Statistical Methodologies with Medical Applications by Poduri S.R.S. Rao


3059841e8c79aba-261x361.jpg Author Poduri S.R.S. Rao
Isbn 9781119258490
File size 2.97mb
Year 2016
Pages 288
Language English
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Category biology



 

Statistical Methodologies with Medical Applications Poduri S.R.S. Rao Professor of Statistics University of Rochester Rochester, New York, USA This edition first published 2017 © 2017 John Wiley & Sons, Ltd Registered Office John Wiley & Sons, Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the author shall be liable for damages arising herefrom. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloging-in-Publication Data Names: Rao, Poduri S.R.S., author. Title: Statistical methodologies with medical applications / Poduri S.R.S. Rao. Description: Chichester, West Sussex, United Kingdom ; Hoboken : John Wiley & Sons Inc., 2016. | Includes bibliographical references and index. Identifiers: LCCN 2016022669| ISBN 9781119258490 (cloth) | ISBN 9781119258483 (Adobe PDF) | ISBN 9781119258520 (epub) Subjects: | MESH: Statistics as Topic Classification: LCC RA409 | NLM WA 950 | DDC 610.2/1–dc23 LC record available at https://lccn.loc.gov/2016022669 A catalogue record for this book is available from the British Library. Cover Image: Gun2becontinued/Gettyimages Set in 10/12pt Times by SPi Global, Pondicherry, India 10 9 8 7 6 5 4 3 2 1 To my grandchildren Asha, Sita, Maya and Wyatt Contents Topics for illustrations, examples and exercises xv Preface xvii List of abbreviations xix 1 Statistical measures 1.1 Introduction 1.2 Mean, mode and median 1.3 Variance and standard deviation 1.4 Quartiles, deciles and percentiles 1.5 Skewness and kurtosis 1.6 Frequency distributions 1.7 Covariance and correlation 1.8 Joint frequency distribution 1.9 Linear transformation of the observations 1.10 Linear combinations of two sets of observations Exercises 1 1 2 3 4 5 6 7 9 10 10 11 2 Probability, random variable, expected value and variance 2.1 Introduction 2.2 Events and probabilities 2.3 Mutually exclusive events 2.4 Independent and dependent events 2.5 Addition of probabilities 2.6 Bayes’ theorem 2.7 Random variables and probability distributions 2.8 Expected value, variance and standard deviation 2.9 Moments of a distribution Exercises 14 14 14 15 15 16 16 17 17 18 18 3 Odds ratios, relative risk, sensitivity, specificity and the ROC curve 3.1 Introduction 3.2 Odds ratio 3.3 Relative risk 19 19 19 20 viii CONTENTS 3.4 Sensitivity and specificity 3.5 The receiver operating characteristic (ROC) curve Exercises 21 22 22 4 Probability distributions, expectations, variances and correlation 4.1 Introduction 4.2 Probability distribution of a discrete random variable 4.3 Discrete distributions 4.3.1 Uniform distribution 4.3.2 Binomial distribution 4.3.3 Multinomial distribution 4.3.4 Poisson distribution 4.3.5 Hypergeometric distribution 4.4 Continuous distributions 4.4.1 Uniform distribution of a continuous variable 4.4.2 Normal distribution 4.4.3 Normal approximation to the binomial distribution 4.4.4 Gamma distribution 4.4.5 Exponential distribution 4.4.6 Chisquare distribution 4.4.7 Weibull distribution 4.4.8 Student’s t- and F-distributions 4.5 Joint distribution of two discrete random variables 4.5.1 Conditional distributions, means and variances 4.5.2 Unconditional expectations and variances 4.6 Bivariate normal distribution Exercises Appendix A4 A4.1 Expected values and standard deviations of the distributions A4.2 Covariance and correlation of the numbers of successes x and failures (n – x) of the binomial random variable 24 24 25 25 25 26 27 27 28 29 29 29 30 31 32 33 34 34 34 35 36 37 38 38 38 39 5 Means, standard errors and confidence limits 5.1 Introduction 5.2 Expectation, variance and standard error (S.E.) of the sample mean 5.3 Estimation of the variance and standard error 5.4 Confidence limits for the mean 5.5 Estimator and confidence limits for the difference of two means 5.6 Approximate confidence limits for the difference of two means 5.6.1 Large samples 5.6.2 Welch-Aspin approximation (1949, 1956) 5.6.3 Cochran’s approximation (1964) 5.7 Matched samples and paired comparisons 5.8 Confidence limits for the variance 5.9 Confidence limits for the ratio of two variances 40 40 41 42 43 44 46 46 46 46 47 48 49 CONTENTS 5.10 Least squares and maximum likelihood methods of estimation Exercises Appendix A5 A5.1 Tschebycheff’s inequality A5.2 Mean square error ix 49 51 52 52 53 6 Proportions, odds ratios and relative risks: Estimation and confidence limits 6.1 Introduction 6.2 A single proportion 6.3 Confidence limits for the