It is superior in several respects to other functions used to probe structure, such as magnetization and structure factors. statistical complexity. Finally, we argue that the information theoretic and computational mechanical analyses of spatial patterns capture the intrinsic computational capabilities embedded in spin systems—how they store, transmit, and manipulate configurational information to produce spatial structure.We compare and contrast three different, but complementary views of “structure” and “pattern” in spatial processes. For definiteness and analytical clarity, we apply all three approaches to the simplest class of spatial processes: one-dimensional Ising spin systems with finite-range interactions. These noncritical systems are well-suited for this study since the change in structure as a function of system parameters is more subtle than that found in critical systems where, at a phase transition, many observables diverge, thereby making the detection of change in structure obvious. This survey demonstrates that the measures of pattern from information theory and computational mechanics differ from known thermodynamic and statistical mechanical functions. Moreover, they capture important structural features that are otherwise missed. In particular, a type of mutual information called the  Authors’ Note: What follows is a manuscript completed in 1998 (Santa Fe Institute Working Paper 98-04-026) and that, for various reasons, has not appeared in the peer-reviewed literature until now, although much of the material covered below was included in the first author’s PhD dissertation on which the second author was advisor ..18]. Thocommunication channel consisting of an information source that produces messages which are encoded and passed through the channel. A receiver then decodes the channel’s output in order to recover the original messages. Key to his analysis was the definition of the source’s rate of information production, called the source entropy rate, and the maximum carrying capacity, called the channel capacity, of the (possibly noisy and error-prone) channel.In the late 1940s Shannon founded the field of communication theory , motivatMuch earlier, Hartley had proposed to measure the amount of information from a source via the logarithm of the number of possible source messages . ShannonIt is important to emphasize that the core of information theory concerns not so much the various definitions of information and entropy, but rather the relationship between the source entropy rates that can be sustained through channels and those channels’ capacities. These connections are what makes the similarity between Boltzmann’s notion of thermodynamic entropy and Shannon’s entropy rate so notable. Boltzmann clearly did not anticipate Shannon’s use of entropy.The primary results on which information theory is built and with which it finds its technological applications are Shannon’s two central coding theorems. The first theorem says that information cannot be transmitted error-free through a channel at a rate higher the channel’s capacity. The second theorem says that as long as the source’s rate respects this limit then there exists an encoding and decoding scheme for the source’s messages such that error-free transmission is possible and can occur at rates arbitrarily close to the channel capacity.The mathematical foundations of Shannon’s communication theory followed quickly ,69, as dH of a distribution, the information gain I between two distributions. When adapted and applied to different communication problems, these are the quantities in which the results of the theory are expressed. It is noteworthy that many uses of information theory in statistical physics and in nonlinear dynamics mostly employ its basic quantities and do not use the more characteristic and central aspects of coding.The basic quantities used in information theory are various forms of Shannon entropy: the entropy X that assumes values Shannon entropyX is defined by:x but of the distribution X. X; see App. 2 of Ref. =apparent spatial memory. (We say “apparent” here in order to leave room for an important distinction made later.) If the configurations are perfectly random or periodic with period 1, then Equations  and 44)44) allowbits. This is clear in Equation , in, in43), Equation  have uniEquation  tells usIt follows immediately that any periodic sequence of period L-blocks in order to recover the asymptotic entropy density—that is, to see just how random each spin is. The entropy convergence plot, L, of the entropy growth curve In the previous section we interpreted the excess entropy as the total amount of information that must be extracted from measuring of Refs. ,78 and aL as:We follow Refs. ,78 and dFrom the boundary conditions on rmations ,78.The total excess entropy was used by Crutchfield and Packard in Refs. ,79,80,81 Györgyi , by Csor Györgyi , and by  Györgyi .Excess entropy was recoined “stored information” by Shaw  and subsssberger  and by Lssberger . Excess ssberger . The quassberger .There have been other prior discussions of using information theory to measure a source’s structure. Most germane to the discussion above, Del Junco and Rahe used excess entropy to classify information sources in terms of their memory . More geA number of the above notions have also recurred in more recent discussions of modeling information sources ,91. See It should be emphasized that there are subtle but significant differences in these works’ notions of effective complexity, memory, and information. One of the main subtleties is found in their differing approaches to minimality of the representations used to measure these quantities. Minimality is crucial for being able to conclude that a given quantity estimated from a model actually describes an intrinsic structural property of a process and is not an artifact of some unarticulated representational choice—a key issue to which we shall return repeatedly in the following. In contrast to the above, Refs. ,90 are eThe preceding observations on the nature of entropy convergence stay within the framework of information theory—a largely statistical view of “structure”. These quantities and a number of the preceding observations have been known for at least a decade and a half, if not longer.how the system’s memory is organized and utilized. Computational mechanics [In the previous section we saw that the excess entropy echanics  addresseechanics ,94.) In how a system processes information: How is information stored, transmitted, and transformed? How much memory is needed to statistically reproduce an ensemble of configurations and how is this memory organized? In general, we are interested in inferring the intrinsic computation being performed by the system itself.Like statistical mechanics, computational mechanics is concerned with a large system consisting of many individual components. However, computational mechanics addresses very different issues. The motivating questions of computational mechanics center around intrinsic computation [By putation  we mean al state . For examn→∞xn=a . This itmn→∞xn=a  in the fIn contrast, when we ask about intrinsic computation, we are interested not in manipulating a system to produce an output that is useful to us—which is akin to an engineering stance towards nature. Instead, we are interested in examining the information processing that the system itself performs and the underlying mechanisms that support it—which is more of a scientific stance: exploring how nature works on its own terms.As a concrete example, consider the two-dimensional nearest-neighbor Ising model at the critical temperature. Here, the correlations between spins decay with a power law as a function of distance, yet the total magnetization of the system remains zero. Computational mechanics is concerned with what sorts of effective information processing the system must perform to reach and maintain the critical state. How much historical and spatial memory is required? How is the memory organized internally? What spatial patterns result? Are the critical configurations in any way “harder” to reach than those found at low or high temperatures? More informally, how does the system balance up and down spins so that the correlations decay as a power law, while keeping zero magnetization?Whereas statistical mechanics starts with a system’s Hamiltonian or a description of its constituents’ local space-time behavior and interactions, computational mechanics begins with the joint probability distribution over the state space trajectories. With knowledge of this joint distribution, the intrinsic computation being performed by the system can be determined. By not requiring a Hamiltonian, computational mechanics can be applied in a wide range of contexts, including those where an energy function for the system may not be manifest.In any case, as noted above, the two microscopic starting points in the many-body setting—a Hamiltonian or the joint probabilities of configurations over time—are related (at equilibrium) to each other by the usual canonical ensemble,Rather than launching into the mathematical development, we begin our review of computational mechanics with several very simple examples. These will lead quite naturally to the definitions put forth in the subsequent section.statistically reproduce refers to the ability to generate infinite configurations whose finite-length spin blocks occur with the same probabilities as those in the original, infinite configuration.The questions we shall be addressing for each example are: How can one statistically reproduce a given infinite configuration using the minimal amount of memory? In particular, how much information about the left half must be remembered to produce the right half? Here, optimally predictive model as one that correctly estimates all of a process’s statistics. Suppose we have a process described by the joint distribution L spins K-spin block Another, equivalent way of stating these questions is: How much memory is needed to optimally predict configurations? We define an L block has probability Consider a string of heads (H) or tails (T) generated by a fair coin toss:A moment’s reflection reveals that one does not need to keep track of any variables. Since the coin tosses are independent, knowledge of previous tosses does not reduce the uncertainty about the next toss. As a result, for this particularly simple example no memory is required to optimally predict subsequent variables. Here, the predictions are as good as they can be , which admittedly is not good at all. The uncertainty about the next coin toss is complete. The result could be either heads or tails with equal probability, as reflected by an entropy rate What must one do in order to perform this optimal prediction? Equivalently, how can one statistically reproduce the configuration? The answer to these questions is illustrated by the probabilistic finite-state machine of The machine operates as follows. Start in state It is important to emphasize that no larger machine—i.e., with more states or edges—is required to reproduce all strings in the class of which Consider a string consisting of a sequence of all The finite-state