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Easy Significance Testing for Deep Neural Networks.

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deep-significance: Easy and Better Significance Testing for Deep Neural Networks

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Contents

:interrobang: Why?

Although Deep Learning has undergone spectacular growth in the recent decade, a large portion of experimental evidence is not supported by statistical hypothesis tests. Instead, conclusions are often drawn based on single performance scores.

This is problematic: Neural network display highly non-convex loss surfaces (Li et al., 2018) and their performance depends on the specific hyperparameters that were found, or stochastic factors like Dropout masks, making comparisons between architectures more difficult. Based on comparing only (the mean of) a few scores, we often cannot conclude that one model type or algorithm is better than another. This endangers the progress in the field, as seeming success due to random chance might lead practitioners astray.

For instance, a recent study in Natural Language Processing by Narang et al. (2021) has found that many modifications proposed to transformers do not actually improve performance. Similar issues are known to plague other fields like e.g., Reinforcement Learning (Henderson et al., 2018) and Computer Vision (Borji, 2017) as well.

To help mitigate this problem, this package supplies fully-tested re-implementations of useful functions for significance testing:

  • Statistical Significance tests such as Almost Stochastic Order (del Barrio et al, 2017; Dror et al., 2019), bootstrap (Efron & Tibshirani, 1994) and permutation-randomization (Noreen, 1989).
  • Bonferroni correction methods for multiplicity in datasets (Bonferroni, 1936).
  • Bootstrap power analysis (Yuan & Hayashi, 2003) and other functions to determine the right sample size.

All functions are fully tested and also compatible with common deep learning data structures, such as PyTorch / Tensorflow tensors as well as NumPy and Jax arrays. For examples about the usage, consult the documentation here , the scenarios in the section Examples or the demo Jupyter notebook.

:inbox_tray: Installation

The package can simply be installed using pip by running

pip3 install deepsig

Another option is to clone the repository and install the package locally:

git clone https://github.com/Kaleidophon/deep-significance.git
cd deep-significance
pip3 install -e .

Warning: Installed like this, imports will fail when the clones repository is moved.

:bookmark: Examples


tl;dr: Use aso() to compare scores for two models. If the returned eps_min < 0.5, A is better than B. The lower eps_min, the more confident the result (we recommend to check eps_min < 0.2 and record eps_min alongside experimental results).

:warning: Testing models with only one set of hyperparameters and only one test set will be able to guarantee superiority in all settings. See General Recommendations & other notes.


In the following, we will lay out three scenarios that describe common use cases for ML practitioners and how to apply the methods implemented in this package accordingly. For an introduction into statistical hypothesis testing, please refer to resources such as this blog post for a general overview or Dror et al. (2018) for a NLP-specific point of view.

We assume that we have two sets of scores we would like to compare, $\mathbb{S}\mathbb{A}$ and $\mathbb{S}\mathbb{B}$, for instance obtained by running two models $\mathbb{A}$ and $\mathbb{B}$ multiple times with a different random seed. We can then define a one-sided test statistic $\delta(\mathbb{S}\mathbb{A}, \mathbb{S}\mathbb{B})$ based on the gathered observations. An example of such test statistics is for instance the difference in observation means. We then formulate the following null-hypothesis:

$$ H_0: \delta(\mathbb{S}\mathbb{A}, \mathbb{S}\mathbb{B}) \le 0 $$

That means that we actually assume the opposite of our desired case, namely that $\mathbb{A}$ is not better than $\mathbb{B}$, but equally as good or worse, as indicated by the value of the test statistic. Usually, the goal becomes to reject this null hypothesis using the SST. p-value testing is a frequentist method in the realm of SST. It introduces the notion of data that could have been observed if we were to repeat our experiment again using the same conditions, which we will write with superscript $\text{rep}$ in order to distinguish them from our actually observed scores (Gelman et al., 2021). We then define the p-value as the probability that, under the null hypothesis, the test statistic using replicated observation is larger than or equal to the observed test statistic:

$$ p(\delta(\mathbb{S}\mathbb{A}^\text{rep}, \mathbb{S}\mathbb{B}^\text{rep}) \ge \delta(\mathbb{S}\mathbb{A}, \mathbb{S}\mathbb{B})|H_0) $$

