BARLOW'S TABLES OF SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, RECIPROCALS

By V. MIRIA

CONTENTS 1 PROLOGUE: REGRESSION ANALYSIS AS PROBLEMATIC 2 A GROUNDED INTRODUCTION TO REGRESSION ANALYSIS SOME EXAMPLES OF REGRESSION ANALYSIS ABORTION AND SUBSEQUENT CRIME MANDATORY BASIC EDUCATION FOR WELFARE RECIPIENTS GENDER AND ACADEMIC SALARIES CLIMATE CHANGE AND WATER RESOURCES IN INDIA DEFORESTATION AND SOIL EROSION IN THE YANGTZE RIVER VALLEY EPIDEMICS OF HEPATITIS C ONWARD AND UPWARD WHAT IS REGRESSION ANALYSIS? A SIMPLE ILLUSTRATION CONTROLLING FOR A THIRD VARIABLE IMPOSING A SMOOTHER GETTING FROM DATA TO STORIES 3 SIMPLE LINEAR REGRESSION INTRODUCTION DESCRIBING A CONDITIONAL RELATIONSHIP WITH A STRAIGHT LINE DEFINING THE “BEST” LINE SOME USEFUL FORMULAS STANDARDIZED SLOPES USING TRANSFORMATIONS FOR A NONLINEAR FIT WHAT ABOUT THE VARIANCE FUNCTION? SUMMARY AND CONCLUSIONS 4 STATISTICAL INFERENCE FOR SIMPLE LINEAR REGRESSION THE ROLE OF SAMPLING RANDOM SAMPLING STRATEGY I: TREATING THE DATA AS POPULATION STRATEGY II: TREATING THE DATA AS IF THEY WERE GENERATED BY RANDOM SAMPLING FROM A POPULATION STRATEGY III: INVENTING AN IMAGINARY POPULATION STRATEGY IV: MODEL-BASED SAMPLING—INVENTING A FRIENDLY NATURAL PROCESS RESPONSIBLE FOR THE DATA A NOTE ON RANDOMIZATION INFERENCE SUMMING UP SIMPLE LINEAR REGRESSION UNDER RANDOM SAMPLING ESTIMATING THE POPULATION REGRESSION LINE ESTIMATING THE STANDARD ERRORS ESTIMATION UNDER MODEL-BASED SAMPLING SOME THINGS THAT CAN GO WRONG TESTS AND CONFIDENCE INTERVALS STATISTICAL POWER STOCHASTIC PREDICTORS MEASUREMENT ERROR CAN RESAMPLING TECHNIQUES HELP? PERCENTILE CONFIDENCE INTERVALS HYPOTHESIS TESTING BOOTSTRAPPING REGRESSION POSSIBLE BENEFITS FROM RESAMPLING 5 CAUSAL INFERENCE FOR THE SIMPLE LINEAR MODEL SOME DEFINITIONS: WHAT’S A CAUSAL EFFECT? THE NEYMAN-RUBIN MODEL THINKING ABOUT CAUSAL EFFECTS AS RESPONSE SCHEDULES WHAT’S AN INTERVENTION? STUDYING CAUSE AND EFFECT WITH DATA USING NONSTATISTICAL SOLUTIONS FOR MAKING CAUSAL INFERENCES USING STATISTICAL SOLUTIONS FOR MAKING CAUSAL INFERENCES USING THE SIMPLE LINEAR MODEL FOR MAKING CAUSAL INFERENCES 6 THE FORMALITIES OF MULTIPLE REGRESSION TERMS AND PREDICTORS SOME NOTATION FOR MULTIPLE REGRESSION ESTIMATION HOW MULTIPLE REGRESSION “HOLDS CONSTANT” 7 USING AND INTERPRETING MULTIPLE REGRESSION