DOCSLIB.ORG

Iteratively Reweighted Least Squares for Maximum Likelihood Estimation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Iteratively Reweighted Least Squares for Maximum Likelihood Estimation J.R. Statist.Soc. B (1984). 46,No. 2, pp. 149-192 IterativelyReweighted Least Squaresfor Maximum Likelihood Estimation,and some Robust and ResistantAlternatives By P. J. GREEN UniversityofDurham, UK [Readbefore the Royal Statistical Society at a meetingorganised by the ResearchSection on Wednesday,December 7th,1983, ProfessorJ. B. Copas in the Chair3 SUMMARY The scope of applicationof iterativelyreweighted least squaresto statisticalesti- mation problemsis considerablywider than is generallyappreciated. It extends beyondthe exponential-family-typegeneralized linear models to otherdistributions, to non-linearparameterizations, and to dependentobservations. Various criteria for estimationother than maximumlikelihood, including resistant alternatives, may be used. The algorithmsare generallynumerically stable, easily programmed without the aid of packages,and highlysuited to interactivecomputation. Keywords:NEWTON-RAPHSON; FISHER SCORING;GENERALIZED LINEAR MODELS; QUASI- LIKELIHOOD; ROBUSTREGRESSION; RESISTANTREGRESSION; RESIDUALS 1. PRELIMINARIES 1.1. An IntroductoryExample This paperis concernedwith fitting regression relationships in probabilitymodels. We shallgen- erallyuse likelihood-basedmethods, but willventure far from familiar Normal theory and linear models. As a motivationfor our discussion,let us considerthe familiarexample of logisticregression. We observeYi,Y2, ** ,Ym whichare assumedto be drawnindependently from Binomial distri- butions with known indices nl, n2,. .,n, Covariates{x11, i = 1,2,.... .,m; j= 1,2, . ..,p} are also available and it is postulated that yix' B(ni,{ 1 + exp (- Exil31)} -1), for parameters ,1, 02, . ., ,p whosevalues are to be estimated.The importantingredients of thisexample from thepoint of view of thispaper are: (A) a regressionfunction ql = q (p), whichhere has the formri - { 1 + exp (-1x11t3)}-; and (B) a probabilitymodel, expressed as a log-likelihoodfunction of t , L(q), whichin thiscase is m L = E {y logi7l + (ni-yi) log(1 - )}- 1=1 In commonwith most of the problemswe shallconsider in thispaper notice that, in theusual applicationof this example, (i) q has muchlarger dimension than i (ii) the probabilitymodel (B) is largelyunquestioned, except perhaps for some referenceto goodnessof fit,and (iii) it is theform of theregression function (A) thatis thefocus of ourattention. Typicallywe wouldbe interestedin selectingcovariates of importance,deciding the formof the regressionfunction, and estimatingthe values of the I3s. Presentaddress: Departmentof Mathematical Sciences, The University,Durham DH1 3LE. ? 1984 Royal StatisticalSociety 0035-9246-84-46149 $2.00 This content downloaded from 138.25.2.146 on Mon, 24 Mar 2014 22:24:16 PM All use subject to JSTOR Terms and Conditions 150 GREEN [No. 2, 1.2. GeneralFormulation We considera log-likelihoodL, a functionof an n-vectorq of predictors.TypicaLly n is equal to, or comparablewith, the numberof individualobservations of whichthe likelihoodforms the densityor probabilityfunction, but we shall be concernedalso with cases whereindividual observationsare difficultto define,for example with one or moremultinomial samples. The predictorvector t1 is functionallydependent on thep-vector P of parametersof interest: p is typicallymuch smaller than n. We base ourinference on thefunction q = tj (p) by estimat- ing the parametersp , and derivingapproximate confidence intervals and significancetests. Initiallywe shallconsider only maximum likelihood estimates, and supposethat the modelis sufficientlyregular that we mayrestrict attention to thelikelihood equations aL T= = DTu =0 (1) ap whereu is the n-vector{aL/a h }, and D the n x p matrix{ }/ia i The standardNewton- Raphson method for the iterativesolution of (1) calls for evaluatingu, D and the second derivativesof L foran initialvalue of p and solvingthe linear equations (-a2L DTu (2) tapp ) ( p*p> foran updatedestimate f * Thisprocedure is repeateduntil convergence. Equation (2) is derived fromthe firsttwo termsof a Taylorseries expansion for aLla f fora log-likelihoodquadratic in themethod converges in one step. Commonlythe second derivatives in (2) arereplaced by an approximation.Note that -a2L aL a2 a a T a2L (anl) a pp T anai a pP T ap anTIT ap and we replacethe termson the rightby theirexpectations (at the currentparameter values 11). By thestandard arguments: ( aL //2L aL 1aL\ T E l E E- = IA5 say,and withthis approximation (essentially Fisher's scoring technique) (2) becomes - (DTAD) (p * ?) = DTu. (3) We will assumethat D is of fullrank p, and thatA is positivedefinite throughout the parameter space: thus(3) is a non-singularp x p systemof equationsfor P *. Ratherthan handle their numerical solution directly, note thatthey have the formof normal equationsfor a weightedleast squares regression: P * solves minimize(A -1u+ D(P - p *))T A(A-lu + D( P -p*)), (4) that is, it resultsfrom regressing A1 u + D P onto the columnsof D usingweight matrix A. Thus we use an iterativelyreweighted least squares(IRLS) algorithm(4) to implementthe Newton-Raphsonmethod with Fisherscoring (3), for an iterativesolution to the likelihood equations(1). Thistreatment of the scoringmethod via least squaresgeneralizes some very long- standingmethods, and specialcases are reviewed in thenext Section. Two commonsimplifications are thatthe modelmay be linearlyparameterized, q = X P say, so thatD is constant,or thatL has the formYL1(i1) (e.g. observationsare independent)so that This content downloaded from 138.25.2.146 on Mon, 24 Mar 2014 22:24:16 PM All use subject to JSTOR Terms and Conditions 1984] IterativelyReweighted Least Squares 151 A is diagonal. IRLS algorithmsalso arisein inferencebased on the conceptof quasi-likelihood,which was proposedby Wedderburn(1974) and extendedto the multivariatecase by McCullagh(1983). Suppose that the n-vectorof observationsy has E(y) = l and var(y) = a2 V(ij), where q= (p) as before,a2 is a scalar,and the matrix-valuedfunction V( ) is specified.The log- quasi-likelihoodQ is definedas anyfunction of tj satisfying aQ -QV(tj) (y- n) (5) all whereV- is a generalizedinverse. We estimate P by solvingthe quasi-likelihood equations O=aQ =a aQ DTU a, ap an say. Since E(aQla-/t ) =0 and E(-a2Q/atp T) = V-, the Newton-Raphsonequations with expectedsecond derivatives have the form (3) withA = V -. Questionsof existenceand uniquenessof themaximum likelihood estimates in variouscases are discussedby Wedderburn(1976), Pratt(1981) and Burridge(1981). The large-sampletheory thatjustifies likelihood-ratio tests and confidenceintervals for parameters will be foundin Cox and Hinkley(1974, p. 294-304) and McCullagh(1983). In particular,the asymptoticcovariance matrixfor the estimate of P is cr2[E(-a2L/a ppT)] -1 = a2(DTAD)-' The importantconnection between such theoreticalresults and the numericalproperties of IRLS is thatboth are justifiedby the approximatequadratic behaviour of the log-likelihood near its maximum.Thus it is reasonablethat IRLS shouldwork when maximum likelihood is relevant. 1.3. Historyand SpecialCases Fromthe first,Fisher noted that maximum likelihood estimates would often require iterative calculation.In Fisher(1925), use of Newton'smethod was mentioned, dL / ( d2LN dO / \dO2 / eitherfor one step only,or iteratedto convergence.Implicitly he replacedthe negativesecond derivative,the observedinformation, at each parametervalue by its expectationassuming that valuewere the trueone. Thistechnique, and itsmulti-parameter generalization, became known as "Fisher's method of scoringfor parameters",and was furtherdiscussed by Bailey (1961 Appendix1), Kale (1961, 1962) and Edwards(1972). Use of the scoringmethod in whatwe termregression problems seems to date fromFisher's contributedappendix to Bliss(1935). Thispaper was concernedwith dosage-mortality curves, a quantalresponse problem as in Section 1.1, except forthe use of the probittransformation in place of the logit.The relativemerits of usingobserved or expectedinformation were discussed by Garwood(1941), and the methodhas becomemore generally known from the various editions of Finney'sbook on ProbitAnalysis (1947, 1952,Appendix II). Moore and Zeigler(1967) discussedthese binomialproblems with an arbitraryregression function,and demonstratedthe quite generalconnection with non-linear least-squares regression. Nelder and Wedderburn(1972) introducedthe class of generalizedlinear models to unifya numberof linearlyparameterized problems in exponentialfamily distributions. These models are discussedin Section3.2. The importantconnection between the