D-Optimal Design for Network A/B Testing
A/B testing refers to the statistical procedure of experimental design and analysis to compare two treatments, A and B, applied to different testing subjects. It is widely used by technology companies such as Facebook, LinkedIn, and Netflix, to compare different algorithms, web-designs, and other on...
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| Vydané v: | Journal of statistical theory and practice Ročník 13; číslo 4 |
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01.12.2019
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| Abstract | A/B testing refers to the statistical procedure of experimental design and analysis to compare two treatments, A and B, applied to different testing subjects. It is widely used by technology companies such as Facebook, LinkedIn, and Netflix, to compare different algorithms, web-designs, and other online products and services. The subjects participating in these online A/B testing experiments are users who are connected in different scales of social networks. Two connected subjects are similar in terms of their social behaviors, education and financial background, and other demographic aspects. Hence, it is only natural to assume that their reactions to online products and services are related to their network adjacency. In this paper, we propose to use the conditional auto-regressive model to present the network structure and include the network effects in the estimation and inference of the treatment effect. A D-optimal design criterion is developed based on the proposed model. Mixed integer programming formulations are developed to obtain the D-optimal designs. The effectiveness of the proposed method is shown through numerical results with synthetic networks and real social networks. |
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| AbstractList | A/B testing refers to the statistical procedure of experimental design and analysis to compare two treatments, A and B, applied to different testing subjects. It is widely used by technology companies such as Facebook, LinkedIn, and Netflix, to compare different algorithms, web-designs, and other online products and services. The subjects participating in these online A/B testing experiments are users who are connected in different scales of social networks. Two connected subjects are similar in terms of their social behaviors, education and financial background, and other demographic aspects. Hence, it is only natural to assume that their reactions to online products and services are related to their network adjacency. In this paper, we propose to use the conditional auto-regressive model to present the network structure and include the network effects in the estimation and inference of the treatment effect. A D-optimal design criterion is developed based on the proposed model. Mixed integer programming formulations are developed to obtain the D-optimal designs. The effectiveness of the proposed method is shown through numerical results with synthetic networks and real social networks. |
| ArticleNumber | 61 |
| Author | Kang, Lulu Zhang, Qiong Mays, D’arcy P. Pokhilko, Victoria |
| Author_xml | – sequence: 1 givenname: Victoria surname: Pokhilko fullname: Pokhilko, Victoria organization: Department of Statistical Sciences and Operations Research, Virginia Commonwealth University – sequence: 2 givenname: Qiong surname: Zhang fullname: Zhang, Qiong organization: School of Mathematical and Statistical Sciences, Clemson University – sequence: 3 givenname: Lulu orcidid: 0000-0002-6000-3436 surname: Kang fullname: Kang, Lulu email: lkang2@iit.edu organization: Department of Applied Mathematics, Illinois Institute of Technology – sequence: 4 givenname: D’arcy P. surname: Mays fullname: Mays, D’arcy P. organization: Department of Statistical Sciences and Operations Research, Virginia Commonwealth University |
