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WeiYa's Work Yard

A traveler with endless curiosity, who fell into the ocean of statistics, tries to write down his ideas and notes to save himself.

Combining p-values in Meta Analysis

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Tags: Meta-analysis, p-value

I came across the term meta-analysis in the previous post, and I had another question about nominal size while reading the paper of the previous post, which reminds me Keith’s notes. By coincidence, I also find the topic about meta-analysis in the same notes. Hence, this post is mainly based on Keith’s notes, and reproduce the power curves by myself.

Suppose testing H0:θ=0 requires expensive experiments, so you cannot perform a large scale experiment. Luckily, many researchers have done the same tests before. However, only p-values are reported in their research papers (without the actual datasets). Then we can combine these p-values by using the methods in this post. Generally, this type of multi-source inference is called meta analysis.

Setup

Let p1,,pm be the p-values of m independent size α tests for the same simple null H0 and same alternative H1, where α(0,1). We want to combine these p-values. The aggregated p-value and the aggregated test are defined to be

ˆp=ˆp(p1:n)andˆψ=1(ˆp<c),

respectively for some cIR.

Tippett (1931): ˆpT=min(p1,,pm)

Under H0, pii.i.dU(0,1), then

ˆpTBeta(1,m),

More generally, if X1,,Xni.i.dU(0,1), then X(j)Beta(j,n+j1) for j=1,,n.

and the critical value cT satisfies

α=P0(ˆpT<cT).

Pearson (1933): ˆpP=mj=1log(1pj)

Under H0, for each j, we have

P0(log(1pj)<t)=P0(pj<1et)=(1et)1(t>0),

implying that log(1pj)Exp(1). Hence, under H0,

ˆpP=mj=1log(1pj)Gamma(m,1).

Gamma distribution generalizes exponential distribution. Let θ>0 and nN, then

X1,,Xni.i.dθExp(1)T=ni=1XiθGamma(n).

If also allow “fractional sample size” nIR+, then TθGamma(n) is called the Gamma distribution with scale parameter θ and shape parameter n. This notation can avoid the confusing traditional notations Gamma(n,θ) and Gamma(n,1/θ) in which the second parameter refer to either the scale or rate parameter.

Fisher (1934): ˆpF=mj=1logpj

Under H0, note that

ˆpF=mj=1logpj=[mj=1logpj]W,

where WGamma(m,1) since 1pjd=pj. Under H0, cF can be found by solving

α=P0(ˆpF<cF)=P(W<cF)=P(W>cF),

that is,

1α=P(W<cF).

Power Curves

Suppose that H0:θ0=0 is tested against H1:θ>0 such that

p1,,pmi.i.dBeta(1,1+θ),θIR+,

and actually Beta(1,1)=Unif(0,1), which also includes the case for H0.

Estimate the power functions with Monte Carlo method, and implement in Julia as follows:

function calc_power(n, m; α = 0.05, θs = 0:2.5:25)
    cT = quantile(Beta(1, m), α)
    cP = quantile(Gamma(m, 1), α)
    cF = -quantile(Gamma(m, 1), 1-α)
    power = zeros(length(θs), 3)
    for k = 1:length(θs)
        freqs = zeros(Int, 3)
        for i = 1:n
            p = rand(Beta(1, 1+θs[k]), m)
            # Tippett
            freqs[1] += (minimum(p) < cT)
            # Pearson
            freqs[2] += (sum(-log.(1 .- p)) < cP)
            # Fisher
            freqs[3] += (sum(log.(p)) < cF)
        end
        power[k, :] .= freqs ./ n
    end
    return power
end

The upper-left plot shows that the tests ˆψP and ˆψT are the most and the least powerful tests among the three combined tests for any θ>0, respectively.

The remaining three plots present the power curves for each test when θ=1,2,3. Visually, the test ˆψT does not improve its power with m, which means that the combining method is bas. On the other hand, the test ˆψF and the test ψP increases their powers when m increases. It is a desirable property.

However, that it does not mean ˆψP is the universal best combing method. When the distribution of the p-values p1:m follow other distribution under H1, other combining methods may become more appealing. Refer to Heard, N. A., & Rubin-Delanchy, P. (2018). Choosing between methods of combining p-values. Biometrika, 105(1), 239–246. for more details.


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