Detailed Description

OneSidedFrequentistUpperLimitWithBands

This is a standard demo that can be used with any ROOT file prepared in the standard way. You specify:

name for input ROOT file
name of workspace inside ROOT file that holds model and data
name of ModelConfig that specifies details for calculator tools
name of dataset

With default parameters the macro will attempt to run the standard hist2workspace example and read the ROOT file that it produces.

The first ~100 lines define a new test statistic, then the main macro starts. You may want to control:

double confidenceLevel=0.95;
int nPointsToScan = 12;
int nToyMC = 150;

This uses a modified version of the profile likelihood ratio as a test statistic for upper limits (eg. test stat = 0 if muhat>mu).

Based on the observed data, one defines a set of parameter points to be tested based on the value of the parameter of interest and the conditional MLE (eg. profiled) values of the nuisance parameters.

At each parameter point, pseudo-experiments are generated using this fixed reference model and then the test statistic is evaluated. Note, the nuisance parameters are floating in the fits. For each point, the threshold that defines the 95% acceptance region is found. This forms a "Confidence Belt".

After constructing the confidence belt, one can find the confidence interval for any particular dataset by finding the intersection of the observed test statistic and the confidence belt. First this is done on the observed data to get an observed 1-sided upper limt.

Finally, there expected limit and bands (from background-only) are formed by generating background-only data and finding the upper limit. This is done by hand for now, will later be part of the RooStats tools.

On a technical note, this technique is NOT the Feldman-Cousins technique, because that is a 2-sided interval BY DEFINITION. However, like the Feldman-Cousins technique this is a Neyman-Construction. For technical reasons the easiest way to implement this right now is to use the FeldmanCousins tool and then change the test statistic that it is using.

Building the confidence belt can be computationally expensive. Once it is built, one could save it to a file and use it in a separate step.

We can use PROOF to speed things along in parallel, however, the test statistic has to be installed on the workers so either turn off PROOF or include the modified test statistic in your $ROOTSYS/roofit/roostats/inc directory, add the additional line to the LinkDef.h file, and recompile root.

Note, if you have a boundary on the parameter of interest (eg. cross-section) the threshold on the one-sided test statistic starts off very small because we are only including downward fluctuations. You can see the threshold in these printouts:

[#0] PROGRESS:Generation -- generated toys: 500 / 999
NeymanConstruction: Prog: 12/50 total MC = 39 this test stat = 0
 SigXsecOverSM=0.69 alpha_syst1=0.136515 alpha_syst3=0.425415 beta_syst2=1.08496 [-1e+30, 0.011215]  in interval = 1

this tells you the values of the parameters being used to generate the pseudo-experiments and the threshold in this case is 0.011215. One would expect for 95% that the threshold would be ~1.35 once the cross-section is far enough away from 0 that it is essentially unaffected by the boundary. As one reaches the last points in the scan, the theshold starts to get artificially high. This is because the range of the parameter in the fit is the same as the range in the scan. In the future, these should be independently controlled, but they are not now. As a result the ~50% of pseudo-experiments that have an upward fluctuation end up with muhat = muMax. Because of this, the upper range of the parameter should be well above the expected upper limit... but not too high or one will need a very large value of nPointsToScan to resolve the relevant region. This can be improved, but this is the first version of this script.

Important note: when the model includes external constraint terms, like a Gaussian constraint to a nuisance parameter centered around some nominal value there is a subtlety. The asymptotic results are all based on the assumption that all the measurements fluctuate... including the nominal values from auxiliary measurements. If these do not fluctuate, this corresponds to an "conditional ensemble". The result is that the distribution of the test statistic can become very non-chi^2. This results in thresholds that become very large. This can be seen in the following thought experiment. Say the model is $ Pois(N | s + b)G(b0|b,sigma) $ where $ G(b0|b,sigma) $ is the external constraint and b0 is 100. If N is also 100 then the profiled value of b given s is going to be some trade off between 100-s and b0. If sigma is $ \sqrt(N) $, then the profiled value of b is probably 100 - s/2 So for s=60 we are going to have a profiled value of b~70. Now when we generate pseudo-experiments for s=60, b=70 we will have N~130 and the average shat will be 30, not 60. In practice, this is only an issue for values of s that are very excluded. For values of s near the 95% limit this should not be a big effect. This can be avoided if the nominal values of the constraints also fluctuate, but that requires that those parameters are RooRealVars in the model. This version does not deal with this issue, but it will be addressed in a future version.