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| Keywords | D-optimal design Conditional auto-regressive model Mixed integer programming Social network A/B testing |
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| References | Atkinson AC, Woods DC (2015) Designs for generalized linear models, Chapter 13. In: Handbook of design and analysis of experiments. Chapman & Hall/CRC, Boca Raton, FL, pp 471–514 BasseGWAiroldiEMModel-assisted design of experiments in the presence of network-correlated outcomesBiometrika2018105849858387786910.1093/biomet/asy03606994539 NemhauserGLSavelsberghMWPSigismondiGSConstraint classification for mixed integer programming formulationsCOAL Bull199220812 AtkinsonADonevATobiasROptimum experimental designs, with SAS2007OxfordOxford University Press1183.62129 WuCJHamadaMSExperiments: planning, analysis, and optimization2011HobokenWiley1229.62100 BhatNFariasVFMoallemiCCSinhaDNear optimal AB testing2017New YorkColumbia Business School Eckles D, Karrer B, Ugander J (2017) Design and analysis of experiments in networks: reducing bias from interference. J Causal Infer. https://doi.org/10.1515/jci-2015-0021 KieferJWolfowitzJOptimum designs in regression problemsAnn Math Stat195930227129410432410.1214/aoms/11777062520090.11404 BertsimasDJohnsonMKallusNThe power of optimization over randomization in designing experiments involving small samplesOper Res201563868876337868110.1287/opre.2015.13611329.90078 Schmidt AM, Nobre WS (2014) Conditional autoregressive (CAR) model. Statistics reference online, Wiley StatsRef, pp 1–11 ChenYQiYLiuQChienPSequential sampling enhanced composite likelihood approach to estimation of social intercorrelations in large-scale networksQuant Market Econ20181640944010.1007/s11129-018-9199-z FedorovVOptimal experimental designWiley Interdiscip Rev Comput Stat2010258158910.1002/wics.100 HoreSDewanjiAChatterjeeADesign issues related to allocation of experimental units with known covariates into two treatment groupsJ Stat Plan Inference2014155117126326445910.1016/j.jspi.2014.06.0021307.62220 Ogburn EL, Sofrygin O, Diaz I, van der Laan MJ (2017) Causal inference for social network data. ArXiv preprint arXiv:1705.08527 Wolsey LA, Nemhauser GL (2014) Integer and combinatorial optimization. John Wiley & Sons RubinDBEstimating causal effects of treatments in randomized and nonrandomized studiesJ Educ Psychol19746668810.1037/h0037350 Leskovec J, Mcauley JJ (2012) Learning to discover social circles in ego networks. In: Advances in neural information processing systems, pp 539–547 BasseGWAiroldiEMLimitations of design-based causal inference and A/B testing under arbitrary and network interferenceSociol Methodol20184813615110.1177/0081175018782569 Nandy P, Basu K, Chatterjee S, Tu Y (2019) A/B testing in dense large-scale networks: design and inference. ArXiv preprint arXiv:1901.10505 Gui H, Xu Y, Bhasin A, Han J (2015) Network a/b testing: from sampling to estimation. In: Proceedings of the 24th international conference on world wide web, international world wide web conferences steering committee, pp 399–409 Xu Y, Chen N, Fernandez A, Sinno O, Bhasin A (2015) From infrastructure to culture: A/b testing challenges in large scale social networks. In: Proceedings of the 21th ACM SIGKDD international conference on knowledge discovery and data mining. ACM, pp 2227–2236 Saveski M, Pouget-Abadie J, Saint-Jacques G, Duan W, Ghosh S, Xu Y, Airoldi EM (2017) Detecting network effects: randomizing over randomized experiments. In: Proceedings of the 23rd ACM SIGKDD international conference on knowledge discovery and data mining. ACM, pp 1027–1035 AtwoodCLOptimal and efficient