The `qqconf`

package extends `base`

graphics
capabilities to put simultaneous testing bands on Quantile-Quantile (QQ)
and Probability-Probability (PP) plots. We support plots for any
distribution with a Cumulative Distribution Function (CDF) implemented
in R. For the creation of simultaneous testing bands, we support the
Kolmogorov-Smirnov (Kolmogorov 1941; Smirnov 1944) method as well as
what we refer to as the Equal Local Levels (ELL) method, which conducts
an \(\eta\) level test on each order
statistic of the sample supplied such that the global testing bands have
some desired \(\alpha\) Type I error
rate. For more information on the computation and properties of ELL,
please see our section about ELL Testing Bounds Generation and our paper.

Our package is available on CRAN and can be installed by running:

`install.packages("qqconf")`

The package contains two functions for creating plots,
`qq_conf_plot`

and `pp_conf_plot`

to create QQ
plots and PP plots, respectively. These two functions have identical
interfaces, with the exception that `qq_conf_plot`

requires
the input of a quantile function (e.g. `qnorm`

) and
`pp_conf_plot`

requires the input of a distribution function
(e.g. `pnorm`

). Thus, any information below about parameters
that can be fed to the `qq_conf_plot`

function also apply to
`pp_conf_plot`

, and vice versa.

First, we load in `qqconf`

`require(qqconf)`

Then, we generate data from a standard normal distribution:

```
set.seed(0)
<- rnorm(n = 100, mean = 0, sd = 1) sample
```

Now, if we did not know the distribution that our sample came from but instead wanted to test if it was sampled i.i.d from a \(N(0, 1)\) distribution, we could create a QQ plot to compare the quantiles of the sample to the theoretical quantiles of a \(N(0, 1)\) distribution.

To do this, we call `qq_conf_plot`

as follows:

```
qq_conf_plot(
obs = sample,
distribution = qnorm,
dparams = list(mean = 0, sd = 1)
)
```

By default, this function creates a QQ plot with ELL testing bounds
calculated with a Type I error rate of .05. Note here that
`dparams`

is not a required parameter. If we were to call
`qq_conf_plot`

without specifying `dparams`

, then
our code would estimate the mean and variance of the sample assuming it
comes from a normal distribution as is specified by the
`distribution`

parameter. More generally, for any continuous
distribution that is neither normal nor uniform, we use maximum
liklihood estimation to estimate all parameters of the distribution. For
the uniform distribution, we do not support parameter estimation but
instead default to \(U(0, 1)\) and
allow the user to specify a custom `max`

and
`min`

. For the normal distribution, because it is so commonly
used as the reference distribution in QQ plots, we experimented with
different methods of parameter estimation and found that estimating the
mean of the distribution as the median of the sample and the standard
deviation as \(S_{n}\) as proposed by
Rousseeuw and Croux (Rousseeuw, Crow 1993) had well calibrated Type I
error rates. When we used the MLEs for the normal distribution, we found
that the testing bounds produced were conservative. Note that the
testing bounds produced for other distributions with parameters
estimated via MLE will also likely be conservative. For more information
on parameter estimation, please see section 2.6 of our paper.

Parameter estimation aside, we have now created a QQ plot that
compares our sample to \(N(0, 1)\).
While this plot is informative, there are a number of potential
modifications we may want to make to this plot for ease of viewing.
First, because the top and the bottom of the confidence bands are cut
off in this plot, we may want to expand the y-axis limits. We do this by
simply adding a `ylim`

argument to the function call:

```
qq_conf_plot(
obs = sample,
distribution = qnorm,
dparams = list(mean = 0, sd = 1),
ylim = c(-4, 4)
)
```

In fact, because `qq_conf_plot`

takes `...`

as
an argument, any other parameter normally passed to the
`plot`

function in `base`

can be passed to
`qq_conf_plot`

. If, for instance, we wanted to modify the
axis labels, we could write:

```
qq_conf_plot(
obs = sample,
distribution = qnorm,
dparams = list(mean = 0, sd = 1),
ylim = c(-4, 4),
xlab = "More Informative Title"
)
```

However, while the `...`

argument can be used for more
general features of the plot like axis titles, because there are so many
separate objects in the plot (points, lines, bands), we’ve added
additional arguments so that the user can easily specify the visual
features of any element of the plot.

