key: cord-0629601-3w652qp5
authors: Aliverti, Emanuele; Russo, Massimilano
title: Dynamic modeling of the Italians' attitude towards Covid-19
date: 2021-08-02
journal: nan
DOI: nan
sha: a8bf607e479e29ea863e1602067fd3b283196222
doc_id: 629601
cord_uid: 3w652qp5

We analyze repeated cross-sectional survey data collected by the Institute of Global Health Innovation, to characterize the perception and behavior of the Italian population during the Covid-19 pandemic, focusing on the period that spans from April to November 2020. To accomplish this goal, we propose a Bayesian dynamic latent-class regression model, that accounts for the effect of sampling bias including survey weights into the likelihood function. According to the proposed approach, attitudes towards Covid-19 are described via three ideal behaviors that are fixed over time, corresponding to different degrees of compliance with spread-preventive measures. The overall tendency toward a specific profile dynamically changes across survey waves via a latent Gaussian process regression, that adjusts for subject-specific covariates. We illustrate the dynamic evolution of Italians' behaviors during the pandemic, providing insights on how the proportion of ideal behaviors has varied during the phases of the lockdown, while measuring the effect of age, sex, region and employment of the respondents on the attitude toward Covid-19.

The outbreak of the covid-19 pandemic has impacted our world, with more than 100 million infected people, and more than 2.6 million deaths at March 2021 (who, 2021) . From an economic perspective, the covid-19 outbreak and the national measures to contain the spread of the disease lead to severe economic recessions in many sectors, such as tourism, accommodation, and food services (e.g., Papadimitriou and Cseres-Gergelyne Blasko, 2020) . Psychological and social effects on the population are less immediate to quantify, but some preliminary results suggest that the pandemic has affected also these aspects of human life (e.g., Cullen et al., 2020; Balkhi et al., 2020; apa, 2020) .

In this article, we study the evolution of behaviors in compliance with some measures that prevent the spread of covid-19 (see Table 1 ). We focus on the Italian population, that represents an interesting case study, because Italy was the first European country to introduce a national lockdown to limit the spread of covid-19, imposing behavioral changes in the population. The adopted measures are likely to have had a psychological and social effect proportionate with their severity and duration. In fact, some early reports, based on online surveys, suggest an increased level of distress, anxiety, and fear (e.g., Forte et al., 2020; Moccia et al., 2020; Motta Zanin et al., 2020) . Monitoring these aspects, quantifying their evolution over time and characterizing their impact on individuals, is of primary importance to evaluate the wellbeing of a population. Similarly, it is of interest to evaluate the compliance with covid-19 policies, that can largely depend on personal status (e.g., Carlucci et al., 2020) .

Variations of personal routines and practices have been documented; for example, the reduction of social activities and gatherings and the changes in mobility patterns, such as the increase of work-fromhome practices and the reduction of public transport use (e.g., Google llc, 2021; Bavadekar et al., 2020) . These behaviors reflect the compliance with the government regulations, as well as the internalization of different recommendations to reduce the spread of the virus, that changed day-to-day life. However, compliance with spread reducing measures depends upon personal conditions, psychological status, and many other personal factors (e.g., Wolff et al., 2020) , and it is likely to change at during different stages of the pandemic.

During the first phase of the pandemic (up to March 2020), there is evidence that the Italian population has scrupulously followed government measures (e.g., Graffigna et al., 2020; Barari et al., 2020) ; however, to the best of our knowledge, no analysis has been performed to assess if such compliance is constant over time, across socio-demographic groups, or proportional to the severity of the measures adopted by the Italian government throughout the 2020. To quantify these aspects, we analyze survey data provided by the Institute of Global Health Innovation (ighi) at the Imperial College of London, in collaboration with the company YouGov (Jones, 2020) , described in Section 2.

