Lesson plan
Compositional data
In statistics, compositional data are quantitative descriptions of the parts of some whole, conveying relative information. Mathematically, compositional data is represented by points on a simplex. Measurements involving probabilities, proportions, percentages, and ppm can all be thought of as compositional data. All analyses of real data deals with noisy measurements. The simpler case is in empirical studies such as questionnaire surveys, where each respondent's total allocation is fixed and one must deal solely with the effects of closure, odds and odds ratios serve as invariant statistics. In such cases, associations are determined solely by the magnitude of the odds ratio relative to 1. But in compositional data containing noise, the noise is present in the total sum and is distributed asymmetrically among the individual components. Since noise and signal are inseparable, the data can only be dichotomized into zeros and noise-contaminated components. Therefore, without explicit noise filtering, even statistics based on odds remain mathematically ill-defined. Indeed, odds and their ratios can only be calculated across samples in the absolute absence of any noise other than that induced by the compositionality itself (except for normalization).
Article summary
In statistics, compositional data are quantitative descriptions of the parts of some whole, conveying relative information. Mathematically, compositional data is represented by points on a simplex. Measurements involving probabilities, proportions, percentages, and ppm can all be thought of as compositional data. All analyses of real data deals with noisy measurements. The simpler case is in empirical studies such as questionnaire surveys, where each respondent's total allocation is fixed and one must deal solely with the effects of closure, odds and odds ratios serve as invariant statistics. In such cases, associations are determined solely by the magnitude of the odds ratio relative to 1. But in compositional data containing noise, the noise is present in the total sum and is distributed asymmetrically among the individual components. Since noise and signal are inseparable, the data can only be dichotomized into zeros and noise-contaminated components. Therefore, without explicit noise filtering, even statistics based on odds remain mathematically ill-defined. Indeed, odds and their ratios can only be calculated across samples in the absolute absence of any noise other than that induced by the compositionality itself (except for normalization).
Key claims
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.