If we want to modify the appearance of the points on the plot, we can
use the optional `points_params`

argument. This argument is a
list that takes any arguments that can be passed to the
`points`

function in `base`

. For instance, if we
wanted to make the points smaller, we could do the following.

```
qq_conf_plot(
obs = sample,
distribution = qnorm,
dparams = list(mean = 0, sd = 1),
ylim = c(-4, 4),
points_params = list(cex = .5) # makes points smaller
)
```

For more options, see the documentation of the `points`

function.

If we want to modify the line in the plot that indicates a perfect
fit between the reference distribution and the sample, we can use the
`line_params`

argument. This list can take any parameters
that can be passed to the `lines`

function and will modify
the plot accordingly. If, for instance, we wanted to make the line red,
we would do the following:

```
qq_conf_plot(
obs = sample,
distribution = qnorm,
dparams = list(mean = 0, sd = 1),
ylim = c(-4, 4),
line_params = list(col="red") # makes expectation line red
)
```

Note, that if we don’t want this line to show up on the plot at all,
we can pass in `type = "n"`

to `line_params`

.

```
qq_conf_plot(
obs = sample,
distribution = qnorm,
dparams = list(mean = 0, sd = 1),
ylim = c(-4, 4),
line_params = list(type="n") # removes expectation line
)
```

Again, for more information see the documentation for the
`lines`

function.

By default, `qq_conf_plot`

does not plot pointwise testing
bands on the plot. Pointwise testing bands, in contrast to simultaneous
testing bands, are anti-conservative, as they ignore the multiple
testing problem inherent in QQ plots and simply conduct a test on each
order statistic. To add these pointwise bands, one can simply set the
`plot_pointwise`

parameter to `True`

in the
`qq_conf_plot`

function. To modify their appearance, one can
use the `pointwise_lines_params`

parameter. Because this
parameter list is simply passed to the `lines`

function, it
works exactly the same way as the `line_params`

parameter.

Finally, if we want to modify the appearance of the simultaneous
testing bands, we can use the `polygon_params`

parameter. As
the name suggests, arguments in this parameter list are passed to the
`polygon`

function in `base`

and modify the
testing bands accordingly. Note that by default, `border`

is
set to `NA`

and `col`

is set to `grey`

.
If the user overrides the default option, all defaults of
`polygon`

are used unless otherwise specified, which does not
set `border`

to `NA`

and `col`

to
`grey`

. If, for instance, we wanted to change the color of
the shading, we would do the following:

```
qq_conf_plot(
obs = sample,
distribution = qnorm,
dparams = list(mean = 0, sd = 1),
ylim = c(-4, 4),
polygon_params = list(col = 'powderblue', border = NA) # change shading and keep no border
)
```

In addition to general visual parameters to modify the elements of the plot, we’ve also included a few parameters that we found to be useful in particular cases when making a QQ or PP plot. Especially when \(n\) is large, it can be difficult to see the relevant parts of the plot.

To demonstrate this, we will generate 10,000 points from a \(Beta(1, 1.05)\) distribution and test this sample against a \(U(0, 1)\) distribution.

`<- rbeta(n = 10000, shape1 = 1.0, shape2 = 1.05) sample `

Because a QQ plot and PP plot are equivalent for the uniform distribution, we will switch to PP plots to demonstrate the interface.

Normally in a PP plot, the expected probabilities are plotted on the x-axis and the theoretical probabilities are plotted on the y-axis.

```
pp_conf_plot(
obs = sample,
distribution = punif,
points_params = list(cex=.1)
)
```

However, in this plot it is very difficult to see anything because the individual points are so small and the magnitude of the deviation from the reference distribution is not very large.

Instead, if we plot the expected probabilities on the x-axis and the
observed probabilities - the expected probabilities on the y-axis by
setting the `difference`

parameter to `TRUE`

,
things become much easier to see:

```
pp_conf_plot(
obs = sample,
distribution = punif,
points_params = list(cex=.1),
difference = TRUE, # Make y-axis differenced
ylim = c(-.0225, .0225)
)
```

Sometimes, we are only interested in deviations of the sample from one tail of the reference distribution. For instance, if we have a sample of \(n\) p-values that we would like to compare to the \(U(0, 1)\) distribution for the sake of global null hypothesis testing, we’re generally only interested in p-values that are small. In this case, especially with large \(n\), it can be helpful to transform the probability values onto the \(-log10\) scale.