 
FeldmanCousins: ntoys per point = 499
FeldmanCousins: nEvents per toy will fluctuate about  expectation
will use global observables for unconditional ensemble
RooArgSet:: = (nominalLumi,nom_alpha_syst1,nom_alpha_syst2,nom_alpha_syst3,nom_gamma_stat_channel1_bin_0,nom_gamma_stat_channel1_bin_1)
 
=== Using the following for ModelConfig ===
Observables:             RooArgSet:: = (obs_x_channel1,channelCat)
Parameters of Interest:  RooArgSet:: = (SigXsecOverSM)
Nuisance Parameters:     RooArgSet:: = (alpha_syst2,alpha_syst3,gamma_stat_channel1_bin_0,gamma_stat_channel1_bin_1)
Global Observables:      RooArgSet:: = (nominalLumi,nom_alpha_syst1,nom_alpha_syst2,nom_alpha_syst3,nom_gamma_stat_channel1_bin_0,nom_gamma_stat_channel1_bin_1)
PDF:                     RooSimultaneous::simPdf[ indexCat=channelCat channel1=model_channel1 ] = 0.190787
 
FeldmanCousins: Model has nuisance parameters, will do profile construction
FeldmanCousins: # points to test = 12
lookup index = 0
NeymanConstruction: Prog: 1/12 total MC = 499 this test stat = 0
 SigXsecOverSM=0.125 alpha_syst2=0.619973 alpha_syst3=0.233361 gamma_stat_channel1_bin_0=1.03213 gamma_stat_channel1_bin_1=1.04741 [-1e+30, 0.329527]  in interval = 1
 
NeymanConstruction: Prog: 2/12 total MC = 499 this test stat = 0
 SigXsecOverSM=0.375 alpha_syst2=0.44772 alpha_syst3=0.177828 gamma_stat_channel1_bin_0=1.02318 gamma_stat_channel1_bin_1=1.03602 [-1e+30, 0.886709]  in interval = 1
 
NeymanConstruction: Prog: 3/12 total MC = 499 this test stat = 0
 SigXsecOverSM=0.625 alpha_syst2=0.286415 alpha_syst3=0.123093 gamma_stat_channel1_bin_0=1.01471 gamma_stat_channel1_bin_1=1.02485 [-1e+30, 1.23248]  in interval = 1
 
NeymanConstruction: Prog: 4/12 total MC = 499 this test stat = 0
 SigXsecOverSM=0.875 alpha_syst2=0.135426 alpha_syst3=0.0710021 gamma_stat_channel1_bin_0=1.00682 gamma_stat_channel1_bin_1=1.0134 [-1e+30, 1.51788]  in interval = 1
 
NeymanConstruction: Prog: 5/12 total MC = 499 this test stat = 0.000123945
 SigXsecOverSM=1.125 alpha_syst2=-0.0142596 alpha_syst3=0.0160367 gamma_stat_channel1_bin_0=0.999288 gamma_stat_channel1_bin_1=1.00312 [-1e+30, 1.54201]  in interval = 1
 
NeymanConstruction: Prog: 6/12 total MC = 499 this test stat = 0.0914825
 SigXsecOverSM=1.375 alpha_syst2=-0.157842 alpha_syst3=-0.0395143 gamma_stat_channel1_bin_0=0.992184 gamma_stat_channel1_bin_1=0.993117 [-1e+30, 1.76272]  in interval = 1
 
