This project provides a systematic examination of Item Response Theory (IRT), a psychometric framework used to model the relationship between respondents' latent traits (e.g., ability, attitude) and their responses to test items. The study is structured into five key phases, ensuring a rigorous and progressive understanding of IRT: ⦿ 1. Foundational IRT Models ◈ Introduction to the Logistic Model and Normal Ogive Model ◈ Analysis of Item Characteristic Curves (ICCs) ⦿ 2. Parameter Estimation & Calibration ◈ Estimation of Item Parameters (difficulty, discrimination, guessing) ◈ Maximum Likelihood Estimation (MLE) for model fitting ◈ Item Information Function (IIF) and Test Information Function (TIF) ◈ Standard Error of Measurement (SEM) ⦿ 3. Latent Trait & Test Performance ