Monitoring the COVID-19 epidemic with nationwide telecommunication data
MMonitoring the COVID-19 epidemicwith nationwide telecommunication data
Joel Persson *† Jurriaan F. Parie * Stefan Feuerriegel * Abstract
In response to the novel coronavirus disease (COVID-19), governments have introduced severepolicy measures with substantial effects on human behavior. Here, we perform a large-scale,spatio-temporal analysis of human mobility during the COVID-19 epidemic. We derive hu-man mobility from anonymized, aggregated telecommunication data in a nationwide setting(Switzerland; February 10–April 26, 2020), consisting of ∼ Keywords : COVID-19, epidemiology, human mobility, telecommunication data, Bayesianmodeling * ETH Zurich (Swiss Federal Institute of Technology), 8092 Zurich, Switzerland † [email protected] a r X i v : . [ s t a t . A P ] J a n Introduction
The novel coronavirus disease (COVID-19) has evolved into a global pandemic, which, as of De-cember 15, 2020, has been responsible for more than 70 million reported cases . In response,governments around the world have put policy measures into effect with the aim of reducing trans-mission rates . Examples of policy measures are border closures, school closures, venue closures,and bans on gatherings.Prior literature has suggested the use of human mobility data to model the COVID-19 epi-demic . Mobility patterns have been inferred from point-of-interest (POI) check-ins and fromlocation logs of smartphone apps . Other works have used telecommunication data to modelspreading patterns
26, 27 and for exploratory analysis of mobility patterns but none have yet em-pirically explored the link between mobility and policy measures. Establishing such a link wouldprovide a scalable tool for near real-time disease surveillance under policy measures and, in par-ticular, enable evidence-based policies. Previously, the value of telecommunication data for dis-ease surveillance has been studied in the context of malaria
31, 32 , influenza , and other infectiousdiseases , where the objective was to make spatio-temporal forecasts. In contrast, this paperdemonstrates the utility of telecommunication data for near real-time assessments of COVID-19policies. In fact, nationwide data from mobile telecommunication networks has been used by gov-ernments during the first wave of COVID-19 . However, to the best of our knowledge, empiricalevidence regarding the effectiveness of telecommunication data for epidemic surveillance in thecontext of COVID-19 is absent.In this paper, we analyze human mobility during the COVID-19 epidemic. Our analysis isbased on large-scale, granular data of human movements (anonymized and aggregated) consistingof ∼ Results
Human mobility derived from nationwide telecommunication data
We analyze large-scale data on human mobility during February 10–April 26, 2020 from Switzer-land. For this, human movements were derived from telecommunication data (see Methods).Telecommunication data provide more reliable and extensive information on mobility compared3o alternative data sources (check-ins or location logs from smartphone apps)
7, 38, 39 . In particu-lar, our telecommunication data represents routine signal exchanges (“pings”) exchanged betweenmobile devices and network antennas. These were recorded for all mobile devices in Switzer-land regardless of the mobile service provider. Based on the telecommunication data, granularlocations (longitude, latitude) of individuals carrying a mobile device were inferred. This yieldsdata on micro-level movements from all mobile devices in a nationwide setting. Altogether, thenationwide mobility for a population of ∼ ∼ >
100 people), bans on gatherings ( > ∼
95 million trips in the first week (February 24–March 1, 2020), during which all fivepolicy measures were in effect in all 26 cantons. In comparison, the same time period in 2019 (asa reference period) registered ∼
186 million trips. This amounts to a reduction of 49.1 %. Thereduction occurred in all cantons (Figure 1a,b). The highest decline was observed in Ticino andGeneva, which are located at the borders with Italy and France, respectively. Both cantons alsoreported the highest number of COVID-19 cases.The largest mobility reduction compared to 2019 occurred on Sunday, March 22, 2020 (Fig-ure 1c). In comparison to Sunday, March 24, 2019, the reduction in trip counts ranged between49.3 % and 77.0 % across the 26 cantons (mean: 61.6 % reduction per canton). Overall, thereduction in mobility is of similar magnitude for both rural (e. g., canton of Valais) and urban re-gions (e. g., cantons of Basel-City and Zurich). Furthermore, movements declined for all modesof mobility (Figure 1d) and for all purposes (Figure 1e). After the implementation of the policymeasures, trips by train remained low for the rest of the study period, while highway traffic wason an upward trend (Figure 1d). Similarly, trips by commuters remained at a low level after theimplementation of the policy measures, whereas trips not for commuting (i. e., personal purposes)started increasing in early April (Figure 1e). 5 d TotalTrainRoadHighway
First case Ban>100Closed schools Closed venuesBan>5 Closed borders Weekend
Eastern T r i p c oun t ( x , , ) b e TotalCommutersNon−commuters
First case Ban>100Closed schools Closed venuesBan>5 Closed borders Weekend
Eastern T r i p c oun t ( x , , ) c Genev a Ticino Zurich
NeuchatelAppenzellInnerrhoden AppenzellInnerrhoden
First case Ban>100Closed schools Closed venuesBan>5 Closed borders
Eastern2020 Eastern2019
Weekend −80%−60%−40%−20%0%+20%+40%+60%+80% D i ff e r en c e i n m ob ili t y vs . Figure 1:
Nationwide human mobility in Switzerland during the first wave of the COVID-19 epidemic. Mobility isquantified by movements (“trips”) between different post code areas. (a)
Total number of trips per canton for the firstweek after all policy measures were put in effect in all cantons (March 25–April 1, 2020). ( b ) Total number of trips forthe same week in 2019 (i. e., as reference year). For this week, the total number of trips dropped from ∼
186 million (in2019) to ∼
95 million (in 2020), i. e., a reduction of 49.1 %. (c)
Percentage change in total trips across 26 sub-nationallevels (cantons) for 2020 vs. 2019 (when aligned for day-of-week patterns). The reason for the comparison to 2019is to show the reduction in mobility relative to a reference year, while accounting for seasonal changes in mobility. Ahigher reduction in mobility is observed for cantons that also reported a high number of COVID-19 cases (i. e., Ticinoand Geneva). (d)
Reduction in trip count by mode of mobility. ( e ) Reduction in trip count by purpose of mobility.Annotations show nationwide implementation dates of policy measures (implementation dates at cantonal level arereported in Supplement A). stimating the reduction in human mobility due to policy measures We estimate the reduction in mobility due to the policy measures with a regression model. Theestimates are identified via a difference-in-difference analysis and may thus be given a causalinterpretation under certain assumptions (see Supplement B.1).The most effective policies for reducing trip counts are as follows (Figure 2a). Based on ourmodel, bans on gatherings of more than 5 people reduced total trips by 24.9 % (95 % credibleinterval [CrI]: 22.1–27.6 %), venue closures reduced total trips by 22.3 % (95 % CrI: 15.6–29.0 %),and school closures reduced total trips by 21.6 % (95 % CrI: 17.9–25.0 %). For a precise ranking,the width of the credible intervals must be considered. Here, the aforementioned policy measuresappear more effective at reducing total trips than the other two policy measures (i. e., bans ongatherings of more than 100 and border closures). In particular, bans on gatherings of more than100 people are linked to a comparatively smaller change in total trips than bans on gatherings ofmore than 5 people (i. e., the 95 % CrIs of the estimates are disjoint). For border closures, thecredible interval includes zero. Overall, policy measures are important determinants of mobilityreductions during the COVID-19 epidemic.The estimated mobility reduction depends on the underlying mode (Figure 2b). Across all pol-icy measures, the mobility reduction is more pronounced for highways than for road movements.This observation is to be expected. Highways are often used for long-distance travel, which ismore likely to be suspended during an epidemic, while roads also include movements within closeproximity and are more likely to correspond to routine or essential activities (e. g., grocery shop-ping). For the ban on gatherings and school closures, the largest reduction is seen in trips by train,which can be explained by the widespread use of public transportation in Switzerland. Finally,we observe a wide credible interval for the estimated effect of venue closures on trips by train. Apotential reason for this is that the use of trains (e. g., for visiting stores) varies across cantons,as some cantons (e. g., Zurich) have a high population density with extensive shopping infrastruc-ture, while others (e. g., Appenzell Innerrhoden) have only a few stores due to their low population7ensity, resulting in the need for travel to visit stores.The estimated effect sizes are fairly similar for trips made for commuting versus non-commuting (Figure 2c). This is interesting considering that no policy measure in Switzerlanddirectly prohibited movement to and from work. The efficacy of border closures is uncertain sincethe credible interval for its estimated effect includes zero. In contrast, a negative effect is observedfor commuting. Here, one potential reason is that border closures have reduced the number ofcross-border commuters.The findings are robust to alternative model specifications (see the robustness checks in Sup-plement D). Specifically, changing the specification of time-related control variables still givesparameter estimates for the policy measures that imply decreases in mobility. a b c
Closed bordersBan > 5Closed venuesClosed schoolsBan > 100−30 −20 −10 0
Effect size (%)Trips
Total −40 −30 −20 −10 0
Effect size (%)Mode
Highway Road Train −30 −20 −10 0
Effect size (%)Purpose
Commuter Non−commuter
Figure 2: The estimated effects of policy measures on mobility. Shown are the estimated effectson ( a ) the total number of trips, ( b ) trips by mode, and ( c ) trips by purpose. The dots in (a) – (c) show posterior means; the thick and thin bars represent the 80 % and 95 % credible intervals,respectively. Policy measures are arranged from top to bottom in the order in which they wereimplemented (cf. Supplement A). Estimating the relationship between mobility and COVID-19 cases
The epidemic dynamics during the first wave of COVID-19 in Switzerland are as follows. Theinitial exponential growth rates exhibit considerable heterogeneity across cantons (Figure 3a). Thestrongest initial growth is observed for the cantons Ticino and Geneva, resulting in the largest8umber of cases towards the end of the sample. Moreover, the number of reported cases at the datesthat policy measures were implemented varies greatly across cantons (Figure 3b). This reflectsdifferent responses among cantons to local infection dynamics.We use regression models to estimate the extent to which decreases in mobility predict futurereductions in the reported number of new cases (Figure 3c). The predicted decrease is studied witha forecast window over 7–13 days. The forecast window is set analogous to previous research ,and so that it covers variations in incubation time combined with reporting delay.We find that decreases in mobility at a given day predict decreases in reported new cases 7to 13 days later. For a 7-day ahead forecast, we find that a 1 % decrease in the total number oftrips predicts a 0.88 % (95 % CrI: 0.7–1.1 %) reduction in the reported number of new cases. For a13-day ahead forecast, a 1 % decrease in the total number of trips predicts a 1.11 % (95 % CrI: 0.9–1.6 %) reduction in the reported number of new cases. Overall, mobility predicts decreases in thereported number of new cases over the whole forecast horizon. The predicted decrease is largerfor longer forecasts. This result is to be expected, as a longer time window accommodates thefull distribution of incubation periods (plus reporting delays). Altogether, the regression analysisprovides evidence of that mobility predicts epidemic dynamics.Our analysis also shows that the predicted change in the reported number of new cases variesacross the mode and purpose of trips. In terms of mode, decreases in trips by highway and trainpredict reductions in the reported number of new cases of similar magnitude (Figure 3d). Theirestimates have comparatively narrow credible intervals, reflecting a higher degree of certainty.Trips are also categorized according to their purpose, namely commuting vs. non-commuting. Theresults show that decreases in trips for commuting predict smaller reductions in the number of re-ported new cases compared to decreases in non-commuting trips (Figure 3e). Predicted reductionsare nonetheless found for both modes of mobility (i. e., commuting vs. non-commuting), all cat-egories of purpose (i. e., highway, road, and train), and for the whole 7–13 day forecast window.Again, a larger reduction is predicted for longer forecasts.9he predictive ability of mobility for reported new cases holds with alternative model specifi-cations (see the robustness checks in Supplement D). For most of the 7–13 day forecasts, changinghow we control for time-related factors still results in a predicted decrease in reported cases givendecreases in mobility. a Zurich GeneveTicino
Number of days since 1st reported case C a s e s pe r , i nhab i t an t s b ZHZGVSVDURTITGSZSOSHSGOWNWNELUJUGRGLGEFRBSBLBEARAIAG Ban > 100 Closedschools Closedvenues Ban > 5 Closedborders 050100150200250300350
Cases per100,000inhabitants c d e
Predicted change (%) D a y ahead Trips
Total −1.0 −0.5 0.0
Predicted change (%)Mode
Highway Road Train −1.0 −0.5 0.0
Predicted change (%)Purpose
Commuter Non−commuter
Figure 3: Decreases in human mobility predicts reductions in reported cases of COVID-19 overa forecast window of 7–13 days. (a)
COVID-19 case growth since the 1st reported case for 26Swiss cantons. (b)
Number of COVID-19 cases when the policy measures were implementedfor all 26 Swiss cantons. Shown are abbreviated names. The predicted change in reported newCOVID-19 cases at a given day is based on mobility lagged by 7 days to 13 days. The predictedchange is reported given a 1 % decrease in (c) total trips, (d) mode, and (e) purpose. Posteriormeans are shown as dots, while 80 % and 95 % credible intervals are shown as thick and thin bars,respectively. 10 stimating the mediating role of mobility
In an extended analysis, we study how decreases in reported case growth is explained by reductionsin mobility due to policy measures versus other behavioral changes due to policy measures. Theestimates are obtained from a mediation analysis that decomposes the total effects of the policymeasures on reported case growth into (1) their direct effects not explained by changes in mobilityand (2) their indirect effects through mobility. The mediation analysis is performed by combiningour two regression models into a structural equation model (see Supplement F for details). Resultsfrom mediation analysis are reported for the total number of trips.The mediation analysis shows a large direct effect for bans on gatherings of more than 5 people,bans on gatherings of more than 100 people, and school closures (4). Pronounced indirect effectsare found for all policy measures. In particular, the indirect effect of venue closures makes up abouta third of their total effect at several lags. Moreover, border closures are estimated to only havereduced the reported number of new cases indirectly through mobility. The results are discussed infurther detail in Supplement F. In summary, the results show that mobility is an important mediator:the studied policy measures operate – to a large degree – through mobility. Thus, policy measuresaimed at reducing mobility appear to be effective for reducing COVID-19 case growth.11 D a y ahead Ban > 100 −40 −30 −20 −10 0
Closed schools −30 −20 −10 0 10
Closed venues −50 −40 −30 −20 −10 0
Ban > 5 −10 0 10 20
Closed borders
Direct effect b D a y ahead Ban > 100 −10.0 −7.5 −5.0 −2.5 0.0
Closed schools −10 −5 0
Closed venues −10 −5 0
Ban > 5 −4 −3 −2 −1 0
Closed borders
Indirect effect c Effect size (%) D a y ahead Ban > 100 −40 −30 −20 −10 0
Effect size (%)
Closed schools −40 −30 −20 −10 0
Effect size (%)
Closed venues −40 −20 0
Effect size (%)
Ban > 5 −10 0 10 20
Effect size (%)
Closed borders
Total effect
Figure 4: Mobility mediates the effect of policy measures on the reported number of new cases.Estimated (a) direct effect of policy measures, (b) indirect effect of policy measures via total trips,and (c) total effect of policy measures on the 7th to 13th day ahead. Posterior means are shownas dots, while 80 % and 95 % credible intervals are shown as thick and thin bars, respectively.Policy measures are arranged from top to bottom in the order in which they were implemented (cf.Supplement A).
