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Research Article

Parameter estimation in behavioral epidemic models with endogenous societal risk-response

Roles Data curation, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing – original draft, Writing – review & editing

* E-mail: [email protected]

Affiliation Department of Industrial and Systems Engineering, Virginia Tech, Blacksburg, Virginia, United States of America

Roles Conceptualization, Investigation, Software, Supervision, Validation, Writing – original draft, Writing – review & editing

• Ann Osi, 
• Navid Ghaffarzadegan

• Published: March 29, 2024
• https://doi.org/10.1371/journal.pcbi.1011992
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Behavioral epidemic models incorporating endogenous societal risk - response, where changes in risk perceptions prompt adjustments in contact rates, are crucial for predicting pandemic trajectories. Accurate parameter estimation in these models is vital for validation and precise projections. However, few studies have examined the problem of identifiability in models where disease and behavior parameters must be jointly estimated. To address this gap, we conduct simulation experiments to assess the effect on parameter estimation accuracy of a) delayed risk response, b) neglecting behavioral response in model structure, and c) integrating disease and public behavior data. Our findings reveal systematic biases in estimating behavior parameters even with comprehensive and accurate disease data and a well-structured simulation model when data are limited to the first wave. This is due to the significant delay between evolving risks and societal reactions, corresponding to the duration of a pandemic wave. Moreover, we demonstrate that conventional SEIR models, which disregard behavioral changes, may fit well in the early stages of a pandemic but exhibit significant errors after the initial peak. Furthermore, early on, relatively small data samples of public behavior, such as mobility, can significantly improve estimation accuracy. However, the marginal benefits decline as the pandemic progresses. These results highlight the challenges associated with the joint estimation of disease and behavior parameters in a behavioral epidemic model.

Author summary

Understanding how society’s evolving risk perceptions alter social interactions and disease spread is key to building models that reliably project pandemic trajectories. This research focuses on systematic estimation of parameters in such models where disease dynamics and behavioral responses are intertwined. We find that even with perfect data and models, estimates of behavioral parameters are biased early in the pandemic, and data for at least one full wave of the disease are needed to have reliable estimates. This is mainly related to the time delay between pandemic risks, perceived risks, and public response. Our findings also show that conventional models that ignore risk response dynamics may replicate data well early on but eventually fail to uncover future waves. Additionally, incorporating small amounts of public behavior data (such as data samples from public mobility patterns) for model calibration significantly improves model accuracy, especially early in a pandemic. Moreover, the marginal benefit of additional behavioral data diminishes as the pandemic progresses. By understanding the challenges of estimating key parameters, we can build more reliable models for informed decision making during public health emergencies.

Citation: Osi A, Ghaffarzadegan N (2024) Parameter estimation in behavioral epidemic models with endogenous societal risk-response. PLoS Comput Biol 20(3): e1011992. https://doi.org/10.1371/journal.pcbi.1011992

Editor: Claudio José Struchiner, Fundação Getúlio Vargas: Fundacao Getulio Vargas, BRAZIL

Received: July 13, 2023; Accepted: March 11, 2024; Published: March 29, 2024

Copyright: © 2024 Osi, Ghaffarzadegan. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All simulation files are available on https://github.com/annosi/Parameter-estimation-in-behavioral-epidemic-models-with-endogenous-societal-risk-response.git and we have archived the files on Zenodo ( https://zenodo.org/records/10501198 ).

Funding: NG received funding for this research from the US National Science Foundation, Division of Mathematical Sciences & Division of Social and Economic Sciences, Award 2229819 ( https://www.nsf.gov/awardsearch/showAward?AWD_ID=2229819 ). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

1. Introduction

During the COVID-19 pandemic, peoples’ behavior and reaction to the state of the pandemic played an important role in altering health outcomes across nations and communities [ 1 , 2 ]. As COVID-19 cases and death tolls grew, public risk perception triggered human responses at the individual and governmental levels. These responses, which included non-pharmaceutical interventions (NPIs), helped flatten the curve of the pandemic and decrease the death toll [ 3 – 5 ]. However, as the number of cases declined, risk perception and compliance with NPI measures decreased, leading to more cases and new waves of the disease [ 6 – 8 ]. It is argued that such intertwined behavior-disease dynamics continues as long as an infectious virus is considered a major life-threatening risk [ 9 ].

With the recognition of the importance of human response in the pandemic trajectory, epidemic modeling scholars suggested developing models that incorporate human behavior [ 10 , 11 ]. Such epidemic models peaked in the aftermath of the 2009 UK A/H1N1 pandemic [ 12 ]. These models often extend Susceptible-Exposed-Infected-Removed (SEIR)-like structures [ 13 ], to include interdependencies between the state of the disease, risk perception or fear, and change in average contact rate (e.g., [ 14 – 16 ]). For example, a common approach is to formulate a risk-response mechanism where infectivity rate is an inverse function of recent death to represent human response to change in death rate [ 2 ].

Despite advancements in modeling techniques, and the availability of more granular data from sources such as mobile phones or surveys, the challenge of validation and parameter identifiability for coupled behavior-disease models persists [ 5 , 17 ]. The issue is much more pronounced for parameters used to formulate human response (e.g., human response sensitivity to reported death) than to formulate the spread of the disease (e.g., incubation period). There are several reasons for this challenge. First, while laboratory experiments can provide inputs about disease-related model parameters, we often lack such a luxurious input for behavioral parameters. Second, human response is lagged. For disease-related parameters, effects are generally observed within a week or so (e.g., the delay between exposure, symptom onset, and recovery or death), while behavioral parameters (e.g., changes in public risk perception) take longer, slowly going through several information filters such as public media, social media, and daily conversations. Third, behavioral parameters often vary across different regions and demographics, whereas most disease-related parameters (such as the distribution of incubation period) are similar across different populations. The implication is that the estimated behavioral parameters from one region may not be easily transferable to another setting. These issues introduce several challenges for behavioral epidemic modeling, inhibiting model validation, reliable policy analysis, and accurate projections.

In this study, we systematically examine the challenges of parameter estimation in a specific type of behavior-disease model. Our primary focus is on addressing parameter identifiability challenges, commonly referred to as an inverse problem or model identifiability, rather than developing new models or theories regarding human behavior in epidemic models. Furthermore, behavioral mechanisms can be modeled in various ways and can represent different phenomena. Our focus here is on one specific mechanism, referred to as risk-response [ 9 ]. When modeled endogenously, the risk-response feedback loop represents societal response to change in risk perceptions and influences contact rate or infectivity. This is one of the most common additions in behavior-disease models.

From the existing literature, we select a previously validated behavioral epidemic model that represents risk-response endogenously while having a small number of parameters to be estimated jointly. This model, known as the SEIRb model ( b stands for behavior), is a simple extension on SEIR models and formulates infectivity rate as an inverse function of recent death to represent human response [ 2 ]. The addition of behavioral feedback in the SEIRb model has been shown to substantially enhance forecasting performance of COVID-19 models [ 2 , 9 ].

Using the SEIRb model, we generate synthetic data with a set of assumed parameter values as ground truth. We design experimental setups in which the ground truth (true values of parameters used to generate synthetic data) is concealed from the researcher. Our objective is to evaluate the accuracy of parameter recovery and identify any unique challenges in estimating behavior parameters compared to disease parameters.

2. Background

Behavioral epidemic models.

Epidemic models date back to the works of Kermack and McKendrick [ 13 ] and have been successfully applied to a wide range of infectious diseases [ 18 ]. In many subsequent epidemic models since this foundational work, human behavior, or infectivity rates, remain constant or change by external factors [ 2 ]. Behavioral epidemic models represent a departure from traditional infectious disease models in that they capture the interdependence between the state of the disease and human behavior [ 2 , 16 ]. The primary premise of the behavioral epidemic models revolves around capturing change in human behavior within epidemic models.

There are several systematic literature reviews studying different approaches to incorporate human behavior in epidemic models [ 11 , 12 , 19 ]. One common classification of past efforts is based on modelers’ approach to formulate change in behavior and specifically the source of information that drives behavior change in the models. In reality, such information can be widely available through media or public awareness campaigns [ 20 – 22 ], or it may come from local networks like neighborhoods and friends [ 23 – 25 ]. Another model classification considers how behavior change is formulated. Some models implement exogenous behavior change, controlled by external factors or data input [ 26 – 29 ]. In this approach, change in behavior is not a function of the models’ state variable, but is assumed based on the modelers’ intuition or occasionally feeding time series data. On the other hand, many modelers incorporate behavior change endogenously, where human behavior changes within the model and in response to the system’s states [ 2 , 15 , 30 ]. This latter endogenous approach is essential in capturing potential future changes in human behavior and leads to developing models in which behavior and disease dynamics are coupled.

Among the endogenous behavioral epidemic models, a critical classification lies in identifying the driving factor behind behavior change. In most of the models, behavior changes in response to disease prevalence (infections or mortality) [ 31 – 33 ]. The core idea is that as the disease spreads, people perceive risks, and react by increasing their NPI adherence, which decreases the spread of the disease. There are also studies where change in human behavior is more explicitly modeled by representing opinion dynamics or the spread of fear coupled with the spread of the disease [ 34 – 36 ].

We narrow our focus to a specific type of behavioral epidemic models. We look into a model that incorporates endogenous societal risk-response, where responses change with perceived risks represented by the recent mortality rate. Referred to as SEIRb, this is one of the simplest yet effective approaches to incorporate change in human behavior in epidemic models [ 2 ]. In these models, transmission rates dynamically change in response to society’s perceived risk of deaths. An increase in deaths elevates risk perception, leading to a reduction in transmission rates through the adoption of NPIs. As more people adopt NPIs, disease transmission and the subsequent death rate decreases lowering risk perception. This creates a balancing feedback loop between disease mortality and behavior change. This feedback loop has been shown to be pivotal in the long-term predictive power of multi-wave COVID-19 trajectories [ 2 ].