IRLS algorithmfor maximum likelihood estimation and the Gauss-Newtonmethod for least-squaresfitting of non-linearregressions was further elucidatedby Wedderburn(1974). Jennrichand Moore (1975) consideredmaximum likelihood estimation in a moregeneral This content downloaded from 138.25.2.146 on Mon, 24 Mar 2014 22:24:16 PM All use subject to JSTOR Terms and Conditions 152 GREEN [No. 2, exponentialfamily than did Nelderand Wedderburn;their approach is similarto ours,except thatthe predictors j mustbe theexpected values of the observations. Importantrecent contributions have come fromMcCullagh (1983) and Jorgensen(1983), particularlyregarding the treatment of dependentobservations. 2. PRACTICALITIES 2.1. Multinomialdata Much of our discussionof the detailedproperties of IRLS algorithmsthat followscan be motivatedby exampleswith multinomial data. First consider a single multinomialdistribution with polynomiallyparameterized cell probabilities,such as oftenarises with data on gene frequencies.In
Load more

Recommended publications
  • Curriculum Vitae
    CURRICULUM VITAE Name Ankit Patras Address 111 Agricultural and Biotechnology Building, Department of Agricultural and Environmental Sciences, Tennessee State University, Nashville TN 37209 Phone 615-963-6007, 615-963-6019/6018 Email [email protected], [email protected] EDUCATION 2005- 2009: Ph.D. Biosystems Engineering: School of Biosystems Engineering, College of Engineering & Architecture, Institute of Food and Health, University College Dublin, Ireland. 2005- 2006: Post-graduate certificate (Statistics & Computing): Department of Statistics and Actuarial Science, School of Mathematical Sciences, University College Dublin, Ireland 2003- 2004: Master of Science (Bioprocess Technology): UCD School of Biosystems Engineering, College of Engineering & Architecture, University College Dublin, Ireland 1998- 2002: Bachelor of Technology (Agricultural and Food Engineering): Allahabad Agriculture Institute, India ACADEMIC POSITIONS Assistant Professor, Food Biosciences: Department of Agricultural and Environmental Research, College of Agriculture, Human and Natural Sciences, Tennessee State University, Nashville, Tennessee 2nd Jan, 2014 - Present • Leading a team of scientist and graduate students in developing a world-class food research centre addressing current issues in human health, food safety specially virus, bacterial and mycotoxins contamination • Developing a world-class research program on improving safety of foods and pharmaceuticals • Develop cutting edge technologies (i.e. optical technologies, bioplasma, power Ultrasound,
    [Show full text]
  • Full-Text (PDF)
    Vol. 13(6), pp. 153-162, June 2019 DOI: 10.5897/AJPS2019.1785 Article Number: E69234960993 ISSN 1996-0824 Copyright © 2019 Author(s) retain the copyright of this article African Journal of Plant Science http://www.academicjournals.org/AJPS Full Length Research Paper Adaptability and yield stability of bread wheat (Triticum aestivum) varieties studied using GGE-biplot analysis in the highland environments of South-western Ethiopia Leta Tulu1* and Addishiwot Wondimu2 1National Agricultural Biotechnology Research Centre, P. O. Box 249, Holeta, Ethiopia. 2Department of Plant Sciences, College of Agriculture and Veterinary Science, Ambo University. P. O. Box 19, Ambo, Ethiopia. Received 13 February, 2019; Accepted 11 April, 2019 The objectives of this study were to evaluate released Ethiopian bread wheat varieties for yield stability using the GGE biplot method and identify well adapted and high-yielding genotypes for the highland environments of South-western Ethiopia. Twenty five varieties were evaluated in a randomized complete block design with three replications at Dedo and Gomma during the main cropping season of 2016 and at Dedo, Bedelle, Gomma and Manna during the main cropping season of 2017, generating a total of six environments in location-by-year combinations. Combined analyses of variance for grain yield indicated highly significant (p<0.001) mean squares due to environments, genotypes and genotype-by- environment interaction. Yield data were also analyzed using the GGE (that is, G, genotype + GEI, genotype-by-environment interaction) biplot method. Environment explained 73.2% of the total sum of squares, and genotype and genotype X environment interaction explained 7.16 and 15.8%, correspondingly.