Discussion
This study shows the ability of telecommunication data for near real-time monitoring of theCOVID-19 epidemic. Our analysis is based on nationwide telecommunication data during Febru-ary 10–April 26, 2020 from Switzerland, which were used to infer nationwide mobility patterns.12his supports monitoring of the COVID-19 epidemic as follows: (1) We first studied the link be-tween policy measures and human mobility. In particular, we performed a difference-in-differencequantifying how mobility was reduced due to 5 different policy measures (bans on gatherings,school closures, venue closures, and border closures). The largest reduction in total trips waslinked to bans on gatherings of more than 5 people, followed by venue closures and school clo-sures. Overall, the policy measures resulted in substantial reductions of human mobility. (2) Wethen studied the link between human mobility and reported COVID-19 cases. Reductions in mo-bility predicted decreases in the number of reported new cases. Specifically, a reduction in humanmovement by 1 % predicted a 0.88–1.11 % reduction in the daily number of new cases of COVID-19 over a forecast horizon of 7 days to 13 days. Our modeling approach with telecommunicationdata therefore provides near real-time insights for disease surveillance. Taken together, the find-ings enable quantitative comparisons of the extent to which policy measures reduce mobility and,subsequently, reduce reported cases of COVID-19.The use of telecommunication data for nationwide monitoring has several benefits
7, 40 . First,telecommunication data from mobile networks provide comprehensive coverage. Specifically,such data capture all movements of individuals carrying mobile devices without explicit user in-teraction, including those from non-residential and even foreign individuals. Mobile devices rou-tinely exchange information when searching for signals from adjacent antennas; hence, metadataare retrieved regardless of the underlying mobile service provider. Second, such metadata can becollected in an anonymized manner that is compatible with data privacy laws. Third, movements ata micro-level (e. g., trips to other households, school, and work) can be inferred. Thus, comparedto alternative sources of mobility information such as check-ins or smartphone apps, telecommu-nication data are considered to be more complete
7, 38, 39 . Fourth, unlike smartphone apps, telecom-munication data are also available in low-income countries . Finally, telecommunication dataare measured with high frequency (e. g., daily), thereby enabling regularly updated monitoring asneeded by decision-makers. Based on these benefits, telecommunication data appear to be highly13ffective for policy monitoring during the COVID-19 epidemic.This work is subject to the typical limitations of observational studies. First, the findings de-pend on the accuracy of the data on COVID-19 cases. Second, our models are informed by recom-mendations for COVID-19 modeling and, therefore, follow parsimonious specifications to isolatefeatures of the epidemic for policy-relevant insights. We cannot, however, rule out the possibil-ity that there exist external factors beyond those that are captured by the spatially and temporallyvarying variables in the models. To address this, we use flexible models and conduct extensiverobustness checks (Supplement D). Third, the model linking policy measures to mobility estimateseffects, while the model linking mobility to cases is predictive. The different objectives of themodels address the needs of public decision-makers: the former serves policy assessments and thelatter epidemiological forecasting, respectively. Therefore, the estimates from the former are iden-tified with a difference-in-difference analysis and may thus warrant causal interpretations undercertain assumptions (see Supplement B.1). On the other hand, the estimates from the latter areconditional associations since the model does not control for that policy measures reduce both mo-bility and cases. Therefore, the latter model predicts decreases in reported cases from reductions inmobility when both reductions are driven by policy measures (see Supplement B.2 for a discussionof this approach). Fourth, our findings are limited to our study setting, that is, the first wave inSwitzerland. Future research may confirm the external validity of our findings by analyzing othercountries or time periods.Inferring mobility patterns from telecommunication data is inherently coupled to the coverageof such data and our definition of trips. Only movements for individuals who carry a mobile de-vice with a SIM card are included. In particular, trips are not included for individuals who do notcarry SIM cards. Similarly, trips may be counted several times if an individual carries several SIMcards (e. g., when carrying both a phone and a SIM-based tablet). It is also possible that trips bychildren, elderly, or other parts of the population with less phone usage are underrepresented inthe data. Furthermore, micro-level movements are not observed but recovered via triangulation14etween antennas through the use of a positioning algorithm achieving state-of-the-art accuracy.In spite of these limitations, telecommunication data are considered to be scalable and, in par-ticular, more complete for inferring mobility patterns compared to alternative data sources
7, 38, 39 .Moreover, our objective is not to obtain accurate estimates of mobility in itself, but to evaluatethe predictive ability of telecommunication data for reported case growth. Our analysis confirmstelecommunication data as such a monitoring tool.Our findings are of direct value for public decision-makers. Nationwide mobility data frommobile telecommunication networks can be leveraged for the management of epidemics. Thereby,we fill a previously noted void in the case of COVID-19
38, 39, 43 . Specifically, monitoring mobil-ity supports public decision-makers when managing the COVID-19 epidemic in two ways. First,it helps public decision-makers in assessing the impact of policy measures targeted at mobilitybehavior. Second, by predicting epidemic growth, it provides a scalable tool for near real-time epi-demic surveillance. Such tools are relevant for evidence-based policy-making of public authoritiesin the current COVID-19 epidemic. 15 ethods
The aim of this study is to make population inference from nationwide telecommunication data. Inour study, telecommunication data are collected from routine signal exchanges (i. e., pings) withantennas, regardless of the actual service provider. Based on the telecommunication data, mobilityestimates are inferred as follows (Fig. 5): (1) Telecommunication data are collected at the levelof antenna. (2) Telecommunication data at antenna level are used to infer micro-level movementsof individuals via triangulation. (3) Data on micro-level movements are used to count movementsbetween postal codes (named “trips”) over time. This procedure is performed to capture mobilitylevels in the population. (4) The data are further aggregated at the cantonal level per day in orderto link them to policy measures and COVID-19 case numbers. The procedure is detailed in thefollowing.
Antenna level (input)
Position data
Micro-level movements N A = 29,679 antennas Postal code level
Canton level (modeling) N ZIP = 3,196 postal codes N = 26 cantons S pa t i a l g r anu l a r i t y AggregationTriangulationAggregation
Figure 5: Deriving mobility estimates from nationwide telecommunication data for monitoring theCOVID-19 epidemic.
Nationwide telecommunication data
Telecommunication data are routinely collected from signal exchanges (i. e., pings) between mo-bile devices and adjacent antennas. Such signal exchanges occur for all SIM-based mobile devices(e. g., mobile phones, smartphones), regardless of the actual service provider. In particular, our16ata also include movements of people with a foreign SIM card and, hence, represents nationwidetelecommunication. A network event between a mobile device and mobile network compromisesmetadata as follows: the IMSI number of the SIM card, the date and time the SIM card connectedto the mobile network, and the ID of the mobile antenna to which the SIM card was connected.The IMSI number is available for all SIM cards and thus represents a unique identifier, indepen-dent of the actual service provider. The events from mobile networks are extracted from the mobilecommunications systems every night, and thus the mobile data is available the following day. Inour analysis, we use telecommunication metadata collected according to the above description bySwisscom . Swisscom also ensured that IMSI numbers are stored in an anonymized format (seeRef. for details).Telecommunication data hold advantages over alternative data sources for the purpose of mea-suring human mobility. The advantages become especially clear in comparison to location datafrom check-ins or location logs from smartphone apps . First, compared to smartphoneapplications, SIM-based devices are fairly ubiquitous. This holds for both high-income coun-tries (such as Switzerland) and low-income countries. Second, the use of telecommunicationdata ensures coverage for large parts of society. Specifically, it reduce the risk of an age bias(e. g., check-ins are known to be more frequent among younger, technology-savvy people). Third,telecommunication data avoid the need for user interaction with a device. Hence, many micro-level movements are captured (e. g., school visits, commuting to work, grocery shopping) thatwould otherwise not be subject to monitoring.The telecommunication infrastructure operated by Swisscom has wide coverage . Specifically,it covers 99.9% of the geographic area in Switzerland. The infrastructure records telecommuni-cation metadata via almost 30,000 antennas. Hence, there are multiple antenna per postal code.In particular, approx. 7,000 of the antennas are of the GSM type, 11,000 of the UMTS type, and12,000 of the LTE type .The frequency of pings is determined by the mobile device, occurring every time a mobile17hone connects to a new antenna or, if in between two antennas, every ∼ Inferring positions of SIM cards via triangulation
The locations of SIM cards within antenna areas are determined via triangulation between anten-nas through the use of a positioning algorithm . A high-level description of the algorithm is asfollows. Every signal from a SIM card in the telecommunication data is associated with a prob-ability distribution over locations that represents the uncertainty of its actual location in a givenantenna area at the time. The location is estimated from two random variables: the radius R , givenby the distance of the signal from the origin of the antenna area, and its angle Θ to the antennaazimuth. Here, R is Gaussian distributed with empirical mean and variance estimated via maxi-mum likelihood, and Θ follows a multinomial distribution depending only on the antenna azimuthand its bandwidth. The inferred locations are subject to a delay between signals between antennasand SIM cards. To address this, the location at a given point in time is estimated by marginalizingthe probability distribution of the radius over the empirical distribution of signal delays estimatedfrom all observations. Details are available in . In sum, by tracking the location of SIM cardsover time, we are able to capture micro-level movements of individuals.The accuracy of the positioning algorithm has been empirically validated . The median posi-tioning error is 132 meters, making it highly accurate compared to state-of-the art methods . Theaccuracy was determined by comparing the algorithm’s predicted positions to self-reported actualpositions for more than 6,000 trips with over 12,000 end-points .18 eriving mobility estimates from telecommunication data Mobility has been frequently found to be helpful for understanding urban phenomena
51, 52 . In thisstudy, we derive mobility estimates from nationwide telecommunication data.Trips are computed as follows. A single trip is defined as the movement of a SIM card betweentwo different post code areas after the location has been static for 20 minutes. The trip is thencounted for both post codes. Similarly, trips that cross midnight are counted for both days.Post codes were chosen to define trips as they represent the smallest spatial unit that is offi-cially defined by the federal government. Switzerland has 3,196 post code areas with high spatialgranularity. The exact size varies between urban and rural regions, but, on average, a post codearea in Switzerland covers merely 12.9 km . Moreover, 71 % of the Swiss working populationcommute between different post codes for work and oftentimes even between cantons ( ). The average (one-way) travel distanceto work is 15.0 km (see previous URL), and, the average daily travel distance for leisureactivities is 36.8 km ( ). Forboth work and leisure activities, travel routinely spans several postal code areas. Hence, the use ofnationwide telecommunication data combined with our definition of trips provides comparativelylarge-scale estimates of aggregate mobility.For the analysis, daily trips were further aggregated as follows. First, each trip between postcodes was mapped onto cantons at the sub-national level. Here, we used cantonal shape files fromthe Swiss government and aggregated all daily trips within each canton. Note that the number oftrips into and out of cantons are captured with this aggregation due to the attribution of trips toboth the departure and arrival post codes. The result of the aggregation is a panel (longitudinal)dataset of trip counts across cantons.The reason for the aggregation to the cantonal level per day is twofold. First, policy measures19re implemented within cantons, and, second, COVID-19 case data are only published per day atthe cantonal level. Therefore, data on policy measures and case number are not available on a moregranular level.Trips were further labeled according to both mode and purpose. The mode of tripswas differentiated based on estimating the location of SIM cards with the positioning al-gorithm and the position of antennas along train, highway, and road networks. If sev-eral modes of mobility were used in the same trip, the mode with the longest leg waschosen. For comparison, public transport through train is an important mode of trans-portation in Switzerland ( ) that isrelevant for explaining the results. The purpose of mobility was classified based on trips to/fromwork (called “commuting”) and all other trips (called “non-commuting”). This differentiation wasbased on the home and work locations of individuals (i. e., postal code area). These locationswere derived from the most frequent geographic location of individuals between 8 pm–8 am forhome locations and 8 am–5 pm for work locations. Afterwards, both home and work location werematched against the departure and arrival (postal code area) of a trip to determine whether the tripwas to/from work, and hence labeled “commuting”. The classification or trips into mode of trans-port is highly accurate. A validation against self-reported data was performed, showing that 90 %of all trips were correctly classified . Modeling overview