machine for Now consider an infinite, alternating-spin configuration:How can we use the machine of does matter. Suppose we choose to start always in state However, in another sense, the state in which we start most definitely start state to the model. This is shown in There is an easy remedy for this situation: start in state In this example, the start state is a transient state. It is never revisited after the machine outputs the first spin value and moves to state must follow, since the configuration never exhibits two adjacent 1’s. After seeing the 0 that follows the first 1, a 0 or a 1 can occur with equal probability—the only rule is that every other symbol is 0. The probabilistic finite-state machine for this configuration is shown in Finally, consider an infinite binary string in which every other symbol sampled from the alphabet Note that the uncertainty associated with predicting the next symbol changes as one moves back and forth between state Despite the examples’ simplicity, a few summarizing remarks are in order before moving on to formalize the notion of “effective” state that we just used implicitly.First, note that the coin-toss configuration Second, note that entropy rate synchronize. For the period-2 example, we argued that the transient state Third, transient states tell one how the machines Last, note that we have taken care to construct minimal machines. That is, the machines put forth are such that if one removes any state or transition then one can no longer exactly statistically reproduce the configuration. This notion of minimality will be made more precise below. In complementary fashion, in each example one gains no predictability by elaborating any of the machines by adding states or transitions.The preceding section considered models capable of reproducing the distribution of all finite length blocks observed in several translationally invariant configurations. This section presents a general procedure for constructing such models—minimal, optimally predictive, probabilistic state machines.First, we need to formalize the intuitive arguments through which the “effective” states of the four example systems were discovered. The key step is identifying the notion of effective state with the conditional probability distribution over right-half configurations. Furthermore, a central property is that the resulting representation be minimal. When constructing an optimally-predictive, minimal state-machine description, there is no need to distinguish between different left-half configurations that give rise to an identical state of knowledge about the right-half configurations that follow it. Maintaining a distinction between two such states adds to the model’s size without increasing its predictive ability. Therefore, we will be looking for the smallest set of predictive states.L at site i: To make these ideas precise, consider the probability distribution of all possible right halves We now use this form of a conditional probability to define an equivalence relation ∼ on the space of all left halves. We say that two left-half configurations at different lattice sites are equivalent (under ∼) if and only if they give rise to identical distributions over right-half configurations, conditioned on those left-halves. Formally, we define the relation ∼ by:In a setting in which the conditional probabilities Equation . Implemecausal states and denoted The equivalence classes over:We shall use the convention that causal states predecessor set L spin blocks that lead to causal state The causal states, as determined by the equivalence classes induced by Equation , give trsee Ref. ). AlthouIn contrast, recurrent states are those visited infinitely often and that have large (positive measure) predecessor sets in the same limit. By considering how the process synchronizes—i.e., how it reaches the recurrent states as successively longer blocks are generated—it is possible to construct the transient states and their transitions from knowledge of the recurrent states alone. The procedure by which this is done is given in We denote the set of causal states by  setting ,41. Let There is a mirror image definition of causal states obtained by scanning the lattice in the opposite direction (right to left); which thus uses distributions conditioned on right-half configurations. Since we will study a restricted class of systems that respect this symmetry, the causal states will be the same regardless of the scanning direction. In the general case, in which this reversal symmetry need not hold, it is possible to find different causal states if one scans rections .t, we have:Assuming scan-direction invariance, the causal states factor the joint distribution over configurations into conditionally independent distributions. That is, at time learning ,101.As we saw above, for the period-2 system there are 3 causal states, denoted in s uniquely determines the subsequent causal state s determines the next causal state Once the set  to find:Tαβ(s)≡Prdeterministic. A transition from state s is uniquely determined by s. That is, not determine what symbol was produced as the transition was made. For example, either a 0 or a 1 can be produced upon a transition from state This is the sense in which the causal state representation is Equation , assumins. In other words, Equation  indicates, we obtain the stochastic connection matrix Summing over the spin values For the period-2 machine of T(s=↑)=010010andT(s=↓)=00ϵ-machine [The set -machine —of the os. Recall that we demand that the derived and not added on as an axiom to force a desired feature.An Equation , [25,102quation (Pr(s|Sα)≠quation (Pr(s|Sα)≠The “l states . For exareconstruction. In either case, the goal is to factor the joint distribution over spin configurations into the causal state conditional distributions. The result is an In the following, we determine It is instructive to compare the representation class of automata ,94. UnliA further restriction on these nonprobabilistic Unlike discrete-state automata, however, functions of Markov chains [probabilistic discrete-state automata [stochastic deterministic finite automata [hidden Markov models [The v chains , probabiautomata , or stocautomata ,41, sincv models .For more general processes than 1D finite-range spin systems l states ,41. Thusleast powerful model class. Probabilistic finite-memory machines are employed first; that is, we simply assume Given that ontinuum ,41. If tontinuum . The latontinuum ,94.It was suggested in the introduction that, in a statistical mechanics context, using the most compact mathematical entity that provides a complete description of a system is an important way to distinguish between systems that are structured in different ways. The determination of an is the algebraic structure determined by the causal states and their transitions. If, for example, the Furthermore, an le, Ref. . Despitele, Ref. . ExampleIn summary, an An X as the average minimal code size, the informational size of the distribution statistical complexity [different interpretations. The excess entropy measures the apparent memory stored in the configurations, since it is determined directly from the spin-block distribution; that is, from the spin observables. In contrast, not equal.To predict successive spins as one scans a configuration from left to right, one must track in which causal state the process is, since knowledge of the causal state gives the required conditional distribution for optimal prediction. Thus, by the same arguments given earlier that motivate and interpret the Shannon entropy mplexity :(69)Cμ≡HEquation . Like thtopological complexity Another, coarser measure of the l states ;(70)C0= of Ref. .)We claimed above that an First, note that the sequence of causal states is Markovian. The probability of a transition from recurrent state s. Each word matrix However, we are interested in the distribution of spin blocks, as well as sequences of causal states. Recall that Given the joint distribution over spins blocks, Equation , the bloEquation .Recall that using the translation invariance of the spin configurations, the two-spin mutual information of Equation . Thus, tT matrices.Equation :(75)Pr Thus, the memory needed to perform optimal prediction of the right-half configurations can exceed the mutual information between the left and right halves themselves. This relationship reflects the fact that, in the general setting, a process’s internal state sequences are not in one-to-one correspondence with L-block or even ∞-length configurations.As remarked above, the excess entropy and the statistical complexity are different measures of a system’s memory. However, it turns out that the excess entropy sets a lower bound on the statistical complexity:guration ,109.  nvariant . That isAs noted above, an te model . In contNote that ty is hμ ,36. So, average value of the growth rate of the Kolmogorov–Chaitin complexity can be calculated and is equal to the Shannon entropy rate One unfortunate shortcoming of KC complexity, and its framework, is that it is in general uncomputable ,36. Thatlogical depth of Bennett [A quantity more closely related to statistical complexity and excess entropy is the  Bennett . WhereasEquation , is randAs a result, like excess entropy and statistical complexity, the logical depth captures a property—being low for both simple and random configurations—that is distinct from randomness and from those properties captured by the entropy rate and Kolmogorov–Chaitin complexity. While there are superficial similarities, however, mputable ,36. Thatsophistication of Koppel [Another quantity, related to logical depth and even closer to the statistical complexity, is the f Koppel . The sopProbably the closest alternative to statistical complexity is found in the semigroup theoretic approach introduced by Rhodes . The focFor other approaches to statistical complexity and correlational structure see Refs. ,114,115 As a final note, we mention that ) curves ,116,117.) curves ,119, to In Ref.  it was sEquation , with thIn the previous sections we reviewed the tools used by statistical mechanics, information theory, and computational mechanics to measure correlation and structure. The main quantities from statistical mechanics are correlation functions However, information theory tells us little about how a system utilizes its memory nor whether the apparent memory , one caquation 86)Pr(x→Pr(x→(86)R neighboring spins. This observation was made in a different context by Baker [Notice that Equation  indicateEquation  are membby Baker .R interactions in the Hamiltonian. The probability of a right-half configuration This somewhat surprising result can be explained physically as a direct consequence of the range-K system with range R interactions there are at most L from 0, Equation η, assumiR-spin block distribution. As already noted, R to give a density, is not the entropy density Equation  is equivEquation ,(99)Cμ=Since the probability of a spin block depends only on the value of that block’s nearest neighbor, Equation  for the ussed in hμ=1Rlog2Equation  has alsoR interactions also lead to a compact expression for the excess entropy. Recall that R-spin block of one