Parameter estimation

In disease transmission models, it is a common practice to estimate epidemiological parameters that cannot be directly observed by fitting a model to disease data such as infection cases, deaths, hospitalizations, and mobility. Whether or not model parameters can be reliably estimated by this practice is a topic of model identifiability research. For example, when the model is identifiable, infectivity rate (often shown by β) can be estimated by finding a value that minimizes the difference between data and simulation, measured by sum of squared errors. Prior to the COVID-19 pandemic, only a few behavior-disease models utilized real-life data for parameterization or validation [ 11 , 12 , 19 ]. While the pandemic has spurred the development of data-driven models, to the best of our knowledge, none have studied the topic of identifiability.

Parameter identifiability tests are typically recommended as the first step in parameter estimation to ensure reliable estimation of model parameters from observed data [ 37 – 39 ]. These tests take two forms: structural identifiability examines parameter estimation under ideal conditions, and practical identifiability examines realistic conditions of noisy and limited data and improper model structures [ 39 – 41 ]. Parameter identifiability of SIR models with latency, seasonal forcing, immunity, and various other features are extensively covered in epidemic literature [ 42 – 44 ]. However, there is a notable gap for parameter identifiability in behavioral epidemic models. This shortage of theory hinders accurate parameter estimation in these models.

The challenge of parameter identifiability in coupled behavior-disease models arises from increasing model complexity from the inclusion of more compartments, variables, and feedback relationships to describe the behavior side of the model [ 16 ]. The uncertainties and biases accompanying parameter identifiability escalate as both the number of jointly estimated parameters and the model’s complexity increase [ 42 , 45 ].

Even for simpler models, due to computational complexities and mathematical challenges, modelers tend to bypass parameter identifiability analysis [ 39 , 43 ]. Instead, they proceed directly to fitting models to available data, employing methods such as least squares fitting [ 46 ], maximum likelihood estimation [ 47 ], or Bayesian estimation [ 48 ], then measure how well simulation results fit data [ 38 , 49 ]. This approach might not significantly impact modeling outcomes because researchers are already cognizant of identifiability challenges when estimating disease parameters in SIR-like models. However, the same cannot be asserted when modelers must estimate both disease and behavior parameters.

Hence, our study designs practical experiments to systematically investigate the challenges of parameter estimation in behavioral epidemic models. As stated, we focus on models with endogenous societal risk-response where disease and behavior parameters must be jointly estimated. We use the SEIRb model to generate synthetic data and then try to uncover model parameters using a simplistic, yet commonly employed, least squares approach. We compare the accuracy of parameter estimation, the model’s fit to data, and its prediction accuracy across different stages of the pandemic.

3. Study hypotheses

We offer three major hypotheses related to delay in behavioral response, neglecting behavioral response, and availability of behavior data.

Effect of a delayed risk-response

Studies have shown that disease parameters in SEIR models are structurally identifiable [ 44 , 50 ]. This means that under ideal conditions where modelers have accurate disease data (e.g. cases and deaths) and the model structure closely represents reality, one can accurately estimate parameters. In models with endogenous societal risk-response, both behavioral and disease parameters need to be jointly estimated. One can assume that if the disease parameters are identifiable with accurate disease data and model structure that closely represents reality, the same extends to behavioral parameters. Thus, we present the following hypothesis:

• H1a: In the presence of an accurate model and disease data , disease and behavior parameters can be jointly estimated throughout the pandemic accurately .

While we intuitively expect the above to hold, one may argue that early in the pandemic people have not reacted to the state of the disease and are still learning about the situation. Thus, the available data, even if accurate, does not fully incorporate human sensitivity to possible change in risk. If that holds, H1a will not be fully supported. The implications are that the disease-related parameters would be estimated accurately earlier in the pandemic but for behavioral parameters a longer time series will be needed. As more data are collected over the course of an outbreak, estimates for behavioral parameters should also become more precise, and uncertainty should decrease. Thus, we propose to investigate the following competing hypothesis:

• H1b : Estimation of behavioral parameters is unreliable before the peak of the first wave of the pandemic even with accurate model and disease data .

Effect of neglecting risk-response in an epidemic model

All models are simplified representations of the real world, but still having relevant and reasonable assumptions are crucial for accurate parameter estimation [ 51 , 52 ]. Structural uncertainties arise when models include unrealistic scenarios or fail to consider essential aspects of the outbreak, leading to inability of the model to replicate the disease trajectory [ 53 , 54 ]. Such uncertainties cannot be eliminated by refining data; instead, model assumptions that are responsible for failure must be corrected [ 42 ]. To test the importance of the risk-response assumption, we use a conventional SEIR model that lacks behavioral feedback to estimate parameters from a dataset produced by a behavioral model with endogenous risk-response. This is technically a structural sensitivity test [ 55 ] and represents the situation where a modeler neglects behavioral phenomena of a system by using a model that lacks such mechanisms. If supported, the implication is that even if extensive data are available, a model that is not properly representing behavioral responses will fail in parameter estimation and, consequently, in projection. Thus, we propose this hypothesis:

• H2 : Neglecting risk-response leads to unreliable parameter estimation , even with extensive data .

Effect of integrating disease and public behavior data

Various data sources can be used to validate a coupled model. Specifically, a coupled model that generates disease data endogenously can be compared with the pandemic data for validation and parameter estimation, and it is expected that such models should also generate risk response accurately even if they are not compared with risk perception data. Numerous studies have explored data types that can serve as proxies for behavioral responses during outbreaks within target populations [ 56 , 57 ]. Despite this, coupled behavior-disease models that have estimated parameters from data often have focused on minimizing the difference between the disease data and simulation, without trying to replicate the behavioral data as well [ 15 ]. Moreover, data on risk perception, or change in mobility, or any other behavioral reaction is argued to be critical for further model validation (refer to the topic of partial model tests by Homer [ 58 ] and Oliva’s discussion on the importance of partial model tests [ 59 ]. Availability of such behavioral data is hypothesized to significantly improve the estimation of behavior parameters and increase the accuracy of disease forecasting [ 10 – 12 , 19 , 24 ].

Building on these ideas, one can assume that if all data, including data on human behavior such as mobility, are observable and accurate, and model structure closely represents reality, then joint utilization of disease and behavior data for parameter estimation should improve parameter estimation accuracy. Thus, we investigate the following hypothesis:

• H3: In the presence of an accurate model and data , the addition of public behavior data improves accuracy of parameter estimation throughout the pandemic .

Based on the above hypotheses, we also propose that, with better parameter estimation, the accuracy of model projections will improve throughout the pandemic if a model structure with assumptions that closely align with reality is used and the estimation process incorporates public behavior data in addition to accurate data on the state of the disease (e.g., daily death).

4.1. Procedure

We use a three-step process to explore the challenges of parameter estimation under various experimental conditions. First, we create synthetic data from a model with assumed parameter values to serve as “ground truth.” Second, in an inverse process, by providing a fraction of the data, we try to estimate the assumed parameter values through model calibration. By comparing the estimated parameter values from the second step with the ground truth values that were used in the first step to create the synthetic data, we evaluate the accuracy of the estimated parameters. This process is repeated for different conditions to examine uncertainties in parameter estimation for different delays in behavioral response at different periods of the pandemic (H1), when considering incorrect model assumptions (H2) and when utilizing data on public behavior (H3). Finally, to evaluate the potential significance of errors made during calibration, we compare death projection with the estimated parameters with the ground truth data for a period of 365 days.

4.2. Synthetic data generation

We use a stochastic SEIRb model with assumed parameter values to generate synthetic data. SEIRb builds upon the traditional SEIR model by incorporating a balancing feedback loop to capture behavioral responses endogenously [ 2 ]. In short, the model is consistent with the SEIR model and adds dynamic human response to risk; the fact that an increase in the death rate can lead to an increase in risk perception, resulting in more non-pharmaceutical intervention (NPI) initiatives and a decrease in cases.

We make slight modifications to the SEIRb model to make our experiments more realistic. We include autocorrelated noise affecting exposure rate to depict intrinsic stochastic factors that cause real-world data to deviate from expected values such as weather patterns and holiday movements [ 60 ]. While the stochasticity can be introduced to other parts of the model too, such as directly to death, mathematically, we do not expect any meaningful difference with the current formulation as all stochastic elements can be summed up in one variable. This noise is derived from exponential smoothing of white noise, with a mean of 1, standard deviation of 0.3, and a correlation time of 15 days. The correlation time reflects how past noise values influence the current value. It follows an exponential decay, with the time constant matching the correlation time (refer to the topic of correlated noise in [ 61 ]).

Fig 1 depicts the model structure. See S1 Text for detailed model equations.

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Our method does not aim to replicate any specific disease progression with synthetic data. However, we employ parameter values within realistic ranges associated with COVID-19. S1 Text contains a list of the parameter values used in generating the data. The model generates two major outcomes of daily deaths and contact rate which we use to estimate model parameters. The contact rate serves as data on public behavior, and it represents observations of how people are adjusting their contacts during the progression of the disease. We assume that both daily deaths and contact rate are precisely measured.

To generate synthetic data of daily death and contact rate, we simulate the SEIRb model using 100 noise seeds. This results in 100 stochastic realizations of both daily death and contact rate. Fig 2 shows a sample of simulation outcomes for daily death.

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4.3. Experimental conditions

For all experimental setups, only a portion of data is utilized to estimate parameter values. Specifically, the available data is either from the first 60 days of the pandemic (early stage), 120 days (following the first wave), or 365 days (after one year). This design helps us investigate our hypothesis at different periods of the pandemic.

First, we use a “perfect model” (SEIRb, the deterministic version of the same model that created the synthetic data) and daily deaths data to perform model calibration. We record the estimated parameters for each of the different data periods.

Second, we repeat the first experiment using an "imperfect model" (SEIR, a model that does not account for human behavior response). This allows us to determine the impact of using a better model for parameter estimation.

And finally, we perform model calibration using the "perfect model" (SEIRb) but also incorporating both daily deaths data and public behavior data represented by contact rate. This enables us to investigate the benefit of utilizing data on public behavior in improving model calibration for various periods of the pandemic.

The entire process for each experiment is summarized in Table 1 .

https://doi.org/10.1371/journal.pcbi.1011992.t001

4.4. Parameter estimation process

We use the synthetic data generated, depending on the experimental condition being evaluated, to estimate the model parameters. We consider three parameters of the model to be unknown and grouped into two sets: the "disease set" consisting of the infectivity rate (β o ) and the "behavior set" consisting of sensitivity to risk (α) and time to perceive risk (λ p ). In Experiment 2, where we examine the effect of neglecting behavioral response using the "imperfect model" (SEIR), we only need to estimate the infectivity rate (β o ).

Parameter estimation is done using a calibration process that involves minimizing the difference between model output and data [ 59 ]. Detailed information about the parameter estimation method is provided in S1 Text . Since we have 100 separate outcomes of data, each experiment results in 100 estimates of each unknown parameter. For each experimental setup, we compare results obtained from different periods (early 60 days, mid-term 120 days, and late 365 days) and then compare all three experimental conditions.

We assess the performance of parameter estimation by comparing the estimated values to the true parameter values, in terms of bias (closeness to the true value) and variance (dispersion of estimates around the median). A good estimation performance is characterized by low bias (with the median of the estimates close to true values), and low variance (with most estimates tightly clustered).

4.5. Deaths prediction

Using the estimated parameter values, we generate simulated deaths data for 365 days into the future using the estimated parameters obtained from each experiment. The purpose is to see to what extent the estimation errors in parameters affect projection performance. The prediction error is calculated as the daily mean absolute percentage error between simulated deaths and actual deaths cumulated over 365 days (see S1 Text for equation).

4.6. Sensitivity analysis

We test the robustness of our findings through several sensitivity analyses. Specifically, first we examine the sensitivity of our results to change in the delay period in human responses to risk. To that end we test a wide range of perception delay values from 20 to 200 days. Then, we conduct a set of tests to examine identifiability in the presence of three additional mechanisms: seasonality, variant emergence, and loss of immunity after infection. This specific test helps to see if oscillations generated from behavior can be distinguished from other oscillatory mechanisms. Third, we increase the number of unknown disease parameters adding to the complexity of the process of parameter estimation. Fourth, we conduct experiments with different parameter values expanding the range of tests. Finally, we examine the sensitivity of the results to the extent of available behavior data by testing the incremental impact of adding more behavior data. The results are summarized in the results section and reported in detail in S1 Text .

5.1. Effect of delay in behavioral response on parameter estimation

Before analyzing the results from calibration of 100 different realizations of data, we focus on one single estimation as an example. Fig 3 compares actual daily deaths with simulated deaths from one calibration outcome of the SEIRb model. Simulated deaths accurately replicate data for all stages of the disease outbreak during the periods for which data are available. However, during the projection period, calibration in the early stage leads to poorly fitted outcomes. The fit improves in the mid stage and is best at the late stage.

The shaded area is the area for which data were available, and the border between shaded and unshaded area is the time for model calibration (observation time). The unshaded portion is projection for the next 365 days.

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Next, we systematically compare results from calibration with the ground truth values i.e., the assumed values of the parameters during data generation. Fig 4 illustrates the distribution of parameter estimates from calibrating the SEIRb model with daily deaths data. Fig 4 shows that efficient estimation of β o is achieved with low bias and variance at all stages of the outbreak. In contrast, estimation of behavior parameters (sensitivity to death and time to perceive) is unreliable with high bias and variance in the early days of the outbreak but improves at later stages.

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We then statistically compare the estimation errors during different stages of the outbreak (see S1 Text ). The estimates of β o (Infectivity rate) remain relatively consistent, with no significant difference in errors between earlier and later stages of the outbreak. On the other hand, estimation errors of behavior parameters (sensitivity to death and time to perceive) are significantly higher in earlier stages of the outbreak compared to later stages (p<0.001), indicating less accuracy in the former. Overall, the smallest parameter estimation error was achieved using 365 days of data, which explains the better fit of simulated deaths to actual deaths obtained with this observation time.

We conclude that even when using the same model that generated the data, efficient parameter estimation could not be achieved in the early days of disease outbreak for behavioral parameters. Interestingly this is not the case for the disease-related parameter, infectivity rate. Thus, researchers with enough accurate disease data can estimate the initial reproduction rate of R 0 . By not finding support for H1a, we note that the challenge of estimating behavioral parameters during the early periods of the pandemic is beyond data and model accuracy.

Overall, the findings suggest that even a "perfect" model requires data that includes at least one wave of the disease to achieve accurate behavior parameter estimation, and by extension, a good model fit. As the estimations improve after the first wave, we can argue that the difficulty of parameter estimation arises from the delay in public perception and reaction to the changes in the state of the disease.

5.2. Effect of neglecting behavioral response on parameter estimation

Would using a model that insufficiently represents human behavior provide reasonable projections? We start by analyzing a single simulation run. Fig 5 compares actual deaths with simulated deaths from calibrating the SEIRb and SEIR models with daily deaths data. Simulated deaths from the SEIR model only fit well with actual data in the early stage of disease outbreak, and only for the period that data are available (note early 60-days, depicted by grey area in Fig 5 ). However, during the projection period, the SEIR model fails to replicate the oscillations in actual deaths data and produces a single, overestimated wave. As more data are used for calibration, the overestimation decreases, but the wave lags significantly, and simulation does not match data. This behavior is consistent across all SEIR calibration outcomes.

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Next, we systematically investigate, over all 100 realizations, the effect of using an improper model structure on the accuracy of estimated parameters. Fig 6 displays the distribution of parameter estimates obtained from calibrating the SEIR model.

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Interestingly, the improper SEIR model yields comparable β o estimates to those of the SEIRb model in the pandemic’s early stage. The SEIR model produced more accurate estimates, with a smaller β o estimation error (p<0.001; see S1 Text ). However, as calibration periods lengthen, β o estimation accuracy declines with the SEIR model. The estimates become more precise but significantly deviate from true values. This suggests underfitting and explains why the SEIR model fails to capture the data in the long run.

Overall, we find support for H2 that even with extensive data, and even for non-behavioral parameters (such as infectivity rate), parameter estimation would be unreliable if models that neglect behavioral response, such as SEIR, are used. This stresses the fact that data availability when models are not proper for the purpose of analysis will not help. The fact that early in the pandemic, the disease parameters might be accurately estimated with an improper model (SEIR) raises concerns that such early estimations can mislead modelers to rely overconfidently on such models for a long period before time eventually proves that such models substantially fail to represent the reality.

5.3. Effect of data about public behavior on parameter estimation

Can availability of additional data on public behavior help? We first check a single calibration effort as an example. Fig 7 compares actual daily deaths with simulated daily deaths from calibrating the SEIRb model with and without the use of contact rate data. Irrespective of the use of contact rate data, simulated deaths effectively replicate actual deaths in the period for which data is available. Differences in fit to data emerge when projecting deaths for the next 365 days. Fig 7 shows that using contact rate data results in better fit to data in the early stages of disease outbreak, with differences in fit becoming less apparent in the mid-stage, and negligible in the late stage. These findings apply to all 100 calibration outcomes.

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Fig 8 displays the distribution of parameter estimates obtained from calibrating the SEIRb model with and without the use of contact rate data.

Actual parameter values are indicated by broken lines.

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We observe that the accuracy of behavior parameter estimation improves over time with the incorporation of contact rate data in model calibration. Early stage estimates of behavior parameters appear more precise when contact rate data is used than otherwise. However, only mid-stage estimates of sensitivity to death had improved estimation accuracy. In late stage calibration, no notable difference is observed when contact rate data is used.

Additionally, the estimation errors for behavior parameters are significantly smaller in the early stage (p<0.05) when contact rate data is used, and estimation errors for sensitivity to death are significantly smaller in mid stage calibration (p<0.001) with the use of contact rate data. In late stage calibration, the inclusion of contact rate data does not yield significant differences in estimation errors (see S1 Text ).

Overall, these findings partially support H3, suggesting that using data on public behavior improves accuracy of parameter estimation. This effect is most pronounced during the early stages of pandemics, when disease data is insufficient in capturing the impact of behavioral responses to the disease outbreak. While we see statistically an improvement in estimation, Fig 8 shows that the values are still far from the ground truth.

5.4. Variability in model trajectory

In order to compare projections across different experimental conditions, Fig 9 depicts all individual simulations. We see that model fit improves over time for all conditions examined. However, when behavioral response is disregarded in the SEIR model, the model fit is notably the poorest, as it fails to replicate the oscillatory trends observed in the data. For early-stage projections, utilizing contact rate data with a perfect model yields the most favorable results. On the other hand, for late-stage projections, having a perfect model alone is sufficient.

The black lines are 100 synthetic deaths data used for calibration. Green, red, and blue lines are 100 simulated deaths from calibrating the SEIRb model, SEIR model, and SEIRb model with contact rate data, respectively.

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5.5. Future death rate projection

If the purpose is outbreak projection, a reasonable question is how much does the estimated errors in parameter values matter? Fig 10 shows the change in cumulative errors in the next 365-day death projection of different models and data availability conditions from the previous stage. All values are normalized to the projection error of an SEIRb model that uses the early 60-day data (i.e., the first bar on the left is at zero percent).

