---
title: "Outlier Testing"
output: rmarkdown::html_vignette
author: "Jonas Kurle"
date: "11 June 2022"
vignette: >
  %\VignetteIndexEntry{Outlier Testing}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---
<style>
body {
text-align: justify}
</style>

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```
```{r, include = FALSE, echo = FALSE}
Sys.setenv(LANGUAGE="en")
```

# Introduction

This vignette introduces the testing suite implemented in version 0.2.0 that
offers several ways of testing for presence of outliers in the data. We will use
artificial data to illustrate the usage of the different functions and how they
can be combined.

See the vignette *Introduction to the robust2sls Package* for a general overview
over the 2SLS setting and the package itself and the vignette *Monte Carlo
Simulations* for more details about how to simulate data and register various
back-ends for parallel or sequential execution.

# Artificial Data

First, we generate the parameters of the 2SLS model randomly and create 
artificial data.

```{r, data}
library(robust2sls)
# create parameters
p <- generate_param(3, 2, 3, sigma = 2, intercept = TRUE, seed = 42)
# draw random sample of 1000 observations following the model
d <- generate_data(parameters = p, n = 1000)$data
```

# Outlier Detection

First, we use one of the algorithms to detect outliers in our sample. Suppose we
use *Robustified 2SLS*, iterate the algorithm five times, and choose a
significance level $\gamma_{c}$ of 0.05 (cut-off 1.96 under normality) to
classify an observation as an outlier or not.

Running the algorithm can be done as follows. The returned object is an object
of class *robust2sls*. This is a list storing, inter alia, the model estimates
and classifications as outliers for each iteration.

```{r, single_alg}
model <- outlier_detection(data = d, formula = p$setting$formula, ref_dist = "normal",
                           sign_level = 0.05, initial_est = "robustified", iterations = 5)
print(model)
```

For some of the tests, we might want to use the same algorithm but different
cut-off values. Under the null hypothesis of no outliers, we expect a 
proportional relationship between $\gamma_{c}$ and the share of detected 
outliers. For example, for $\gamma_{c} = 0.01$, we expect to classify 1% of the
observations as outliers; for $\gamma_{c} = 0.05$, we expect to classify 5% as
outliers. The utility function `multi_cutoff()` allows the user to estimate
several such models by giving the function a vector of multiple $\gamma$ values.

The function can be executed in parallel by registering a parallel back-end. We
choose to run the expression sequentially because it is already fast. The 
function simply returns a list of *robust2sls* objects.

```{r, multiple_alg}
# choose which gamma values to use
gammas <- seq(0.01, 0.05, 0.01)
# register backend
library(doFuture)
registerDoFuture()
plan(sequential)
models <- multi_cutoff(gamma = gammas, data = d, formula = p$setting$formula, ref_dist = "normal",
                       initial_est = "robustified", iterations = 5)
length(models)
```

# Outlier Testing

This section introduces four different tests:

* proportion test
* count test
* scaling sum test
* scaling supremum test

Each subsection quickly explains the intuition, the formula for the test
statistic, and provides a small example how to implement it.

As usual, all tests require a significance level to decide whether to reject or
not. Note that there are two significance levels at play: We use $\gamma_{c}$ to
refer to the significance level that determines the cut-off value beyond which
an error is classified as an outlier. It is a tuning parameter in the 
`outlier_detection()` algorithm. In contrast, $\alpha$ refers to the 
significance level of the tests, such that we reject the null hypothesis when
the p-value is smaller than the significance level.

## Proportion Test

The idea of the proportion test is to check whether the detected share of
outliers deviates significantly from the expected share of outliers under the
null hypothesis of no outliers. The test statistic is a simple t-test based on
the asymptotic distribution of the false outlier detection rate (FODR) and is
hence given as
$$t = \frac{\hat\gamma_{c} - \gamma_{c}}{se},$$
where $se$ is the standard error.

The function `proptest()` implements the test. It can either be given a 
*robust2sls* object or a list thereof, as returned by `multi_cutoff()`. 

```{r, proptest}
# using a single robust2sls object
proptest(model, alpha = 0.05, iteration = 5, one_sided = FALSE)

# using a list of robust2sls objects
proptest(models, alpha = 0.05, iteration = 5, one_sided = TRUE)
```

The function returns a data frame and each row corresponds to one setting of
$\gamma_{c}$. Note that the value of the test statistic is the same for the
first call of `proptest()` and the last row of the second call. They refer to
the same model.

