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356 CHAPTER 17. RELATIONS AMONG VARIABLES: ADVANCED The point is that (17.40) looks like our no-interaction ANOVA models, e.g. (??). On the other hand, if we assume instead that Education is independent of IDE and Language but that IDE and Language are not independent of each other, our model would be log(pijk) = P X (1) = i and X (2) = j · P X (3) = k (17.43) = ai + bj + dij + ck (17.44) Here we have written P X (1) = i and X (2) = j as a sum of “main effects” ai and bj, and “interaction effects,” dij, analogous to ANOVA. Another possible model would have IDE and Language conditionally independent, given Education, meaning that at any level of education, a programmer’s preference to use IDE or not, and his choice of programming language, are not related. We’d write the model this way: log(pijk) = P X (1) = i and X (2) = j · P X (3) = k (17.45) = ai + bj + fik + hjk + ck (17.46) Note carefully that the type of independence in (17.46) has a quite different interpretation than that in (17.44). The full model, with no independence assumptions at all, would have three two-way interaction terms, as well as a three-way interaction term. 17.4.4.4 Parameter Estimation Remember, whenever we have parametric models, the statistician’s “Swiss army knife” is maximum likelihood estimation. That is what is most often used in the case of log-linear models. How, then, do we compute the likelihood of our data, the Nijk? It’s actually quite straightforward, because the Nijk have a multinomial distribution. Then L = n! Πi,j,kNijk! pNijk ijk (17.47) We then write the pijk in terms of our model parameters. Take for example (17.44), where we write pijk = e ai+bj+dij+ck (17.48)
17.5. SIMPSON’S (NON-)PARADOX 357 We then substitute (17.48) in (17.47), and maximize the latter with respect to the ai, bj, dij and ck, subject to constraints such as (17.42). The maximization may be messy. But certain cases have been worked out in closed form, and in any case today one would typically do the computation by computer. In R, for example, there is the loglin() function for this purpose. 17.4.4.5 The Goal: Parsimony Again Again, we’d like “the simplest model possible, but not simpler.” This means a model with as much independence between factors as possible, subject to the model being accurate. Classical log-linear model procedures do model selection by hypothesis testing, testing whether various interaction terms are 0. The tests often parallel ANOVA testing, with chi-square distributions arising instead of F-distributions. 17.5 Simpson’s (Non-)Paradox Suppose each individual in a population either possesses or does not possess traits A, B and C, and that we wish to predict trait A. Let Ā, ¯ B and ¯ C denote the situations in which the individual does not possess the given trait. Simpson’s Paradox then describes a situation in which and yet P (A|B) > P (A| ¯ B) (17.49) P (A|B, C) of trait B seems to have a positive predictive power for A by itself, but when in addition trait C is held constant, the relation between B and A turns negative. An example is given by Fabris and Freitas, 7 concerning a classic study of tuberculosis mortality in 1910. Here the attribute A is mortality, B is city (Richmond, with ¯ B being New York), and C is race (African-American, with ¯ C being Caucasian). In probability terms, the data show that (these of course are sample estimates) 7 C.C. Fabris and A.A. Freitas. Discovering Surprising Patterns by Detecting Occurrences of Simpson’s Paradox. In Research and Development in Intelligent Systems XVI (Proc. ES99, The 19th SGES Int. Conf. on Knowledge-Based Systems and Applied Artificial Intelligence), 148-160. Springer-Verlag, 1999
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From Algorithms to Z-Scores: Probab
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Contents 1 Time Waste Versus Empowe
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CONTENTS iii 3.7 A Combinatorial Ex
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CONTENTS v 5.5.1.3 Example: Modelin
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CONTENTS vii 7.3.1 Properties of Me
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CONTENTS ix 10.1 Sampling Distribut
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CONTENTS xi 11.9.4 What to Do Inste
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CONTENTS xiii 15.2 Example Applicat
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CONTENTS xv 17.2.3 Logistic Regress
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CONTENTS xvii 19.2 Simulation of Ra
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CONTENTS xix 21.4 Loss Models . . .
