Bayesian estimation of multilevel item response model with missing data

This study proposed a novel statistical method that modeled the multiple outcomes and missing data process jointly using item response theory. This method follows the "intent-to-treat" principle in clinical trials and accounts for the correlation between outcomes and missing data process. This method may provide a good solution to chronic mental disorder study. ^ The simulation study demonstrated that if the true model is the proposed model with moderate or strong correlation, ignoring the within correlation may lead to overestimate of the treatment effect and result in more type I error than specified level. Even if the within correlation is small, the performance of proposed model is as good as naïve response model. Thus, the proposed model is robust for different correlation settings if the data is generated by the proposed model.^

Regression with autocorrelated data: A study of time trend of global infant mortality rate

The infant mortality rate (IMR) is considered to be one of the most important indices of a country's well-being. Countries around the world and other health organizations like the World Health Organization are dedicating their resources, knowledge and energy to reduce the infant mortality rates. The well-known Millennium Development Goal 4 (MDG 4), whose aim is to archive a two thirds reduction of the under-five mortality rate between 1990 and 2015, is an example of the commitment. ^ In this study our goal is to model the trends of IMR between the 1950s to 2010s for selected countries. We would like to know how the IMR is changing overtime and how it differs across countries. ^ IMR data collected over time forms a time series. The repeated observations of IMR time series are not statistically independent. So in modeling the trend of IMR, it is necessary to account for these correlations. We proposed to use the generalized least squares method in general linear models setting to deal with the variance-covariance structure in our model. In order to estimate the variance-covariance matrix, we referred to the time-series models, especially the autoregressive and moving average models. Furthermore, we will compared results from general linear model with correlation structure to that from ordinary least squares method without taking into account the correlation structure to check how significantly the estimates change.^

Mixture modeling for joint analysis of survival, discrete, and continuous data

Mixture modeling is commonly used to model categorical latent variables that represent subpopulations in which population membership is unknown but can be inferred from the data. In relatively recent years, the potential of finite mixture models has been applied in time-to-event data. However, the commonly used survival mixture model assumes that the effects of the covariates involved in failure times differ across latent classes, but the covariate distribution is homogeneous. The aim of this dissertation is to develop a method to examine time-to-event data in the presence of unobserved heterogeneity under a framework of mixture modeling. A joint model is developed to incorporate the latent survival trajectory along with the observed information for the joint analysis of a time-to-event variable, its discrete and continuous covariates, and a latent class variable. It is assumed that the effects of covariates on survival times and the distribution of covariates vary across different latent classes. The unobservable survival trajectories are identified through estimating the probability that a subject belongs to a particular class based on observed information. We applied this method to a Hodgkin lymphoma study with long-term follow-up and observed four distinct latent classes in terms of long-term survival and distributions of prognostic factors. Our results from simulation studies and from the Hodgkin lymphoma study demonstrated the superiority of our joint model compared with the conventional survival model. This flexible inference method provides more accurate estimation and accommodates unobservable heterogeneity among individuals while taking involved interactions between covariates into consideration.^

Data Accuracy in Medical Record Abstraction

Clinical Research Data Quality Literature Review and Pooled Analysis We present a literature review and secondary analysis of data accuracy in clinical research and related secondary data uses. A total of 93 papers meeting our inclusion criteria were categorized according to the data processing methods. Quantitative data accuracy information was abstracted from the articles and pooled. Our analysis demonstrates that the accuracy associated with data processing methods varies widely, with error rates ranging from 2 errors per 10,000 files to 5019 errors per 10,000 fields. Medical record abstraction was associated with the highest error rates (70–5019 errors per 10,000 fields). Data entered and processed at healthcare facilities had comparable error rates to data processed at central data processing centers. Error rates for data processed with single entry in the presence of on-screen checks were comparable to double entered data. While data processing and cleaning methods may explain a significant amount of the variability in data accuracy, additional factors not resolvable here likely exist. Defining Data Quality for Clinical Research: A Concept Analysis Despite notable previous attempts by experts to define data quality, the concept remains ambiguous and subject to the vagaries of natural language. This current lack of clarity continues to hamper research related to data quality issues. We present a formal concept analysis of data quality, which builds on and synthesizes previously published work. We further posit that discipline-level specificity may be required to achieve the desired definitional clarity. To this end, we combine work from the clinical research domain with findings from the general data quality literature to produce a discipline-specific definition and operationalization for data quality in clinical research. While the results are helpful to clinical research, the methodology of concept analysis may be useful in other fields to clarify data quality attributes and to achieve operational definitions. Medical Record Abstractor’s Perceptions of Factors Impacting the Accuracy of Abstracted Data Medical record abstraction (MRA) is known to be a significant source of data errors in secondary data uses. Factors impacting the accuracy of abstracted data are not reported consistently in the literature. Two Delphi processes were conducted with experienced medical record abstractors to assess abstractor’s perceptions about the factors. The Delphi process identified 9 factors that were not found in the literature, and differed with the literature by 5 factors in the top 25%. The Delphi results refuted seven factors reported in the literature as impacting the quality of abstracted data. The results provide insight into and indicate content validity of a significant number of the factors reported in the literature. Further, the results indicate general consistency between the perceptions of clinical research medical record abstractors and registry and quality improvement abstractors. Distributed Cognition Artifacts on Clinical Research Data Collection Forms Medical record abstraction, a primary mode of data collection in secondary data use, is associated with high error rates. Distributed cognition in medical record abstraction has not been studied as a possible explanation for abstraction errors. We employed the theory of distributed representation and representational analysis to systematically evaluate cognitive demands in medical record abstraction and the extent of external cognitive support employed in a sample of clinical research data collection forms. We show that the cognitive load required for abstraction in 61% of the sampled data elements was high, exceedingly so in 9%. Further, the data collection forms did not support external cognition for the most complex data elements. High working memory demands are a possible explanation for the association of data errors with data elements requiring abstractor interpretation, comparison, mapping or calculation. The representational analysis used here can be used to identify data elements with high cognitive demands.

Pathway Semantics: An Algebraic Data Driven Algorithm to Generate Hypotheses about Molecular Patterns Underlying Disease Progression

The overarching goal of the Pathway Semantics Algorithm (PSA) is to improve the in silico identification of clinically useful hypotheses about molecular patterns in disease progression. By framing biomedical questions within a variety of matrix representations, PSA has the flexibility to analyze combined quantitative and qualitative data over a wide range of stratifications. The resulting hypothetical answers can then move to in vitro and in vivo verification, research assay optimization, clinical validation, and commercialization. Herein PSA is shown to generate novel hypotheses about the significant biological pathways in two disease domains: shock / trauma and hemophilia A, and validated experimentally in the latter. The PSA matrix algebra approach identified differential molecular patterns in biological networks over time and outcome that would not be easily found through direct assays, literature or database searches. In this dissertation, Chapter 1 provides a broad overview of the background and motivation for the study, followed by Chapter 2 with a literature review of relevant computational methods. Chapters 3 and 4 describe PSA for node and edge analysis respectively, and apply the method to disease progression in shock / trauma. Chapter 5 demonstrates the application of PSA to hemophilia A and the validation with experimental results. The work is summarized in Chapter 6, followed by extensive references and an Appendix with additional material.

DEVELOPMENT OF NOVEL METHODS TO MINIMIZE THE IMPACT OF SEQUENCING ERRORS IN THE NEXT-GENERATION SEQUENCING DATA ANALYSIS

Next-generation sequencing (NGS) technology has become a prominent tool in biological and biomedical research. However, NGS data analysis, such as de novo assembly, mapping and variants detection is far from maturity, and the high sequencing error-rate is one of the major problems. . To minimize the impact of sequencing errors, we developed a highly robust and efficient method, MTM, to correct the errors in NGS reads. We demonstrated the effectiveness of MTM on both single-cell data with highly non-uniform coverage and normal data with uniformly high coverage, reflecting that MTM’s performance does not rely on the coverage of the sequencing reads. MTM was also compared with Hammer and Quake, the best methods for correcting non-uniform and uniform data respectively. For non-uniform data, MTM outperformed both Hammer and Quake. For uniform data, MTM showed better performance than Quake and comparable results to Hammer. By making better error correction with MTM, the quality of downstream analysis, such as mapping and SNP detection, was improved. SNP calling is a major application of NGS technologies. However, the existence of sequencing errors complicates this process, especially for the low coverage (

The SanaViz: Human Centered Geovisualization to facilitate visual exploration of Public Health data