    [Show full text]
  • Multivariate Data Analysis in Sensory and Consumer Science
    MULTIVARIATE DATA ANALYSIS IN SENSORY AND CONSUMER SCIENCE Garmt B. Dijksterhuis, Ph. D. ID-DLO, Institute for Animal Science and Health Food Science Department Lely stad The Netherlands FOOD & NUTRITION PRESS, INC. TRUMBULL, CONNECTICUT 06611 USA MULTIVARIATE DATA ANALYSIS IN SENSORY AND CONSUMER SCIENCE MULTIVARIATE DATA ANALYSIS IN SENSORY AND CONSUMER SCIENCE F NP PUBLICATIONS FOOD SCIENCE AND NUTRITIONIN Books MULTIVARIATE DATA ANALYSIS, G.B. Dijksterhuis NUTRACEUTICALS: DESIGNER FOODS 111, P.A. Lachance DESCRIPTIVE SENSORY ANALYSIS IN PRACTICE, M.C. Gacula, Jr. APPETITE FOR LIFE: AN AUTOBIOGRAPHY, S.A. Goldblith HACCP: MICROBIOLOGICAL SAFETY OF MEAT, J.J. Sheridan er al. OF MICROBES AND MOLECULES: FOOD TECHNOLOGY AT M.I.T., S.A. Goldblith MEAT PRESERVATION, R.G. Cassens S.C. PRESCOlT, PIONEER FOOD TECHNOLOGIST, S.A. Goldblith FOOD CONCEPTS AND PRODUCTS: JUST-IN-TIME DEVELOPMENT, H.R.Moskowitz MICROWAVE FOODS: NEW PRODUCT DEVELOPMENT, R.V. Decareau DESIGN AND ANALYSIS OF SENSORY OPTIMIZATION, M.C. Gacula, Jr. NUTRIENT ADDITIONS TO FOOD, J.C. Bauernfeind and P.A. Lachance NITRITE-CURED MEAT, R.G. Cassens POTENTIAL FOR NUTRITIONAL MODULATION OF AGING, D.K. Ingram ef al. CONTROLLEDlMODIFIED ATMOSPHERENACUUM PACKAGING, A. L. Brody NUTRITIONAL STATUS ASSESSMENT OF THE INDIVIDUAL, G.E. Livingston QUALITY ASSURANCE OF FOODS, J.E. Stauffer SCIENCE OF MEAT & MEAT PRODUCTS, 3RD ED., J.F. Price and B.S. Schweigert HANDBOOK OF FOOD COLORANT PATENTS, F.J. Francis ROLE OF CHEMISTRY IN PROCESSED FOODS, O.R. Fennema et al. NEW DIRECTIONS FOR PRODUCT TESTING OF FOODS, H.R. Moskowitz ENVIRONMENTAL ASPECTS OF CANCER: ROLE OF FOODS, E.L. Wynder et al.
    [Show full text]
  • Relevance of Eating Pattern for Selection of Growing Pigs
    RELEVANCE OF EATING PATTERN FOR SELECTION OF GROWING PIGS CENTRALE LANDB OUW CA TA LO GU S 0000 0489 0824 Promotoren: Dr. ir. E.W. Brascamp Hoogleraar in de Veefokkerij Dr. ir. M.W.A. Verstegen Buitengewoon hoogleraar op het vakgebied van de Veevoeding, in het bijzonder de voeding van de eenmagigen L.C.M, de Haer RELEVANCE OFEATIN G PATTERN FOR SELECTION OFGROWIN G PIGS Proefschrift ter verkrijging van de graad van doctor in de landbouw- en milieuwetenschappen, op gezag van de rector magnificus, dr. H.C. van der Plas, in het openbaar te verdedigen op maandag 13 april 1992 des namiddags te vier uur in de aula van de Landbouwuniversiteit te Wageningen m st/eic/, u<*p' Cover design: L.J.A. de Haer BIBLIOTHEEK! LANDBOUWUNIVERSIIBtt WAGENINGEM De Haer, L.C.M., 1992. Relevance of eating pattern for selection of growing pigs (Belang van het voeropnamepatroon voor de selektie van groeiende varkens). In this thesis investigations were directed at the consequences of testing future breeding pigs in group housing, with individual feed intake recording. Subjects to be addressed were: the effect of housing system on feed intake pattern and performance, relationships between feed intake pattern and performance and genetic aspects of the feed intake pattern. Housing system significantly influenced feed intake pattern, digestibility of feed, growth rate and feed conversion. Through effects on level of activity and digestibility, frequency of eating and daily eating time were negatively related with efficiency of production. Meal size and rate of feed intake were positively related with growth rate and backfat thickness.
