Common Statistical Methods
In this section, the most commonly used statistical methods from combine will be covered including specific instructions on how to obtain limits, significances and likelihood scans. For all of these methods, the assumed parameters of interest (POI) is the overall signal strength r (i.e the default PhysicsModel). In general however, the first POI in the list of POIs (as defined by the PhysicsModel) will be taken instead of r which may or may not make sense for a given method ... use your judgment!
This section will assume that you are using the default model unless otherwise specified.
Asymptotic Frequentist Limits
The AsymptoticLimits
method allows to compute quickly an estimate of the observed and expected limits, which is fairly accurate when the event yields are not too small and the systematic uncertainties don't play a major role in the result.
The limit calculation relies on an asymptotic approximation of the distributions of the LHC teststatistic, which is based on a profile likelihood ratio, under signal and background hypotheses to compute two pvalues p_{\mu}, p_{b} and therefore CL_s=p_{\mu}/(1p_{b}) (see the (see the FAQ section for a description of these)  i.e it is the asymptotic approximation of computing limits with frequentist toys using the LHC teststatistic for limits:
 The test statistic is defined using the ratio of likelihoods q_{r} = 2\ln[\mathcal{L}(\mathrm{data}r,\hat{\theta}_{r})/\mathcal{L}(\mathrm{data}r=\hat{r},\hat{\theta})] , in which the nuisance parameters are profiled separately for r=\hat{r} and r. The value of q_{r} set to 0 when \hat{r}>r giving a one sided limit. Furthermore, the constraint r>0 is enforced in the fit. This means that if the unconstrained value of \hat{r} would be negative, the test statistic q_{r} is evaluated as 2\ln[\mathcal{L}(\mathrm{data}r,\hat{\theta}_{r})/\mathcal{L}(\mathrm{data}r=0,\hat{\theta}_{0})]
This method is so commonly used that it is the default method (i.e not specifying M
will run AsymptoticLimits
)
A realistic example of datacard for a counting experiment can be found in the HiggsCombination package: data/tutorials/counting/realisticcountingexperiment.txt
The method can be run using
combine M AsymptoticLimits realisticcountingexperiment.txt
The program will print out the limit on the signal strength r (number of signal events / number of expected signal events) e .g. Observed Limit: r < 1.6297 @ 95% CL
, the median expected limit Expected 50.0%: r < 2.3111
and edges of the 68% and 95% ranges for the expected limits.
<<< Combine >>>
>>> including systematics
>>> method used to compute upper limit is AsymptoticLimits
[...]
 AsymptoticLimits ( CLs ) 
Observed Limit: r < 1.6281
Expected 2.5%: r < 0.9640
Expected 16.0%: r < 1.4329
Expected 50.0%: r < 2.3281
Expected 84.0%: r < 3.9800
Expected 97.5%: r < 6.6194
Done in 0.01 min (cpu), 0.01 min (real)
By default, the limits are calculated using the CL_{s} prescription, as noted in the output, which takes the ratio of pvalues under the signal plus background and background only hypothesis. This can be altered to using the strict pvalue by using the option rule CLsplusb
(note that CLsplusb
is the jargon for calculating the pvalue p_{\mu}). You can also change the confidence level (default is 95%) to 90% using the option cl 0.9
or any other confidence level. You can find the full list of options for AsymptoticLimits
using help M AsymptoticLimits
.
Warning
You may find that combine issues a warning that the best fit for the backgroundonly Asimov dataset returns a nonzero value for the signal strength for example;
WARNING: Best fit of asimov dataset is at r = 0.220944 (0.011047 times
rMax), while it should be at zero
If this happens, you should check to make sure that there are no issues with the datacard or the Asimov generation used for your setup. For details on debugging it is recommended that you follow the simple checks used by the HIG PAG here.
The program will also create a rootfile higgsCombineTest.AsymptoticLimits.mH120.root
containing a root tree limit
that contains the limit values and other bookeeping information. The important columns are limit
(the limit value) and quantileExpected
(1 for observed limit, 0.5 for median expected limit, 0.16/0.84 for the edges of the 65% interval band of expected limits, 0.025/0.975 for 95%).
$ root l higgsCombineTest.AsymptoticLimits.mH120.root
root [0] limit>Scan("*")
************************************************************************************************************************************
* Row * limit * limitErr * mh * syst * iToy * iSeed * iChannel * t_cpu * t_real * quantileE *
************************************************************************************************************************************
* 0 * 0.9639892 * 0 * 120 * 1 * 0 * 123456 * 0 * 0 * 0 * 0.0250000 *
* 1 * 1.4329109 * 0 * 120 * 1 * 0 * 123456 * 0 * 0 * 0 * 0.1599999 *
* 2 * 2.328125 * 0 * 120 * 1 * 0 * 123456 * 0 * 0 * 0 * 0.5 *
* 3 * 3.9799661 * 0 * 120 * 1 * 0 * 123456 * 0 * 0 * 0 * 0.8399999 *
* 4 * 6.6194028 * 0 * 120 * 1 * 0 * 123456 * 0 * 0 * 0 * 0.9750000 *
* 5 * 1.6281188 * 0.0050568 * 120 * 1 * 0 * 123456 * 0 * 0.0035000 * 0.0055123 * 1 *
************************************************************************************************************************************
Blind limits
The AsymptoticLimits
calculation follows the frequentist paradigm for calculating expected limits. This means that the routine will first fit the observed data, conditionally for a fixed value of r and set the nuisance parameters to the values obtained in the fit for generating the Asimov data, i.e it calculates the postfit or aposteriori expected limit. In order to use the prefit nuisance parameters (to calculate an apriori limit), you must add the option noFitAsimov
or bypassFrequentistFit
.
For blinding the results completely (i.e not using the data) you can include the option run blind
.
Warning
You should never use t 1
to get blind limits!
Splitting points
In case your model is particularly complex, you can perform the asymptotic calculation by determining the value of CL_{s} for a set grid of points (in r
) and merging the results. This is done by using the option singlePoint X
for multiple values of X, hadding the output files and reading them back in,
combine M AsymptoticLimits realisticcountingexperiment.txt singlePoint 0.1 n 0.1
combine M AsymptoticLimits realisticcountingexperiment.txt singlePoint 0.2 n 0.2
combine M AsymptoticLimits realisticcountingexperiment.txt singlePoint 0.3 n 0.3
...
hadd limits.root higgsCombine*.AsymptoticLimits.*
combine M AsymptoticLimits realisticcountingexperiment.txt getLimitFromGrid limits.root
Asymptotic Significances
The significance of a result is calculated using a ratio of profiled likelihoods, one in which the signal strength is set to 0 and the other in which it is free to float, i.e the quantity is 2\ln[\mathcal{L}(\textrm{data}r=0,\hat{\theta}_{0})/\mathcal{L}(\textrm{data}r=\hat{r},\hat{\theta})], in which the nuisance parameters are profiled separately for r=\hat{r} and r=0.
The distribution of this teststatistic can be determined using Wilke's theorem provided the number of events is large enough (i.e in the Asymptotic limit). The significance (or pvalue) can therefore be calculated very quickly and uses the Significance
method.
It is also possible to calculate the ratio of likelihoods between the freely floating signal strength to that of a fixed signal strength other than 0, by specifying it with the option signalForSignificance=X
Info
This calculation assumes that the signal strength can only be positive (i.e we are not interested in negative signal strengths). This can be altered by including the option uncapped
Compute the observed significance
The observed significance is calculated using the Significance
method, as
combine M Significance datacard.txt
The printed output will report the significance and the pvalue, for example, when using the realisticcountingexperiment.txt datacard, you will see
<<< Combine >>>
>>> including systematics
>>> method used is Significance
[...]
 Significance 
Significance: 0
(pvalue = 0.5)
Done in 0.00 min (cpu), 0.01 min (real)
which is not surprising since 0 events were observed in that datacard.
The output root file will contain the significance value in the branch limit. To store the pvalue instead, include the option pval
. These can be converted between one another using the RooFit functions RooFit::PValueToSignificance
and RooFit::SignificanceToPValue
.