To demonstrate this, we will generate 10,000 points from a mixture of a \(Beta(.25, 1)\) and a \(U(0, 1)\) and test this sample against a \(U(0, 1)\) reference distribution.

```
<- distr::UnivarMixingDistribution(
mix ::Beta(shape1 = .25, shape2 = 1),
distr::Unif(),
distrmixCoeff=c(.01, .99)
)
<- distr::r(mix)
sampler
<- sampler(10000) sample
```

Again, if we make a standard PP plot the relevant parts of the plot are very difficult to make out.

```
pp_conf_plot(
obs = sample,
distribution = punif,
points_params = list(cex=.1)
)
```

Even if we set `difference`

to `TRUE`

, it’s
still difficult to see the left tail of the distribution.

```
pp_conf_plot(
obs = sample,
distribution = punif,
points_params = list(cex=.1),
difference = TRUE
)
```

However, setting `log10`

to `TRUE`

makes the
plot much easier to see.

```
pp_conf_plot(
obs = sample,
distribution = punif,
points_params = list(cex=.1),
log10 = TRUE
)
```

Note, that if we want to highlight the right tail of the distribution
as opposed to the left tail, we can set `log10`

to
`TRUE`

and the `right_tail`

parameter to
`TRUE`

. This results in a plot with the data transformed onto
the `log10`

scale instead of the `-log10`

scale.

In addition to the plotting interfaces explained above,
`qqconf`

provides functions to directly produce testing
bounds generated by the equal local levels method. Functions exist for
both two-sided and one-sided testing. For information on the generation
of testing bounds using equal local levels, please see section 2 of our
paper.

Given a desired global Type I error rate \(\alpha\) and a sample size \(n\), we can generate two-sided bounds using
the `get_bounds_two_sided`

function. Below, we generate such
bounds for \(n = 100\) and \(\alpha = .05\):

`<- get_bounds_two_sided(alpha = .05, n = 100) bounds `

In this case, the `bounds`

object is a list with a number
of objects (for more information see the documentation), but the lower
bounds can be extracted with `bounds$lower_bounds`

and the
upper bounds with `bounds$upper_bounds`

.

In addition to two-sided testing, `qqconf`

also provides
functionality for one-sided testing.

Given a desired global Type I error rate \(\alpha\) and a sample size \(n\), we can generate such bounds using the
`get_bounds_one_sided`

function. Below, we generate such
bounds for \(n = 100\) and \(\alpha = .05\):

`<- get_bounds_one_sided(alpha = .05, n = 100) bounds `

From the `bounds`

object, the upper bounds can be
extracted via `bounds$bound`

. Again, this function returns a
list and more information can be extracted from `bounds`

.

All plots above were generated with the following session information:

```
sessionInfo()
#> R version 4.2.2 (2022-10-31)
#> Platform: x86_64-apple-darwin17.0 (64-bit)
#> Running under: macOS Big Sur ... 10.16
#>
#> Matrix products: default
#> BLAS: /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRblas.0.dylib
#> LAPACK: /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRlapack.dylib
#>
#> locale:
#> [1] C/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
#>
#> attached base packages:
#> [1] stats graphics grDevices utils datasets methods base
#>
#> other attached packages:
#> [1] qqconf_1.3.2
#>
#> loaded via a namespace (and not attached):
#> [1] Rcpp_1.0.10 digest_0.6.31 MASS_7.3-58.3 distr_2.9.1
#> [5] R6_2.5.1 jsonlite_1.8.4 evaluate_0.20 highr_0.10
#> [9] rlang_1.1.0 cachem_1.0.7 cli_3.6.1 jquerylib_0.1.4
#> [13] bslib_0.4.2 rmarkdown_2.21 tools_4.2.2 sfsmisc_1.1-14
#> [17] xfun_0.38 yaml_2.3.7 fastmap_1.1.1 compiler_4.2.2
#> [21] htmltools_0.5.5 knitr_1.42 startupmsg_0.9.6 sass_0.4.5
```

[Rousseeuw, Peter J., and Christophe Croux. “Alternatives to the median absolute deviation.” Journal of the American Statistical association 88.424 (1993): 1273-1283.]

[Kolmogorov A (1941). “Confidence limits for an unknown distribution function.” The annals of mathematical statistics, 12(4), 461–463.]

[Smirnov NV (1944). “Approximate laws of distribution of random variables from em- pirical data.” Uspekhi Matematicheskikh Nauk, (10), 179–206.]

University of Chicago, [email protected]↩︎

University of Chicago, [email protected]↩︎

University of Chicago, [email protected]↩︎