In this section, we present the regression models used to estimate the relationship between (1) pol-icy measures and mobility and (2) mobility and reported cases. Here, the first model estimates thereduction in mobility due to policy measures. The estimates are identified with a difference-in-difference analysis and may therefore be given a causal interpretation under certain assumptions(see Supplement B.1). The second model, in turn, estimates the extent to which reductions in20obility predict decreases in the reported number of new cases as policy measures are being im-plemented.The models have parsimonious specifications recommended for isolating policy-relevant in-sights and are informed by epidemiology. In particular, they are formalized as Bayesian hier-archical negative binomial regression models. Rather than modeling the disease dynamics them-selves (as with a compartment model), our focus is on estimating the relative effect of other de-terminants, namely, policy measures and mobility. The use of negative binomial distributions iscommon in epidemiological modeling, as it allows for overdispersion in dependent variable (i. e.,the number of trips and the reported number of new cases). Furthermore, each model uses a log-link between the dependent and explanatory variables. For the model of the reported number ofnew cases, it enables us to capture the exponential growth in cases during the initial stages of anepidemic, also observed in our data. For the model of mobility, it makes the estimates relative tothe observed levels of mobility. Both dependent variables were found to follow negative binomialdistributions (with overdispersion). See Appendix C.4 for an analysis of overdispersion.The models include further controls for (1) population size per canton, (2) unobserved het-erogeneity between cantons, and (3) time effects as follows. (1) We control for differences inpopulation size among cantons with an offset term. This is motivated by the fact that the magni-tude of the estimated effects depend on the population size. Hence, the model estimates are relativeto the number of inhabitants per canton. (2) Unobserved heterogeneity is estimated with a cantonrandom effect. We thereby account for unobserved factors that affect both policy measures andmobility (for the former model) and mobility and cases (for the latter model). (3) Time effects aremodeled in two ways. On the one hand, we include weekday fixed effects to control for variationsin the implementation of policy measures, levels of mobility, and reporting/testing across week-days (e. g., mobility is higher on weekdays, whereas testing is lower on weekends; thus, reportedcases are lower on weekends and tests conducted on weekends may be reported on Mondays orTuesdays). On the other hand, we incorporate a trend variable that controls for changes in case21ynamics or behavioral adaptations towards social distancing that occur over time since a cantonfirst reported a case. This could for instance occur due to unobserved changes in adherence toother policy measures (e. g., wearing masks and keeping physical distance of at least 1.5 meters)over time. Here, we model the variation in when cantons reported their first case as potentiallydependent on the unobserved canton heterogeneity.The results with additional controls (e. g., test frequency, spatial correlation between cantons,and dependence between the different trip variables) are part of the robustness checks. Model for estimating the reduction in human mobility due to policy measures
A multiple time period, multiple group difference-in-difference (DiD) analysis is conducted toestimate the effect of each policy measure on mobility. We restrict the analysis to the time periodbetween February 24 and April 5, 2020; that is, starting before the first reported COVID-19 casein Switzerland and ending prior to Easter holidays. With this time period, the initial observationsact as a control group in which mobility is at the baseline level (as individuals may not yet havevoluntarily reduced their mobility as a response to reported cases). Furthermore, by ending atApril 5, there can be no confounding of the effects of policy measures due to Easter holidays. Suchconfounding would be caused by that during holidays, mobility generally changes from regularlevels and, as a consequence, policy measures are more or less likely to be implemented relativenon-holidays.Let M itk denote the trip count on mobility variable k = 1 , , . . . , K (i. e., total trips, road trips,train trips, etc.) in canton i = 1 , , . . . , N on day t = 1 , , . . . , T . The variable M itk is derived asexplained in the previous section and represents the dependent variable in regression model k forthe DiD analysis. The values of the model parameters depend on which mobility variable is thedependent variable of the regression; hence, we index all model parameters with k .22e model M itk to follow a negative binomial distribution with conditional mean function E [ M itk | η ( k ) it , E i ] = µ ( M itk ) = E i exp η ( k ) it (1)where E i denotes the population size of canton i . Then, exp η ( k ) it = ( µ ( M itk )) /E i is the expectednumber of daily trips per inhabitant in canton i . The estimates of this model are therefore adjustedaccording the variation in canton population sizes. The term η ( k ) it is the linear predictor, specifiedin hierarchical form as η ( k ) it = α ( k ) i + δ ( k ) w ( t ) + γ ( k ) log z it + L (cid:88) l =1 β ( k ) l d itl , (2) α ( k ) i = α ( k ) + θ ( k ) i + γ ( k ) B Ę log z i , (3)whose variables and parameters are explained in the following.The first term α ( k ) i is a time-invariant effect specific to canton i . We model α ( k ) i as a functionof several variables that vary across cantons; see Equation (3). Here, α ( k ) is the intercept amongall cantons, which represents the overall baseline relative mobility on Mondays before any policymeasure was implemented and any COVID-19 cases were reported. The term θ ( k ) i is a randomeffect that captures unobserved time-invariant factors for canton i (e. g., population density) thatconfound the effect of policy measures on mobility. The final variable Ę log z i is discussed in de-tail below. The subscript B on the associaed parameter denotes that it measures between-cantoneffects; that is, the parameter only measures the effect of increases in the variable across cantons.The variable d itl is a dummy variable that takes a value of if policy measure l ∈ { , , . . . , L } is implemented by canton i at day t and otherwise. Hence, exp( β ( k ) l ) measures the multiplicativeeffect of policy measure l on the expected number of daily trips on mobility variable k per cantoninhabitant. Note that all policy measure variables are included in (2). Hence, the effect of eachpolicy measure is conditional on the other policy measures being held fixed. Figure 1c shows that23he reduction in mobility is similar across cantons; therefore, we do not estimate the heterogeneityin the effect of the policy measures on mobility across cantons.The term δ ( k ) w ( t ) represents the fixed effect of weekday w on the relative (log-transformed) mo-bility compared to the reference weekday (here: Monday). The term controls for the confoundingfactor that aggregate mobility and the probability of implementing a policy measure is likely higheron, e. g., Mondays than Sundays.The variable z it captures other sources of time-related confounding and is derived as follows.Let q it be the number of days since the first reported COVID-19 case in canton i . The variable iscalculated as q it = t − t (cid:48) i , if t > t (cid:48) i , , if t ≤ t (cid:48) i , (4)where t (cid:48) i is the date the first case was reported in canton i . Since the logarithm of zero is undefined,we then set z it = q it + 1 and include the logarithm of z it in the model. The associated parameter γ ( k ) is therefore interpreted as the percentage increase in relative mobility given a 1 % increasein the number of days since a canton first reported a case. The rationale for including z it is thatindividuals may adapt their mobility behavior over time irrespective of social distancing policies.Therefore, the variable captures how mobility would trend over time even if the policy measureswere not implemented.The variable Ę log z i = T − (cid:80) Tt =1 log z it is the time average of log z it in canton i . The barover the expression denotes an average value. The variable is included in the model to allow thecanton-specific effect θ ( k ) i to be correlated with log z it over the cantons. Such correlation would,for instance, arise if the date that the first COVID-19 case is reported in each canton dependson the unobserved canton-specific factors. As an example, the date the first case is reported ina canton could depend on the (unobserved) adherence of inhabitants to social distancing recom-24endations. If such correlation exists but is ignored, it would instead enter the error term of themodel, leading to a violation of the exogeneity assumption and incorrect parameter estimates. Byincluding Ę log z i , we essentially make θ ( k ) i a Mundlak-type correlated random effect . The benefitof correlated random effects over fixed effects or standard random effects is that they use onlythe within-unit variation to estimate parameters (and, hence, give identical estimates to those offixed effects models) while also having the random effects property of estimating the variation inthe unobserved heterogeneity via partial pooling. Note that time averages of the policy measurevariables or weekday effects are not included in the model since those are exogenously determinedand, therefore, uncorrelated with the model errors. We refer to for a detailed discussion of theunderlying benefits of this approach relative to fixed effects.By substituting the linear predictor (2) into the conditional mean function (1) and expanding α ( k ) i , the full model of mobility variable k becomes log E [ M itk | η ( k ) it , E i ] = log E i + α ( k ) + θ ( k ) i + δ ( k ) w ( t ) + γ ( k ) log z it + γ ( k ) B Ę log z i + L (cid:88) l =1 β ( k ) l d itl . (5)The conditional variance of M itk is given by V [ M itk | η ( k ) it , E i , ζ ( k,M ) ] = µ ( M itk ) (cid:18) µ ( M itk ) ζ ( k,M ) (cid:19) , (6)where ζ ( k,M ) is the overdispersion parameter (the superscript M distinguishes the overdispersionparameter of the mobility model from the model of reported cases).We specify one regression equation in the form of (5) for each mobility variable and estimatethem separately. Each regression has the same explanatory variables but a different mobility vari-able as the dependent variable (i. e., total trips or one of the mobility variables based on mode orpurpose). 25 odel for estimating the relationship between mobility and COVID-19 cases The model for estimating the relationship between mobility and reported COVID-19 cases is simi-lar in structure to the model used to link policy measures to mobility. To accommodate the forecasthorizon, we lag the mobility variables to estimate how a decrease in mobility at a given day pre-dicts reductions on the reported number of new cases at a later day. This enables forecasting offuture reported case growth by evaluating the model at daily observed mobility levels.Let C it denote the cumulative number of reported cases in canton i = 1 , , . . . , N until andincluding day t = 1 , , . . . , T i . Then, Y it = C it − C i,t − is the number of new cases that arereported in canton i on day t . Note that the time frame (i. e., total number of days T i ) varies acrosscantons i = 1 , . . . , N . The reason for this is that the dependent variable of this regression modelis the reported number of new cases, and as such we restrict the data to start at the date of eachcantons first reported case. Hence, the data for this model starts between February 24 and March 16(depending on the canton) and ends at April 5 (for all cantons). Our modeling approach accountsfor the resulting unbalanced number of observations per canton.We model Y it as following a negative binomial distribution with a conditional mean function E [ Y it | η ( k,s ) it , E i ] = µ ( k,s ) ( Y it ) = E i exp η ( k,s ) it , (7)where η ( k,s ) it = α ( k,s ) i + δ ( k,s ) w ( t ) + γ ( k,s ) log z it + ξ ks log m i,t − s,k , (8) α ( k,s ) i = α ( k,s ) + θ ( k,s ) i + γ ( k,s ) B Ę log z i + ξ ks,B Ğ log m ik (9)is the hierarchical linear predictor. In this model, exp η ( k,s ) it = ( µ ( k,s ) ( Y it )) /E i is the expected num-ber of reported positive cases in canton i on day t relative to the canton population size (sometimescalled the relative risk in spatial epidemiological modeling