half influences the distribution over the other half, it follows that:R spins after seeing The range-R-range one-dimensional spin systems for which Equation  a aE usingation Equation  holds.In terms of the transfer matrix, an expression for Equation  into EquEquation  and simpV and using Equation  a a101) anEquation  is satis; namely:Cμ=E+Rhμ,; namely:Cμ=E+Rhμ,; namely:Cμ=E+Rhμ,Equation .Equation  shows thlarger than the information available (on average) from individual spin observations. However, Finally, since mutual information, and thus Equation . It mighStarting with the transfer matrix, the preceding section developed a general method for determining the causal states, constructing an Equation . The maiFor the special case of a spin-Equation  holds, tThe transition matrices The stochastic connection matrix quations  and  composite variables; it does not break them into pairwise interactions. This is the sense in which we say that q, The excess entropy is capable of detecting structure at any wavelength because it is a much more “global” function than the structure factors. Although calculation of the structure factor involves summing over all the variables in the chain, the correlations are considered in pairs, since the two-point correlation functions The differences between Simply put, excess entropy and structure factors measure different things: Since lsewhere , it emphThe fact that q’s. A response to this objection might be to sum all q’s. However, if one does this the different phases destructively interfere. As a result:Looking at All we’re left with is Summing over the absolute value of the structure factors, as inOne also might be tempted to add together all the connected correlation functions. That is, combine the two-spin, three-spin, four-spin, etc., connected correlation functions.  Howevei, where F.More specifically, one can show that the connected correlation functions are, up to factors of e energy . Thus, aIn summary, Since any periodic 1D spin configuration has More generally, “pattern” is not synonymous with memory as measured by Lastly, note that, like Equation  hμ=0 impIn the previous section we saw that B increases, the ground state for the system changes. This can be seen by considering the excess entropy and the entropy density, which are plotted along with To illustrate this, we consider a spin-B is varied, the system makes transitions between three different spatially periodic ground states.The previous section noted that a 1D spin system with B. The period-2 structure factor B falls below 2.  or empirically reconstructing M period-2 blocks are inserted is A similar structural analysis of the configurations in the In summary, the preceding subsections illustrated how The dip in the excess entropy and the peak in the entropy density at the transition regions give a general indication of high-entropy, low-apparent-memory configurations. These structures are not periodic and, thus, are not compactly described by the structure factors that implicitly assume the system has strong periodic components. However, the configurations most certainly are not structureless. The Lastly, recall that our description of spin configurations began with a Hamiltonian with nearest and next-nearest neighbor interactions, which in turn led to a is the underlying pattern in the sense that it is a minimal representation of all the (group and semigroup theoretic) regularities in a configuration ensemble. Excess entropy and At this point we have reached our two main conclusions. First, the entropy density Before we conclude, however, there are a few remaining issues that need addressing. Specifically, we need to explicitly compare the excess entropy with some of the statistical mechanical functions defined in We begin by comparing excess entropy with the correlation length Equation . QualitaT while However, we can compare their qualitative behavior as the temperature is varied. Looking at rate at which correlations between spins decay as a function of increasing distance. The decay rate provides little or no information about how much total correlation or memory is present. The excess entropy, on the other hand, measures the mutual information between two semi-infinite halves of the configuration and thus provides a measure of the total spatial memory of the system.A more important difference, though, is that C and the excess entropy C reaches a maximum at C does in C cannot be viewed as providing any general indication of spatial structure.Despite the similarities, these two quantities measure very different properties of the system. As mentioned in discussing Equation , the sper. Hence, In the low temperature limit T goes to infinity and as the correlations vanish. However, recall that In contrast, the high-temperature behavior of In between these temperature extremes, there is a region where the correlation between spins is largest. Here, the system is neither random, as it is at high temperatures, nor is it trivially ordered, as it is at low temperatures. Not surprisingly, both However, note that The shape of the curve in T. Like the structure factor, As our last phenomenological comparison, That Statistical mechanics possesses several functions that are similar to the excess entropy, but none can be interpreted as measures of spatial memory as Moreover, this section demonstrated that each of these quantities reaches a maximum at different parameter values. This means that the statistical mechanical functions cannot be used to determine the parameter setting at which a given system’s spatial memory is the largest. Simply put, to measure (apparent) spatial memory, one must use H and then focusing on the excess entropy In summary, we reviewed three complementary approaches to correlational structure. The next three sections developed a direct comparison of statistical-mechanical, information-theoretic, and computational-mechanical approaches to structural complexity. There were three main conclusions that emerged as a result of these comparisons.First, Second, Equation  between Finally, Several ancillary observations, based on the foregoing results, are now in order. First, we have seen that for 1D spin systems the number of causal states and their connectivity typically does not change as spin system parameters are varied. What does change, however, are the probabilities of the causal states and their transitions. In contrast, for deterministic dynamical systems it is typically the number of causal states that change as the system parameters are varied ,43. ThusSecond, approaches to structural complexity, such as those of Refs. ,34,129, First, by using a universal Turing machine (UTM), the most powerful discrete computational model, one loses the ability to distinguish between systems that can be described by different computational models less powerful than a UTM ,25.Second, and perhaps more importantly, the KC complexity is uncomputable in that there exists no general algorithm for its computation. Thus, approaches focusing on KC complexity, including logical depth  and sophThird, KC complexity-based approaches inherit a fundamental relativity—a relativity that is built into how regularity and structure are accounted for and that derives from the UTM’s lack of uniqueness and minimality. Computational mechanics takes a completely different approach and makes a specific commitment to causal states and Thus, given the problems arising from KC complexity being based on UTMs, it seems to us that these approaches to structure and pattern will continue to find few empirical applications. Significant supplemental assumptions would have to be introduced to make these approaches viable. In contrast, due to its specificity of representation, computational mechanics is testable and its hypotheses—e.g., linking pattern, organization, and computation—are refutable.We conclude by discussing some open questions and possible areas of application. It remains an interesting open question as to how mbiguous ,130,131 mbiguous ,78,127, mbiguous ,132,133 Another important set of open questions concerns the development of better techniques for estimating the excess entropy and reconstructing Finally, there are a number of issues that parallel our comparisons with elementary statistical mechanics, but that relate to phase transitions and critical phenomena. Are the intrinsic computational properties reflected in the excess entropy or in infinite


Three themes and six subthemes were identified to underscore Latinx attitudes toward COVID-19 vaccine uptake or hesitancy. These were: (1) Attitudes regarding vaccine intake; (2) Sources of Information; and (3) Science Education. The degree to which each of these themes exercised influence on vaccine intake or hesitancy varied. The multi origin Latinx participation in the focus groups strengthened findings by broadening representation and discussion. In the end and despite the various national origins, all participants indicated receiving most of their information on COVID-19 related topics from their family, physicians, social networks, and some form of media.During the Spring of 2021 in Miami-Dade County, four virtual focus groups were held with 31 participants from four diverse local Latinx communities as part of the Florida Community Engagement Alliance (FL-CEAL) Against COVID-19 Disparities project. The main objective was to explore attitudes about COVID-19 information and prevention strategies among South Florida's diverse Latinx populations, across a broad geographical area. The study used a semi-structured focus group qualitative design and chose participants from four well established Latinx neighborhoods. Participants were mostly women, diversity was strong with birth regions including the Caribbean, North, Central and South America. Though a third ( Yo creo en la ciencia y creo que puede haber la capacidad de desarrollar una vacuna en muy poco tiempo. Sin embargo, han sido tantas las noticias que hemos escuchado, la información que nos bombardean y nos tienen confundidos… pero aún no estoy convencida que sea lo mejor y que esté completamente probado que es lo que debemos hacer. Yo todavía no estoy a punto de convencida de que debemos colocarnos la vacuna .At the time that Angela expressed the views above, COVID-19 related deaths in the United States (US) had surpassed 483,000 with 27,600,000 confirmed cases of infection  Against COVID-19 Disparities project . The groAt the start of the FL-CEAL project, the only CDC recommended prevention approaches based on research consisted of mitigation strategies such as social distancing, wearing masks, washing hands regularly, and isolation when or if infected . During It was in this highly politicized context regarding the COVID-19 pandemic and vaccine that we held focus groups and community observations in MDC, Florida, in early Spring 2021. The purpose was to explore how diverse Latinx residents—of various national origins, socio-economic status, political beliefs, and sexual orientation—viewed the COVID-19 pandemic and recently developed vaccines, while also identifying potential barriers or motivators to receiving the vaccine and promoting it to family members and friends.N = 31 overall) were conducted