The first column of the output stores the iteration that was tested and the
second column refers to the actual iteration that was tested. This only differs
when the user tests the convergenct distribution, which may differ across
different settings of $\gamma_{c}$. The one-sided test only rejects for positive
deviations from the expected value while the two-sided test rejects in both
directions.

When testing several settings of the algorithms, we have a multiple testing 
issue. We can apply the Simes (1986) (DOI: 10.1093/biomet/73.3.751) 
procedure to fix the significance level for the global null hypothesis, which is
rejected if any of the individual null hypotheses is rejected at the adjusted
significance level.

```{r, proptest simes}
proptests <- proptest(models, alpha = 0.05, iteration = 5, one_sided = TRUE)
a <- globaltest(tests = proptests, global_alpha = 0.05)

# decision for global hypothesis test
a$reject

# details for the Simes procedure
a$tests[, c("iter_test", "iter_act", "gamma", "t", "pval", "alpha_adj", "reject_adj")]

```

## Count Test

The idea is similar to the proportion test. Instead of comparing the share of
detected outliers to its expected value, we now compare the expected number of
detected outliers to the expected number. The test statistic asymptotically
follows a Poisson distribution.
$$n \hat  \gamma_{c},$$
where $n$ is the sample size.

The function `counttest()` implements the test. As `porptest()`, it can either
take a *robust2sls* object or a list thereof.

```{r, counttest}
# using a single robust2sls object
counttest(model, alpha = 0.05, iteration = 5, one_sided = FALSE)

# using a list of robust2sls objects
counttest(models, alpha = 0.05, iteration = 5, one_sided = TRUE)
```

As before, we can use the Simes (1986) (DOI: 10.1093/biomet/73.3.751) 
procedure to account for multiple hypothesis testing.

```{r, counttest simes}
counttests <- counttest(models, alpha = 0.05, iteration = 5, one_sided = TRUE)
b <- globaltest(tests = counttests, global_alpha = 0.05)

# decision for global hypothesis test
b$reject

# details for the Simes procedure
b$tests[, c("iter_test", "iter_act", "gamma", "num_act", "num_exp", "pval", "alpha_adj", "reject_adj")]
```

## Scaling Sum Test

This test directly exploits the information we get from running several 
configurations of the outlier detection algorithm with varying $\gamma_{c}$. We
therefore need a list of *robust2sls* objects, as for example returned by
`multi_cutoff()`.

The test statistic is constructed by summing up the deviations across different
cut-offs / values of $\gamma_{c}$:
$$t = \sum_{k = 1}^{K} \sqrt n (\hat \gamma_{c_{k}} - \gamma_{c_{k}})/se,$$
where $K$ is the number of different cut-off values that we have tried and $se$
is the standard error of this sum.

```{r, sumtest}
c <- sumtest(models, alpha = 0.05, iteration = 1, one_sided = FALSE)

attr(c, "gammas")
```

Note that we tested iteration 1 instead of iteration 5, which was the number of 
iterations we applied the algorithm. In general, as long as all *robust2sls* 
model objects contain that iteration, it can be tested - even if the model 
objects themselves contain further iterations.

The returned data frame has an attribute called `"gammas"`, which stores the
$\gamma_{c}$ values that were used in constructing the test.

## Scaling Supremum Test

As in the previous test, we combine several deviations across different cut-offs
/ values of $\gamma_{c}$. In this case, we take the supremum / maximum, since we
have a finite number of $\gamma$ values:
$$t = \sup_{k = 1,...,K} |\sqrt n (\hat \gamma_{c_{k}} - \gamma_{c_{k}})|$$
We simulate the asymptotic distribution of this object to derive the critical
values and the p-value.

The function `suptest()` implements this test. The user can specify which 
critical values should be returned. The default is to report the values 
corresponding to the 90th, 95th, and 99th quantile.

```{r, suptest}
d <- suptest(models, alpha = 0.05, iteration = 5)

attr(d, "gammas")
attr(d, "critical")
```

As for `sumtest()`, the returned data frame has an attribute called `"gammas"`,
which stores the $\gamma_{c}$ values that were used in constructing the test. In
addition, there is an attribute `"critical"`, which stores the critical values
of the simulated asymptotic distribution against which the test statistic was
compared.