    [Show full text]
  • General Linear Models - Part II
    Introduction to General and Generalized Linear Models General Linear Models - part II Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby October 2010 Henrik Madsen Poul Thyregod (IMM-DTU) Chapman & Hall October 2010 1 / 32 Today Test for model reduction Type I/III SSQ Collinearity Inference on individual parameters Confidence intervals Prediction intervals Residual analysis Henrik Madsen Poul Thyregod (IMM-DTU) Chapman & Hall October 2010 2 / 32 Tests for model reduction Assume that a rather comprehensive model (a sufficient model) H1 has been formulated. Initial investigation has demonstrated that at least some of the terms in the model are needed to explain the variation in the response. The next step is to investigate whether the model may be reduced to a simpler model (corresponding to a smaller subspace),. That is we need to test whether all the terms are necessary. Henrik Madsen Poul Thyregod (IMM-DTU) Chapman & Hall October 2010 3 / 32 Successive testing, type I partition Sometimes the practical problem to be solved by itself suggests a chain of hypothesis, one being a sub-hypothesis of the other. In other cases, the statistician will establish the chain using the general rule that more complicated terms (e.g. interactions) should be removed before simpler terms. In the case of a classical GLM, such a chain of hypotheses n corresponds to a sequence of linear parameter-spaces, Ωi ⊂ R , one being a subspace of the other. n R ⊆ ΩM ⊂ ::: ⊂ Ω2 ⊂ Ω1 ⊂ R ; where Hi : µ 2
    [Show full text]
  • Introducing SPM® Infrastructure
    Introducing SPM® Infrastructure © 2019 Minitab, LLC. All Rights Reserved. Minitab®, SPM®, SPM Salford Predictive Modeler®, Salford Predictive Modeler®, Random Forests®, CART®, TreeNet®, MARS®, RuleLearner®, and the Minitab logo are registered trademarks of Minitab, LLC. in the United States and other countries. Additional trademarks of Minitab, LLC. can be found at www.minitab.com. All other marks referenced remain the property of their respective owners. Salford Predictive Modeler® Introducing SPM® Infrastructure Introducing SPM® Infrastructure The SPM® application is structured around major predictive analysis scenarios. In general, the workflow of the application can be described as follows. Bring data for analysis to the application. Research the data, if needed. Configure and build a predictive analytics model. Review the results of the run. Discover the model that captures valuable insight about the data. Score the model. For example, you could simulate future events. Export the model to a format other systems can consume. This could be PMML or executable code in a mainstream or specialized programming language. Document the analysis. The nature of predictive analysis methods you use and the nature of the data itself could dictate particular unique steps in the scenario. Some actions and mechanisms, though, are common. For any analysis you need to bring the data in and get some understanding of it. When reviewing the results of the modeling and preparing documentation, you can make use of special features embedded into charts and grids. While we make sure to offer a display to visualize particular results, there’s always a Summary window that brings you a standard set of results represented the same familiar way throughout the application.
    [Show full text]
  • Measuring the Outcome of Psychiatric Care Using Clinical Case Notes
    MEASURING THE OUTCOME OF PSYCHIATRIC CARE USING CLINICAL CASE NOTES Marco Akerman MD, MSc A report submitted as part of the requirements for the degree of Doctor of Philosophy in the Faculty of Medicine, University of London Department of Epidemiology and Public Health University College London 1993 ProQuest Number: 10016721 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest. ProQuest 10016721 Published by ProQuest LLC(2016). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. Microform Edition © ProQuest LLC. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 ABSTRACT This thesis addresses the question: ‘Can case notes be used to measure the outcome of psychiatric treatment?’ Patient’s case notes from three psychiatric units (two in-patient and one day hospital) serving Camden, London (N=225) and two units (one in-patient and one day- hospital) in Manchester (N=34) were used to assess outcome information of psychiatric care in tenns of availability (are comparable data present for both admission and discharge?), reliability (can this data be reliably extracted?), and validity (are the data true measures?). Differences in outcome between units and between diagnostic groups were considered in order to explore the possibility of auditing the outcome of routine psychiatric treatment using case notes.