proportion 6.4 Difference of two proportions or percentages 6.5 Combining proportions from independent samples 6.6 More than two classes or categories 6.7 Odds ratio 6.8 Relative risk Exercises Appendix A6 A6.1 Approximation to the variance of lnp1 54 54 54 55 56 56 57 58 59 59 60 60 7 Tests of hypotheses: Means and variances 7.1 Introduction 7.2 Principle steps for the tests of a hypothesis 7.2.1 Null and alternate hypotheses 7.2.2 Decision rule, test statistic and the Type I & II errors 7.2.3 Significance level and critical region 7.2.4 The p-value 7.2.5 Power of the test and the sample size 7.3 Right-sided alternative, test statistic and critical region 7.3.1 The p-value 7.3.2 Power of the test 7.3.3 Sample size required for specified power 7.3.4 Right-sided alternative and estimated variance 7.3.5 Power of the test with estimated variance 7.4 Left-sided alternative and the critical region 7.4.1 The p-value 7.4.2 Power of the test 7.4.3 Sample size for specified power 7.4.4 Left-sided alternative with estimated variance 7.5 Two-sided alternative, critical region and the p-value 7.5.1 Power of the test 7.5.2 Sample size for specified power 7.5.3 Two-sided alternative and estimated variance 7.6 Difference between two means: Variances known 7.6.1 Difference between two means: Variances estimated 62 62 63 63 63 64 64 65 65 66 66 67 68 69 69 70 70 71 71 72 73 74 74 75 76 x CONTENTS 7.7 Matched samples and paired comparison 7.8 Test for the variance 7.9 Test for the equality of two variances 7.10 Homogeneity of variances Exercises 8 Tests of hypotheses: Proportions and percentages 8.1 A single proportion 8.2 Right-sided alternative 8.2.1 Critical region 8.2.2 The p-value 8.2.3 Power of the test 8.2.4 Sample size for specified power 8.3 Left-sided alternative 8.3.1 Critical region 8.3.2 The p-value 8.3.3 Power of the test 8.3.4 Sample size for specified power 8.4 Two-sided alternative 8.4.1 Critical region 8.4.2 The p-value 8.4.3 Power of the test 8.4.4 Sample size for specified power 8.5 Difference of two proportions 8.5.1 Right-sided alternative: Critical region and p-value 8.5.2 Right-sided alternative: Power and sample size 8.5.3 Left-sided alternative: Critical region and p-value 8.5.4 Left-sided alternative: Power and sample size 8.5.5 Two-sided alternative: Critical region and p-value 8.5.6 Power and sample size 8.6 Specified difference of two proportions 8.7 Equality of two or more proportions 8.8 A common proportion Exercises 9 The Chisquare statistic 9.1 Introduction 9.2 The test statistic 9.2.1 A single proportion 9.2.2 Specified proportions 9.3 Test of goodness of fit 9.4 Test of independence: (r x c) classification 9.5 Test of independence: (2x2) classification 9.5.1 Fisher’s exact test of independence 9.5.2 Mantel-Haenszel test statistic 77 77 78 79 80 82 82 82 83 84 84 84 85 85 86 86 86 87 87 88 88 89 90 90 91 92 93 93 94 95 95 96 97 99 99 99 100 100 101 101 104 105 106 CONTENTS Exercises Appendix A9 A9.1 A9.2 Derivations of 9.4(a) Equality of the proportions 10 Regression and correlation 10.1 Introduction 10.2 The regression model: One independent variable 10.2.1 Least squares estimation of the regression 10.2.2 Properties of the estimators 10.2.3 ANOVA (Analysis of Variance) for the significance of the regression 10.2.4 Tests of hypotheses, confidence limits and prediction intervals 10.3 Regression on two independent variables 10.3.1 Properties of the estimators 10.3.2 ANOVA for the significance of the regression 10.3.3 Tests of hypotheses, confidence limits and prediction intervals 10.4 Multiple regression: The least squares estimation 10.4.1 ANOVA for the significance of the regression 10.4.2 Tests of hypotheses, confidence limits and prediction intervals 10.4.3 Multiple correlation, adjusted R2 and partial correlation 10.4.4 Effect of including two or more independent variables and the partial F-test 10.4.5 Equality of two or more series of regressions 10.5 Indicator variables 10.5.1 Separate regressions 10.5.2 Regressions with equal slopes 10.5.3 Regressions with the same intercepts 10.6 Regression through the origin 10.7 Estimation of trends 10.8 Logistic regression and the odds ratio 10.8.1 A single continuous predictor 10.8.2 Two continuous predictors 10.8.3 A single dichotomous predictor 10.9 Weighted Least Squares (WLS) estimator 10.10 Correlation 10.10.1 Test of the hypothesis that two random variables are uncorrelated 10.10.2 Test of the hypothesis that the correlation coefficient takes a specified value 10.10.3 Confidence limits for the correlation coefficient 10.11 Further topics in regression xi 107 109 109 109 110 110 110 112 113 114 116 118 120 121 122 124 126 