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CMS-CAT-23-001 ; CERN-EP-2024-078
The CMS statistical analysis and combination tool: Combine
Accepted for publication in Computing and Software for Big Science
Abstract: This paper describes the Combine software package used for statistical analyses by the CMS Collaboration. The package, originally designed to perform searches for a Higgs boson and the combined analysis of those searches, has evolved to become the statistical analysis tool presently used in the majority of measurements and searches performed by the CMS Collaboration. It is not specific to the CMS experiment, and this paper is intended to serve as a reference for users outside of the CMS Collaboration, providing an outline of the most salient features and capabilities. Readers are provided with the possibility to run Combine and reproduce examples provided in this paper using a publicly available container image. Since the package is constantly evolving to meet the demands of ever-increasing data sets and analysis sophistication, this paper cannot cover all details of Combine. However, the online documentation referenced within this paper provides an up-to-date and complete user guide.
Figures & Tables Summary References CMS Publications
Link to the Combine manual

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Figure 1:
Histograms used to define the pdfs for Datacard dc:template. The red and blue histograms show the nominal yields in each bin $ \omega_{b}^{0} $ for the background and signal processes, respectively. The dotted and dashed lines show the histograms that provide the values of $ \omega_{b}^{+} $ and $ \omega_{b}^{-} $, respectively for each of the systematic uncertainties that modify the shape of the signal and background pdfs. The red dashed and dotted lines are associated with the effect of the nuisance parameter alpha on the background process, while the blue dashed and dotted lines are associated with the effect of the nuisance parameter sigma affecting the signal process. The black points show the observed number of events in data in each bin. The error bars indicate the statistical uncertainty.

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Figure 2:
Distributions of the invariant mass observable for the signal and background processes defined in Datacard dc:param. The red and blue curves show the parametric functions used to define the probability density for the invariant mass for the background and signal processes, respectively, at the default values of the nuisance parameters, normalized to their expected total yields. The blue shaded band shows the variation of the signal pdf when sigma is varied between 0.7 and 1.3. The red shaded region shows the variation of the background pdf when alpha is varied within 10% of its default value of-0.1. The black points show the distribution of the observed data. The binning and error bars are only for visualization and neither are used by Combine to build the likelihood function.

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Figure 3:
Distributions of $ \widetilde{q}_{\mathrm{LHC}}(\mu=0.4) $ from 100,000 pseudo-data sets for $ \mu= $ 0 (red histogram) and $ \mu= $ 0.4 (blue histogram) using the analysis described in Datacard dc:template. The observed value of the test statistic is indicated by the black vertical line and the regions used to determine 1 $ -p_{b} $ and $ p_{\mu} $ are indicated by the pink hatched and light blue shaded regions, respectively.

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Figure 4:
Calculated $ \text{CL}_\text{s} $ as a function of $ \mu $, used to determine the 95% CL upper limit for Datacard dc:template. The solid red line is used to interpolate the $ \text{CL}_\text{s} $ values to find the crossing at 0.05, and the shaded band indicates the uncertainty in the interpolation that is used to estimate an uncertainty in the upper limit. The vertical dashed blue lines show the derived upper limit and the estimated uncertainty due to the number of pseudo-data sets used in the calculation.

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Figure 5:
Distribution of $ q_{0} $ in 100,000 pseudo-data sets from Datacard dc:param. The observed value of the test statistic is indicated by the black vertical line and the region used to determine $ p_{0} $ is indicated by the light gray shaded region.

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Figure 6:
Values of $ q(r_{\mathrm{g}\mathrm{g}\mathrm{H}},r_{\mathrm{q}\mathrm{q}\mathrm{H}}) $ for Datacard dc:multisig in a model with two parameters of interest $ r_{\mathrm{g}\mathrm{g}\mathrm{H}} $ and $ r_{\mathrm{q}\mathrm{q}\mathrm{H}} $. The orange scale shows the values obtained in Combine at the set of points indicated by the black dots, using the grid algorithm. The blue box is constructed using the cross algorithm with $ (1-\alpha)= $ 0.68. The white cross and white dots indicate, respectively, the maximum likelihood estimates for $ r_{\mathrm{g}\mathrm{g}\mathrm{H}} $ and $ r_{\mathrm{q}\mathrm{q}\mathrm{H}} $ from the best fit, and the 68% CL confidence region obtained using the contour2d algorithm defined as the values of $ (r_{\mathrm{g}\mathrm{g}\mathrm{H}},r_{\mathrm{q}\mathrm{q}\mathrm{H}}) $ for which $ q(r_{\mathrm{g}\mathrm{g}\mathrm{H}},r_{\mathrm{q}\mathrm{q}\mathrm{H}})= $ 2.3.