    [Show full text]
  • Types of Sums of Squares
    Types of Sums of Squares With flexibility (especially unbalanced designs) and expansion in mind, this ANOVA package was implemented with general linear model (GLM) approach. There are different ways to quantify factors (categorical variables) by assigning the values of a nominal or ordinal variable, but we adopt binary coding for each factor level and all applicable interactions into dummy (indicator) variables. An ANOVA can be written as a general linear model: Y = b0 + b1X1 + b2X2 + ... + bkXk+e With matrix notation, it is reduced to a simple form Y = Xb + e The design matrix for a 2-way ANOVA with factorial design 2X3 looks like Data Design Matrix A B A*B A B A1 A2 B1 B2 B3 A1B1 A1B2 A1B3 A2B1 A2B2 A2B3 1 1 1 1 0 1 0 0 1 0 0 0 0 0 1 2 1 1 0 0 1 0 0 1 0 0 0 0 1 3 1 1 0 0 0 1 0 0 1 0 0 0 2 1 1 0 1 1 0 0 0 0 0 1 0 0 2 2 1 0 1 0 1 0 0 0 0 0 1 0 2 3 1 0 1 0 0 1 0 0 0 0 0 1 After removing an effect of a factor or an interaction from the above full model (deleting some columns from matrix X), we obtain the increased error due to the removal as a measure of the effect. And the ratio of this measure relative to some overall error gives an F value, revealing the significance of the effect.
    [Show full text]
  • Chapter 1: Introduction
    GEO4-4420 Stochastic Hydrology Prof. dr. Marc F.P. Bierkens Department of Physical Geography Utrecht University 2 Contents 1. Introduction 5 2. Descriptive statistics 15 3. Probablity and random variables 27 4. Hydrological statistics and extremes 51 5. Random functions 71 6. Time series analysis (by Martin Knotters, Alterra) 97 7. Geostatistics 117 8. Forward stochastic modelling 147 9. State estimation and data-assimilation (handouts) 183 References 185 3 4 Chapter 1: Introduction 1.1 Why stochastic hydrology? The term “stochastic” derives from the Greek word στοχαστιχηζ which is translated with “a person who forecasts a future event in the sense of aiming at the truth”, that is a seer or soothsayer. So, “stochastic” refers to predicting the future. In the modern sense “stochastic” in stochastic methods refers to the random element incorporated in these methods. Stochastic methods thus aim at predicting the value of some variable at non- observed times or at non-observed locations, while also stating how uncertain we are when making these predictions. But why should we care so much about the uncertainty associated with our predictions? The following example (Figure 1.1) shows a time series of observed water table elevations in a piezometer and the outcome of a groundwater model at this location. Also plotted are the differences (residuals) between the data and the model results. We can observe two features. First, the model time series seems to vary more smoothly then the observations. Secondly, there are noisy differences between model results and observations. These differences, which are called residuals, have among others the following causes: • observation errors.
    [Show full text]
  • Evaluation of Left Ventricular Diastolic Dysfunction in Type Ii Diabetes
    DOI: 10.14260/jemds/2014/2222 ORIGINAL ARTICLE EVALUATION OF LEFT VENTRICULAR DIASTOLIC DYSFUNCTION IN TYPE II DIABETES MELLITUS – THE ROLE OF VALSALVA MANEUVER Uday G1, Jayakumar S2, Jnaneshwari M3, Arun Kumar N4 HOW TO CITE THIS ARTICLE: Uday G, Jayakumar S, Jnaneshwari M, Arun Kumar N. “Evaluation of Left Ventricular Diastolic Dysfunction in Type II Diabetes Mellitus – The Role of Valsalva Maneuver”. Journal of Evolution of Medical and Dental Sciences 2014; Vol. 3, Issue 11, March 17; Page: 2898-2906, DOI: 10.14260/jemds/2014/2222 ABSTRACT: BACKGROUND & OBJECTIVES: Left ventricular diastolic dysfunction is known to occur in early stages of diabetic cardiomyopathy but its exact prevalence is not known. The present study was conducted to assess the prevalence of diastolic dysfunction in diabetic patients in the absence of hypertension or CAD. The role of valsalva maneuver in diagnosis of diastolic dysfunction was also studied. AIMS: To study left ventricular diastolic dysfunction in asymptomatic normotensive patients with Type 2 Diabetes. To study the role of valsalva maneuver in diagnosing diastolic dysfunction. METHODOLOGY: 100 consecutive asymptomatic normotensive patients (mean age 52.34 ±8.6 yrs.) with Type 2 diabetes free of any major clinical diabetic complications and having no evidence of CAD on non-invasive testing were studied for LV diastolic functions, using pulsed Doppler at the tip of mitral valve. The peak velocities of LV filling during the early rapid (E wave) and atrial contraction (A wave) phases, the ratio of the 2 filling velocities (E/A ratio) were recorded at end expiration at baseline and again during phase II of valsalva maneuver.