56, 57 ). The model for the daily growth26n reported cases can then be written as log E [ Y it | η ( k,s ) it , E i ] = log E i + α ( k,s ) + θ ( k,s ) i + δ ( k,s ) w ( t ) + γ ( k,s ) log z it + γ ( k,s ) B Ę log z i + ξ ks log m i,t − s,k + ξ ks,B Ğ log m ik . (10)The conditional variance of Y it given mobility variable k lagged by s days is given by V [ Y it | η ( k,s ) it , E i , ζ ( k,s,Y ) ] = µ ( k,s ) ( Y it ) (cid:18) µ ( k,s ) ( Y it ) ζ ( k,s,Y ) (cid:19) (11)where ζ ( k,s,Y ) is the overdispersion parameter.For simplicity, we use the same notation for variables and parameters in this model as in themodel used for the effect of policy measures on mobility (but the estimated parameters have, ofcourse, a different interpretation and values). The superscript ( k, s ) is attached to parameters toindicate that their values depend on the choice of mobility variable k and its lag s .The parameter of interest is ξ ks . It measures the expected percentage change in the reportednumber of new cases per canton inhabitant s days after a 1 % increase in mobility variable k .Hence, the parameter shows how the relative growth rate in reported cases changes as a function oflagged mobility, after adjusting for relevant factors but where mobility varies according to whichpolicy measures are implemented. Note that we intentionally include only a single lag s (andthen refit the model for different lags) rather than including multiple lags at the same time. Wefollow this approach because infection disease dynamics imply that there are carry-on effects ininfection rates between consecutive days that would be modeled as being fixed if several lags areincluded, resulting in biased estimates. We also only include only a single lagged mobility variable k . The reason is that every trip by mode or reason contributes to the total number of trips; hence,to avoid counting single trips several times, both the variable for the total number of trips and anyvariable for trips by mode or reason should not be included in the same model. Moreover, since27rips by mode or purpose are different subsets of total trips, every trip by one mode must be a tripof one purpose (e. g., a train trip is either made for the purpose of commuting or non-commuting);hence, for the same reason, the model should not include mobility variables for both a mode and apurpose. Supplement B.2 further explains our rationale.The intercept α ( k,s ) gives the baseline number of reported cases relative to the canton populationfor Mondays.The parameter δ ( k,s ) w ( t ) is the effect of weekday w relative to the Monday effect. By includingweekday effects in the model, we control for confounding differences in the number of trips andthe number of reported COVID-19 cases between weekdays that would bias the parameter estimateon the lagged mobility variable. For instance, people travel to schools and work primarily onweekdays and, similarly, there are fewer COVID-19 tests on weekends and thus fewer reportedcases on Monday/Tuesday (due to reporting delays).The variable log z it is also included in this model. It now controls for the fact that mobility andthe reported case growth both depend on when the first case was reported in a canton.The Mundlak-style random effects θ ( k,s ) i estimate the impact of unobserved canton-specificfactors that may be correlated with both the variation in mobility across cantons and the loga-rithm of the number of days since the first case was reported in each canton. We achieve thisby including variables of the time averages of log z it and lagged mobility, that is, Ğ log m ik = T − (cid:80) t − s log m i,t − s,k . Then, any potential cross-canton correlation between the random effectand log m i,t − s,k or log z it via the models error term is controlled for.The above regression model is fitted separately for each ( k, s ) , that is, each pair of mobilityvariable and lag. This allows us to investigate to what extent different lags of each mobility variablepredict the number of reported cases. 28 stimation details We estimate our models in a fully Bayesian framework. We run 4 Markov chains for every model,each with 2000 warm-up samples and another 2000 samples from the posterior distributions. Sinceour models are fitted with a log-link, we transform the posterior parameter samples so that theygive estimates for the original scale of the dependent variable. For each parameter, we reportin our plots the posterior mean and the associated 80 % and 95 % credible intervals (CrI) of thetransformed posterior distribution.The software used for estimation is the R package brms
58, 59 version 2.11.1 built upon thestatistical modeling platform Stan . Parameter estimates are obtained by Markov chain MonteCarlo sampling in Stan version 2.19.2 using the Hamiltonian Monte Carlo algorithm
61, 62 and theNo-U-Turn sampler (NUTS) .Table 1 presents our choices of priors for the variables in the models. We use weakly infor-mative priors to stabilize the computations and provide some regularization. Our prior on each β l reflects that we expect each policy measure to reduce the logarithm of expected mobility with25 %, on average, but that effects between 0–50 % are relatively probable. The prior on each ξ ks implies that we expect that a 1 % decrease in the lagged mobility variable predicts a 1 % decreasein reported cases for each of the considered lags, with negative effect sizes or effect sizes exceeding2 % being unlikely. The prior on γ implies that we expect the relative outcome to increase with 1 %for each 1 % increase in the number of days since the first reported case. The intercept, overdisper-sion parameter, and standard deviation of the canton random effects are given weakly informativepriors. The prior on δ w ( t ) states that the effect of a given weekday that is not Monday shouldfall within 50–150 % of the Monday effect. The parameters for the variables of between-cantonaverages are assigned vague priors since we have no a priori belief of their effects.29 arameter Description Prior Model β l Policy measure l N ( − . , . (5) ξ ks Log mobility variable k with a lag of s N (1 , (10) α Intercept
Half - t (3 , . , . (5), (10) θ i Canton random effect N (0 , σ θ ) (5), (10) σ θ Standard deviation for canton random effect
Half - t (3 , , . (5), (10) δ w ( t ) Weekday w (compared to Monday) N (0 , . (5), (10) γ Log no. of days since 1st reported case N (1 , (5), (10) γ B Between-canton average of log no. of days since 1st reported case N (0 , (5), (10) ξ ks,B Between-canton average of log mobility with a lag of s N (0 , (10) ζ Overdispersion in dependent variable
Gamma (0 . , . (5), (10) Note:
The superscripts ( k ) and ( k, s ) are omitted as the same priors are assigned to each model. The column“Description” states what effect the associated parameter represent (except for the overdispersion parameter). Table 1: Choice of priors
Model diagnostics
We followed common practice for model diagnostics of Bayesian models . For each of the mod-els, we inspected (1) posterior predictive checks, (2) divergent transitions, (3) effective samplesize and convergence of the Markov chains, (4) overdispersion in the dependent variables, (5) in-fluential observations, and (6) correlation between the policy parameters. All model diagnosticsindicate a good fit. Details are provided in Supplement C. Robustness checks
First, we checked the robustness of the model estimates against alternative specifications of timeeffects: (a) A model specified as in the main paper but where the logarithmic trend is replacedwith corresponding linear and quadratic trends (of the number of days since the first reported case30n each canton) to capture nonlinearities in both the reported case dynamics and general behaviortowards social distancing. (b) A model with additional week fixed effects (i. e., a weekday fixedeffect, a week fixed effect, and a trend variable in logarithmic form). This model allows us tocontrol for weekly exogenous shocks (e. g., media reports about the shortage of critical care inItaly) but acknowledge that such fixed effects would be unknown at the time of forecasting and,therefore, cannot be used to predict reported case growth at a future date. All models yield similarestimates and hence confirm the explanatory power of the mobility variables (see Appendix D.1).Second, the number of reported cases could potentially depend on the number of conductedtests per canton and day. When controlling for this, we obtain similar estimates (Appendix D.2).Third, we extend the models by including a spatial random effect, as commonly used in thespatial epidemiology and disease mapping literature
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Bayesian Analysis , 515–534 (2006).40 ata availability Human mobility data presented in this work are available from the Swisscom Mobility InsightsPlatform ( https://mip.swisscom.ch ). Cantonal geographic boundaries can be foundas shape files at the official portal of the Swiss government ( https://shop.swisstopo.admin.ch/en/products/landscape/boundaries3D ).Data on cumulative reported COVID-19 cases per canton and relative to the cantonal popu-lation size (i. e., cases per 100,000 inhabitants) come from the Federal Office of Public Healthof the Swiss Confederation; BAG ( https://covid-19-schweiz.bagapps.ch/de-1.html ). We also use this source to obtain data on the total number of daily tests conductedin Switzerland. Additional information on the Swiss population come from the Swiss FederalStatistical Office; BFS ( ). Data on the population size per Swiss canton are obtained from the SwissFederal Statistical Office ( ). Weuse the 2018 permanent resident population statistics by canton as these are the most recently pub-lished official numbers. Details on policy measure data collection are provided in Appendix A.When referring to cantons, we use abbreviations instead of full canton names ( ). Ethics declarations
Competing interests.
S.F. declares membership in a
COVID-19 Working Group by the
WorldHealth Organization (WHO) but without competing interests. Furthermore, S.F. acknowledgesfunding from the
Swiss National Science Foundation (SNSF) on data-driven health management,yet outside of the submitted work. The funding bodies had no control over design, conduct, data,analysis, review, reporting, or interpretation of the research conducted.41 thics approval.
Ethics approval (2020-N-41) was obtained from the institutional review board atETH Zurich.
Author contributions.
J.P. and S.F. contributed to conceptualization, data collection, data analysis(modeling), results interpretation, and manuscript writing. J.F.P. contributed to conceptualization,data collection, data analysis (exploratory), results interpretation, and manuscript writing.
Acknowledgments.
We thank Dominik Hangartner and Achim Ahrens for the invaluable feed-back. We also thank Swisscom for their extensive support.
Correspondence.
Joel Persson ( [email protected] )42 upplements
A Data
A.1 Policy measures
We collected data on policy measures implemented at both national level directly from the offi-cial resources of the federal government in Switzerland ( ). Implementations ofpolicy measures at sub-national (cantonal) level were collected from official resources of the can-tonal authorities (e. g., , ). These policy measures were implementedthroughout the complete canton (i. e., there is no “partial” implementation). We then checkedour data on policy measures against benchmark datasets, namely the government response eventdataset CoronaNet , the Government Response Tracker and the Swiss National COVID-19 Sci-ence Task Force ( https://ncs-tf.ch/en/situation-report ). As our goal is to studythe response and impact of mobility, we excluded policy measures that were primarily used totarget physical distance and not mobility. Examples of such policy measures is the requirementto wear a mask or the recommendation to keep a physical distance of at least 1.5 meters betweenpeople.The policy measures were encoded as follows. We encoded “school closures” such that theclosure falls on a weekday. That is, when school closures were put into effect on a Saturday, weencoded “school closures” as being closed from Monday onwards. The reason is that both primaryand secondary schools are, by default, closed on weekends and, hence, movements from/to schoolcan only be in effect on the next weekday. Different from other countries, schools were closed not“partially”, e. g., for specific age groups, but for all. We encoded “closed border” such that thispolicy measure is in effect when any side of the border is closed. As an example, when Italy closedits border, we encoded the borders for the adjacent cantons as closed. The rationale is that trips will43e cancelled if people cannot return. For all cantons without borders to other countries, we encoded“border closures” as implemented when the national government decided put travel restrictions inplace. Consider Zurich as an example. The cantonal government of Zurich did not put travelrestrictions into effect, and, hence, we set “border closures” for the canton of Zurich to March 25,2020, which is the date when the national government enforced travel restrictions. Before thatdate, travel into the canton was possible through Zurich airport and it was only restricted fromMarch 25, 2020 onward due to the national (and not the cantonal) government.The resulting list of policy measures is shown in Table 2. All of these policy measures re-mained in effect from the day of their implementation until the end of our study period (i. e.,through April 26, 2020). Furthermore, note the cross-canton variation in implementation dates forsome of the policy measures. The difference in timing for a given policy measure is one of theaspects that enables us identify the effects of the policy measures with the difference-in-differenceanalysis. These comparisons better approximate differences between a treatment and control groupwhen the cantons are comparable by having similar values on unobservable and included controlvariables in the regression model (see Model for estimating the reduction in human mobility dueto policy measures). An example of a comparable difference is that between St. Gallen (SG) andLucerne (LU). These cantons are similar in geographical size, population size, and population den-sity, but whereas SG implemented border closures on March 14, LU was not affected by borderclosures until March 25 (Table 2). Hence, between these dates, LU provide control observationsfor SG that enable us to identify the effect of borders closures in the whole country.44 mplementationdate Policy measure Description Cantons10-03-2020 Border closures Italian border closed GR, TI, VS13-03-2020 Ban >
100 Ban on gatherings with >
100 peo-ple AG,AR,AI,BL,BS,BE,FR,GE,GL,GR,JU,LU,NE,NW,OW,SH,SZ,SO,SG,TI,TG,UR,VD,VS,ZG,ZH14-03-2020 Venue closures Closure of venues TI14-03-2020 Border closures Austrian border closed GR, SG16-03-2020 School closures Closure of primary and highschools AG,AR,AI,BL,BS,BE,FR,GE,GL,GR,JU,LU,NE,NW,OW,SH,SZ,SO,SG,TI,TG,UR,VD,VS,ZG,ZH17-03-2020 Venue closures Closure of non-essential stores(all businesses except supermar-kets, food suppliers and pharma-cies), museums, zoos, hairdressers,garden centers, restaurants, night-clubs and bars AG,AR,AI,BL,BS,BE,FR,GE,GL,GR,JU,LU,NE,NW,OW,SH,SZ,SO,SG,TI,TG,UR,VD,VS,ZG,ZH17-03-2020 Border closures German border closed AG, BL, BS, SH, TG, ZH18-03-2020 Ban > > > > Table 2: Timeline of policy measure implementations across sub-national levels (cantons). Whenreferring to cantons, we use abbreviations from .45 .2 Variation in timing of policy measures across cantons
We summarize the spatio-temporal heterogeneity across cantons of when policy measures wereput into effect. For this, we first study the difference in days between cantons of when any pair oftwo policy measures were implemented. We take the absolute value of the difference in days asthe difference should not depend on which canton implemented a policy measure first. We thencalculate the mean and median of the absolute differences to obtain easily interpretable summarystatistics of the variation in timing. Table 3 shows that the policy measures are rarely implementedclose in time; instead, there is a substantial delay between implementation dates across cantons.This variation among policy measures between cantons enables us to disentangle the effects of thedifferent policy measures and affects the precision of their estimated effects. See Appendix C.6and Figure 18 for a further discussion. B a n > S c hoo l c l o s u r e s V e nu ec l o s u r e s B a n > B o r d e r c l o s u r e s Ban >
100 — 3.0 3.9 6.9 7.8School closures 3.0 — 1.0 3.9 5.6Venue closures 3.9 1.0 — 3.0 4.8Ban > B a n > S c hoo l c l o s u r e s V e nu ec l o s u r e s B a n > B o r d e r c l o s u r e s Ban >
100 — 3.0 4.0 7.0 5.0School closures 3.0 — 1.0 4.0 6.0Venue closures 4.0 1.0 — 3.0 7.0Ban > Table 3: Variation in implementation dates of policy measures across cantons (absolute time dif-ference in days): mean (left) and median (right).46
Identification strategy
This section explains how the model estimates are identified from the data. In doing so we alsoexplain the assumptions underlying the modeling.
