Piecewise exponential models to assess the influence of jobspecific experience on the hazard of acute injury for hourly factory workers
 Jessica Kubo^{1}Email author,
 Mark R Cullen^{2},
 Linda Cantley^{3},
 Martin Slade^{3},
 Baylah TessierSherman^{3},
 Oyebode Taiwo^{3} and
 Manisha Desai^{1}
DOI: 10.1186/147122881389
© Kubo et al.; licensee BioMed Central Ltd. 2013
Received: 21 December 2012
Accepted: 7 July 2013
Published: 10 July 2013
Abstract
Background
An inverse relationship between experience and risk of injury has been observed in many occupations. Due to statistical challenges, however, it has been difficult to characterize the role of experience on the hazard of injury. In particular, because the time observed up to injury is equivalent to the amount of experience accumulated, the baseline hazard of injury becomes the main parameter of interest, excluding Cox proportional hazards models as applicable methods for consideration.
Methods
Using a data set of 81,301 hourly production workers of a global aluminum company at 207 US facilities, we compared competing parametric models for the baseline hazard to assess whether experience affected the hazard of injury at hire and after later job changes. Specific models considered included the exponential, Weibull, and two (a hypothesisdriven and a datadriven) twopiece exponential models to formally test the null hypothesis that experience does not impact the hazard of injury.
Results
We highlighted the advantages of our comparative approach and the interpretability of our selected model: a twopiece exponential model that allowed the baseline hazard of injury to change with experience. Our findings suggested a 30% increase in the hazard in the first year after job initiation and/or change.
Conclusions
Piecewise exponential models may be particularly useful in modeling risk of injury as a function of experience and have the additional benefit of interpretability over other similarly flexible models.
Keywords
Piecewise exponential models Weibull models Baseline hazard Time to event data Occupational health Censored data Frailty models Survival analysisBackground
There is abundant evidence that traumatic injuries occur more commonly in newly hired workers, suggesting that inexperience may be a risk factor for occupational injury. The relationship between experience and risk of injury has been evaluated in several occupations including aluminum manufacturing [1, 2], farming [3], nursing [4, 5], steel manufacturing [6], railway work [7] and through analysis of larger injury databases [8, 9]. There are several methodological challenges, however, that can greatly influence interpretation of results examining this association. A common approach to addressing this question has been to compare rates of injury between newly hired employees and longerterm employees by making use of crosssectional case–control studies [3–5, 7, 9–13]. The potential for confounding, however, is great. New hires are unlikely to be representative of workers in general. In addition to being younger, various “negative” characteristics, such as personal instability, poor work attendance, lesser health, and poor work ethic among others are likely to be overrepresented in this pool. Moreover, “first jobs” are also not representative of all jobs; it is commonplace for newer hires to assume the least desirable of the tasks, and “graduate” into safer and more palatable jobs over time.
Importantly, a crosssectional assessment does not provide the newly hired employee an opportunity to be observed as an experienced employee or vice versa, allowing each employee to serve as his/her own control. Longitudinal studies are equipped for this purpose and many have made use of such designs [6, 8, 9, 14–16]. These studies, however, did not exploit the longitudinal nature of the data, but rather dealt with it as a nuisance [6], if at all. An additional challenge occurs when employees are followed for varying lengths of time, which affects the probability of observing injury during the period studied. Finally, none of the studies have addressed the complication encountered when employees change jobs within the same industry, often with exposure to new and different tasks.
The overall goal of this study is to assess whether experience affects the risk of acute injury in a population of hourly factory workers of a global aluminum company by paying close attention to the statistical issues raised above. For example, the length of time observed until injury needs to be considered as well as whether the employee was injured while observed. As individuals may or may not be injured during their period of study, employees may be censored, bringing to mind survival analytic tools to handle such data. A complication, however, is that the length of time to injury also defines the level of experience, making standard semiparametric models like Cox proportional hazards (Cox PH) models inapplicable. To that end, we argue that the parameter of interest is the baseline hazard of injury itself. In this paper, we demonstrate the use of comparing competing parametric survival models for the baseline hazard when it is the main parameter of interest. In particular we illustrate the use of piecewise exponential models that enable changes in the baseline hazard in order to evaluate the influence of experience on the hazard of injury [17]. While we are not the first to utilize such models [18, 19], this is a novel application of these methods.
Methods
Study population
Our study population included 81,301 hourly production workers of a global aluminum company at 207 US facilities (90 with >250 employees) employed during the time period spanning January 1, 1996 through December 31, 2007. Data were obtained from a variety of administrative sources including a real time incident management system capturing every injury since 1989, and the electronic human resources data. These datasets are described in greater detail in previous publications [1, 2, 21]. Notably, there is no reliance on any selfreport or survey data.
Definition of acute injury
Acute injuries included events such as chemical/thermal burns, bruises, cuts/abrasions, fractures, instantaneous musculoskeletal injuries, and other acute incidents. Chronic conditions or events that were cumulative in nature, classified as “nearmisses” or not directly related to work did not meet the criteria for acute injury.
Statistical methods
Primary outcome
The primary outcome was time in months from the start of a particular job to injury on that job, where multiple injuries may occur. Noninjured employees were censored at the time corresponding to the earliest occurrence of the following: change of job within the corporation, death, termination from the company, or the end of the observation period, December 31, 2007. Individuals with multiple jobs (ie, who changed jobs during the observation period) and/or multiple injuries on a particular job had multiple outcomes reflecting the length of time from the start of a particular job to observed injury on that job. Once a job change was initiated, the employee was newly considered inexperienced, where a job change involved a significant change in job title. In sensitivity analyses, we considered restricting injuries to those of greater severity.
Potential confounders
We considered the following important confounders: age and overall company tenure at job initiation, gender, race, physical demand of the job, union status of the plant, plant type (e.g., whether it included a smelting facility), and sociodemographic characteristics.
Comparison of competing models for investigating research question
To evaluate the association between the hazard of injury and experience, we modeled the hazard of injury as a function of time on the job (i.e., experience) and other relevant confounders using the AndersenGill approach for incorporating recurrent events and timevarying covariates [22, 23]. Cox PH models were not applicable for our purposes, as our main parameter of interest was the baseline hazard itself; assessing whether the hazard for injury changes with experience was equivalent to addressing whether the baseline hazard of injury, λ _{ 0 } (t), was constant over the time on the job (i.e., whether λ _{ 0 } (t)=λ _{ 0 }, the form of an exponential baseline hazard).
Our approach to addressing the research question, therefore, was to compare competing baseline hazards. As described above, a parametric model that assumes a constant baseline hazard is the exponential model, where λ _{ i } (t,x _{ i } )=λ _{ 0 }exp(x’ _{ i } β). We assumed this model under the null hypothesis (i.e., that experience does not affect the hazard of injury).
Three models are considered under the alternative hypothesis: the Weibull and two piecewise exponential models. These models were chosen for their flexibility and convenience, as the null model is nested within each alternative. Specifically, a Weibull distribution has a twoparameter baseline hazard that allows the baseline hazard to change over time, and reduces to the exponential distribution when the scale parameter is equal to one. A KaplanMeier loglog survival curve plot was utilized to gauge appropriateness of the Weibull as a baseline hazard. Kpiece exponential models allow the hazard to change at K distinct time points, but constrain the hazard to be constant within each interval.
The two twopiece exponential models were formulated for different purposes and were not to be directly compared. The first was hypothesisdriven; we set the cut point τ _{1} at 12 months to address the specific hypothesis that the hazard changes after one year of experience. The second was datadriven and estimated the optimal time of change (${\widehat{\tau}}_{1}$) under a 2piece exponential model, for example, to address up to what time point would a hypothetical intervention be useful. To estimate ${\widehat{\tau}}_{1}$ we utilized an approach similar to one employed by Tarres et al. [18] where the cut point that yielded the largest likelihood value was chosen over all possible cut points. Bootstrap methods were used to estimate the variability of ${\widehat{\tau}}_{1}$ by resampling with replacement 400 data sets each with 10,000 employees [24].
For computational purposes, piecewise exponential models can be rewritten as Poisson models [25], greatly simplifying estimation. Specifically, the term representing change in baseline hazard in the model can be subsumed into the vector of parameters as a dummy variable that represents change in hazard [25], making these models straightforward to fit.
Correlation of outcomes within employee and among employees within plant
As workers contributed multiple outcomes, we applied frailty models, which included a random employeespecific multiple of the baseline hazard, to adjust for the expected correlation of responses within the same worker. We assumed the frailty terms followed a Gamma distribution. Similarly, employees from the same plant may be more similar in their outcomes than employees across plants. To address this issue we assessed the robustness of the results to a variety of models that 1) considered plant a random effect, 2) considered plant a fixed effect, with inclusion of plantlevel covariates or 3) solely included plantlevel covariates as fixed effects. Results were examined for heterogeneity. If results did not differ, the simplest method (3) was chosen.
Handling of missing data
The physical demand of a particular job was a key confounder of the relationship between injury and experience. This variable was missing for a large proportion of workers; these data were previously measured by the employer for jobs at only 11 of the 207 plants. Availability of physical demand data was related to a number of plantlevel and individuallevel characteristics. For example, those with physical demand data were 90% male, 87% Caucasian, and at the start of a job, are approximately 43 years old and employed by the company for about 10 years. In contrast, the full cohort was 78% male, 70% Caucasian, and at the start of a job, approximately 39 years old and employed by the company for about 3.5 years. In addition a higher proportion of severe injuries are observed among those in jobs with physical demand data, where these employees are more likely to work at a union plant that performs smelting.
Excluding subjects missing at least one variable included in the model (or a completecase (CC) analysis) would exclude 84% of the data set. Further, validity of results from a CC analysis relies on an assumption that the data are missing completely at random, which we have demonstrated is violated. In order to minimize bias in describing how the hazard of injury varies with experience, we employed multiple imputation (MI) techniques, which provide statistically valid results when the data are missing at random (MAR), a less stringent assumption about missingness that allows missingness to be correlated with observed variables only (e.g., gender) [26]. Conditional on relevant covariates (e.g., an indicator for whether the employee worked for a union plant or whether the employee worked at one of the smelting facilities), this is a reasonable assumption here. We employed the fully conditional specification approach for MI, one of the two main approaches for doing MI, described in detail by van Buuren [27]. We used m = 4 imputations, and as suggested by the missing data literature, confirmed the appropriateness of this number by examining stability of estimates.
Model selection
Comparisons across models were performed using crossvalidation techniques. More specifically, the data were randomly partitioned into two pieces where twothirds of the employees are randomly selected to be in the “learning set”, while the remaining onethird serves as the “test set”, on which all models are compared. Such an approach allowed for model development such as estimation of τ _{1} for the optimally chosen twopiece exponential model and fair comparisons on an untouched data set. Final results, however, were presented on the entire data set. Detailed steps for model selection and fitting were provided in Appendix.
Data cleaning was performed using SAS v9.3. Model selection, model fitting and multiple imputation were performed using STATA 11.2, specifically with STATA’s ice() program [28]. All analyses were conducted after receiving approval from the Institutional Administrative Panels on Human Subjects in Medical Research (IRB).
Results
Demographics of hourly production workers at 207 U.S. facilities of a global aluminum company employed between January 1, 1996 and December 31, 2007
Entire cohort  Physical demand cohort  

Cohort demographics  Count  Percent / Mean (SD)Median  Count  Percent / Mean (SD)Median 
Total jobs  191 692  33 427  
Total unique employees  81 301  13 427  
Number of jobs per employee  2.36 (2.30) 2  3.99 (2.88) 3  
Number of injuries per employee  0.39 (0.96) 0  0.98 (1.49) 1  
Gender  
Male  63 233  77.78  12 105  90.15 
Female  18 006  22.15  1 319  9.82 
Unknown/Missing  72  0.09  3  0.00 
Race/Ethnicity  
White  55 709  70.46  11 669  86.92 
Black  12 371  15.65  1242  9.25 
Hispanic  8340  10.55  386  2.88 
Asian  1786  2.26  46  0.34 
American Indian  610  0.77  66  0.49 
Mixed (more than one reported)  247  0.31  16  0.12 
Unknown/Missing  2238  2.75  2  0.01 
Age (years) at start of job  39.37 (11.32) 39.23  42.61 (10.19) 43.24  
Tenure (years) at start of job  8.43 (10.25) 3.62  13.27 (11.96) 9.60 
Characteristics of injuries in hourly production workers at 207 U.S. facilities of a global aluminum company employed between January 1, 1996 and December 31, 2007
Entire cohort  Physical demand cohort  