You may find it useful to resort to a bruteforce fitting algorithm when calculating the significance which scans the nll (repeating fits until a tolerance is reached), bypassing MINOS, which can be activated with the option bruteForce
. This can be tuned using the options setBruteForceAlgo
, setBruteForceTypeAndAlgo
and setBruteForceTolerance
.
Computing the expected significance
The expected significance can be computed from an Asimov dataset of signal+background. There are two options for this
 aposteriori expected: will depend on the observed dataset.
 apriori expected (the default behavior): does not depend on the observed dataset, and so is a good metric for optimizing an analysis when still blinded.
The apriori expected significance from the Asimov dataset is calculated as
combine M Significance datacard.txt t 1 expectSignal=1
In order to produced the aposteriori expected significance, just generate a postfit Asimov (i.e add the option toysFreq
in the command above).
The output format is the same as for observed signifiances: the variable limit in the tree will be filled with the significance (or with the pvalue if you put also the option pvalue
)
Bayesian Limits and Credible regions
Bayesian calculation of limits requires the user to assume a particular prior distribution for the parameter of interest (default r). You can specify the prior using the prior
option, the default is a flat pior in r.
Since the Bayesian methods are much less frequently used, the tool will not build the default prior. For running the two methods below, you should include the option noDefaultPrior=0
.
Computing the observed bayesian limit (for simple models)
The BayesianSimple
method computes a Bayesian limit performing classical numerical integration; very fast and accurate but only works for simple models (a few channels and nuisance parameters).
combine M BayesianSimple simplecountingexperiment.txt noDefaultPrior=0
[...]
 BayesianSimple 
Limit: r < 0.672292 @ 95% CL
Done in 0.04 min (cpu), 0.05 min (real)
The output tree will contain a single entry corresponding to the observed 95% upper limit. The confidence level can be modified to 100*X% using cl X
.
Computing the observed bayesian limit (for arbitrary models)
The MarkovChainMC
method computes a Bayesian limit performing a montecarlo integration. From the statistics point of view it is identical to the BayesianSimple
method, only the technical implementation is different. The method is slower, but can also handle complex models. For this method, you can increase the accuracy of the result by increasing the number of markov chains at the expense of a longer running time (option tries
, default is 10). Let's use the realistic counting experiment datacard to test the method
To use the MarkovChainMC method, users need to specify this method in the command line, together with the options they want to use. For instance, to set the number of times the algorithm will run with different random seeds, use option tries
:
combine M MarkovChainMC realisticcountingexperiment.txt tries 100 noDefaultPrior=0
[...]
 MarkovChainMC 
Limit: r < 2.20438 +/ 0.0144695 @ 95% CL (100 tries)
Average chain acceptance: 0.078118
Done in 0.14 min (cpu), 0.15 min (real)
Again, the resulting limit tree will contain the result. You can also save the chains using the option saveChain
which will then also be included in the output file.
Exclusion regions can be made from the posterior once an ordering principle is defined to decide how to grow the contour (there's infinite possible regions that contain 68% of the posterior pdf...)
Below is a simple example script which can be used to plot the posterior distribution from these chains and calculate the smallest such region. Note that in this example we are ignoring the burnin (but you can add it by just editing for i in range(mychain.numEntries()):
to for i in range(200,mychain.numEntries()):
eg for a burnin of 200.
import ROOT
rmin = 0
rmax = 30
nbins = 100
CL = 0.95
chains = "higgsCombineTest.MarkovChainMC.blahblahblah.root"
def findSmallestInterval(hist,CL):
bins = hist.GetNbinsX()
best_i = 1
best_j = 1
bd = bins+1
val = 0;
for i in range(1,bins+1):
integral = hist.GetBinContent(i)
for j in range(i+1,bins+2):
integral += hist.GetBinContent(j)
if integral > CL :
val = integral
break
if integral > CL and ji < bd :
bd = ji
best_j = j+1
best_i = i
val = integral
return hist.GetBinLowEdge(best_i), hist.GetBinLowEdge(best_j), val
fi_MCMC = ROOT.TFile.Open(chains)
# Sum up all of the chains (or we could take the average limit)
mychain=0
for k in fi_MCMC.Get("toys").GetListOfKeys():
obj = k.ReadObj
if mychain ==0:
mychain = k.ReadObj().GetAsDataSet()
else :
mychain.append(k.ReadObj().GetAsDataSet())
hist = ROOT.TH1F("h_post",";r;posterior probability",nbins,rmin,rmax)
for i in range(mychain.numEntries()):
#for i in range(200,mychain.numEntries()): burnin of 200
mychain.get(i)
hist.Fill(mychain.get(i).getRealValue("r"), mychain.weight())
hist.Scale(1./hist.Integral())
hist.SetLineColor(1)
vl,vu,trueCL = findSmallestInterval(hist,CL)
histCL = hist.Clone()
for b in range(nbins):
if histCL.GetBinLowEdge(b+1) < vl or histCL.GetBinLowEdge(b+2)>vu: histCL.SetBinContent(b+1,0)
c6a = ROOT.TCanvas()
histCL.SetFillColor(ROOT.kAzure3)
histCL.SetFillStyle(1001)
hist.Draw()
histCL.Draw("histFsame")
hist.Draw("histsame")
ll = ROOT.TLine(vl,0,vl,2*hist.GetBinContent(hist.FindBin(vl))); ll.SetLineColor(2); ll.SetLineWidth(2)
lu = ROOT.TLine(vu,0,vu,2*hist.GetBinContent(hist.FindBin(vu))); lu.SetLineColor(2); lu.SetLineWidth(2)
ll.Draw()
lu.Draw()
print " %g %% (%g %%) interval (target) = %g < r < %g "%(trueCL,CL,vl,vu)
Running the script on the output file produced for the same datacard (including the saveChain
option) will produce the following output
0.950975 % (0.95 %) interval (target) = 0 < r < 2.2
along with a plot of the posterior shown below. This is the same as the output from combine but the script can also be used to find lower limits (for example) or credible intervals.
An example to make contours when ordering by probability density is in bayesContours.cxx, but the implementation is very simplistic, with no clever handling of bin sizes nor any smoothing of statistical fluctuations.
The MarkovChainMC
algorithm has many configurable parameters, and you're encouraged to experiment with those because the default configuration might not be the best for you (or might not even work for you at all)
Iterations, burnin, tries
Three parameters control how the MCMC integration is performed:
 the number of tries (option
tries
): the algorithm will run multiple times with different ransom seeds and report as result the truncated mean and rms of the different results. The default value is 10, which should be ok for a quick computation, but for something more accurate you might want to increase this number even up to ~200.  the number of iterations (option
i
) determines how many points are proposed to fill a single Markov Chain. The default value is 10k, and a plausible range is between 5k (for quick checks) and 2030k for lengthy calculations. Usually beyond 30k you get a better tradeoff in time vs accuracy by increasing the number of chains (optiontries
)  the number of burnin steps (option
b
) is the number of points that are removed from the beginning of the chain before using it to compute the limit. The default is 200. If your chain is very long, you might want to try increase this a bit (e.g. to some hundreds). Instead going below 50 is probably dangerous.
Proposals
The option proposal
controls the way new points are proposed to fill in the MC chain.
 uniform: pick points at random. This works well if you have very few nuisance parameters (or none at all), but normally fails if you have many.
 gaus: Use a product of independent gaussians one for each nuisance parameter; the sigma of the gaussian for each variable is 1/5 of the range of the variable (this can be controlled using the parameter
propHelperWidthRangeDivisor
). This proposal appears to work well for a reasonable number of nuisances (up to ~15), provided that the range of the nuisance parameters is reasonable, like ±5σ. It does not work without systematics.  ortho (default): This proposalis similar to the multigaussian proposal but at every step only a single coordinate of the point is varied, so that the acceptance of the chain is high even for a large number of nuisances (i.e. more than 20).
 fit: Run a fit and use the uncertainty matrix from HESSE to construct a proposal (or the one from MINOS if the option
runMinos
is specified). This sometimes work fine, but sometimes gives biased results, so we don't recommend it in general.
