Understanding advanced statistical methods

Introduction: Probability, Statistics, and ScienceReality, Nature, Science, and ModelsStatistical Processes: Nature, Design and Measurement, and DataModelsDeterministic ModelsVariabilityParametersPurely Probabilistic Statistical ModelsStatistical Models with Both Deterministic and Probabilistic ComponentsStatistical InferenceGood and Bad ModelsUses of Probability ModelsRandom Variables and Their Probability DistributionsIntroductionTypes of Random Variables: Nominal, Ordinal, and ContinuousDiscrete Probability Distribution FunctionsContinuous Probability Distribution FunctionsSome Calculus-Derivatives and Least SquaresMore Calculus-Integrals and Cumulative Distribution FunctionsProbability Calculation and SimulationIntroductionAnalytic Calculations, Discrete and Continuous CasesSimulation-Based ApproximationGenerating Random NumbersIdentifying DistributionsIntroductionIdentifying Distributions from Theory AloneUsing Data: Estimating Distributions via the HistogramQuantiles: Theoretical and Data-Based EstimatesUsing Data: Comparing Distributions via the Quantile-Quantile PlotEffect of Randomness on Histograms and q-q PlotsConditional Distributions and IndependenceIntroductionConditional Discrete DistributionsEstimating Conditional Discrete DistributionsConditional Continuous DistributionsEstimating Conditional Continuous DistributionsIndependenceMarginal Distributions, Joint Distributions, Independence, and Bayes' TheoremIntroductionJoint and Marginal DistributionsEstimating and Visualizing Joint DistributionsConditional Distributions from Joint DistributionsJoint Distributions When Variables Are IndependentBayes' TheoremSampling from Populations and ProcessesIntroductionSampling from PopulationsCritique of the Population Interpretation of Probability ModelsThe Process Model versus the Population ModelIndependent and Identically Distributed Random Variables and Other Models Checking the iid AssumptionExpected Value and the Law of Large NumbersIntroductionDiscrete CaseContinuous CaseLaw of Large NumbersLaw of Large Numbers for the Bernoulli DistributionKeeping the Terminology Straight: Mean, Average, Sample Mean, Sample Average, and Expected ValueBootstrap Distribution and the Plug-In PrincipleFunctions of Random Variables: Their Distributions and Expected ValuesIntroductionDistributions of Functions: The Discrete CaseDistributions of Functions: The Continuous CaseExpected Values of Functions and the Law of the Unconscious StatisticianLinearity and Additivity PropertiesNonlinear Functions and Jensen's InequalityVarianceStandard Deviation, Mean Absolute Deviation, and Chebyshev's InequalityLinearity Property of VarianceSkewness and KurtosisDistributions of TotalsIntroductionAdditivity Property of VarianceCovariance and CorrelationCentral Limit TheoremEstimation: Unbiasedness, Consistency, and EfficiencyIntroductionBiased and Unbiased EstimatorsBias of the Plug-In Estimator of VarianceRemoving the Bias of the Plug-In Estimator of VarianceThe Joke Is on Us: The Standard Deviation Estimator Is Biased after AllConsistency of EstimatorsEfficiency of EstimatorsLikelihood Function and Maximum Likelihood EstimatesIntroductionLikelihood FunctionMaximum Likelihood EstimatesWald Standard ErrorBayesian StatisticsIntroduction: Play a Game with Hans!Prior Information and Posterior KnowledgeCase of the Unknown SurveyBayesian Statistics: The OverviewBayesian Analysis of the Bernoulli ParameterBayesian Analysis Using SimulationWhat Good Is Bayes?Frequentist Statistical MethodsIntroductionLarge-Sample Approximate Frequentist Confidence Interval for the Process MeanWhat Does Approximate Really Mean for an Interval Range?Comparing the Bayesian and Frequentist ParadigmsAre Your Results Explainable by Chance Alone?IntroductionWhat Does by Chance Alone Mean?The p-ValueThe Extremely Ugly "pv ≤ 0.05" Rule of ThumbChi-Squared, Student's t, and F-Distributions, with ApplicationsIntroductionLinearity and Additivity Properties of the Normal DistributionEffect of Using an Estimate of sChi-Squared DistributionFrequentist Confidence Interval for sStudent's t-DistributionComparing Two Independent Samples Using a Confidence IntervalComparing Two Independent Homoscedastic Normal Samples via Hypothesis TestingF-Distribution and ANOVA TestF-Distribution and Comparing Variances of Two Independent GroupsLikelihood Ratio TestsIntroductionLikelihood Ratio Method for Constructing Test StatisticsEvaluating the Statistical Significance of Likelihood Ratio Test StatisticsLikelihood Ratio Goodness-of-Fit TestsCross-Classification Frequency Tables and Tests of IndependenceComparing Non-Nested Models via the AIC StatisticSample Size and PowerIntroductionChoosing a Sample Size for a Prespecified Accuracy MarginPowerNoncentral DistributionsChoosing a Sample Size for Prespecified PowerPost Hoc Power: A Useless StatisticRobustness and Nonparametric MethodsIntroductionNonparametric Tests Based on the Rank TransformationRandomization TestsLevel and Power RobustnessBootstrap Percentile-t Confidence IntervalFinal WordsIndexVocabulary, Formula Summaries, and Exercises appear at the end of each chapter.