throughout Miami-Dade County (MDC) from January 21 to February 15, 2021. Participants were recruited from high density, multi-origin Latinx communities—specifically, North Miami-Dade County (North-MDC) (N = 8), Central Miami-Dade County  (N = 8), South Miami-Dade County (South-MDC) (N = 8), and an additional county-wide sexual gender minority group (SGM) (N = 7). Participants were recruited by an outreach worker and are active community residents with no formal roles in the agencies that facilitated their recruitment. Outreach protocols established by (FL-CEAL) and the Center for Latino Health Research Opportunities (CLaRO)  were instrumental in obtaining community support to remotely contact and recruit participants. Specifically, the study utilized the protocols directing community health workers to target minority communities with outreach focused on education and information regarding COVID-19 research and prevention efforts. The design leveraged South Florida's multi-origin Latinx population to conduct outreach and recruit diverse Latinx participants.The current study used a semi-structured focus group qualitative design. Four virtual focus groups  North-MDC where Latinx populations compose 29.5% of the population; (2) Central-MDC, including the neighborhoods of Allapattah, composed of over 76% Latinxs, predominantly of Dominican Republic origin and Little Havana, composed of 95% Latinxs, historically of Cuban descent; however, in recent years the area has become more diverse and home to immigrants of Mexico, Central America, South America, and the Caribbean , 16. FinFor all focus groups, the following inclusion criteria were used: being (i) an adult aged 18 years or older, who is (ii) a leader or active member of the target community that (iii) consented to complete a one-on-one pre- and post-survey and (iv) consented to attend a 90-min Zoom focus group and was (v) able to understand and effectively communicate in English or Spanish. Although measures were taken to ensure a diverse range of ages and genders, participants in the three community focus groups consisted of nearly 88% females for all three geographic groups , while the sexual and gender minority group (SGM) consisted of six males and one transgender female See , 2. Datavia the Zoom online conferencing platform) during separate days for each neighborhood and the SGM group. All focus groups were held within a 3-week period to limit participants' exposure to media messages and frequently changing sources of information. Groups were conducted in the early evening to accommodate participants' schedules and help facilitate participation. Before each focus group started, participants completed a short demographic survey. Each ~90-min focus group was facilitated by a study co-investigator and assisted by a co-facilitator. Whenever clarity was needed during the course of the focus group, the moderator rephrased the question or asked from a different point of view, as suggested by Krueger and Casey  group in which both languages were used, as needed. While developing the focus group guide and preparing for the focus groups, the first author watched local Spanish language daily television news to gain insight into the community's exposure to the COVID-19 related information provided by these outlets. Concurrently and in an effort to contextualize place, fieldwork observations were conducted at retail pharmacies and food markets located in targeted neighborhoods.As indicated in this paper's opening quotation—and supported by our research team's fieldwork observations—confusion about the development of the vaccine and its possible deleterious effects was extensive in these communities at the time focus groups were conducted. Spanish media focusing on COVID-19 and vaccine-related information, varied according to the different audiences to which it reached. For example, most radio programs in Spanish were guided by countries of origins, political orientations, and most frequently, religious affiliations. The media's influence on the Latinx community's response to the vaccine was notable at the time focus groups were held because local media broadcasts transmitted widely different rumors and stories. For example, popular during late Fall 2020 and early 2021—immediately preceding and concurrent with our focus groups—were media stories about a pastor at one of the largest Spanish language mega-churches in MDC who discouraged uptake of the vaccine and instead urged “taking believe in divine immunity”  still don't even understand well what Coronavirus is. In the above comments, Carmen shares knowledge about vaccine development, which informs her behavioral capability and subsequent exercise of self-efficacy when deciding on whether to get vaccinated or not. Alternatively, others expressed hesitancy because of fears of side effects. Amelia from Central-MDC suggested: “I think it's best to maintain good health and wait… to see what's going to happen to those people with horrible side effects.” Similarly, one participant from the South-MDC farm workers focus group reminded fellow participants that all medications have side effects and continued by noting, “If it [the vaccine] affects other organs, it may protect you, but it's affecting something else. So, I think we have to wait a little longer to see what the effects of the vaccine are.”Vaccine hesitancy was common even among participants who expressed support of science. One such participant, as presented in the opening quotation of this paper, expressed reluctance resulting from overwhelming media information:…so, the more news you listen to, the more questions arise with less answers. I believe in science; I believe in the companies that develop it [vaccine], and I believe it can be a good thing, but I'm still not convinced that it's the best thing for us at this time, and that it has been completely proven that it's what we should do… I'm still not at the point where I'm convinced that we should get the vaccine. (CMDC)Similar to the participant above, Jesus, a participant from the SGM group indicated: “What you most hear in the community is the uncertainty, or that, ‘I don't know what can happen, rather than it's good or bad.”Those who expressed hesitancy unanimously expressed their concern for the speed at which the vaccine was developed—similar to findings reported from a recent study by Bateman et al. . Though Participants' statements frequently revealed fears rooted in past events and confusion by what they considered overwhelming daily information from multiple sources, such as television news coverage, radio talk shows, internet and social media, and word of mouth. Fernando, a participant from the SGM group, indicated that media messaging surrounding the vaccine “lacked transparency and fueled issues of mistrust and hesitancy.”I didn't look up more details, but I remember... the Florida doctor, that he took the vaccine, and then 16 days later, he died. And then, in the news, the wife said that ‘Oh. It's due to the vaccine.' And Pfizer … said, ‘No. It has nothing to do with the vaccine.' But then... the CDC is investigating the case. So, what I don't like is how the media will blast the whole article... kind of to put fear into the people about the vaccine.…Vaccine hesitancy and vaccine acceptance are not mutually exclusive and participants comments reflect both. Indeed, this was the case for Fernando. In response to the facilitator's probe on whether he perceived any personal consequences to receiving the vaccine, he responded: “Oh, yeah, I have no cons against it, for sure. When it is available for me, I'm the first person to go.” Unlike the above participant, others admitted they were “not ready” or “still searching” to receive the vaccine. However, they were not opposed to it, indicating their willingness to receive it at a later time when the vaccine had been more adequately tested, or when more was known about its side effects. Some made exceptions, however, to their “wait-and-see” opinion. One participant in the SMDC group, who was told that it could be required to travel to a foreign country, almost immediately modified her earlier “wait-and-see” approach to the vaccine and expressed that she would get the vaccine within “a moment's notice,” if required to travel abroad to visit her loved ones. A woman of Central American origin from the same group added, “if for international travel the vaccine is mandated, then I will get it immediately to visit my family.” Others who had expressed hesitancy earlier agreed to vaccination, if necessary to travel, almost immediately modifying their earlier position resulting from the recently received information.Among those who said that they would get vaccinated were several participants in the SGM group who acknowledged that their motivations were related to their comorbidities, which increased their risk and fear of infecting others. These included references to living with an older family member, attending a social event, being a “vector” of the virus to others or a desire to return to “normal”.Additionally, one participant in the SGM focus group, Camille, drew a parallel between the COVID-19 pandemic and the experience of the SGM community with the HIV epidemic. She ended her comment by advising others to get the vaccine:I want to say I came through another epidemic a long, long, long time ago. So, yes, there was a lot of stigmas, probably just as much stigma as is going on with the pandemic today, but I educated not only me, but I educated other people, and that helped diffuse the fear a lot. I've seen that epidemic progress. I'm talking about way from ‘80s, I've seen the epidemic progress. So, when it comes to this pandemic, I can't emphasize how important it is to educate them people. Me, personally, if they're in a category that are a lot of—I think I heard people say underlying conditions—that they need to be vaccinated, my advice would be to get it.Moreover, while there were many who expressed hesitancy, of the 31 focus group participants, only three gave an adamant “never.” Interestingly, among the three who expressed the most hesitancy, two, soon after, received the first dose and shared the news with the outreach coordinator. Despite earlier discussions on hesitancy, mistrust, and other concerns, over one-third of participants expressed willingness to get vaccinated “as soon as possible”—not finding it necessary to wait for additional information on the vaccine and indicating strong efficacy. A woman from South-MDC expressed satisfaction about receiving the vaccine as early as mid-January 2021, during a time when the majority of the US population had not been vaccinated. She said:Well, I called my primary physician, I also have several other specialists, since I have a chronic health condition that requires frequent medical visits, and my physician told me that his practice was recommending intake of the vaccine. I went ahead and got it. A day later, I heard my daughter say, “the person I most wanted to get vaccinated was my mother, and she already got it.” When I heard her say this, I felt so much better. I already have my appointment for my second shot.The above woman shared that her daughter, a nurse, had been heavily