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Paper text:
the ihw package is available from bioconductor at http://www.bioconductor.org/packages/ihw . it comes with detailed documentation and a vignette that showcases the application of ihw to a real dataset .
the vignette also provides guidance on the choice of informative covariates and suggests diagnostic plots , so that users can determine if their covariate satisfies the required conditions .
executable documents ( rmarkdown ) reproducing all analyses shown here can be downloaded at http://bioconductor.org/packages/ihwpaper .
the hypothesis tests are divided into g different groups based on the covariate , typically of about equal size .
the following optimization problem is solved : find the weight vector w = ( w1 
 ,  , wg ) that maximizes the number of rejections of the weighted bh method at level . this method , naive ihw , is modified by the following three extensions .
in statistical terms this corresponds to replacing the empirical cumulative distribution functions ( ecdf ) of the p - values with the grenander estimators ( least concave majorant of the ecdf ) .
the resulting problem is convex and can be efficiently solved even for large numbers of hypotheses .
, we apply convex ihw to the other k 1 folds and assign the learned weights to the remaining fold .
thus the weight assigned to a given hypothesis does not directly depend on its p - value , but only on its covariate . e3 .
the performance of the algorithm can be further improved by ensuring that the 
 weights learned with k 1 folds generalize to the 
 held - out fold .
therefore , we introduce a regularization parameter 
   0 , and the optimization is done over a 
 constrained subset of the weights . for an ordered covariate , we require that , 
 g=2gwgwg1 i. e. , weights of successive groups should not be 
 too different . for an unordered covariate , we use instead the constraint 
 g=1gwg1 i.e. , deviations from 1 are penalized . in the 
 limit case
 = 0 , all weights are the same , so 
 ihw with  = 0 is just the bh method .
choice of  is a model selection problem , so within each 
 split in e2 we apply a second nested layer of cross - validation .
e3 is optional ; 
 whether or not to apply it will depend on the data
. it will be most beneficial 
 if the number of hypotheses per group is relatively small .
a complete description of the algorithm , including an efficient computational implementation of the optimization task , is provided in supplementary note 2 .
the hypothesis tests are divided into g different groups based on the covariate , typically of about equal size .
the following optimization problem is solved : find the weight vector w = ( w1 
 ,  , wg ) that maximizes the number of rejections of the weighted bh method at level . this method , naive ihw , is modified by the following three extensions .
in statistical terms this corresponds to replacing the empirical cumulative distribution functions ( ecdf ) of the p - values with the grenander estimators ( least concave majorant of the ecdf ) .
the resulting problem is convex and can be efficiently solved even for large numbers of hypotheses .
, we apply convex ihw to the other k 1 folds and assign the learned weights to the remaining fold .
thus the weight assigned to a given hypothesis does not directly depend on its p - value , but only on its covariate . e3 .
the performance of the algorithm can be further improved by ensuring that the 
 weights learned with k 1 folds generalize to the 
 held - out fold .
therefore , we introduce a regularization parameter 
   0 , and the optimization is done over a 
 constrained subset of the weights . for an ordered covariate , we require that , 
 g=2gwgwg1 i. e. , weights of successive groups should not be 
 too different . for an unordered covariate , we use instead the constraint 
 g=1gwg1 i.e. , deviations from 1 are penalized . in the 
 limit case
 = 0 , all weights are the same , so 
 ihw with  = 0 is just the bh method .
choice of  is a model selection problem , so within each 
 split in e2 we apply a second nested layer of cross - validation .
e3 is optional ; 
 whether or not to apply it will depend on the data
. it will be most beneficial 
 if the number of hypotheses per group is relatively small .
a complete description of the algorithm , including an efficient computational implementation of the optimization task , is provided in supplementary note 2 .
hypothesis weighting improves the power of large - scale multiple testing . 
 we describe a method that uses covariates independent of the p - values under the null hypothesis , but informative of each test s power or prior probability of the null hypothesis . 
 independent hypothesis weighting ( ihw ) increases power while controlling the false discovery rate ( fdr ) . 
 ihw is a practical approach to discover associations in large datasets as encountered in genomics and high - throughput biology . 
 availability : www.bioconductor.org/packages/ihw