If you believe there's something going wrong, e.g. if your chain remains stuck after accepting only a few events, the option debugProposal
can be used to have a printout of the first N proposed points to see what's going on (e.g. if you have some region of the phase space with probability zero, the gaus and fit proposal can get stuck there forever)
Computing the expected bayesian limit
The expected limit is computed by generating many toy mc observations and compute the limit for each of them. This can be done passing the option t
. E.g. to run 100 toys with the BayesianSimple
method, just do
combine M BayesianSimple datacard.txt t 100 noDefaultPrior=0
The program will print out the mean and median limit, and the 68% and 95% quantiles of the distributions of the limits. This time, the output root tree will contain one entry per toy.
For more heavy methods (eg the MarkovChainMC
) you'll probably want to split this in multiple jobs. To do this, just run combine
multiple times specifying a smaller number of toys (can be as low as 1
) each time using a different seed to initialize the random number generator (option s
if you set it to 1, the starting seed will be initialized randomly at the beginning of the job), then merge the resulting trees with hadd
and look at the distribution in the merged file.
Multidimensional bayesian credible regions
The MarkovChainMC
method allows the user to produce the posterior pdf as a function of (in principle) any number of parameter of interest. In order to do so, you first need to create a workspace with more than one parameter, as explained in the physics models section.
For example, lets use the toy datacard test/multiDim/toyhgg125.txt (counting experiment which vaguely resembles the H→γγ analysis at 125 GeV) and convert the datacard into a workspace with 2 parameters, ggH and qqH cross sections using text2workspace
with the option P HiggsAnalysis.CombinedLimit.PhysicsModel:floatingXSHiggs PO modes=ggH,qqH
.
Now we just run one (or more) MCMC chain(s) and save them in the output tree.By default, the nuisance parameters will be marginalized (integrated) over their pdfs. You can ignore the complaints about not being able to compute an upper limit (since for more than 1D, this isn't well defined),
combine M MarkovChainMC workspace.root tries 1 saveChain i 1000000 m 125 s seed noDefaultPrior=0
The output of the markov chain is again a RooDataSet of weighted events distributed according to the posterior pdf (after you cut out the burn in part), so it can be used to make histograms or other distributions of the posterior pdf. See as an example bayesPosterior2D.cxx.
Below is an example of the output of the macro,
$ root l higgsCombineTest.MarkovChainMC....
.L bayesPosterior2D.cxx
bayesPosterior2D("bayes2D","Posterior PDF")
Computing Limits with toys
The HybridNew
method is used to compute either the hybrid bayesianfrequentist limits popularly known as "CL_{s} of LEP or Tevatron type" or the fully frequentist limits which are the current recommended method by the LHC Higgs Combination Group. Note that these methods can be resource intensive for complex models.
It is possible to define the criterion used for setting limits using rule CLs
(to use the CL_{s} criterion) or rule CLsplusb
(to calculate the limit using p_{\mu}) and as always the confidence level desired using cl=X
The choice of teststatistic can be made via the option testStat
and different methodologies for treatment of the nuisance parameters are available. While it is possible to mix different teststatistics with different nuisance parameter treatments, this is highly notreccomended. Instead one should follow one of the following three procedures,

LEPstyle:
testStat LEP generateNuisances=1 fitNuisances=0
 The test statistic is defined using the ratio of likelihoods q_{\mathrm{LEP}}=2\ln[\mathcal{L}(\mathrm{data}r=0)/\mathcal{L}(\mathrm{data}r)].
 The nuisance parameters are fixed to their nominal values for the purpose of evaluating the likelihood, while for generating toys, the nuisance parameters are first randomized within their pdfs before generation of the toy.

TEVstyle:
testStat TEV generateNuisances=0 generateExternalMeasurements=1 fitNuisances=1
 The test statistic is defined using the ratio of likelihoods q_{\mathrm{TEV}}=2\ln[\mathcal{L}(\mathrm{data}r=0,\hat{\theta}_{0})/\mathcal{L}(\mathrm{data}r,\hat{\theta}_{r})], in which the nuisance parameters are profiled separately for r=0 and r.
 For the purposes of toy generation, the nuisance parameters are fixed to their postfit values from the data (conditional on r), while the constraint terms are randomized for the evaluation of the likelihood.

LHCstyle:
LHCmode LHClimits
, which is the shortcut fortestStat LHC generateNuisances=0 generateExternalMeasurements=1 fitNuisances=1
 The test statistic is defined using the ratio of likelihoods q_{r} = 2\ln[\mathcal{L}(\mathrm{data}r,\hat{\theta}_{r})/\mathcal{L}(\mathrm{data}r=\hat{r},\hat{\theta})] , in which the nuisance parameters are profiled separately for r=\hat{r} and r.
 The value of q_{r} set to 0 when \hat{r}>r giving a one sided limit. Furthermore, the constraint r>0 is enforced in the fit. This means that if the unconstrained value of \hat{r} would be negative, the test statistic q_{r} is evaluated as 2\ln[\mathcal{L}(\mathrm{data}r,\hat{\theta}_{r})/\mathcal{L}(\mathrm{data}r=0,\hat{\theta}_{0})]
 For the purposes of toy generation, the nuisance parameters are fixed to their postfit values from the data (conditionally on the value of r), while the constraint terms are randomized in the evaluation of the likelihood.
Warning
The recommended style is the LHCstyle. Please note that this method is sensitive to the observation in data since the postfit (after a fit to the data) values of the nuisance parameters (assuming different values of r) are used when generating the toys. For completely blind limits you can first generate a prefit asimov toy dataset (described in the toy data generation section) and use that in place of the data.
While the above shortcuts are the common variants, you can also try others. The treatment of the nuisances can be changed to the socalled "HybridBayesian" method which effectively integrates over the nuisance parameters. This can be achieved (with any teststatistic which is not profiled over the nuisances) by setting generateNuisances=1 generateExternalMeasurements=0 fitNuisances=0
.
Info
Note that (observed and toy) values of the test statistic stored in the instances of RooStats::HypoTestResult
when the option saveHybridResult
has been specified, are defined without the factor 2 and therefore are twice as small as the values given by the formulas above. This factor is however included automatically by all plotting script supplied within the Combine package.
Simple models
For relatively simple models, the observed and expected limits can be calculated interactively. Since the LHCstyle is the reccomended procedure for calculating limits using toys, we will use that in this section but the same applies to the other methods.
combine realisticcountingexperiment.txt M HybridNew LHCmode LHClimits
Show output
<<< Combine >>>
>>> including systematics
>>> using the Profile Likelihood test statistics modified for upper limits (Q_LHC)
>>> method used is HybridNew
>>> random number generator seed is 123456
Computing results starting from observation (aposteriori)
Search for upper limit to the limit
r = 20 +/ 0
CLs = 0 +/ 0
CLs = 0 +/ 0
CLb = 0.264 +/ 0.0394263
CLsplusb = 0 +/ 0
Search for lower limit to the limit
Now doing proper bracketing & bisection
r = 10 +/ 10
CLs = 0 +/ 0
CLs = 0 +/ 0
CLb = 0.288 +/ 0.0405024
CLsplusb = 0 +/ 0
r = 5 +/ 5
CLs = 0 +/ 0
CLs = 0 +/ 0
CLb = 0.152 +/ 0.0321118
CLsplusb = 0 +/ 0
r = 2.5 +/ 2.5
CLs = 0.0192308 +/ 0.0139799
CLs = 0.02008 +/ 0.0103371
CLs = 0.0271712 +/ 0.00999051
CLs = 0.0239524 +/ 0.00783634
CLs = 0.0239524 +/ 0.00783634
CLb = 0.208748 +/ 0.0181211
CLsplusb = 0.005 +/ 0.00157718
r = 2.00696 +/ 1.25
CLs = 0.0740741 +/ 0.0288829
CLs = 0.0730182 +/ 0.0200897
CLs = 0.0694474 +/ 0.0166468
CLs = 0.0640182 +/ 0.0131693
CLs = 0.0595 +/ 0.010864
CLs = 0.0650862 +/ 0.0105575
CLs = 0.0629286 +/ 0.00966301
CLs = 0.0634945 +/ 0.00914091
CLs = 0.060914 +/ 0.00852667
CLs = 0.06295 +/ 0.00830083
CLs = 0.0612758 +/ 0.00778181
CLs = 0.0608142 +/ 0.00747001
CLs = 0.0587169 +/ 0.00697039
CLs = 0.0591432 +/ 0.00678587
CLs = 0.0599683 +/ 0.00666966
CLs = 0.0574868 +/ 0.00630809
CLs = 0.0571451 +/ 0.00608177
CLs = 0.0553836 +/ 0.00585531
CLs = 0.0531612 +/ 0.0055234
CLs = 0.0516837 +/ 0.0052607
CLs = 0.0496776 +/ 0.00499783
CLs = 0.0496776 +/ 0.00499783
CLb = 0.216635 +/ 0.00801002
CLsplusb = 0.0107619 +/ 0.00100693
Trying to move the interval edges closer
r = 1.00348 +/ 0
CLs = 0.191176 +/ 0.0459911
CLs = 0.191176 +/ 0.0459911
CLb = 0.272 +/ 0.0398011
CLsplusb = 0.052 +/ 0.00992935
r = 1.50522 +/ 0
CLs = 0.125 +/ 0.0444346
CLs = 0.09538 +/ 0.0248075
CLs = 0.107714 +/ 0.0226712
CLs = 0.103711 +/ 0.018789
CLs = 0.0845069 +/ 0.0142341
CLs = 0.0828468 +/ 0.0126789
CLs = 0.0879647 +/ 0.0122332
CLs = 0.0879647 +/ 0.0122332
CLb = 0.211124 +/ 0.0137494
CLsplusb = 0.0185714 +/ 0.00228201
r = 1.75609 +/ 0
CLs = 0.0703125 +/ 0.0255807
CLs = 0.0595593 +/ 0.0171995
CLs = 0.0555271 +/ 0.0137075
CLs = 0.0548727 +/ 0.0120557
CLs = 0.0527832 +/ 0.0103348
CLs = 0.0555828 +/ 0.00998248
CLs = 0.0567971 +/ 0.00923449
CLs = 0.0581822 +/ 0.00871417
CLs = 0.0588835 +/ 0.00836245
CLs = 0.0594035 +/ 0.00784761
CLs = 0.0590583 +/ 0.00752672
CLs = 0.0552067 +/ 0.00695542
CLs = 0.0560446 +/ 0.00679746
CLs = 0.0548083 +/ 0.0064351
CLs = 0.0566998 +/ 0.00627124
CLs = 0.0561576 +/ 0.00601888
CLs = 0.0551643 +/ 0.00576338
CLs = 0.0583584 +/ 0.00582854
CLs = 0.0585691 +/ 0.0057078
CLs = 0.0599114 +/ 0.00564585
CLs = 0.061987 +/ 0.00566905
CLs = 0.061836 +/ 0.00549856
CLs = 0.0616849 +/ 0.0053773
CLs = 0.0605352 +/ 0.00516844
CLs = 0.0602028 +/ 0.00502875
CLs = 0.058667 +/ 0.00486263
CLs = 0.058667 +/ 0.00486263
CLb = 0.222901 +/ 0.00727258
CLsplusb = 0.0130769 +/ 0.000996375
r = 2.25348 +/ 0
CLs = 0.0192308 +/ 0.0139799
CLs = 0.0173103 +/ 0.00886481
CLs = 0.0173103 +/ 0.00886481
CLb = 0.231076 +/ 0.0266062
CLsplusb = 0.004 +/ 0.001996
r = 2.13022 +/ 0
CLs = 0.0441176 +/ 0.0190309
CLs = 0.0557778 +/ 0.01736
CLs = 0.0496461 +/ 0.0132776
CLs = 0.0479048 +/ 0.0114407
CLs = 0.0419333 +/ 0.00925719
CLs = 0.0367934 +/ 0.0077345
CLs = 0.0339814 +/ 0.00684844
CLs = 0.03438 +/ 0.0064704
CLs = 0.0337633 +/ 0.00597315
CLs = 0.0321262 +/ 0.00551608
CLs = 0.0321262 +/ 0.00551608
CLb = 0.230342 +/ 0.0118665
CLsplusb = 0.0074 +/ 0.00121204
r = 2.06859 +/ 0
CLs = 0.0357143 +/ 0.0217521
CLs = 0.0381957 +/ 0.0152597
CLs = 0.0368622 +/ 0.0117105
CLs = 0.0415097 +/ 0.0106676
CLs = 0.0442816 +/ 0.0100457
CLs = 0.0376644 +/ 0.00847235
CLs = 0.0395133 +/ 0.0080427
CLs = 0.0377625 +/ 0.00727262
CLs = 0.0364415 +/ 0.00667827
CLs = 0.0368015 +/ 0.00628517
CLs = 0.0357251 +/ 0.00586442
CLs = 0.0341604 +/ 0.00546373
CLs = 0.0361935 +/ 0.00549648
CLs = 0.0403254 +/ 0.00565172
CLs = 0.0408613 +/ 0.00554124
CLs = 0.0416682 +/ 0.00539651
CLs = 0.0432645 +/ 0.00538062
CLs = 0.0435229 +/ 0.00516945
CLs = 0.0427647 +/ 0.00501322
CLs = 0.0414894 +/ 0.00479711
CLs = 0.0414894 +/ 0.00479711
CLb = 0.202461 +/ 0.00800632
CLsplusb = 0.0084 +/ 0.000912658
 HybridNew, before fit 
Limit: r < 2.00696 +/ 1.25 [1.50522, 2.13022]
Warning in : Could not create the Migrad minimizer. Try using the minimizer Minuit
Fit to 5 points: 1.91034 +/ 0.0388334
 Hybrid New 
Limit: r < 1.91034 +/ 0.0388334 @ 95% CL
Done in 0.01 min (cpu), 4.09 min (real)
Failed to delete temporary file roostatsSprxsw.root: No such file or directory
The result stored in the limit branch of the output tree will be the upper limit (and its error stored in limitErr). The default behavior will be, as above, to search for the upper limit on r however, the values of p_{\mu}, p_{b} and CL_{s} can be calculated for a particular value r=X by specifying the option singlePoint=X
. In this case, the value stored in the branch limit will be the value of CL_{s} (or p_{\mu}) (see the FAQ section).
Expected Limits
For the simple models, we can just run interactively 5 times to compute the median expected and the 68% and 95% interval boundaries. Use the HybridNew
method with the same options as per the observed limit but adding a expectedFromGrid=<quantile>
where the quantile is 0.5 for the median, 0.84 for the +ve side of the 68% band, 0.16 for the ve side of the 68% band, 0.975 for the +ve side of the 95% band, 0.025 for the ve side of the 95% band.
The output file will contain the value of the quantile in the branch quantileExpected which can be used to separate the points.
Accuracy
The search for the limit is performed using an adaptive algorithm, terminating when the estimate of the limit value is below some limit or when the precision cannot be futher improved with the specified options. The options controlling this behaviour are:
rAbsAcc
,rRelAcc
: define the accuracy on the limit at which the search stops. The default values are 0.1 and 0.05 respectively, meaning that the search is stopped when Δr < 0.1 or Δr/r < 0.05.clsAcc
: this determines the absolute accuracy up to which the CLs values are computed when searching for the limit. The default is 0.5%. Raising the accuracy above this value will increase significantly the time to run the algorithm, as you need N^{2} more toys to improve the accuracy by a factor N, you can consider enlarging this value if you're computing limits with a larger CL (e.g. 90% or 68%). Note that if you're using theCLsplusb
rule then this parameter will control the uncertainty on p_{\mu} rather than CL_{s}.T
ortoysH
: controls the minimum number of toys that are generated for each point. The default value of 500 should be ok when computing the limit with 9095% CL. You can decrease this number if you're computing limits at 68% CL, or increase it if you're using 99% CL.
Note, to further improve the accuracy when searching for the upper limit, combine will also fit an exponential function to several of the points and interpolate to find the crossing.
Complex models
For complicated models, it is best to produce a grid of test statistic distributions at various values of the signal strength, and use it to compute the observed and expected limit and bands. This approach is good for complex models since the grid of points can be distributed across any number of jobs. In this approach we will store the distributions of the teststatistic at different values of the signal strength using the option saveHybridResult
. The distribution at a single value of r=X can be determined by
combine datacard.txt M HybridNew LHCmode LHClimits singlePoint X saveToys saveHybridResult T 500 clsAcc 0
Warning
We have specified the accuracy here by including clsAcc=0
which turns off adaptive sampling and specifying the number of toys to be 500 with the T N
option. For complex models, it may be necessary to split the toys internally over a number of instances of HybridNew
using the option iterations I
. The total number of toys will be the product I*N.
The above can be repeated several times, in parallel, to build the distribution of the teststatistic (giving the random seed option s 1
). Once all of the distributions are finished, the resulting output files can be merged into one using hadd and read back to calculate the limit, specifying the merged file with grid=merged.root
.
The observed limit can be obtained with
combine datacard.txt M HybridNew LHCmode LHClimits readHybridResults grid=merged.root
and similarly, the median expected and quantiles can be determined using
combine datacard.txt M HybridNew LHCmode LHClimits readHybridResults grid=merged.root expectedFromGrid <quantile>
substituting <quantile>
with 0.5 for the median, 0.84 for the +ve side of the 68% band, 0.16 for the ve side of the 68% band, 0.975 for the +ve side of the 95% band, 0.025 for the ve side of the 95% band.
The splitting of the jobs can be left to the user's preference. However, users may wish to use the combineTool for automating this as described in the section on combineTool for job submission
Plotting
A plot of the CL_{s} (or p_{\mu}) as a function of r, which is used to find the crossing, can be produced using the option plot=limit_scan.png
. This can be useful for judging if the grid was sufficient in determining the upper limit.
If we use our realisticcountingexperiment.txt datacard and generate a grid of points r\varepsilon[1.4,2.2] in steps of 0.1, with 5000 toys for each point, the plot of the observed CL_{s} vs r should look like the following,
You should judge in each case if the limit is accurate given the spacing of the points and the precision of CL_{s} at each point. If it is not sufficient, simply generate more points closer to the limit and/or more toys at each point.
The distributions of the teststatistic can also be plotted, at each value in the grid, using the simple python tool,
python test/plotTestStatCLs.py input mygrid.root poi r val all mass MASS
The resulting output file will contain a canvas showing the distribution of the test statistic background only and signal+background hypothesis at each value of r.
Info
If you used the TEV or LEP style test statistic (using the commands as described above), then you should include the option doublesided
, which will also take care of defining the correct integrals for p_{\mu} and p_{b}.
Computing Significances with toys
Computation of expected significance with toys is a two step procedure: first you need to run one or more jobs to construct the expected distribution of the test statistic. As with setting limits, there are a number of different configurations for generating toys but we will use the preferred option using,
 LHCstyle:
LHCmode LHCsignificance
, which is the shortcut fortestStat LHC generateNuisances=0 generateExternalMeasurements=1 fitNuisances=1 significance
 The test statistic is defined using the ratio of likelihoods q_{0} = 2\ln[\mathcal{L}(\textrm{data}r=0,\hat{\theta}_{0})/\mathcal{L}(\textrm{data}r=\hat{r},\hat{\theta})], in which the nuisance parameters are profiled separately for r=\hat{r} and r=0.
 The value of the test statistic is set to 0 when \hat{r}<0
 For the purposes of toy generation, the nuisance parameters are fixed to their postfit values from the data assuming no signal, while the constraint terms are randomized for the evaluation of the likelihood.
Observed significance
To construct the distribution of the test statistic run as many times as necessary,
combine M HybridNew datacard.txt LHCmode LHCsignificance saveToys fullBToys saveHybridResult T toys i iterations s seed
with different seeds, or using s 1
for random seeds, then merge all those results into a single root file with hadd
.
The observed significance can be calculated as
combine M HybridNew datacard.txt LHCmode LHCsignificance readHybridResult grid=input.root [pvalue ]
where the option pvalue
will replace the result stored in the limit branch output tree to be the pvalue instead of the signficance.
Expected significance, assuming some signal
The expected significance, assuming a signal with r=X can be calculated, by including the option expectSignal X
when generating the distribution of the test statistic and using the option expectedFromGrid=0.5
when calculating the significance for the median. To get the ±1σ bands, use 0.16 and 0.84 instead of 0.5, and so on...
You need a total number of background toys large enough to compute the value of the significance, but you need less signal toys (especially if you only need the median). For large significance, you can then run most of the toys without the fullBToys
option (about a factor 2 faster), and only a smaller part with that option turned on.
As with calculating limits with toys, these jobs can be submitted to the grid or batch systems with the help of the combineTool
as described in the section on combineTool for job submission
Goodness of fit tests
The GoodnessOfFit
method can be used to evaluate how compatible the observed data are with the model pdf.
The module can be run specifying an algorithm, and will compute a goodness of fit indicator for that algorithm and the data. The procedure is therefore to first run on the real data
combine M GoodnessOfFit datacard.txt algo=<somealgo>
and then to run on many toy mc datasets to determine the distribution of the goodness of fit indicator
combine M GoodnessOfFit datacard.txt algo=<somealgo> t <numberoftoys> s <seed>
When computing the goodness of fit, by default the signal strength is left floating in the fit, so that the measure is independent from the presence or absence of a signal. It is possible to instead keep it fixed to some value by passing the option fixedSignalStrength=<value>
.
The following algorithms are supported:

saturated
: Compute a goodnessoffit measure for binned fits based on the saturated model method, as prescribed by the StatisticsCommittee (note). This quantity is similar to a chisquare, but can be computed for an arbitrary combination of binned channels with arbitrary constraints. 
KS
: Compute a goodnessoffit measure for binned fits using the KolmogorovSmirnov test. It is based on the highest difference between the cumulative distribution function and the empirical distribution function of any bin. 
AD
: Compute a goodnessoffit measure for binned fits using the AndersonDarling test. It is based on the integral of the difference between the cumulative distribution function and the empirical distribution function over all bins. It also gives the tail ends of the distribution a higher weighting.
The output tree will contain a branch called limit
which contains the value of the teststatistic in each toy. You can make a histogram of this teststatistic t and from this distribution (f(t)) and the single value obtained in the data (t_{0}) you can calculate the pvalue $$p = \int_{t=t_{0}}^{\mathrm{+inf}} f(t) dt $$. Note: in rare cases the test statistic value for the toys can be undefined (for AS and KD), and in this case we set the test statistic value to 1. When plotting the test statistic distribution, those toys should be excluded. This is automatically taken care of if you use the GoF collection script in CombineHarvester described below.
When generating toys, the default behavior will be used. See the section on toy generation for options on how to generate/fit nuisance parameters in these tests. It is recomended to use the frequentist toys (toysFreq
) when running the saturated
model, and the default toys for the other two tests.
Further goodness of fit methods could be added on request, especially if volunteers are available to code them. The output limit tree will contain the value of the teststatistic in each toy (or the data)
Warning
The above algorithms are all concerned with onesample tests. For twosample tests, you can follow an example CMS HIN analysis described in this Twiki
Masking analysis regions in the saturated model
For searches that employs a simultaneous fit across signal and control regions, it may be useful to mask one or more analysis regions either when the likelihood is maximized (fit) or when the teststatistic is computed. This can be done by using the options setParametersForFit
and setParametersForEval
, respectively. The former will set parameters before each fit while the latter is used to set parameters after each fit, but before the NLL is evauated. Note of course that if the parameter in the list is floating, it will be still floating in each fit so will not effect the results when using setParametersForFit
.
A realistic example for a binned shape analysis performed in one signal region and two control samples can be found in this directory of the Higgscombine package Datacardsshapeanalysismultipleregions.