We describe the attitude of the Italian population toward covid-19 throughout the pandemic with a dynamic Bayesian latent class model (e.g., Lazarsfeld, 1950; Dayton and Macready, 1988) . These models assume that the population is composed of H groups corresponding to ideal behaviors, that can represent different attitudes towards covid-19. At any given time, each subject composing the population is associated with one of these ideal profiles, and the categorical variables characterizing behaviors are modeled as conditionally independent given the profile memberships. Latent class models are conceptually simple and have been used as a building block for several methods to analyze categorical variables; for example, in problems including survey weights (Vermunt and Magidson, 2007; Patterson et al., 2002) or when interest is on characterizing temporal dependence across contingency tables (e.g., Kunihama and Dunson, 2013) or differences among groups of subjects (e.g., Russo et al., 2018) . See also Fruhwirth-Schnatter et al. (2019) , Chapters 9 and 11.5, for further references.

The paper is structured as follows: Section 2 describes the data, Section 3 introduces the proposed Bayesian latent-class regression model including survey weights, Section 4 illustrates the results, and, finally, Section 5 provides a brief discussion.

We analyze repeated cross-sectional survey data provided by the Institute of Global Health Innovation (ighi) at the Imperial College of London, in collaboration with the company YouGov (Jones, 2020) . Data are publicly available for research purposes at the repository https://github.com/YouGov-Data/ covid-19-tracker, along with a brief description of the collected variables. This survey aims to investigate how different populations responded to covid-19, gathering self-reported data on several aspects of the pandemic, including objective measurements, such as testing results and observed symptoms, and subjective measurements, such as daily behaviors.

We are interested in subjective measurements describing the compliance with national preventive regulations. This measurements include questions on self-isolation, avoidance of social gatherings, frequency of hand washing, use of hand sanitizer and contact with other people, among many others. The complete list of variables used in this study is reported in Table 1 . Subjects respond to these questions with "Not at all", "Rarely", "Sometimes", "Frequently" and "Always", according to their level of agreement. Table 1 includes 17 out of the 20 subjective measurements available in the survey. The removed items were considered uninformative for the population behavior in relation with the covid-19 measures adopted by the Italian government. For example, one removed question asked whether children living in Table 1 . List of analyzed survey items with code, label, and description. Subjects can respond to each item with "Not at all", "Rarely", "Sometimes", "Frequently" and "Always", according to their level of agreement with each survey item.

Label Description i12 health 1 ih1 Worn a face mask outside your home (e.g. when on public transport going to a supermarket or going to a main road) i12 health 2 ih2

Washed hands with soap and water i12 health 3 ih3 Used hand sanitiser i12 health 4 ih4 Covered your nose and mouth when sneezing or coughing i12 health 5 ih5 Avoided contact with people who have symptoms or you think may have been exposed to the coronavirus i12 health 6

ih6 Avoided going out in general i12 health 7

ih7 Avoided going to hospital or other healthcare settings i12 health 8

ih8 Avoided taking public transport i12 health 11 ih11

Avoided having guests to your home i12 health 12 ih12

Avoided small social gatherings (not more than 2 people) i12 health 13 ih13

Avoided medium-sized social gatherings (between 3 and 10 people) i12 health 14 ih14

Avoided large-sized social gatherings (more than 10 people) i12 health 15 ih15

Avoided crowded areas i12 health 16 ih16

Avoided going to shops i12 health 17 ih17

Slept in separate bedrooms at home when normally you would share a bedroom i12 health 18 ih18

Eaten separately at home when normally you would eat a meal with others i12 health 19 ih19

Cleaned frequently touched surfaces in the home (e.g. doorknobs, toilets, taps) the same household were going to school; however, schools have been closed from March to September and students could only attend lectures remotely.

We focus on T = 20 survey waves conducted from April to November 2020. Notably, the survey waves are not administered at regular time-intervals; for example, 4 survey waves were submitted in April and May, and only 1 in July and August; see Table 2 for details on the waves administration dates. Without loss of generality, we will indicate this time grid with T = {t 1 , . . . , t T }. In each wave t ∈ T , cross-sectional data were collected for a representative sample of n = 1000 statistical units, indexed as i = 1, . . . , n. Each unit is associated with a sampling weight w i (t) and a vector of covariates x i (t) = [x i1 (t), . . . , x im (t)] , including age, sex, region of residence ("North-West", "North-East", "Center", "South", "Islands") and employment status ("Full-time employment", "Part-time employment", "Not working", "Student" and "Retired"). Note that unit i at wave t indicates a different subject than unit i at wave t = t; for this reason, we include the wave index t in the covariate vector as x i (t), even if the covariates x i (t) can be considered fixed over time. Refer to Jones (2020) for further information.