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Fig 10 shows that calibrations utilizing the SEIRb model demonstrate a decreasing trend in projection errors over time. These results indicate that, given accurate data and model assumptions, prediction accuracy improves over time, and early-stage projections of deaths in a pandemic are likely to be unreliable.

The incorporation of behavior data led to an 8% reduction in projection errors during early stage calibration. Mid-stage calibration saw a 79% reduction in errors when behavior data was used, compared to a 36% reduction when it was not used. In late stage calibration, using behavior data led to an 82% reduction in errors, but this reduction was identical to that achieved without behavior data. These findings highlight that the impact of behavior data on projection accuracy is dependent on the availability of sufficient disease data.

In contrast, calibrations using the improper SEIR model exhibit non-monotonic change in projection errors over time. Early stage calibration shows an 11% reduction in errors, but mid stage calibration is 93% higher error than SEIRb’s early calibration. The SEIR model produces better predictions in the early days when data is limited, but its structural deficiencies start affecting projections as time progresses. Late-stage calibration with the SEIR model shows a 36% smaller error, worse than SEIRb’s 82% reduction in error. Despite smaller errors in later stages of the pandemic, the SEIR model’s projections remain less accurate than those of the SEIRb model.

Overall, late-stage SEIRb calibration produced the smallest reduction in projection error (82%) compared to early-stage SEIRb calibration.

5.6. Results of sensitivity analysis

Our set of sensitivity tests shows that, overall, the main results are robust in a wide range of conditions (see S1 Text for detailed results). First, the argument that at least one full wave of disease data is required for accurate estimations holds for a wide range of perception delay periods. This is consistently observable by varying delay periods from 20 to 200 days. The main reason is that extending the delay to observe human response also influences the period of first wave of the disease. Second, our three major structural changes also confirm that the behavioral model is still fairly identifiable. The tests included adding seasonality, waning immunity, and variant emergence in the behavioral epidemic model. This is especially helpful given that these sub-structures can create additional oscillations in data, which can overlap the effect of risk-response.

We also test the effects of masking more disease-related parameters in four sets of tests. The results are in line with our main argument that during the early phases, estimating behavioral parameters is prone to error. Having more than one unknown on the disease side of the model makes early estimation of disease parameters also challenging. We extend the tests to different parameter values by changing the values of infectivity rate, sensitivity to death, and time to perceive. Results remain consistent. Finally, we test the effect of having a smaller sample size of contact rate data. The goal was to examine the extent to which behavioral data would be sufficient and useful. Notably, smaller samples of contact rate data from the first wave still provide considerable improvements, showing the significant gain of having any information about public reaction to the disease.

6. Discussion and conclusion

We conducted a set of simulation experiments to assess challenges and uncertainties of parameter estimation in behavioral epidemic models with endogenous societal risk-response. Our aim was to examine model identifiability and accurate joint estimation of disease and behavior parameters. Specifically, we hypothesized that the reliability of behavioral parameter estimation is limited before the first wave. Additionally, we investigated the accuracy of parameter estimation under models that overlook behavioral responses. Furthermore, we tested the hypothesis that the addition of data on public behavior, such as observed changes in contact rates in response to the disease, enhances the accuracy of parameter estimates. We projected deaths using the parameter estimates derived from all experiments to examine if the impact is meaningful on projection errors, and to determine which experimental condition yielded the lowest projection error.

Our findings reveal the difficulty in accurately estimating behavioral parameters prior to the first pandemic wave, even in the presence of sufficient disease data and a sound model structure. This challenge arises from the delayed human response to pandemic risks; before the first wave, human reactions are insufficiently triggered, and the data are unaffected by these responses. The societal risk-response takes longer to get triggered compared to other system delays such as the incubation period or disease period [ 62 ]. The first waves of the pandemic are mainly driven by changes in risk responses, further complicating the calibration process prior to experiencing the wave. In other words, pre-wave data offers limited insights into people’s response to risks, hindering model calibration. However, a sufficiently extended observation period capturing risk responses enables feasible parameter estimation even in complex systems with additional oscillatory features like seasonality, waning immunity, or variant emergence. This underscores the importance of the timing of societal risk-response delays, setting a minimum data requirement for identifying behavioral parameters. Thus, modelers should exercise patience in projections and explore alternative sources to obtain more direct parameter information.

Furthermore, we find that while the accuracy of predictions improves over time when using an accurate model that better represents reality, this is not the case with a model that improperly neglects change in behavior. In the early days of a disease outbreak, before public reactions have a significant impact on the disease’s state, the SEIR model produces smaller errors than the SEIRb model, despite being an improper model (p<0.001). This is likely due to the fact that the SEIR model has lower degrees of freedom which allows for reasonable estimates of parameter values despite structural deficiencies. This finding suggests that, under certain conditions, an improper model can produce reasonable results. However, the issue is that this can misleadingly create overconfidence for modelers thinking that they can rely on SEIR-like structures that lack behavioral feedback for too long. As time progresses and public behavior changes in response to the disease, the SEIR model’s projection errors increase considerably. The SEIR model cannot generate multiple waves of a pandemic without fine-tuning and changing infectivity rate exogenously. We observe that numerical methods or using larger quantities of data cannot mitigate the impact of ignoring behavioral feedback in models’ structure. As the behavioral feedback in models with endogenous societal risk-response only becomes apparent over time, its exclusion can lead to short-term prediction success but long-term failure at later periods of disease outbreak. These findings emphasize the insufficiency of solely relying on extensive data when models are not properly formulated to incorporate change in human behavior.

While the use of observed data on public behavior has been suggested to enhance the accuracy of pandemic forecasts, our findings indicate that its impact is dependent on the availability of sufficient disease data. Specifically incorporating behavior data in the early stages of a pandemic results in improvements to projection accuracy, but the marginal benefits of having behavioral data decline as the pandemic progresses. These findings emphasize the potential of behavior data, even in limited quantities, for improving early-stage projection accuracy in pandemics.

This study contributes to the growing body of research on behavioral system dynamics [ 63 ] and specifically the integration of behavior into dynamic models of infectious diseases [ 11 ]. Our analysis, while resonating with several calls for improving validation of behavioral epidemic models [ 19 , 24 , 64 ], shows that the challenges go beyond data availability and include model identifiability limitations.

We recognize several limitations to our approach that provide avenues for further research. Our focus was on a single behavioral epidemic model, and we suggest future studies to analyze parameter estimation in more complex behavioral epidemic models including agent-based structures. Importantly, our study uses a known, true model, whereas in reality, structural uncertainties are greater than what we have assumed, and the set of unknown parameters to be estimated could be larger, posing a greater challenge for accurate parameter estimation. Additionally, inaccuracies in data arising from measurement errors, underreporting or overreporting of cases and deaths were not considered. While we employed the least squares estimation method, using scaled Gaussian and negative binomial methods may provide less biased estimates [ 49 ]. However, the choice of estimation method is unlikely to affect the comparison of errors across experimental conditions, which is determined by the model and data rather than the fitting process [ 50 ].

Overall, this study points to the challenge of joint estimation of disease and behavioral parameters in a behavioral epidemic model with endogenous societal risk-response. The challenges are related to the observation period, where in early stages it might be infeasible to provide accurate estimation of the parameter values, and to the use of improper models for model calibration. While gathering extensive and accurate data is required, it is not sufficient for providing accurate projections.

Supporting information

S1 text. additional model details and results..

S1 Text comprises ten sections: 1) model equations; 2) list of parameter values used to generate synthetic data; 3) parameter estimation method used in experiments; 4) performance metrics for deaths projection; 5) statistical test results for main experiments; 6) sensitivity analysis to length of delay in human response to risk; 7) sensitivity analysis to other oscillatory phenomena; 8) sensitivity analysis for other disease parameters; 9) sensitivity analysis to different parameter values, and 10) sensitivity analysis to amount of contact rate data.

https://doi.org/10.1371/journal.pcbi.1011992.s001

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Title: hyper-parameter optimization: a review of algorithms and applications.

Abstract: Since deep neural networks were developed, they have made huge contributions to everyday lives. Machine learning provides more rational advice than humans are capable of in almost every aspect of daily life. However, despite this achievement, the design and training of neural networks are still challenging and unpredictable procedures. To lower the technical thresholds for common users, automated hyper-parameter optimization (HPO) has become a popular topic in both academic and industrial areas. This paper provides a review of the most essential topics on HPO. The first section introduces the key hyper-parameters related to model training and structure, and discusses their importance and methods to define the value range. Then, the research focuses on major optimization algorithms and their applicability, covering their efficiency and accuracy especially for deep learning networks. This study next reviews major services and toolkits for HPO, comparing their support for state-of-the-art searching algorithms, feasibility with major deep learning frameworks, and extensibility for new modules designed by users. The paper concludes with problems that exist when HPO is applied to deep learning, a comparison between optimization algorithms, and prominent approaches for model evaluation with limited computational resources.

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Parameter and Statistics – A Comparison Based Guide

Published by Owen Ingram at August 31st, 2021 , Revised On September 20, 2023

We deal with these two terms in statistics a lot and often have difficulty differentiating them. If you are one of those people, you have come to the perfect place. This guide is on statistics vs. parameters is the perfect rejoinder to all your sceptical thoughts.

What are Parameter and Statistics in Research?

A parameter  in statistics implies a summary description of the characteristic of an entire population based on all the elements within it.  For instance, all the children in one city, all female workers in a country, all the items in grocery stores globally, and so on.

Suppose you ask the students in a class what kind of lunch they prefer. Half of them might say fries, and half would say a burger or sandwich. Now the parameter here would be 50% . But can you count how many children in the whole world like fries or sandwiches? You sure cannot do that. If it is impossible to count or measure a value in such cases, you take a sample and assess the entire population based on it. So the sample here (children in the class) represents the entire population (children in the world) .