We can interpret this expression as follows: Assuming that $\mathbb{A}$ is not better than $\mathbb{B}$, the test assumes a corresponding distribution of statistics that $\delta$ is drawn from. So how does the observed test statistic $\delta(\mathbb{S}\mathbb{A}, \mathbb{S}\mathbb{B})$ fit in here? This is what the $p$-value expresses: When the probability is high, $\delta(\mathbb{S}\mathbb{A}, \mathbb{S}\mathbb{B})$ is in line with what we expected under the null hypothesis, so we can not reject the null hypothesis, or in other words, we \emph{cannot} conclude $\mathbb{A}$ to be better than $\mathbb{B}$. If the probability is low, that means that the observed $\delta(\mathbb{S}, \mathbb{S}_\mathbb{B})$ is quite unlikely under the null hypothesis and that the reverse case is more likely - i.e. that it is likely larger than - and we conclude that $\mathbb{A}$ is indeed better than $\mathbb{B}$. Note that the $p$-value does not express whether the null hypothesis is true. To make our decision about whether or not to reject the null hypothesis, we typically determine a threshold - the significance level $\alpha$, often set to 0.05 - that the p-value has to fall below. However, it has been argued that a better practice involves reporting the p-value alongside the results without a pidgeonholing of results into significant and non-significant (Wasserstein et al., 2019).

Intermezzo: Almost Stochastic Order - a better significance test for Deep Neural Networks

Deep neural networks are highly non-linear models, having their performance highly dependent on hyperparameters, random seeds and other (stochastic) factors. Therefore, comparing the means of two models across several runs might not be enough to decide if a model A is better than B. In fact, even aggregating more statistics like standard deviation, minimum or maximum might not be enough to make a decision. For this reason, del Barrio et al. (2017) and Dror et al. (2019) introduced Almost Stochastic Order (ASO), a test to compare two score distributions.

It builds on the concept of stochastic order: We can compare two distributions and declare one as stochastically dominant by comparing their cumulative distribution functions:

Here, the CDF of A is given in red and in green for B. If the CDF of A is lower than B for every $x$, we know the algorithm A to score higher. However, in practice these cases are rarely so clear-cut (imagine e.g. two normal distributions with the same mean but different variances). For this reason, del Barrio et al. (2017) and Dror et al. (2019) consider the notion of almost stochastic dominance by quantifying the extent to which stochastic order is being violated (red area):

ASO returns a value $\epsilon_\text{min}$, which expresses (an upper bound to) the amount of violation of stochastic order. If $\epsilon_\text{min} < \tau$ (where \tau is 0.5 or less), A is stochastically dominant over B in more cases than vice versa, then the corresponding algorithm can be declared as superior. We can also interpret $\epsilon_\text{min}$ as a confidence score. The lower it is, the more sure we can be that A is better than B. Note: ASO does not compute p-values. Instead, the null hypothesis formulated as

$$ H_0: \epsilon_\text{min} \ge \tau $$

If we want to be more confident about the result of ASO, we can also set the rejection threshold to be lower than 0.5 (see the discussion in this section). Furthermore, the significance level $\alpha$ is determined as an input argument when running ASO and actively influence the resulting $\epsilon_\text{min}$.

Scenario 1 - Comparing multiple runs of two models

In the simplest scenario, we have retrieved a set of scores from a model A and a baseline B on a dataset, stemming from various model runs with different seeds. We want to test whether our model A is better than B (higher scores = better)- We can now simply apply the ASO test:

import numpy as np
from deepsig import aso

seed = 1234
np.random.seed(seed)

# Simulate scores
N = 5  # Number of random seeds
my_model_scores = np.random.normal(loc=0.9, scale=0.8, size=N)
baseline_scores = np.random.normal(loc=0, scale=1, size=N)

min_eps = aso(my_model_scores, baseline_scores, seed=seed)  # min_eps = 0.225, so A is better

Note that ASO does not make any assumptions about the distributions of the scores. This means that we can apply it to any kind of test metric, as long as a higher score indicates a better performance (to apply ASO to cases where lower scores indicate better performances, just multiple your scores by -1 before feeding them into the function). The more scores of model runs is supplied, the more reliable the test becomes, so try to collect scores from as many runs as possible to reject the null hypothesis confidently.

