Biostatistics

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268 CHAPTER 7 HYPOTHESIS TESTING the respective variances are the variances computed from samples taken from the populations. We would like to know if the difference that, undoubtedly, will exist between the sample variances is indicative of a real difference in population variances, or if the difference is of such magnitude that it could have come about as a result of chance alone when the population variances are equal. Two methods of chemical analysis may give the same results on the average. It may be, however, that the results produced by one method are more variable than the results of the other. We would like some method of determining whether this is likely to be true. Variance Ratio Test Decisions regarding the comparability of two population variances are usually based on the variance ratio test, which is a test of the null hypothesis that two population variances are equal. When we test the hypothesis that two population variances are equal, we are, in effect, testing the hypothesis that their ratio is equal to 1. We learned in the preceding chapter that, when certain assumptions are met, the quantity s 2 1 1 =s2 = s 2 2 =s2 2 is distributed as F with n1 1 numerator degrees of freedom and n 2 1 denominator degrees of freedom. If we are hypothesizing that s 2 1 ¼ s2 2 , we assume that the hypothesis is true, and the two variances cancel out in the above expression leaving s 2 1 =s2 2 , which follows the same F distribution. The ratio s2 1 =s2 2 will be designated V.R. for variance ratio. For a two-sided test, we follow the convention of placing the larger sample variance in the numerator and obtaining the critical value of F for a=2 and the appropriate degrees of freedom. However, for a one-sided test, which of the two sample variances is to be placed in the numerator is predetermined by the statement of the null hypothesis. For example, for the null hypothesis that s 2 1 =s2 2 , the appropriate test statistic is V:R: ¼ s2 1 =s2 2 . The critical value of F is obtained for a (not a=2) and the appropriate degrees of freedom. In like manner, if the null hypothesis is that s 2 1 s2 2 , the appropriate test statistic is V:R: ¼ s2 2 =s2 1 . In all cases, the decision rule is to reject the null hypothesis if the computed V.R. is equal to or greater than the critical value of F. EXAMPLE 7.8.1 Borden et al. (A-30) compared meniscal repair techniques using cadaveric knee specimens. One of the variables of interest was the load at failure (in newtons) for knees fixed with the FasT-FIX technique (group 1) and the vertical suture method (group 2). Each technique was applied to six specimens. The standard deviation for the FasT-FIX method was 30.62, and the standard deviation for the vertical suture method was 11.37. Can we conclude that, in general, the variance of load at failure is higher for the FasT-FIX technique than the vertical suture method? Solution: 1. Data. See the statement of the example. 2. Assumptions. Each sample constitutes a simple random sample of a population of similar subjects. The samples are independent. We assume the loads at failure in both populations are approximately normally distributed.
7.8 HYPOTHESIS TESTING: THE RATIO OF TWO POPULATION VARIANCES 269 .05 0 5.05 F (5, 5) Nonrejection region Rejection region FIGURE 7.8.1 Rejection and nonrejection regions, Example 7.8.1. 3. Hypotheses. 4. Test statistic. H 0 : s 2 1 s2 2 H A : s 2 1 > s2 2 V:R: ¼ s2 1 s 2 2 (7.8.1) 5. Distribution of test statistic. When the null hypothesis is true, the test statistic is distributed as F with n 1 1 numerator and n 2 1 denominator degrees of freedom. 6. Decision rule. Let a ¼ :05. The critical value of F, from Appendix Table G, is 5.05. Note that if Table G does not contain an entry for the given numerator degrees of freedom, we use the column closest in value to the given numerator degrees of freedom. Reject H 0 if V:R: 5:05. The rejection and nonrejection regions are shown in Figure 7.8.1. 7. Calculation of test statistic. V:R: ¼ ð30:62Þ2 ð11:37Þ 2 ¼ 7:25 8. Statistical decision. We reject H 0 , since 7:25 > 5:05; that is, the computed ratio falls in the rejection region. 9. Conclusion. The failure load variability is higher when using the FasT- FIX method than the vertical suture method. 10. p value. Because the computed V.R. of 7.25 is greater than 5.05, the p value for this test is less than 0.05. Excel calculates this p value to be .0243. & Several computer programs can be used to test the equality of two variances. Outputs from these programs will differ depending on the test that is used. We saw in Figure 7.3.3,
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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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8.2 THE COMPLETELY RANDOMIZED DESIG
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Dependent Variable: Selenium Tukey
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EXERCISES 329 Elbow Angle (Degrees)
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EXERCISES 331 Age Groups (Years) Ag
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EXERCISES 333 Group 1 Group 2 Group
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Objective The objective in using th
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8.3 THE RANDOMIZED COMPLETE BLOCK D
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EXERCISES 343 The SAS System Analys
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EXERCISES 345 group were randomly a
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8.4 THE REPEATED MEASURES DESIGN 34
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EXERCISES 357 Is there sufficient e
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8.5 THE FACTORIAL EXPERIMENT 359 In
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8.5 THE FACTORIAL EXPERIMENT 361 In
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8.5 THE FACTORIAL EXPERIMENT 363 Be
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8.5 THE FACTORIAL EXPERIMENT 365 TA
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8.5 THE FACTORIAL EXPERIMENT 367 Ro
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EXERCISES 369 The SAS System Analys
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EXERCISES 371 Physical Psychiatric
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8.6 SUMMARY 373 Percent Change in C
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REFERENCES 409 Applications Referen
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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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EXERCISES 685 Dialog box: Session c
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13.5 THE MEDIAN TEST 687 TABLE 13.5
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EXERCISES 689 Dialog box: Session c
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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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EXERCISES 717 Subject Area Student
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13.10 THE SPEARMAN RANK CORRELATION
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EXERCISES 725 Rank by Rank by Area
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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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REFERENCES 777 REFERENCES Methodolo
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A-2 APPENDIX STATISTICAL TABLES
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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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Biostatistics

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