involved with COVID-19 patients and at “the center of the crisis.” Another participant from the same community indicated that she would get vaccinated as soon as she could make an appointment, even when questions remained about possible side effects. Still, referring to a well-known Spanish proverb, she shared, “it is always easier to prevent rather than to treat.”In general, most participants who expressed hesitancy also observed that it was not whether they will get it or not, but rather when they would feel or think “sufficiently secured getting it.” Only three participants appeared reticent at the time, and one said that “it would take a lot of information [from those she most trusted] for me to be ready for the vaccine,” illustrating constructs, such as expectations, expectancies, and expanding her behavioral capability. While some ignored the negative comments and were ready to be vaccinated, as a participant from Central-MDC (Little Havana) said, “the sooner the better.” Others, who expressed hesitancy in terms of a “wait-and-see” attitude, also acknowledged the possibility of getting “the vaccine sometime in the future.” When carefully analyzing the data line-by-line, two groups were identified: one group who said they were ready for the vaccine, and a second group that professed a “wait-and-see” attitude, with most noting that, “not getting it now, does not mean we will never get it.” This second group expressed dissatisfaction with the information being received at the time from mass media, indicating that “at a future time when more scientific information became available and when the numbers of the vaccinated increased, they were more likely to get it.”Beginning with the first focus group, sources of information emerged as a significant theme in participants' narratives, and several participants found the source to be of utmost importance when making decisions about COVID-19 exposure, mitigation practices, or vaccination. A participant from the N-MDC focus group referred to the CDC as a “most trusted source of information on COVID and the vaccine.” Data analysis revealed that participants from all targeted MDC communities unanimously trusted information from their physicians, immediate family, or both. Those who gave higher priorities to trust in their physicians, were those who also indicated having chronic conditions that required long term relationships with their physicians. Fernando, a participant in the SGM group made a particularly strong reference to his physician as the person most trusted when stating: “I suffer from chronic conditions, diabetes, asthma, among others… I trust my physician.” Similarly, a woman with chronic conditions from South-MDC indicated that she had consulted with her physicians and, following their advice, she elected to be vaccinated. Others, not reporting major chronic conditions, were more likely to indicate close family members as their most trusted sources, especially when the relative was in the health field. Ranked close to their physicians and family members, were other health professionals (such as pharmacists), and reputable websites  and university websites . Over half of all community participants indicated that reputable and university websites were also important resources; the latter was frequently mentioned (5+ times) during focus group sessions when participants sought to justify their various positions by citing sources of information and informants beyond their close friends and family members.More distant, yet important, sources of information frequently reported by participants were media personalities  and recognized television networks, such as US CNN, FOX News, and local and Spanish language broadcasting networks. For example, Ana Maria, a participant from North-MDC, whose daughter lives in Spain, was adamant in her opinion that Spain's major national television network, TVE (Television Española), was her preferred source of COVID-19 media coverage. Ana Maria noted, “I watch TVE regularly and their hosts are very clear when speaking about COVID.” When asked why she prefers TVE for COVID-19 news and information, rather than US networks, she noted, “they are more trustworthy, they provide clarity to the topic, their presenters stay away from politics; actually, they refer to science not politics when conveying news and mitigation practices.”Participants were also frequent users of online sources of information, among which YouTube was the most frequently cited. Angela from Central-MDC observed:Well, I have membership in various channels, and in a YouTube channel, there is a host called Gary Burg who is a medical doctor and also very young, he is fabulous because he explains all illnesses very detailed, and he tells us what needs to be done in order to eat better, live better, the connection between mind and body, how to sustain stable health. I also watch Dr. Hyman who has a clinic in New York... [and] there are lots of good medical doctors on television that present programs that help with health that are free and these are the programs that I follow and help me in making my health decisions.cosas in Spanish). Here, “things” (cosas) was a word frequently mentioned during focus groups when participants were referring to the information they heard and watched on COVID-19 news. When further probed about the meaning of “things,” there was hesitation from Ana Maria and those in the group who agreed with her; participants then explained that, when referring to news broadcasts, “things” indicated noise, such as information not to be taken seriously or trusted, whether from television, radio, or print. When referring to “things,” Ana Maria clarified, “I speak about mistrust in terms of local media coverage of COVID news and information, I am cynical about all the “things” that I listen on a daily basis.” She adds:Within the broad topic of media as a source of COVID-19 information, participants in the North-MDC focus group observed that US media messages appeared contradictory and confusing. When asked, “What comes to mind when you listen to COVID-19 news, read about COVID-19 , or talk about it to your friends and family?” Ana Maria, a Central-MDC participant noted: “Listening to contradictory “things” , who indicated that they played a role in informing, advising, and sharing a sense of responsibility for older family members. For example, Fernando notes:My biggest fear especially with my family was right before Thanksgiving when my partner got COVID 19, and I feared for my grandmother. So luckily, she was not living with me, but if she needs my help, or anything, I try to limit my exposure to her. So, she got the vaccine now…. then obviously when I get it (the vaccine) it eliminates the fear of like Oh, I can potentially give it to her.Fernando, further indicated: “For example, my grandmother, the second I heard about the elderly people getting the vaccine, I already had her phone number, I was willing to do the appointment for her, but she had an appointment already.”Another participant from the SGM group, Jesus, also noted that when asked about the vaccine by his mother-in-law, his suggestion was: “…go for it. …that was my suggestion because that happened to be my mother-in-law.” The above comments indicate that this younger group, plays the role of informers and advisors partaking in culture and guiding their older family members.Participants' views on science, scientific information and scientists emerged as a third major theme in focus groups discussions. At the time focus groups were conducted, while the pandemic was at its height, most participants, across all focus groups, expressed trusting views of science and scientists. References to scientific information, knowledge, and scientists were common throughout all focus groups. As in the opening quotation, most participants acknowledged respect for science, frequently referring to the CDC website and to “Dr. Fauci,” who most, if not all, knew from the various news outlets they watched.Upon concluding a second line-by-line review of focus group data, no refutations of science were found in the transcripts; even when expressing hesitancy about the vaccine, no participant openly or directly expressed a disbelief in science or scientific methods. When vaccine hesitancy was expressed, it was in relation to doubts about the methods of science not being appropriately or sufficiently applied during development and testing. For example, a participant from South-MDC noted: “I keep reiterating to friends and family that we need to be super informed of the latest scientific information, this virus is all over the world; in the months ahead, we will learn a lot more about it.”When referring to scientists, a woman also from South-MDC, Victoria, complained by saying:The problem is, they first said it was an unknown virus… then suddenly they make a vaccine. How can they make a vaccine for a virus they don't know? …in other words, so much has been said that not even they [scientists] really understand the virus, yet they already have a vaccine for a virus they don't understand? …I will not be convinced.In response, participant Maria added, “When she travels, then she'll have to take it,” but Victoria replied with, “Not even, I'm willing to not travel...”When asked, when was “the right moment to get the vaccine,” Maria from South-MDC, responded:me, then I'll get it (vaccine).When scientists produce a medical journal that says: “People who were vaccinated have become immunologically protected, they are now immune to the virus”; because I want to know, who are these people participating in vaccine trials? Is it a Puerto Rican, a Colombian? A US born Colombian or a foreign-born Colombian? A Puerto Rican like myself? born there… or here? You know what I mean? So… when I hear that they did trials with a group like Jesus from the SGM focus group shared his trust in science, noting:What you mostly hear in the community is the uncertainty, or that “I don't know what can happen,” … But it seems like everything is running fine and I trust science. There was research before, so that's it, Thank you.Rebeca from North-MDC repeatedly said that she never watches the news; however, she followed, “… what they [scientists] say especially in scientific journals which are peer reviewed. If I find the articles interesting, I read it; if not, I pass.”Finally, only three participants made references to alternative health choices: two participants described the YouTube doctors they watch regularly, who endorse a holistic curative approach to illness through nutrition; the third participant, a woman, shared her observations about an Asian couple who used herbal teas to treat COVID-19. Others in all groups demonstrated strong support and respect for science, scientists, and health professionals, despite reservations about the vaccine rollout.Participants felt strongly about education, and phrases such as “getting educated” and “becoming educated” were expressed frequently during focus group discussions, especially when referring to following CDC recommended mitigation practices or choosing to ignore them.Angela from the South-MDC group made a case for education.I want to share with the group that above