2008ÊÊÊ季季季 PPP母母母數數數統統統計計計講講講LLL ù福Ê 中山大.T用數.系 [email protected] PRACTICAL NONPARAMETRIC STATISTICS Third Edition (1999) W. J. Conover 2008-02-18 ∼ 2008-06-15 This page intentionally left blank CONTENTS i CONTENTS Preface iii 課程+紹 ........................................ iii ®¼ .......................................... iv b用網頁 ........................................ iv I Lecture Notes 1 1 Probability Theory 2 1.1 Counting ..................................... 2 1.2 Probability ................................... 5 1.3 Random variables ................................ 8 1.4 Some properties of random variables ..................... 12 1.5 Continuous random variables ......................... 19 1.6 Summary .................................... 24 2 Statistical Inference 32 2.1 Populations, samples, and statistics ...................... 32 2.2 Estimation .................................... 35 2.3 Hypothesis testing ............................... 42 2.4 Some properties of hypothesis tests ...................... 46 2.5 Some comments on nonparametric statistics ................. 50 2.6 Summary .................................... 53 3 Some Tests Based on the Binomial Distribution 57 3.1 The binomial test and estimation of p ..................... 58 3.2 The quantile test and estimation of xp .................... 63 3.3 Tolerance limits ................................. 69 3.4 The sign test .................................. 72 3.5 Some variations of the sign test ........................ 77 3.6 Summary .................................... 83 4 Contingency Tables 84 4.1 The 2 £ 2 contingency table .......................... 85 4.2 The r £ c contingency table .......................... 94 4.3 The median test ................................. 103 4.4 Measures of dependence ............................ 107 4.5 The chi-squared goodness-of-¯t test ...................... 112 4.6 Cochran's test for related observations .................... 117 CONTENTS ii 4.7 Some comments on alternative methods of analysis ............. 120 4.8 Summary .................................... 121 5 Some Methods Based on Ranks 126 5.1 Two independent samples ........................... 127 5.2 Several independent samples .......................... 135 5.3 A test for equal variances ............................ 140 5.4 Measures of rank correlation .......................... 145 5.5 Nonparametric linear regression methods ................... 154 5.6 Methods for monotonic regression ....................... 162 5.7 The one-sample or matched-pairs case .................... 166 5.8 Several related samples ............................. 175 5.9 The balanced incomplete block design ..................... 186 5.10 Tests with A.R.E. of 1 or more ........................ 191 5.11 Fisher's method of randomization ....................... 192 5.12 Summary .................................... 192 II Appendices 200 A Preface 201 B Introduction 203 C PPP母母母數數數統統統計計計報報報告告告 205 C.1 報告¶®的注意¯4 ............................... 205 C.2 報告P» ..................................... 207 D What is Statistics? 210 D.1 Introduction ................................... 210 D.2 History ...................................... 210 D.3 Statistical methods ............................... 211 D.4 Tabulation and presentation of data ...................... 212 D.5 Measures of central tendency .......................... 213 D.6 Measures of variability ............................. 213 D.7 Correlation ................................... 214 D.8 Mathematical models .............................. 214 D.9 Tests of reliability ................................ 215 D.10 Higher statistics ................................. 215 D.11 Di®erence between statistics and probability ................. 216 Index 217 CONTENTS iii PREFACE Contents 課課課程程程+++紹紹紹 .................................... iii ®®®¼¼¼ ...................................... iv bbb用用用網網網頁頁頁 .................................... iv PPP母母母數數數統統統計計計課課課程程程+++紹紹紹（（（2008ÊÊÊ季季季））） 上課` 週一10:10∼12:00、週ë11:10∼12:00 上課2F §4009-1 D¡` 週四、" 10:00∼12:00 0課>/ ù福Ê >0 §.o 3002-4 Tel: (O) (07) 525-2000 ext 3823 Email: [email protected] 講0]P 課Ä講0 >C課Í Conover, W. J. (1999). Practical Nonparametric Statistics, 3rd edition. http://as.wiley.com/WileyCDA/WileyTitle/ productCd-0471160687.html ISBN: 0471160687 Publisher: Wiley 中山大.ÆZh局 Tel: (07) 525-0930 Wº評量 ®¼(Na2一g): 25%; 二次考試：60%; 報告: 15% Ï一次考試(Ï一a∼Ïëa) 30% Ï二次考試(Ï四a∼Ï"a) 30% 8體 R, Splus, SPSS, SAS, Statistica,. >./容 Chap. 1: Probability Theory Chap. 2: Statistical Inference Chap. 3: Some Tests Based on the Binomial Distribution Chap. 4: Contingency Tables Chap. 5: Some Tests Based on Ranks CONTENTS iv >.'n {體：é\、單槍7Å^ 8體：Yap ®®®¼¼¼ ÏÏÏ一一一aaa E1.10, P1.1, E2.12, E2.16, E3.6, P3.1, E4.10, E4.12, E5.4, E5.10 ÏÏÏ二二二aaa E1.4, E1.6, P1.2, E2.2, E2.6, E2.8, E3.2, E3.6, E4.4, E4.6 ÏÏÏëëëaaa E1.2, E1.8, E2.2, E2.6, E3.4, E3.10, E4.2, E4.4, E5.4, E5.6 ÏÏÏ四四四aaa E1.5, E1.8, E2.4, E2.8, E3.2, E3.6, E4.4, E5.2, E5.6, E6.2 ÏÏÏ"""aaa E1.1, E1.5, E2.2, E2.6, E3.1, E3.3, E4.2, E4.4, E5.1, E5.2 哥林9Gh10:23 ñ¯K可行，¬不KbÇ處；ñ¯K可行，¬不K建C人。 