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360 CHAPTER 8 ANALYSIS OF VARIANCE TABLE 8.5.2 Data of Table 8.5.1 Altered to Show the Effect of One Type of Interaction Factor B—Drug Dosage Factor A—Age j ¼ 1 j ¼ 2 j ¼ 3 Young ði ¼ 1Þ m 11 ¼ 5 m 12 ¼ 10 m 13 ¼ 20 Old ði ¼ 2Þ m 21 ¼ 15 m 22 ¼ 10 m 23 ¼ 5 Reduction in reaction time 30 Age 25 20 a 1 15 10 5 a 2 0 b 1 b 2 b 3 Drug dosage FIGURE 8.5.2 Age and drug effects, interaction present. Reduction in reaction time 30 25 20 15 10 5 0 a 1 a 2 Age Drug dosage b 1 b 2 b 3 The presence of interaction between two factors can affect the characteristics of the data in a variety of ways depending on the nature of the interaction. We illustrate the effect of one type of interaction by altering the data of Table 8.5.1 as shown in Table 8.5.2. The important characteristics of the data in Table 8.5.2 are as follows. 1. The difference between means for any two levels of factor B is not the same for both levels of factor A. We note in Table 8.5.2. for example, that the difference between levels 1 and 2 of factor B is 5 for the young age group and þ5 for the old age group. 2. The difference between means for both levels of factor A is not the same at all levels of factor B. The differences between factor A means are 10, 0, and 15 for levels 1, 2, and 3, respectively, of factor B. 3. The factor level curves are not parallel, as shown in Figure 8.5.2. When population data exhibit the characteristics illustrated in Table 8.5.2 and Figure 8.5.2, we say that there is interaction between the two factors. We emphasize that the kind of interaction illustrated by the present example is only one of many types of interaction that may occur between two factors. & In summary, then, we can say that there is interaction between two factors if a change in one of the factors produces a change in response at one level of the other factor different from that produced at other levels of this factor. Advantages The advantages of the factorial experiment include the following. 1. The interaction of the factors may be studied. 2. There is a saving of time and effort.
8.5 THE FACTORIAL EXPERIMENT 361 In the factorial experiment all the observations may be used to study the effects of each of the factors under investigation. The alternative, when two factors are being investigated, would be to conduct two different experiments, one to study each of the two factors. If this were done, some of the observations would yield information only on one of the factors, and the remainder would yield information only on the other factor. To achieve the level of accuracy of the factorial experiment, more experimental units would be needed if the factors were studied through two experiments. It is seen, then, that 1 two-factor experiment is more economical than 2 one-factor experiments. 3. Because the various factors are combined in one experiment, the results have a wider range of application. The Two-Factor Completely Randomized Design A factorial arrangement may be studied with either of the designs that have been discussed. We illustrate the analysis of a factorial experiment by means of a two-factor completely randomized design. 1. Data. The results from a two-factor completely randomized design may be presented in tabular form as shown in Table 8.5.3. Here we have a levels of factor A, b levels of factor B, and n observations for each combination of levels. Each of the ab combinations of levels of factor A with levels of factor B is a treatment. In addition to the totals and means shown in Table 8.5.3, we note that the total and mean of the ijth cell are T ij ¼ Xn k¼1 x ijk and x ij ¼ T ij=n TABLE 8.5.3 Table of Sample Data from a Two-Factor Completely Randomized Experiment Factor B Factor A 1 2 . . . b Totals Means 1 x 111 x 121 ... x 1b1 . . . . T 1:: x 1:: x 11n x 12n ... x 1bn 2 x 211 x 221 ... x 2b1 . . . . T 2:: x 2:: . . x 21n x 22n ... x 2bn . . . . . . . . . . . . a x a11 x a21 ... x ab1 . . . . T a:: x a:: x a1n x a2n ... x abn Totals T :1: T :2: ... T :b: T ... Means x :1: x :2: ... x :b: x ...
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TENTH EDITION BIOSTATISTICS A Found
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VP & EXECUTIVE PUBLISHER: ACQUISITI
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PREFACE This 10th edition of Biosta
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PREFACE ix SUPPLEMENTS Instructor
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BRIEF CONTENTS 1 INTRODUCTION TO BI
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xiv CONTENTS 6.3 The t Distribution
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CHAPTER1 INTRODUCTION TO BIOSTATIST
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1.2 SOME BASIC CONCEPTS 3 Sources o
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1.3 MEASUREMENT AND MEASUREMENT SCA
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1.4 SAMPLING AND STATISTICAL INFERE
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1.5 THE SCIENTIFIC METHOD AND THE D
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1.6 COMPUTERS AND BIOSTATISTICAL AN
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REVIEW QUESTIONS AND EXERCISES 17 a
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CHAPTER2 DESCRIPTIVE STATISTICS CHA
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2.2 THE ORDERED ARRAY 21 TABLE 2.2.
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2.3 GROUPED DATA: THE FREQUENCY DIS
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EXERCISES 31 3 ¼ an abundance of s
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EXERCISES 33 25.21 30.49 27.38 36.4
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EXERCISES 35 2.3.7 In a study of ph
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EXERCISES 37 0.6870 0.6900 0.6910 0
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2.4 DESCRIPTIVE STATISTICS: MEASURE
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EXERCISES 53 2.5.1 Porcellini et al
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SUMMARY OF FORMULAS FOR CHAPTER 2 5
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REVIEW QUESTIONS AND EXERCISES 57 R
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REVIEW QUESTIONS AND EXERCISES 59 1
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REFERENCES 63 DRINK TOUNCES TGRAMS
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CHAPTER3 SOME BASIC PROBABILITY CON
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3.2 TWO VIEWS OF PROBABILITY: OBJEC
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3.4 CALCULATING THE PROBABILITY OF
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EXERCISES 77 (d) If we pick a subje
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3.5 BAYES’ THEOREM, SCREENING TES
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3.5 BAYES’ THEOREM, SCREENING TES
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EXERCISES 83 EXERCISES 3.5.1 A medi
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REVIEW QUESTIONS AND EXERCISES 87 (
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REVIEW QUESTIONS AND EXERCISES 89 i
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REFERENCES 91 Applications Referenc
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4.2 PROBABILITY DISTRIBUTIONS OF DI
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4.3 THE BINOMIAL DISTRIBUTION 99 TA
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4.3 THE BINOMIAL DISTRIBUTION 101 E
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4.3 THE BINOMIAL DISTRIBUTION 103 f
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EXAMPLE 4.3.4 According to a June 2
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EXERCISES 107 Data: C1: 0 1 2 3 4 5
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4.4 THE POISSON DISTRIBUTION 109 pr
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4.4 THE POISSON DISTRIBUTION 111 or
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4.5 CONTINUOUS PROBABILITY DISTRIBU
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4.5 CONTINUOUS PROBABILITY DISTRIBU
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4.6 THE NORMAL DISTRIBUTION 117 .68
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4.6 THE NORMAL DISTRIBUTION 119 σ
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4.6 THE NORMAL DISTRIBUTION 121 _ 2
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4.7 NORMAL DISTRIBUTION APPLICATION
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TableD,wefindthattheareatotheleftof
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EXERCISES 127 Data: C1:-3-2-10123 D
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SUMMARY OF FORMULAS FOR CHAPTER 4 S
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REVIEW QUESTIONS AND EXERCISES 131
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REFERENCES 133 REFERENCES Methodolo
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5.2 SAMPLING DISTRIBUTIONS 135 voca
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5.3 DISTRIBUTION OF THE SAMPLE MEAN
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5.4 DISTRIBUTION OF THE DIFFERENCE
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EXERCISES 149 FIGURE 5.4.2 Sampling
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5.5 DISTRIBUTION OF THE SAMPLE PROP
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EXERCISES 153 51 percent had adequa
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CHAPTER6 ESTIMATION CHAPTER OVERVIE
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6.1 INTRODUCTION 163 We will find t
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6.2 CONFIDENCE INTERVAL FOR A POPUL
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6.3 THE t DISTRIBUTION 171 someone
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6.3 THE t DISTRIBUTION 173 Normal d
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EXERCISES 175 Yes Population normal
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6.4 CONFIDENCE INTERVAL FOR THE DIF
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EXERCISES 183 EXERCISES R Code: > t
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6.7 DETERMINATION OF SAMPLE SIZE FO
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6.8 DETERMINATION OF SAMPLE SIZE FO
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6.9 CONFIDENCE INTERVAL FOR THE VAR
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6.9 CONFIDENCE INTERVAL FOR THE VAR
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EXERCISES 197 x 2 1 ða=2Þ ¼ 14:4
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6.10 CONFIDENCE INTERVAL FOR THE RA
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6.10 CONFIDENCE INTERVAL FOR THE RA
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REFERENCES 211 3. W. V. BEHRENS,
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REFERENCES 213 A-30. ELENI KOUSTA,
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7.1 INTRODUCTION 215 7.1 INTRODUCTI
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7.1 INTRODUCTION 217 question: Can
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7.1 INTRODUCTION 219 DEFINITION The
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7.1 INTRODUCTION 221 Evaluate data
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7.2 HYPOTHESIS TESTING: A SINGLE PO
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EXERCISES 233 Dialog box: Session c
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EXERCISES 235 7.2.10 Following a we
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7.3 HYPOTHESIS TESTING: THE DIFFERE
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EXAMPLE 7.3.2 The purpose of a stud
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EXERCISES 245 The SAS System The TT
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EXERCISES 247 Primary Hypertensive
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7.4 PAIRED COMPARISONS 249 7.3.10 R
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7.4 PAIRED COMPARISONS 251 where d
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7.4 PAIRED COMPARISONS 253 a = .05
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EXERCISES 255 7.4.1 Ellen Davis Jon
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EXERCISES 263 MINITAB Output Test a
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7.8 HYPOTHESIS TESTING: THE RATIO O
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7.8 HYPOTHESIS TESTING: THE RATIO O
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EXERCISES 271 EXERCISES In the foll
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7.9 THE TYPE II ERROR AND THE POWER
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7.9 THE TYPE II ERROR AND THE POWER
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7.10 DETERMINING SAMPLE SIZE TO CON
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EXERCISES 279 We set the right-hand
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REFERENCES 301 A-17. JOHN S. MORLEY
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REFERENCES 303 A-56. LUIGI BENINI,
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8.1 INTRODUCTION 305 8.1 INTRODUCTI
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8.1 INTRODUCTION 307 who received d
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REFERENCES 411 Postmenopausal Women
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CHAPTER9 SIMPLE LINEAR REGRESSION A
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9.2 THE REGRESSION MODEL 415 some e
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9.3 THE SAMPLE REGRESSION EQUATION
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EXERCISES EXERCISES 423 The Least-S
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EXERCISES 425 CoaguChek Hospital Co
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9.4 EVALUATING THE REGRESSION EQUAT
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9.5 USING THE REGRESSION EQUATION I
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9.5 USING THE REGRESSION EQUATION 4
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9.6 THE CORRELATION MODEL 445 9.6 T
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9.7 THE CORRELATION COEFFICIENT 447
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EXERCISES 455 is employed. We first
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EXERCISES 457 9.7.3 In the study by
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9.8 SOME PRECAUTIONS 459 X Y X Y 5.
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9.9 SUMMARY 461 variance, which tes
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9.7.3 t statistic for correlation c
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REFERENCES 487 A-4. ROBERT B. PARKE
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CHAPTER10 MULTIPLE REGRESSION AND C
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Assumptions follows. 10.2 THE MULTI
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10.3 OBTAINING THE MULTIPLE REGRESS
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10.3 OBTAINING THE MULTIPLE REGRESS
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EXERCISES 497 EXERCISES Obtain the
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EXERCISES 499 10.3.3 In a study of
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10.4 EVALUATING THE MULTIPLE REGRES
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10.5 USING THE MULTIPLE REGRESSION
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EXERCISES 509 and has an education
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10.6 THE MULTIPLE CORRELATION MODEL
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This equation may be used for estim
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EXERCISES 521 HIV DNA Blood (Y) HIV
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SUMMARY OF FORMULAS FOR CHAPTER 10
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10.6.7 t statistic for testing hypo
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REFERENCES 537 4. Refer to the data
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CHAPTER11 REGRESSION ANALYSIS: SOME
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11.1 INTRODUCTION 541 TABLE 11.1.1
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11.2 QUALITATIVE INDEPENDENT VARIAB
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EXERCISES 557 11.2.1 For subjects u
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EXERCISES 559 11.2.4 Refer to Exerc
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11.3 VARIABLE SELECTION PROCEDURES
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11.3 VARIABLE SELECTION PROCEDURES
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EXERCISES 565 CGAGE CGINCOME CGDUR
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EXERCISES 567 11.3.2 Machiel Naeije
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11.4 LOGISTIC REGRESSION 569 REACTI
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11.4 LOGISTIC REGRESSION 571 betwee
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11.4 LOGISTIC REGRESSION 573 The LO
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11.4 LOGISTIC REGRESSION 575 Standa
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11.4 LOGISTIC REGRESSION 577 Parame
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11.4 LOGISTIC REGRESSION 579 Model
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EXERCISES EXERCISES 581 Further Rea
