The winner’s curse is the statistical and epistemological phenomenon by which the (quantitative) importance of a given factor seems exaggerated at first before being re-evaluated (through replication) to more modest values. This is also part of a wider phenomenon known as the decline effect. In a genomics setup, this could mean that a marker, when it is first discovered, may seem to be far more important than it truly is actually: the odds ratio may be inflated. Another example is provided by the search of quantitative trait loci (QTLs): QTLs are genetic polymorphisms that are statistically associated (and perhaps causally associated) with differential transcription levels. It may appear that the transcription level is more stronger influenced by an allele than is really the case.

Suppose that some physiological value \(Y\) is normally distributed with variance \(\sigma^2 = 1\). The normal value of \(Y\) is centred on zero but some drug \(X\) shifts it by \(\mu = 0.4\). If each arm of a trial contains \(N = 50\) people, the statistical power can be calculated using power.t.test():

mu = .4
n = 50
alpha = .05
power.t.test(n=n, delta=mu, sd=1, sig.level=alpha)

     Two-sample t test power calculation 

              n = 50
          delta = 0.4
             sd = 1
      sig.level = 0.05
          power = 0.5081451
    alternative = two.sided

NOTE: n is number in *each* group

It can also be estimated through simulation (with a confidence interval thanks to prop.test()):

set.seed(123)

R = 1000

f = sum(replicate(R, {
    y1 = rnorm(n)
    y2 = rnorm(n, mean=mu)
    t = t.test(y1, y2)
    t$p.value < alpha
}))

prop.test(f, R)

    1-sample proportions test with continuity correction

data:  f out of R, null probability 0.5
X-squared = 4.489, df = 1, p-value = 0.03411
alternative hypothesis: true p is not equal to 0.5
95 percent confidence interval:
 0.5025106 0.5652249
sample estimates:
    p 
0.534 

Suppose you run many independent tries following an identical protocol (imagine that many countries or pharmaceutical firms are testing the drug at the same time). What’s the expected effect size that will be reported when the trial is conclusive (ie, the \(p\) value is below the significance level \(\alpha = 0.05\)).