everything else there is education [Pause]. I think that the community needs to be educated on a daily basis and addressed with a great deal of sensibility. I believe that first we need education to make people aware that despite whatever our needs, we should not be out shopping without protection, especially those with COVID symptoms should not go to public spaces without being tested first, that, when necessary, they follow mitigation practices.A respondent from the same group agreed:I think that education is a fundamental part. We must strengthen education programs in our communities, make people more aware, conscientize the community, so they engage in mitigation behaviors.Rosa added:I believe that as time goes by, we will see this illness developing and further infecting others in large scale. This is the time when we are going to have to run and gain confidence on the vaccine and learn how to educate families, like mine.Cecilia from North-MDC observed, “people need to make educated decisions, to share information.” Alicia, also from North-MDC, followed: “people need to read, share information, search databases, so they become more educated.” Susana agreed: “studying is important, community leaders should educate the community and provide information.”Participants in the SGM focus group also felt strongly about education. Camille emphasized the need for education when speaking on the vaccine, noting: “education is the key to a lot of things. Find out as much as you can about the vaccine, and don't go at it just because it's fear of something.” Similarly, another participant—when referring to the absence of mitigation practices among sex workers—commented: “…so there might be another way to allow them to access education, or to access the workforce, education, to get more dignifying degrees, then there is a greater sense of worth.” Another participant, also from the SGM group, added that “education decreases the stigma associated with partner violence in our communities.”Three themes and six subthemes were identified to underscore Latinx attitudes toward COVID-19 vaccine uptake or hesitancy. The three major themes that emerged were: (1) Attitudes Regarding Vaccine Intake, with two subthemes: (a) vaccine hesitancy and (b) vaccine acceptance; (2) Sources of Information, with two subthemes, (a) the media and other sources of public information, and (b) information and support across generations; and (3) Science and Education, with two subthemes: (a) trust in science and (b) education, studying, and following mitigation recommendations. The degree to which each of these themes exercised influence on vaccine intake or hesitancy varied.Data analysis from the four focus groups provided the opportunity to reach, identify, and report on multi-origin Latinx participants' attitudes toward the vaccine, including themes on science and medicine while also highlighting reasons for vaccine hesitancy. We expect findings from these groups to assist in establishing the foundation for an improved and wider understanding of Latinx vaccine behaviors in general and their openness to vaccination.Within the theme of “valued sources of information,” participants from all focus groups viewed their physicians, immediate family, or both as their major and trusted source of vaccine information and inclination. Physicians and families, whether nuclear or semi extended, or both, were their major and most frequently solicited source of information on all aspects of the vaccine, including not only recommendations or rejections, but also information and discussion on the scientific merits of the vaccine and/or getting vaccinated. Participants expressed less engagement with neighbors or distant family members, even when in close neighborhood proximity.Findings suggest that participants held strong views on science. They were often eager to find evidence in support or rejection of vaccine intake based on their access to the readily available scientific information. Those participants frequently searched through online sources or local media. All participants were familiar with using virtual modes of communication and demonstrated familiarity and reliance on websites that promoted different vaccines perspectives.Findings presented here have to be interpreted within the multi-origin, broadly diverse Latinx population of Miami Dade County, not only diverse in national origin but also in socio-economic characteristics. However, above findings provide a preliminary outline of the groups' attitudes and behaviors on the vaccine, as well as their views on science and respected sources of information. Participants acknowledged respect for science, professional expertise, and information. However, vertical networks of family members in health professions and horizontal networks of friends and neighbors were also important sources of information. Vertical family networks were valued sources of information and support. Preferably, advice and information were sought from family members who were health professionals . Some participants in the SGM group who acknowledged close proximity to physicians and other health workers, many in second generation vertical family dyads, advised and encouraged other family members, most often a grandmother or mother-in-law, to become vaccinated. Current findings are similar to studies that have shown Latinxs are likely to get their health information from sources such as physicians, family, friends, and social networks, and some form of media . In a moIn contrast, between 2006 and 2016, there was a 37% increase (74%) of Latinxs reported use of the Internet , of which 66% of those used it to search for health information. Moreover, 41% reported their decisions to treat health conditions were influenced by what they saw in media . As suchn = 10) and obviously not affected. Second, the demographics of Miami-Dade County (MDC), where 54% of the population is foreign born  and National Institute of Minority Health and Health Disparities (U54MD002266) and approved by a joint IRB Florida International University and University of Miami .The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.


Indoor air pollution is injurious to human health, even worse than outdoor air pollution. However, there is a lack of empirical evidence using large samples in developing countries regarding whether indoor air purification can improve human health by reducing indoor air pollutants. Using the data from the China Health and Nutrition Survey in 2015, this study analyzes the relationship between indoor air purification and residents’ self-rated health. We apply the generalized ordered logit model and find that indoor air purification has a significantly positive effect on residents’ self-rated health. This positive effect is limited to improving the probability of residents’ health level being rated “good”, and there is no significant movement between the two levels of “bad” and “fair”. The results also show that, as an important source of indoor air pollutants, solid fuels used in cooking significantly reduced residents’ self-rated health level. Additional results show the heterogeneity of the relationship between indoor air purification and resident health among groups with different characteristics. This study provides empirical evidence for further optimizing the indoor air environment. PeopThere are various ways to reduce indoor air pollution. These ways include: removing pollution at source ; reducinAir purifiers can significantly reduce indoor pollution concentrations through filtration, adsorption, sterilization, oxidation, etc. This is one of the most commonly used effective methods and has received increasing attention ,33,34. HAir purifiers are widely used to reduce indoor air pollution . Most reHowever, some scholars believe that the effect of air purifiers on health is not clear . Due to Mannan and Al-Ghamdi , by revi3 over an average 24-h period are common, but the peak PM10 exposures have been reported to be over 30,000 μg/m3 during cooking . We removed those who answered “Unknown” from the sample. We also exchanged the signs of 0 and 1 in order to measure the objective health level. Since the objective health level index is a binary variable, we used the basic logit model. The results in the last column of The impact of indoor air purification on residents’ health has been of interest to scholars, but empirical evidence from large samples is lacking. Based on 2015 CHNS data and a generalized ordinal logit model, we investigated the causal relationship between indoor air purification in housing and residents’ self-rated health. The results show that indoor air purification has a positive effect on residents’ self-rated health, but this effect is limited to improving the probability of residents’ health level to “Good”, and there is no significant movement between the two levels of “Bad” and “Fair”. If the number of air purifiers increases by one unit, the odds ratio of self-rating becoming “Good” is 1.236 times higher than before. Our research also found that solid fuels used in cooking, as a significant source of indoor air pollutants, significantly reduces residents’ self-rated health.In addition, we also found that heterogeneity between different characteristic groups also affects the relationship between indoor air purification and residents’ self-rated health. The results of the heterogeneity analysis showed that, due to differences in housing conditions, indoor air purification has a significant positive impact on the self-rated health of rural residents, but the effect on urban residents is not statistically significant. The use of air purifiers has a greater impact on the health of low-education individuals. Due to going out for work, the impact of indoor air purification on the health of non-working residents is greater than that for working residents. Furthermore, non-smokers pay more attention to indoor air purification and personal health than smokers.From a policy perspective, this study demonstrates the need for greater popularization and subsidy incentives for indoor air purification in order to control the health risks caused by indoor pollutants. Furthermore, considering the heterogeneity between groups with different characteristics, the government and other departments can formulate targeted policies to guide them to pay attention to indoor air purification, such as providing a cleaner working environment for urban residents who are busy with work. Furthermore, rural residents should be given more subsidies for indoor air purification, as the use of indoor air purifiers is more beneficial.This study has some limitations. As the data from the China Health and Nutrition Survey are only updated to 2015, we are not able to obtain the latest data and further discuss the possible effect of indoor air purification on the infectivity of COVID-19. Moreover, due to limited data, we did not examine the impact of more sources of indoor pollutants and particulate matter, and/or gaseous pollutant concentrations on residents’ health; thus, this study cannot provide a better examination of the impacts of this potential mechanism. These limitations can be used as the main direction for future research.