The repeated cross-sectional nature of the data (also referred to as "pseudo-panel" in economoetrics; e.g., Verbeek, 2008) does not allow to follow the attitude of specific individuals across the pandemic. Moreover, researchers from ighi have completed different privacy assessments to avoid re-identification of the respondents from the released data, safeguarding the anonymity of respondents (Jones, 2020) . Therefore, we focus on the global attitude of Italian population toward covid-19, providing an interpretable low-dimensional representation of the data. 

Let y i (t) = (y i1 (t), . . . , y ip (t)) denote the responses of subject i to the survey items outlined in Table 1 during survey wave t; without loss of generality, we encode these responses as y ij (t) ∈ {1, . . . , d} for j = 1, . . . , p, with d = 5 and p = 17. We indicate with y(t) = {y i (t), i = 1, . . . , n} the observed responses for the n interviewed subjects during the survey wave t. Our main goal is to study the evolution of the Italian population's behaviors over time and during different stages of the covid-19 pandemic. To accomplish this goal, we specify a low-dimensional dynamic model for multivariate categorical data, that characterizes the time-varying probability mass function of y i (t) in terms of a set of static latent classes and dynamic class-memberships. Since at each survey wave we observe n statistical units chosen according to a survey design, we adjust the proposed model likelihood to obtain an approximation of the population parameters.

In order to describe the evolution of y(t) over time, we assume that the population can be divided into H ideal behaviors (or profiles) that express different attitudes toward covid-19. These ideal profiles can correspond to different behavioral patterns, such as people that rigidly observed all rules and directives to avoid the spread of the disease, or people that kept behaving as usual, ignoring the emergency. The interpretation and the structure of the ideal behaviors is considered fixed over time, while their proportion can change dynamically. For example, it is likely that before seeing the effect of covid-19 several people were not concerned about using public transport or having guests at home. However, as the disease spread and the effects became more evident, these inclinations could potentially have reversed for a subset of individuals. This evolution is modeled relying on a dynamic latent-class regression model, graphically described in Figure 1 . For each wave t, subject i is associated to one of the H profiles via the discrete latent variable z i (t) ∈ {1, . . . , H} with probability ν ih (t) = pr(z i (t) = h | x i (t)) that depends on the observed covariate vector x i (t). Given the profile membership z i (t) = h, the probability that subject i responds with c j to the survey item j is denoted as ψ

. . , d} and j = 1, . . . , p. Therefore, we assume that the profiles' attributes are constant over time, and can be characterized by the conditional probabilities of responding with a certain category to the different items, namely

hcj , c j = 1, . . . , d, j = 1, . . . , p}. Instead, the profile-specific membership varies across the survey waves, and according to subject-specific covariates. Specifically, the evolution of the profile memberships are modeled trough the vector ν i (t) = (ν i1 (t), . . . , ν iH (t)) , with ν ih (t) denoting the probability that subject i of survey wave t belongs to the h-th profile and H h=1 ν ih (t) = 1. This parameter is decomposed into two quantities: a profile-specific component η h (t) and a subject-specific effect β h x i (t) that linearly depends on their observed covariates, with β h = (β 1h , . . . , β mh ) . The dynamic component η h (t) characterizes the temporal evolution of the profiles' memberships, while the linear term for the effect of subject-specific covariates.

Isolating the individual and the profile-specific component-which is shared across the populationis particularly relevant in our setting, where we analyze repeated cross-sectional data, including different subjects across survey waves. Indeed, this structure allows to estimate the shared dynamic component η h (t), adjusting for the different composition of the cross-sectional population, and measuring the effect of demographic information on the probability of belonging to a certain profile.