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REFERENCES 597 KNOW ¼ Knowledge. S
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REFERENCES 599 A-26. EITHNE MULLOYa
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12.1 INTRODUCTION 12.2 THE MATHEMAT
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12.2 THE MATHEMATICAL PROPERTIES OF
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12.3 TESTS OF GOODNESS-OF-FIT 605 W
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12.3 TESTS OF GOODNESS-OF-FIT 607 x
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12.3 TESTS OF GOODNESS-OF-FIT 609 9
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that the first expected frequency i
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12.3 TESTS OF GOODNESS-OF-FIT 613 T
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12.3 TESTS OF GOODNESS-OF-FIT 615 V
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EXERCISES 617 EXERCISES 12.3.1 The
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12.4 TESTS OF INDEPENDENCE 619 12.3
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Observed Versus Expected Frequencie
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12.4 TESTS OF INDEPENDENCE 623 Comp
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12.4 TESTS OF INDEPENDENCE 625 Note
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12.4 TESTS OF INDEPENDENCE 627 9. C
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EXERCISES 629 Performed Baseline Te
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12.5 TESTS OF HOMOGENEITY 631 TABLE
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12.5 TESTS OF HOMOGENEITY 633 8. St
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EXERCISES 635 12.5.2 Coughlin et al
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Hypotheses alternatives. 12.6 THE F
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12.6 THE FISHER EXACT TEST 639 TABL
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12.7 RELATIVE RISK, ODDS RATIO, AND
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EXERCISES 653 Smoking_status * Obse
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12.8 SUMMARY 655 collected data on
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REFERENCES 667 12. R. LATSCHA, “T
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REFERENCES 669 A-35. CRAIG R. COHEN
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13.1 INTRODUCTION 671 2. be able to
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13.3 THE SIGN TEST 673 13.3 THE SIG
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13.3 THE SIGN TEST 675 TABLE 13.3.2
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13.3 THE SIGN TEST 677 EXAMPLE 13.3
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13.3 THE SIGN TEST 679 one sample i
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13.4 THE WILCOXON SIGNED-RANK TEST
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13.4 THE WILCOXON SIGNED-RANK TEST
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13.5 THE MEDIAN TEST 687 TABLE 13.5
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EXAMPLE 13.6.1 A researcher designe
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13.6 THE MANN-WHITNEY TEST 693 to t
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13.6 THE MANN-WHITNEY TEST 695 Dial
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EXERCISES 697 deficit in Sprague-Da
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13.7 THE KOLMOGOROV-SMIRNOV GOODNES
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EXERCISES 703 1.0 .9 Cumulative rel
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13.8 THE KRUSKAL-WALLIS ONE-WAY ANA
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13.8 THE KRUSKAL-WALLIS ONE-WAY ANA
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EXERCISES 709 Data: C1: 12.22 28.44
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EXERCISES 711 Improvement (Control)
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13.9 THE FRIEDMAN TWO-WAY ANALYSIS
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13.9 THE FRIEDMAN TWO-WAY ANALYSIS
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EXERCISES 717 Subject Area Student
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EXERCISES 725 Rank by Rank by Area
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13.11 NONPARAMETRIC REGRESSION ANAL
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13.11 NONPARAMETRIC REGRESSION ANAL
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REFERENCES 747 cardiac operations.
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REFERENCES 749 A-14. WENDY GANTT an
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14.2 TIME-TO-EVENT DATA AND CENSORI
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14.3 THE KAPLAN-MEIER PROCEDURE 757
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14.3 THE KAPLAN-MEIER PROCEDURE 759
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EXERCISES 761 1.0 .8 73% Low-grade
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14.4 COMPARING SURVIVAL CURVES 763
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14.4 COMPARING SURVIVAL CURVES 765
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EXERCISES 767 EXERCISES 14.4.1 If a
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14.5 COX REGRESSION: THE PROPORTION
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14.5 COX REGRESSION: THE PROPORTION
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REFERENCES 777 REFERENCES Methodolo
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A-2 APPENDIX STATISTICAL TABLES
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ANSWERS TO ODD-NUMBERED EXERCISES C
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ANSWERS TO ODD-NUMBERED EXERCISES A
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I-2 INDEX Confidence coefficient, 1
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I-4 INDEX Median test, 686-689 MINI
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I-6 INDEX SAS: and chi-square analy
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