First of all, one needs to combine the individual datacards to build a single model and to introduce the channelmasking variables as follow:
combineCards.py signal_region.txt dimuon_control_region.txt singlemuon_control_region.txt > combined_card.txt
text2workspace.py combined_card.txt channelmasks
More information about the channelmasking can be found in this section Channel Masking. The saturated teststatic value for a simultaneous fit across all the analysis regions can be calculated as:
combine M GoodnessOfFit d combined_card.root algo=saturated n _result_sb
In this case, signal and control regions are included in both the fit and in the evaluation of the teststatic, and the signal strength is freely floating. This measures the compatibility between the signal+background fit and the observed data. Moreover, it can be interesting to assess the level of compatibility between the observed data in all the regions and the background prediction obtained by only fitting the control regions (CRonly fit). This is computed as follow:
combine M GoodnessOfFit d combined_card.root algo=saturated n _result_bonly_CRonly setParametersForFit mask_ch1=1 setParametersForEval mask_ch1=0 freezeParameters r setParameters r=0
where the signal strength is frozen and the signal region is not considered in the fit (setParametersForFit mask_ch1=1
), but it is included in the teststatistic computation (setParametersForEval mask_ch1=0
). To show the differences between the two models being tested, one can perform a fit to the data using the FitDiagnostics method as:
combine M FitDiagnostics d combined_card.root n _fit_result saveShapes saveWithUncertainties
combine M FitDiagnostics d combined_card.root n _fit_CRonly_result saveShapes saveWithUncertainties setParameters mask_ch1=1
By taking the total background, the total signal, and the data shapes from FitDiagnostics output, we can compare the postfit predictions from the S+B fit (first case) and the CRonly fit (second case) with the observation as reported below:
FitDiagnostics S+B fit
FitDiagnostics CRonly fit
To compute a pvalue for the two results, one needs to compare the observed goodnessoffit value previously computed with expected distribution of the teststatistic obtained in toys:
combine M GoodnessOfFit combined_card.root algo=saturated n result_toy_sb toysFrequentist t 500
combine M GoodnessOfFit d combined_card.root algo=saturated n _result_bonly_CRonly_toy setParametersForFit mask_ch1=1 setParametersForEval mask_ch1=0 freezeParameters r setParameters r=0,mask_ch1=1 t 500 toysFrequentist
where the former gives the result for the S+B model, while the latter gives the teststatistic for CRonly fit. The command setParameters r=0,mask_ch1=1
is needed to ensure that toys are thrown using the nuisance parameters estimated from the CRonly fit to the data. The comparison between the observation and the expected distribition should look like the following two plots:
Goodnessoffit for S+B model
Goodnessoffit for CRonly model
Making a plot of the GoF teststatistic distribution
If you have also checked out the combineTool, you can use this to run batch jobs or on the grid (see here) and produce a plot of the results. Once you have the jobs, you can hadd them together and run (e.g for the saturated model),
combineTool.py M CollectGoodnessOfFit input data_run.root toys_run.root m 125.0 o gof.json
plotGof.py gof.json statistic saturated mass 125.0 o gof_plot titleright="my label"
Channel Compatibility
The ChannelCompatibilityCheck
method can be used to evaluate how compatible are the measurements of the signal strength from the separate channels of a combination.
The method performs two fits of the data, first with the nominal model in which all channels are assumed to have the same signal strength multiplier r, and then another allowing separate signal strengths r_{i} in each channel. A chisquarelike quantity is computed as 2 \ln \mathcal{L}(\mathrm{data} r)/L(\mathrm{data}\{r_{i}\}_{i=1}^{N_{\mathrm{chan}}}). Just like for the goodness of fit indicators, the expected distribution of this quantity under the nominal model can be computed from toy mc.
By default, the signal strength is kept floating in the fit with the nominal model. It can however be fixed to a given value by passing the option fixedSignalStrength=<value>
.
In the default models build from the datacards the signal strengths in all channels are constrained to be nonnegative. One can allow negative signal strengths in the fits by changing the bound on the variable (option rMin=<value>
), which should make the quantity more chisquarelike under the hypothesis of zero signal; this however can create issues in channels with small backgrounds, since total expected yields and pdfs in each channel must be positive.
When run with the a verbosity of 1, as the default, the program also prints out the best fit signal strengths in all channels; as the fit to all channels is done simultaneously, the correlation between the other systematical uncertainties is taken into account, and so these results can differ from the ones obtained fitting each channel separately.
Below is an example output from combine,
$ combine M ChannelCompatibilityCheck comb_hww.txt m 160 n HWW
<<< Combine >>>
>>> including systematics
>>> method used to compute upper limit is ChannelCompatibilityCheck
>>> random number generator seed is 123456
Sanity checks on the model: OK
Computing limit starting from observation
 ChannelCompatibilityCheck 
Nominal fit : r = 0.3431 0.1408/+0.1636
Alternate fit: r = 0.4010 0.2173/+0.2724 in channel hww_0jsf_shape
Alternate fit: r = 0.2359 0.1854/+0.2297 in channel hww_0jof_shape
Alternate fit: r = 0.7669 0.4105/+0.5380 in channel hww_1jsf_shape
Alternate fit: r = 0.3170 0.3121/+0.3837 in channel hww_1jof_shape
Alternate fit: r = 0.0000 0.0000/+0.5129 in channel hww_2j_cut
Chi2like compatibility variable: 2.16098
Done in 0.08 min (cpu), 0.08 min (real)
The output tree will contain the value of the compatibility (chisquare variable) in the limit branch. If the option saveFitResult
is specified, the output root file contains also two RooFitResult objects fit_nominal and fit_alternate with the results of the two fits.
This can be read and used to extract the best fit for each channel and the overall best fit using
$ root l
TFile* _file0 = TFile::Open("higgsCombineTest.ChannelCompatibilityCheck.mH120.root");
fit_alternate>floatParsFinal().selectByName("*ChannelCompatibilityCheck*")>Print("v");
fit_nominal>floatParsFinal().selectByName("r")>Print("v");
The macro cccPlot.cxx can be used to produce a comparison plot of the best fit signals from all channels.
Likelihood Fits and Scans
The MultiDimFit
method can do multidimensional fits and likelihood based scans/contours using models with several parameters of interest.
Taking a toy datacard test/multiDim/toyhgg125.txt (counting experiment which vaguely resembles the H→γγ analysis at 125 GeV), we need to convert the datacard into a workspace with 2 parameters, ggH and qqH cross sections
text2workspace.py toyhgg125.txt m 125 P HiggsAnalysis.CombinedLimit.PhysicsModel:floatingXSHiggs PO modes=ggH,qqH
A number of different algorithms can be used with the option algo <algo>
,

none
(default): Perform a maximum likelihood fitcombine M MultiDimFit toyhgg125.root
; The output root tree will contain two columns, one for each parameter, with the fitted values. 
singles
: Perform a fit of each parameter separately, treating the others as unconstrained nuisances:combine M MultiDimFit toyhgg125.root algo singles cl=0.68
. The output root tree will contain two columns, one for each parameter, with the fitted values; there will be one row with the best fit point (andquantileExpected
set to 1) and two rows for each fitted parameter, where the corresponding column will contain the maximum and minimum of that parameter in the 68% CL interval, according to a onedimensional chisquare (i.e. uncertainties on each fitted parameter do not increase when adding other parameters if they're uncorrelated). Note that if you run, for example, withcminDefaultMinimizerStrategy=0
, these uncertainties will be derived from the Hessian, whilecminDefaultMinimizerStrategy=1
will invoke Minos to derive them. 
cross
: Perform joint fit of all parameters:combine M MultiDimFit toyhgg125.root algo=cross cl=0.68
. The output root tree will have one row with the best fit point, and two rows for each parameter, corresponding to the minimum and maximum of that parameter on the likelihood contour corresponding to the specified CL, according to a Ndimensional chisquare (i.e. uncertainties on each fitted parameter do increase when adding other parameters, even if they're uncorrelated). Note that the output of this way of running are not 1D uncertainties on each parameter, and shouldn't be taken as such. 
contour2d
: Make a 68% CL contour a la minoscombine M MultiDimFit toyhgg125.root algo contour2d points=20 cl=0.68
. The output will contain values corresponding to the best fit point (withquantileExpected
set to 1) and for a set of points on the contour (withquantileExpected
set to 1CL, or something larger than that if the contour is hitting the boundary of the parameters). Probabilities are computed from the the ndimensional \chi^{2} distribution. For slow models, you can split it up by running several times with different number of points and merge the outputs (something better can be implemented). You can look at the contourPlot.cxx macro for how to make plots out of this algorithm. 