Injury characteristics  Count  Percent  Count  Percent 
Injuries  31 456  9549  
Case Category  
First Aid  22 748  72.32  6187  64.79 
Medical Treatment  4598  14.62  1861  19.49 
Restricted Work  3585  11.40  1328  13.91 
Lost Work Day  525  1.67  173  1.81 
Injury Description  
Abrasion  2591  8.24  715  7.49 
Burn  4054  12.89  1099  11.51 
Contusion  6983  22.20  2131  22.32 
Foreign body  2270  7.22  625  6.55 
Fracture  940  2.99  334  3.50 
Laceration  5620  17.87  1427  14.94 
Acute musculoskeletal  6283  19.97  2342  24.53 
Other  2696  8.57  875  9.16 
Missing  19  0.06  1  0.01 
Fit statistics and comparison of models by LRT statistic and Bayes factors using the test set
Exponential  Weibull  Datadriven 2piece exponential  Hypothesisdriven 2piece exponential  

Fit statistics  
Number of Parameters  8  9  9  9 
Loglikelihood  −32576.2  −32573.6  −32447.3  −32468.2 
BIC (Smaller is better)  65263.6  65272.2  65019.6  65061.5 
Model comparison  
LRT Statistic  
Bayes Factor  
Exponential  NA  5.21*  257.9***  216.0*** 
1.000  0.996  0.997  
Weibull  5.21*  NA  Not nested  Not nested 
1.000  0.996  0.997  
Datadriven 2piece exponential  257.9***  Not nested  NA  Not nested 
1.004  1.004  1.001  
Hypothesisdriven 2piece exponential  216.0***  Not nested  Not nested  NA 
1.003  1.003  0.999 
Results from CC and MI frailty models for 2piece exponential model with 12 month cut point (hypothesisdriven model)
Complete case analysis (N=33 427 jobs for 13 427 employees)  Multiple imputation (N=191 692 jobs for 81 301 employees)  

HR (95% CI)  Pvalue  HR (95% CI)  Pvalue  
012 Months (inexperienced period)  1.32 (1.26, 1.38)  <.001  1.25 (1.23, 1.28)  <.001 
Male gender  0.62 (0.57, 0.68)  <.001  0.71 (0.68, 0.73)  <.001 
NonWhite  1.08 (1.00, 1.17)  0.061  0.97 (0.93, 1.00)  0.073 
First job at company  0.88 (0.79, 0.98)  0.017  0.90 (0.87, 0.93)  <.001 
Age at start of job  0.98 (0.98, 0.99)  <.001  0.99 (0.98, 0.99)  <.001 
Physical demand  1.25 (1.22, 1.29)  <.001  1.26 (1.24, 1.29)  <.001 
Smelter plant  1.25 (1.18, 1.33)  <.001  1.30 (1.24, 1.29)  <.001 
Union plant  1.15 (1.04, 1.27)  0.008  1.30 (1.23, 1.37)  <.001 
Original plant  1.01 (0.83, 1.22)  0.952  1.58 (1.50, 1.65)  <.001 
Results from CC and MI frailty models for 2piece exponential model with 19 month cut point (datadriven model)
Complete case analysis (N=33 427 jobs for 13 427 employees)  Multiple imputation (N=191 692 jobs for 81 301 employees)  