designs of experimentsAnn Math Stat1969401570160226011910.1214/aoms/11776973740182.51905 Pouget-Abadie J, Saveski M, Saint-Jacques G, Duan W, Xu Y, Ghosh S, Airoldi EM (2017) Testing for arbitrary interference on experimentation platforms. ArXiv preprint arXiv:1704.01190 BrookDOn the distinction between the conditional probability and the joint probability approaches in the specification of nearest-neighbour systemsBiometrika19645148148320531510.1093/biomet/51.3-4.4810129.10703 WoodsDDesigning experiments under random contamination with application to polynomial spline regressionStat Sin20051561922339021086.62080 YangMBiedermannSTangEOn optimal designs for nonlinear models: a general and efficient algorithmJ Am Stat Assoc201310814111420317471710.1080/01621459.2013.8062681283.62161 YatesFSir Ronald fisher and the design of experimentsBiometrics19642030732110.2307/2528399 Basse GW, Airoldi EM (2015) Optimal model-assisted design of experiments for network correlated outcomes suggests new notions of network balance. ArXiv preprint arXiv:1507.00803 BesagJSpatial interaction and the statistical analysis of lattice systemsJ R Stat Soc Ser B (Methodol)19743621922253732080327.60067 DraperNSmithHApplied regression analysis1966New YorkWiley108116 MorganKLRubinDBRerandomization to improve covariate balance in experimentsAnn Stat20124012631282298595010.1214/12-AOS10081274.62509 PukelsheimFOptimal design of experiments1993New DelhiSIAM0834.62068 Bivand R, Bernat A, Carvalho M, Chun Y, Dormann C, Dray S, Halbersma R, Lewin-Koh N, Ma J, Millo G et al (2005) The spdep package. Comprehensive R Archive Network, Version 05–83 KieferJOptimum experimental designsJ R Stat Soc Ser B (Methodol)1959212723041132630108.15303 WallMMA close look at the spatial structure implied by the CAR and SAR modelsJ Stat Plan inference2004121311324203882410.1016/S0378-3758(03)00111-31036.62097 J Kiefer (58_CR19) 1959; 30 KL Morgan (58_CR21) 2012; 40 D Brook (58_CR11) 1964; 51 58_CR29 J Kiefer (58_CR18) 1959; 21 GW Basse (58_CR6) 2018; 105 J Besag (58_CR8) 1974; 36 S Hore (58_CR17) 2014; 155 58_CR23 DB Rubin (58_CR28) 1974; 66 58_CR25 58_CR26 58_CR20 CL Atwood (58_CR3) 1969; 40 58_CR22 GW Basse (58_CR5) 2018; 48 58_CR16 MM Wall (58_CR31) 2004; 121 V Fedorov (58_CR15) 2010; 2 A Atkinson (58_CR1) 2007 Y Chen (58_CR12) 2018; 16 GL Nemhauser (58_CR24) 1992; 20 D Woods (58_CR32) 2005; 15 58_CR2 N Bhat (58_CR9) 2017 N Draper (58_CR13) 1966 M Yang (58_CR35) 2013; 108 58_CR34 58_CR14 CJ Wu (58_CR33) 2011 58_CR4 58_CR30 58_CR10 D Bertsimas (58_CR7) 2015; 63 F Pukelsheim (58_CR27) 1993 F Yates (58_CR36) 1964; 20 |
| References_xml | – reference: WoodsDDesigning experiments under random contamination with application to polynomial spline regressionStat Sin20051561922339021086.62080 – reference: YatesFSir Ronald fisher and the design of experimentsBiometrics19642030732110.2307/2528399 – reference: BrookDOn the distinction between the conditional probability and the joint probability approaches in the specification of nearest-neighbour systemsBiometrika19645148148320531510.1093/biomet/51.3-4.4810129.10703 – reference: Schmidt AM, Nobre WS (2014) Conditional autoregressive (CAR) model. Statistics reference online, Wiley StatsRef, pp 1–11 – reference: BesagJSpatial interaction and the statistical analysis of lattice systemsJ R Stat Soc Ser B (Methodol)19743621922253732080327.60067 – reference: Pouget-Abadie J, Saveski M, Saint-Jacques G, Duan W, Xu Y, Ghosh S, Airoldi EM (2017) Testing for arbitrary interference on experimentation platforms. ArXiv preprint arXiv:1704.01190 – reference: AtwoodCLOptimal and efficient designs of experimentsAnn Math Stat1969401570160226011910.1214/aoms/11776973740182.51905 – reference: YangMBiedermannSTangEOn optimal designs for nonlinear models: a general