Scenario 2 - Comparing multiple runs across datasets

When comparing models across datasets, we formulate one null hypothesis per dataset. However, we have to make sure not to fall prey to the multiple comparisons problem: In short, the more comparisons between A and B we are conducting, the more likely gets is to reject a null-hypothesis accidentally. That is why we have to adjust our significance threshold $\alpha$ accordingly by dividing it by the number of comparisons, which corresponds to the Bonferroni correction (Bonferroni et al., 1936):

import numpy as np
from deepsig import aso 

seed = 1234
np.random.seed(seed)

# Simulate scores for three datasets
M = 3  # Number of datasets
N = 5  # Number of random seeds
my_model_scores_per_dataset = [np.random.normal(loc=0.3, scale=0.8, size=N) for _ in range(M)]
baseline_scores_per_dataset  = [np.random.normal(loc=0, scale=1, size=N) for _ in range(M)]

# epsilon_min values with Bonferroni correction 
eps_min = [aso(a, b, confidence_level=0.95, num_comparisons=M, seed=seed) for a, b in zip(my_model_scores_per_dataset, baseline_scores_per_dataset)]
# eps_min = [0.006370113450148568, 0.6534772728574852, 0.0]

Scenario 3 - Comparing sample-level scores

In previous examples, we have assumed that we compare two algorithms A and B based on their performance per run, i.e. we run each algorithm once per random seed and obtain exactly one score on our test set. In some cases however, we would like to compare two algorithms based on scores for every point in the test set. If we only use one seed per model, then this case is equivalent to scenario 1. But what if we also want to use multiple seeds per model?

In this scenario, we can do pair-wise comparisons of the score distributions between A and B and use the Bonferroni correction accordingly:

from itertools import product 

import numpy as np
from deepsig import aso 

seed = 1234
np.random.seed(seed)

# Simulate scores for three datasets
M = 40   # Number of data points
N = 3  # Number of random seeds
my_model_scored_samples_per_run = [np.random.normal(loc=0.3, scale=0.8, size=M) for _ in range(N)]
baseline_scored_samples_per_run = [np.random.normal(loc=0, scale=1, size=M) for _ in range(N)]
pairs = list(product(my_model_scored_samples_per_run, baseline_scored_samples_per_run))

# epsilon_min values with Bonferroni correction 
eps_min = [aso(a, b, confidence_level=0.95, num_comparisons=len(pairs), seed=seed) for a, b in pairs]
# eps_min = [0.3831678636198528, 0.07194780234194881, 0.9152792807128325, 0.5273463008857844, 0.14946944524461184, 1.0, 
# 0.6099543280369378, 0.22387448804041898, 1.0]

Scenario 4 - Comparing more than two models

Similarly, when comparing multiple models (now again on a per-seed basis), we can use a similar approach like in the previous example. For instance, for three models, we can create a $3 \times 3$ matrix and fill the entries with the corresponding $\epsilon_\text{min}$ values.

The package implements the function multi_aso() exactly for this purpose. It has the same arguments as aso(), with a few differences. First of all, the function takes a single scores argument, which can be a list of lists (of scores), or a nested NumPy array or Tensorflow / PyTorch / Jax tensor or dictionary (more about that later). Let's look at an example:

import numpy as np 
from deepsig import multi_aso 

seed = 1234
np.random.seed(seed)

N = 5  # Number of random seeds
M = 3  # Number of different models / algorithms

# Simulate different model scores by sampling from normal distributions with increasing means
# Here, we will sample from N(0.1, 0.8), N(0.15, 0.8), N(0.2, 0.8)
my_models_scores = np.array([np.random.normal(loc=loc, scale=0.8, size=N) for loc in np.arange(0.1, 0.1 + 0.05 * M, step=0.05)])

eps_min = multi_aso(my_models_scores, confidence_level=0.95, seed=seed)

# eps_min =
# array([[1.       , 0.92621655, 1.        ],
#       [1.        , 1.        , 1.        ],
#       [0.82081635, 0.73048716, 1.        ]])

In the example, eps_min is now a matrix, containing the $\epsilon_\text{min}$ score between all pairs of models (for the same model, it set to 1 by default). The matrix is always to be read as ASO(row, column).

The function applies the bonferroni correction for multiple comparisons by default, but this can be turned off by using use_bonferroni=False.

