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How can you perform hypothesis testing for non-Gaussian data?

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Hypothesis testing is a powerful tool for evaluating the validity of a claim or a prediction based on data. However, many traditional methods of hypothesis testing rely on the assumption that the data follows a normal or Gaussian distribution, which is often not the case in real-world scenarios. For example, data from natural phenomena, social networks, or machine learning models may exhibit skewed, heavy-tailed, or multimodal distributions that violate the normality assumption. How can you perform hypothesis testing for non-Gaussian data? In this article, you will learn about some alternative approaches that can handle different types of non-Gaussian data.

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Sharat Achanta

Machine Learning | Deep Learning | NLP | Data Analytics
Miguel Angel Maga?a Fuentes

?? Machine Learning Engineer | ?? Data Scientist | ?? Analytics Engineer | ?? Deep Learning | ?? Quantum Materials…

1 Parametric tests

Parametric tests are hypothesis tests that use a specific probability distribution to model the data and calculate the test statistic and the p-value. For example, the t-test, the ANOVA, and the chi-square test are parametric tests that use the normal, the F, and the chi-square distributions, respectively. However, parametric tests are not suitable for non-Gaussian data, as they may produce inaccurate or misleading results. To use parametric tests, you need to check the normality of your data using graphical methods (such as histograms or Q-Q plots) or numerical methods (such as the Shapiro-Wilk test or the Kolmogorov-Smirnov test). If your data is not normal, you need to either transform it to make it more normal (such as using log or Box-Cox transformations) or use a different distribution that fits your data better (such as the exponential, the Poisson, or the beta distributions).

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Sharat Achanta

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Types of parametric tests: 1. Z-test 2. Z-test for proportions 3. One-sample t-test 4. Paired t-test (or dependent samples t-test) 5. Two-sample t-test with equal variance (independent samples t-test) 6. Two-sample t-test with unequal variance (Welch's t-test) 7. One-way ANOVA 8. Repeated-measures ANOVA 9. Pearson correlation coefficient 10. Simple Linear regression

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2 Nonparametric tests

Nonparametric tests are hypothesis tests that do not rely on any specific probability distribution to model the data. Instead, they use the ranks, the signs, or the frequencies of the data to calculate the test statistic and the p-value. For example, the Mann-Whitney U test, the Wilcoxon signed-rank test, and the Kruskal-Wallis test are nonparametric tests that compare the medians of two or more groups. Nonparametric tests are more robust and flexible than parametric tests, as they can handle non-Gaussian data without any transformation or assumption. However, nonparametric tests are also less powerful and sensitive than parametric tests, as they lose some information by using ranks or signs instead of values. To use nonparametric tests, you need to check the assumptions of each test, such as the independence, the symmetry, or the homogeneity of the data.

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Sharat Achanta

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Types of non-parametric tests: 1. Wilcoxon Matched Pair Signed Test (or Wilcoxon signed-rank test; these are essentially the same, so I've listed them together) 2. Mann-Whitney U test (non-parametric alternative to the two-sample t-test) 3. Chi-Square Test (for independence or goodness of fit) 4. Fischer's exact test (used when expected frequencies are small) 5. Kruskal-Wallis test (non-parametric equivalent to one-way ANOVA) 6. Friedman test (non-parametric equivalent to repeated-measures ANOVA) 7. Mood's median test (another non-parametric alternative to one-way ANOVA) 8. Spearman’s Rank Correlation 9. Levene's Test (to check the equality of variances)

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3 Bootstrap tests

Bootstrap tests are hypothesis tests that use resampling techniques to generate multiple samples from the original data and estimate the distribution of the test statistic and the p-value. For example, the bootstrap t-test, the bootstrap ANOVA, and the bootstrap chi-square test are bootstrap tests that use the same test statistic as their parametric counterparts, but use bootstrap samples instead of theoretical distributions. Bootstrap tests are more adaptive and reliable than parametric or nonparametric tests, as they can handle any type of non-Gaussian data without any transformation or assumption. However, bootstrap tests are also more computationally intensive and complex than parametric or nonparametric tests, as they require a large number of bootstrap samples and a suitable method to calculate the p-value. To use bootstrap tests, you need to check the validity and the accuracy of the bootstrap method, such as the choice of the sample size, the resampling scheme, or the confidence interval.

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Miguel Angel Maga?a Fuentes

?? Machine Learning Engineer | ?? Data Scientist | ?? Analytics Engineer | ?? Deep Learning | ?? Quantum Materials Researcher | ?? Quantum Computing Enthusiast
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For non-Gaussian data, various bootstrapping methods can be applied: Standard Bootstrapping: Ideal for independent, identically distributed (i.i.d.) data; it resamples with replacement. Stratified Bootstrapping: Targets imbalanced datasets, ensuring proportional representation in resamples. Cluster Bootstrapping: Designed for clustered data; resamples entire clusters instead of individual entries. Smooth Bootstrapping: Adds random noise to the resampled data, which is beneficial for discrete datasets. Block Bootstrapping: Tailored for time series data, maintaining temporal structures by resampling in sequential blocks. Spatial Bootstrapping: For spatial data, it factors in spatial correlations for sampling.

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4 Bayesian tests

Bayesian tests are hypothesis tests that use the Bayesian framework to update the prior beliefs about the parameters of interest based on the data and calculate the posterior probability of the hypotheses. For example, the Bayesian t-test, the Bayesian ANOVA, and the Bayesian chi-square test are Bayesian tests that use the same test statistic as their parametric counterparts, but use the posterior distribution instead of the p-value. Bayesian tests are more informative and intuitive than parametric or nonparametric tests, as they can handle non-Gaussian data with different levels of prior knowledge and provide a direct measure of the evidence for or against the hypotheses. However, Bayesian tests are also more subjective and controversial than parametric or nonparametric tests, as they require a choice of the prior distribution and the likelihood function, which may affect the results and the interpretation. To use Bayesian tests, you need to check the appropriateness and the sensitivity of the Bayesian model, such as the choice of the prior, the likelihood, or the posterior.

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5 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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How can you perform hypothesis testing for non-Gaussian data?

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3

4

5

1 Parametric tests

2 Nonparametric tests

3 Bootstrap tests

4 Bayesian tests

5 Here’s what else to consider

Machine Learning

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How can you perform hypothesis testing for non-Gaussian data?

1

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3

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5

1 Parametric tests

2 Nonparametric tests

3 Bootstrap tests

4 Bayesian tests

5 Here’s what else to consider

Machine Learning

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