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Preface Why is this book different
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Chapter 1 Time Waste Versus Empower
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Chapter 2 Basic Probability Models
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2.2. THE CRUCIAL NOTION OF A REPEAT
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2.3. OUR DEFINITIONS 7 2009, cannot
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2.4. “MAILING TUBES” 9 but in m
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2.5. BASIC PROBABILITY COMPUTATIONS
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2.6. BAYES’ RULE 13 Note by the w
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2.8. SOLUTION STRATEGIES 15 2.8 Sol
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2.10. EXAMPLE: A SIMPLE BOARD GAME
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2.11. EXAMPLE: BUS RIDERSHIP 19 Aga
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2.12. SIMULATION 21 1 # roll d dice
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2.12. SIMULATION 23 So, in evaluati
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2.12. SIMULATION 25 3 count
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2.13. COMBINATORICS-BASED PROBABILI
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Chapter 3 Discrete Random Variables
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3.4. EXPECTED VALUE 37 3.4.1.1 What
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3.4. EXPECTED VALUE 39 So It turns
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3.4. EXPECTED VALUE 41 • For rand
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3.4. EXPECTED VALUE 43 of two rando
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3.5. VARIANCE 45 ance of U is defin
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3.5. VARIANCE 47 for any constant d
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3.7. A COMBINATORIAL EXAMPLE 49 You
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3.8. A USEFUL FACT 51 Note carefull
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3.10. EXPECTED VALUE, ETC. IN THE A
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3.11. DISTRIBUTIONS 55 3.11.1 Examp
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3.12. PARAMETERIC FAMILIES OF PMFS
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3.13. RECOGNIZING SOME PARAMETRIC D
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3.13. RECOGNIZING SOME PARAMETRIC D
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3.15. A CAUTIONARY TALE 73 T has a
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3.16. WHY NOT JUST DO ALL ANALYSIS
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3.18. RECONCILIATION OF MATH AND IN
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Chapter 4 Introduction to Discrete
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4.3. EXAMPLE: 3-HEADS-IN-A-ROW GAME
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4.4. EXAMPLE: ALOHA 87 The quantity
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4.6. AN INVENTORY MODEL 89 4.6 An I
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Chapter 5 Continuous Probability Mo
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5.3. BUT EQUATION (??) PRESENTS A P
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5.3. BUT EQUATION (??) PRESENTS A P
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5.4. DENSITY FUNCTIONS 97 2(0.1)fX(
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5.4. DENSITY FUNCTIONS 99 5.4.2 Pro
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5.5. FAMOUS PARAMETRIC FAMILIES OF
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5.8. “HYBRID” CONTINUOUS/DISCRE
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5.8. “HYBRID” CONTINUOUS/DISCRE
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Chapter 6 Stop and Review: Probabil
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• famous parametric families of d
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Chapter 7 Covariance and Random Vec
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7.1. MEASURING CO-VARIATION OF RAND
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7.2. SETS OF INDEPENDENT RANDOM VAR
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7.2. SETS OF INDEPENDENT RANDOM VAR
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7.3. MATRIX FORMULATIONS 139 this l
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7.3. MATRIX FORMULATIONS 141 consis
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7.3. MATRIX FORMULATIONS 143 import
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7.3. MATRIX FORMULATIONS 145 since
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7.3. MATRIX FORMULATIONS 147 minimi
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Chapter 8 Multivariate PMFs and Den
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8.1. MULTIVARIATE PROBABILITY MASS
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8.2. MULTIVARIATE DENSITIES 153 So,
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8.2. MULTIVARIATE DENSITIES 155 we
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8.3. MORE ON SETS OF INDEPENDENT RA
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8.4. EXAMPLE: FINDING THE DISTRIBUT
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8.5. PARAMETRIC FAMILIES OF MULTIVA
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Chapter 9 Introduction to Continuou
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9.1. MEMORYLESS PROPERTY OF EXPONEN
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9.3. HOLDING-TIME DISTRIBUTION 187
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Chapter 10 Introduction to Confiden
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10.1. SAMPLING DISTRIBUTIONS 195 Wh
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10.1. SAMPLING DISTRIBUTIONS 197 Ap
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10.3. CONFIDENCE INTERVALS FOR MEAN
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10.4. MEANING OF CONFIDENCE INTERVA