    [Show full text]
  • Use of Statistical Packages for Designing and Analysis of Experiments in Agriculture Latika Sharma and Nitu Mehta* (Ranka) Dept
    Popular Article Popular Kheti Volume -2, Issue-1 (January-March), 2014 Available online at www.popularkheti.info © 2014 popularkheti.info ISSN:2321-0001 Use of Statistical Packages for Designing and Analysis of Experiments in Agriculture Latika Sharma and Nitu Mehta* (Ranka) Dept. of Agricultural Economics & Management, Rajasthan College of Agriculture, MPUAT, Udaipur-313001, India *Email of corresponding author: [email protected] As long as analysis of data generated is concerned there are many statistical software packages that come very handy. But as far as generation of design is concerned there are not many software packages available for this purpose. And when we talk of generation of the design, we mean an algorithm that is capable of generating the design already available in the literature. Using computer algorithms, it has not been possible to generate new designs. This is an open area of research. So the purpose of the present article is to give an exposure to various statistical software packages that are useful for designing and analysis of experiments. Introduction Statistical Science is concerned with the twin aspect of theory of design of experiments and sample surveys and drawing valid inferences there from using various statistical techniques/methods. The art of drawing valid conclusions depends on how the data have been collected and analyzed. Depending upon the objective of the study, one has to choose an appropriate statistical procedure to test the hypothesis. When the number of observations is large or when the researcher is interested in multifarious aspects or some time series study, such calculations are very tedious and time consuming on a desk calculator.
    [Show full text]
  • Assessment of the Effects of Liquid and Granular Fertilizers on Maize Yield in Rwanda
    Afr. J. Food Agric. Nutr. Dev. 2021; 21(3): 17787-17800 https://doi.org/10.18697/ajfand.98.19670 ASSESSMENT OF THE EFFECTS OF LIQUID AND GRANULAR FERTILIZERS ON MAIZE YIELD IN RWANDA Hatungimana JC1*, Srinivasan RT1 and RR Vetukuri2 Hatungimana Jean Claude *Corresponding author email: [email protected] 1Department of Crop Science, School of Agriculture and Food Science, College of Agriculture and Veterinary Medicine, University of Rwanda (UR - CAVM), Rwanda 2Department of Plant Breeding, Swedish University of Agricultural Sciences (SLU), Sweden https://doi.org/10.18697/ajfand.98.19670 17787 ABSTRACT Maize (Zea mays L.) is the most widely grown cereal in the world, accounting for 1,116.34 MT of production in 2019/2020. In Africa, this crop represented approximately 56% of the total cultivated area from 1990 to 2005. About 50% of the African population depends on maize as a staple food and source of carbohydrates, protein, iron, vitamin B, and minerals. Lately, maize has become a cash crop which contributes to the improvement of farmers' livelihoods. For example, the Strategic Plan for Agricultural Transformation (SPAT) III outlined that fertilizer availability in Rwanda should increase to 55,000 MT per year, while fertilizer use should increase from 30 kg/ha in 2013 to 45 kg/ha for the 2017/18 cropping season. Only inorganic fertilizers are currently being used in maize production in Rwanda. This research was conducted to assess the effects of liquid (CBX: Complete Biological Extract) and granular fertilizers on maize crop yields in Rwanda. The study was conducted in the fields of the Rwanda Agriculture and Animal Resources Development Board (Rubona Station) during the 2018/2019 cropping season.
    [Show full text]