NeymanConstruction: Prog: 7/12 total MC = 499 this test stat = 0.348977
 SigXsecOverSM=1.625 alpha_syst2=-0.293106 alpha_syst3=-0.0887909 gamma_stat_channel1_bin_0=0.985748 gamma_stat_channel1_bin_1=0.98241 [-1e+30, 1.57284]  in interval = 1
 
NeymanConstruction: Prog: 8/12 total MC = 499 this test stat = 0.767852
 SigXsecOverSM=1.875 alpha_syst2=-0.422663 alpha_syst3=-0.140541 gamma_stat_channel1_bin_0=0.979593 gamma_stat_channel1_bin_1=0.972417 [-1e+30, 1.52915]  in interval = 1
 
NeymanConstruction: Prog: 9/12 total MC = 499 this test stat = 1.34349
 SigXsecOverSM=2.125 alpha_syst2=-0.544235 alpha_syst3=-0.191156 gamma_stat_channel1_bin_0=0.973827 gamma_stat_channel1_bin_1=0.96257 [-1e+30, 1.20669]  in interval = 0
 
NeymanConstruction: Prog: 10/12 total MC = 499 this test stat = 2.07144
 SigXsecOverSM=2.375 alpha_syst2=-0.657512 alpha_syst3=-0.240962 gamma_stat_channel1_bin_0=0.968396 gamma_stat_channel1_bin_1=0.952936 [-1e+30, 1.85229]  in interval = 0
 
NeymanConstruction: Prog: 11/12 total MC = 499 this test stat = 2.94737
 SigXsecOverSM=2.625 alpha_syst2=-0.763058 alpha_syst3=-0.290526 gamma_stat_channel1_bin_0=0.963223 gamma_stat_channel1_bin_1=0.943652 [-1e+30, 1.1503]  in interval = 0
 
NeymanConstruction: Prog: 12/12 total MC = 499 this test stat = 3.9668
 SigXsecOverSM=2.875 alpha_syst2=-0.861405 alpha_syst3=-0.338707 gamma_stat_channel1_bin_0=0.958362 gamma_stat_channel1_bin_1=0.934519 [-1e+30, 1.2709]  in interval = 0
 
[#1] INFO:Eval -- 8 points in interval
 
95% interval on SigXsecOverSM is : [0.125, 1.875] 
[#1] INFO:Minimization -- p.d.f. provides expected number of events, including extended term in likelihood.
[#1] INFO:Minimization --  Including the following constraint terms in minimization: (lumiConstraint,alpha_syst1Constraint,alpha_syst2Constraint,alpha_syst3Constraint,gamma_stat_channel1_bin_0_constraint,gamma_stat_channel1_bin_1_constraint)
[#1] INFO:Minimization -- The global observables are not defined , normalize constraints with respect to the parameters (Lumi,SigXsecOverSM,alpha_syst1,alpha_syst2,alpha_syst3,gamma_stat_channel1_bin_0,gamma_stat_channel1_bin_1)
[#1] INFO:Minimization -- RooProfileLL::evaluate(nll_simPdf_obsData_with_constr_Profile[SigXsecOverSM]) Creating instance of MINUIT
[#1] INFO:Fitting -- RooAddition::defaultErrorLevel(nll_simPdf_obsData_with_constr) Summation contains a RooNLLVar, using its error level
[#1] INFO:Minimization -- RooProfileLL::evaluate(nll_simPdf_obsData_with_constr_Profile[SigXsecOverSM]) determining minimum likelihood for current configurations w.r.t all observable
RooAbsTestStatistic::initSimMode: creating slave calculator #0 for state channel1 (2 dataset entries)
[#1] INFO:Fitting -- RooAbsTestStatistic::initSimMode: created 1 slave calculators.
[#1] INFO:Minimization -- RooProfileLL::evaluate(nll_simPdf_obsData_with_constr_Profile[SigXsecOverSM]) minimum found at (SigXsecOverSM=1.11589)
.
Will use these parameter points to generate pseudo data for bkg only
  1) 0x55cde64b6fa0 RooRealVar::               alpha_syst2 = 0.711146 +/- 0.91413  L(-5 - 5)  "alpha_syst2"
  2) 0x55cde64684a0 RooRealVar::               alpha_syst3 = 0.261455 +/- 0.929175  L(-5 - 5)  "alpha_syst3"
  3) 0x55cde6471ce0 RooRealVar:: gamma_stat_channel1_bin_0 = 1.03677 +/- 0.0462907  L(0 - 1.25)  "gamma_stat_channel1_bin_0"
  4) 0x55cde643f7f0 RooRealVar:: gamma_stat_channel1_bin_1 = 1.05319 +/- 0.0761269  L(0 - 1.5)  "gamma_stat_channel1_bin_1"
  5) 0x55cde6441840 RooRealVar::             SigXsecOverSM = 0 +/- 0  L(0 - 3) B(12)  "SigXsecOverSM"
-2 sigma  band 4.94066e-324
-1 sigma  band 0.105 [Power Constraint)]
median of band 0.855
+1 sigma  band 1.605
+2 sigma  band 2.085
 