Lastly, when the scores argument is a dictionary and the function is called with return_df=True, the resulting matrix is given as a pandas.DataFrame for increased readability:

import numpy as np 
from deepsig import multi_aso 

seed = 1234
np.random.seed(seed)

N = 5  # Number of random seeds
M = 3  # Number of different models / algorithms

# Same setup as above, but use a dict for scores
my_models_scores = {
  f"model {i+1}": np.random.normal(loc=loc, scale=0.8, size=N) 
  for i, loc in enumerate(np.arange(0.1, 0.1 + 0.05 * M, step=0.05))
}

# my_model_scores = {
#   "model 1": array([...]),
#   "model 2": array([...]),
#   ...
# }

eps_min = multi_aso(my_models_scores, confidence_level=0.95, return_df=True, seed=seed)

# This is now a DataFrame!
# eps_min =
#          model 1   model 2  model 3
# model 1  1.000000  0.926217      1.0
# model 2  1.000000  1.000000      1.0
# model 3  0.820816  0.730487      1.0

:newspaper: How to report results

When ASO used, two important details have to be reported, namely the confidence level $\alpha$ and the $\epsilon_\text{min}$ score. Below lists some example snippets reporting the results of scenarios 1 and 4:

Using ASO with a confidence level $\alpha = 0.05$, we found the score distribution of algorithm A based on three 
random seeds to be stochastically dominant over B ($\epsilon_\text{min} = 0$).

We compared all pairs of models based on five random seeds each using ASO with a confidence level of 
$\alpha = 0.05$ (before adjusting for all pair-wise comparisons using the Bonferroni correction). Almost stochastic 
dominance ($\epsilon_\text{min} < \tau$ with $\tau = 0.2$) is indicated in table X.

:control_knobs: Sample size

It can be hard to determine whether the currently collected set of scores is large enough to allow for reliable significance testing or whether more scores are required. For this reason, deep-significance also implements functions to aid the decision of whether to collect more samples or not.

First of all, it contains Bootstrap power analysis (Yuan & Hayashi, 2003): Given a set of scores, it gives all of them a uniform lift to create an artificial, second sample. Then, the analysis runs repeated analyses using bootstrapped versions of both samples, comparing them with a significance test. Ideally, this should yield a significant result: If the difference between the re-sampled original and the lifted sample is non-significant, the original sample has too high of a variance. The analyses then returns the percentage of comparisons that yielded significant results. If the number is too low, more scores should be collected and added to the sample.

The result of the analysis is the statistical power: The higher the power, the smaller the risk of falling prey to a Type II error - the probability of mistakenly accepting the null hypothesis, when in fact it should actually be rejected. Usually, a power of ~ 0.8 is recommended (although that is sometimes hard to achieve in a machine learning setup).

The function can be used in the following way:

import numpy as np
from deepsig import bootstrap_power_analysis

scores = np.random.normal(loc=0, scale=20, size=5)  # Create too small of a sample with high variance
power = bootstrap_power_analysis(scores, show_progress=False)  # 0.081, way too low

scores2 = np.random.normal(loc=0, scale=20, size=50)  # Let's collect more samples
power2 = bootstrap_power_analysis(scores2, show_progress=False)  # Better power with 0.2556

By default, bootstrap_power_analysis() uses a one-sided Welch's t-test. However, this can be modified by passing a function to the significance_test argument, which expects a function taking two sets of scores and returning a p-value.

Secondly, if the Almost Stochastic Order test (ASO) is being used, there is a second function available. ASO estimates the violation ratio of two samples using bootstrapping. However, there is necessarily some uncertainty around that estimate, given that we only possess a finite number of samples. Using more samples decreases the uncertainty and makes the estimate tighter. The degree to which collecting more samples increases the tightness can be computed using the following function:

import numpy as np
from deepsig import aso_uncertainty_reduction

scores1 = np.random.normal(loc=0, scale=0.3, size=5)  # First sample with five scores
scores2 = np.random.normal(loc=0.2, scale=5, size=3)  # Second sample with three scores

red1 = aso_uncertainty_reduction(m_old=len(scores1), n_old=len(scores2), m_new=5, n_new=5)  # 1.1547005383792515
red2 = aso_uncertainty_reduction(m_old=len(scores1), n_old=len(scores2), m_new=7, n_new=3)  # 1.0583005244258363

