## What is an Asymptotic Test?

In hypothesis testing, you generally have two choices: an exact test or an

**asymptotic test**. You can think of an asymptotic test as an approximation and an exact test as “the exact result.” For example, the chi-square test is an asymptotic test; the exact version is the binomial test, which creates approximations for p-values. The more data points you have, the better the asymptotic test approximation. However, large samples come with a computational cost—you may simply not have enough computer resources to be able to run one. In that case, an exact test might be the better option.

As a specific example, Let’s say you wanted to test marginal homogeneity with a sample size of 6. The usual test to run is the asymptotic McNemar’s test. However, the small sample means you can’t use the asymptotic version of McNemars test; Instead, you can use the exact version, which uses binomial probabilities [1].

## Why Use Asymptotic Test Statistics?

Sometimes it isn’t possible to calculate exact statistics for a test due to hardware of software constraints. Instead, we can calculate an approximation of the “true” statistic, an **asymptotic test value**. Many statistical tests used asymptotic test values, especially for large samples. For example, you might see a p-value reported as .001 when in fact the exact p-value might have dozens of decimal places.

It is computationally efficient to give these approximations and it usually makes no difference to the results of a statistical test. However, for small samples or sparse data, exact values are usually recommended as approximations can be significantly different than the true values.

## References

[1] Park, T. Is the exact test better than the asymptotic test for testing marginal homogeneity? Retrieved November 17, 2021 from: https://www.stat.fi/isi99/proceedings/arkisto/varasto/park0373.pdf