Research

Research Interests

Check out my current research statement to get a feel for what I spend most of my time on.

Preprints

Fritz, C., Mehrl, M., Thurner, P. W., & Kauermann, G. (2022). Exponential Random Graph Models for Dynamic Signed Networks: An Application to International Relations. ArXiv.
```
@article{fritzmehrl_signed_2022,
  author = {Fritz, Cornelius and Mehrl, Marius and Thurner, Paul W. and Kauermann, Göran},
  title = {Exponential Random Graph Models for Dynamic Signed Networks: An Application to International Relations},
  keywords = {preprint},
  primaryclass = {cs.SI},
  journal = {arXiv},
  eprint = {2205.13411},
  year = {2022}
}
```
Substantive research in the Social Sciences regularly investigates signed networks, where edges between actors are either positive or negative. For instance, schoolchildren can be friends or rivals, just as countries can cooperate or fight each other. This research often builds on structural balance theory, one of the earliest and most prominent network theories, making signed networks one of the most frequently studied matters in social network analysis. While the theorization and description of signed networks have thus made significant progress, the inferential study of tie formation within them remains limited in the absence of appropriate statistical models. In this paper we fill this gap by proposing the Signed Exponential Random Graph Model (SERGM), extending the well-known Exponential Random Graph Model (ERGM) to networks where ties are not binary but negative or positive if a tie exists. Since most networks are dynamically evolving systems, we specify the model for both cross-sectional and dynamic networks. Based on structural hypotheses derived from structural balance theory, we formulate interpretable signed network statistics, capturing dynamics such as "the enemy of my enemy is my friend". In our empirical application, we use the SERGM to analyze cooperation and conflict between countries within the international state system.

Publications

2023

De Nicola, G., Fritz, C., Mehrl, M., & Kauermann, G. (2023). Dependence matters: Statistical models to identify the drivers of tie formation in economic networks. Journal of Economic Behavior & Organization, 215, 351–363. https://doi.org/https://doi.org/10.1016/j.jebo.2023.09.021
```
@article{de_nicola_statistical_2022,
  title = {Dependence matters: Statistical models to identify the drivers of tie formation in economic networks},
  author = {De Nicola, Giacomo and Fritz, Cornelius and Mehrl, Marius and Kauermann, G{\"o}ran},
  year = {2023},
  volume = {215},
  pages = {351-363},
  journal = {Journal of Economic Behavior \& Organization},
  doi = {https://doi.org/10.1016/j.jebo.2023.09.021},
  keywords = {publications}
}
```
Networks are ubiquitous in economic research on organizations, trade, and many other areas. However, while economic theory extensively considers networks, no general framework for their empirical modeling has yet emerged. We thus introduce two different statistical models for this purpose – the Exponential Random Graph Model (ERGM) and the Additive and Multiplicative Effects network model (AME). Both model classes can account for network interdependencies between observations, but differ in how they do so. The ERGM allows one to explicitly specify and test the influence of particular network structures, making it a natural choice if one is substantively interested in estimating endogenous network effects. In contrast, AME captures these effects by introducing actor-specific latent variables affecting their propensity to form ties. This makes the latter a good choice if the researcher is interested in capturing the effect of exogenous covariates on tie formation without having a specific theory on the endogenous dependence structures at play. After introducing the two model classes, we showcase them through real-world applications to networks stemming from international arms trade and foreign exchange activity. We further provide full replication materials to facilitate the adoption of these methods in empirical economic research.

Schweinberger, M., & Fritz, C. (2023). Discussion of “A tale of two datasets: Representativeness and generalisability of inference for samples of networks” by Pavel N. Krivitsky, Pietro Coletti, and Niel Hens. Journal of the American Statistical Association, (OnlineFirst), 1–5. https://doi.org/10.1080/01621459.2023.2223680

@article{fritz_discussion,
  title = {Discussion of “{A} tale of two datasets: {Representativeness} and generalisability of inference for samples of networks” by {P}avel {N}. {Krivitsky}, {P}ietro {C}oletti, and {N}iel {H}ens},
  volume = {(OnlineFirst)},
  doi = {10.1080/01621459.2023.2223680},
  journal = {Journal of the American Statistical Association},
  author = {Schweinberger, Michael and Fritz, Cornelius},
  year = {2023},
  keywords = {publications},
  pages = {1--5}
}