127 128 129 130 132 132 133 134 135 136 138 139 139 140 141 142 143 143 144 144 xii CONTENTS 10.11.1 Linearity of the regression model and the lack of fit test 144 10.11.2 The assumption that V εi Xi = σ 2 , same at each Xi 146 10.11.3 Missing observations 146 10.11.4 Transformation of the regression model 147 147 10.11.5 Errors of measurements of (Xi, Yi) Exercises 148 Appendix A10 149 149 A10.1 Square of the correlation of Yi and Ŷi A10.2 Multiple regression 149 A10.3 Expression for SSR in (10.38) 151 11 Analysis of variance and covariance: Designs of experiments 11.1 Introduction 11.2 One-way classification: Balanced design 11.3 One-way random effects model: Balanced design 11.4 Inference for the variance components and the mean 11.5 One-way classification: Unbalanced design and fixed effects 11.6 Unbalanced one-way classification: Random effects 11.7 Intraclass correlation 11.8 Analysis of covariance: The balanced design 11.8.1 The model and least squares estimation 11.8.2 Tests of hypotheses for the slope coefficient and equality of the means 11.8.3 Confidence limits for the adjusted means and their differences 11.9 Analysis of covariance: Unbalanced design 11.9.1 Confidence limits for the adjusted means and the differences of the treatment effects 11.10 Randomized blocks 11.10.1 Randomized blocks: Random and mixed effects models 11.11 Repeated measures design 11.12 Latin squares 11.12.1 The model and analysis 11.13 Cross-over design 11.14 Two-way cross-classification 11.14.1 Additive model: Balanced design 11.14.2 Two-way cross-classification with interaction: Balanced design 11.14.3 Two-way cross-classification: Unbalanced additive model 11.14.4 Unbalanced cross-classification with interaction 11.14.5 Multiplicative interaction and Tukey’s test for nonadditivity 11.15 Missing observations in the designs of experiments 152 152 153 155 155 157 159 160 161 161 163 164 165 167 168 170 170 172 172 174 175 176 178 179 183 184 184 CONTENTS Exercises Appendix A11 A11.1 Variance of σ 2α in (11.25) from Rao (1997, p. 20) A11.2 The total sum of squares (Txx, Tyy) and sum of products (Txy) can be expressed as the within and between components as follows xiii 186 189 189 189 12 Meta-analysis 12.1 Introduction 12.2 Illustrations of large-scale studies 12.3 Fixed effects model for combining the estimates 12.4 Random effects model for combining the estimates 12.5 Alternative estimators for σ 2α 12.6 Tests of hypotheses and confidence limits for the variance components Exercises Appendix A12 190 190 190 191 193 194 13 Survival analysis 13.1 Introduction 13.2 Survival and hazard functions 13.3 Kaplan-Meier Product-Limit estimator 13.4 Standard error of Ŝ(tm) and confidence limits for S(tm) 13.5 Confidence limits for S(tm) with the right-censored observations 13.6 Log-Rank test for the equality of two survival distributions 13.7 Cox’s proportional hazard model Exercises Appendix A13 Expected value and variance of Ŝ(tm) and confidence limits for S(tm) 197 197 198 198 199 199 201 202 203 14 Nonparametric statistics 14.1 Introduction 14.2 Spearman’s rank correlation coefficient 14.3 The Sign test 14.4 Wilcoxon (1945) Matched-pairs Signed-ranks test 14.5 Wilcoxon’s test for the equality of the distributions of two non-normal populations with unpaired sample observations 14.5.1 Unequal sample sizes 14.6 McNemer’s (1955) matched pair test for two proportions 14.7 Cochran’s (1950) Q-test for the difference of three or more matched proportions 14.8 Kruskal-Wallis one-way ANOVA test by ranks Exercises 205 205 205 206 208 194 195 196 203 209 210 210 211 212 213 xiv CONTENTS 15 Further topics 15.1 Introduction 15.2 Bonferroni inequality and the Joint Confidence Region 15.3 Least significant difference (LSD) for a pair of treatment effects 15.4 Tukey’s studentized range test 15.5 Scheffe’s simultaneous confidence intervals 15.6 Bootstrap confidence intervals 15.7 Transformations for the ANOVA Exercises Appendix A15 A15.1 Variance stabilizing transformation 215 215 215 217 217 218 219 220 221 221 221 Solutions to exercises 222 Appendix tables 249 References 261 Index 264 Topics for illustrations, examples and exercises Heights, weights and BMI (Body Mass Index) of sixteen and twenty-year-old boys from growth charts Immunization coverage of one-year-olds: Measles, DTP3 and HEP B3 from WHO reports Medical insurance for children Sudden Infant Death Syndrome (SIDS) Population growth rates and fertility Age, family size, income and health insurance Healthcare expenditure in Africa, Asia and Europe Vaccination for flu for different age groups Emergency department visits for cold symptoms, injuries and other reasons. Overweight