With great interest, we have read the article by Mansour et al. , reportiDeep learning networks show poor classification results and tend to be overfitted when trained on a very small data set . A medicNo justification was given for why a VGG19 network was chosen, although it has a significantly worse accuracy in the analysis of CT data than, for instance, ResNet-50 or DenseNet-201 networks . At the Furthermore, it was not mentioned whether regularization methods such as transformations of the raw data , weight constraints, or activity regularizations were used for reducing overfitting . It remaAnother aspect is the use of PCA. Because PCA is a linear algorithm for dimensionality reduction, the question arises on which basis a linear relationship between the detected features can be assumed. Given the complexity of the present data in terms of possible blurring or degradation due to fluctuating contrast, it is problematic to make such assumptions on the basis of the representation of shapes and images using smooth manifolds. Nonlinear methods (manifold learning), such as kernel PCA, t-distributed Stochastic Neighbor Embedding, or Multidimensional Scaling, could be applied instead. Moreover, the authors write “single-scan testing was repeated so that each of the 54 scans served as the test scan exactly one time” . AlthougAs the authors reported that early HIBI signs were due to “subtle changes that evade the detection threshold of the human eye” , it wouln = 16) for validation. On this data set, the positive predictive value was 0.5, indicating that in the validation set a prediction of severe HIBI from early HCT had a 50% chance of being correct.The authors used a very small data set (In conclusion, we agree that machine learning is an attractive new tool that may help to better predict severe HIBI from early HCT scans in cardiac arrest survivors in the future. The study by Mansour et al.  is a fir


Recent contributions to the literature on small firm growth have been marked by a growingsense of frustration with the state-of-the-art and what it implicates in both theory andpolicy. In short, while growth episodes appear relatively common, a tiny proportion offirms sustain growth and ‘scale’. This calls into question the very basis upon whichpolicies seeking to target high growth firms (HGFs) rest. In addition, it cautions againstperspectives that view growth as the essence of entrepreneurship. In this paper, we arguethat understanding the frequency of growth episodes and the rarity of sustained growthrequires a better understanding of growth consequences. To this end, we describe casestudy evidence from ambitious entrepreneurs whose firms experienced an episode of highgrowth followed by longer periods of mixed performance. Our goal is to shed light on howthe experience of growing affects further growth. Our data provide initial insights intothe mechanisms linking past growth to growth motivations and into the ways in which pastgrowth lays the foundations for future performance. Inc. Magazine began ranking high-growth companies, only 69have made it onto the list two or more times implies at least two things. First, that highgrowth is astonishingly difficult to replicate; and second, that many more firms experiencean episode of high growth than observations at a specific point in time approaches mightsuggest.Our interest in the consequences of growth in smaller firms was initially occasioned byrereading The first is reinforced by recent evidence on growth persistence. Simply put, past growthdoes not seem to be a useful predictor of future growth in samples of small and young firms.Indeed, high growth firms are as likely to be found among the previous period’s worstperformers as they are among the best performers . InshorAccepting that many firms grow, but that very few enjoy sustained periods of growth, oughtto have us thinking about the consequences of growth at least as much as the causes . Of courThis, of course, is not a new idea. The organisational lifecycle models of firm growth thatenjoyed considerable prominence in the early entrepreneurship literature often explicitlyrecognised the organisational challenges that accompany growth see  for a re. A focus on consequences is consistent with a view of growth as an intermediateoutcome  sustaining growth.And, second, to elaborate a general case for the further study of growth consequences. Inthe next section, we introduce our cases, outline their selection and our processes of datacollection and analysis. In the following sections, we explore patterns in the case studydata that appear to inform our research question. We conclude with suggestions for futureresearch and with reflections on the potential implications for policy and practice of abetter understanding of growth consequences.What is a good firm? Of course, there are likely to be many reasonable answers to thatquestion. For our current purposes, we consider a ‘good firm’ to be one that had enjoyed atleast one period of high growth and that continued to trade profitably for several yearsafter the growth episode. These are not the MUPPETS  provocatively identified by Specifically, our purposeful sampling  sought pNonetheless, in selecting our sample, we were conscious of longstanding debates in theliterature regarding suitable growth metrics . EmploymWe also use the terms ‘rapid growth’ and ‘high growth’ colloquially and interchangeably.While the former is explicitly concerned with time, the latter is more generally concernedwith extent. However, in practice, a temporal dimension is typical in studies of high growth, withwi, but the essence of each is set out in the vignettes below.Finally, our cases are highly varied. Three had their origins in family businesses thatpredated the involvement of our entrepreneurs, while three were de novo ventures. Two of thecompanies are engaged in manufacturing, three are in Business-to-Business (B2B) services,and one is in retail. At the time we interviewed the entrepreneurs, venture sales rangedfrom £875,000 to over £14,000,000. Employment ranged from 18 to 99. However, ourentrepreneurs had in common an experience of high growth and a prior commitment to furthergrowth and development. Some small effort has been made to disguise the companiesCase AGed Martin and Safety First. The ‘business had trundled along for six or seven years, doing next to nothingwith regards to turnover, and very little on the profit side’. However, sales almostdoubled between 2009 and 2010 and doubled again in 2011. The growth was largely drivenby client acquisition; especially the acquisition of two major clients within a threeweek period in 2010. As the data in Safety First was started by Ged Martin in late 2002. The business delivers electricalsafety, fire safety and health and safety assessments and services to businessesthroughout the UK. Having resigned from his previous employment, disillusioned withlabour practices in a large, unionised environment, Ged set up on his own, providingsafety training ‘until I worked out what was next…[because]…bills still needed paying’Case BTom Johnson and Food ServicesAfter a health scare forced Tom Johnson to retire from the police force in 1990, hejoined the family food services business. The business entered insolvency in 2007. Thiswas Tom’s entrepreneurial moment, acquiring assets from the administrator and, ‘justwith the bits that I wanted’, starting over again. The new business consisted of aproduction facility and three retail sites. The first couple of years of the businesswere concerned with settling debts and rebuilding reputation ‘because one of the thingsyou learn when you go into liquidation is no one wants to trade with you’. Growth camein 2009–10 with a shift from fixed to mobile retail. As Tom explained, ‘retail is forme, but not through a shop. Because a shop’s got leases and rents and rates and it’sfixed and I can’t move it’. Sales reach £862,000 in 2011, up 32% from the previous yearand more than double 2008 sales. Employment peaked at 37 employees. As Case CPeter Gray and PGS LtdPeter Gray started PGS in 1986 in partnership with his father. Peter had left schooland was working for a large UK engineering company. His job was beginning to become acareer when his father launched PGS. ‘Then my dad said, “I could really do with somehelp in the first couple of years. Can you, you know?”, so I said, “Yeah. Okay. Soundsinteresting”. I’m still here’. PGS provides power generation servicing for business andinstitutional clients. The early years were concerned with establishing the business:‘…we just cracked on and everything the company made, we ploughed back into the businessfor quite a while. We didn't pay ourselves a lot for at least probably the first 10years’. The company grew gradually until the mid-2000s, moving to new premises andemploying 35 people. Gradual growth had the firm ‘running like a sewingmachine…[meaning]…people at the top had spare capacity’. It is at this point that rapidsales growth occurred, opening a second location in another part of the UK and expandinginto equipment rental and sales. The business nearly doubled in size through 2011 and2012 before contracting a little through the remainder of the decade see . This inCase DRebecca Maier and Industrial DairiesRebecca Maier and her husband joined the family dairy business in 1989, injecting£80,000 to alleviate cash flow problems following an investment in new facilities.Initial growth came a year later, with the opportunity to shift from small scalefarmhouse production, selling to a distributor, to production and packaging directly fora national retailer. Establishing themselves as a reliable supplier to large retailprovided the platform for further growth. As Rebecca told the story: ‘when we took itover in 1989, it was turning over £250,000, and five people. Very soon went up toeight…and very soon grew to £400,000. And then with packing, it obviously just keptgoing, so by the end of the ’90s, we were seven-and-a-half million, so that was justevery year. Grow, grow, grow, grow, grow’. The growth through the 1990s required largerpremises and the firm acquired a second production facility away from the farm. The move‘doubled our overheads overnight, because you got two sites…and didn't double sales, sowe had a few lean years’. With slower sales growth in the first half of the 2000s, and ashifting customer focus from quality to cost putting pressure on margins, they made twoacquisitions to meet anticipated new demand from a large retailer. The company struggledfor a number of years following the acquisition before ‘it all came together for us in2012–13’. Steady growth followed over the next five or six years . At the Case E, James McDonald and The People PeopleAt 18 years old, James McDonald ‘bunked out of university’, worked various jobs ‘butwanted to always have more’. After spending 18 months working as a consultant for arecruitment agency, James concluded that ‘I don't like how this industry is and I canchange it’. In 1998, in partnership with his father, who had sold his own smallbusiness, and with a £30,000 overdraft facility, James launched The People People. Thevision was ‘to change the way in which people recruit in the UK’. Early growth wasfrenetic. Turnover was over one million pounds in the first year, over two million inthe second year, and £4.5 m by the end of year three; changing premises twice throughthis time. This early growth was driven by sales, ‘…on the phone, out and about, in thecar, putting thousands of miles on my Volkswagen Polo…Going out and winning newbusiness’. Following this initial success, performance plateaued as James’ focuswaivered. However, as he reengaged and started to recruit his own staff ‘just to takesome of the time from me’, the business modified its business model and enjoyed severalyears of steady growth, acquiring ‘some great clients’ and moving to ‘swanky offices’.The 2008 financial crisis hit the firm hard, with revenues dropping from £12m to £9m‘pretty much