observed 95% upper-limit 1.875
CLb strict [P(toy>obs|0)] for observed 95% upper-limit 0.966667
CLb inclusive [P(toy>=obs|0)] for observed 95% upper-limit 0.966667

 
#include "TFile.h"
#include "TROOT.h"
#include "TH1F.h"
#include "TCanvas.h"
#include "TSystem.h"
 
#include "RooWorkspace.h"
#include "RooSimultaneous.h"
#include "RooAbsData.h"
 
#include "RooStats/ModelConfig.h"
#include "RooStats/FeldmanCousins.h"
#include "RooStats/ToyMCSampler.h"
#include "RooStats/PointSetInterval.h"
#include "RooStats/ConfidenceBelt.h"
 
#include "RooStats/RooStatsUtils.h"
#include "RooStats/ProfileLikelihoodTestStat.h"
 
using namespace RooFit;
using namespace RooStats;
 
bool useProof = false; // flag to control whether to use Proof
int nworkers = 0;      // number of workers (default use all available cores)
 
// -------------------------------------------------------
// The actual macro
 
void OneSidedFrequentistUpperLimitWithBands(const char *infile = "", const char *workspaceName = "combined",
                                            const char *modelConfigName = "ModelConfig",
                                            const char *dataName = "obsData")
{
 
   double confidenceLevel = 0.95;
   int nPointsToScan = 12;
   int nToyMC = 150;
 
   // -------------------------------------------------------
   // First part is just to access a user-defined file
   // or create the standard example file if it doesn't exist
   const char *filename = "";
   if (!strcmp(infile, "")) {
      filename = "results/example_combined_GaussExample_model.root";
      bool fileExist = !gSystem->AccessPathName(filename); // note opposite return code
      // if file does not exists generate with histfactory
      if (!fileExist) {
#ifdef _WIN32
         cout << "HistFactory file cannot be generated on Windows - exit" << endl;
         return;
#endif
         // Normally this would be run on the command line
         cout << "will run standard hist2workspace example" << endl;
         gROOT->ProcessLine(".! prepareHistFactory .");
         gROOT->ProcessLine(".! hist2workspace config/example.xml");
         cout << "\n\n---------------------" << endl;
         cout << "Done creating example input" << endl;
         cout << "---------------------\n\n" << endl;
      }
 
   } else
      filename = infile;
 
   // Try to open the file
   TFile *file = TFile::Open(filename);
 
   // if input file was specified byt not found, quit
   if (!file) {
      cout << "StandardRooStatsDemoMacro: Input file " << filename << " is not found" << endl;
      return;
   }
 
   // -------------------------------------------------------
   // Now get the data and workspace
 