According to this specification, the model depicted in Figure 1 can be formalized as

(1)

Using the first latent profile as a reference, we let η 1 (t) = 0 and β k1 = 0 for k = 1, . . . , m and interpret each β kh as the effect of the k-th covariate on the log-odds of belonging to profile h, instead of the first one, as in multinomial logit regression (e.g., Azzalini and Scarpa, 2012) . The conditional independence assumption among the p categorical variables, and the inclusion of the temporal component in the mixture weights, leads to substantial dimensionality reduction in the number of model parameters, while incorporating borrowing of information across survey waves. Following Equation 1, the likelihood contribution for subject i in the survey wave t can be expressed as

where 1[A] denotes the indicator function for the event A. To mitigate the effect of potential bias due to the survey design, we follow the approach described in Vermunt and Magidson (2007) and Patterson et al. (2002) , and incorporate the survey weights into the model via

exponentiating each likelihood contribution in (2) to the corresponding survey weight w i (t). The pseudo likelihood in (3) is used as building-block of several likelihood-based procedures that include survey weights, for example Godambe and Thompson (1986) , Rabe-Hesketh and Skrondal (2006) , and Skinner and Mason (2012) , and recently in some Bayesian methods (e.g., Gunawan et al., 2020) . Alternatively, in a Bayesian setting, one can use the survey weights to approximate the population generative mechanism, and inferring characteristics of the non-sampled units (e.g., Si et al., 2015) , or, when available, use strong prior information to correct for the sampling bias (Gao et al., 2021) . However, a weighted likelihood approach is conceptually simpler and computationally more efficient when the sampling mechanism is unknown, as in our application (Gunawan et al., 2020) .

We consider a Bayesian approach to inference, using the pseudo-likelihood outlined in Equation (3). An advantage of this approach is that prior regularization can avoid convergence issues of maximization algorithms, such as Expectation-Maximization (em), when used in latent-class regression (e.g., Linzer et al., 2011; Durante et al., 2019; Bolck et al., 2004) . Also, a Bayesian approach facilitates the modeling of temporal dependencies across survey waves leveraging a hierarchical dynamic model. To describe the evolution of the profiles over time, we rely on a Gaussian Process (gp) specification for the profile-specific parameters ν h (·); refer, for example, to Rasmussen and Williams (2006) for an introduction to gp. This choice allows to deal with unequally spaced time grids, to flexibly characterize the dynamic evolution of profile memberships, and to directly model values of ν i (t) for t / ∈ T . Recalling that the first group is fixed as a reference, we let for each latent group h = 2, . . . , H

denoting a squared-exponential covariance function and ϑ 1h ∈ R + , ϑ 2h ∈ R + and ϑ 1h ∈ R + corresponding to the variance, length-scale and noise variance parameters, respectively (e.g., Rasmussen and Williams, 2006, Chapter 2.3) . The squared exponential function favors smooth transitions across time points, with the parameters ϑ h = {ϑ 1h , ϑ 2h , ϑ 3h } controlling the overall structure of the covariance function. The gp prior for η h (·) evaluated on the survey waves grid T results in a multivariate Gaussian distribution,

with 0 T denoting a T -dimensional zero vector, and Σ h a T × T covariance matrix with elements (Σ h ) ls . Prior specification is completed selecting: independent log-normal distributions with log-mean 0 and log-standard deviation 10 for the components of ϑ h , standard Gaussian distribution for the coefficients β lh , and symmetric Dirichlet distributions for the profile-specific conditional probabilities ψ 

Posterior inference relies on 4000 iterations collected after a burn-in of 1000. Convergence was assessed via graphical inspection of the trace-plots, auto-correlation function and convergence diagnostics. All chains mixed well, with an effective sample size larger than 3500 for all parameters. We did not observe label switching across the chains; see Appendix A.2 for further details. Simulating the 5000 draws from 91  86  62  89  93  50  89  94  92  80  95  98  95  49  28  21  47   4  12  23  8  4  38  6  3  7  16  5  2  5  36  3  5  31   1  1  8  1  1  8  2  1  1  2  0  0  0  11  3  7  13   1  0  3  1  0  2  1  0  0  1  0  0  0  2  4  7  5   3  0  5  1  2  2  2  1  0  1  0  0  0  2  62 

In this Section, we describe the composition of the three considered latent profiles and their response pattern Ψ h , for h = 1, 2, 3, estimated via posterior mean and reported in Figure 2 . The first profile is composed of individuals that scrupulously followed spread-preventive measures. Individuals in this group, with a probability of approximately 90%, always wore masks outside home (ih1), washed their hands frequently (ih2), and avoided going out in general (ih6). With probability close to 30%, they always slept in separate bedrooms when normally sharing the room (ih17), and with probability 21% always ate in separate rooms even when normally sharing the meals (ih18). Additionally, with a probability of 89% they avoided going to an hospital or health care institution (ih7). Also, they avoided any small gathering with probability of 80% (ih12). We will refer to this group as "meticulous" in the rest of the article.