These three manuscripts are presented as a PhD dissertation for the study of using GeoVis application to evaluate telehealth programs. The primary reason of this research was to understand how the GeoVis applications can be designed and developed using combined approaches of HC approach and cognitive fit theory and in terms utilized to evaluate telehealth program in Brazil. First manuscript The first manuscript in this dissertation presented a background about the use of GeoVisualization to facilitate visual exploration of public health data. The manuscript covered the existing challenges that were associated with an adoption of existing GeoVis applications. The manuscript combines the principles of Human Centered approach and Cognitive Fit Theory and a framework using a combination of these approaches is developed that lays the foundation of this research. The framework is then utilized to propose the design, development and evaluation of “the SanaViz” to evaluate telehealth data in Brazil, as a proof of concept. Second manuscript The second manuscript is a methods paper that describes the approaches that can be employed to design and develop “the SanaViz” based on the proposed framework. By defining the various elements of the HC approach and CFT, a mixed methods approach is utilized for the card sorting and sketching techniques. A representative sample of 20 study participants currently involved in the telehealth program at the NUTES telehealth center at UFPE, Recife, Brazil was enrolled. The findings of this manuscript helped us understand the needs of the diverse group of telehealth users, the tasks that they perform and helped us determine the essential features that might be necessary to be included in the proposed GeoVis application “the SanaViz”. Third manuscript The third manuscript involved mix- methods approach to compare the effectiveness and usefulness of the HC GeoVis application “the SanaViz” against a conventional GeoVis application “Instant Atlas”. The same group of 20 study participants who had earlier participated during Aim 2 was enrolled and a combination of quantitative and qualitative assessments was done. Effectiveness was gauged by the time that the participants took to complete the tasks using both the GeoVis applications, the ease with which they completed the tasks and the number of attempts that were taken to complete each task. Usefulness was assessed by System Usability Scale (SUS), a validated questionnaire tested in prior studies. In-depth interviews were conducted to gather opinions about both the GeoVis applications. This manuscript helped us in the demonstration of the usefulness and effectiveness of HC GeoVis applications to facilitate visual exploration of telehealth data, as a proof of concept. Together, these three manuscripts represent challenges of combining principles of Human Centered approach, Cognitive Fit Theory to design and develop GeoVis applications as a method to evaluate Telehealth data. To our knowledge, this is the first study to explore the usefulness and effectiveness of GeoVis to facilitate visual exploration of telehealth data. The results of the research enabled us to develop a framework for the design and development of GeoVis applications related to the areas of public health and especially telehealth. The results of our study showed that the varied users were involved with the telehealth program and the tasks that they performed. Further it enabled us to identify the components that might be essential to be included in these GeoVis applications. The results of our research answered the following questions; (a) Telehealth users vary in their level of understanding about GeoVis (b) Interaction features such as zooming, sorting, and linking and multiple views and representation features such as bar chart and choropleth maps were considered the most essential features of the GeoVis applications. (c) Comparing and sorting were two important tasks that the telehealth users would perform for exploratory data analysis. (d) A HC GeoVis prototype application is more effective and useful for exploration of telehealth data than a conventional GeoVis application. Future studies should be done to incorporate the proposed HC GeoVis framework to enable comprehensive assessment of the users and the tasks they perform to identify the features that might be necessary to be a part of the GeoVis applications. The results of this study demonstrate a novel approach to comprehensively and systematically enhance the evaluation of telehealth programs using the proposed GeoVis Framework.

Baseline evaluation of environmental physical activity and nutrition support of Texas Medical Center hospital worksites