Fritz, C., De Nicola, G., Kevorg, S., Harhoff, D., & Kauermann, G. (2023). Modelling the large and dynamically growing bipartite network of German patents and inventors. Journal of the Royal Statistical Society. Series A (Statistics in Society), 186(3), 557–576. https://doi.org/10.1093/jrsssa/qnad009
```
@article{fritz2023,
  author = {Fritz, Cornelius and De Nicola, Giacomo and Kevorg, Sevag and Harhoff, Dietmar and Kauermann, G{\"{o}}ran},
  title = {Modelling the large and dynamically growing bipartite network of German patents and inventors},
  volume = {186},
  number = {3},
  pages = {557–576},
  doi = {10.1093/jrsssa/qnad009},
  journal = {Journal of the Royal Statistical Society. Series A (Statistics in Society)},
  keywords = {publications},
  year = {2023}
}
```
To explore the driving forces behind innovation, we analyse the dynamic bipartite network of all inventors and patents registered within the field of electrical engineering in Germany in the past two decades. To deal with the sheer size of the data, we decompose the network by exploiting the fact that most inventors tend to only stay active for a relatively short period. We thus propose a Temporal Exponential Random Graph Model with time-varying actor set and sufficient statistics mirroring substantial expectations for our analysis. Our results corroborate that inventor characteristics and team formation are essential to the dynamics of invention.
Rügamer, D., Kolb, C., Fritz, C., Pfisterer, F., Bischl, B., Shen, R., Bukas, C., de Andrade e Sousa, L. B., Thalmeier, D., Baumann, P., Klein, N., & Müller, C. L. (2023). deepregression: a Flexible Neural Network Framework for Semi-Structured Deep Distributional Regression. Journal of Statistical Software, 105(2), 1–31. https://doi.org/10.18637/jss.v105.i02
```
@article{ruegamer2023,
  title = {deepregression: a Flexible Neural Network Framework for Semi-Structured Deep Distributional Regression},
  author = {Rügamer, David and Kolb, Chris and Fritz, Cornelius and Pfisterer, Florian and Bischl, Bernd and Shen, Ruolin and Bukas, Christina and de Andrade e Sousa, Lisa Barros and Thalmeier, Dominik and Baumann, Philipp and Klein, Nadja and Müller, Christian L.},
  year = {2023},
  journal = {Journal of Statistical Software},
  volume = {105},
  number = {2},
  pages = {1–31},
  doi = {10.18637/jss.v105.i02 },
  keywords = {publications}
}
```
In this paper we describe the implementation of semi-structured deep distributional regression, a flexible framework to learn conditional distributions based on the combination of additive regression models and deep networks. Our implementation encompasses (1) a modular neural network building system based on the deep learning library TensorFlow for the fusion of various statistical and deep learning approaches, (2) an orthogonalization cell to allow for an interpretable combination of different subnetworks, as well as (3) pre-processing steps necessary to set up such models. The software package allows to define models in a user-friendly manner via a formula interface that is inspired by classical statistical model frameworks such as mgcv. The package’s modular design and functionality provides a unique resource for both scalable estimation of complex statistical models and the combination of approaches from deep learning and statistics. This allows for state-of-the-art predictive performance while simultaneously retaining the indispensable interpretability of classical statistical models.
Fritz, C., Mehrl, M., Thurner, P. W., & Kauermann, G. (2023). All that Glitters is not Gold: Relational Events Models with Spurious Events. Network Science, 11(SI 2). https://doi.org/https://doi.org/10.48550/arXiv.2109.10348
```
@article{fritzmehrl2023,
  title = {All that Glitters is not Gold: Relational Events Models with Spurious Events},
  author = {Fritz, Cornelius and Mehrl, Marius and Thurner, Paul W. and Kauermann, Göran},
  year = {2023},
  journal = {Network Science},
  doi = {https://doi.org/10.48550/arXiv.2109.10348},
  volume = {11},
  number = {SI 2},
  keywords = {publications}
}
```
As relational event models are an increasingly popular model for studying relational structures, the reliability of large-scale event data collection becomes more and more important. Automated or human-coded events often suffer from non-negligible false-discovery rates in event identification. And most sensor data is primarily based on actors’ spatial proximity for predefined time windows; hence, the observed events could relate either to a social relationship or random co-location. Both examples imply spurious events that may bias estimates and inference. We propose the Relational Event Model for Spurious Events (REMSE), an extension to existing approaches for interaction data. The model provides a flexible solution for modeling data while controlling for spurious events. Estimation of our model is carried out in an empirical Bayesian approach via data augmentation. Based on a simulation study, we investigate the properties of the estimation procedure. To demonstrate its usefulness in two distinct applications, we employ this model to combat events from the Syrian civil war and student co-location data. Results from the simulation and the applications identify the REMSE as a suitable approach to modeling relational event data in the presence of spurious events.

