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406 Buried treasuresmeasurement technology and on different tissues. The findings lead to a newhypothesis about how HPV+/− tumors differentially deregulate the cell-cycleprocesses during tumorigenesis as well as to biomarkers for HPV−associatedcancers (Pyeon et al., 2011). Figure 36.1 shows a summary of gene-level differentialexpression scores between HPV+ and HPV− cancers (so-called logfold changes), for all genes in the genome (left), as well as for m = 99 genesfrom a cell-cycle regulatory pathway.A key statistical issue in this case was how to standardize a sample variancestatistic. The gene-level data were first reduced to the log-scale foldchange between HPV+ and HPV− cell types; these x g ,forgenesg, werethenconsidered fixed in subsequent calculations. For a known functional categoryc ⊆{1,...,G} of size m, thestatisticu(x, c) measured the sample varianceof the x g ’s within c. This statistic was standardized by imagining the distributionof u(x, C), for random sets C, consideredtobedrawnuniformlyfromamong all ( Gm)possible size-m subsets of the genome. Well forgetting aboutall the genomics, the statistical question concerned the distribution of thesample variance in without-replacement finite-population sampling; in particular,I needed an expected value and variance of u(x, C) underthissampling.Not being especially well versed in the findings of finite-population sampling,I approached these moment questions from first principles and with a novice’svigor, figuring that something simple was bound to emerge. I did not makemuch progress on the variance of u(x, C), but was delighted to discover a beautifulsolution in Tukey (1950, p. 517), which had been developed far from thecontext of genomics and which was not widely cited. Tukey’s buried treasureused so-called K functions, which are set-level statistics whose expected valueequals the same statistic computed on the whole population. SubsequentlyIlearnedthatearlierR.A.Fisherhadalsoderivedthisvariance;seealsoChoet al. (2005). In any case, I was glad to have gained some insight from Tukey’sgeneral framework.36.1.2 Bootstrapping and rank statisticsResearchers were actively probing the limits of bootstrap theory when I beganmy statistics career. A case of interest concerned generalized bootstrap means.From a real-valued random sample X 1 ,...,X n , one studied the conditionaldistribution of the randomized statistic¯X W n= 1 nn∑W n,i X i ,i=1conditional on the data X i , and where the random weights W n,i were generatedby the statistician to enable the conditional distribution of ¯XW n toapproximate the marginal sampling distribution of ¯X n .Efron’sbootstrapcorrespondsto weights having a certain multinomial distribution, but indicationswere that useful approximations were available for beyond the multinomial.