CMS-CAT-23-001

CMS-CAT-23-001 ; CERN-EP-2024-078
The CMS statistical analysis and combination tool: Combine
CMS Collaboration
10 April 2024
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.
Links: e-print arXiv:2404.06614 [hep-ex] (PDF) ; CDS record ; inSPIRE record ; Physics Briefing ; CADI line (restricted) ;

Figures & Tables	Summary	References	CMS Publications

Link to the Combine manual

Figures
png pdf	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.
png pdf	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.
png pdf	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.
png pdf	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.
png pdf	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.
png pdf	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.
png pdf	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.
png pdf	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.
png pdf	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].
png pdf	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.

Tables

png pdf

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.

Summary

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.

References
1	R. Brun and F. Rademakers	ROOT: An object oriented data analysis framework	NIM A 389 (1997) 81
2	W. Verkerke and D. P. Kirkby	The RooFit toolkit for data modeling	in Proceedings of the 13th International Conference for Computing in High-Energy and Nuclear Physics (CHEP03), 2003 [eConf C0303241, MOLT007]	physics/0306116
3	ATLAS and CMS Collaborations, and LHC Higgs Combination Group	Procedure for the LHC Higgs boson search combination in Summer 2011	Technical Report CMS-NOTE-2011-005, ATL-PHYS-PUB-2011-11, 2011
4	CMS Collaboration	Combined results of searches for the standard model Higgs boson in pp collisions at $ \sqrt{s}= $ 7 TeV	PLB 710 (2012) 26	CMS-HIG-11-032 1202.1488
5	CMS Collaboration	Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC	PLB 716 (2012) 30	CMS-HIG-12-028 1207.7235
6	CMS Collaboration	Observation of a new boson with mass near 125 GeV in pp collisions at $ \sqrt{s} = $ 7 and 8 TeV	JHEP 06 (2013) 081	CMS-HIG-12-036 1303.4571
7	CMS Collaboration	Combined measurements of Higgs boson couplings in proton-proton collisions at $ \sqrt{s}= $ 13 TeV	EPJC 79 (2019) 421	CMS-HIG-17-031 1809.10733
8	CMS Collaboration	Combined searches for the production of supersymmetric top quark partners in proton-proton collisions at $ \sqrt{s} = $ 13 TeV	EPJC 81 (2021) 970	CMS-SUS-20-002 2107.10892
9	CMS Collaboration	Measurement of the top quark pole mass using $ \textrm{t}\overline{\textrm{t}} $+jet events in the dilepton final state in proton-proton collisions at $ \sqrt{s} $ = 13 TeV	JHEP 07 (2023) 077	CMS-TOP-21-008 2207.02270
10	ATLAS and CMS Collaborations	Combined measurement of the Higgs boson mass in $ pp $ collisions at $ \sqrt{s}= $ 7 and 8 TeV with the ATLAS and CMS experiments	PRL 114 (2015) 191803	1503.07589
11	ATLAS and CMS Collaborations	Measurements of the Higgs boson production and decay rates and constraints on its couplings from a combined ATLAS and CMS analysis of the LHC $ pp $ collision data at $ \sqrt{s} $ = 7 and 8 TeV	JHEP 08 (2016) 45	1606.02266
12	CMS Collaboration	Combine	HiggsAnalysis-CombinedLimit package, 2004 github
13	D. Piparo, V. Innocente, and T. Hauth	Speeding up HEP experiment software with a library of fast and auto-vectorisable mathematical functions	J. Phys. Conf. Ser 513 (2014) 052027
14	M. Galassi et al.	GNU Scientific Library Reference Manual	3rd edition, ISBN~978-0-9546120-7-8, 2009
15	B. Jacob, G. Guennebaud	Eigen	2024 link
16	The Boost C++ community	Boost C++ libraries	link
17	Free Software Foundation	GNU Gzip	link
18	D. Merkel	Docker: Lightweight Linux containers for consistent development and deployment	Linux J. 2014 (2014)
19	F. E. James	Statistical methods in experimental physics	World Scientific, Singapore, 2nd edition, ISBN~978-981-270-527-3, 2006 link
20	Particle Data Group , R. L. Workman et al.	Review of particle physics	Prog. Theor. Exp. Phys. 2022 (2022) 083C01
21	CMS Collaboration	Observation of Higgs boson decay to bottom quarks	PRL 121 (2018) 121801	CMS-HIG-18-016 1808.08242
22	CMS Collaboration	Observation of four top quark production in proton-proton collisions at $ \sqrt{s}= $ 13 TeV	PLB 847 (2023) 138290	CMS-TOP-22-013 2305.13439
23	CMS Collaboration	Search for the standard model Higgs boson decaying to $ \mathrm{W}^{+}\mathrm{W}^{-} $ in the fully leptonic final state in pp collisions at $ \sqrt{s}= $ 7 TeV	PLB 710 (2012) 91	CMS-HIG-11-024 1202.1489
24	K. Cranmer	Practical statistics for the LHC	in Proc. 2011 European School of High-Energy Physics, 2014 link	1503.07622
25	R. Barlow and C. Beeston	Fitting using finite Monte Carlo samples	Comput. Phys. Commun. 77 (1993) 219