2022

Fritz, C., De Nicola, G., Rave, M., Weigert, M., Berger, U., Küchenhoff, H., & Kauermann, G. (2022). Statistical modelling of COVID-19 data: Putting Generalised Additive Models to work. Statistical Modelling, (OnlineFirst). https://doi.org/10.1177/1471082X221124628
```
@article{fritz2022,
  author = {Fritz, Cornelius and De Nicola, Giacomo and Rave, Martje and Weigert, Maximilian and Berger, Ursula and Küchenhoff, Helmut and Kauermann, G{\"{o}}ran},
  title = {Statistical modelling of COVID-19 data: Putting Generalised Additive Models to work},
  keywords = {publications},
  year = {2022},
  journal = {Statistical Modelling},
  volume = {(OnlineFirst)},
  doi = {10.1177/1471082X221124628}
}
```
Over the course of the COVID-19 pandemic, Generalized Additive Models (GAMs) have been successfully employed on numerous occasions to obtain vital data-driven insights. In this article we further substantiate the success story of GAMs, demonstrating their flexibility by focusing on three relevant pandemic-related issues. First, we examine the interdepency among infections in different age groups, concentrating on school children. In this context, we derive the setting under which parameter estimates are independent of the (unknown) case-detection ratio, which plays an important role in COVID-19 surveillance data. Second, we model the incidence of hospitalizations, for which data is only available with a temporal delay. We illustrate how correcting for this reporting delay through a nowcasting procedure can be naturally incorporated into the GAM framework as an offset term. Third, we propose a multinomial model for the weekly occupancy of intensive care units (ICU), where we distinguish between the number of COVID-19 patients, other patients and vacant beds. With these three examples, we aim to showcase the practical and ‘off-the-shelf’ applicability of GAMs to gain new insights from real-world data.

Fritz, C., De Nicola, G., Günther, F., Rügamer, D., Rave, M., Schneble, M., Bender, A., Weigert, M., Brinks, R., Hoyer, A., Berger, U., Küchenhoff, H., & Kauermann, G. (2022). Challenges in Interpreting Epidemiological Surveillance Data - Experiences from Germany. Journal of Computational and Graphical Statistics, to appear.

@article{fritzetal2022,
  title = {Challenges in Interpreting Epidemiological Surveillance Data - Experiences from Germany},
  author = {Fritz, Cornelius and De Nicola, G. and Günther, F. and Rügamer, D. and Rave, M. and Schneble, M. and Bender, A. and Weigert, M. and Brinks, R. and Hoyer, A. and Berger, U. and Küchenhoff, H. and Kauermann, G.},
  year = {2022},
  volume = {to appear},
  journal = {Journal of Computational and Graphical Statistics},
  keywords = {publications}
}