The second profile is composed of individuals that moderately followed preventative measure. Compared to the previous group, a smaller fraction of individuals avoided to go out (5% probability of responding "Always", to item ih6). In particular, subjects in this group completely avoided going to shops with probability 6% (compared to the 49% of the meticulous group, item ih16). A larger fraction of individuals considered small gathering as safe; in fact, the probability of completely avoiding small gatherings is of 16%, compared with the 80%, of the meticulous group (ih12). Large gatherings are avoided also in this group, with a probability of replying "Always" equal to 73% (ih14). Compared to the meticulous group, the probability of always sleeping in separate bedrooms decreased to 10% (compared to 30%, item ih17), and the probability of always having separate meals decreased to 6% (compared to 21% item ih18). Finally, the probability of avoiding to go to an hospital decreased to 49%. We will refer to this group as "moderate".

The third profile is composed of individuals with a more lenient attitude towards covid-19 measures. In this group the probability of always wearing a mask outside home is 64% compared to 91% and 88% for meticulous and moderate (ih1); in particular, the probability of using a mask outside home "Sometimes" or "Rarely" is 20%. The probability of responding "Always" to "Avoided going out in general" drops to 2%, while it increases to 50% for the response "Not at all" (ih6). This profile is characterized by more permissive behaviors also for gatherings. For example, the probability of responding "Not at all" to items referring to avoiding gatherings is 36%, 24% and 16% for small, medium and large gatherings, respectively (items ih12, ih13 and ih14). Notably, the probability of responding "Always" to "avoiding crowded places" is only 17%, compared to 57% and 95% of the moderate and meticulous group (ih15). We refer to this group as "permissive".

To summarize, the three estimated latent profiles can be interpreted in terms of level of compliance with covid-19 preventive measures. Some behaviors are similar in the three profiles; for example the modal class for sleeping in the same bed or eating in the same room is "Not at all" in all the profiles, even if the distribution of the responses is different across groups. Other behaviors switch across profiles; for example in the item "Avoiding going out in general" (ih6), we observe modal category "Always", "Sometimes" and "Not at all" for the meticulous, moderate, and permissive group, respectively. Figure 3 illustrates the effects of the covariates on the log-odds of being assigned to the moderate or permissive group against the meticulous group, that is used as baseline. Our empirical findings suggest that older respondents are more likely to be in the meticulous group rather than in the moderate or permissive ones. Males are more frequently associated with the permissive group compared to females. We also observe a clear regional effect: individuals living in the south of Italy or in the Islands report higher probabilities to be associated with the meticulous group. Lastly, students, retired and people that are not currently working show an higher probability to be associated with the meticulous group rather than moderate or permissive ones. These results are in line with what reported in Carlucci et al. (2020) , and references therein.

To characterize the evolution of the attitudes towards covid-19, we rely on the properties of the gp, providing daily predictions based on a new set of locations T , with equally spaced points between the first and last survey wave. These predictions are mapped to the proportion of Italians associated to each profile described in Section 4.1, at any given time, by setting the values of the covariates in Equation 1 at the Italian national averages. Posterior estimates are displayed in Figure 4 .

Each panel in Figure 4 is divided in three areas, separated by dashed lines that correspond to different phases in Italian covid-19 policy. The first phase in Italian lockdown lasted to June 3, and included several preventative measures such as avoiding leaving the house for non-essential reasons. In the second phase (June 3-October 13) shops, bars and restaurants were open to the public, although appropriate social distance was still required. The third phase (after October 13) correspond to the second lockdown, after covid-19 cases increased over Europe.

According to our analysis, in the interval from April 2 to May 18, about 86% of the Italian population observed a meticulous behavior, following most of covid-19 preventive measures. In the same period, the proportion of individuals in the moderate profile was 11%, and only 3% for the indulgent group. This is in agreement with what reported in Graffigna et al. (2020) and Barari et al. (2020) , although with different methodologies and data.