Background: Obesity is a major health problem in the United States that has reached epidemic proportions. With most U.S adults spending the majority of their waking hours at work, the influence of the workplace environment on obesity is gaining in importance. Recent research implicates worksites as providing an 'obesogenic' environment as they encourage overeating and reduce the opportunity for physical activity. Objective: The aim of this study is to describe the nutrition and physical activity environment of Texas Medical Center (TMC) hospitals participating in the Shape Up Houston evaluation study to develop a scoring system to quantify the environmental data collected using the Environmental Assessment Tool (EAT) survey and to assess the inter-observer reliability of using the EAT survey. Methods: A survey instrument that was adapted from the Environmental Assessment Tool (EAT) developed by Dejoy DM et al in 2008 to measure the hospital environmental support for nutrition and physical activity was used for this study. The inter-observer reliability of using the EAT survey was measured and total percent agreement scores were computed. Most responses on the EAT survey are dichotomous (Yes and No) and these responses were coded with a '0' for a 'no' response and a '1' for a 'yes' response. A summative scoring system was developed to quantify these responses. Each hospital was given a score for each scale and subscale on the EAT survey in addition to a total score. All analyses were conducted using Stata 11 software. Results: High inter-observer reliability is observed using EAT. The percentage agreement scores ranged from 94.4%–100%. Only 2 of the 5 hospitals had a fitness facility onsite and scores for exercise programs and outdoor facilities available for hospital employees ranged from 0–62% and 0–37.5%, respectively. The healthy eating percentage for hospital cafeterias range from 42%–92% across the different hospitals while the healthy vending scores were 0%–40%. The total TMC 'healthy hospital' score was 49%. Conclusion: The EAT survey is a reliable instrument for measuring the physical activity and nutrition support environment of hospital worksites. The study results showed a large variability among the TMC hospitals in the existing physical activity and nutrition support environment. This study proposes cost effective policy changes that can increase environmental support to healthy eating and active living among TMC hospital employees.^

New methods for quantification and analysis of quantitative real-time polymerase chain reaction data

Quantitative real-time polymerase chain reaction (qPCR) is a sensitive gene quantitation method that has been widely used in the biological and biomedical fields. The currently used methods for PCR data analysis, including the threshold cycle (CT) method, linear and non-linear model fitting methods, all require subtracting background fluorescence. However, the removal of background fluorescence is usually inaccurate, and therefore can distort results. Here, we propose a new method, the taking-difference linear regression method, to overcome this limitation. Briefly, for each two consecutive PCR cycles, we subtracted the fluorescence in the former cycle from that in the later cycle, transforming the n cycle raw data into n-1 cycle data. Then linear regression was applied to the natural logarithm of the transformed data. Finally, amplification efficiencies and the initial DNA molecular numbers were calculated for each PCR run. To evaluate this new method, we compared it in terms of accuracy and precision with the original linear regression method with three background corrections, being the mean of cycles 1-3, the mean of cycles 3-7, and the minimum. Three criteria, including threshold identification, max R2, and max slope, were employed to search for target data points. Considering that PCR data are time series data, we also applied linear mixed models. Collectively, when the threshold identification criterion was applied and when the linear mixed model was adopted, the taking-difference linear regression method was superior as it gave an accurate estimation of initial DNA amount and a reasonable estimation of PCR amplification efficiencies. When the criteria of max R2 and max slope were used, the original linear regression method gave an accurate estimation of initial DNA amount. Overall, the taking-difference linear regression method avoids the error in subtracting an unknown background and thus it is theoretically more accurate and reliable. This method is easy to perform and the taking-difference strategy can be extended to all current methods for qPCR data analysis.^

Application of the general linear model to assess the effect of missing data on bone marrow mononuclear cell therapy with infusion timing and follow-up MRI

With most clinical trials, missing data presents a statistical problem in evaluating a treatment's efficacy. There are many methods commonly used to assess missing data; however, these methods leave room for bias to enter the study. This thesis was a secondary analysis on data taken from TIME, a phase 2 randomized clinical trial conducted to evaluate the safety and effect of the administration timing of bone marrow mononuclear cells (BMMNC) for subjects with acute myocardial infarction (AMI).^ We evaluated the effect of missing data by comparing the variance inflation factor (VIF) of the effect of therapy between all subjects and only subjects with complete data. Through the general linear model, an unbiased solution was made for the VIF of the treatment's efficacy using the weighted least squares method to incorporate missing data. Two groups were identified from the TIME data: 1) all subjects and 2) subjects with complete data (baseline and follow-up measurements). After the general solution was found for the VIF, it was migrated Excel 2010 to evaluate data from TIME. The resulting numerical value from the two groups was compared to assess the effect of missing data.^ The VIF values from the TIME study were considerably less in the group with missing data. By design, we varied the correlation factor in order to evaluate the VIFs of both groups. As the correlation factor increased, the VIF values increased at a faster rate in the group with only complete data. Furthermore, while varying the correlation factor, the number of subjects with missing data was also varied to see how missing data affects the VIF. When subjects with only baseline data was increased, we saw a significant rate increase in VIF values in the group with only complete data while the group with missing data saw a steady and consistent increase in the VIF. The same was seen when we varied the group with follow-up only data. This essentially showed that the VIFs steadily increased when missing data is not ignored. When missing data is ignored as with our comparison group, the VIF values sharply increase as correlation increases.^

