Statistical Computing In R

A licence is granted for personal study and classroom use. Redistribution in any other form is prohibited. Languages shape the way we think, and determine what we can think about (Benjamin Whorf.). This latest revision has corrected several errors. I plan, in due course, to post a new document that will largely replace this now somewhat dated document, taking more adequate account of recent changes and enhancements to the R system and its associated packages since 2002.

The immunization program must be sustained yearly because new babies are born every year new vaccines are
being introduced, and a high level of population immunity to illnesses must be maintained. OBJECTIVES: To assess
immunization among Children under 6 years in urban and rural for the years 2020-2022. The study was conducted to
statistical and analyse the number of children in a primary care centre, and to evaluate the use of geographic information
system technology as a tool for spatial mapping of the vaccine coverage method and analysis by interpolation in two ways
and statistical analysis in R language programming, as well as the analysis vaccinations table in Excel. Data were gathered
from Health Department Dhi Qar. The governorate is distinguished by its huge size (13738.67 km²). divided into 15
administrative units. The number of health centres' immunization services available is about 95 centres and the number of
children is about 308207. Outcome: the study found that the immunization services were (4 -7 % in 5 health sectors) class
weak, (8-11% in 2 health sectors) class below moderate, and (12-15 % in 1 health sector) class moderate, and (18-22 % in
2 health sector) class strong. The common reasons for non-immunization it was a lack of information and lack of attention
from the parents and Child disease added to that (COVID-19). Identify urban areas with low immunization coverage thanks
to the spatial mapping of immunization coverage. Conclusions: Vaccine coverage in the region needs to be improved
further, which can be accomplished by educating mothers, the exclusion of unvaccinated children from school is a critical
factor in ensuring the success of the program.

R topics documented: plotrix-package....................................... 5 ablineclip.......................................... 6 add.ps............................................ 7 addtable2plot........................................ 9 arctext............................................ 10 axis.break.......................................... 12 axis.mult........................................... 13 barlabels........................................... 14 barNest........................................... 15 barp............................................. 19 battleship.plot........................................ 22 1 2 R topics documented: bin.wind.records...................................... 23

R topics documented: ablineclip.......................................... 4 addtable2plot........................................ 5 arctext............................................ 6 axis.break.......................................... 7 axis.mult........................................... 8 barNest........................................... 10 barNest.svymean...................................... 12 barNest.svyprop....................................... 14 barp............................................. 15 bin.wind.records...................................... 18 boxed.labels......................................... 19

Supplemental material, sj-pdf-2-sgo-10.1177_21582440211040121 for Student Perceptions of Teaching Quality in Five Countries: A Partial Credit Model Approach to Assess Measurement Invariance by Rikkert M. van der Lans, Ridwan Maulana, Michelle Helms-Lorenz, Carmen-María Fernández-García, Seyeoung Chun, Thelma de Jager, Yulia Irnidayanti, Mercedes Inda-Caro, Okhwa Lee, Thys Coetzee, Nurul Fadhilah, Meae Jeon and Peter Moorer in SAGE Open

Machine learning blends concepts and techniques from many different traditional fields, such as mathematics, statistics, and computer science. As such, there are many ways to learn the discipline. Considering its theoretical foundations in mathematics and statistics, newcomers would do well to attain some degree of mastery of the formal specifications of basic machine learning techniques. There are many excellent books that focus on the fundamentals, the classic work being Hastie, Tibshirani, and Friedman's The Elements of Statistical Learning ([HTF09]; full references can be found in the Works Cited). 1 But another important part of the hacker mantra is to learn by doing. Many hackers may be more comfortable thinking of problems in terms of the process by which a solution is attained, rather than the theoretical foundation from which the solution is derived. From this perspective, an alternative approach to teaching machine learning would be to use "cookbook"-style examples. To understand how a recommendation system works, for example, we might provide sample training data and a version of the model, and show how the latter uses the former. There are many useful texts of this kind as well, and Segaran's Programming Collective Intelligence is one recent example [Seg07]. Such a discussion would certainly address the how of a hacker's method of learning, but perhaps less of the why. Along with understanding the mechanics of a method, we may also want to learn why it is used in a certain context or to address a specific problem. 1. The Elements of Statistical Learning can now be downloaded free of charge at http://www-stat.stanford .edu/~tibs/ElemStatLearn/.

Depends mvtnorm, lattice, R (> = 2.4.1) Description This package implements a methodology for the design and analysis of dose-response studies that combines aspects of multiple comparison procedures and modeling approaches (Bretz,Pinheiro and Branson, 2005, Biometrics 61, 738-748). The package provides tools for the analysis of dose finding trials as well as a variety of tools necessary to plan a trial to be conducted with the MCPMod methodology. ***NOTE** * The MCPMod package will not be further developed, all future development of the MCP-Mod methodology will be done in the DoseFinding R-package. ***NOTE***

Declaration of Originality This dissertation presents the results of research conducted at the Royal Botanic Gardens, Sydney, of the Botanic Gardens and Domain Trust and the Faulty of Agriculture and Environment, The University of Sydney. The content is, to the best of my knowledge, original and contains no work formally published or written by another person except where dully acknowledged. This thesis does

To understand how statistical and other types of reasoning are coordinated with actions to reduce uncertainty, we conducted a case study in vocational education that involved statistical hypothesis testing. We analyzed an intern's research project in a hospital laboratory in which reducing uncertainties was crucial to make a valid statistical inference. In his project, the intern, Sam, investigated whether patients' blood could be sent through pneumatic post without influencing the measurement of particular blood components. We asked, in the process of making a statistical inference, how are reasons and actions coordinated to reduce uncertainty? For the analysis, we used the semantic theory of inferentialism, specifically, the concept of webs of reasons and actions-complexes of interconnected reasons for facts and actions; these reasons include premises and conclusions, inferential relations, implications, motives for action, and utility of tools for specific purposes in a particular context. Analysis of interviews with Sam, his supervisor and teacher as well as video data of Sam in the classroom showed that many of Sam's actions aimed to reduce variability, rule out errors, and thus reduce uncertainties so as to arrive at a valid inference. Interestingly, the decisive factor was not the outcome of a t test but of the reference change value, a clinical chemical measure of analytic and biological variability. With insights from this case study, we expect that students can be better supported in connecting statistics with context and in dealing with uncertainty.

HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A simulation study is a useful tool in examining how validly item response theory (IRT) models can be applied in various settings. Typically, a large number of replications are required to obtain the desired precision. However, many standard software packages in IRT, such as MULTILOG and BILOG, are not well suited for a simulation study requiring a large number of replications because they were developed as a stand-alone software package that is best suited for a single run. This article demonstrated how built-in R functions can be used to automate the simulation study using the stand-alone software packages in IRT. For a demonstration purpose, MULTILOG was used in the example codes in the appendices, but the overall framework of a simulation study and the builtin R functions used in this article can be applied for a simulation study using other stand-alone software packages as well.

Models to predict ramp meter queue length from traffic detector data are potentially useful tools in improving traffic operations and safety. Existing research, however, has been based on microscopic simulation or relied on extensive calibration of Kalman filter and related models to produce reliable queue length estimates. This research seeks to develop methodologies for improving and simplifying the calibration process of existing queue length models by applying loop detector data including volume, occupancy, and the metering rate data for ramp meters along I-15 in Utah. A conservation model and several variations of a Kalman filter model generated estimated queues that were compared to observed queue lengths in 60 s bins. A modified Kalman filter model and a new heuristic model derived from cluster analysis—the models that yielded the best results—provided queue length estimates that were generally within approximately eight vehicles of the observed queue length. Using the ramp m...

16. ABSTRACT A major challenge for implementing queue control has been to accurately estimate on-ramp queue lengths, particularly during saturated onramp conditions, when the vehicle queue extends around or beyond the ramp entrance. The main outcome of the research was the development and field test verification of an accurate on-ramp queue length estimation algorithm based on vehicle re-identification, which performs reliably even under severe on-ramp saturated conditions. Four different queue estimation methodologies were studied using wireless magnetic sensors installed on a single lane loop on-ramp. Queue length estimation based on (i) occupancy measurements at ramp entrance, (ii) vehicle counts at on-ramp entrance and exit, (iii) speed measurements at ramp entrance, and (iv) vehicle re-identification were considered. These queue estimation methods were evaluated using available raw and processed sensor data, retrieved from the test site through mobile data communication and dow...

Scatterplot3d is an R package for the visualization of multivariate data in a three dimensional space. R is a "language for data analysis and graphics". In this paper we discuss the features of the package. It is designed by exclusively making use of already existing functions of R and its graphics system and thus shows the extensibility of the R graphics system. Additionally some examples on generated and real world data are provided, as well as the source code and the help page of scatterplot3d.

The study investigated the knowledge of Ghanaian mothers on birth spacing, how this knowledge influenced their attitudes and birth spacing practices and possible implications for maternal health and child nutrition. A total of 200 mothers attending antenatal and/or postnatal clinics at a hospital in Accra participated in the study. A structured questionnaire with both open and close ended questions was used to collect information on socio-demographic characteristics of respondents, knowledge on birth spacing, attitudes towards birth spacing and the practices of birth spacing. The results revealed that a greater proportion of women (98%) had heard about birth spacing, however only a few (10%) had in depth knowledge. The top three birth spacing methods known by most respondents were pills (83%), followed by the male or female condom (72%) and abstinence (56%). The study sample used a variety of birth spacing methods with majority using various short-term contraceptives e.g. the pill, ...

Supplemental material, Beauducel_Kersting_Supplement_R1 for Identification of Facet Models by Means of Factor Rotation: A Simulation Study and Data Analysis of a Test for the Berlin Model of Intelligence Structure by André Beauducel and Martin Kersting in Educational and Psychological Measurement

In Ghana, diarrhoea is one of the leading causes of childhood morbidity and mortality. The objective of this study was to examine maternal and child level factors associated with diarrhoea among children in Ghana. A weighted group of 4,821 mother-child dyads was used for the analysis. The analysis was restricted to mothers with children under 2 years of age. Multivariate logistic regression analysis was performed on the pooled data. The pooled prevalence of diarrhoea among the children was 18 per cent. It was found that children older than five months, rural children, children whose mothers had 24 months or more preceding birth interval, and those houses' floors made of unimproved materials were more likely to have had experienced diarrhoea. While children with mothers who obtained secondary or higher education and those had access to unimproved sources of drinking water reported less occurrence of diarrhoea. Maternal and child factors have some effects on the occurrence of diarrhoea among children less than two years in Ghana. To decrease the prevalence of childhood diarrhoea in the country, more educative programmes are required for mothers whose children are more likely to having frequent episodes of diarrhoea.

Supplemental Material, sj-pdf-1-std-10.1177_0956462420968998 for Quantifying geographical diversity in sexually transmitted infections using population attributable risk: results from HIV prevention trials in South Africa by Handan Wand, Gita Ramjee and Tarylee Reddy in International Journal of STD & AIDS

Smart cities, participatory sensing as well as location data available in communication systems and social networks generates a vast amount of heterogeneous mobility data that can be used for traffic management. This chapter gives an overview of the different data sources and their characteristics and describes a framework for utilizing the various sources efficiently in the context of traffic management. Furthermore, different types of traffic models and algorithms are related to both the different data sources as well as some key functionalities of active traffic management, for example short-term prediction and control.

This paper develops an automatic freeway bottleneck identification and visualization algorithm using a combination of image processing techniques and traffic flow theory. Unlike previous studies that are based solely on loop detector data, the proposed method can use traffic measurements from various sensing technologies. Four steps are included in the proposed algorithm. First, the raw spatiotemporal speed data are transformed into binary matrices using image binarization techniques. Second, two post-processer filters are developed to clean the binary matrices by filtering scattered noise cells and localized congested regions. Subsequently, the roadway geometry information is used to remove the impact of acceleration zones downstream of bottlenecks and thus locate bottlenecks more precisely. Finally, the major characteristics of bottlenecks including activation and deactivation points, shockwave speeds and traffic delay caused by bottleneck are automatically extracted and visualize...

L'analyse et la mesure de la qualite de vie peuvent se faire via deux approches differentes et complementaires. La premiere est tournee vers l'analyse des niveaux de satisfaction de la vie a l'aide d'enquetes aupres des individus. La seconde, a laquelle nous nous interessons ici, vise a analyser les conditions de vie des personnes et s'appuie sur des donnees nationales. L'enjeu de cette approche consiste a creer des indices composites des conditions de vie. Dans cet objectif, les methodes de reduction de dimension sont particulierement adaptees car elles permettent de construire de nouvelles variables qui resument "au mieux" l'information contenue dans les variables initiales. Selon Noll [1], les composantes de la qualite de vie sont reliees a differents themes (groupes de variables) dont les plus souvent cites sont "Les conditions familiales", "Les conditions economiques", "Les conditions de logement", "L'...

This paper discusses the methodologies and datasets used in the cartographic study of morphological change in a 19th century residential scheme in Mulhouse, France. It examines (1) the availability and comparability of existing datasets (spatial, historical and archival), (2) the collection of new data from field surveys to historical maps, building permits, photographs, and planning documents by means of visual mapping, drawing, text review, and recording, and (3) their pre-processing including their transformation, cleansing, and integration using specialised tools, such as GIS, Rhino, Grasshopper, and RStudio.
The final outcome is a detailed and georeferenced three-dimensional model for the entire housing scheme of Cité Ouvrière that includes all 1,253 buildings (in volumes) and plots (in polygons) in their original (circa 1897) and current state. The produced model combines the accuracy of smaller-size data with the extent of bigger-size data. Its extent, level of aggregation and detail make it possible to use for cartographic research and spatial analysis at different scales and link it to available geographic information employed for municipal and national planning. Specifically, the model has already been proven valuable in revealing patterns of change and measuring the impact of the building densification process at the microlevel (Kostourou, 2021). This is because it can support a granular, longitudinal historical study that spans over 150 years and cover different spatial scales (from the neighbourhood to the building) while serving as a tool to systematically document transformations in the built form across scales and illustrating how local individual processes contribute to the formation and transformation of the larger whole.
The paper introduces an integrated framework for mapping physical changes of buildings and plots, by joining historical research, spatial analytics, and architectural modelling. Proposed as a solution to the inadequacy of current frameworks and datasets, the new methodology is able to collect and combine available information from online sources, archival documents, and field work; create and process high resolution and aggregate data; overcome limitations of current cartographic studies and open up possibilities for researchers to exploit two-dimensional and three-dimensional, categorical and numerical data for the analysis of spatial structures.

Description The Function performs a parallel analysis using simulated polychoric correlation matri-ces. The nth-percentile of the eigenvalues distribution obtained from both the randomly gener-ated and the real data polychoric correlation matrices is returned. A plot compar-ing the two types of eigenvalues (real and simulated) will help determine the number of real eigen-values that outperform random data. The function is based on the idea that if real data are non-normal and the polychoric correlation matrix is needed to perform a Factor Analysis, then the Par-allel Analysis method used to choose a non-random number of factors should also be based on ran-domly generated polychoric correlation matrices and not on Pearson correlation matrices. Ran-dom data sets are simulated assuming or a uniform or a multinomial distribution or via the boot-strap method of resampling (i.e., random permutations of cases). Also Multigroup Parallel analy-sis is made available for random (uniform and mu...

In this study we aim to examine the latent profiles of teacher candidates’ 21st century learner skills. In other words, we intend to discover the profiles of participant groups who hang together in terms of 21st century learner skill scores. The data are gathered from students who enrolled in either undergraduate programs in the school of education or pedagogical formation certificate programs. For data collection purposes, we used the 21 st century learner skills usage inventory. In order to achieve our goal, profile analysis via multidimensional scaling approach is employed. Data analysis resulted in two-dimensional solution indicating two major profile patterns. Students whose observed patterns are similar to the first major profile tend to have lower ability in cognitive processing and coding of information and realization of the products. They tend to have higher abilities in self-management, self-control, designing more flexible learning environment, and adaptation of new tech...

Scatterplot3d is an R package for the visualization of multivariate data in a three dimensional space. R is a "language for data analysis and graphics". In this paper we discuss the features of the package. It is designed by exclusively making use of already existing functions of R and its graphics system and thus shows the extensibility of the R graphics system. Additionally some examples on generated and real world data are provided, as well as the source code and the help page of scatterplot3d.