and efficient algorithmJ Am Stat Assoc201310814111420317471710.1080/01621459.2013.8062681283.62161 – reference: BasseGWAiroldiEMModel-assisted design of experiments in the presence of network-correlated outcomesBiometrika2018105849858387786910.1093/biomet/asy03606994539 – reference: NemhauserGLSavelsberghMWPSigismondiGSConstraint classification for mixed integer programming formulationsCOAL Bull199220812 – reference: Basse GW, Airoldi EM (2015) Optimal model-assisted design of experiments for network correlated outcomes suggests new notions of network balance. ArXiv preprint arXiv:1507.00803 – reference: Wolsey LA, Nemhauser GL (2014) Integer and combinatorial optimization. John Wiley & Sons – reference: DraperNSmithHApplied regression analysis1966New YorkWiley108116 – reference: Leskovec J, Mcauley JJ (2012) Learning to discover social circles in ego networks. In: Advances in neural information processing systems, pp 539–547 – reference: RubinDBEstimating causal effects of treatments in randomized and nonrandomized studiesJ Educ Psychol19746668810.1037/h0037350 – reference: FedorovVOptimal experimental designWiley Interdiscip Rev Comput Stat2010258158910.1002/wics.100 – reference: Bivand R, Bernat A, Carvalho M, Chun Y, Dormann C, Dray S, Halbersma R, Lewin-Koh N, Ma J, Millo G et al (2005) The spdep package. Comprehensive R Archive Network, Version 05–83 – reference: KieferJWolfowitzJOptimum designs in regression problemsAnn Math Stat195930227129410432410.1214/aoms/11777062520090.11404 – reference: Eckles D, Karrer B, Ugander J (2017) Design and analysis of experiments in networks: reducing bias from interference. J Causal Infer. https://doi.org/10.1515/jci-2015-0021 – reference: WallMMA close look at the spatial structure implied by the CAR and SAR modelsJ Stat Plan inference2004121311324203882410.1016/S0378-3758(03)00111-31036.62097 – reference: WuCJHamadaMSExperiments: planning, analysis, and optimization2011HobokenWiley1229.62100 – reference: Xu Y, Chen N, Fernandez A, Sinno O, Bhasin A (2015) From infrastructure to culture: A/b testing challenges in large scale social networks. In: Proceedings of the 21th ACM SIGKDD international conference on knowledge discovery and data mining. ACM, pp 2227–2236 – reference: PukelsheimFOptimal design of experiments1993New DelhiSIAM0834.62068 – reference: Ogburn EL, Sofrygin O, Diaz I, van der Laan MJ (2017) Causal inference for social network data. ArXiv preprint arXiv:1705.08527 – reference: Gui H, Xu Y, Bhasin A, Han J (2015) Network a/b testing: from sampling to estimation. In: Proceedings of the 24th international conference on world wide web, international world wide web conferences steering committee, pp 399–409 – reference: AtkinsonADonevATobiasROptimum experimental designs, with SAS2007OxfordOxford University Press1183.62129 – reference: BhatNFariasVFMoallemiCCSinhaDNear optimal AB testing2017New YorkColumbia Business School – reference: Saveski M, Pouget-Abadie J, Saint-Jacques G, Duan W, Ghosh S, Xu Y, Airoldi EM (2017) Detecting network effects: randomizing over randomized experiments. In: Proceedings of the 23rd ACM SIGKDD international conference on knowledge discovery and data mining. ACM, pp 1027–1035 – reference: Atkinson AC, Woods DC (2015) Designs for generalized linear models, Chapter 13. In: Handbook of design and analysis of experiments. Chapman & Hall/CRC, Boca Raton, FL, pp 471–514 – reference: BasseGWAiroldiEMLimitations of design-based causal inference and A/B testing under arbitrary and network interferenceSociol Methodol20184813615110.1177/0081175018782569 – reference: HoreSDewanjiAChatterjeeADesign issues related to allocation of experimental units with known covariates into two treatment groupsJ Stat Plan Inference2014155117126326445910.1016/j.jspi.2014.06.0021307.62220 – reference: ChenYQiYLiuQChienPSequential sampling enhanced composite likelihood approach to estimation of social intercorrelations in large-scale networksQuant Market Econ20181640944010.1007/s11129-018-9199-z – reference: BertsimasDJohnsonMKallusNThe power of optimization over randomization in designing experiments involving small samplesOper Res201563868876337868110.1287/opre.2015.13611329.90078 – reference: Nandy P, Basu K, Chatterjee S, Tu Y (2019) A/B testing in dense large-scale networks: design and inference. ArXiv preprint arXiv:1901.10505 – reference: MorganKLRubinDBRerandomization to improve covariate balance in experimentsAnn Stat20124012631282298595010.1214/12-AOS10081274.62509 – reference: KieferJOptimum experimental designsJ R Stat Soc Ser B (Methodol)1959212723041132630108.15303 – ident: 58_CR4 – ident: 58_CR34 doi: 10.1145/2783258.2788602 – ident: 58_CR2 – volume-title: Optimal design of experiments year: 1993 ident: 58_CR27 – ident: 58_CR16 doi: 10.1145/2736277.2741081 – volume: 30 start-page: 271 issue: 2 year: 1959 ident: 58_CR19 publication-title: Ann Math Stat doi: 10.1214/aoms/1177706252 – volume: 108 start-page: 1411 year: 2013 ident: 58_CR35 publication-title: J Am Stat Assoc doi: 10.1080/01621459.2013.806268 – volume-title: Optimum experimental designs, with SAS year: 2007 ident: 58_CR1 doi: 10.1093/oso/9780199296590.001.0001 – ident: 58_CR25 – ident: 58_CR10 – volume: 15 start-page: 619 year: 2005 ident: 58_CR32 publication-title: Stat Sin – volume: 66 start-page: 688 year: 1974 ident: 58_CR28 publication-title: J Educ Psychol doi: 10.1037/h0037350 – ident: 58_CR14 doi: 10.1515/jci-2015-0021 – volume: 40 start-page: 1570 year: 1969 ident: 58_CR3 publication-title: Ann Math Stat doi: 10.1214/aoms/1177697374 – volume-title: Experiments: planning, analysis, and optimization year: 2011 ident: 58_CR33 – volume: 20 start-page: 307 year: 1964 ident: 58_CR36 publication-title: Biometrics doi: 10.2307/2528399 – volume: 63 start-page: 868 year: 2015 ident: 58_CR7 publication-title: Oper Res doi: 10.1287/opre.2015.1361 – ident: 58_CR22 – start-page: 108 volume-title: Applied regression analysis year: 1966 ident: 58_CR13 – volume: 40 start-page: 1263 year: 2012 ident: 58_CR21 publication-title: Ann Stat doi: 10.1214/12-AOS1008 – volume: 2 start-page: 581 year: 2010 ident: 58_CR15 publication-title: Wiley Interdiscip Rev Comput Stat doi: 10.1002/wics.100 – ident: 58_CR23 doi: 10.1002/9781118627372.ch1 – volume: 155 start-page: 117 year: 2014 ident: 58_CR17 publication-title: J Stat Plan Inference doi: 10.1016/j.jspi.2014.06.002 – volume: 20 start-page: 8 year: 1992 ident: 58_CR24 publication-title: COAL Bull – ident: 58_CR29 doi: 10.1145/3097983.3098192 – ident: 58_CR20 – ident: 58_CR26 – volume: 51 start-page: 481 year: 1964 ident: 58_CR11 publication-title: Biometrika doi: 10.1093/biomet/51.3-4.481 – volume: 105 start-page: 849 year: 2018 ident: 58_CR6 publication-title: Biometrika doi: 10.1093/biomet/asy036 – volume: 36 start-page: 192 issue: 2 year: 1974 ident: 58_CR8 publication-title: J R Stat Soc Ser B (Methodol) doi: 10.1111/j.2517-6161.1974.tb00999.x – volume: 21 start-page: 272 year: 1959 ident: 58_CR18 publication-title: J R Stat Soc Ser B (Methodol) doi: 10.1111/j.2517-6161.1959.tb00338.x – volume: 48 start-page: 136 year: 2018 ident: 58_CR5 publication-title: Sociol Methodol doi: 10.1177/0081175018782569 – ident: 58_CR30 – volume-title: Near optimal AB testing year: 2017 ident: 58_CR9 – volume: 121 start-page: 311 year: 2004 ident: 58_CR31 publication-title: J Stat Plan inference doi: 10.1016/S0378-3758(03)00111-3 – volume: 16 start-page: 409 year: 2018 ident: 58_CR12 publication-title: Quant Market Econ doi: 10.1007/s11129-018-9199-z |
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