ANOTHER FORMAL PERSPECTIVE ON HOLDING CONSTANT WHEN DOES HOLDING CONSTANT MAKE SENSE? STANDARDIZED REGRESSION COEFFICIENTS: ONCE MORE WITH FEELING VARIANCES OF THE COEFFICIENT ESTIMATES 8 SOME POPULAR EXTENSIONS OF MULTIPLE REGRESSION MODEL SELECTION AND STEPWISE REGRESSION MODEL SELECTION BY REMOVING TERMS TESTS TO COMPARE MODELS SELECTING TERMS WITHOUT TESTING STEPWISE SELECTION METHODS SOME IMPLICATIONS USING CATEGORICAL TERMS: ANALYSIS OF VARIANCE AND ANALYSIS OF COVARIANCE AN EXTENDED EXAMPLE BACK TO THE VARIANCE FUNCTION: WEIGHTED LEAST SQUARES VISUALIZING LACK OF FIT WEIGHTED LEAST SQUARES AS A POSSIBLE FIX EVALUATING THE MEAN FUNCTION LOCALLY WEIGHTED REGRESSION SMOOTHER 9 SOME REGRESSION DIAGNOSTICS TRANSFORMATIONS OF THE RESPONSE VARIABLE BOX-COX PROCEDURES INVERSE FITTED VALUE RESPONSE PLOTS LEVERAGE AND INFLUENCE INFLUENTIAL CASES AND COOK’S DISTANCE CROSS-VALIDATION MISSPECIFICATION TESTS INSTRUMENTAL VARIABLES TESTS FOR EXOGENEITY CONCLUSIONS 10 FURTHER EXTENSIONS OF REGRESSION ANALYSIS 10 1 REGRESSION MODELS FOR LONGITUDINAL DATA 10 1 1 MULTIPLE LINEAR REGRESSION FOR TIME SERIES DATA 10 2 REGRESSION ANALYSIS WITH MULTIPLE TIME SERIES DATA 10 2 1 FIXED EFFECTS MODELS 10 2 2 RANDOM EFFECTS MODELS 10 2 3 ESTIMATION 10 3 MULTILEVEL MODELS 10 4 THE GENERALIZED LINEAR MODEL 10 4 1 GLM STRUCTURE 10 4 2 NORMAL MODELS 10 4 3 POISSON MODELS 10 4 4 POISSON MODELS FOR CONTINGENCY TABLES 10 4 5 BINOMIAL REGRESSION 10 5 MULTIPLE EQUATION MODELS 10 5 1 CAUSAL INFERENCE ONCE AGAIN 10 5 2 A FINAL OBSERVATION 10 6 META-ANALYSIS 10 7 CONCLUSIONS 11 WHAT TO DO 11 1 HOW DID WE GET INTO THIS MESS? 11 2 THREE CHEERS FOR DESCRIPTION 11 2 1 WHAT’S DESCRIPTION? 11 2 2 ADVOCACY SETTINGS 11 2 3 DESCRIPTIVE REGRESSIONS AS PART OF A BROAD RESEARCH PROGRAM 11 2 4 SPOTTING PROVOCATIVE ASSOCIATIONS 11 2 5 SOME OTHER BENEFITS OF DESCRIPTION 11 3 TWO CHEERS FOR STATISTICAL INFERENCE 11 3 1 WORKING WITH NEAR-RANDOM SAMPLES 11 3 2 WORKING WITH DATA FROM NATURE 11 3 3 WORKING WITH A NEARLY CORRECT MODEL 11 4 ONE CHEER FOR CAUSAL INFERENCE 11 4 1 SPECIAL-PURPOSE ESTIMATORS 11 4 2 PROPENSITY SCORES 11 4 3 SENSITIVITY ANALYSIS OF THE SELECTION PROCESS 11 4 4 BOUNDING TREATMENT EFFECTS 11 4 5 SOME FORECASTS 11 5 SOME FINAL OBSERVATIONS 11 5 1 A POLICE STORY 11 5 2 REGRESSION ANALYSIS AS TOO LITTLE, TOO LATE REFERENCES