The mathematics of the last century worth of experiment design. This is about the classical framing, where you think about designing and running experiments and deciding if can reasonably be construed to be true or not, then go home. There are many elaborations of this approach in the modern world. For example, we examine large numbers of hypotheses at once under multiple testing. It can be considered as part of model selection question, or maybe even made particularly nifty using sparse model selection. Probably the most interesting family of tests are tests of conditional independence, especially multiple version fo those.

But the classic simplest case has things to teach us. Probably the least sexy thing in statistics and as such, usually taught by the least interesting professor in the department, or at least one who couldn’t find an interesting enough excuse to get out of it, which is a strong correlate. Said professor will then teach it to you as if you were in turn the least interesting student in the school, and so the spiral of boredom winds on. Anyhow, it turns out there are powerful tools within this area, and also instructive examples.

**tl;dr** classic statistical tests are linear models where your goal decide if a coefficient should be regarded as non-zero or not.
Jonas Kristoffer Lindeløv’s explains this perspective: Common statistical tests are linear models.

Daniel Lakens asks Do You Really Want to Test a Hypothesis?:

The lecture “Do You Really Want to Test a Hypothesis?” aims to explain which question a hypothesis tests asks, and discusses when a hypothesis tests answers a question you are interested in. It is very easy to say what not to do, or to point out what is wrong with statistical tools. Statistical tools are very limited, even under ideal circumstances. It’s more difficult to say what you

cando. If you follow my work, you know that this latter question is what I spend my time on. Instead of telling you optional stopping can’t be done because it isp-hacking, I explain how you can do it correctly through sequential analysis. Instead of telling you it is wrong to conclude the absence of an effect fromp> 0.05, I explain how to use equivalence testing. Instead of telling youp-values are the devil, I explain how they answer a question you might be interested in when used well. Instead of saying preregistration is redundant, I explain from which philosophy of science preregistration has value. And instead of saying we should abandon hypothesis tests, I try to explain in this video how to use them wisely. This is all part of my ongoing #JustifyEverything educational tour. I think it is a reasonable expectation that researchers should be able to answer at least a simple ‘why’ question if you ask why they use a specific tool, or use a tool in a specific manner.

Lucile Lu, Robert Chang and Dmitriy Ryaboy of Twitter have a practical guide to risky testing at scale: Power, minimal detectable effect, and bucket size estimation in A/B tests

Bob Sturm recommends, Bailey (2008) for discussion of hypothesis testing in terms of linear subspaces.

(side note: the proportional odds model generalises K-W/WMW. Huh.)

- Multiplicitous – Put A Number On It!
- Experiment power calculator tells you how many data points you need to have and thus whether it is likely you can (dis)prove the thing with the budget you have.

Everything so far has been in a frequentist framing.
The entire question of what hypothesist testing is much more likely to be be vacuous in Bayesian settings (although Bayes model selection is a thing).
See also Thomas Lumley on a Bayesian *t*-test which ends up being a kind of bootstrap in an elegant way.

I cannot decide if tea-lang is a passive-aggressive joke or not. It is a compiler for statistical tests.

Tea is a domain specific programming language that automates statistical test selection and execution… Users provide 5 pieces of information:

- the dataset of interest,
- the variables in the dataset they want to analyze,
- the study design (e.g., independent, dependent variables),
- the assumptions they make about the data based on domain knowledge(e.g., a variable is normally distributed), and
- a hypothesis.
Tea then “compiles” these into logical constraints to select valid statistical tests. Tests are considered valid if and only if all the assumptions they make about the data (e.g., normal distribution, equal variance between groups, etc.) hold. Tea then finally executes the valid tests.

## Goodness-of-fit tests

Also a useful thing to have; the hypothesis here is kind-of more interesting, along the lines of it-is-unlikely-that-the-model-you-propose-contains-this-data.

## Design of experiments

TBD

## References

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*Handbook of Combinatorial Designs, Second Edition*. CRC Press.

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*Resampling Methods: A Practical Guide to Data Analysis*. Birkhäuser Basel. https://doi.org/10.1007/978-1-4757-3049-4.

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*Electronic Journal of Statistics*9: 643–78. https://doi.org/10.1214/15-EJS1005.

*Testing Statistical Hypotheses*. 3. ed. Springer Texts in Statistics. New York, NY: Springer.

*Annual Review of Public Health*23 (1): 151–69. https://doi.org/10.1146/annurev.publhealth.23.100901.140546.

*Quantitative Finance*3 (4): 231–50. https://doi.org/10.1088/1469-7688/3/4/301.

*The American Statistician*73 (March): 235–45. https://doi.org/10.1080/00031305.2018.1527253.

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