   // get the workspace out of the file
   RooWorkspace *w = (RooWorkspace *)file->Get(workspaceName);
   if (!w) {
      cout << "workspace not found" << endl;
      return;
   }
 
   // get the modelConfig out of the file
   ModelConfig *mc = (ModelConfig *)w->obj(modelConfigName);
 
   // get the modelConfig out of the file
   RooAbsData *data = w->data(dataName);
 
   // make sure ingredients are found
   if (!data || !mc) {
      w->Print();
      cout << "data or ModelConfig was not found" << endl;
      return;
   }
 
   // -------------------------------------------------------
   // Now get the POI for convenience
   // you may want to adjust the range of your POI
 
   RooRealVar *firstPOI = (RooRealVar *)mc->GetParametersOfInterest()->first();
   /*  firstPOI->setMin(0);*/
   /*  firstPOI->setMax(10);*/
 
   // --------------------------------------------
   // Create and use the FeldmanCousins tool
   // to find and plot the 95% confidence interval
   // on the parameter of interest as specified
   // in the model config
   // REMEMBER, we will change the test statistic
   // so this is NOT a Feldman-Cousins interval
   FeldmanCousins fc(*data, *mc);
   fc.SetConfidenceLevel(confidenceLevel);
   fc.AdditionalNToysFactor(
      0.5); // degrade/improve sampling that defines confidence belt: in this case makes the example faster
   /*  fc.UseAdaptiveSampling(true); // speed it up a bit, don't use for expected limits*/
   fc.SetNBins(nPointsToScan); // set how many points per parameter of interest to scan
   fc.CreateConfBelt(true);    // save the information in the belt for plotting
 
   // -------------------------------------------------------
   // Feldman-Cousins is a unified limit by definition
   // but the tool takes care of a few things for us like which values
   // of the nuisance parameters should be used to generate toys.
   // so let's just change the test statistic and realize this is
   // no longer "Feldman-Cousins" but is a fully frequentist Neyman-Construction.
   /*  ProfileLikelihoodTestStatModified onesided(*mc->GetPdf());*/
   /*  fc.GetTestStatSampler()->SetTestStatistic(&onesided);*/
   /* ((ToyMCSampler*) fc.GetTestStatSampler())->SetGenerateBinned(true); */
   ToyMCSampler *toymcsampler = (ToyMCSampler *)fc.GetTestStatSampler();
   ProfileLikelihoodTestStat *testStat = dynamic_cast<ProfileLikelihoodTestStat *>(toymcsampler->GetTestStatistic());
   testStat->SetOneSided(true);
 
   // Since this tool needs to throw toy MC the PDF needs to be
   // extended or the tool needs to know how many entries in a dataset
   // per pseudo experiment.
   // In the 'number counting form' where the entries in the dataset
   // are counts, and not values of discriminating variables, the
   // datasets typically only have one entry and the PDF is not
   // extended.
   if (!mc->GetPdf()->canBeExtended()) {
      if (data->numEntries() == 1)
         fc.FluctuateNumDataEntries(false);
      else
         cout << "Not sure what to do about this model" << endl;
   }
 
   // We can use PROOF to speed things along in parallel
   // However, the test statistic has to be installed on the workers
   // so either turn off PROOF or include the modified test statistic
   // in your `$ROOTSYS/roofit/roostats/inc` directory,
   // add the additional line to the LinkDef.h file,
   // and recompile root.
   if (useProof) {
      ProofConfig pc(*w, nworkers, "", false);
      toymcsampler->SetProofConfig(&pc); // enable proof
   }
 
   if (mc->GetGlobalObservables()) {
      cout << "will use global observables for unconditional ensemble" << endl;
      mc->GetGlobalObservables()->Print();
      toymcsampler->SetGlobalObservables(*mc->GetGlobalObservables());
   }
 
   // Now get the interval
   PointSetInterval *interval = fc.GetInterval();
   ConfidenceBelt *belt = fc.GetConfidenceBelt();
 
