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Extensions of Fisher's method

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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 χ-distribution with 2k degrees of freedom:

X = − 2 ∑ i = 1 k log e ⁡ ( p i ) ∼ χ 2 ( 2 k ) . {\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 χ-distribution, cχ(k’), with k’ degrees of freedom.

The mean and variance of this scaled χ variable are:

E ⁡ [ c χ 2 ( k ′ ) ] = c k ′ , {\displaystyle \operatorname {E} =ck',}
Var ⁡ [ c χ 2 ( k ′ ) ] = 2 c 2 k ′ . {\displaystyle \operatorname {Var} =2c^{2}k'.}

where c = Var ⁡ ( X ) / ( 2 E ⁡ [ X ] ) {\displaystyle c=\operatorname {Var} (X)/(2\operatorname {E} )} and k ′ = 2 ( E ⁡ [ X ] ) 2 / Var ⁡ ( X ) {\displaystyle k'=2(\operatorname {E} )^{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.

Kost's method: t approximation

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

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:

X = ∑ i = 1 k ω i tan ⁡ [ ( 0.5 − p i ) π ] , {\displaystyle X=\sum _{i=1}^{k}\omega _{i}\tan,}

where ω i {\displaystyle \omega _{i}} are non-negative weights, subject to ∑ i = 1 k ω i = 1 {\displaystyle \sum _{i=1}^{k}\omega _{i}=1} . Under the null, p i {\displaystyle p_{i}} are uniformly distributed, therefore tan ⁡ [ ( 0.5 − p i ) π ] {\displaystyle \tan} are Cauchy distributed. Under some mild assumptions, but allowing for arbitrary dependency between the p i {\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:

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

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

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.
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