Fritz, C., Dorigatti, E., & Rügamer, D. (2022). Combining Graph Neural Networks and Spatio-temporal Disease Models to Predict COVID-19 Cases in Germany. Scientific Reports, 3930(12), 1–18. https://doi.org/10.1038/s41598-022-07757-5
```
@article{fritz2021,
  title = {Combining Graph Neural Networks and Spatio-temporal Disease Models to Predict COVID-19 Cases in Germany},
  author = {Fritz, Cornelius and Dorigatti, Emilio and Rügamer, David},
  year = {2022},
  pages = {1-18},
  volume = {3930},
  number = {12},
  journal = {Scientific Reports},
  doi = {10.1038/s41598-022-07757-5},
  keywords = {publications}
}
```
During 2020, the infection rate of COVID-19 has been investigated by many scholars from different research fields. In this context, reliable and interpretable forecasts of disease incidents are a vital tool for policymakers to manage healthcare resources. In this context, several experts have called for the necessity to account for human mobility to explain the spread of COVID-19. Existing approaches often apply standard models of the respective research field, frequently restricting modeling possibilities. For instance, most statistical or epidemiological models cannot directly incorporate unstructured data sources, including relational data that may encode human mobility. In contrast, machine learning approaches may yield better predictions by exploiting these data structures yet lack intuitive interpretability as they are often categorized as black-box models. We propose a combination of both research directions and present a multimodal learning framework that amalgamates statistical regression and machine learning models for predicting local COVID-19 cases in Germany. Results and implications: the novel approach introduced enables the use of a richer collection of data types, including mobility flows and colocation probabilities, and yields the lowest mean squared error scores throughout the observational period in the reported benchmark study. The results corroborate that during most of the observational period more dispersed meeting patterns and a lower percentage of people staying put are associated with higher infection rates. Moreover, the analysis underpins the necessity of including mobility data and showcases the flexibility and interpretability of the proposed approach.
Berger, U., Fritz, C., & Kauermann, G. (2022). Reihentestungen an Schulen können die Dunkelziffer vonCOVID-19 Infektionen unter Schülern signifikant senken. Das Gesundheitswesen. https://doi.org/10.1055/a-1813-9778
```
@article{Berger,
  title = {Reihentestungen an Schulen können die Dunkelziffer vonCOVID-19 Infektionen unter Schülern signifikant senken},
  author = {Berger, Ursula and Fritz, Cornelius and Kauermann, Göran},
  doi = {10.1055/a-1813-9778},
  year = {2022},
  journal = {Das Gesundheitswesen},
  keywords = {publications}
}
```
Aim of the study The aim of this was to study investigate the effectiveness of mandatory Covid-19 tests for in-classroom teaching in reopened schools as a containment measure in the pandemic. In Bavaria, mandatory testing at schools was implemented directly after the Easter vacations in 2021. For the first week after the vacations, this resulted in a natural experiment that allowed us to quantify the impact of the new testing strategy on reported Covid-19 cases.Methods We compared changes in the reported 7-day incidence of new infections between districts with in-classroom teaching at school and districts with closed schools. During the calendar week 15, districts with reported incidences below 100 were allowed to reopen schools and have in-classroom teaching if mandatory COVID-19 testing was performed at school with rapid antigen tests. We do not have data on the rapid test results; however, positive test results in the rapid antigen test were verified by a PCR test, and cases of positive PCR test results were reported at the district level by age groups. In the calendar weeks 13 and 14, all schools in Bavaria were closed due to Easter vacations. Taking into account a latency period of about 3–4 days and a reporting period of 1–2 days, this means that any additional increase in reported incidences for districts with in-class teaching and mandatory testing in the week after the vacation cannot be attributed to transmissions at schools, but reflects the reduction of underreporting due to the newly implemented testing strategy.Results Reported incidence increased by a factor of 6.6 for 5–11 year old and by 1.7 for 12–20 year old pupils in districts with in-classroom teaching and mandatory testing at schools. This increase was accompanied by a reduction in underreporting and was significant compared to districts with school closure. Given the situation of a natural experiment, this increase in the reported incidence among school children can be attributed to the testing strategy. For the same time period, no differences in reported incidences were found for the other age groups.Conclusion In-class teaching with mandatory testing in reopened schools changes the role of schools in the pandemic. Our analyses show that reopening schools with a mandatory testing approach is beneficial from an epidemiologic perspective as it can strongly reduce the dark figure of COVID-19 cases among children.
Fritz, C., & Kauermann, G. (2022). On the Interplay of Regional Mobility, Social Connectedness, and the Spread of COVID-19 in Germany. Journal of the Royal Statistical Society. Series A (Statistics in Society), 185(1). https://doi.org/10.1111/rssa.12753
```
@article{Fritz2020,
  author = {Fritz, Cornelius and Kauermann, G{\"{o}}ran},
  journal = {Journal of the Royal Statistical Society. Series A (Statistics in Society)},
  volume = {185},
  number = {1},
  keywords = {publications},
  title = {On the Interplay of Regional Mobility, Social Connectedness, and the Spread of COVID-19 in Germany},
  doi = {10.1111/rssa.12753},
  year = {2022}
}
```
Since the primary mode of respiratory virus transmission is person-to-person interaction, we are required to reconsider physical interaction patterns to mitigate the number of people infected with COVID-19. While research has shown that non-pharmaceutical interventions (NPI) had an evident impact on national mobility patterns, we investigate the relative regional mobility behaviour to assess the effect of human movement on the spread of COVID-19. In particular, we explore the impact of human mobility and social connectivity derived from Facebook activities on the weekly rate of new infections in Germany between March 3rd and June 22nd, 2020. Our results confirm that reduced social activity lowers the infection rate, accounting for regional and temporal patterns. The extent of social distancing, quantified by the percentage of people staying put within a federal administrative district, has an overall negative effect on the incidence of infections. Additionally, our results show spatial infection patterns based on geographic as well as social distances.
Fritz, C., Mehrl, M., Thurner, P. W., & Kauermann, G. (2022). The Role of Governmental Weapons Procurements in Forecasting Monthly Fatalities in Intrastate Conflicts: A Semiparametric Hierarchical Hurdle Model. International Interactions, 8(4), 778–799. https://doi.org/10.1080/03050629.2022.1993210
```
@article{FritzMehrl2022,
  author = {Fritz, Cornelius and Mehrl, Marius and Thurner, Paul W. and Kauermann, G{\"{o}}ran},
  journal = {International Interactions},
  number = {4},
  title = {The Role of Governmental Weapons Procurements in Forecasting Monthly Fatalities in Intrastate Conflicts: A Semiparametric Hierarchical Hurdle Model},
  doi = {10.1080/03050629.2022.1993210},
  year = {2022},
  volume = {8},
  pages = {778-799},
  keywords = {publications}
}
```
Accurate and interpretable forecasting models predicting spatially and temporally fine-grained changes in the numbers of intrastate conflict casualties are of crucial importance for policymakers and international non-governmental organizations (NGOs). Using a count data approach, we propose a hierarchical hurdle regression model to address the corresponding prediction challenge at the monthly PRIO-grid level. More precisely, we model the intensity of local armed conflict at a specific point in time as a three-stage process. Stages one and two of our approach estimate whether we will observe any casualties at the country- and grid-cell-level, respectively, while stage three applies a regression model for truncated data to predict the number of such fatalities conditional upon the previous two stages. Within this modeling framework, we focus on the role of governmental arms imports as a processual factor allowing governments to intensify or deter from fighting. We further argue that a grid cell’s geographic remoteness is bound to moderate the effects of these military buildups. Out-of-sample predictions corroborate the effectiveness of our parsimonious and theory-driven model, which enables full transparency combined with accuracy in the forecasting process.