and obesity Trends of adult obesity BMI and mortality Smoking, heart disease and cancer risk Air pollution and cancer risk Hypertension, systolic and diastolic blood pressures (SBP, DBP) of males and females. Cholesterol levels: LDL and HDL Effects of overweight on LDL Low-dose aspirin and reduction of certain types of cancer Celiac disease and the benefits of gluten-free diet Statins and the reduction of LDL Exercise and its benefits for blood pressure levels Weight loss with diets of combinations of low and high-levels of fatty acids and protein xvi TOPICS FOR ILLUSTRATIONS, EXAMPLES AND EXERCISES Medical rehabilitation of stroke patients Functional independence measures of stroke patients from medical rehabilitation Sources: Reports of WHO, CDC, U.S. Health Statistics; Journal of the American Medical Association (JAMA); New England Journal of Medicine (NEJM), Lancet and other published literature. Preface Statistical analysis, evaluation and inference are essential for every type of medical study and clinical experiment. Physicians and medical clinics and laboratories routinely record the blood pressures, cholesterol levels and other relevant diagnostic measurements of patients. Clinical experiments evaluate and compare the effects of medical treatments and procedures. Medical journals report the research findings on the relative risks and odds ratios related to hypertension, abnormal cholesterol levels, obesity, harmful effects of smoking habits and excessive alcohol consumption and similar topics. Estimation of the means, standard deviations, proportions, odds ratios, relative risks and related statistical measures of health-related characteristics are of importance for the above types of medical studies. Evaluation of the errors of estimation, ascertaining the confidence limits for the population characteristics of interest, tests of hypotheses and statistical inference, and Chisquare tests for independence and association of categorical variables are important aspects of many medical studies and clinical experiments. Statistical inference is employed, for instance, to assess the relationship between obesity and hypertension and the association between air pollution and bronchial problems. A variety of similar problems require statistical investigations and inference. Regression analysis is widely used to determine the relationship of clinical outcomes and physical attributes. In several clinical investigations, correlations between diagnostic observations are examined to search for the causal factors. Analysis of Variance and Covariance procedures are extensively employed to examine the differences between the effects of medical treatments. All the above types of statistical methods, procedures and techniques required for medical studies, research and evaluations are presented in the following chapters. Topics such as the Meta-analysis, Survival Analysis and Hazard Ratios, and nonparametric statistics are also included. Following the descriptive statistical measures in the first chapter, definitions of probability, odds ratios and relative risk appear in Chapters 2 and 3. Binomial, normal, Chisquare and related probability distributions essential for the statistical methods and applications are presented in Chapter 4. Estimation of the means, variances, proportions and percentages, odds ratios and relative risks, Standard Errors (S.E.) of the estimators and confidence intervals appear in Chapters 5 and 6. Tests of hypotheses of means, proportions and variances, p-values, power of a test, sample size required for a specified power are the topics for Chapters 7 and 8. The Chisquare tests for goodness of fit and independence are presented in Chapter 9. Linear, multiple and logistic regressions and correlation are the topics for Chapter 10. Chapter 11 presents the Analysis of Variance (ANOVA) and Covariance procedures, Randomized bocks, xviii PREFACE Latin square designs, fixed and random effects models, and two-way crossclassification with and without interaction. Meta-analysis and Survival Analysis in Chapters 12 and 13 are followed by the nonparametric statistics in Chapter 14. The final chapter contains topics in ANOVA and tests of hypotheses including the Simultaneous Confidence Intervals and Bootstrap Confidence Intervals. Examples, illustrations and exercises with solutions are presented in each chapter. They are constructed from the observations of practical situations, research studies appearing in The New England Journal of Medicine (NEJM), Journal of the American Medical Association (JAMA), Lancet and other medical journals, and the summaries presented in the Health