26	J. S. Conway	Incorporating nuisance parameters in likelihoods for multisource spectra	in PHYSTAT : Workshop on statistical issues related to discovery claims in search experiments and unfolding, 2011 link	1103.0354
27	CMS Collaboration	Measurements of production cross sections of the Higgs boson in the four-lepton final state in proton-proton collisions at $ \sqrt{s} = $ 13 TeV	EPJC 81 (2021) 488	CMS-HIG-19-001 2103.04956
28	CMS Collaboration	Evidence for Higgs boson decay to a pair of muons	JHEP 01 (2021) 148	CMS-HIG-19-006 2009.04363
29	R. Barlow	Asymmetric errors	in PHYSTAT : Statistical problems in particle physics, astrophysics, and cosmology, [eConf C030908, WEMT002], 2003	physics/0401042
30	CDF Collaboration	A measurement of $ \sigma B (\mathrm{W} \to \mathrm{e} \nu) $ and $ \sigma B (\mathrm{Z^0} \to \mathrm{e}^+ \mathrm{e}^-) $ in $ \overline{\mathrm{p}}\mathrm{p} $ collisions at $ \sqrt{s} = $ 1800 GeV	PRD 44 (1991) 29
31	CMS Collaboration	Measurement of the inclusive and differential Higgs boson production cross sections in the decay mode to a pair of $ \tau $ leptons in pp collisions at $ \sqrt{s} = $ 13 TeV	PRL 128 (2022) 081805	CMS-HIG-20-015 2107.11486
32	P. D. Dauncey, M. Kenzie, N. Wardle, and G. J. Davies	Handling uncertainties in background shapes: the discrete profiling method	JINST 10 (2015) P04015	1408.6865
33	F. James and M. Roos	Minuit: A system for function minimization and analysis of the parameter errors and correlations	Comput. Phys. Commun. 10 (1975) 343
34	G. Cowan, K. Cranmer, E. Gross, and O. Vitells	Asymptotic formulae for likelihood-based tests of new physics	EPJC 71 (2011) 1554	1007.1727
35	C. Langenbruch	Parameter uncertainties in weighted unbinned maximum likelihood fits	EPJC 82 (2022) 393	1911.01303
36	B. Efron	Bootstrap methods: Another look at the jackknife	See ``Remark K'', 1979 Ann. Statist. 7 (1979) 1
37	S. M. S. Lee and G. A. Young	Parametric bootstrapping with nuisance parameters	Stat. Probab. Lett. 71 (2005) 143
38	G. J. Feldman and R. D. Cousins	Unified approach to the classical statistical analysis of small signals	PRD 57 (1998) 3873	physics/9711021
39	LEP Working Group for Higgs boson searches, ALEPH, DELPHI, L3 and OPAL Collaborations	Search for the standard model Higgs boson at LEP	PLB 565 (2003) 61	hep-ex/0306033
40	CMS Collaboration	Constraints on the spin-parity and anomalous HVV couplings of the Higgs boson in proton collisions at 7 and 8 TeV	PRD 92 (2015) 012004	CMS-HIG-14-018 1411.3441
41	J. O. Berger, B. Liseo, and R. L. Wolpert	Integrated likelihood methods for eliminating nuisance parameters	Stat. Scien. 14 (1999) 1
42	R. D. Cousins and V. L. Highland	Incorporating systematic uncertainties into an upper limit	NIM A 320 (1992) 331
43	CMS Collaboration	Search for the lepton flavor violating $ \tau \to $ 3$ \mu $ decay in proton-proton collisions at $ \sqrt{s} $ = 13 TeV	Submitted to Phys. Lett. B, 2023	CMS-BPH-21-005 2312.02371
44	T. Junk	Confidence level computation for combining searches with small statistics	NIM A 434 (1999) 435	hep-ex/9902006
45	A. L. Read	Presentation of search results: The CL$ _{\text{s}} $ technique	JPG 28 (2002) 2693
46	CMS Collaboration	Measurement of the properties of a Higgs boson in the four-lepton final state	PRD 89 (2014) 092007	CMS-HIG-13-002 1312.5353
47	H. Jeffreys	Theory of Probability	Oxford University Press, Oxford, 3rd edition, ISBN~978003682, 1961
48	L. Moneta et al.	The RooStats project	in 13th International Workshop on Advanced Computing and Analysis Techniques in Physics Research (ACAT). SISSA, 2010 link	1009.1003
49	CMS Collaboration	A portrait of the Higgs boson by the CMS experiment ten years after the discovery	Nature 607 (2022) 60	CMS-HIG-22-001 2207.00043
50	S. S. Wilks	The large-sample distribution of the likelihood ratio for testing composite hypotheses	Ann. Math. Statist. 9 (1938) 60
51	A. Wald	Tests of statistical hypotheses concerning several parameters when the number of observations is large	Trans. Amer. Math. Soc. 54 (1943) 426
52	R. F. Engle	Chapter 13: Wald, likelihood ratio, and Lagrange multiplier tests in econometrics	in Handbook of Econometrics, volume 2, Elsevier, 1984 link
53	R. D. Cousins	Lectures on statistics in theory: Prelude to statistics in practice		1807.05996
54	A. N. Kolmogorov	Sulla determinazione empirica di una legge di distribuzione	Giorn. Ist. Ital. Attuari 4 (1933) 83
55	N. Smirnov	Table for estimating the goodness of fit of empirical distributions	Ann. Math. Statist. 19 (1948) 279
56	T. W. Anderson and D. A. Darling	Asymptotic theory of certain `goodness of fit' criteria based on stochastic processes	Ann. Math. Statist. 23 (1952) 193
57	M. A. Stephens	EDF statistics for goodness of fit and some comparisons	J. Am. Stat. Assoc. 69 (1974) 730
58	ATLAS Collaboration	Search for the $ b\bar{b} $ decay of the standard model Higgs boson in associated $ (W/Z)H $ production with the ATLAS detector	JHEP 01 (2015) 069	1409.6212