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 principal 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]}

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

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:

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

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

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

Unknown covariance

Harmonic mean p-value

Main article: 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]

Cauchy combination test

This is conceptually similar to Fisher's method: it computes a sum of transformed p-values. Unlike Fisher's method, which uses a log transformation to obtain a test statistic which has a chi-squared distribution under the null, the Cauchy combination test uses a tan transformation to obtain a test statistic whose tail is asymptotic to that of a Cauchy distribution under the null. The test statistic is:

${\displaystyle X=\sum _{i=1}^{k}\omega _{i}\tan[(0.5-p_{i})\pi ],}$

where ${\displaystyle \omega _{i}}$ are non-negative weights, subject to ${\displaystyle \sum _{i=1}^{k}\omega _{i}=1}$. Under the null, ${\displaystyle p_{i}}$ are uniformly distributed, therefore ${\displaystyle \tan[(0.5-p_{i})\pi ]}$ are Cauchy distributed. Under some mild assumptions, but allowing for arbitrary dependency between the ${\displaystyle p_{i}}$, the tail of the distribution of X is asymptotic to that of a Cauchy distribution. More precisely, letting W denote a standard Cauchy random variable:

${\displaystyle \lim _{t\to \infty }{\frac {P[X>t]}{P[W>t]}}=1.}$

This leads to a combined hypothesis test, in which X is compared to the quantiles of the Cauchy distribution. ^[5]

References

1. Brown, M. (1975). "A method for combining non-independent, one-sided tests of significance". Biometrics. 31 (4): 987–992. doi:10.2307/2529826. JSTOR   2529826.
2. Kost, J.; McDermott, M. (2002). "Combining dependent P-values". Statistics & Probability Letters. 60 (2): 183–190. doi:10.1016/S0167-7152(02)00310-3.
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.
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 . PMID   30610179.
5. Liu Y, Xie J (2020). "Cauchy combination test: a powerful test with analytic p-value calculation under arbitrary dependency structures". Journal of the American Statistical Association. 115 (529): 393–402. doi:10.1080/01621459.2018.1554485. PMC   7531765 .