Maximum likelihood estimation and hypothesis testing - applications overview 3502-KONW1

Introduction to statistical inferenceClassic methods of statistical inference (repetition)Basic conceptsRandom variables relevant in statistical inference: exponential, chi-square, normal, Poisson.Random experiment: random sampling. Sample space, sample statistics, sample statistics distribution.Sample mean and its distribution. Central limit theorem on sample mean distribution.Interval estimationThe problem of interval estimation. Confidence level. Confidence interval.Confidence interval for population mean and for population proportion. Limit theorem for binary variable distribution. Exact distribution for Bernoulli experiment statistics.Hypotheses testingNeyman-Pearson method of hypotheses testing. First and second type of error. Decision function, rejection area, acceptance area. Population mean test. Typical tests in survey data analysis. Nonparametric tests. Stochastic independence test with chi-square statistics.Maximum likelihood based inference - introductionBasic conceptsRandom experiment. Random events space. Probability function over random events space. Sufficient statistics. Likelihood function. Log-likelihood function and entropy.Simple stochastic models Coin throw - binomial distribution. Drawing balls from the box - multinomial distribution. Item response - probabilistic versions of the Guttman model of scaling. Mokken scale. Rasch scaling model. Social Network - Bernoulli model.Maximum likelihood estimation.Interval versus point estimation - comparison. Maximum likelihood derivation methods. Vector parameters of the model and their maximum likelihood estimators derivation. Binomial, multinomial and normal distribution parameters and their maximum likelihood estimators. Likelihood ratio tests Likelihood ratio statistics and the problem of its distribution. G-square statistics and its properties. G-square and Chi-square statistics - relationships. Population proportion test. Stochastic independence of categorical variables - likelihood ratio and entropy.ApplicationsLogistic regressionMaximum likelihood regression of the logistic regression parameters.Likelihood ratio test for logistic regression coefficient significance test.Social Network analysisDyadic model of the network - estimation and hypothesis testing: binomial distribution case. Likelihood ratio and structure crystallization coefficient. Entropy reduction and likelihood ratio in a Simple blockmodel parameter estimation and testing. Exponential network models: p-star model, stochastic block model - estimation and likelihood ratio hypotheses testing.Latent structure analysis and stochastic segmentation modelsLazarsfeld model and the problem of parameter's estimation. Segmentation methods based on the probabilistic models of the structure - LatentGold and SPSS algorithms. Categorical variable factor analysis (M. Styczeń): estimation and goodness of fit test. Probabilistic cumulative scalingProbabilistic Guttman scaling: Leo Goodman model, Mokken scale, Rasch model - latent variable estimator and the problem of sufficiency. Pseudolikelihood function and conditional maximum likelihood estimation - problem of vector parameters in Rasch scaling.Factor analysisDistribution assumptions in factor analysis. Maximum likelihood factor extraction method. Maximum likelihood based goodness of fit statistics in factor analysis.Structural equation models Distribution assumptions in structural equation models: case of AMOS and LISREL packages. Structural equation model parameters - maximum likelihood estimators. Hypothesis testing in structural modeling - G-square statistics.