   // print out the interval on the first Parameter of Interest
   cout << "\n95% interval on " << firstPOI->GetName() << " is : [" << interval->LowerLimit(*firstPOI) << ", "
        << interval->UpperLimit(*firstPOI) << "] " << endl;
 
   // get observed UL and value of test statistic evaluated there
   RooArgSet tmpPOI(*firstPOI);
   double observedUL = interval->UpperLimit(*firstPOI);
   firstPOI->setVal(observedUL);
   double obsTSatObsUL = fc.GetTestStatSampler()->EvaluateTestStatistic(*data, tmpPOI);
 
   // Ask the calculator which points were scanned
   RooDataSet *parameterScan = (RooDataSet *)fc.GetPointsToScan();
   RooArgSet *tmpPoint;
 
   // make a histogram of parameter vs. threshold
   TH1F *histOfThresholds =
      new TH1F("histOfThresholds", "", parameterScan->numEntries(), firstPOI->getMin(), firstPOI->getMax());
   histOfThresholds->GetXaxis()->SetTitle(firstPOI->GetName());
   histOfThresholds->GetYaxis()->SetTitle("Threshold");
 
   // loop through the points that were tested and ask confidence belt
   // what the upper/lower thresholds were.
   // For FeldmanCousins, the lower cut off is always 0
   for (Int_t i = 0; i < parameterScan->numEntries(); ++i) {
      tmpPoint = (RooArgSet *)parameterScan->get(i)->clone("temp");
      // cout <<"get threshold"<<endl;
      double arMax = belt->GetAcceptanceRegionMax(*tmpPoint);
      double poiVal = tmpPoint->getRealValue(firstPOI->GetName());
      histOfThresholds->Fill(poiVal, arMax);
   }
   TCanvas *c1 = new TCanvas();
   c1->Divide(2);
   c1->cd(1);
   histOfThresholds->SetMinimum(0);
   histOfThresholds->Draw();
   c1->cd(2);
 
   // -------------------------------------------------------
   // Now we generate the expected bands and power-constraint
 
   // First: find parameter point for mu=0, with conditional MLEs for nuisance parameters
   std::unique_ptr<RooAbsReal> nll{mc->GetPdf()->createNLL(*data)};
   std::unique_ptr<RooAbsReal> profile{nll->createProfile(*mc->GetParametersOfInterest())};
   firstPOI->setVal(0.);
   profile->getVal(); // this will do fit and set nuisance parameters to profiled values
   RooArgSet *poiAndNuisance = new RooArgSet();
   if (mc->GetNuisanceParameters())
      poiAndNuisance->add(*mc->GetNuisanceParameters());
   poiAndNuisance->add(*mc->GetParametersOfInterest());
   w->saveSnapshot("paramsToGenerateData", *poiAndNuisance);
   RooArgSet *paramsToGenerateData = (RooArgSet *)poiAndNuisance->snapshot();
   cout << "\nWill use these parameter points to generate pseudo data for bkg only" << endl;
   paramsToGenerateData->Print("v");
 
   RooArgSet unconditionalObs;
   unconditionalObs.add(*mc->GetObservables());
   unconditionalObs.add(*mc->GetGlobalObservables()); // comment this out for the original conditional ensemble
 
   double CLb = 0;
   double CLbinclusive = 0;
 
   // Now we generate background only and find distribution of upper limits
   TH1F *histOfUL = new TH1F("histOfUL", "", 100, 0, firstPOI->getMax());
   histOfUL->GetXaxis()->SetTitle("Upper Limit (background only)");
   histOfUL->GetYaxis()->SetTitle("Entries");
   for (int imc = 0; imc < nToyMC; ++imc) {
 
      // set parameters back to values for generating pseudo data
      //    cout << "\n get current nuis, set vals, print again" << endl;
      w->loadSnapshot("paramsToGenerateData");
      //    poiAndNuisance->Print("v");
 