overnight’. Following this, they diversified into training and‘outplacement’ work. The next decade was marked by jumps and falls in revenue as largeclients were acquired or lost. At the time we interviewed James, The People People hadreported sales of £14m and employed 55 people.Case FStephen Bechelli and Industrial FilmsStephen Bechelli left university in the late 1970s with a degree in geography.Determined to stay close to friends and family, he found work with a large US-ownedindustrial textiles company in his hometown. Stephen’s father owned a small textile firm‘making work wear, boiler suits, chef’s aprons, these types of things’. Stephen hadworked during school holidays ‘earning and bit of pocket money’ and, in 1980, joined thebusiness full-time. They expanded into industrial textiles but, through the 1980s, ‘wasbumping along a lot…doing very little’. As Stephen recalled, ‘the company was not makingregular profits, it was scraping along. The accounting? We knew last month’s figures sixmonths after the year end’. For Stephen, ‘this was my future’. Shortly after Christmas1991, Stephen convinced his father to retire and took over the company. The fortunesimproved and ‘through the 1990s we began to make a bit of profit and become regular’.Growth was steady through the 2000s, with sales growth of over 20% in 2006 and 2007,taking turnover to £2.7 m. This was driven by ‘taking the opportunities and trying tomake the best out of opportunities when they arrive’. These opportunities included someacquisitions and new ventures in Eastern Europe and Canada. Following these moves, salesgrew by 57% in 2011. However, these expansions met with mixed success and the remainderof the decade saw some ups and downs . Stephen, this has clear implications for how well the patterns in our data may providethe foundation for models or propositions that are suitable for general testing. Thatis, how well our data allow us to make ‘analytical generalisations’  is a key antecedent to growing is firmly established in the literature . Howeverformance .anticipated consequences of growth on motivation ’ (pp. 405-6). Exhibit A presents further case datathat we believe illustrate the role of satisficing in changing growth motivation  . In a sitivation –4.Table The second factor that appears to bear upon changing growth motivations is how difficultthe experience of growing was. When researchers note that ‘growth is hard’, they typicallymean that ‘growth is hard to achieve’, reflecting on the rarity of growth .Howeverexpectations of growth challenges on growth motivation, notes that‘fear of reduced control and employee-wellbeing stand out as the most powerful growthdeterrents’ . But, having experienced both rapid growth and poorer performance,they are now focused on ‘trying to build something which is sustainable’ (Ged).Recognising that what may be good for growth, may not be good for profitability , ourentWhile income and experience had made our entrepreneurs more cautious about growth, theyremained ambitious. However, the extent to which ambition is translated into businessdevelopment is affected both by the complexity of growth and by the entrepreneur’s ‘degreeof volitional control’ , p. 439.The first of these patterns concerned the centrality of the entrepreneur to the initialhigh growth episode. This is a frequent observation, with the size and simplicity of thefirm allowing the entrepreneur to take centre stage . HoweverHowever, as sales grew, weaknesses in systems were revealed. Ged offers a particularlyegregious example: ‘we found one day, by accident, there was an excess of £50,000 notbilled out…We got paid and everything. But, to this day, I know we will not have billedeverything, we won't have found it all because we just weren't set up right’. As Peterobserved, ‘all this growth needed extra resources to actually process all the paperworkand to pack and ship all the goods that were coming in and going out of business. Themargins were just evaporating with all the extra overheads that we were needing’. Theinadequacy of existing systems and the challenges of introducing new systems whilegrowing, is a familiar theme in the growth lifecycle literature . ExhibitThe experiences of our entrepreneurs are consistent with the notion of a ‘curse of fastgrowth’ ; with raeitherexploitation or exploration .As our entrepreneurs devoted more time to exploitation or recognised the increasing needfor better exploitation, they set out to hire explorers or exploiters, respectively. Fourof our entrepreneurs reduced their focus on entrepreneurial behaviours – on opportunityseeking – to focus on managerial tasks, such as improving systems and processes,prioritising profitability over sales. These individuals tried to hire people withentrepreneurial skills to fill the gaps their changing attentions had left behind. Two ofour entrepreneurs continued to focus on opportunities. In these cases, the goal was tohire people with managerial skills, to ensure that systems and processes kept pace withgrowing sales. Regardless of whether they were trying to hire entrepreneurs or managers,explorers or exploiters, our entrepreneurs experienced mixed success. James’ reflectionson a former senior hire are illustrative:‘The technical manager couldn’t be a technical director because I was there as atechnical director, so he didn’t have any power. I didn’t know how to empower him. Themarketing man couldn’t have his own say because, again, I was there saying “No, youcan’t say £19.32. It’s £19.69.”…And I was going that direction, this other man wasgoing that direction. And he wasn’t empowered, so he wasn’t successful’.These missteps in hiring appear to be driven by two factors. The first of these was thereluctance of our entrepreneurs to fully give up former roles. Where an entrepreneur’sformer role becomes a role identity ‘this can create friction between who entrepreneursare and who their ventures need them to be’ , p. 264.Another pattern in our cases was the adhocracy of hiring, especially for senior hires.What little research that exists around hiring patterns in growth firms suggests thatsuccessful firms ‘dedicate extraordinary attention’ to recruiting and developing managers,p. 40. ‘A SME owner knows about all sorts of stuff because they’ve just got to. They may notbe an expert in it, but my God most SME owners can probably tell a lawyer somethingabout employment law that they don’t know if they’re not an employment lawyer becausethey don’t have the niche. They’ve got this wide, huge, and they’ve got to be reallycreative in their thinking and they’ve got to learn themselves because nobody else isgoing to do it for them’.Given this evidence, it would be tempting to echo the lament that static perspectives onhuman capital have dominated the growth literature . HoweverHowever, as the firm grows it becomes more complex and specialist skills are required. Itis growth that triggers the consideration of ‘which hats to keep wearing, which to removeand which new hats to adopt’ , p. 263. it does seem likely that sustaining growth requires that the entrepreneur find ways‘to delegate responsibility and detach themselves…and be happy with it’  and growth motivation willexhibit initially increasing returns, followed by diminishing returns. The point ofinflection on this curve is likely to be a function of aspiration levels. Aspiration levelswill be set through an iterative process of social referencing and by entrepreneurialopportunity costs . They maWe also propose that growth motivations will be moderated by the extent of ‘growing pains’.The more difficult the initial experience of high growth – physically, socially, emotionally– the less likely entrepreneurs are to seek to repeat it. Of course, changing motivationsresulting from growing pains may be more tractable. Where these ‘pains’ are related tosystems failures or human resources, as was most common in our cases, it ought not to bebeyond the capabilities of the various ‘policy’ actors to devise interventions that betterprepare entrepreneurs for these challenges.Certainly, income  and growing pains are not the only factors that may affectattitudes towards growth. In our cases, changing family circumstances  were frequently discussed in relation tocommitment to the business. In many ways, this recalled early social developmentperspectives on entrepreneurship  and more employees during their initial growth, but attract older individuals, already inemployment, in later stages of growth episodes , it isremarkable how little attention is given to human resources and human resource management.Rather, these courses continue to be dominated by discussions of innovation, opportunityrecognition, venture capital, and the like.Of course, our intention is not to suggest that ambidexterity, fostered by considered humanresource management, is the only strategic influence on firm growth. Or, indeed, onsustained or repeated growth. Much as the general literature on the causes of firm growthhas identified numerous influences, of inconsistent impact, we anticipate that researcherswill be able to identify a similar cafeteria of factors that encourage or constrain secondor continued growth , ourgoal was to nudge growth conversations towards a consideration of consequences and to helpus ask better questions.Our exploratory study has some obvious limitations. It rests on a small number of cases,purposively sampled. Our cases are older (both the entrepreneurs and their firms) and intraditional sectors. They are also located in the north of England, with all that entailsThe limitations in the current study suggest avenues for future research. Most directly, wehope to encourage testing of the propositions developed here in larger and more diversesamples. For instance, most empirical research linking incomes to growth intentions haseither been concerned with pre-entrepreneurial incomes and opportunity costs e.g.  or with Beyond this, we do not consider that our propositions are exhaustive. Rather, we hope toencourage further work that acknowledges the episodic nature of growth and recognises that,in addition to causes, episodes have consequences that are likely to be critical tounderstanding the non-linearity of growth paths. In this vein, we anticipate that two linesof research may be particularly promising. The first concerns the multiplicity ofmotivations that entrepreneurs pursue, how these interact, and how they are altered by priorperformance. As discussed earlier, past work has viewed growth as an ‘acquired taste’, withpast positive outcomes reinforcing growth motivations, and vice versa . Howeverth quantile of the growth distribution andhad a negative effect on firm growth in the lowest quantiles. The implication of this workis that what influences performance is likely to vary across the performance distribution.As an extension, we believe that the factors driving growth will also vary across thepast performance distribution. It seems reasonable to speculate thatgrowing out of decline, or from a period of stability, will require different sets ofcapabilities and activities than those required to sustain or repeat high growth.A second line of research concerns the contingent role of past performance on futurebusiness development. Although ‘bouncing back’ may be typical for firms that haveexperienced high growth episodes, there are a small number of firms who are able to repeattheir good performance . In muchWhile our specific goal was to develop testable propositions that would help us in our ownresearch on firm growth, our more general goal was to encourage consideration of growthconsequences, at least as a complement to the large volume of work on growth causes.Specifically, our data suggest that entrepreneurial motivations and behaviours are alteredby past performance and are likely, in their turn, to bear on future performance.We illustrate this in Recognising that growth has proximate consequences that bear directly on future performancepositions growth processes as path dependent . Specifi