On the other hand , statistics   is a measure of a characteristic of a fraction or sample of the population under study . It is basically a part or portion of a population. While the parameter is drawn from the measurements of units in the entire population, a sample is drawn from the measurement of elements from the sample.

For instance, if you are to find out the mean income of all the subscribers of a YouTube channel-a parameter of a population. You might simply draw some 150 subscribers and determine their mean income (a statistic). You would conclude that the average income of the entire population is the same value as the sample.

Here are a few more keynotes to make the differentiation between statistics and parameters:

Below is a list of examples of both for further clarification.

Examples of Parameters:

• The median income of all the females in Atlanta
• Standard deviation of all the rice farms in the city
• Proportion of all the schools in the world

Examples of Statistics:

• The median income of 1050 females in Georgia
• Standard deviation of one rice farm in the city
• The proportion of all the schools in one region

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Statistical Notation of Parameter and Statistics

Interpreting study results with statistics and parameters.

When assessing the outcomes of research or study, you sure can be confident that parameters influence factors relevant to every person, object, and circumstance within a given population.

However, with statistics, it is imperative to keep in mind how the sample was collected in the first place. You must see if the sample is an accurate representation of the entire population. Only then would you be able to generalize conclusions for the whole population. A random sample here is most probably a good representative of a population.

Another thing to take into consideration while interpreting data is understanding what the numbers represent instead of just knowing what numerical calculations you have at hand. If the data is parameters, you must ensure that they are descriptive of the given population.  And, if the data you are provided with is statistics, you must pay attention more than required and take as much time as you need because this is the tricky one.

In the meantime, if you have any confusion or queries, please let us know by commenting below

FAQs About Parameters and Statistics

What is a parameter.

A parameter is a number that describes the whole population. For example, all the boys in the world.

What is a statistic?

A statistic is a number that describes a sample from a population under study.

Why are samples used in research?

Sample are used to make inferences or conclusions about different populations. Sample are a lot easier to collect information/data because they are manageable and cost effective.

What are two commonly used parameters?

The sign μ refers to a population mean while σ refers to the standard deviation of a population.

What are two commonly used statistics?

s2 is used for representing sample variance and X ~ is used to show the distribution of random variable X

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Numerical data is a kind of data expressed in numbers. It is also sometimes referred to as quantitative data, which is always collected in the form of numbers.

A random variable deduced from sample data and used in a hypothesis test is a test statistic. Its primary role is to find out if the null hypothesis is plausible or not.

This comprehensive guide introduces what mode is, how it’s calculated and represented and its importance, along with some simple examples.

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• Inferential Statistics | An Easy Introduction & Examples

Inferential Statistics | An Easy Introduction & Examples

Published on September 4, 2020 by Pritha Bhandari . Revised on June 22, 2023.

While descriptive statistics summarize the characteristics of a data set, inferential statistics help you come to conclusions and make predictions based on your data.

When you have collected data from a sample , you can use inferential statistics to understand the larger population from which the sample is taken.

Inferential statistics have two main uses:

• making estimates about populations (for example, the mean SAT score of all 11th graders in the US).
• testing hypotheses to draw conclusions about populations (for example, the relationship between SAT scores and family income).

Table of contents

Descriptive versus inferential statistics, estimating population parameters from sample statistics, hypothesis testing, other interesting articles, frequently asked questions about inferential statistics.

Descriptive statistics allow you to describe a data set, while inferential statistics allow you to make inferences based on a data set.

• Descriptive statistics

Using descriptive statistics, you can report characteristics of your data:

• The distribution concerns the frequency of each value.
• The central tendency concerns the averages of the values.
• The variability concerns how spread out the values are.

In descriptive statistics, there is no uncertainty – the statistics precisely describe the data that you collected. If you collect data from an entire population, you can directly compare these descriptive statistics to those from other populations.

Inferential statistics

Most of the time, you can only acquire data from samples, because it is too difficult or expensive to collect data from the whole population that you’re interested in.

While descriptive statistics can only summarize a sample’s characteristics, inferential statistics use your sample to make reasonable guesses about the larger population.

With inferential statistics, it’s important to use random and unbiased sampling methods . If your sample isn’t representative of your population, then you can’t make valid statistical inferences or generalize .

Sampling error in inferential statistics

Since the size of a sample is always smaller than the size of the population, some of the population isn’t captured by sample data. This creates sampling error , which is the difference between the true population values (called parameters) and the measured sample values (called statistics).

Sampling error arises any time you use a sample, even if your sample is random and unbiased. For this reason, there is always some uncertainty in inferential statistics. However, using probability sampling methods reduces this uncertainty.

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The characteristics of samples and populations are described by numbers called statistics and parameters :

• A statistic is a measure that describes the sample (e.g., sample mean ).
• A parameter is a measure that describes the whole population (e.g., population mean).

Sampling error is the difference between a parameter and a corresponding statistic. Since in most cases you don’t know the real population parameter, you can use inferential statistics to estimate these parameters in a way that takes sampling error into account.

There are two important types of estimates you can make about the population: point estimates and interval estimates .

• A point estimate is a single value estimate of a parameter. For instance, a sample mean is a point estimate of a population mean.
• An interval estimate gives you a range of values where the parameter is expected to lie. A confidence interval is the most common type of interval estimate.

Both types of estimates are important for gathering a clear idea of where a parameter is likely to lie.

Confidence intervals

A confidence interval uses the variability around a statistic to come up with an interval estimate for a parameter. Confidence intervals are useful for estimating parameters because they take sampling error into account.

While a point estimate gives you a precise value for the parameter you are interested in, a confidence interval tells you the uncertainty of the point estimate. They are best used in combination with each other.

Each confidence interval is associated with a confidence level. A confidence level tells you the probability (in percentage) of the interval containing the parameter estimate if you repeat the study again.

A 95% confidence interval means that if you repeat your study with a new sample in exactly the same way 100 times, you can expect your estimate to lie within the specified range of values 95 times.

Although you can say that your estimate will lie within the interval a certain percentage of the time, you cannot say for sure that the actual population parameter will. That’s because you can’t know the true value of the population parameter without collecting data from the full population.

However, with random sampling and a suitable sample size, you can reasonably expect your confidence interval to contain the parameter a certain percentage of the time.

Your point estimate of the population mean paid vacation days is the sample mean of 19 paid vacation days.

Hypothesis testing is a formal process of statistical analysis using inferential statistics. The goal of hypothesis testing is to compare populations or assess relationships between variables using samples.

Hypotheses , or predictions, are tested using statistical tests . Statistical tests also estimate sampling errors so that valid inferences can be made.

Statistical tests can be parametric or non-parametric. Parametric tests are considered more statistically powerful because they are more likely to detect an effect if one exists.

Parametric tests make assumptions that include the following:

• the population that the sample comes from follows a normal distribution of scores
• the sample size is large enough to represent the population
• the variances , a measure of variability , of each group being compared are similar

When your data violates any of these assumptions, non-parametric tests are more suitable. Non-parametric tests are called “distribution-free tests” because they don’t assume anything about the distribution of the population data.

Statistical tests come in three forms: tests of comparison, correlation or regression.

Comparison tests

Comparison tests assess whether there are differences in means, medians or rankings of scores of two or more groups.

To decide which test suits your aim, consider whether your data meets the conditions necessary for parametric tests, the number of samples, and the levels of measurement of your variables.

Means can only be found for interval or ratio data , while medians and rankings are more appropriate measures for ordinal data .

Correlation tests

Correlation tests determine the extent to which two variables are associated.

Although Pearson’s r is the most statistically powerful test, Spearman’s r is appropriate for interval and ratio variables when the data doesn’t follow a normal distribution.

The chi square test of independence is the only test that can be used with nominal variables.

Regression tests

Regression tests demonstrate whether changes in predictor variables cause changes in an outcome variable. You can decide which regression test to use based on the number and types of variables you have as predictors and outcomes.

Most of the commonly used regression tests are parametric. If your data is not normally distributed, you can perform data transformations.

Data transformations help you make your data normally distributed using mathematical operations, like taking the square root of each value.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Confidence interval
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

Descriptive statistics summarize the characteristics of a data set. Inferential statistics allow you to test a hypothesis or assess whether your data is generalizable to the broader population.

A statistic refers to measures about the sample , while a parameter refers to measures about the population .

A sampling error is the difference between a population parameter and a sample statistic .

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

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A Simple 3-Parameter Model for Cancer Incidences

Xiaoxiao zhang.

1 Bonn-Aachen International Center for Information Technology, Dahlmannstraße 2, Bonn, 53113 Germany

2 Department of Medicine II, Klinikum Rechts der Isar, Technische Universität München, München, 81675 Germany

3 German Cancer Consortium (DKTK), German Cancer Research Center (DKFZ), Heidelberg, 69120 Germany

Holger Fröhlich

4 UCB Biosciences GmbH, Alfred-Nobel-Straße 10, Monheim, 40789 Germany

Dima Grigoriev

5 CNRS, Mathématiques, Université de Lille, Villeneuve d’Ascq, 59655 France

Sergey Vakulenko

6 Institute for Mechanical Engineering Problems, Russian Academy of Sciences, Saint Petersburg, Russia

7 Saint Petersburg National Research University of Information Technologies, Mechanics and Optics, Saint Petersburg, Russia

Jörg Zimmermann

Andreas günter weber.

8 Institut für Informatik II, Universität Bonn, Friedrich-Ebert-Allee 144, Bonn, Germany

Associated Data

All data used in this study are publicly available. The sources are detailed in the section on methods.

We propose a simple 3-parameter model that provides very good fits for incidence curves of 18 common solid cancers even when variations due to different locations, races, or periods are taken into account. From a data perspective, we use model selection (Akaike information criterion) to show that this model, which is based on the Weibull distribution, outperforms other simple models like the Gamma distribution. From a modeling perspective, the Weibull distribution can be justified as modeling the accumulation of driver events, which establishes a link to stem cell division based cancer development models and a connection to a recursion formula for intrinsic cancer risk published by Wu et al . For the recursion formula a closed form solution is given, which will help to simplify future analyses. Additionally, we perform a sensitivity analysis for the parameters, showing that two of the three parameters can vary over several orders of magnitude. However, the shape parameter of the Weibull distribution, which corresponds to the number of driver mutations required for cancer onset, can be robustly estimated from epidemiological data.

Introduction

Cancers arise after accumulating epigenetic and genetic aberrations 1 – 3 . Earlier studies established a power law model on the basis of multi-stage somatic mutation theory to explain age-dependent incidences 4 – 6 for several cancer types. As noted by Hornsby et al . 7 in the context of classical epidemiological studies most cancers occur with the same characteristic pattern of incidence, and the simplicity of this pattern is in contrast to the perceived complexity of carcinogenesis. Orthogonal to these age stratification of different cancer types, Tomasetti and Vogelstein 8 (with follow-ups 9 , 10 ) reported a significant association between life time caner risk and stem cell divisions and concluded the latter substantially contributes to the former. Challenging the conclusion of Tomasetti and Vogelstein 8 of a high-intrinsic cancer risk Wu et al . 11 subdivided cancer risk into extrinsic and intrinsic risk, arguing extrinsic factors contribute more to cancer risks than intrinsic factors do. Based on a mechanistic model of accumulated mutations, these authors provided a recursion formula for theoretical life time intrinsic risk (tLIR) parameterized by age a . This recursion formula has the closed form solution tLIR ( a ) = 1 − (1 − (1 − (1 − r ) log 2 S + d ⋅ a ) k ) S , where S can be interpreted as the numbers of stem cell, d as the stem cell division rate, k as number of driver events required for cancer onset and r as the mutation rate per division. They reported that tLIR goes outside of the plausible range of empirical cancer risks by studying several pairs of values for two parameters (mutation rate and driver gene mutations) concluding that there is a substantial contribution of extrinsic risk factors to cancer development. However, this conclusion only holds in the studied parameter space and when parameters for all cancer types are treated uniformly. By performing a systematic grid search in the space of biologically plausible parameter values we showed that tLIR can be close to empirical risk for different cancer types ( R 2 > 0.85). If the extrinsic risk factor is computed by simply setting it to a complement of 1 for the intrinsic risk factor as performed by Wu et al . it will be concluded that there is a possibility of high intrinsic risk, so that one of the presented arguments by Wu et al . 11 is fallacious.

On a pure mathematical side, we show that a scaled Weibull function with 3 parameters approximates the 4-parameter mechanistic tLIR model. On an epidemiological data analytical side, this simple 3-parameter model excellently agrees with age-dependent cancer incidence curves among 18 common solid cancers even when variations due to different locations, races, or periods are taken into account. With this model, we study the relationship between cancer risk and stem cell divisions, the high correlation between the two entities reported by previous studies 8 , 10 breaks down when considering age stratified data.

Approximation of tLIR model by a scaled Weibull function

As is derived in the Materials and Methods the 4-parameter mechanistic tLIR model can be approximated by a scaled Weibull function with 3 parameters:

assuming that λ is defined by

Here Weibull( λ ,  k )( a ) = 1 −  e −( a / λ ) k is the cumulative distribution function of the Weibull distribution, and P is the number of independent parallel processes, which e.g. can be interpreted as cell population at risk 12 . Whether the total tissue cells or only a fraction of stem cells are susceptible for cancer risk is unclear 13 , 14 . If one sets P = S then rd = λ −1 . However, other possible choices for P allow to account for other factors such as the selection of mutations 15 , 16 , the stem cell microenvironment 17 , 18 , and tissue architecture 19 – 22 , or effects of clonal expansion 23 – 25 . Models incorporating clonal expansion have additional parameters such as the number of clonal copies. Reducing the dimensions of such complexed models results in tLIR, in which S is interpreted as number of independent clusters after clonal expansion rather than the number of stem cells, r and d denote “net” mutation and division rate of independent clusters at average level rather than those of single cells. Whereas a precise analysis of models for clonal expansion will be the topic of future work, these considerations show that when using the scaled Weibull distribution, prior knowledge on the parameter ranges is not necessary. This is indeed one benefit of scaled Weibull function comparing to tLIR model which requires a biologically reasonable guessing on stem cell numbers, mutation rate, cell division rate and number of driver mutations. The Weibull distribution is a special case of the generalized extreme value distribution (GEV) 26 . The GEV distribution plays the same role within extreme value statistics as the normal distribution does in average value statistics. It results in the limit distribution being maximized over many independent and identically distributed random variables, thus becoming the default model for the accumulation of micro events which finally leads to a macro event. The GEV is the limit distribution when one takes the maximum (and not the sum) of many independent and identically distributed random variables, thus being the default model for the accumulation of micro events which finally lead to a macro event. Accordingly, the Weibull distribution is not just a distribution providing a good empirical fit, but can be seen as justifiable for use in a plausible causative model of cancer genesis.

Fitting empirical incidence rates with scaled Weibull function

We performed extensive simulations and parameter fittings for the empirical incidence cuminc c ( a ) of cancer type c at age a using the scaled Weibull function: cuminc c ( a ) ≈ P c ⋅ Weibull( λ c , k c )( a ). The model agrees excellently with age-dependent age incidences of 18 common solid cancers ( R 2 > 0.99, Fig.  1 ).

Empirical cumulative cancer incidence data are consistent with the Weibull cumulative probability function in 18 cancers (data for ages up to 85 years old). Empirical (blue line) and Weibull function-fitted (red line) cancer cumulative incidence curves for 18 tissues, goodness of fit is reported in each subplot. The 18 cancers exhibit a good goodness of fit when using R 2 between model-reported age incidence and the empirical cumulative cancer incidence are used as metrics.

Goodness of fit maintains when parameters P c and λ c , varying roughly two orders of magnitudes (Fig.  2 ). This finding suggests that many parameter combinations provide similar dynamics that are consistent with empirical data. So any interpretations of P c and λ c have to take into account this considerable uncertainty. Nevertheless, the estimates for P c are several orders of magnitude smaller than the realistic number of stem cells provided by Tomasetti and Vogelstein 8 , yielding evidence supporting the above statement that the number of independent local processes is not equal to the number of stem cells.

Sensitivity analysis of parameter estimates using the scaled Weibull function for exemplary 14 cancer types. Whereas the estimates P c for the cell population at risk and the scale parameter λ c can vary over two order of magnitude, the estimates of the shape parameter k c are within about ±1. Notice that the shape parameter allows interpretation as the number of limiting events.

The estimates of parameter k c , which corresponds to the number of driver events in the mechanistic model, are robust against variations of parameters P c and λ c (Fig.  2 ). Moreover, the estimates of k c are robust against race, sex, period and location (Fig.  3 ). In Supplemental Fig.  2 the best fits of shape parameters are plotted against the best fits of scale parameters for 694 time series.

Shape parameters estimated by fitting empirical cancer incidence data using the Weibull function (data with ages for up to 85 years). Cancer patients are grouped by year of diagnosis, race and registry. Cancers are ordered by median values of shape. Shapes are uniform regardless of risk factors, which is consistent with intuitive expectations: race and environmental changes are less likely to alter the number of driver events for cancer onsets.

Relationship between cancer incidence and stem cell divisions

Tomasetti and Vogelstein 8 suggested that the variation in cancer risk among tissues can be explained by the number of stem cell divisions. They reported that the tissue-specific cancer risk is strongly correlated (0.81) with life-time stem cell divisions (LSCD). These authors stated that the total number of stem cell divisions is a causative factor of cancer risk. This assumption yields a prediction on age structured data: for tissue type c the number of stem cell divisions up to age a , which we will denote by LCSD c ( a ), should then be strongly correlated with cuminc c ( a ). However, using age incidence data obtained from the SEER-database 27 we found that the regression lines for most tissue types c for age data of 40, 50, 60, 70, and 80 years of cuminc c ( a ) plotted against LCSD c ( a ) in a log-log-scale are much steeper than the ones of the regression lines for different c and cuminc c (80)—using 80 as average life span as was done by Tomasetti and Vogelstein 8 (see Fig.  4 ).

Relationship between cancer incidence and stem cell divisions among 30 cancer types. The lifetime cancer risk regression line is conceptually the same as that used by Wu et al . 11 .

This “life time cancer risk” moderately associates with age-dependent stem cell divisions, if one takes a life-time a that is less than 70 years (Fig.  5 ).

Relationship between cumulative cancer incidences up to age 40, 50, 60, 70, 80 years old and life time stem cell divisions.

Overall, age-dependent stem cell divisions (using ages 40–80 years) is modestly correlated to age-dependent cancer risk for the 31 cancer types considered by Tomasetti and Vogelstein 8 using the SEER database and the estimates of stem cell divisions given therein (Pearson correlation coefficient ρ = 0.51).

Hence, the strong correlation for (life-time) tissue-specific cancer risk with life-time stem cell divisions (LSCD) cannot be explained by the simple causative factor (involving the product of the number of stem cells and the number of divisions of each stem cell) suggested by Tomasetti and Vogelstein 8 . A causal explanation on cancer risk should at least shows that the association between cancer risk and risk factor observed at overall level is reproducible on age stratified data. However, one caveat to such explanation, co-factors of risk factors might not be appreciated.

In our 3-parameter model, which gives good fits for age dependent cancer risks, several relations between the model parameters and cancer risks at a certain age can be observed. For instance in our parameter estimates good fits are possible when taking the inverse of the lifetime cancer risk P c ≈ 1/cuminc c (85). However, we will not suggest that the number P c of cell population at risk is an explanation for the variation of cancer risks among tissues: as the range of P c yielding good fits varies by two orders of magnitude and independently determining this number is difficult to achieve, a corresponding hypothesis is difficult to verify or to falsify.

As the sensitivity analysis for the scale parameter λ c (Fig.  2 ) shows that this parameter varies over several order of magnitudes, still yielding very good fits ( R 2 > 0.99), the corresponding estimates for the mutation rate r in the tLIR model using the approximation ( 1 ) and relation ( 2 ) are also very uncertain, even when fixing values of S and d and leaving out the considerable uncertainty of these. Nevertheless, when using estimates of S and d taken from the literature 8 the obtained ranges of values of r using relation ( 2 ) for several cancer types do not intersect the range [10 −10 , 10 −6 ] of “plausible values” of r suggested by Wu et al . 11 . If we extend the analysis to allow “good fits” by setting a threshold R 2 > 0.85, then good fit of the tLIR model with r ∈ [10 −10 , 10 −6 ] are possible to achieve (Table  1 ).

One possible combination of parameters with which the tLIR model of 11 fits empirical data well. We are restricting r to be in the range [10 −10 , 10 −6 ] as was done by 11 .

1 Assuming lifetime is 85 years old, stem cells go through log 2 S +  d  ⋅ 85 generations.

*Cancers of which parameter estimates are impossible because division rate is 0.

Testing performance of our 3-parameter against other simpler model

For testing the performance of our 3-parameter model against other simpler models, we compared the fitting performance of the scaled Weibull function to that of 2-parameter power law model arising as the simplest instance from multistage theory 5 , 7 , 12 , 13 . The empirical time series for different locations, periods and races were fitted (694 time series all together) using both models, the power law model had a goodness of fit of R 2 < 0.90 for 90 time series (13.0%), R 2 < 0.95 for 257 time series (37.0%), and R 2 < 0.98 for 366 time series (52.7%). In contrast, our 3-parameter scaled Weibull model resulted in R 2 > 0.9 for all time series, R 2 > 0.98 for 686 (=98.8%) of the time series, and R 2 > 0.99 for 679 (=97.8%) of them (Fig.  6(a) ).

Goodness of fit for scaled Weibull function versus that of power law function ( a ), and scaled Gamma function ( b ). Each dot represents R 2 for one cancer subtype defined by the combination of cancer type and one factor such as diagnosis year, race, location and sex. Cancer types are color coded.

We compare the fitting performance of the scaled Weibull function against that of the scaled Gamma function. Although both functions fit data equivalently well in most cases, the scaled Weibull function outperforms the scaled Gamma function in several time series. (Fig.  6(b) ) displays R 2 reporting goodness of fit for the two functions. We also calculate the Akaike information ciriterion(AIC) 28 , a likelihood based measurement. A lower AIC value indicates a better fit. Table  2 reports the AIC for 18 cancer types, the AIC values for Weibull function are lower than those of Gamma function in 15 cancer types.

AIC of the scaled Gamma function and the scaled Weibull function.

Estimating the Number of Driver Mutations for Cancer Onset

In our model the shape parameter k c reflects the number of mutations required for cancer onset. The values of this parameter are, however, higher than the number of mutations estimated from sequencing data by Vogelstein et al . 29 . Vogelstein et al . suggested technical issues as an explanation for the inconsistency between estimates from epidemiological data and sequencing data. Notably, our k c estimates and the number of driver mutations estimated from a classical power law model are roughly in the same numerical range (Fig.  7 ). Since we obtain better and more robust fits than the power law model, we believe that our estimated driver mutation numbers are more trustworthy.

Number of driver mutations required for cancer onset estimated by classical power law model (red) and our scaled Weibull model (blue).

In this study we connected the mechanism-based cancer development tLIR model to the Weibull distribution function. We tested its validity by fitting a 3-parameter Weibull function to data from 18 common solid cancer types, consisting of more than 600 time series. The scaled Weibull function fits well with age dependent incidence curves of all studied cancers and outperforms other models, such as the commonly used 2-parameter power law model and a 3-parameter scaled Gamma function model. With the scaled Weibull function, we can estimate the number of driver mutations required for cancer onset in individual cancer types. To our knowledge, this is the first work matching pan-cancer incidence curves with a statistical distribution function that is partially biologically informative.

Compared to the tLIR model developed by Wu et al . 11 we see two technical benefits of our suggested approach: First, the scaled Weibull function involves less parameters than tLIR, but it remains to be biologically interpretable. The tLIR model includes several details of the multi-staged process of cancer development, e.g. the number of steps required for transforming a normal cell to malignancy, the number of stem cells in a tissue and division rate of stem cells. Although the tLIR model indeed provides useful insights into linking age-dependent somatic mutations to cancer risk, it has also limitations. For example, it ignores the effects of clonal expansion 25 . Another issue is that most parameters in the tLIR model are difficult to measure accurately in practice. Following Tomasetti and Vogelstein 8 the number of stem cell divisions can be estimated, but the accuracy has been criticized 30 . In contrast, our suggested model requires less specific assumptions about the parameters to be measured in practice. Moreover, the Weibull distribution is a special case of the generalized extreme value distribution (GEV) which is well connected to classical statistical approaches to describe rare events 26 .

Our analysis results mostly agree with those provided by Wu et al . 11 . They defined intrinsic cancer risk as the probability that one tissue transforms from normal to tumor because of accumulated mutations, and extrinsic cancer risk as 1–intrinsic cancer risk. They quantified upper bounds to intrinsic cancer risk by tLIR but did not properly fit tLIR to epidemiological data, concluding intrinsic factor insignificantly contributes to cancer. According to our understanding their argument mainly results from insufficient exploration of parameter space and implicitly assumes that all tissues require the same number of driver mutation to initiate cancer. Our results suggest that the contribution of extrinsic factors to cancer is overestimated by Wu et al . 11 . However, one should note that the excellent agreement between the scaled Weibull distribution function and empirical data does not necessarily exclude that in addition to intrinsic there are further extrinsic and unknown risk factors. In that context it is worthwhile to mention that our estimated number of driver mutations required for cancer onset differs from tissue to tissue. Although the exact number is not validated by biological experiment, this observation is consistent with findings in genetic studies 29 .

One interesting observation is that all non-reproductive tissues have a similar cancer risk accumulation pattern. Cancer incidence rates increase dramatically at about 40–50 years, peaking at about 55–70 years and then decrease. This pattern matches findings reported by Podolskiy et al . 31 . A question for future work is whether mutation load agrees with the scaled Weibull function or age-specific mutational signatures 32 – 34 . Another interesting observation is that testicular germ cell cancer incidence peaks at younger age compared to other cancer types, which might be explained by accelerated aging of testis 31 . Altogether we believe that our suggested approach provides insights into cancer development by providing a link between empirical data and a mechanism-based model.

Fitting cumulative cancer incidence with a model for theoretical intrinsic cancer risk (tLIR)

Wu et al . 11 provided the following recursion formula to compute the chance that a single stem cell acquires k mutation hits after g divisions given a mutation rate r .

given the initial cell state at generation 0:

Here X g is accumulated driver mutations at generation g , i and j represents accumulated driver mutations at generation g and g + 1, respectively. A fully developed tissue with S stem cells must go through n = log 2 S +  d  ⋅  a rounds of divisions, assuming division rate is d and age a . With this transition probability (3), the theoretical lifetime intrinsic cancer risk (tLIR) is formulated as

Although the recursion formula being dependent on more than one parameter cannot directly be solved in closed form by standard algorithmic techniques, it has nevertheless a simple closed form solution, which was derived by hand computations and verified by standard symbolic computations (using the computer algebra system Maple 2015.2):

The formula for the age-parameterized theoretical lifetime intrinsic cancer risk (tLIR) hence has the following simple closed form solution, which allows much faster and hence more extensive computations and extends the range of admissible values of k from the positive integers to the positive real numbers:

Notice that our result basically coincides with the one obtained by Calabrese and Shibata 35 that was obtained by a direct probabilistic reasoning.

Relating the tLIR model to a scaled Weibull function

We found a connection between

and the scaled Weibull function

where P is the cell population at risk.

To see this connection, we assume that r  ≪ 1 . Then

and using the Taylor series for log and exp, we obtain

Comparing ( 7 ) and ( 9 ) we observe that these relations coincide if

Since for small f 0 > 0 we have 1 − f 0 = exp(− f 0 ), the last equation can rewritten as

Using that relation and ( 8 ) one finds

So we have obtained a shifted Weibull distribution. However, if we remove d −1 log 2 S from the left hand side of the last equality assuming that

we obtain an unshifted one. This condition admits a transparent interpretation, namely, the number of stem cell divisions (for a fixed cell) should be more than the logarithm of stem cell number. Then we have that the tLIR incidence approximately equals to the scaled Weibull incidence if the parameters satisfy

Notice that using a Poisson approximation [ 12 , p. 104] we finally obtain

Stem cell data

Tomasetti and Vogelstein 8 collected stem cell information for 31 cancer types, including stem cell division rate, stem cell number, tissue total cell number. We excluded 6 from 31 cancer types due to lack of age incidence data: colorectal adenocarcinoma in familial adenomatous polyposis (FAP) patients, colorectal adenocarcinoma in patients with hereditary non-polyposis colorectal cancer (HNPCC, also called lynch syndrome), duodenal adenocarcinoma in FAP patients, head and neck squamous cell carcinoma with human papillomavirus (HPV), hepatocellular carcinoma with hepatitis C virus infection (HCV), lung adenocarcinoma in smokers. Among the 25 remaining cancer types, stem cell information were obtained from supplementary materials of Tomasetti and Vogelstein 8 . We discuss life time stem cell division (LSCD) hypothesis and extrinsic risk factor hypothesis for 25 remained cancers: AML, acute myeloid leukemia; BCC, basal cell carcinoma; CLL, chronic lymphocytic leukemia; COAD, colorectal adenocarcinoma; DUAD, duodenum adenocarcinoma; ESCA, esophageal squamous cell carcinoma; GBNPAD, gallbladder non papillary adenocarcinoma; GBM, glioblastoma; HNSC, head and neck squamous cell carcinoma; LHCA, hepatocellular carcinoma; LUAD, lung adenocarcinoma; MBM, medulloblastoma; SKCM, melanoma; OSARC, osteosarcoma; OSARCA, osteosarcoma of the arms; OSARCH, osteosarcoma of the head; OSARCL, osteosarcoma of the legs; OSARCP, osteosarcoma of the pelvis; OVGC, ovarian germ cell; PDAD, pancreatic ductal adenocarcinoma; PECA, pancreatic endocrine (islet cell) carcinoma; SIAD, small intestine adenocarcinoma; TGCC, testicular germ cell cancer; TPFC, thyroid papillary or follicular carcinoma; TMCA, thyroid medullary carcinoma.