random
: Scan N random points and compute the probability out of the profile likelihoodcombine M MultiDimFit toyhgg125.root algo random points=20 cl=0.68
. Again, best fit will havequantileExpected
set to 1, while each random point will havequantileExpected
set to the probability given by the profile likelihood at that point. 
fixed
: Compare the loglikelihood at a fixed point compared to the best fit.combine M MultiDimFit toyhgg125.root algo fixed fixedPointPOIs r=r_fixed,MH=MH_fixed
. The output tree will contain the difference in the negative loglikelihood between the points (\hat{r},\hat{m}_{H}) and (\hat{r}_{fixed},\hat{m}_{H,fixed}) in the branchdeltaNLL
. 
grid
: Scan on a fixed grid of points not with approximately N points in total.combine M MultiDimFit toyhgg125.root algo grid points=10000
. You can partition the job in multiple tasks by using options
firstPoint
andlastPoint
, for complicated scans, the points can be split as described in the combineTool for job submission section. The output file will contain a columndeltaNLL
with the difference in negative log likelihood with respect to the best fit point. Ranges/contours can be evaluated by filling TGraphs or TH2 histograms with these points.  By default the "min" and "max" of the POI ranges are not included and the points which are in the scan are centered , eg
combine M MultiDimFit algo grid rMin 0 rMax 5 points 5
will scan at the points r=0.5, 1.5, 2.5, 3.5, 4.5. You can instead include the optionalignEdges 1
which causes the points to be aligned with the endpoints of the parameter ranges  egcombine M MultiDimFit algo grid rMin 0 rMax 5 points 6 alignEdges 1
will now scan at the points r=0, 1, 2, 3, 4, 5. NB  the number of points must be increased by 1 to ensure both end points are included.
 You can partition the job in multiple tasks by using options
With the algorithms none
and singles
you can save the RooFitResult from the initial fit using the option saveFitResult
. The fit result is saved into a new file called muiltidimfit.root
.
As usual, any floating nuisance parameters will be profiled which can be turned of using the freezeParameters
option.
For most of the methods, for lower precision results you can turn off the profiling of the nuisances setting option fastScan
, which for complex models speeds up the process by several orders of magnitude. All nuisance parameters will be kept fixed at the value corresponding to the best fit point.
As an example, lets produce the 2\Delta\ln{\mathcal{L}} scan as a function of r_ggH
and r_qqH
from the toy H→γγ datacard, with the nuisance parameters fixed to their global best fit values.
combine toyhgg125.root M MultiDimFit algo grid points 2000 setParameterRanges r_qqH=0,10:r_ggH=0,4 m 125 fastScan
Show output
<<< Combine >>>
>>> including systematics
>>> method used is MultiDimFit
>>> random number generator seed is 123456
ModelConfig 'ModelConfig' defines more than one parameter of interest. This is not supported in some statistical methods.
Set Range of Parameter r_qqH To : (0,10)
Set Range of Parameter r_ggH To : (0,4)
Computing results starting from observation (aposteriori)
POI: r_ggH= 0.88152 > [0,4]
POI: r_qqH= 4.68297 > [0,10]
Point 0/2025, (i,j) = (0,0), r_ggH = 0.044444, r_qqH = 0.111111
Point 11/2025, (i,j) = (0,11), r_ggH = 0.044444, r_qqH = 2.555556
Point 22/2025, (i,j) = (0,22), r_ggH = 0.044444, r_qqH = 5.000000
Point 33/2025, (i,j) = (0,33), r_ggH = 0.044444, r_qqH = 7.444444
Point 55/2025, (i,j) = (1,10), r_ggH = 0.133333, r_qqH = 2.333333
Point 66/2025, (i,j) = (1,21), r_ggH = 0.133333, r_qqH = 4.777778
Point 77/2025, (i,j) = (1,32), r_ggH = 0.133333, r_qqH = 7.222222
Point 88/2025, (i,j) = (1,43), r_ggH = 0.133333, r_qqH = 9.666667
Point 99/2025, (i,j) = (2,9), r_ggH = 0.222222, r_qqH = 2.111111
Point 110/2025, (i,j) = (2,20), r_ggH = 0.222222, r_qqH = 4.555556
Point 121/2025, (i,j) = (2,31), r_ggH = 0.222222, r_qqH = 7.000000
Point 132/2025, (i,j) = (2,42), r_ggH = 0.222222, r_qqH = 9.444444
Point 143/2025, (i,j) = (3,8), r_ggH = 0.311111, r_qqH = 1.888889
Point 154/2025, (i,j) = (3,19), r_ggH = 0.311111, r_qqH = 4.333333
Point 165/2025, (i,j) = (3,30), r_ggH = 0.311111, r_qqH = 6.777778
Point 176/2025, (i,j) = (3,41), r_ggH = 0.311111, r_qqH = 9.222222
Point 187/2025, (i,j) = (4,7), r_ggH = 0.400000, r_qqH = 1.666667
Point 198/2025, (i,j) = (4,18), r_ggH = 0.400000, r_qqH = 4.111111
Point 209/2025, (i,j) = (4,29), r_ggH = 0.400000, r_qqH = 6.555556
Point 220/2025, (i,j) = (4,40), r_ggH = 0.400000, r_qqH = 9.000000
[...]
Done in 0.00 min (cpu), 0.02 min (real)
The scan, along with the best fit point can be drawn using root,
$ root l higgsCombineTest.MultiDimFit.mH125.root
limit>Draw("2*deltaNLL:r_ggH:r_qqH>>h(44,0,10,44,0,4)","2*deltaNLL<10","prof colz")
limit>Draw("r_ggH:r_qqH","quantileExpected == 1","P same")
TGraph *best_fit = (TGraph*)gROOT>FindObject("Graph")
best_fit>SetMarkerSize(3); best_fit>SetMarkerStyle(34); best_fit>Draw("p same")
To make the full profiled scan just remove the fastScan
option from the combine command.
Similarly, 1D scans can be drawn directly from the tree, however for 1D likelihood scans, there is a python script from the CombineHarvester/CombineTools
package plot1DScan.py which can be used to make plots and extract the crossings of the 2*deltaNLL
 e.g the 1σ/2σ boundaries.
Useful options for likelihood scans
A number of common, useful options (especially for computing likelihood scans with the grid algo) are,
autoBoundsPOIs arg
: Adjust bounds for the POIs if they end up close to the boundary. This can be a comma separated list of POIs, or "*" to get all of them.autoMaxPOIs arg
: Adjust maxima for the POIs if they end up close to the boundary. Can be a list of POIs, or "*" to get all.autoRange X
: Set to any X >= 0 to do the scan in the \hat{p} \pm Xσ range, where \hat{p} and σ are the best fit parameter value and uncertainty from the initial fit (so it may be fairly approximate). In case you do not trust the estimate of the error from the initial fit, you can just centre the range on the best fit value by using the optioncenteredRange X
to do the scan in the \hat{p} \pm X range centered on the best fit value.squareDistPoiStep
: POI step size based on distance from midpoint ( either (maxmin)/2 or the best fit if used withautoRange
orcenteredRange
) rather than linear separation.skipInitialFit
: Skip the initial fit (saves time if for example a snapshot is loaded from a previous fit)
Below is a comparison in a likelihood scan, with 20 points, as a function of r_qqH
with our toyhgg125.root
workspace with and without some of these options. The options added tell combine to scan more points closer to the minimum (bestfit) than with the default.
You may find it useful to use the robustFit=1
option to turn on robust (bruteforce) for likelihood scans (and other algorithms). You can set the strategy and tolerance when using the robustFit
option using the options setRobustFitAlgo
(default is Minuit2,migrad
), setRobustFitStrategy
(default is 0) and setRobustFitTolerance
(default is 0.1). If these options are not set, the defaults (set using cminDefaultMinimizerX
options) will be used.
If running robustFit=1
with the algo singles, you can tune the accuracy of the routine used to find the crossing points of the likelihood using the option setCrossingTolerance
(default is set to 0.0001)
If you suspect your fits/uncertainties are not stable, you may also try to run custom HESSEstyle calculation of the covariance matrix. This is enabled by running MultiDimFit
with the robustHesse=1
option. A simple example of how the default behaviour in a simple datacard is given here.