On May 18, the prime minister Conte introduced an easing of the lockdown restrictions, allowing most businesses to open to the public. Commuting across Regions was still banned until the official announcement of phase 2, on June 3. In this period, we notice a rapid variation in the composition of the profiles, with the proportions at June 3 corresponding to 43%, 47%, and 10% for the meticulous, moderate, and permissive group, respectively.

In the period ranging from July to October the proportion of Italians in the three profiles is essentially constant across time, with values close to 14%, 57% and 29%, respectively. In this phase, also the number of confirmed cases is approximately constant (see Figure 5 ), and most of the population has a moderate attitude towards covid-19. Although almost one third of the population is in the permissive group, it is worth noting that most of their behaviors, such as not wearing a mask outside, were allowed in this period.

After a new lockdown was enforced on October 13, we observe significant changes in the proportions of Italians associated with each profile. For example, the proportion of Italians in the meticulous group in the last three observed waves (October 15, 23 and November 11) correspond to 19%, 29% and 47%. For the moderate group we observe values of 62%, 60% and 46% while the permissive drop to 18%, 10%, and 6%. It is worth noting that despite the reported number of cases of covid-19 on November 11 was comparable to the early phase of the pandemic, only half of the population shows a meticulous attitude toward covid-19, compared to 86% of the first lockdown. The permissive group also presents an higher proportion: 6%, compared to 3% of the first phase.

These results suggest that, although the increase in the number of cases was comparable between November and April, the behaviors and the reactions of the population were different, with the second phase characterized by less meticulous behaviors than the previous lockdown. This result is partly coherent with the different rules adopted by the Italian government and their regional differences, but can be also due to a slow transition towards more meticulous behaviors after the easing of the restrictions during the summer. Another possible explanation involves different level of awareness on the disease compared to the first lockdown. For example, in the first period of the pandemic, it was recommended to clean streets and surfaces with disinfectant to avoid the infection; this practice was later flagged as an exaggerate behavior (e.g., Goldman, 2020).

We analyzed Italian attitude towards covid-19 leveraging a dynamic Bayesian latent class regression model for survey data. The estimated latent profiles can be interpreted as different degrees of precaution that determine the compliance with the national rules. At the population level, the proportion of Italians associated with each of these profiles follows the various phases of the lockdown. This suggests that Italians have followed the national rules.

There are several potential future directions for this work. When more recent data will be released, it would be interesting to analyze how the composition of the profiles changes in relation to the Christmas's holidays, and to include in the analysis information on availability of vaccines. A further research direction would be to simultaneously study the evolution of attitude towards covid-19 in several countries, comparing different nations and highlighting the main differences. However, such an extension is not immediate, since the interpretation of the profiles in different country might be substantially different and highly influenced by cultural aspects. M a r 3 0 A p r 0 6 A p r 1 3 A p r 2 0 A p r 2 7 M a y 0 4 M a y 1 1 M a y 1 8 M a y 2 5 J u n 0 1 J u n 0 8 J u n 1 5 J u n 2 2 J u n 2 9 J u l 0 6 J u l 1 3 J u l 2 0 J u l 2 7 A u g 0 3 A u g 1 0 A u g 1 7 A u g 2 4 A u g 3 1 S e p 0 7 S e p 1 4 S e p 2 1 S e p 2 8 O c t 0 5 O c t 1 2 O c t 1 9 O c t 2 6 N o v 0 2 N o v 0 9 N o v 1 6