HR (95% CI)  Pvalue  HR (95% CI)  Pvalue  
019 Months (inexperienced period)  1.41 (1.35, 1.47)  <.001  1.33 (1.29, 1.36)  <.001 
Male gender  0.62 (0.57, 0.68)  <.001  0.71 (0.68, 0.73)  <.001 
NonWhite  1.08 (0.99, 1.17)  0.070  0.97 (0.93, 1.00)  0.067 
First job at company  0.88 (0.79, 0.98)  0.017  0.90 (0.88, 0.93)  <.001 
Age at start of job  0.98 (0.98, 0.99)  <.001  0.99 (0.98, 0.99)  <.001 
Physical demand  1.25 (1.21, 1.29)  <.001  1.26 (1.24, 1.29)  <.001 
Smelter plant  1.24 (1.17, 1.32)  <.001  1.30 (1.24, 1.36)  <.001 
Union plant  1.16 (1.04, 1.28)  0.006  1.30 (1.23, 1.38)  <.001 
Original plant  1.01 (0.83, 1.22)  0.942  1.57 (1.50, 1.65)  <.001 
Our findings are consistent regarding previously identified correlates of injury risk [1, 2]. Previous studies have indicated that females and Caucasians are at an increased risk of injury. As anticipated, a 1unit increase in the scale of physical demand increased risk of injury by 25%. Similarly, those working at a smelter plant had hazards that are 1.2 times as great as those in nonsmelter plants and those at union plants had hazards that are 1.2 times those at nonunion plants. For every 1year increase in age, we observed a decrease in hazard, as previously reported. Importantly, the reduction in hazard of injury by experience persisted even after adjustment for age.
A sensitivity analysis that included calendar year in the model to account for secular trends in injury during the study period did not affect inference regarding experience. A second sensitivity analysis restricting the outcome to injuries reportable to the Occupational Safety and Health Administration (OSHA) supported the conclusion that less than one year of jobspecific experience yielded an increase in the hazard of injury; here, the magnitude of association was increased (HR=1.62 vs. 1.32).
Discussion
This paper illustrates the novel use of a flexible statistical approach for evaluating the association between job experience and hazard of injury. By identifying the baseline hazard as the main parameter of interest, we were able to address the research question by comparing competing models for the baseline hazard. We found the piecewise exponential model to be particularly useful in this context with respect to goodness of fit, interpretability, as well as computational ease. Further our approach is applicable to other contexts where the baseline hazard is the main focus. After formulating our general analytic approach, we still faced a number of methodological challenges. These included correlation of observations and missing data. We addressed these issues by including a frailty term in the model to account for correlated responses among employees and through multiple imputation techniques to handle the missing data. Using our selected model, our findings demonstrated a 30% increase in the hazard of injury prior to accumulating one year of experience and even higher when only more serious, OSHA reportable injuries are included.
Models for consideration
Because the outcome, the length of time to injury, was equivalent to the predictor of interest, experience, this precluded the use of the commonly applied Cox PH model to evaluate how experience affected hazard of injury and directed the focus to the baseline hazard. For this reason, we compared various competing parametric models for the baseline hazard. Relevant comparisons were drawn between models under the alternative (Weibull and twopiece exponential models) and the model under the null (exponential model). Favoring the more flexible model was an indication that the hazard varied by time on the job and thus, that experience mattered. Other models that we could have considered for the baseline hazard include several in the accelerated failure time family such as the loglogistic, lognormal, and generalized gamma models [29]. Other flexible approaches involving splines could have been considered as well [30]. Our approach however was to identify alternative models to the exponential that were both more flexible and that included the exponential model as a special case.
Consideration of two different twopiece exponential models
The two twopiece exponential models were considered for slightly different purposes. The twopiece exponential that allowed the baseline hazard to change at 1 year directly addressed whether the hazard changed significantly after 1 year (an a priori specified time point of interest), suggesting that targeting an intervention prior to this period may reduce risk. The optimallydefined twopiece exponential model was fit to assess where the maximized difference in hazard under a twopiece model occurred. This provided insight into when one could be considered optimally experienced from an injuryhazard perspective, and/or up to what time point a hypothetical intervention could be considered useful. A Kpiece exponential model with K > 2 may have better fit the data, as it would have allowed for even greater flexibility. Our questions were addressed, however, with these 2piece models. Another approach would be to use crossvalidation measures to optimize K and then estimate changes in hazard given the optimal K. This would answer a slightly different question, although still a potentially interesting one.
Comparing competing models
Comparisons between the Weibull and exponential models demonstrated the Weibull model as the better fit, indicating sufficient evidence to reject the null that the hazard is constant over time. Similarly comparisons between the exponential and each twopiece demonstrated the need for such flexibility. These comparisons addressed our first question of whether the hazard was nonconstant over time (i.e., whether there is a relationship between experience and injury risk). We did not formally compare the Weibull to each twopiece model using a LRT as neither model is nested within the other and thus, the behavior of the LRT statistic is not known in this case. Graphical evidence, however, suggests the superior fit of each twopiece model to the Weibull. Additionally, the likelihood values corresponding to the twopiece models were considerably larger than that of the Weibull, while the number of parameters estimated was the same. From an interpretability standpoint, the twopiece models were more appealing as they yielded parameter estimates that directly corresponded to changes in hazard at the specified time point, which may be useful in targeting interventions (while the parameter estimates for a Weibull model describe the shape of the baseline hazard, which is not directly meaningful).
