The procedure examines the power of various statistics using a portion of the data in an exploratory first stage and then applies this most powerful test to the rest of the data in the second stage. The statistics from the two stages are combined to make full use of the information. This approach of combining the results of the two stages is equivalent to a more general method of combining *p*-values. For the procedure of combining these *p*-values to be valid, however, we need to specify before the analysis which statistic will be used to obtain the *p*-value (*p*_{1}) from the exploratory stage in the combination. The *p*-value from the second stage (*p*_{2}) is calculated based on the most powerful statistic found at the first stage. Under the null hypothesis, each *p* value is, at least asymptotically, distributed uniformly on *U*(0, 1). The final decision then depends on a combining function *f*(*p*_{1}, *p*_{2}). The most common such function may be Fisher's combination test [5], which is defined by

*f*(*p*_{1}, *p*_{2}) = -2log(*p*_{1}*p*_{2}),

where under the null hypothesis Fisher's statistic will be distributed as a *χ*^{2} with 4 degrees of freedom. Another example is the weighted inverse normal method,

*f*(*p*_{1}, *p*_{2}) = 1 - Φ[*w*_{1}Φ^{-1}(1 - *p*_{1}) + *w*_{2}Φ^{-1}(1 - *p*_{2})],

where Φ is the cumulative distribution function of a standard normal distribution, 0 <*w*_{
i
}< 1 and *w*_{1}^{2} + *w*_{2}^{2} = 1. This statistic will be distributed as a standard normal distribution.

To obtain *p*_{2}, we have to estimate the power of the various statistics at the exploratory first stage. Traditional power calculation methods require the trait distribution to be known, which is not the case here. A bootstrap method of using the data from the exploratory stage can be adopted to approximate the power [6, 7]. The bootstrap and permutation are two often used nonparametric procedures. It is often desired to obtain "exact" *p*-values by employing a permutation procedure to generate the null distribution of the statistic that is used for a test. Here, on the other hand, we want to estimate the power of a statistic, and for this we need the distribution of the statistics under the alternative hypothesis; a permutation procedure cannot be directly applied for this purpose. Let the trait values of individuals with genotype *g* be denoted *x*_{
g
}, where g = 0, 1, 2 for an additive SNP marker and g = 0, 1 for a recessive/dominant marker. For this example, we assume a dominant model for the rarer allele. We denote the sample size for each genotype *n*_{
g
}. We assume the distribution of trait values for different genotypes have similar shape, but the locations of the distributions are shifted by *d*_{
g
}. The hypothesis to detect association between a marker and the trait is then defined as *H*_{0}:*d* = 0. The power function of the statistic *T* for d = *δ* at the significance level *α* is then given by *P*(*T*; *δ*, *α*) The method of Collings and Hamilton [6] to approximate *P*(*T*; *δ*, *α*) by a nonparametric bootstrap procedure is as follows:

1. For each genotype group *g*, a random sample of n={\displaystyle {\sum}_{g}{n}_{g}}, *n*_{2} <*n*_{0}, trait values is drawn with replacement. The sampled trait values are denoted *X*_{
g
}^{b}= (*x*_{1g}^{b},...,*x*_{
ng
}^{b}). A simulation sample of trait values, *Y*_{
g
}^{b}, is then obtained by adding *X*_{
g
}^{b}to (**0**, *δ*), where **0** is a row vector of *n*_{0} elements each of which is 0 and *δ*is row vector of *n*_{1} + *n*_{2} elements, each of which is *δ*. The corresponding genotype groups are set to be *G*^{b}= (**0**, **1**).

2. Different statistics are calculated on the simulated sample values *Y*_{
g
}^{b}and *G*^{b}, and the corresponding *p*-values (*p*_{
g
}^{b}) are recorded.

3. Steps 1 and 2 are repeated *B* times. The estimated power function of {\widehat{P}}_{g}(T;\delta ,\alpha ) is given by \frac{{\displaystyle \sum {I}_{\{{p}_{g}^{b}<\alpha \}}}}{B}.

4. Finally, we estimate the power of the different statistics using the weighted average estimates of the different genotype groups, given by \frac{{\displaystyle {\sum}_{g}({n}_{g}{\widehat{P}}_{g})}}{n}.

We compared non-adaptive methods and this adaptive method using the simulated data of Problem 3 in GAW15, which has 100 replicates. For an adaptive method, we considered different proportions of samples at the exploratory stage (*π*_{1}), different methods of combining tests (Fisher's and the Inverse normal methods) and two statistics (Hotelling's *T*^{2} [1] and the nonparametric Wilcoxon statistic [8]). These statistics were calculated using the R package (version 2.4.1). In each replicate, we sampled 200 independent individuals to map the IgM QTL. To examine the validity of the various tests, we randomly selected from each of the 100 replicates 10 SNPs that are not associated with IgM and therefore from these results the type I error rate is given by

\frac{\#\{p-value<\alpha \}}{1000}.