Cancer incidence data

SEER-9 registries (1973–2013), SEER-4 registries (1992–2013), SEER-5 registries (2000–2013) data were downloaded from Surveillance, Epidemiology, and End Results Program (SEER) database 27 . SEER database covers about 28% USA population, involving more than 100 features such as race, sex, period, location, histology and ICD (international classification of disease) code. These data were stored in ASCII file, we used the SEERaBomb R package to parse them into sqlite file facilitating data manipulation.

Cancer names provided by Tomasetti and Vogelstein 8 can not be directly mapped into those in SEER database. We addressed this difficulty by two steps: first, annotate tumor primary site to (international classification of disease-oncology 3) ICD-O-3 code based on the literal sense of site in Tomasetti and Vogelstein 8 ; second, annotate histology to ICD-O-3 code based on the literal sense of cancer histology by Tomasetti and Vogelstein 8 . For instance, primary site of lung adenocarcinoma is lung, corresponding to ICD-O-3 site code: C340, C341, C342, C343, C348, C349; adenocarcinoma of lung cancer corresponds to ICD-O-3 histology code 8140, 8141, 8143, 8147, 8570, 8571, 8572, 8573, 8574, 8575, 8576. The dictionary needed for mapping step (we call it ICD dictionary) can be found in http://seer.cancer.gov/icd-o-3/ . Osteosarcoma definition can be found in ICD dictionary, it is a subtype of malignant bone neoplasm, corresponding ICD-O-3 histology code: 9180–9189. However, the ICD dictionary does not differentiate between osteosarcoma detected in the head, leg, or arm. The ICD9Data database ( http://www.icd9data.com/ ) defines bone cancer using ICD9 code 1700–1709, bone cancer in head, arms, legs, pelvis using ICD9 code 1700, 1704–1705, 1707–1708, 1706 respectively. Head and neck squamous cell carcinoma involves tumors located in many sites, ICD dictionary fails to provide its definition. Liao et al . 36 provided ICD9 site code: 1400–1419, 1430–1499, 1600–1619, we then used ICD-O-3 histology code: 8070–8076, 8078 to select squamous cell carcinoma. More detailed cancer definitions using ICD code can be found in Table  3 . Two hematopoietic cancers: acute myeloid leukemia and chronic lymphocytic leukemia, are defined using site recode ICD-O-3/WHO 2008 definition ( http://seer.cancer.gov/siterecode/icdo3_dwhoheme/index.html ).

Manually curated cancer definitions.

1 Either ICD-O-3 site code or ICD9 code describing tumor primary site is provided.

2ICD-O-3 histology code.

Although we carefully annotated 25 cancer definitions using ICD code, we can not avoid misclassifications. because annotation needs several data sources of which information confidential levels differ from each other. The Cancer Genome Atlas (TCGA) program 37 is a flag project of cancer research hosted by National Institutes of Health, it provides comprehensive, high-quality molecular and clinical data. Cancer definitions are well annotated using ICD code in TCGA clinic documents. We therefore assume TCGA cancer definitions are precise and extracted definitions of 18 solid tumors (Table  4 ). With 18 cancer definitions, we selected patients who were diagnosed with cancer after 2000 from SEER-9 registries, SEER-4 registries, SEER-5 registries data to form SEER-18 registries data. As the highest time resolution of SEER data is 1 year, for each year, we took middle age for fitting models, for example, 0 year-old is modified as 0.5 years-old.

TCGA cancer definitions for 18 cancer types.

*KICH, kidney chromophobe; KIRC, kidney renal clear cell carcinoma; KIRP, kidney renal papillary cell carcinoma.

For robustness analysis of parameter estimates we classified each cancer into subgroups based on location, period and race, data of subgroups were separately fitted to the mathematical models.

Fitting the models to empirical cancer incidence data

As was done in previous work 38 , empirical cancer incidence I ( a ) was calculated by

where p i is frequency of people diagnosed with caner at age i .

We performed grid search on an extensive parameter space to fit the tLIR model using R 2 = ( ∑ ( x i − x ¯ ) ( y i − y ¯ ) ∑ ( x i − x ¯ ) 2 ∑ ( y i − y ¯ ) 2 ) 2 as the metrics for goodness of fit, where x i and y i is empirical and model-derived cancer incidence respectively, x ¯ and y ¯ respectively denotes mean value of x and y . Results of fits are given in Table  1 showing that there are biologically reasonable parameter combinations that can yield good fits of the tLIR model for most cancer types.

Data availability

Electronic supplementary material

Author Contributions

X.Z., H.F. and A.W. conceived the experiments. All authors developed the models. X.Z. programmed the analysis software and performed the experiments. D.G., S.V. and J.Z. performed the mathematical analysis of the models. All authors analysed the results and reviewed the manuscript.

Competing Interests

The authors declare no competing interests.

Supplementary information accompanies this paper at 10.1038/s41598-018-21734-x.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Underfilled blood collection tubes as pathologizing factor for measured laboratory parameters in older patients.

Author(s): Rosada A, Prpic M, Spieß E, Friedrich K, Neuwinger N, Müller-Werdan U, Kappert K

Publication: J Am Geriatr Soc , 2024, Vol. , Page

PubMed ID: 38314861 PubMed Review Paper? No

Purpose of Paper

This paper evaluated the impact of partially filled tubes on the levels of 14 clinical chemistry analytes in plasma from geriatric patients. Effects of partial filling were further analyzed in groups stratified by patient gender and age (≤80 years and >80 years).

Conclusion of Paper

Relative to matched tubes that were completely filled during blood collection, plasma from collection tubes that were half-filled had significantly higher levels of lactate dehydrogenase (LDH, 51%), glutamic oxaloacetic transaminase (GOT, 19.4%), potassium (12%) and hemolysis index values (408%). The increased levels of affected analytes was consistent, with a increase in underfilled tubes observed in approximately 90% of matched pairs and in all subgroups when the data was stratified by patient gender or age (≤80 years and >80 years). Such increases translated to a higher percentage of specimens with a result above the reference limit than observed for match tubes that were filled completely. Small statistically significant differences in levels of sodium, chloride, inorganic phosphate (PO4), C-reactive protein (CRP), and icterus index were also observed, but not further discussed. Levels of creatinine kinase, creatinine, and urea and the estimated glomerular filtration rate were comparable in filled and underfilled tubes.

Study Purpose

This study compared the levels of fourteen clinical chemistry analytes in plasma from matched half-filled and properly filled blood collection tubes using blood collected from geriatric patients. Effects of partial filling were further analyzed in groups stratified by patient gender and age (≤80 years and >80 years). Blood from 156 geriatric patients (>60 years if age) was collected from the antecubital vein into paired lithium heparin tubes during a single venipuncture. While not all diagnoses were specified, 62.9% of the patients were anemic, 8.2% had heart failure, 5% had hyperkalemia, 3.8% had cancer, and 1.3% had kidney failure. From each patient, one tube was properly and completely filled and one tube was half-filled. Plasma was obtained by centrifugation at 3000 g for 10 min at 4°C. Levels of sodium, potassium, chloride, GOT, LDH, creatine kinase, inorganic PO4, urea, and CRP were quantified on a Roche cobas 8000 autoanalyzer. The icterus, hemolysis, and lipemia indices were determined by an autoanalyzer.

Summary of Findings:

Compared to matched completely filled tubes, plasma from tubes that were half-filled during blood collection had significantly higher levels of LDH (51%, P<0.001), GOT (19%, P<0.001), potassium (12%, P<0.001), and hemolysis index values (408%, P<0.001, all). The increases in LDH, GOT, potassium, and hemolysis index that occurred because of underfilling were observed in approximately 90% of the matched pairs examined. When the data was stratified by patient gender or age (≤80 years and >80 years), the increases in the levels of affected analytes in underfilled tubes were observed in all subgroups. Levels that were above reference limit values were observed more often in plasma from underfilled than completely filled tubes for LDH (89% versus 45%), GOT (24% versus 12%), potassium (40% versus 11%), and hemolysis (56% versus 4%).  Small statistically significant differences in the levels of sodium (-1%, P<0.001), chloride (0.3%, P<0.001), inorganic PO4 (-1.9%, P<0.001), CRP (-2.6%, P<0.001) and icterus index (-9.4%, P<0.001) were also observed, but not discussed further. Levels of creatinine kinase, creatinine, and urea and the estimated glomerular filtration rate were comparable in completely filled and underfilled tubes.

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Biospecimens

• Bodily Fluid - Plasma
• Bodily Fluid - Blood

Preservative Types

• None (Fresh)
• Other diagnoses
• Not specified
• Cardiovascular Disease
• Neoplastic - Not specified

Pre-analytical Factors:

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