Statistics of the Center for Disease Control (CDC) in the United States and the World Health Organization (WHO). They are related to a variety of medical topics of general interest including the following: (a) heights, weights and Body Mass Index (BMI) of ten-to-twenty-year-old boys and girls; (b) immunization of children; (c) overweight, obesity, hypertension and high cholesterol levels of adults; (d) benefits of fat-free and gluten-free diets and exercise, and (e) healthcare expenditures and medical insurance. BMI is the ratio of the weight in kilograms to the square of the height in meters. A person is considered to be of normal weight if the BMI is 18.5–24, overweight if it is 25–29, and obese if it is 30 or more. For the blood cholesterol levels of adults, LDL less than 100 mg/dL and HDL higher than 40 mg/dL are considered optimal. Systolic and diastolic blood pressures, SBP and DBP of 120/80 mmHg are considered desirable. Illustrations and examples and exercises throughout the chapters are related to these medical measurements and other health-related topics. Readily available software programs in Excel, Minitab and R are utilized for the solutions of the illustrations, examples and exercises. The various topics in these chapters are presented at the level of comprehension of the students pursuing statistics, biostatistics, medicine, biological, physical and natural sciences and epidemiological studies. Each topic is illustrated through examples. More than one hundred exercises with solutions are included. This book can be recommended for a one-semester or two-quarter course for the above types of students, and also for self-study. One or two semesters of training in the principles and applications of statistical methods provides adequate preparation to pursue the different topics. The various statistical methods for medical studies presented in this manuscript can also be of interest to clinicians, physicians, and medical students and residents. I would like to thank the editor, Ms. Kathryn Sharples, for her interest in this project. Thanks to Charles Heckler, Kevin Rader and Nicholas Zaino for their expert reviews of the manuscript. Thanks also to Sarah Briscoe, Isabelle Weir and Patricia Digiorgio for their assistance in assembling the manuscript on the word processor. Special thanks to my wife and daughter, Drs. K.R. Poduri, MD and Ann Hug Poduri, MD, MPH for sharing their medical expertise in selecting the various topics and illustrations throughout the chapters. Poduri S.R.S. Rao Professor of Statistics University of Rochester List of abbreviations WHO: World Health Organization CDC: Center for Disease Control LDL: Low Density Lipoprotein HDL: High Density Lipoprotein LDL and HDL are measures of cholesterol levels in units of milligrams for Deciliter (mg/dl) SBP : Systolic Blood Pressure DBP: Diastolic Blood Pressure SBP and DBP are measures of pressure in the blood vessels in units of millimeters of mercury (mmHg) BMI: Body Mass Index 1 Statistical measures 1.1 Introduction Medical professionals, hospitals and healthcare centers record heights, weights and other relevant physical measurements of patients along with their blood pressures cholesterol levels and similar diagnostic measurements. National organizations such as the Center for Disease Control (CDC) in the United States, the World Health Organization (WHO) and several national and international organizations record and analyze various aspects of the healthcare status of the citizens of all age groups. Epidemiological studies and surveys collect and analyze health-related information of the people around the globe. Clinical trials and experiments are conducted for the development of effective and improved medical treatments. Statistical measures are utilized to analyze the various diagnostic measurements as well as the outcomes of clinical experiments. The mean, mode and median described in the following sections locate the centers of the distributions of the above types of observations. The variance, standard deviation (S.D.) and the related coefficient of variation (C.V.) are the measures of dispersion of a set of observations. The quartiles, deciles and percentiles divide the data respectively into four, ten and one hundred equal parts. The skewness coefficient exhibits the departure of the data from its symmetry, and the kurtosis coefficient its peakedness. The measurements on the heights, weights and Body Mass Indexes (BMIs) of a sample of twentyyear-old boys obtained from the Chart Tables of the CDC (2008) are presented in Table 1.1. These measurements for the ten and sixteen- year old boys and girls are presented in Appendix Tables T1.1–T1.4. Statistical Methodologies with Medical Applications, First Edition. Poduri S.R.S. Rao. © 2017 John Wiley & Sons, Ltd. Published 2017 by John Wiley & Sons, Ltd. 2 STATISTICAL MEASURES Table 1.1 Heights (cm), weights (kg) and BMIs of twenty-year old boys. Height Weight BMI 54 55 58 59 60 62 63 66 68 72 75 75 78 80 82 84 86 88 95 102 20.58 20.70 20.80 20.90 20.76 20.96 21.30 22.05 22.46 23.24 24.21 24.21 24.90 25.25 25.88 25.93 25.40 25.99 27.46 28.86 162 163 167 168 170 172 172 173 174 176 176 176 177 178 178 180 184 184 186 188 BMI = Weight/(Height)2. 