Daily variations of the total number of cases

Daily variations in the total number of covid-19 cases in Italy. The data used for the plot can be downloaded at https://lab24.ilsole24ore.com/coronavirus/. Fig. 6 . Natural logarithm of the BIC for separate Latent class regressions, estimated at each survey wave. The number of profiles ranges from 2 to 5, and the models include age, gender, region of residence, and employment status as covariates for the latent profiles. Figure 6 shows the logarithm of BIC for separate latent class regression models, estimated separately at each survey wave. The model with H = 2 shows an higher BIC for the waves from July 25 to October 15, while it is comparable with the other models for the remaining waves. This is coherent with the results in Figure 4 , described in Section 4.2, since the proportion of subjects associated with the permissive group is very low outside this range; therefore, this group could be unnecessary in that time range. With the exception of the wave on July 25, the model with H = 3 does not show significantly higher BIC compared with models including more profiles. In most waves, the model with H = 3 is the one with lower BIC, therefore representing a good candidate model. Figure 7 shows the probabilities for the item "Worn a face mask outside your home" estimated with separate latent class regressions, and H = 3. We can notice that most of the probabilities are similar over time, with overlapping error bands. The other variables described in Table 1 present similar results. Therefore, we can conclude that the profile-specific structure can be reasonably modeled as constant across survey waves. Figure 8 shows traceplots of the marginal posterior distributions for some parameters of the model described in Section 3. The chains exhibit good mixing, showing no jumps between profiles, that would indicate label switching. Traceplots for the remaining model parameters behave similarly, and can be reproduced using the code available at url-availble-upon-acceptance. 

Data analysis and data mining: An introduction

Psychological and behavioral response to the coronavirus (COVID-19) pandemic. Cureus

Evaluating COVID-19 public health messaging in italy: Selfreported compliance and growing mental health concerns

Google COVID-19 search trends symptoms dataset: Anonymization process description (version 1.0)

Estimating latent structure models with categorical variables: One-step versus three-step estimators

Demographic and attitudinal factors of adherence to quarantine guidelines during COVID-19: The italian model

Mental health in the Covid-19 pandemic

Concomitant-variable latent-class models

A nested expectation-maximization algorithm for latent class models with covariates

The enemy which sealed the world: Effects of COVID-19 diffusion on the psychological state of the italian population

Handbook of Mixture Analysis

Improving Multilevel Regression and Poststratification with Structured Priors

Parameters of superpopulation and survey population: Their relationships and estimation. International Statistical Review / Revue Internationale de Statistique

Exaggerated risk of transmission of COVID-19 by fomites. The Lancet Infectious Diseases

Google llc (2021) Google COVID-19 community mobility reports

Measuring Italian citizens' engagement in the first wave of the COVID-19 pandemic containment measures: A cross-sectional study

Bayesian weighted inference from surveys

Imperial College London YouGov Covid data hub, v 1.0. Imperial College London Big Data Analytical Unit and YouGov Plc

Bayesian modeling of temporal dependence in large sparse contingency tables

The logical and mathematical foundation of latent structure analysis

polca: An R package for polytomous variable latent class analysis

Affective temperament, attachment style, and the psychological impact of the COVID-19 outbreak: an early report on the Italian general population

A preliminary evaluation of the public risk perception related to the COVID-19 health emergency in Italy

Economic sectors at risk due to COVID-19 disruptions: will men and women in the EU be affected similarly?

Latent class analysis of complex sample survey data: application to dietary data

Multilevel modelling of complex survey data

Gaussian Processes for Machine Learning. Adaptive Computation and Machine Learning

Bayesian inference on group differences in multivariate categorical data

Bayesian Nonparametric Weighted Sampling Inference

Weighting in the regression analysis of survey data with a crossnational application

RStan: the R interface to Stan

Patients with depression and anxiety surge as psychologists respond to the coronavirus pandemic

Pseudo-panels and repeated cross-sections

Latent class analysis with sampling weights: A maximumlikelihood approach

High boredom proneness and low trait self-control impair adherence to social distancing guidelines during the COVID-19 pandemic

Profile 2Profile 1 M a r 2 3 M a r 3 0 A p r 0 6 A p r 1 3 A p r 2 0 A p r 2 7 M a y 0 4 M a y 1 1 M a y 1 8 M a y 2 5

The model presented in Section 3 relies on two main assumptions: (i) the data can be represented via a mixture of H = 3 multinomial distributions; (ii) the three mixture components are approximately constant across waves. To assess if assumption (i) is realistic in our application, we estimate separate models for each of the survey wave, with a number of profiles H ranging from 2 to 5. For each model we compute the BIC as a measure of model fit. For assumption (ii), for each of the model with H = 3 components, we inspected the probabilities for each item and each profile, checking if they were significantly different across survey waves.