Limitations
There are some limitations to the study that merit mention. First, we have not accounted for the variable number of hours worked each month (hence opportunity for an injury) across individuals. We expect variability across employees because of business cycles, overtime and vacation schedules [4, 20]. Incorporating this variability is challenging because of data limitations. Another potential concern is that we did not adjust for changes in injury risk over secular time during the study period, given that nationally and over the course of this study, injury rates have improved at the company. We have explored the implications of this by refitting our candidate models with the inclusion of timevarying indicators for calendar year. While the relationship between calendar year and the hazard of injury is as expected, after adjusting for secular trends, the influence of experience on the hazard of injury did not change (data not shown).
In addition, physical demand, an established risk factor for injury in this workforce [2], was measured in a nonrepresentative sample of plants inducing missing data in our cohort of employees eligible for study. We utilized multiple imputation techniques to address this issue that relied on assumptions plausible for our data set.
Finally, we acknowledge that this employer and workforce may be unique; aluminum manufacturing is intrinsically more dangerous than average for the manufacturing sector while the studied employer is observably more injuryadverse than the sector average. Generalizing to manufacturing more broadly, or beyond to such domains as construction, mining or service work, while tempting, may be unjustified.
Strengths
Although there were several limitations to our study, our goal was to illustrate the application of flexible parametric methods for studying this association in the presence of several statistical challenges. In addition our approach has application to other contexts where the focus is the baseline hazard [17–19]. As our findings are demonstrated on this large longitudinal population using rich highquality administrative data, our analysis lends confidence that previous concerns about higher injury risk among inexperienced workers are well founded, at least in our setting. Our result is highly consistent with the prevailing literature, but is the first to characterize the reduction in hazard while managing many of the methodological issues previously ignored. Moreover, our findings are not confined to newly hired workers, but extend also to those longer tenured who change jobs to ones demanding new tasks. Previous studies typically equate “new” with newlyhired, whereas we considered employees as newly inexperienced for subsequent job change. As job changes are typically more frequent in the manufacturing workplace than new hires, especially in the present economic climate, we would propose that efforts to control the risks of inexperience be focused not just on new hires but on all workers “new to the job”.
Conclusions
In summary, we have demonstrated that piecewise exponential models offer the flexibility of modeling changes in the hazard of injury with ease in interpretation. They are particularly useful in the context in which the baseline hazard is primarily of interest and where Cox PH models are inapplicable. While the Weibull model demonstrated that experience mattered, it did not provide a simple characterization of the change in hazard. In particular, the reduction at a specific time point was not described, nor was the ideal time point at which change occurred. Both of these features were easily accessible through the two formulated (hypothesisdriven and datadriven) twopiece exponential models. We encourage analysts to consider use of piecewise exponential models, which allow flexibility for changes in the hazard, are easy to fit, and provide meaningful interpretation.
Appendix
Step 1. Randomly partition the employees into the learning (twothirds of subjects) and test sets (remaining onethird of subjects). The Full Cohort contains the same respective employees.
Step 2. Estimate τ _{1} for the twopiece exponential model using the learning set.
Step 3: From the learning set, resample with replacement 400 data sets of 10,000 employees and for each, estimate τ _{1} to obtain an estimated variance for ${\widehat{\tau}}_{1}$.
Step 4. Make relevant comparisons among proposed models on the test set using the likelihood ratio test (LRT) statistic for nested models (e.g., Weibull and exponential) and the Bayesian Information Criterion (BIC) for nonnested models (e.g., Weibull and twopiece exponential models). The twopiece models are not compared as they address different hypotheses.
Step 5. Fit the chosen model(s) with a frailty term on the Full Cohort performing a completecase and MIbased analysis adjusting for prespecified confounders.
Abbreviations
 Cox PH:

Cox proportional hazards
 CC:

Complete case
 MI:

Multiple imputation
 MAR:

Missing at random
 KM:

KaplanMeier
 LRT:

Likelihood ratio test
 OSHA:

Occupational Safety and Health Administration
 BIC:

Bayesian information criterion.
Declarations
Acknowledgments
This work was supported by the National Institute on Aging (R01AG026291) and the Alcoa Foundation.
Authors’ Affiliations
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