How do you deal with non-normal data that cannot be transformed or normalized?

Normality matters because most of the standard statistical models that we run (e.g., linear regression, t-test) depend on our variables of interest being normally distributed. However, as researchers we know that data can be skewed or non-normally distributed. We need to first decide if whether it's essential to have normally distributed variables because our statistical models depend on it. If this is the case, then one must take precautionary steps to collect the most normal data possible and this starts as soon as we begin thinking about starting a research (data) project.

Siendo la realidad multivariable, enfrentar la no normalidad de los datos puede ser complicado. Utilizo transformaciones como el análisis de componentes principales para reducir dependencias y acercar los componentes a una distribución más normal. También aplico modelos de distribución mixta multivariados para manejar datos con colas pesadas. En regresión, prefiero métodos robustos como la regresión cuantílica multivariada, que minimiza el impacto de los outliers. Las copulas son útiles para modelar dependencias complejas, permitiendo unir variables modeladas independientemente en una distribución conjunta. Empleo bootstrap multivariado para evaluar la incertidumbre sin asumir normalidad,para pruebas de hipótesis y intervalos de confianza.

Non-normal variables affect the sample size required to achieve representativeness. This happens, for example, when conducting a study related to average income and your sample includes a billionaire. It skews the mean, and to achieve representativeness, you need a much larger sample size.

When dealing with non-normal data that cannot be transformed or normalized, alternative methods such as non-parametric statistical tests can be employed. These tests, such as the Mann-Whitney U test or the Kruskal-Wallis H test, do not assume normality and are suitable for ordinal data or non-normally distributed interval data. Normality matters because many statistical techniques assume normal distribution of data to accurately estimate parameters and make inferences. However, non-parametric methods offer a robust alternative, allowing for effective analysis without the need for normal distribution.

Data Follows a Different Distribution There are many data types that follow a non-normal distribution by nature. Examples include: Weibull distribution, found with life data such as survival times of a product Log-normal distribution, found with length data such as heights Largest-extreme-value distribution, found with data such as the longest down-time each day Exponential distribution, found with growth data such as bacterial growth Poisson distribution, found with rare events such as number of accidents Binomial distribution, found with “proportion” data such as percent defectives If data follows one of these different distributions, it must be dealt with using the same tools as with data that cannot be “made” normal.

To check for normality, statistical tests and visual methods are commonly used. The Shapiro-Wilk test is a popular statistical test for assessing normality, with a null hypothesis that data is normally distributed. If the p-value is less than the chosen significance level, the data is not normally distributed. Visual methods include plotting the data using histograms or Q-Q (quantile-quantile) plots, where deviations from the linearity in a Q-Q plot suggest deviations from normality. Both methods provide insights into the data's distribution, aiding in deciding the appropriate analytical approach.

When dealing with non-normal data, transforming or normalizing it can enhance its suitability for statistical analyses. Identify skewness or outliers and choose appropriate transformations like logarithmic or square root functions. Alternatively, opt for normalization by scaling data to standard ranges such as 0 to 1 or -1 to 1. Apply the chosen method across relevant variables and features. After transformation or normalization, assess normality using tests like Shapiro-Wilk or visual tools. Iteratively refine the process if necessary, considering different methods. Exercise caution in interpreting results, acknowledging that transformation or normalization may not entirely resolve underlying issues.

To transform or normalize non-normal data, several techniques can be applied. Log transformation is common for data with right-skewed distributions, making the data more normally distributed. Square root and cube root transformations can also be used for right-skewed data, albeit to a lesser extent. For left-skewed data, square or cubic transformations may help. The Box-Cox transformation offers a systematic way of identifying an appropriate transformation that makes the data as close to normal as possible. These techniques modify the scale or distribution of data, facilitating the application of parametric statistical methods that assume normality.

When faced with non-normal data resistant to transformation, explore robust statistical methods like median or quantile-based techniques. Alternatively, consider non-parametric approaches such as bootstrapping or permutation tests. Tailor normalization strategies to data characteristics for effective handling of non-normal distributions.