Data quality in trauma transfusion studies and the impact of missing data on predicting massive transfusion

Maximizing data quality may be especially difficult in trauma-related clinical research. Strategies are needed to improve data quality and assess the impact of data quality on clinical predictive models. This study had two objectives. The first was to compare missing data between two multi-center trauma transfusion studies: a retrospective study (RS) using medical chart data with minimal data quality review and the PRospective Observational Multi-center Major Trauma Transfusion (PROMMTT) study with standardized quality assurance. The second objective was to assess the impact of missing data on clinical prediction algorithms by evaluating blood transfusion prediction models using PROMMTT data. RS (2005-06) and PROMMTT (2009-10) investigated trauma patients receiving ≥ 1 unit of red blood cells (RBC) from ten Level I trauma centers. Missing data were compared for 33 variables collected in both studies using mixed effects logistic regression (including random intercepts for study site). Massive transfusion (MT) patients received ≥ 10 RBC units within 24h of admission. Correct classification percentages for three MT prediction models were evaluated using complete case analysis and multiple imputation based on the multivariate normal distribution. A sensitivity analysis for missing data was conducted to estimate the upper and lower bounds of correct classification using assumptions about missing data under best and worst case scenarios. Most variables (17/33=52%) had <1% missing data in RS and PROMMTT. Of the remaining variables, 50% demonstrated less missingness in PROMMTT, 25% had less missingness in RS, and 25% were similar between studies. Missing percentages for MT prediction variables in PROMMTT ranged from 2.2% (heart rate) to 45% (respiratory rate). For variables missing >1%, study site was associated with missingness (all p≤0.021). Survival time predicted missingness for 50% of RS and 60% of PROMMTT variables. MT models complete case proportions ranged from 41% to 88%. Complete case analysis and multiple imputation demonstrated similar correct classification results. Sensitivity analysis upper-lower bound ranges for the three MT models were 59-63%, 36-46%, and 46-58%. Prospective collection of ten-fold more variables with data quality assurance reduced overall missing data. Study site and patient survival were associated with missingness, suggesting that data were not missing completely at random, and complete case analysis may lead to biased results. Evaluating clinical prediction model accuracy may be misleading in the presence of missing data, especially with many predictor variables. The proposed sensitivity analysis estimating correct classification under upper (best case scenario)/lower (worst case scenario) bounds may be more informative than multiple imputation, which provided results similar to complete case analysis.^

Analyzing left-truncated right-censored data with uncertain onset time with parametric models

Prevalent sampling is an efficient and focused approach to the study of the natural history of disease. Right-censored time-to-event data observed from prospective prevalent cohort studies are often subject to left-truncated sampling. Left-truncated samples are not randomly selected from the population of interest and have a selection bias. Extensive studies have focused on estimating the unbiased distribution given left-truncated samples. However, in many applications, the exact date of disease onset was not observed. For example, in an HIV infection study, the exact HIV infection time is not observable. However, it is known that the HIV infection date occurred between two observable dates. Meeting these challenges motivated our study. We propose parametric models to estimate the unbiased distribution of left-truncated, right-censored time-to-event data with uncertain onset times. We first consider data from a length-biased sampling, a specific case in left-truncated samplings. Then we extend the proposed method to general left-truncated sampling. With a parametric model, we construct the full likelihood, given a biased sample with unobservable onset of disease. The parameters are estimated through the maximization of the constructed likelihood by adjusting the selection bias and unobservable exact onset. Simulations are conducted to evaluate the finite sample performance of the proposed methods. We apply the proposed method to an HIV infection study, estimating the unbiased survival function and covariance coefficients. ^

A simulation study of the effects of small center enrollment in multi -center clinical trials