For a full list of options use combine M MultiDimFit help
Fitting only some parameters
If your model contains more than one parameter of interest, you can still decide to fit a smaller number of them, using the option parameters
(or P
), with a syntax like this:
combine M MultiDimFit [...] P poi1 P poi2 ... floatOtherPOIs=(01)
If floatOtherPOIs
is set to 0, the other parameters of interest (POIs), which are not included as a P
option, are kept fixed to their nominal values. If it's set to 1, they are kept floating, which has different consequences depending on algo
:
 When running with
algo=singles
, the other floating POIs are treated as unconstrained nuisance parameters.  When running with
algo=cross
oralgo=contour2d
, the other floating POIs are treated as other POIs, and so they increase the number of dimensions of the chisquare.
As a result, when running with floatOtherPOIs
set to 1, the uncertainties on each fitted parameters do not depend on what's the selection of POIs passed to MultiDimFit, but only on the number of parameters of the model.
Info
Note that poi
given to the the option P
can also be any nuisance parameter. However, by default, the other nuisance parameters are left floating, so you do not need to specify that.
You can save the values of the other parameters of interest in the output tree by adding the option saveInactivePOI=1
. You can additionally save the postfit values any nuisance parameter, function or discrete index (RooCategory) defined in the workspace using the following options;
saveSpecifiedNuis=arg1,arg2,...
will store the fitted value of any specified constrained nuisance parameter. Useall
to save every constrained nuisance parameter. Note that if you want to store the values offlatParams
(or floating parameters which are not defined in the datacard) orrateParams
, which are unconstrained, you should instead use the generic optiontrackParameters
as described here.saveSpecifiedFunc=arg1,arg2,...
will store the value of any function (egRooFormulaVar
) in the model.saveSpecifiedIndex=arg1,arg2,...
will store the index of anyRooCategory
object  eg adiscrete
nuisance.
Using best fit snapshots
This can be used to save time when performing scans so that the bestfit needs not be redone and can also be used to perform scans with some nuisances frozen to the bestfit values. Sometimes it is useful to scan freezing certain nuisances to their bestfit values as opposed to the default values. To do this here is an example,
 Create a workspace workspace for a floating r,m_{H} fit
text2workspace.py hgg_datacard_mva_8TeV_bernsteins.txt m 125 P HiggsAnalysis.CombinedLimit.PhysicsModel:floatingHiggsMass PO higgsMassRange=120,130 o testmass.root`
 Perfom the fit, saving the workspace
combine m 123 M MultiDimFit saveWorkspace n teststep1 testmass.root verbose 9
Now we can load the bestfit \hat{r},\hat{m}_{H} and fit for r freezing m_{H} and lumi_8TeV to the bestfit values,
combine m 123 M MultiDimFit d higgsCombineteststep1.MultiDimFit.mH123.root w w snapshotName "MultiDimFit" n teststep2 verbose 9 freezeParameters MH,lumi_8TeV
Feldman Cousins
The FeldmanCousins (FC) procedure for computing confidence intervals for a generic model is,
 use the profile likelihood as the teststatistic q(x) =  2 \ln \mathcal{L}(\mathrm{data}x,\hat{\theta}_{x})/\mathcal{L}(\mathrm{data}\hat{x},\hat{\theta}) where x is a point in the (Ndimensional) parameter space, and \hat{x} is the point corresponding to the best fit. In this teststatistic, the nuisance parameters are profiled, separately both in the numerator and denominator.
 for each point x:
 compute the observed test statistic q_{\mathrm{obs}}(x)
 compute the expected distribution of q(x) under the hypothesis of x as the true value.
 accept the point in the region if p_{x}=P\left[q(x) > q_{\mathrm{obs}}(x) x\right] > \alpha
With a critical value \alpha.
In combine
, you can perform this test on each individual point (param1, param2,...) = (value1,value2,...) by doing,
combine workspace.root M HybridNew LHCmode LHCfeldmancousins clsAcc 0 singlePoint param1=value1,param2=value2,param3=value3,... saveHybridResult [Other options for toys, iterations etc as with limits]
The point belongs to your confidence region if p_{x} is larger than \alpha (e.g. 0.3173 for a 1σ region, 1\alpha=0.6827).
Warning
You should not use this method without the option singlePoint
. Although combine will not complain, the algorithm to find the crossing will only find a single crossing and therefore not find the correct interval. Instead you should calculate the FeldmanCousins intervals as described above.
Physical boundaries
Imposing physical boundaries (such as requiring \mu>0 for a signal strength) is achieved by setting the ranges of the physics model parameters using
setParameterRanges param1=param1_min,param1_max:param2=param2_min,param2_max ....
The boundary is imposed by restricting the parameter range(s) to those set by the user, in the fits. Note that this is a trick! The actual fitted value, as one of an ensemble of outcomes, can fall outside of the allowed region, while the boundary should be imposed on the physical parameter. The effect of restricting the parameter value in the fit is such that the teststatistic is modified as follows ;
q(x) =  2 \ln \mathcal{L}(\mathrm{data}x,\hat{\theta}_{x})/\mathcal{L}(\mathrm{data}\hat{x},\hat{\theta}), if \hat{x} in contained in the bounded range
and,
q(x) =  2 \ln \mathcal{L}(\mathrm{data}x,\hat{\theta}_{x})/\mathcal{L}(\mathrm{data}x_{B},\hat{\theta}_{B}), if \hat{x} is outside of the bounded range. Here x_{B} and \hat{\theta}_{B} are the values of x and \theta which maximise the likelihood excluding values outside of the bounded region for x  typically, x_{B} will be found at one of the boundaries which is imposed. For example, if the boundary x>0 is imposed, you will typically expect x_{B}=0, when \hat{x}\leq 0, and x_{B}=\hat{x} otherewise.
This can sometimes be an issue as Minuit may not know if has successfully converged when the minimum lies outside of that range. If there is no upper/lower boundary, just set that value to something far from the region of interest.
Info
One can also imagine imposing the boundaries by first allowing Minuit to find the minimum in the unrestricted region and then setting the teststatistic to that above in the case that minimum lies outside the physical boundary. This would avoid potential issues of convergence  If you are interested in implementing this version in combine, please contact the development team.
As in general for HybridNew
, you can split the task into multiple tasks (grid and/or batch) and then merge the outputs, as described in the combineTool for job submission section.
Extracting contours
As in general for HybridNew
, you can split the task into multiple tasks (grid and/or batch) and then merge the outputs with hadd
. You can also refer to the combineTool for job submission section for submitting the jobs to the grid/batch.
1D intervals
For onedimensional models only, and if the parameter behaves like a crosssection, the code is somewhat able to do interpolation and determine the values of your parameter on the contour (just like it does for the limits). As with limits, read in the grid of points and extract 1D intervals using,
combine workspace.root M HybridNew LHCmode LHCfeldmancousins readHybridResults grid=mergedfile.root cl <1alpha>
The output tree will contain the values of the POI which crosses the critical value (\alpha)  i.e, the boundaries of the confidence intervals,
You can produce a plot of the value of p_{x} vs the parameter of interest x by adding the option plot <plotname>
.
2D contours
There is a tool for extracting 2D contours from the output of HybridNew
located in test/makeFCcontour.py
provided the option saveHybridResult
was included when running HybridNew
. It can be run with the usual combine output files (or several of them) as input,
./test/makeFCcontour.py toysfile1.root toysfile2.root .... [options] out outputfile.root
To extract 2D contours, the names of each parameter must be given xvar poi_x yvar poi_y
. The output will be a root file containing a 2D histogram of value of p_{x,y} for each point (x,y) which can be used to draw 2D contours. There will also be a histogram containing the number of toys found for each point.
There are several options for reducing the running time (such as setting limits on the region of interest or the minimum number of toys required for a point to be included) Finally, adding the option storeToys
in this python tool will add histograms for each point to the output file of the teststatistic distribution. This will increase the momory usage however as all of the toys will be stored in memory.