# Adding two runs to scores1 increases tightness of estimate by 1.15
# But adding two runs to scores2 only increases tightness by 1.06! So spending two more runs on scores1 is better

:sparkles: Other features

:rocket: For the impatient: ASO with multi-threading

Waiting for all the bootstrap iterations to finish can feel tedious, especially when doing many comparisons. Therefore, ASO supports multithreading using joblib via the num_jobs argument.

from deepsig import aso
import numpy as np
from timeit import timeit

a = np.random.normal(size=1000)
b = np.random.normal(size=1000)

print(timeit(lambda: aso(a, b, num_jobs=1, show_progress=False), number=5))  # 616.2249192680001
print(timeit(lambda: aso(a, b, num_jobs=4, show_progress=False), number=5))  # 208.05637107000007

If you want to select the maximum number of jobs possible on your device, you can set num_jobs=-1:

print(timeit(lambda: aso(a, b, num_jobs=-1, show_progress=False), number=5))  # 187.26257274800003

:electric_plug: Compatibility with PyTorch, Tensorflow, Jax & Numpy

All tests implemented in this package also can take PyTorch / Tensorflow tensors and Jax or NumPy arrays as arguments:

from deepsig import aso 
import torch

a = torch.randn(5, 1)
b = torch.randn(5, 1)

aso(a, b)  # It just works!

:woman_farmer: Setting seeds for replicability

In order to ensure replicability, both aso() and multi_aso() supply as seed argument. This even works when multiple jobs are used!

:game_die: Permutation and bootstrap test

Should you be suspicious of ASO and want to revert to the good old faithful tests, this package also implements the paired-bootstrap as well as the permutation randomization test. Note that as discussed in the next section, these tests have less statistical power than ASO. Furthermore, a function for the Bonferroni-correction using p-values can also be found using from deepsig import bonferroni_correction.

import numpy as np
from deepsig import bootstrap_test, permutation_test

a = np.random.normal(loc=0.8, size=10)
b = np.random.normal(size=10)

print(permutation_test(a, b))  # 0.16183816183816183
print(bootstrap_test(a, b))    # 0.103

General recommendations & other notes

  • Naturally, the CDFs built from scores_a and scores_b can only be approximations of the true distributions. Therefore, as many scores as possible should be collected, especially if the variance between runs is high. If only one run is available, comparing sample-wise score distributions like in scenario 3 can be an option, but comparing multiple runs will always be preferable. Ideally, scores should be obtained even using different sets of hyperparameters per model. Because this is usually infeasible in practice, Bouthilier et al. (2020) recommend to vary all other sources of variation between runs to obtain the most trustworthy estimate of the "true" performance, such as data shuffling, weight initialization etc.

  • num_bootstrap_iterations can be reduced to increase the speed of aso(). However, this is not recommended as the result of the test will also become less accurate. Technically, $\epsilon_\text{min}$ is a upper bound that becomes tighter with the number of samples and bootstrap iterations (del Barrio et al., 2017). Thus, increasing the number of jobs with num_jobs instead is always preferred.

  • While we could declare a model stochastically dominant with $\epsilon_\text{min} < 0.5$, we found this to have a comparatively high Type I error (false positives). Tests in our paper have shown that a more useful threshold that trades of Type I and Type II error between different scenarios might be $\tau = 0.2$.

  • Bootstrap and permutation-randomization are all non-parametric tests, i.e. they don't make any assumptions about the distribution of our test metric. Nevertheless, they differ in their statistical power, which is defined as the probability that the null hypothesis is being rejected given that there is a difference between A and B. In other words, the more powerful a test, the less conservative it is and the more it is able to pick up on smaller difference between A and B. Therefore, if the distribution is known or found out why normality tests (like e.g. Anderson-Darling or Shapiro-Wilk), something like a parametric test like Student's or Welch's t-test is preferable to bootstrap or permutation-randomization. However, because these test are in turn less applicable in a Deep Learning setting due to the reasons elaborated on in Why?, ASO is still a better choice.