Multi-center clinical trials are very common in the development of new drugs and devices. One concern in such trials, is the effect of individual investigational sites enrolling small numbers of patients on the overall result. Can the presence of small centers cause an ineffective treatment to appear effective when treatment-by-center interaction is not statistically significant?^ In this research, simulations are used to study the effect that centers enrolling few patients may have on the analysis of clinical trial data. A multi-center clinical trial with 20 sites is simulated to investigate the effect of a new treatment in comparison to a placebo treatment. Twelve of these 20 investigational sites are considered small, each enrolling less than four patients per treatment group. Three clinical trials are simulated with sample sizes of 100, 170 and 300. The simulated data is generated with various characteristics, one in which treatment should be considered effective and another where treatment is not effective. Qualitative interactions are also produced within the small sites to further investigate the effect of small centers under various conditions.^ Standard analysis of variance methods and the "sometimes-pool" testing procedure are applied to the simulated data. One model investigates treatment and center effect and treatment-by-center interaction. Another model investigates treatment effect alone. These analyses are used to determine the power to detect treatment-by-center interactions, and the probability of type I error.^ We find it is difficult to detect treatment-by-center interactions when only a few investigational sites enrolling a limited number of patients participate in the interaction. However, we find no increased risk of type I error in these situations. In a pooled analysis, when the treatment is not effective, the probability of finding a significant treatment effect in the absence of significant treatment-by-center interaction is well within standard limits of type I error. ^

Decomposition of $\rm R\times C$ contingency table chi -square: Application to binned DNA fragment size data and population structure analysis

The purpose of this research is to develop a new statistical method to determine the minimum set of rows (R) in a R x C contingency table of discrete data that explains the dependence of observations. The statistical power of the method will be empirically determined by computer simulation to judge its efficiency over the presently existing methods. The method will be applied to data on DNA fragment length variation at six VNTR loci in over 72 populations from five major racial groups of human (total sample size is over 15,000 individuals; each sample having at least 50 individuals). DNA fragment lengths grouped in bins will form the basis of studying inter-population DNA variation within the racial groups are significant, will provide a rigorous re-binning procedure for forensic computation of DNA profile frequencies that takes into account intra-racial DNA variation among populations. ^

Prevalence and risk factors of asthma and work -related asthma among U.S. adults: Data from the Third National Health and Nutrition Examination Survey, 1988--1994

This cross-sectional analysis of the data from the Third National Health and Nutrition Examination Survey was conducted to determine the prevalence and determinants of asthma and wheezing among US adults, and to identify the occupations and industries at high risk of developing work-related asthma and work-related wheezing. Separate logistic models were developed for physician-diagnosed asthma (MD asthma), wheezing in the previous 12 months (wheezing), work-related asthma and work-related wheezing. Major risk factors including demographic, socioeconomic, indoor air quality, allergy, and other characteristics were analyzed. The prevalence of lifetime MD asthma was 7.7% and the prevalence of wheezing was 17.2%. Mexican-Americans exhibited the lowest prevalence of MD asthma (4.8%; 95% confidence interval (CI): 4.2, 5.4) when compared to other race-ethnic groups. The prevalence of MD asthma or wheezing did not vary by gender. Multiple logistic regression analysis showed that Mexican-Americans were less likely to develop MD asthma (adjusted odds ratio (ORa) = 0.64, 95%CI: 0.45, 0.90) and wheezing (ORa = 0.55, 95%CI: 0.44, 0.69) when compared to non-Hispanic whites. Low education level, current and past smoking status, pet ownership, lifetime diagnosis of physician-diagnosed hay fever and obesity were all significantly associated with MD asthma and wheezing. No significant effect of indoor air pollutants on asthma and wheezing was observed in this study. The prevalence of work-related asthma was 3.70% (95%CI: 2.88, 4.52) and the prevalence of work-related wheezing was 11.46% (95%CI: 9.87, 13.05). The major occupations identified at risk of developing work-related asthma and wheezing were cleaners; farm and agriculture related occupations; entertainment related occupations; protective service occupations; construction; mechanics and repairers; textile; fabricators and assemblers; other transportation and material moving occupations; freight, stock and material movers; motor vehicle operators; and equipment cleaners. The population attributable risk for work-related asthma and wheeze were 26% and 27% respectively. The major industries identified at risk of work-related asthma and wheeze include entertainment related industry; agriculture, forestry and fishing; construction; electrical machinery; repair services; and lodging places. The population attributable risk for work-related asthma was 36.5% and work-related wheezing was 28.5% for industries. Asthma remains an important public health issue in the US and in the other regions of the world. ^