      RooDataSet *toyData = 0;
      // now generate a toy dataset
      if (!mc->GetPdf()->canBeExtended()) {
         if (data->numEntries() == 1)
            toyData = mc->GetPdf()->generate(*mc->GetObservables(), 1);
         else
            cout << "Not sure what to do about this model" << endl;
      } else {
         //      cout << "generating extended dataset"<<endl;
         toyData = mc->GetPdf()->generate(*mc->GetObservables(), Extended());
      }
 
      // generate global observables
      // need to be careful for simpdf
      //    RooDataSet* globalData = mc->GetPdf()->generate(*mc->GetGlobalObservables(),1);
 
      RooSimultaneous *simPdf = dynamic_cast<RooSimultaneous *>(mc->GetPdf());
      if (!simPdf) {
         RooDataSet *one = mc->GetPdf()->generate(*mc->GetGlobalObservables(), 1);
         const RooArgSet *values = one->get();
         std::unique_ptr<RooArgSet> allVars{mc->GetPdf()->getVariables()};
         allVars->assign(*values);
         delete values;
         delete one;
      } else {
 
         // try fix for sim pdf
         for (auto const& tt : simPdf->indexCat()) {
            auto const& catName = tt.first;
 
            // Get pdf associated with state from simpdf
            RooAbsPdf *pdftmp = simPdf->getPdf(catName.c_str());
 
            // Generate only global variables defined by the pdf associated with this state
            std::unique_ptr<RooArgSet> globtmp{pdftmp->getObservables(*mc->GetGlobalObservables())};
            std::unique_ptr<RooDataSet> tmp{pdftmp->generate(*globtmp, 1)};
 
            // Transfer values to output placeholder
            globtmp->assign(*tmp->get(0));
         }
      }
 
      //    globalData->Print("v");
      //    unconditionalObs = *globalData->get();
      //    mc->GetGlobalObservables()->Print("v");
      //    delete globalData;
      //    cout << "toy data = " << endl;
      //    toyData->get()->Print("v");
 
      // get test stat at observed UL in observed data
      firstPOI->setVal(observedUL);
      double toyTSatObsUL = fc.GetTestStatSampler()->EvaluateTestStatistic(*toyData, tmpPOI);
      //    toyData->get()->Print("v");
      //    cout <<"obsTSatObsUL " <<obsTSatObsUL << "toyTS " << toyTSatObsUL << endl;
      if (obsTSatObsUL < toyTSatObsUL) // not sure about <= part yet
         CLb += (1.) / nToyMC;
      if (obsTSatObsUL <= toyTSatObsUL) // not sure about <= part yet
         CLbinclusive += (1.) / nToyMC;
 
      // loop over points in belt to find upper limit for this toy data
      double thisUL = 0;
      for (Int_t i = 0; i < parameterScan->numEntries(); ++i) {
         tmpPoint = (RooArgSet *)parameterScan->get(i)->clone("temp");
         double arMax = belt->GetAcceptanceRegionMax(*tmpPoint);
         firstPOI->setVal(tmpPoint->getRealValue(firstPOI->GetName()));
         //   double thisTS = profile->getVal();
         double thisTS = fc.GetTestStatSampler()->EvaluateTestStatistic(*toyData, tmpPOI);
 
         //   cout << "poi = " << firstPOI->getVal()
         // << " max is " << arMax << " this profile = " << thisTS << endl;
         //      cout << "thisTS = " << thisTS<<endl;
         if (thisTS <= arMax) {
            thisUL = firstPOI->getVal();
         } else {
            break;
         }
      }
 