1.2 Mean, mode and median The diagnostic measurements of a sample of n individuals can be represented by xi , i = 1,2,…, n . Their mean or average is n xi n = x1 + x2 + … + xn n x= (1.1) i=1 For the heights of the boys in Table 1.1, the mean becomes x = 162 + 163 + … + 188 20 = 175 2 cm. Similarly, the mean of their weights is 73.1 kg. For the BMI, which is (Weight/Height2), the mean becomes 23.59. The mode is the observation occurring more frequently than the remaining observations. For the heights of the boys, it is 176 cm. The median is the middle value of the observations. If the number of observations n is odd, it is the (n + 1)th observation. If n is an even number, it is the average of the (n/2)th and the next observation. Both the mode and median of the twenty heights of the boys in Table 1.1 are equal to 176 cm, which is slightly larger than the mean of 175.2 cm. STATISTICAL MEASURES 2 4 9 (6) 5 2 16 16 17 17 18 18 3 23 78 02234 666788 044 68 Figure 1.1 Stem and leaf display of the heights of the twenty boys. Leaf unit = 1.0. The median class has (6) observations. The cumulative number of observations below and above the median class are (2, 4, 9) and (5, 2). The mean, mode and median locate the center of the observations. The mean is also known as the first moment m1 of the observations. For the healthcare policies, for instance, it is of importance to examine the average amount of the medical expenditures incurred by families of different sizes or specified ranges of income. At the same time, useful information is provided by the median and modal values of their expenditures. Figure 1.1 is the Stem and Leaf display of the heights in Table 1.1. The cumulative number of observations below and above the median appear in the first column. The second and third columns are the stems, with the attached leaves. 1.3 Variance and standard deviation The variance is a measure of the dispersion among the observations, and it is given by n xi − x s2 = 2 n−1 i=1 = xi − x 2 + x2 − x 2 + … xn − x 2 n−1 (1.2) The divisor (n – 1) in this expression represents the degrees of freedom (d.f.). If (n – 1) of the observations and the sum or mean of the n observations are known, the remaining observation is automatically determined. The expression in (1.2) can also 2 be expressed as xi − xj n n − 1 , which is the average of the squared differences i j of the n(n – 1) pairs of the observations. The standard deviation (S.D.) is given by s, the positive square root of the variance. The second central moment of the observations m2 = xi − x 2 n is the same as n − 1 s2 n. For the twenty heights of boys in

Author Poduri S.R.S. Rao Isbn 9781119258490 File size 2.97mb Year 2016 Pages 288 Language English File format PDF Category Biology Book Description: FacebookTwitterGoogle+TumblrDiggMySpaceShare This book presents the methodology and applications ofa range ofimportant topics in statistics, and is designed forgraduate students in Statistics and Biostatistics and for medical researchers.Illustrations and more than ninety exercises with solutions are presented. They are constructed from the research findings of the medical journals, summary reports of the Centre for Disease Control (CDC) and the World Health Organization (WHO), and practical situations. The illustrations and exercises are related to topics such as immunization, obesity, hypertension, lipid levels, diet and exercise, harmful effects of smoking and air pollution, and the benefits of gluten free diet. Thisbook can be recommended for a one or two semester graduate level course forstudents studying Statistics, Biostatistics, Epidemiology and Health Sciences. Itwill also be useful asa companion for medical researchers and research oriented physicians.     Download (2.97mb) Childhood Obesity (mymodernhealth Faqs) Illustrated Reviews: Cell & Molecular Biology The Fight Against Hunger And Malnutrition New Research On Antioxidants (nova Biomedical) Gut: The Inside Story of Our Body’s Most Underrated Organ Load more posts

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