There are testing context dependent options beyond the mentioned use of non-parametric methods and robust methods to address non-normal data. Additionally if statistical testing is in reference to percentiles other than the median, non-parametric tests can be defined for any desired percentile. If statistical testing is in reference to control limits there is an adaptable general purpose skew correction methodology for tail values provided in "Control Charts for Skewed and Left-Censored Data that Approximately Control the Family Type I Error" (Quality Engineering 2009). The referenced approach provides control limits as a function of n, mean, standard deviation and skew.

When dealing with non-normal data that cannot be transformed or normalized, consider using non-parametric methods. These methods do not assume normal distribution and are suitable for analyzing ordinal or non-normally distributed interval data. Examples include the Mann-Whitney U test for comparing two independent samples, the Kruskal-Wallis H test for comparing more than two independent samples, and the Spearman rank correlation for assessing relationships between variables. Non-parametric methods offer a robust alternative, allowing for valid statistical analysis without the requirement of normal distribution, making them a practical solution for analyzing non-normal data.

Dealing with non-normal data that resists transformation or normalization requires alternative statistical techniques such as robust statistical methods or non-parametric tests. These approaches accommodate skewed or heavy-tailed distributions without requiring data transformation, ensuring valid analysis in diverse data scenarios.

When dealing with non-normal data that cannot be transformed or normalized, consider the following approaches: Non-parametric Tests: Use statistical methods that do not assume a normal distribution, such as the Mann-Whitney U test or Kruskal-Wallis test. Robust Methods: Employ robust statistical techniques that are less sensitive to non-normality, like median regression or robust standard errors. Resampling Methods: Utilize bootstrapping to assess the sampling distribution without assuming normality. Generalized Linear Models (GLMs): Apply GLMs, which are flexible and can handle various distributions, not just the normal distribution. These methods help mitigate the impact of non-normality on your analysis.

Here are some things to consider when reporting non-normally distributed data that can not be normalized: 1) Be honest with your reader about the data you have even if it is non-normally distributed. 2) Describe your data using descriptive statistics, visuals, etc. Let the reader know what measures you took to salvage the data you collected (e.g., data cleaning, etc.). 3) Acknowledge the limitations of how you collected your data and see if you can replicate your study. If you encounter non-normality again that is okay, it is still insightful to you and the reader because it can help to understand the sample, and thus the population. Overall, non-normality is a good way to help us formulate new hypothesis/predictions.

When reporting non-normal data that cannot be transformed or normalized, clearly describe the nature of the data and the statistical methods used for analysis. Include a rationale for choosing non-parametric methods or alternative approaches, highlighting their suitability for the data characteristics. Present descriptive statistics such as median and interquartile range (IQR) instead of mean and standard deviation, as they provide a more accurate representation of central tendency and variability in non-normal distributions. Additionally, graphical representations like box plots can effectively illustrate the data's distribution. Transparency about the data's distribution and analytical methods enhances the credibility and interpretability

When handling non-normal data that defies transformation, focus shifts to robust statistical methods like non-parametric tests. Reporting such data entails transparency about its distribution, using descriptive statistics like median and interquartile range, or employing alternative visualizations to convey the data's nature accurately.

Here are some steps to take when you find non-normality in your data, notice the order: 1) Ask yourself what variable in your data is non-normally distributed. Is it your dependent or independent variable? 2) Identify what research question you are trying to answer to determine the type of statistical model you will run. 3) Once you understand the statistical model, determine if whether you need to transform your variable or not. In other words, decide if whether converting your variable will help you make the most of your statistical model and that you're not violating any model assumptions. 4) If possible, collect more data and/or run further models that can help assimilate resampling (e.g., Bootstrap).

When faced with non-normal data resistant to transformation, alternative statistical methods such as non-parametric tests or robust statistical techniques can be employed. These methods are less reliant on data distribution assumptions, providing reliable analysis despite non-normality.