2021

Fritz, C., Thurner, P. W., & Kauermann, G. (2021). Separable and Semiparametric Network-based Counting Processes applied to the International Combat Aircraft Trades. Network Science, 9(3), 291–311. https://doi.org/10.1017/nws.2021.9
```
@article{Fritz2021a,
  author = {Fritz, Cornelius and Thurner, Paul W. and Kauermann, G{\"{o}}ran},
  doi = {10.1017/nws.2021.9},
  journal = {Network Science},
  keywords = {publications},
  number = {3},
  pages = {291--311},
  title = {{Separable and Semiparametric Network-based Counting Processes applied to the International Combat Aircraft Trades}},
  volume = {9},
  year = {2021}
}
```
We propose a novel tie-oriented model for longitudinal event network data. The generating mechanism is assumed to be a multivariate Poisson process that governs the onset and repetition of yearly observed events with two separate intensity functions. We apply the model to a network obtained from the yearly dyadic number of international deliveries of combat aircraft trades between 1950 and 2017. Based on the trade gravity approach, we identify economic and political factors impeding or promoting the number of transfers. Extensive dynamics as well as country heterogeneities require the specification of semiparametric time-varying effects as well as random effects. Our findings reveal strong heterogeneous as well as time-varying effects of endogenous and exogenous covariates on the onset and repetition of aircraft trade events.