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Figure 7:
Example of $ q(r_{\mathrm{g}\mathrm{g}\mathrm{H}},\hat{r}_{\mathrm{q}\mathrm{q}\mathrm{H}}) $ and $ q(r_{\mathrm{q}\mathrm{q}\mathrm{H}},\hat{r}_{\mathrm{g}\mathrm{g}\mathrm{H}}) $ obtained from Combine with Datacard dc:multisig. The points indicate the values at which the functions are evaluated using the grid algorithm, and the shaded region indicates the 68% CL intervals on each parameter obtained using the singles algorithm. The horizontal dashed lines indicate the values of $ q(\mu) $ used to define 68% and 95% CL intervals.

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Figure 8:
Distribution of the saturated test statistic $ t $ in 10,000 pseudo-data sets using Datacard dc:template. The observed value of the test statistic is indicated by the black vertical line and the region used to determine $ p $ is indicated by the light gray shaded region.

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Figure 9:
Example of nuisance parameter uncertainties and impacts calculated in Combine for the observation of four top quark production. Each row gives the name of the nuisance parameter, the difference in its maximum likelihood estimate $ \hat{\nu} $ with respect to its default value $ \nu_{0} $ relative to its uncertainty $ \Delta\nu $, and the impact with respect to the default physics model parameter $ \Delta r $. The nuisance parameter constraints and impacts are calculated using the observed data set (obs.) and an Asimov dataset constructed assuming standard model production of four top quarks (exp.). The red and blue lines in each row represent the positive impact $ \Delta r^{+} $ and negative impact $ \Delta r^{-} $, respectively, for the observed data. Similarly, the red and blue shaded boxes represent the same quantities for the Asimov dataset. The error bars on the fit constraint values indicate the ratio of $ \Delta^{-}\nu $ or $ \Delta^{+}\nu $, to their default values. The two numerical values displayed in the figure give the value of $ \hat{\nu}^{+ \Delta^{+}\nu}_{- \Delta^{-}\nu} $ for two rate parameters, which do not have well-defined default uncertainty values. Figure adapted from Ref. [22].

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Figure 10:
Distributions of the observable $ x $ for the data and background process in Datacard dc:template and their uncertainties. The upper panel shows the distribution for the default values of the nuisance parameters (red solid line, pre-fit) and for the maximum likelihood estimates assuming no signal (blue dashed line, post-fit). The pink hatched and blue shaded bands show the estimate of the uncertainty in each bin for the pre-fit and post-fit distributions, respectively. The middle panel shows the difference between the expected number of events in the background processes ($ \lambda $) and the data ($ n $) in the pre-fit (red solid line) and post-fit (blue dashed line) cases, and the lower panel shows the ratios of the estimated uncertainties of the post-fit distribution $ \Delta\lambda^{\text{Post-fit}} $ to the pre-fit $ \Delta\lambda^{\text{Pre-fit}} $ in each bin.

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Table 1:
Available uncertainty types for counting experiments. The second and third columns indicate the entries for the datacard required to specify the type, and the relative effect on the yield of each process in each channel. The fourth and fifth columns indicate the resulting multiplicative factor by which Combine scales the normalization of the relevant process in the specified channel, and the term $ p(y;\nu) $ that is included in Eq. (1). Finally, the last column indicates the default values of $ \nu $ and $ y $. Where relevant, the value of $ \kappa- $ 1 can be interpreted as the relative uncertainty in the process normalization in a given channel.
After a decade of development, the Combine package has become the main tool used for statistical analysis of data by the CMS Collaboration. The tool is based on the ROOT [1], ROOFIT [2], and RooStats [2] software packages to provide a command-line interface to several common statistical workflows used in high-energy physics. The statistical model is constructed from a text file provided by the user and a configurable physics model that encodes the parameters of interest and the nuisance parameters that model systematic uncertainties. The Combine package can perform a variety of statistical procedures including calculating confidence or credible intervals, evaluating profile likelihoods, and performing goodness of fit tests. The online documentation [12] contains comprehensive information on the capabilities and instructions for running the Combine package, as well as detailed instructions for its installation.
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Compact Muon Solenoid