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10.5. GENERAL FORMATION OF CONFIDEN
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10.7. CONFIDENCE INTERVALS FOR PROP
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10.7. CONFIDENCE INTERVALS FOR PROP
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10.8. CONFIDENCE INTERVALS FOR DIFF
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10.10. R COMPUTATION 215 algebra, w
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10.13. OTHER CONFIDENCE LEVELS 217
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10.14. ONE MORE TIME: WHY DO WE USE
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Chapter 11 Introduction to Signific
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11.2. GENERAL TESTING BASED ON NORM
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11.5. ONE-SIDED HA 225 By checking
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11.6. EXACT TESTS 227 It is natural
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11.8. THE POWER OF A TEST 229 11.8
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11.9. WHAT’S WRONG WITH SIGNIFICA
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Chapter 12 General Statistical Esti
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12.1. GENERAL METHODS OF PARAMETRIC
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12.2. BIAS AND VARIANCE 247 people,
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12.2. BIAS AND VARIANCE 249 Moreove
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12.3. MORE ON THE ISSUE OF INDEPEND
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12.4. NONPARAMETRIC DISTRIBUTION ES
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12.5. BAYESIAN METHODS 259 his plan
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12.5. BAYESIAN METHODS 261 it now b
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12.5. BAYESIAN METHODS 263 number
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12.5. BAYESIAN METHODS 265 (b) ˆp,
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Chapter 13 Simultaneous Inference M
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13.2. SCHEFFE’S METHOD 269 You ca
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13.4. OTHER METHODS FOR SIMULTANEOU
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Chapter 14 Introduction to Model Bu
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14.1. “DESPERATE FOR DATA” 275
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14.1. “DESPERATE FOR DATA” 277
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14.2. ASSESSING “GOODNESS OF FIT
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14.4. ROBUSTNESS 281 bin width, or
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14.5. REAL POPULATIONS AND CONCEPTU
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Chapter 15 Relations Among Variable
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15.3. ADJUSTING FOR COVARIATES 287
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15.6. ESTIMATING THAT RELATIONSHIP
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15.6. ESTIMATING THAT RELATIONSHIP
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15.7. EXAMPLE: BASEBALL DATA 293
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15.9. EXAMPLE: BASEBALL DATA (CONT
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15.11. PREDICTION 297 matter, it ma
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15.12. PARAMETRIC ESTIMATION OF LIN
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15.12. PARAMETRIC ESTIMATION OF LIN
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15.14. DUMMY VARIABLES 303 15.14 Du
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Page 407 and 408: 19.3. MIXTURE MODELS 383 Now that w
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Page 413 and 414: 19.4. TRANSFORM METHODS 389 transfo
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20.1. DISCRETE-TIME MARKOV CHAINS 4
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20.2. SIMULATION OF MARKOV CHAINS 4
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20.4. CONTINUOUS-TIME MARKOV CHAINS
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20.5. HITTING TIMES ETC. 419 least
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20.5. HITTING TIMES ETC. 421 First
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20.5. HITTING TIMES ETC. 423 valued
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20.5. HITTING TIMES ETC. 425 summon
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Chapter 21 Introduction to Queuing
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21.2. M/M/1 429 busy for approximat
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21.2. M/M/1 431 • Due to the memo
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21.3. MULTI-SERVER MODELS 433 Recal
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21.4. LOSS MODELS 435 1 = i,j,k π
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21.5. NONEXPONENTIAL SERVICE TIMES
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21.6. REVERSED MARKOV CHAINS 439 So
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21.6. REVERSED MARKOV CHAINS 441 21
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21.6. REVERSED MARKOV CHAINS 443 Re
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21.7. NETWORKS OF QUEUES 445 • Gi
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21.7. NETWORKS OF QUEUES 447 Let Li
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Appendix A Review of Matrix Algebra
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A.2. MATRIX TRANSPOSE 451 • Matri
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A.6. EIGENVALUES AND EIGENVECTORS 4
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Appendix B R Quick Start Here we pr
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B.3. FIRST SAMPLE PROGRAMMING SESSI
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B.4. SECOND SAMPLE PROGRAMMING SESS
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B.4. SECOND SAMPLE PROGRAMMING SESS
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B.6. COMPLEX NUMBERS 463 B.6 Comple
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