:mortar_board: Cite

Using this package in general, please cite the following:

@article{ulmer2022deep,
  title={deep-significance-Easy and Meaningful Statistical Significance Testing in the Age of Neural Networks},
  author={Ulmer, Dennis and Hardmeier, Christian and Frellsen, Jes},
  journal={arXiv preprint arXiv:2204.06815},
  year={2022}
}

If you use the ASO test via aso() or `multi_aso, please cite the original works:

@inproceedings{dror2019deep,
  author    = {Rotem Dror and
               Segev Shlomov and
               Roi Reichart},
  editor    = {Anna Korhonen and
               David R. Traum and
               Llu{\'{\i}}s M{\`{a}}rquez},
  title     = {Deep Dominance - How to Properly Compare Deep Neural Models},
  booktitle = {Proceedings of the 57th Conference of the Association for Computational
               Linguistics, {ACL} 2019, Florence, Italy, July 28- August 2, 2019,
               Volume 1: Long Papers},
  pages     = {2773--2785},
  publisher = {Association for Computational Linguistics},
  year      = {2019},
  url       = {https://doi.org/10.18653/v1/p19-1266},
  doi       = {10.18653/v1/p19-1266},
  timestamp = {Tue, 28 Jan 2020 10:27:52 +0100},
}

@incollection{del2018optimal,
  title={An optimal transportation approach for assessing almost stochastic order},
  author={Del Barrio, Eustasio and Cuesta-Albertos, Juan A and Matr{\'a}n, Carlos},
  booktitle={The Mathematics of the Uncertain},
  pages={33--44},
  year={2018},
  publisher={Springer}
}

For instance, you can write

In order to compare models, we use the Almost Stochastic Order test \citep{del2018optimal, dror2019deep} as 
implemented by \citet{ulmer2022deep}.

:medal_sports: Acknowledgements

This package was created out of discussions of the NLPnorth group at the IT University Copenhagen, whose members I want to thank for their feedback. The code in this repository is in multiple places based on several of Rotem Dror's repositories, namely this, this and this one. Thanks also go out to her personally for being available to answer questions and provide feedback to the implementation and documentation of this package.

The commit message template used in this project can be found here. The inline latex equations were rendered using readme2latex.

:people_holding_hands: Papers using deep-significance

In this last section of the readme, I would like to refer to works already using deep-significance. Open an issue or pull request if you would like to see your work added here!

:books: Bibliography

Del Barrio, Eustasio, Juan A. Cuesta-Albertos, and Carlos Matrán. "An optimal transportation approach for assessing almost stochastic order." The Mathematics of the Uncertain. Springer, Cham, 2018. 33-44.

Bonferroni, Carlo. "Teoria statistica delle classi e calcolo delle probabilita." Pubblicazioni del R Istituto Superiore di Scienze Economiche e Commericiali di Firenze 8 (1936): 3-62.

Borji, Ali. "Negative results in computer vision: A perspective." Image and Vision Computing 69 (2018): 1-8.

Bouthillier, Xavier, et al. "Accounting for variance in machine learning benchmarks." Proceedings of Machine Learning and Systems 3 (2021).

Dror, Rotem, et al. "The hitchhiker’s guide to testing statistical significance in natural language processing." Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers). 2018.

Dror, Rotem, Shlomov, Segev, and Reichart, Roi. "Deep dominance-how to properly compare deep neural models." Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics. 2019.

Efron, Bradley, and Robert J. Tibshirani. "An introduction to the bootstrap." CRC press, 1994.

Andrew Gelman, John B Carlin, Hal S Stern, David B Dunson, Aki Vehtari, Donald B Rubin, John Carlin, Hal Stern, Donald Rubin, and David Dunson. Bayesian data analysis third edition, 2021.

Henderson, Peter, et al. "Deep reinforcement learning that matters." Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 32. No. 1. 2018.

Hao Li, Zheng Xu, Gavin Taylor, Christoph Studer, Tom Goldstein. "Visualizing the Loss Landscape of Neural Nets." NeurIPS 2018: 6391-6401

Narang, Sharan, et al. "Do Transformer Modifications Transfer Across Implementations and Applications?." arXiv preprint arXiv:2102.11972 (2021).

Noreen, Eric W. "Computer intensive methods for hypothesis testing: An introduction." Wiley, New York (1989).

Ronald L Wasserstein, Allen L Schirm, and Nicole A Lazar. Moving to a world beyond “p< 0.05”, 2019

Yuan, Ke‐Hai, and Kentaro Hayashi. "Bootstrap approach to inference and power analysis based on three test statistics for covariance structure models." British Journal of Mathematical and Statistical Psychology 56.1 (2003): 93-110.

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