      /*
      // loop over points in belt to find upper limit for this toy data
      double thisUL = 0;
      for(Int_t i=0; i<histOfThresholds->GetNbinsX(); ++i){
         tmpPoint = (RooArgSet*) parameterScan->get(i)->clone("temp");
         cout <<"----------------  "<<i<<endl;
         tmpPoint->Print("v");
         cout << "from hist " << histOfThresholds->GetBinCenter(i+1) <<endl;
         double arMax = histOfThresholds->GetBinContent(i+1);
         // cout << " threhold from Hist = aMax " << arMax<<endl;
         // double arMax2 = belt->GetAcceptanceRegionMax(*tmpPoint);
         // cout << "from scan arMax2 = "<< arMax2 << endl; // not the same due to TH1F not TH1D
         // cout << "scan - hist" << arMax2-arMax << endl;
         firstPOI->setVal( histOfThresholds->GetBinCenter(i+1));
         //   double thisTS = profile->getVal();
         double thisTS = fc.GetTestStatSampler()->EvaluateTestStatistic(*toyData,tmpPOI);
 
         //   cout << "poi = " << firstPOI->getVal()
         // << " max is " << arMax << " this profile = " << thisTS << endl;
         //      cout << "thisTS = " << thisTS<<endl;
 
         // NOTE: need to add a small epsilon term for single precision vs. double precision
         if(thisTS<=arMax + 1e-7){
            thisUL = firstPOI->getVal();
         } else{
            break;
         }
      }
      */
 
      histOfUL->Fill(thisUL);
 
      // for few events, data is often the same, and UL is often the same
      //    cout << "thisUL = " << thisUL<<endl;
 
      delete toyData;
   }
   histOfUL->Draw();
   c1->SaveAs("one-sided_upper_limit_output.pdf");
 
   // if you want to see a plot of the sampling distribution for a particular scan point:
   /*
   SamplingDistPlot sampPlot;
   int indexInScan = 0;
   tmpPoint = (RooArgSet*) parameterScan->get(indexInScan)->clone("temp");
   firstPOI->setVal( tmpPoint->getRealValue(firstPOI->GetName()) );
   toymcsampler->SetParametersForTestStat(tmpPOI);
   SamplingDistribution* samp = toymcsampler->GetSamplingDistribution(*tmpPoint);
   sampPlot.AddSamplingDistribution(samp);
   sampPlot.Draw();
      */
 
   // Now find bands and power constraint
   Double_t *bins = histOfUL->GetIntegral();
   TH1F *cumulative = (TH1F *)histOfUL->Clone("cumulative");
   cumulative->SetContent(bins);
   double band2sigDown, band1sigDown, bandMedian, band1sigUp, band2sigUp;
   for (int i = 1; i <= cumulative->GetNbinsX(); ++i) {
      if (bins[i] < RooStats::SignificanceToPValue(2))
         band2sigDown = cumulative->GetBinCenter(i);
      if (bins[i] < RooStats::SignificanceToPValue(1))
         band1sigDown = cumulative->GetBinCenter(i);
      if (bins[i] < 0.5)
         bandMedian = cumulative->GetBinCenter(i);
      if (bins[i] < RooStats::SignificanceToPValue(-1))
         band1sigUp = cumulative->GetBinCenter(i);
      if (bins[i] < RooStats::SignificanceToPValue(-2))
         band2sigUp = cumulative->GetBinCenter(i);
   }
   cout << "-2 sigma  band " << band2sigDown << endl;
   cout << "-1 sigma  band " << band1sigDown << " [Power Constraint)]" << endl;
   cout << "median of band " << bandMedian << endl;
   cout << "+1 sigma  band " << band1sigUp << endl;
   cout << "+2 sigma  band " << band2sigUp << endl;
 
   // print out the interval on the first Parameter of Interest
   cout << "\nobserved 95% upper-limit " << interval->UpperLimit(*firstPOI) << endl;
   cout << "CLb strict [P(toy>obs|0)] for observed 95% upper-limit " << CLb << endl;
   cout << "CLb inclusive [P(toy>=obs|0)] for observed 95% upper-limit " << CLbinclusive << endl;
}

Authors: Kyle Cranmer, Haichen Wang, Daniel Whiteson

Definition in file OneSidedFrequentistUpperLimitWithBands.C.