d1", errorbar = "none",
plotStyle="line",
variable = "score",
errorbarParams = list(width=0) # so that the null-width error bar is invisible
)+ ylab("Cohen's d_1") +
labs(title="d_1 with no error bars") +
coord_cartesian( ylim = c(-0.5,+1.5) )
```

## Creating a custom-made interval function to be used in `superbPlot`

It is also possible to create custom-made confidence interval functions.

Hereafter, we add a confidence interval for the Cohen's \$d_1\$. We use the predictive approach documented in Lecoutre [@l99; @cg20] which requires the lambda-prime distribution provided in the `sadists` library [@p20].

```    library(sadists)

CI.d1 <- function(X, gamma = .95) {
n    <- length(X)
dlow <- qlambdap(1/2-gamma/2, df = n-1, t = d1(X) * sqrt(n) )
dhig <- qlambdap(1/2+gamma/2, df = n-1, t = d1(X) * sqrt(n) )
c(dlow, dhig) / sqrt(n)
}

# we test as an example the data from group 1
CI.d1(grp1)

# or check that it is a valid interval function
superb:::is.errorbar.function("CI.d1")
```

We have all we need to make a plot with error bars!

```    superbPlot(dataFigure1,
BSFactors = "grp",
statistic = "d1", errorbar = "CI",
plotStyle="line",
variable = "score"
)+ ylab("Cohen's d_1") +
labs(title="d_1 with 95% confidence interval of d_1") +
coord_cartesian( ylim = c(-0.5,+1.5) )
```

Note that this difference-adjusted confidence interval is fully compatible with the Cohen's \$d_p\$ (between the two groups) as \$d_p\$ is 0.498 with a 95% confidence interval of [-0.068, +1.059] [all the formulas are from @gc18;@cg20].

```    # compute the Cohen's dp
dp <- (mean(grp1)-mean(grp2))/ sqrt((var(grp1)+var(grp2))/2)
dp

# get the confidence interval of this
lecoutre2007 <- function(dp, n, gamma = .95) {
dlow <- qlambdap(1/2-gamma/2, df = 2*(n-1), t = dp * sqrt(n/2) )
dhig <- qlambdap(1/2+gamma/2, df = 2*(n-1), t = dp * sqrt(n/2) )
limits <- c(dlow, dhig) / sqrt(n/2)
limits
}
lecoutre2007(dp, length(grp1) )
```

## Creating bootstrap-based confidence intervals

It is also possible to create bootstrap estimates of confidence intervals and integrate these into `superb`.

The general idea is to subsample with replacement the sample and compute on this subsample the descriptive statistics. This process is repeated a large number of times (here, 10,000) and the quantiles containing, say, 95% of the results is a 95% confidence interval.

Here, we illustrate this process with the mean. The function must be name "interval function.mean", so we choose to call it `myBootstrapCI.mean`.

```    # we define myBootstrapCI which subsample the whole sample, here called X
myBootstrapCI.mean <- function(X, gamma = 0.95) {
res = c()
for (i in 1:10000) {
res[i] <- mean(sample(X, length(X), replace = T))
}
quantile(res, c(1/2 - gamma/2, 1/2 + gamma/2))
}

# we check that it is a valid interval function
superb:::is.errorbar.function("myBootstrapCI.mean")
```

This is all we need to make the plot which we can compare with the parametric CI

```    plt1 <- superbPlot(dataFigure1,
BSFactors = "grp",
variable = c("score"),
plotStyle="line",
statistic = "mean", errorbar = "myBootstrapCI",
) +
xlab("Group") + ylab("Score") +
labs(title="means and difference-adjusted\n95% confidence intervals") +
coord_cartesian( ylim = c(85,115) ) +
theme_gray(base_size=10) +
scale_x_discrete(labels=c("1" = "Collaborative games", "2" = "Unstructured activity"))

plt2 <- superbPlot(dataFigure1,
BSFactors = "grp",
variable = c("score"),
plotStyle="line",
statistic = "mean", errorbar = "CI",
) +
xlab("Group") + ylab("Score") +
labs(title="means and difference-adjusted\n95% bootstrap confidence intervals") +
coord_cartesian( ylim = c(85,115) ) +
theme_gray(base_size=10) +
scale_x_discrete(labels=c("1" = "Collaborative games", "2" = "Unstructured activity"))

library(gridExtra)
plt <- grid.arrange(plt1, plt2, ncol=2)
```

As seen, there is not much difference between the two. This was expected: when the normality assumption and the homogeneity of variances assumption are not invalid, the parametric approach (based on these assumptions) is identical on average to the bootstrap approach.

## In summary

The function `superbPlot()` is entirely customizable: you can put any descriptive statistic function and any interval function into `superbPlot()`. In a sense, `superbPlot()` is simply a proxy that manage the dataset and produces standardized dataframes apt to be transmitted to a `ggplot()` specification. It is also possible to obtain the summary dataframe by issuing the argument `showPlot=FALSE` or by using the related function `superbData()`.

The function `superbPlot` is also customizable with regards to the plot produced. Included in the package are

• `superbPlot.line` : shows the results as points and lines,

• `superbPlot.point`: shows the results as points only,

• `superbPlot.bar` : shows the results using bars,

• `superbPlot.pointjitter`: shows the results with points, and the raw data with jittered points,

• `superbPlot.pointjitterviolin`: also shows violin plot behind the jitter points, and

• `superbPlot.pointindividualline`: show the results with fat points, and individual results with thin lines,

• `superbPlot.raincloud`: show the results along with clouds (violin distributions) and rain drops (jittered raw data),

Vignette 5 shows how to create new layouts. Proposals are welcome!

# References

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superb documentation built on April 22, 2021, 1:06 a.m.