This paper presents the development of a four-regime speed–flow relationship for highway work zones and the effects of police presence (police) and speed photo enforcement (SPE) on the speed–flow relationship and capacity. The base data were collected when signage typical of that shown in the Manual on Uniform Traffic Control Devices for an Interstate highway work zone with no lane reduction was present. Police and SPE data were collected, respectively, when a police patrol car and an SPE van were added to the typical work zone. From each data set, the step-by-step progression from a three-regime model based on least-squares regression to a four-regime speed–flow curve is presented. The four-regime model comprises four equations: (a) horizontal line for free-flow regime that covered the volume levels up to 900 passenger cars per hour per lane, (b) fourth-degree spline for the upper transition part of the speed–flow curve, (c) another fourth-degree spline for the lower transitions pa...

This vignette presents the R package RationalExp associated to D’Haultfœuille, Gaillac, and Maurel (2018) (DGM hereafter). This package implements a test of the rational expectations hypothesis based on the marginal distributions of realizations and subjective beliefs. This test can be used in cases where realizations and subjective beliefs are observed in two different datasets that cannot be matched, or when they are observed in the same dataset. The test can be implemented with covariates and survey weights. The package also computes the estimator of the minimal deviations from rational expectations than can be rationalized by the data. R and the package RationalExp are open-source software projects and can be freely downloaded from CRAN: http://cran.r-project.org.

Located in Western North Carolina, Linville Gorge Wilderness Area has an extensive fire history that has been characterized by burns of variable size and severity. Beginning in the 1940s, an era of burn suppression policies curtailed the Gorge's established fire regime. Local stakeholders, including The Wilderness Society (TWS) and the U.S. Forest Service, are concerned about the effects of burn suppression on fire dependent species and communities. As such, these groups are interested in following a modern resurgence of anthropogenic wildfires with a prescribed fire program. Many sampling efforts since 1992 have studied permanent vegetation plots spread throughout Linville Gorge to characterize local plant communities and their relationship with five recent fires. With my Master's Project, done for The Wilderness Society, two objectives related to fire in the Gorge are addressed. First, structural and compositional trends in Linville Gorge forests have been identified, and those trends have been overlain with geospatial environmental variables as well as remotely sensed fire severity estimations. Second, wildfires have been evaluated for their meeting of restoration goals. TWS's restoration targets include a reduction in the importance of ericaceous and fire intolerant species, an increase in the importance of fire dependent species, and a lack of invasion from nonnative species following fire events. Multivariate statistical methods have been implemented to analyze the Linville vegetation dataset for structural and compositional trends, and paired t-tests have been utilized to evaluate changes in target species' importance with fire. Fire has emerged as a major driver of change and compositional heterogeneity in the Wilderness Area. However, burns have produced variable success in meeting restoration goals. While ericaceous species have been reduced in importance, fire dependent species also have experienced declines. Fire intolerant species have increased in abundance with fire; similar increases have not been observed on unburned plots. Invasive species may be a concern, particularly in twice burned forests. If prescribed fire is pursued as a restoration tool in Linville Gorge Wilderness Area, managers should be careful in planning the frequency and severity of fires, and continue to monitor results for the achievement of goals.

Backfitted values of the independent variable of Richards curves are computed and plotted. References Richards, F.J.: A flexible growth function for empirical use. J. Exp. Bot. 1959; 10:290-300. Findlay JW, Dillard RF.: Appropriate calibration curve fitting in ligand binding assays. AAPS J. 2007 Jun 29;9(2). Finney DJ.: Statistical methods in biological assay, 3rd ed. London: Griffin; 1978. Finney DJ.: Response curves for radioimmunoassay. Clinical Chemistry 1983; 29:1562-1566. Schofield T.: Assay Validation. In: Encyclopedia of Biopharmaceutical Statistics. New York: Marcel Dekker 2000; p. 21-30. http://en.wikipedia.org/wiki/ELISA

At a high level, the tidyverse is a language for solving data science challenges with R code. Its primary goal is to facilitate a conversation between a human and a computer about data. Less abstractly, the tidyverse is a collection of R packages that share a high-level design philosophy and low-level grammar and data structures, so that learning one package makes it easier to learn the next.

Background Under-five mortality in Chad reached a minimum value of 119 deaths per 1000 live births in 2018, compared with a maximum of 250 in 1972. Despite this decline in the  mortality trend, for every six children in Chad, one dies before the age of five. This study, therefore, investigated the proximate, intermediate, and distal determinants of under-five mortality in Chad. Methods We used data from the 2014–15 Chad's Demographic and Health Survey. Data of 7782 children below 5 years were used for the study. Both descriptive and multivariable hierarchical logistic regression analyses were performed. Statistical significance was declared at p 

R est un logiciel d'analyse statistique qui fournit toutes les procédures usuelles (t-tests, anova, tests non paramétriques...) et possède des possibilités graphiques performantes pour visualiser les données. Pouvant être utilisé aussi bien en mode interactif qu'en mode « batch », R est disponible pour Linux (et autres unixes), Windows et Macintosh. De plus R est un logiciel libre, dont le code source est disponible et qui peut être recopié et diffusé gratuitement. Le but de cette note est de permettre au lecteur de découvrir les possibilités du programme en matière d'analyse de variance et de faciliter sa prise en main par de nouveaux utilisateurs. Pour cela, nous détaillons quelques exemples d'analyses de données structurées selon des plans classiques. * www.pallier.org. INSERM U562, SHFJ CEA, Orsay. Ce document n'est pas dans un état définitif. Tout commentaire pour l'améliorer ou me signaler des erreurs est bienvenu.

R est un logiciel d'analyse statistique qui fournit toutes les procédures usuelles (t-tests, anova, tests non paramétriques...) et possède des possibilités graphiques performantes pour visualiser les données. Pouvant être utilisé aussi bien en mode interactif qu'en mode « batch », R est disponible pour Linux (et autres unixes), Windows et Macintosh. De plus R est un logiciel libre, dont le code source est disponible et qui peut être recopié et diffusé gratuitement. Le but de cette note est de permettre au lecteur de découvrir les possibilités du programme en matière d'analyse de variance et de faciliter sa prise en main par de nouveaux utilisateurs. Pour cela, nous détaillons quelques exemples d'analyses de données structurées selon des plans classiques. * www.pallier.org. INSERM U562, SHFJ CEA, Orsay. Ce document n'est pas dans un état définitif. Tout commentaire pour l'améliorer ou me signaler des erreurs est bienvenu.

Cite this vignette as
H. Ishwaran, M. Lu, and U. B. Kogalur. 2021. “randomForestSRC: hybrid parallel processing vignette.” http://randomforestsrc.org/articles/hybrid.html.

@misc{HemantHybrid,
    author = "Hemant Ishwaran and Min Lu and Udaya B. Kogalur",
    title = {{randomForestSRC}: hybrid parallel processing vignette},
    year = {2021},
    url = {http://randomforestsrc.org/articles/hybrid.html}
}

Provides readers with a comprehensive and thorough guide to the evidential approach in statistics. The approach uses likelihood ratios, rather than the probabilities used by other statistical inference approaches. The evidential approach is conceptually easier to grasp, and the calculations more straightforward to perform. This book explains how to express data in terms of the strength of statistical evidence for competing hypotheses. The evidential approach is currently underused, despite its mathematical precision and statistical validity. Evidence-Based Statistics is an accessible and practical text filled with examples, illustrations and exercises. Additionally, the companion website complements and expands on the information contained in the book. While the evidential approach is unlikely to replace probability-based methods of statistical inference, it provides a useful addition to any statistician’s “bag of tricks.” In this book: It explains how to calculate statistical evidence for commonly used analyses, in a step-by-step fashion. Analyses include: t tests, ANOVA (one-way, factorial, between- and within-participants, mixed), categorical analyses (binomial, Poisson, McNemar, rate ratio, odds ratio, data that’s ‘too good to be true’, multi-way tables), correlation, regression and nonparametric analyses (one sample, related samples, independent samples, multiple independent samples, permutation and bootstraps). Equations are given for all analyses, and R statistical code provided for many of the analyses. Sample size calculations for evidential probabilities of misleading and weak evidence are explained. Useful techniques, like Matthews’s critical prior interval, Goodman’s Bayes factor, and Armitage’s stopping rule are described. Recommended for undergraduate and graduate students in any field that relies heavily on statistical analysis, as well as active researchers and professionals in those fields, belongs on the bookshelf of anyone who wants to amplify and empower their approach to statistical analysis.