Extensions of Fisher's method

In statistics, extensions of Fisher's method are a group of approaches that allow approximately valid statistical inferences to be made when the assumptions required for the direct application of Fisher's method are not valid. Fisher's method is a way of combining the information in the p-values from different statistical tests so as to form a single overall test: this method requires that the individual test statistics (or, more immediately, their resulting p-values) should be statistically independent.

Dependent statistics

A principle limitation of Fisher's method is its exclusive design to combine independent p-values, which renders it an unreliable technique to combine dependent p-values. To overcome this limitation, a number of methods were developed to extend its utility.

Known covariance

Brown's method

Fisher's method showed that the log-sum of k independent p-values follow a χ²-distribution with 2k degrees of freedom:[1][2]

X=-2\sum _{i=1}^{k}\log _{e}(p_{i})\sim \chi ^{2}(2k).

In the case that these p-values are not independent, Brown proposed the idea of approximating X using a scaled χ²-distribution, cχ²(k’), with k’ degrees of freedom.

The mean and variance of this scaled χ² variable are:

\operatorname {E} [c\chi ^{2}(k')]=ck',

\operatorname {Var} [c\chi ^{2}(k')]=2c^{2}k'.

where $c=\operatorname {Var} (X)/(2\operatorname {E} [X])$ and $k'=2(\operatorname {E} [X])^{2}/\operatorname {Var} (X)$ . This approximation is shown to be accurate up to two moments.

Unknown covariance

Harmonic mean p-value

The harmonic mean p-value offers an alternative to Fisher's method for combining p-values when the dependency structure is unknown but the tests cannot be assumed to be independent.[3][4]

Kost's method: t approximation

This method requires the test statistics' covariance structure to be known up to a scalar multiplicative constant.[2]

gollark: - it makes assumptions about any universes which might be embedding ours which we have ~zero evidence on- you can probably get "good enough" behavior by approximating heavily, although people will eventually notice

gollark: > checkmate simulation theory 😎If this is meant unironically, then no.

gollark: (Almost) nobody analyses a computer program by simulating every atom in the CPU or something.

gollark: There are, still, apparently reasonably good and useful-for-predictions models of what people do in stuff like behavioral economics and psychology, even if exactly how stuff works isn't known.

gollark: We cannot, yet, just spin up a bunch of test societies with and without [CONTENTIOUS THING REDACTED] to see if this is actually true.

References

Brown, M. (1975). "A method for combining non-independent, one-sided tests of significance". Biometrics. 31: 987–992. doi:10.2307/2529826.
Kost, J.; McDermott, M. (2002). "Combining dependent P-values". Statistics & Probability Letters. 60: 183–190. doi:10.1016/S0167-7152(02)00310-3.
Good, I J (1958). "Significance tests in parallel and in series". Journal of the American Statistical Association. 53 (284): 799–813. doi:10.1080/01621459.1958.10501480. JSTOR 2281953.
Wilson, D J (2019). "The harmonic mean p-value for combining dependent tests". Proceedings of the National Academy of Sciences USA. 116 (4): 1195–1200. doi:10.1073/pnas.1814092116. PMC 6347718.

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[1] Brown, M. (1975). "A method for combining non-independent, one-sided tests of significance". Biometrics. 31: 987–992. doi:10.2307/2529826.

[:2-2] Kost, J.; McDermott, M. (2002). "Combining dependent P-values". Statistics & Probability Letters. 60: 183–190. doi:10.1016/S0167-7152(02)00310-3.

[:0-3] Good, I J (1958). "Significance tests in parallel and in series". Journal of the American Statistical Association. 53 (284): 799–813. doi:10.1080/01621459.1958.10501480. JSTOR 2281953.

[:1-4] Wilson, D J (2019). "The harmonic mean p-value for combining dependent tests". Proceedings of the National Academy of Sciences USA. 116 (4): 1195–1200. doi:10.1073/pnas.1814092116. PMC 6347718.