B.1 Estimating the reduction in human mobility due to policy measuresthrough a a difference-in-difference analysis
The parameter estimates for the policy measures can be interpreted as causal effects under the fol-lowing standard assumptions for difference-in-difference (DiD) analysis: (1) the policy measureswere implemented (a) “as if” independently of the level of mobility or, (b) in the absence of anypolicy measure, all cantons would have parallel trends in mobility; (2) the canton population dis-tributions were stable over time; and (3) the implementation of each policy measure in any cantondid not affect the mobility in any other canton.Assumption (1a) implies that there are no omitted confounders of the relation between pol-icy measures and mobility. This seems credible conditional on unobserved canton-specific andweekday-specific factors, the time since the a canton’s first reported case, and a canton’s popula-tion size. We therefore adjust for these factors in our model. Note that assumption 1a is strongerthan assumption 1b but that both are sufficient for the causal interpretation together with assump-tion (2) and (3). Hence if assumption (1a) does not hold, the DiD estimates can still be interpretedas causal effects under assumptions (1b), (2), and (3) together. In our context, assumption 1b im-plies that, if no policy measures was implemented, mobility would trend (or not trend) in the sameway for all cantons. Then for a given day, the cantons that have not yet implemented a policy mea-sure provide valid control observations for the cantons that have implemented the policy measure.This assumption is inherently untestable, but likely holds since the cantons had parallel levels ofmobility prior to the policy measures (see Fig. 1 in the main paper).Assumptions (2) and (3) are together known as the stable unit treatment value assumption
69, 70 ,47hich is a standard assumption in the causal inference literature and for difference-in-differenceanalysis. Assumption 2 seems plausible given the official recommendations to limit long-distancetravel, the lockdown of many European countries at the time, and that our data cover close to thewhole Swiss population. The credibility of Assumption (3) is debatable since some people may,for precautionary reasons, reduce their mobility when a policy measure that is not yet implementedin their canton is implemented in another canton. The assumption could be relaxed by estimatingeffects of the policy measures across cantons, but this would increase the modeling complexitysubstantially and lead to more difficult interpretation of the results. Note that even if the aboveassumptions for causality do not hold, the estimates from the DiD analysis are still useful as theythen give the conditional association between mobility and policy measures.The analysis also requires the variables to follow a specific measurement order in time. Tomeasure the effect of policy measures on mobility within days, they must – for each day – beimplemented earlier than mobility is measured. This holds true in our study. The policy measureswere put into effect at midnight at their date of implementation in Switzerland, while movementswere then collected for subsequent days. Recall that, for each mobility variable, we calculate thenumber of trips by aggregating the total number over the full day. Hence, the measurement orderis such that the policy measures precede mobility.Altogether, the above elaboration points to assumptions under which the estimations can beinterpreted to be causal effects.
B.2 Estimating how reductions in mobility predict COVID-19 cases
We estimate the conditional relationship between lagged mobility and the reported number ofCOVID-19 cases using a regression model. We control for variables that confound the relationshipbetween mobility and reported cases but let the policy measures vary. The regression therebyidentifies the extent to which the level of mobility at a given day predicts the number of new casesreported in a later day when both depend on policy measures. By lagging mobility, we rule out48he possibility that the parameter estimates for mobility are biased by simultaneity or reversedcausality, meaning that there are feedback loops between mobility and reported cases or that thenumber of reported cases affect mobility.Two factors determine the delay of mobility in predicting the reported number of new cases:(1) the incubation period, and (2) the reporting delay. Previously, the mean incubation periodwas estimated to 5.1 days, while 97.5 % of infected people had symptoms within 11.5 days afterexposure . Adding a 2 day reporting delay (cf. the case data from the Federal Office of PublicHealth of the Swiss Confederation; BAG) leads to the chosen lags of mobility of 7–13 days.We fit the regression model separately for each lag and mobility variable. It would also bepossible to fit the regression with all lags of one mobility variable included. However, this approachwas discarded because of the following reason. Infection dynamics imply that the growth rate inreported cases at a given day depends on earlier growth rates. Hence, the relationship betweenmobility and the reported number of new cases at a given date is partially an outcome of theirrelationship at an earlier date. Including all lags of a mobility variable in the model would restrictthis dependence. The parameter estimate associated with each lag of mobility would then reflecthow mobility at a given day predicts the number of cases reported in the number of days aheadgiven by the lag, after the predicted change of the other lags of mobility has been removed. Toaddress this concern, our model specification includes only a single lag.For a similar reason as above, the regression model was specified to include only a singlemobility variable (and not all mobility variables at the same time). Here the rationale is as follows.Trips by mode and purpose are different subsets of total trips. A trip of one purpose is thereforealso a trip of one mode. For instance, every trip by train is either a commuter trip or non-commutertrip. Including all mobility variables in the same regression would artificially hold the variables inthe subsets fixed against each other, resulting in incorrect estimates. This is ruled out in our modelspecification where a only single mobility variable is included but where the model is re-estimatedfor different ones. 49 Model diagnostics
This section presents an analysis of model diagnostics. For reasons of brevity, we only showdiagnostics for the models with the total trips variable. The diagnostics for the models of trips byeach mode and purpose also showed good fit.The R package loo version 2.3.1 is used to estimate the Pareto tail shape parameter and,based on it, check for influential observations. The other diagnostic plots are obtained with the R package bayesplot
73, 74 version 1.7.2.
C.1 Checking posterior predictive ability
We assess model fit through posterior predictive checks. For each model, we simulate 10 drawsfrom its posterior predictive distributions with the same values of the explanatory variables as thoseused to fit the model, resulting in simulated replicates of the response variable. Consistent withthe notation in we denote these y rep . Figure 6 and Figure 7 shows kernel density estimates ofthe replicated responses and the observed responses for the models of mobility and cases. Thereplicated response densities fit the observed response density well, indicating good model fit. C.2 Checking for divergent transitions
Figure 8 shows Kernel density and time series of the MCMC samples of the parameters for themobility model. Figure 11 shows the autocorrelation among the MCMC samples of the sameparameters. Figure 9 and Figure 11 shows the same information for the models with cases asdependent variables. For brevity we omit the corresponding plots for control variables, interceptsand random and fixed effects. For each lag of total trips, the posterior density of the parameteris approximately Gaussian and that their chains have good mixing with no signs of divergence orsubstantial autocorrelation. 50 .3 Checking the effective sample size and convergence of Markov chains
Figure 12 and Figure 14 shows ratios of effective sample size to total sample size ( ˆ n eff /N ) and ˆ R values, respectively, for the MCMC samples of all parameters in the regression models. Thecorresponding plots for the models of cases are available in Figure 13 and Figure 15. The ratioof effective sample size to total sample size is between around 0.4 and 1 for most parameters,indicating a sufficient number of independent draws from the posterior distributions . The ˆ R values are close to 1 for all parameters and models. Hence, this indicates convergence of theMarkov chains. C.4 Checking for overdispersion
The estimated overdispersion parameter (for the models with total trips) is as follows:• The first model (for estimating the reduction in human mobility due to policy measures) hasan estimated overdispersion parameter of 51.33 (95 % CrI: 47.02–55.88).• The second model (for estimating the relationship between mobility and cases) has an es-timated overdispersion parameter of 3.43 (95 % CrI: 2.92–3.99) at a lag of 7 days. Theparameter estimate amounts to 3.95 (95 % CrI: 3.34–4.65) at a lag of 13 days.• In the mediation analysis, the estimated overdispersion parameter of the mediation modelamounts to 54.40 (95 % CrI: 49.49–59.54) for a lag of 7 days. The estimate increases to61.83 (95 % CrI: 56.14–67.71) for a lag of 13 days. For the outcome model, the estimatedoverdispersion parameter increases from 4.29 (95 % CrI: 3.60–5.06) at a lag 7 to an estimateof 5.42 (95 % CrI: 4.48–6.49) at lag 13.Overall, there is substantial overdispersion in the dependent variables. Because this, the use of anegative binomial distribution for the dependent variables is recommended.51 .5 Checking for influential observations
Influential observations can have a negative effect on model fit. To check for the presence ofinfluential observations, we plot the Pareto tail shape parameter k against the observation indicesfor each of the models. The results for the model of total trips are shown in Figure 16 while theresults for the model of cases on total trips are available in Figure 17. A value of k less than 0.5is generally considered good whereas a value between 0.5–0.7 is considered okay
75, 76 . No modelhas many observations with an estimated value of k above 0.5. C.6 Checking for correlations between parameters
A similar timing of the policy measures across cantons could make it difficult to distinguish theirindividual effects. To investigate this issue, Figure 18 depicts scatterplots of the pairwise bivariateposterior distributions of the policy measure parameters. There is a negative correlation betweenthe posterior samples of the parameters for some of the NPI pairs, reflecting a difficulty to distin-guish the effects of the two policy measures if they are introduced close in time. It seems to besomewhat difficult to distinguish between the effects of the ban on gatherings and school closures,as they were implemented close in time in many cantons (Table 3). The same holds for the bans ofgatherings of more than five people and border closures. In contrast, the effects of border closures,school closures, and venue closures can be distinguished, as judged by the even spread of theirpairwise scatters (Figure 18). 52 e+06 2e+06 3e+06 4e+06 5e+06 yy rep Total trips
Figure 6: Posterior predictive plot for the model of the effect of the policy measures on total trips. y and y rep are kernel density estimates of the observed response values and 10 simulated replicates,respectively. 53
100 200 300 400 500 yy rep Cases ~ Lag 7 total trips yy rep Cases ~ Lag 8 total trips yy rep Cases ~ Lag 9 total trips yy rep Cases ~ Lag 10 total trips yy rep Cases ~ Lag 11 total trips yy rep Cases ~ Lag 12 total trips yy rep Cases ~ Lag 13 total trips
Figure 7: Posterior predictive plot of the model for the effect of each lag of total trips on thereported number of new cases. y and y rep are kernel density estimates of the observed responsevalues and 10 simulated replicates, respectively.54 Total trips ~ Ban > 100 −0.15−0.10−0.05 0 200 400 600 800 1000
Chain
Total trips ~ Closed schools −0.30−0.25−0.20 0 200 400 600 800 1000
Chain
Total trips ~ Closed venues −0.4−0.3−0.2−0.1 0 200 400 600 800 1000
Chain
Total trips ~ Ban > 5 −0.32−0.28−0.24 0 200 400 600 800 1000
Chain
Total trips ~ Closed borders −0.050−0.0250.0000.0250.050 0 200 400 600 800 1000
Chain
Figure 8: Kernel density and time series plots of the MCMC samples of the parameter for the effectof each policy measure and total trips. 55
Cases ~ Lag 7 total trips
Chain
Cases ~ Lag 8 total trips
Chain
Cases ~ Lag 9 total trips
Chain
Cases ~ Lag 10 total trips
Chain
Cases ~ Lag 11 total trips
Chain
Cases ~ Lag 12 total trips
Chain
Cases ~ Lag 13 total trips
Chain
Figure 9: Kernel density and time series plots of the MCMC samples of the parameter for the effectof each lag of total trips on the reported number of new cases.56 an > 100
Lag A u t o c o rr e l a ti on Total trips ~ Ban > 100
Closed schools
Lag A u t o c o rr e l a ti on Total trips ~ Closed schools
Closed venues
Lag A u t o c o rr e l a ti on Total trips ~ Closed venues
Ban > 5
Lag A u t o c o rr e l a ti on Total trips ~ Ban > 5
Closed borders
Lag A u t o c o rr e l a ti on Total trips ~ Closed borders
Figure 10: Autocorrelation plots of MCMC samples for each chain of the parameter for the effectof each policy measure on total trips. 57 ases ~ Lag 7 total trips
Lag A u t o c o rr e l a ti on Cases ~ Lag 8 total trips
Lag A u t o c o rr e l a ti on Cases ~ Lag 9 total trips
Lag A u t o c o rr e l a ti on Cases ~ Lag 10 total trips
Lag A u t o c o rr e l a ti on Cases ~ Lag 11 total trips
Lag A u t o c o rr e l a ti on Cases ~ Lag 12 total trips
Lag A u t o c o rr e l a ti on Cases ~ Lag 13 total trips
Lag A u t o c o rr e l a ti on Figure 11: Autocorrelation plots of MCMC samples for each chain of the parameter for the effectof each lag of total trips on the reported number of new cases.58 N eff N N eff N £ eff N £ eff N > Total trips ~ Ban > 100 N eff N N eff N £ eff N £ eff N > Total trips ~ Closed schools N eff N N eff N £ eff N £ eff N > Total trips ~ Closed venues N eff N N eff N £ eff N £ eff N > Total trips ~ Ban > 5 N eff N N eff N £ eff N £ eff N > Total trips ~ Closed borders
Figure 12: Ratios of effective sample size to total sample size for MCMC samples of the parameterfor the effect of each policy measure on total trips.59 N eff N N eff N £ eff N £ eff N > Cases ~ Lag 7 total trips N eff N N eff N £ eff N £ eff N > Cases ~ Lag 8 total trips N eff N N eff N £ eff N £ eff N > Cases ~ Lag 9 total trips N eff N N eff N £ eff N £ eff N > Cases ~ Lag 10 total trips N eff N N eff N £ eff N £ eff N > Cases ~ Lag 11 total trips N eff N N eff N £ eff N £ eff N > Cases ~ Lag 12 total trips N eff N N eff N £ eff N £ eff N > Cases ~ Lag 13 total trips
Figure 13: Ratios of effective sample size to total sample size for MCMC samples of the parameterfor the effect of each lag of total trips on the reported number of new cases.60 .00 1.05 R^ R^ £ £ > Total trips ~ Ban > 100 R^ R^ £ £ > Total trips ~ Closed schools R^ R^ £ £ > Total trips ~ Closed venues R^ R^ £ £ > Total trips ~ Ban > 5 R^ R^ £ £ > Total trips ~ Closed borders
Figure 14: ˆ R values for MCMC samples of the parameter for the effect of each policy measure ontotal trips. 61 .00 1.05 R^ R^ £ £ > Cases ~ Lag 7 total trips R^ R^ £ £ > Cases ~ Lag 8 total trips R^ R^ £ £ > Cases ~ Lag 9 total trips R^ R^ £ £ > Cases ~ Lag 10 total trips R^ R^ £ £ > Cases ~ Lag 11 total trips R^ R^ £ £ > Cases ~ Lag 12 total trips R^ R^ £ £ > Cases ~ Lag 13 total trips
Figure 15: ˆ R values for MCMC samples of the parameter for the effect of each lag of total trips onthe reported number of new cases. 62
200 400 600 800 1000
Total trips ~ Ban > 100
Data point P a r e t o s hape k −0.20.00.20.40.6 0 200 400 600 800 1000 Total trips ~ Closed schools
Data point P a r e t o s hape k −0.20.00.20.40.60 200 400 600 800 1000 Total trips ~ Closed venues
Data point P a r e t o s hape k −0.20.00.20.40.6 0 200 400 600 800 1000 Total trips ~ Ban > 5
Data point P a r e t o s hape k −0.20.00.20.40.60 200 400 600 800 1000 Total trips ~ Closed borders
Data point P a r e t o s hape k −0.20.00.20.40.6 Figure 16: Estimated Pareto tail shape parameter k against the observation indices for the modelof the effect of policy measures on total trips. 63
200 400 600 800
Cases ~ Lag 7 total trips
Data point P a r e t o s hape k −0.20.00.20.4 0 200 400 600 800 Cases ~ Lag 8 total trips
Data point P a r e t o s hape k −0.20.00.20.40 200 400 600 800 Cases ~ Lag 9 total trips
Data point P a r e t o s hape k −0.20.00.20.40.6 0 200 400 600 800 Cases ~ Lag 10 total trips
Data point P a r e t o s hape k −0.10.00.10.20.30.40 200 400 600 800 Cases ~ Lag 11 total trips
Data point P a r e t o s hape k −0.20.00.20.40.6 0 200 400 600 800 Cases ~ Lag 12 total trips
Data point P a r e t o s hape k −0.20.00.20.40.60 200 400 600 800 Cases ~ Lag 13 total trips
Data point P a r e t o s hape k −0.20.00.20.40.6 Figure 17: Estimated Pareto tail shape parameter k against the observation indices for the modelsof the effect of each lag of total trips on the reported number of new cases.64igure 18: Pairwise bivariate posterior distributions of the policy measure parameters in the modelof total trips. 65 Robustness checks
In this section, we test the robustness of our model estimates against alternative specifications.