2020

Baumann, S. A., Fritz, C., & Mueller, R. S. (2020). Food antigen-specific IgE in dogs with suspected food hypersensitivity. Tierarztliche Praxis. Ausgabe K, Kleintiere/Heimtiere, 48(6), 395–402. https://doi.org/10.1055/A-1274-9210/ID/R12749210-0044
```
@article{Baumann2020,
  author = {Baumann, Sandra A. and Fritz, Cornelius and Mueller, Ralf S.},
  doi = {10.1055/A-1274-9210/ID/R12749210-0044},
  journal = {Tierarztliche Praxis. Ausgabe K, Kleintiere/Heimtiere},
  number = {6},
  pages = {395--402},
  title = {{Food antigen-specific IgE in dogs with suspected food hypersensitivity}},
  volume = {48},
  year = {2020},
  keywords = {publications}
}
```
Objective: Knowledge of cross-reactions in food-sensitive dogs will influence the choice of elimination diets and the long-term management of those patients. The objective of this study was to evaluate food allergen-specific IgE tests of suspected allergic dogs for concurrent positive reactions as possible evidence for cross reactions between allergens. Material and methods: Results of serum IgE tests from 760 suspected allergic dogs submitted to 2 laboratories were evaluated statistically. After the tested allergens were grouped by their phylogenetic relationship, odds ratios as well as a sensitivity analysis of the odds ratios were performed to evaluate if concurrent positive IgE results to 2 allergens occurred more often than expected. Results: Within related allergen pairs 27% (laboratory 1) and 72% (laboratory 2) of the pairs could be considered as associated. For the unrelated allergen pairs only 6.8% and 10.6% of the analyzed pairs were considered associated respectively. Strong correlations were shown in the group of ruminant allergens, especially beef and lamb, and grain allergens. High rates of concurrent reactions were also detected in the poultry group, especially between chicken and duck, as well as between pork and ruminant allergens, and soy and grain allergens. Conclusion: As our results showed not only correlations within related but also between non-related allergens, the possible relevance of carbohydrate moieties as well as panallergens for canine hypersensitivities warrants further study. Further investigations are necessary to distinguish co-sensitization from cross-reactions and determine the clinical relevance of food-specific IgE reactivity. Clinical relevance: Due to possible cross reactivity related allergens, especially beef and lamb as well as grain allergens, should not be used for an elimination diet to avoid false results.
Fritz, C., Lebacher, M., & Kauermann, G. (2020). Tempus Volat, Hora Fugit: A survey of Tie‐oriented Dynamic Network Models in Discrete and Continuous Time. Statistica Neerlandica, 74(3), 275–299. https://doi.org/10.1111/stan.12198
```
@article{Fritz2019,
  author = {Fritz, Cornelius and Lebacher, Michael and Kauermann, G{\"{o}}ran},
  doi = {10.1111/stan.12198},
  journal = {Statistica Neerlandica},
  keywords = {publications},
  number = {3},
  pages = {275--299},
  title = {{Tempus Volat, Hora Fugit: A survey of Tie‐oriented Dynamic Network Models in Discrete and Continuous Time}},
  volume = {74},
  year = {2020}
}
```
Given the growing number of available tools for modeling dynamic networks, the choice of a suitable model becomes central. The goal of this survey is to provide an overview of tie-oriented dynamic network models. The survey is focused on introducing binary network models with their corresponding assumptions, advantages, and shortfalls. The models are divided according to generating processes, operating in discrete and continuous time. First, we introduce the Temporal Exponential Random Graph Model (TERGM) and the Separable TERGM (STERGM), both being time-discrete models. These models are then contrasted with continuous process models, focusing on the Relational Event Model (REM). We additionally show how the REM can handle time-clustered observations, i.e., continuous time data observed at discrete time points. Besides the discussion of theoretical properties and fitting procedures, we specifically focus on the application of the models on two networks that represent international arms transfers and email exchange. The data allow to demonstrate the applicability and interpretation of the network models.

Theses

Statistical Approaches to Dynamic Networks in Society. Dissertation, LMU Munich
Dynamic Social Network Models for Time-Stamped Data. Master Thesis, LMU Munich
Explorative Datenvisualisierung mit Shiny in R.. Bachelor Thesis, LMU Munich