1 Corinthians 10:23 All things are lawful, but not all things are pro¯table; all things are lawful, but not all things build up. bbb用用用網網網頁頁頁 1. Statistical Computing (UCLA Academic Technology Services) http://www.ats.ucla.edu/stat/ 1 Part I Lecture Notes CHAPTER 1. PROBABILITY THEORY 2 Chapter 1 PROBABILITY THEORY Contents 1.1 Counting ................................ 2 1.2 Probability .............................. 5 1.3 Random variables .......................... 8 1.4 Some properties of random variables ............... 12 1.5 Continuous random variables ................... 19 1.6 Summary ............................... 24 Preliminary remarks Nonparametric statistical methods: Not necessary to be an expert in probability theory to understand the theory behind the methods. With a few easily learned, elementary concepts, the basic fundamentals underlying most nonparametric statistical methods become quite accessible. Recommended procedure for study: Read the text, pencil through the examples, work the exercises and problems. 1.1 Counting Process of computing probabilities often depends on being able to count. Some sophisti- cated methods of counting are developed to handle those complicated situations. I Toss a coin once: H or T I Toss a coin twice: HH; HT; T H or TT I Toss a coin n times: 2n possible outcomes Experiment: A process of following a well-de¯ned set of rules, where the result of following those rules is not known prior to the experiment. Model: I The value of coin tossing is that it serves as a prototype for many di®erent models in many di®erent situations. 1.1. COUNTING 3 I Good models: Tossing coins, rolling dice, drawing chips from a jar, placing balls into boxes. I They serve as useful and simple prototypes of many more complicated models arising from experimentation in diverse areas. I Excellent study of the diversity of models above is given by Feller (1968). Event: Possible outcomes of an experiment. Rule 1.1.1 If an experiment consists of n trials where each trial may result in one of k possible outcomes, there are kn possible outcomes of the entire experiment. Example 1.1.1 Suppose an experiment is composed of seven trials, where each trial con- sist of throwing a ball into one of the three boxes. I First throw: 3 di®erent outcomes. I First two throws: 32 = 9 outcomes. I Seven throws: 37 = 2187 di®erent outcomes. ¤ Rule 1.1.2 (Permutation) There are n! ways of arranging n distinguishable objects into a row. Example 1.1.2 Consider the number of ways of arranging the letters A; B and C in a row. I First letter can be any of the three letters. I Second letter can be chosen two di®erent ways once the ¯rst letter is selected. I The remaining letter becomes the ¯nal letter selected. I Total: (3)(2)(1) = 6 di®erent arrangements: ABC; ACB; BAC; BCA; CAB; CBA: ¤ Example 1.1.3 Suppose that in a horse race there are eight horses. If you correctly predict which horse will win the race and which horse will come in second and wager to that e®ect, you are said to \win the exacta". I Win the exacta: Need to purchase (8)(7) = 56 betting tickets. I Outcomes of all eight positions: 8! = 40320 di®erent ways. ¤ Rule 1.1.3 (Multinomial coe±cient) If a group of n objects is composed of n1 objects of type 1, n2 identical objects of type 2; : : : ; nr; identical objects of type r, the number of distinguishable arrangements into a row, denoted by µ ¶ n n! = : n1; : : : ; nr n1! : : : nr! ¡ ¢ n n! In particular, k = k!(n¡k)! if n1 = k and n2 = n ¡ k. CHAPTER 1. PROBABILITY THEORY 4 Example 1.1.4 (In example 2) Suppose A and B are identical. We will denote them by the letter X, then I Original 3! = 6 arrangements. ¡3¢ I Reduce to 2 = 3 distinguishable arrangements, XXC; XCX and CXX. ¤ Example 1.1.5 In a coin tossing experiment where a coin is tossed ¯ve times, the result is two heads and three tails. I The number of di®erent sequences of two heads and three tails equals the number of distinguishable arrangements of two objects of one kind and three objects of another, ¡5¢ which is 2 = 10. HHTTT;THHTT;TTHHT;HTHTT;THTHT; TTHTH;HTTHT;THTTH;TTTHH;HTTTH: ¡n¢ I How many di®erent groups of k objects may be formed from n objects? k ¤ Example 1.1.6 Consider again the three letters A; B and C. The number of ways of ¡3¢ selecting two of these letters is 2 = 3, that is, AB; BC and BC. I To see how this relates to the previous discussion, we will \tag" two of the three letters with an asterisk (*) denoting the tag. A¤B¤C gives AB A¤BC¤ gives AC and AB¤C¤ gives BC ¤ Binomial coe±cient ¡n¢ I Binomial coe±cient: i I Binomial expansion: µ ¶ Xn n (x + y)n = xiyn¡i i i=0 ¡ ¢ I Multinomial coe±cient: n n1;:::;nr I Multinomial expansion: X µ ¶ n n n1 nr (x1 + ¢ ¢ ¢ + xr) = x1 ¢ ¢ ¢ xr n1; : : : ; nr n1+¢¢¢+nr=n I Evaluate (2 + 3)4 by binomial expansion: µ ¶ X4 4 (2 + 3)4 = 2i34¡i = 625 i i=0 1.2. PROBABILITY 5 1.2 Probability De¯nition 1.2.1 (Sample space) The sample space is the collection of all possible dif- ferent outcomes of an experiment. De¯nition 1.2.2 (Sample point) A point in the sample space is a possible outcome of an experiment.