D.1 Alternative specifications of time effects
Because the data cover daily observations over a relatively long time period, there are many possi-ble ways of controlling for different time effects. We therefore perform a robustness check of ourestimates by re-fitting the models with two alternative specifications of time effects.Table 4 summarises how the main and alternative model specifications differ. Like the mainmodels, both alternative specifications include weekday fixed effects. The alternative specifica-tion (a) uses trend variables of the non-transformed and squared number of days since the first casein each canton was reported (the main specification instead uses the logarithm). Compared to themain specification, it simply postulates a linear and quadratic effect of the number of days sincethe first case. The alternative specification (b) adds week fixed-effects to the main specification.Thus, it also controls heterogeneity between weeks.A prior of N (0 , is assigned to the parameters for the linear and quadratic time trends in thesecond alternative specification. The week fixed-effects in the third alternative specification areassigned a weakly informative prior of Half-t (3 , , . . All other priors are the same as in themain specification.The model with the first alternative specification of time effects needed 6000 iterations duringMCMC estimation to converge (3000 for samples for warm-up and 3000 samples from its posteriordistributions). 66odel specificationTime-effect Main (a) (b)Weekday fixed effect (cid:88) (cid:88) (cid:88) Week fixed-effect (cid:88)
Number of days since 1st reported case (cid:88)
Squared number of days since 1st reported case (cid:88)
Log number of days since 1st reported case (cid:88) (cid:88)
Table 4: Specifications of time effects in the models of mobility on policy measures and reportednumber of new cases on lagged mobility. The symbol " (cid:88) " means that a term for the effect isincluded in the model.The models of mobility with alternative specifications of time effects give similar estimatesas the main specification (Figure 19). Specifically, replacing the logarithmic trend variable withthe corresponding linear and squared trends leaves the effects of the policy measures intact (Fig-ure 19a). As expected, adding week fixed effects to the main specification reduces the estimatedeffects of some policy measures. It also widens the credible intervals for school closures (Fig-ure 19b). Apart from this the estimates are very similar to those of the main model specification.Hence, the estimated policy effects are robust against a range of alternative controls for time ef-fects. 67
Closed bordersBan > 5Closed venuesClosed schoolsBan > 100 −30 −20 −10 0
Effect size (%)Trips
Total −40 −30 −20 −10 0 10
Effect size (%)Mode
Highway Road Train −30 −20 −10 0
Effect size (%)Purpose
Commuter Non−commuter b Closed bordersBan > 5Closed venuesClosed schoolsBan > 100 −40 −20 0 20
Effect size (%)Trips
Total −50 −25 0 25
Effect size (%)Mode
Highway Road Train −40 −20 0 20
Effect size (%)Purpose
Commuter Non−commuter
Figure 19: Estimated effect of policy measures on total trips, trips by mode and trip by purposefor three alternative model specifications: (a) weekday fixed effects and linear and quadratic trendof the number of days since the first reported case in each canton, and (b) weekday fixed effects,week fixed effects, and a trend trend of the log number of days since the first reported case in eachcanton. Posterior means are shown as dots, while 80 % and 95 % credible intervals are shown asthick and thin bars, respectively. Policy measures are arranged in the order in which they wereimplemented, shown in Appendix A.The alternative specifications of time effects are also applied to the model of the reportednumber of new cases on lagged mobility. Figure 20 shows their predictions of the percentagechange in the reported number of new cases given a 1 % decrease in mobility lagged by 7–13days. Replacing the logarithmic trend variable with linear and quadratic trends does not alterthe estimated relationship between the mobility variables and reported case growth substantially(Figure 20a). The predicted reduction in case growth given decrease in total trips is stronger than68n the main specification whereas the predicted reduction given decreases in commuter trips isslightly weaker. Adding week fixed-effects to the main specification leads to a smaller predictedchange in reported case growth given reductions in mobility (Figure 20b). For some lags of themobility variables is the predicted reduction not statistically distinguishable from zero. However,decreases in mobility still predicts reductions in reported case growth for the larger lags.Altogether, the robustness checks show that the predicted reductions in reported cases givendecreases in mobility are robust to alternative specifications of time effects. a Predicted change (%) D a y ahead Trips
Total −1.25 −1.00 −0.75 −0.50 −0.25 0.00
Predicted change (%)Mode
Highway Road Train −1.0 −0.5 0.0
Predicted change (%)Purpose
Commuter Non−commuter b Predicted change (%) D a y ahead Trips
Total −0.5 0.0 0.5
Predicted change (%)Mode
Highway Road Train −1.0 −0.5 0.0
Predicted change (%)Purpose
Commuter Non−commuter
Figure 20: Predicted change in reported COVID-19 cases given a 1 % reduction of lagged totaltrips, trips by mode, and trips by purpose for alternative specifications of time effects: (a) weekdayfixed effects and linear and quadratic trend of the number of days since the first reported case ineach canton, and (b) weekday fixed effects, week fixed effects, and a trend trend of the log numberof days since the first reported case in each canton. Posterior means are shown as dots, while 80 %and 95 % credible intervals are shown as thick and thin bars, respectively. Policy measures arearranged in the order in which they were implemented, shown in Appendix A.69 .2 Controlling for the number of tests
The number of positive reported cases depends on the number of tests performed, which can varyover the course of the pandemic. We therefore re-estimate our models of the reported number ofnew cases on lagged mobility while controlling for the number of tests performed every day. Thisis based on the assumption that cantons might not conduct equally many tests each day. Instead,the number of tests that each cantons carry out might be determined by their population size andother relevant risk factors in their population. Since test availability determines the number ofpositive tests that can be detected, we extend the model of the relation between mobility and thereported case growth with the number of daily tests per canton.To the best of our knowledge, there exists no data on the number of tests conducted per day ineach canton of Switzerland but only in the country as a whole. We therefore estimate the numberof daily tests conducted per canton with the the number of daily tests in the whole of Switzerlandstandardized to the canton population shares. That is, we divide the number of daily tests inSwitzerland as a whole with each cantons population share.Formally, let u it be the true unknown number of tests reported in canton i in day t and let u t = (cid:80) Ni =1 u it be the total and observed number of reported tests in the country in day t . Weestimate the number of tests in canton i in day t with ˆ u it = u t × E i (cid:80) Ni =1 E i , (12)where E i is the population in canton i , and (cid:80) Ni =1 E i is the population in the country. The estimate ˆ u it thereby approximates the number of COVID-19 tests conducted in each canton and day underthe assumption that the number of daily tests per canton is proportionate to the share of the pop-ulation that lives in each canton. We include the logarithm of ˆ u it in the model so that it estimatesthe percentage change in the reported number of new cases when the estimated number of dailytests increases by 1 %. We assign the log-transformed variable a prior of N (1 , , meaning that we70xpect the effect size to be 1 % on average, but likely not negative or larger than 2 %.After controlling for the number of daily tests per canton, decreases in mobility lagged by 10–13 days still predict reductions in the reported number of new cases. Our findings are thereforequalitatively the same for the majority of lags. However, the above analysis should be interpretedwith caution as it may be subject to post-treatment bias. The reason is as follows. Given anincrease in mobility, hospitals may in anticipation of a growth in cases in the near future start tocarry out more tests. Then, for a given day, the number of tests is (partially) an outcome of previouslevels of mobility. As a consequence, variation in the reported number of new cases that should beattributed to changes in mobility gets incorrectly attributed to changes in the number of tests. As aconsequence, the parameter estimate for lagged mobility is biased. Such post-treatment bias mightexplain the near-zero and positive estimates observed for some lags of mobility after controllingfor the number of tests. 71 Modeling extensions
This section includes extensions of the models featured in the paper. Estimation details are inAppendix G.
E.1 Accounting for the spatial dependence in mobility among neighboringcantons
Our main models treat mobility as being independent across cantons. Modeling cross-sectionalunits as independent is a conventional approach in panel and longitudinal data analysis. However,one would reasonably expect that the level of mobility is similar for neighboring cantons but notso similar for geographically distant cantons. If a dependence in mobility between cantons is notmodeled, it will enter the error term, violating the assumption that error terms are independentacross cantons. For this purpose, we extend the mobility model with a spatial random effect thatcaptures the similarity in mobility among neighboring cantons prior to any policy measure.Let φ ( k ) i be a spatial random effect for canton i that has an additive effect on mobility variable k on the log scale. The model for the effect of the policy measure on mobility variable k is thengiven by log E [ M itk | η ( k ) it , E i ] = log E i + α ( k ) + θ ( k ) i + φ ( k ) i + δ ( k ) w ( t ) + γ ( k ) log z it + γ ( k ) B Ę log z i + L (cid:88) l =1 β ( k ) l d itl . (13)The spatially-extended model may be viewed as a longitudinal version of the Besag-York-Mollié(BYM) model commonly used for disease mapping applications in epidemiology. The term“disease mapping” refers to statistical methods that estimates the spatial correlation between ob-servations and where interest lies in the distribution of a disease over geographical units. The BYMmodel, in turn, is a Bayesian hierarchical Poisson regression model with two types of random ef-72ects: (1) random effects that capture spatial correlation between units, and (2) random effectsthat captures unobserved heterogeneity between units . Our spatially extended model adapts theoriginal BYM model with the use of a negative binomial distribution to account for the overdis-persion in the trip counts. It also adapts the original BYM model by treating the random effectsfor unobserved heterogeneity as Mundlak-style correlated random effects. This enables us to getcorrect inference even if the date that each cantons first case is reported depends on unobservedcanton-specific factors.As the original BYM model, we use an intrinsic conditional auto-regressive (ICAR) model asa prior on the spatial random effect φ i . The ICAR model of the conditional distribution of each φ i is φ i | { φ j : j ∈ N i } ∼ N (cid:32) |N i | (cid:88) j ∈N i φ j , τ |N i | (cid:33) , (14)where N i is the set of neighbors of canton i , meaning the cantons that share a border with thecanton, and |N i | denotes the cardinality of N i , that is, its number of neighbours. For practicalimplementation, the sets of neighbors can be constructed by defining a symmetric N × N adjacencymatrix whose element in row i and column j equals 1 if cantons i and j share a common border,and 0 otherwise.The ICAR model states that each φ i is conditionally Gaussian with mean equal to the averageof its neighbors spatial random effects and a variance that decreases with more neighbors. Theterm τ is a hyperparameter for the variance of the composite random effect υ i = θ i + φ i , where θ i is the random effect for unobserved heterogeneity. Based on derivations in , the joint distributionof φ = ( φ , . . . , φ N ) is proportional to a particular pairwise difference form: p ( φ | τ ) ∝ exp (cid:32) − τ (cid:88) i,j : j ∈N i ( φ i − φ j ) (cid:33) = exp (cid:16) − τ φ (cid:62) Qφ (cid:17) . (15)73ere, Q is the precision matrix, e. g., the inverse spatial covariance matrix, that has entries Q i,j = |N i | , if i = j, − , if j ∈ N i , , else . The pairwise difference form (15) shows that the ICAR model penalises large differences in thevalues of neighboring cantons spatial random effects. Hence, finding the values of { φ i } Ni =1 thatminimize these differences leads to local spatial smoothing . However, the distributions in (14)and (15) are improper priors since they define the value of each cantons spatial effect relative to itsneighbors. As a consequence, the priors do not identify the overall mean among { φ i } Ni =1 . To solvethis, the soft constraint N − (cid:80) Ni =1 φ i ∼ N (0 , . is used.Another problem is that the random effects φ i and θ i cannot be separately identified. Thus,heterogeneity across cantons that should be attributed to θ will be modeled as spatial correlationby φ even when no spatial dependence is present. To solve this issue, we use the reformulatedBYM2 model . It involves a scaling and reparameterization of υ = θ + φ , explained next.Let φ ∗ i := φ i / √ κ be the spatial random effect for canton i after scaling. The scaling factor κ is calculated from the neighborhood structure over all cantons and ensures that V [ φ ∗ i ] ≈ V [ θ i ] ≈ for each canton i . The scaled reparameterization of υ i = θ i + φ i is √ τ (cid:16)(cid:0)(cid:112) − ϕ (cid:1) θ i + √ ϕφ ∗ i (cid:17) . (16)The corresponding covariance matrix of υ = ( υ , . . . , υ N ) is V [ υ | τ, ϕ ] = 1 τ (cid:16) (1 − ϕ ) I + ϕ Q −∗ (cid:17) . (17)Here, Q −∗ is the generalized inverse of the scaled precision matrix Q ∗ , ϕ ∈ [0 , is a mixing pa-74ameter, and /τ is the marginal variance of log relative mobility that is explained by the combinedrandom effect υ . The fraction of this variance explained by the canton random effect θ and thescaled spatial random effect φ ∗ are (1 − ϕ ) and ϕ , respectively .Summarizing, the random effect θ for between-canton heterogeneity has prior distribution MVN ( , I ) and the spatial random effects φ has prior marginal distribution MVN ( , Q −∗ ) andprior conditional distribution given by (14) for each φ i . See Riebler et al. for details. Replacing υ i = θ i + φ i in (13) with (16) gives the explicit equation for the model of mobility variable k withthe BYM2 spatial random effect.Accounting for potential spatial dependence in mobility among neighboring cantons does notalter the estimates of the policies (Figure 21). In other words, the spatial dependence in baselinemobility is low. We therefore do not include the BYM2 spatial random effect in our main models. Closed bordersBan > 5Closed venuesClosed schoolsBan > 100 −30 −20 −10 0
Effect size (%) Spatial
NoYes
Figure 21: Estimated effect of policy measures on total trips with and without a BYM2 ICARrandom effect for spatial dependence between neighboring cantons. Posterior means are shown asdots, while 80 % and 95 % credible intervals are shown as thick and thin bars, respectively. Policymeasures are arranged in the order in which they were implemented, shown in Appendix A.
E.2 Accounting for the dependence between the mobility variables
The variables for trips by mode or reason are different subsets of the total trips. As a result,the trip variables are dependent on each other. One way in which this dependence could ariseis through the variation in baseline trip counts across cantons prior to any reported COVID-1975ase and policy measure. As an example, we would expect the cross-canton variation in baselinecommuting trips to be correlated with the cross-canton variation in baseline non-commuting trips.Since each regression equation’s canton random effect θ ( k ) = ( θ ( k )1 , θ ( k )2 , . . . , θ ( k ) N ) gives the cross-canton variation in mobility variable k at baseline, we can estimate the dependence between themobility variables at baseline with the correlation coefficient between the canton random effects.To do so, we model θ (1) , θ (2) , . . . , θ ( K ) as jointly multivariate Gaussian distributed: θ (1) θ (2) ... θ ( K ) ∼ MVN ( , Ω M ) , Ω M = σ θ (1) I σ θ (2) θ (1) I . . . σ θ ( K ) σ θ (1) I σ θ (1) θ (2) I σ θ (2) I . . . σ θ ( K ) σ θ (2) I ... ... . . . ... σ θ (1) θ ( K ) I σ θ (2) θ ( K ) I . . . σ θ ( K ) I = Σ M ⊗ I , (18)where σ θ ( k ) is the variance of θ ( k ) = ( θ ( k )1 , θ ( k )2 , . . . , θ ( k ) N ) of regression equation k = 1 , , . . . , K ,the term σ θ ( k ) θ ( p ) is the corresponding covariance between θ ( k ) and θ ( p ) from regressions equations k and p , respectively, and the K × K covariance matrix Σ M contains these variance and covarianceterms. Moreover, I is an N × N identity matrix and the symbol ⊗ denotes the Kronecker product.We then obtain the correlations between all pairs of canton random effects for each pair of k, p = 1 , , . . . , K , k (cid:54) = p , by computing the Pearson coefficient ρ θ ( k ) θ ( p ) = σ θ ( k ) θ ( p ) (cid:113) σ θ ( k ) σ θ ( p ) . (19)The correlation coefficients are collected as off-diagonal elements in the K × K correlation matrix R M corresponding to Σ M and can be interpreted as summaries of the dependence between themobility variables’ baseline relative trip counts across cantons in the absence of any policy measureand reported COVID-19 case. By modeling this dependence, we obtain more precise estimatesof uncertainty in parameter values than estimating each mobility variable’s regression equationseparately. The approach is analogous to that of seemingly unrelated regressions , except that76ach regression model is a negative binomial generalized mixed model with a log-link.Figure 22 shows estimates of the effect of the policy measures on mobility from the multivariatemodel (Figure 22a) against the estimates from the separate regressions per mobility variables forwhich the correlation between their canton random effects is not modeled (Figure 22b). Compar-ing the multivariate and separate regressions, we find that the point estimates given by the posteriormeans are practically identical. There is a small tendency of the credible intervals in the multivari-ate model to be tighter. Note that estimating the multivariate model requires considerably morecomputational runtime. 77 Closed bordersBan > 5Closed venuesClosed schoolsBan > 100 −20 −10 0
Effect size (%)Trips
Total −40 −30 −20 −10 0
Effect size (%)Mode
Highway Road Train −30 −20 −10 0
Effect size (%)Purpose
Commuter Non−commuter b Closed bordersBan > 5Closed venuesClosed schoolsBan > 100−30 −20 −10 0
Effect size (%)Trips
Total −40 −30 −20 −10 0
Effect size (%)Mode
Highway Road Train −30 −20 −10 0
Effect size (%)Purpose
Commuter Non−commuter
Figure 22: Estimated effects of policy measures on total trips, trips by mode, trips by purposefrom ( a ) the joint multivariate model over all mobility variables, and ( b ) the separate regressionsper mobility variable. The joint multivariate model estimates the correlation between the baselinelevels of the mobility variables across cantons by modeling the canton random effects as drawnfrom a multivariate Gaussian distribution. Posterior means are shown as dots, while 80 % and 95 %credible intervals are shown as thick and thin bars, respectively. Policy measures are arranged inthe order in which they were implemented, shown in Appendix A.78 Mediation analysis of the role of mobility in the effect of pol-icy measures on reported cases
We perform a mediation analysis to decompose the effects of policy measures on reported COVID-19 cases into (1) their direct effects, meaning the part of their effects that is due to behavioraladaptions not related to mobility, and (2) their indirect effects, meaning the part of their effects thatis explained by changes in mobility. The mediation analysis is performed by combining our tworegression models into a structural equation model. Estimation details are provided in Appendix G.
F.1 Mediation model
The structural equation model consists of the mediation model log E [ M i,t − s,k | η ( k,s,M ) i,t − s , E i ] = µ ( M i,t − s,k )= log E i + α ( k,s,M ) i + δ ( k,s,M ) w ( t − s ) + γ ( k,s,M ) log z i,t − s + L (cid:88) l =1 β ( k,s ) l d i,t − s,l α ( k,s,M ) i = α ( k,s,M ) + θ ( k,s,M ) i + γ ( k,s,M ) B Ę log z i (20)and the outcome model log E [ Y it | η ( k,s,Y ) it , E i ] = µ ( k,s ) ( Y it )= log E i + α ( k,s,Y ) i + δ ( k,s,Y ) w ( t ) + γ ( k,s,Y ) log z it + ψ ks log m i,t − s,k + L (cid:88) l =1 λ ( k,s ) l d i,t − s,l α ( k,s,Y ) i = α ( k,s,Y ) + θ ( k,s,Y ) i + γ ( k,s,Y ) B Ę log z i + ψ ks,B Ğ log m ik . (21)79or notation, the bar again denotes the average value of an expression. The superscripts M and Y are added to indicate that the parameter values are generally different for the mediation model andoutcome model with the same mobility variable k lagged by s days. The conditional variance of M i,t − s,k and Y it is given by V [ M i,t − s,k | η ( k,s,M ) i,t − s , E i , ζ ( k,s,M ) ] = µ ( M i,t − s,k ) (cid:18) µ ( M i,t − s,k ) ζ ( k,s,M ) (cid:19) (22)and V [ M i,t − s,k | η ( k,s,Y ) it , E i , ζ ( k,s,Y ) ] = µ ( k,s ) ( Y it ) (cid:18) µ ( k,s ) ( Y it ) ζ ( k,s,Y ) (cid:19) . (23)Here, ζ ( k,s,M ) and ζ ( k,s,Y ) is the overdispersion parameter from the mediation model and outcomemodel, respectively, in which mobility variable k is lagged by s days.The mediation model is identical to the model that links policy measures to mobility, but haslagged values of all time-varying variables since their effects on the reported number of new casesare delayed. The outcome model is the same as the model that links mobility to reported cases, butnow also includes the policy measures. The reason for this is that the structural equation modelshall estimate (1) the effects of mobility conditional on the policy measures, and (2) the effects ofthe policy measures conditional on mobility. Here, (1) gives the mediation effect of mobility thatenables us to estimate the indirect effect of policy measures via mobility, and (2) gives the directeffect of policy measures that explained by other behavioral adaptions. The added superscripts m and y signifies that the parameters generally take different values in the mediation model andoutcome model.To facilitate the interpretation of our results, we state how the parameters of interest shall beinterpreted in the mediation analysis:• β ( k,s ) l is the effect of policy measure l on the log expected number of trips on mobility80ariable k .• λ ( k,s ) l is the direct effect of policy measure l lagged by s days on the log expected numberof reported cases, conditional on mobility variable k lagged by s days and the other policymeasures lagged by s days . Formally, this is the change that would occur in the reportednumber of new cases if (lagged) policy measure l would have been implemented but mobilitywould not have changed as a consequence of policy measure l .• ψ ks is the expected percentage change in the reported number of new cases as the numberof trips on mobility variable k lagged by s days increases by 1 %, conditional on all policymeasures lagged by s days being held fixed.• β ( k,s ) l × ψ ks is the indirect effect of policy measure l via mobility variable k , both laggedby s days, on the log expected reported number of new cases. The indirect effect gives theexpected change that would occur in the reported number of new cases if policy measure l lagged by s days would always have been implemented but mobility variable k lagged by s days would have changed as if the policy measure would vary as in the data.• λ ( k,s ) l + β ( k,s ) l × ψ ks is the total effect of policy measure l lagged by s days on the log expectednumber of reported cases. It is the effect of policy measure l with the mobility variable k changing as a consequence of implementing the policy measure. Hence, the total effect givesthe change in the reported number of new cases due to any behavioral adaption.As shown in the above, we compute the indirect effect using the product method and sumthe direct and indirect effect to obtain the total effect. This is applicable to linear Bayesian mul-tilevel models with random intercepts and no mediator-treatment interaction (i. e., no treatment-moderated mediation), as the models of this study.We check the assumption of no treatment-mediator interaction by, for every lag of 7–13 days,re-estimating the outcome model (21) with an interaction term between each policy measure and81he mobility variable. We find that the parameters of the interaction terms have point estimatesclose to zero with some of the credible intervals covering zero. We take this as sufficient evidenceof no treatment-mediator interaction and, therefore, proceed with the outcome model without suchterms.The directed acyclic graph in Figure 23 visualises the relationships between the variables in thestructural equation model at a given day. For simplicity, we denote weekday, time and unobservedcanton factors lagged by s days with the vector X i,t − s . The structural relationships among thevariables are as follows: At day t , the weekday, the number of days that has passed since thefirst case was reported in the canton, and unobserved canton-specific factors jointly determinethe number of trips in the canton on each mobility variable and whether each policy measure isimplemented in the canton that day. The weekday, number of days that has passed since the firstreported case, and unobserved canton-specific factors also determine the reported number of newcases, but with a lag of s days due to incubation periods and reporting delay. Whether a policymeasure is implemented in a canton at day t affects the trip count on each mobility variable insame the canton that day (path β ( k,s ) l ). Given the implementation of policy measures at day t , thelevel of mobility affects the number of reported cases s days later (path ψ ks ). Hence, each policymeasure has an immediate effect on mobility, which is transferred into a delayed indirect effecton the reported number of new cases through mobility (the product of paths β ( k,s ) l and ψ ks ). Eachpolicy measures also has a direct effect on the number of reported cases via behavioral adaptionsnot related to mobility. This effect is also subject to a lag of s days (path λ ( k,s ) l ).82 i,t − s,k D i,t − s,l Y it X i,t − s β ( k,s ) l λ ( k,s ) l ψ ks Figure 23: A directed acyclic graph of the structural equation model within a fixed point in time.The vector X i,t − s denotes the weekday, time, and canton effects in the models.Note that every time-varying variable in the mediation model is lagged by s days, but that in theoutcome model, only the mobility variable and the policy measures are subject to this lag. Withthis specification, we estimate how the instantaneous effect of the policy measures on mobilityvariable k in day t − s affect the number of reported new cases in day t directly and indirectly viamobility.For estimation, we allow the canton random effects θ ( k,M ) = ( θ ( k,s,M )1 , . . . , θ ( k,s,M ) N ) and θ ( k,s,Y ) = ( θ ( k,s,Y )1 , . . . , θ ( k,s,Y ) N ) of the mediator model and outcome model, respectively, to becorrelated. We thereby account for possible correlation between the baseline relative mobility andbaseline relative (reported) infection risk across cantons. We achieve this by modeling the random83ffects as drawn from a bivariate Gaussian distribution: θ ( k,s,M ) θ ( k,s,Y ) ∼ MVN ( , Ω M,Y ) , Ω M,Y = σ θ ( k,s,M ) I σ θ ( k,s,M ) θ ( k,s,Y ) I σ θ ( k,s,Y ) θ ( k,s,M ) I σ θ ( k,s,Y ) I = Σ M,Y ⊗ I . (24)Here, σ θ ( k,s,M ) and σ θ ( k,s,Y ) is the cross-canton variance of the random effect of the mediation modeland outcome model ( k, s ) , respectively, and the off-diagonal term is the cross-canton covariancebetween the models’ respective random effects. The × covariance matrix Σ M,Y contains thesevariance and covariance terms. Computing the Pearson correlation coefficient ρ θ ( k,s,M ) θ ( k,s,Y ) = σ θ ( k,s,M ) θ ( k,s,Y ) (cid:113) σ θ ( k,s,M ) σ θ ( k,s,Y ) (25)then gives the correlation between the random effects of mediation model and outcome model ( k, s ) . The correlation coefficient is in turn the off-diagonal element in the × correlation matrix R M,Y that corresponds to the covariance matrix Σ M,Y .We estimate the structural equation model separately for mobility and the policy measureslagged by 7–13 days to see how the direct, indirect and total effects vary across the delay of theeffects. To save space, however, we only estimate the models with the total trips mobility variableas a mediator.
F.2 Identification
We now state the identification conditions and assumptions required for the mediation analysis.Imai et al.
84, 85 propose a “sequential ignorability” assumption under which the direct and indirecteffects are identified if the structural equation model is linear and contains no treatment-mediatorinteraction. The assumption consist of two parts. In the setting of this study, the assumptions are:841) each policy measure is implemented independently of mobility and the reported number of newcases, and (2) within observations with the same status on the policy measures, the number of tripson the mobility variable is independent of the potential number of reported cases. The first partof the assumption means that the control variables, the fixed effects, and the random effects in themodels are sufficient to remove any confounding in the relationships between the policy measuresand mobility as well as any confounding in the relationships between policy measures and thereported number of new cases. The second part of the assumption implies that, if we also conditionon the policy measures, there is no confounding of the relationship between mobility and thereported number of new cases. This is a stronger assumption since it must hold for every observedcombination of the policy measures in the data. Another assumption necessary for identifyingindirect effects is that the mediator is measured without error. Since our data cover every trip madewith a mobile device the assumption likely holds in this study.
F.3 Results of mediation analysis
The results from the mediation analysis with total trips as a mediator are shown in Figure 24.
Direct effects.
The ban on gatherings of more than 5 people has the strongest estimated directeffect on the reduction in the reported number of new cases. Its estimated 95 % credible intervalsrange between a reduction of 18.8–38.7 % (at a lag of 13 days) to a reduction of 35.9–50.5 % (ata lag of 8 days). The credible interval of the direct effect of bans on gatherings of more than100 people includes zero for lags 7–8 days. Hence, bans on larger gatherings may have a longerdelayed direct effects than bans on smaller gatherings. Border closures do not appear to havedirectly reduced the reported number of new cases.
Indirect effects.
The estimated indirect effects show that mobility mediates the effects ofpolicy measures on the reported number of new cases. The largest indirect effects are found forvenue closures and bans on gatherings of more than 5 people. Both policy measures are estimatedto have reduced the reported case growth with around 6–8 % at the higher order lags indirectly85ia mobility. Of particular interest is the estimated indirect effect for border closures. Recall thatthe estimated direct effects of border closures include zero. Hence, this implies that the effect ofborder closures occurs exclusively through mobility. This is to be expected considering that borderclosures may not affect other types of behavior than mobility.
Total effects.
Combining the direct effects and indirect effects gives the total effects. Again,bans on gatherings of more than 5 people, bans on gatherings of more than 100 people, and schoolclosures have the largest estimated total effects, in part due to their pronounced indirect effects.For any of the days that schools were closed, the estimates imply that there would on average be21.2 % (95 % CrI: 9.48–32.3 %) more reported cases at the 10th day ahead and 34.1 % (95 % CrI:24.3–42.9 %) more reported cases at the 13th day ahead if schools would instead have remainedopen. Note that for several lags, the indirect effect of venue closures makes up more than a thirdof its total effect.Overall, the mediation analysis demonstrates that the effects of social distancing policies op-erate – to a large degree – through mobility. Hence, telecommunication data provides valuableinformation for estimating effects of policy measures aimed at reducing mobility.86 D a y ahead Ban > 100 −40 −30 −20 −10 0
Closed schools −30 −20 −10 0 10
Closed venues −50 −40 −30 −20 −10 0
Ban > 5 −10 0 10 20
Closed borders
Direct effect b D a y ahead Ban > 100 −10.0 −7.5 −5.0 −2.5 0.0
Closed schools −10 −5 0
Closed venues −10 −5 0
Ban > 5 −4 −3 −2 −1 0
Closed borders
Indirect effect c Effect size (%) D a y ahead Ban > 100 −40 −30 −20 −10 0
Effect size (%)
Closed schools −40 −30 −20 −10 0
Effect size (%)
Closed venues −40 −20 0
Effect size (%)
Ban > 5 −10 0 10 20
Effect size (%)
Closed borders
Total effect
Figure 24: Mobility mediates the effect of policy measures on the reported number of new cases.Estimated (a) direct effect of policy measures, (b) indirect effect of policy measures via total trips,and (c) total effect of policy measures on the 7th to 13th day ahead. Posterior means are shownas dots, while 80 % and 95 % credible intervals are shown as thick and thin bars, respectively.Policy measures are arranged from top to bottom in the order in which they were implemented (cf.Supplement A).
F.4 Sensitivity analysis for the assumption of no unmeasured confounding
The mediation analysis rests on the sequential ignorability assumption of no unmeasured con-founding in both the mediation model and the outcome model. Now, if there exists unobservedconfounders that affect both the mediator and the outcome, then those variables will be part of87oth models’ error terms. Hence, under the assumption of no unmeasured confounding, thereshould be no correlation between the models’ respective errors. Based on this observation, Imaiet al. propose a sensitivity analysis based on the correlation between the residuals in the twomodels.To conduct the sensitivity analysis, we draw 4000 samples (e. g., as many as the length of theparameters’ Markov chains, including the warm-up) from the models’ predictive posterior distri-butions, each of the same size as the number of observations in the data. For each of the models,we subtract these values from the observed responses to get posterior samples of the model’s pre-dictive errors. We then compute the Pearson correlation coefficient between the mediation model’spredictive errors and the outcome model’s predictive errors for each of the 4000 posterior errorsamples.Figure 25 shows kernel density estimates of the Pearson correlation coefficient distributions.The mean correlation coefficient is for each lag close to zero with the 95 % credible intervalscovering zero. Hence, the sensitivity analysis suggests that there is no unmeasured confounding inthe mediation analysis. 88 Lag 7
Pearson correlation coefficient D en s i t y −0.20 −0.10 0.00 0.10 Lag 8
Pearson correlation coefficient D en s i t y −0.20 −0.10 0.00 0.10 Lag 9
Pearson correlation coefficient D en s i t y −0.2 −0.1 0.0 0.1 0.2 Lag 10
Pearson correlation coefficient D en s i t y −0.2 −0.1 0.0 0.1 0.2 Lag 11
Pearson correlation coefficient D en s i t y −0.2 −0.1 0.0 0.1 Lag 12
Pearson correlation coefficient D en s i t y −0.20 −0.10 0.00 0.10 Lag 13
Pearson correlation coefficient D en s i t y Figure 25: Kernel density estimates of the distribution of Pearson correlation coefficients betweenposterior samples of the mediation model’s and outcome model’s respective predictive errors. Theshaded regions are 95 % credible intervals computed as the the area between the 2.5th and 97.5thpercentile Pearson correlation coefficient. 89
Estimation details for modeling extensions and mediationanalysis
Table 5 shows our choices of prior distributions for the parameters in the extended models. Thenew parameters are (1) the parameters for the direct effects in the mediation analysis, (2) theICAR parameters in the spatial model, (3) the correlation coefficients for the dependence betweenthe canton random effects in the outcome model and mediation model, and (4) the correlationcoefficients for the dependence between the canton random effects in the multivariate mobilitymodel. Below we explain the priors for the new parameters. The remaining priors are explained inSection 1.As in the main paper, the prior on β l reflects that we expect each policy measure to reducelog expected trips by on average 25 %. The prior on the direct effect parameter λ l means that weexpect each policy measure to reduce the log expected number of reported cases with on average25 % conditional on mobility, with a variance such that increases in reported cases due to the policymeasures are unlikely. Moreover, the prior on the mean and variance of ψ ks implies that we expectmobility to predict half as large reduction in cases once we condition on the policy measures,but that negative parameter estimates are still unlikely. Note that in the spatially extended model,the canton random effects are assigned a prior of N (0 , due to the BYM2 reparametrization andscaling of the combined canton and spatial random effects. Moreover, the prior on the spatialrandom effects is technically on its conditional distribution given its neighbouring cantons spatialrandom effects, not its marginal distribution as it is for simplicity written in Table 5. The ICARparameters are assigned priors according to the recommendations by Morris and Gelman et al. .The prior on the canton random effects correlations is, per the standard procedure in Stan, assignedon the Cholesky factor of the correlation matrix. We assign a LKJ correlation distribution with ashape parameter of 2 which makes extreme correlations less likely. All variables that are alsoincluded in the non-extended models are given the same priors as in those models, see Table 1 in90he paper. The ICAR model is estimated via the implementation available in the brms package.See for details. Parameter Description Prior Model β l Policy measure l N ( − . , . (13), (20) λ l Policy measure l N ( − . , . (21) ψ ks Log mobility variable k with a lag of s N (0 . , . (21) θ i Canton random effect N (0 , σ θ ) (20), (21), (18) σ θ Standard deviation for canton random effect
Half - t (3 , , . (20), (21), (18) θ i Canton random effect in the spatial model N (0 , (13) α Intercept
Half - t (3 , . , . All δ w ( t ) Weekday w compared to Monday N (0 , . All γ Log no. of days since 1st reported case N (1 , All γ B Between-canton average of log no. of days since 1st reported case N (0 , All ψ ks,B Between-canton average of log mobility with a lag of s N (0 , (21) ζ Overdispersion in dependent variable
Gamma (0 . , . All φ i Spatial random effect ICAR model (14) (13) ϕ ICAR mixing parameter
Beta (0 . , . (13) τ − . ICAR standard deviation
Half - t , , . (13) R M,Y
Cross-mediation-outcome random effect correlation matrix
LKJ (2) (24) R M Cross-mobility variable random effect correlation matrix
LKJ (2) (18)
Note:
The superscripts ( k ) and ( k, s ) are omitted as the same priors are assigned to each model. The column“Description” states what effect the associated parameter represent (except for the overdispersion parameter). Table 5: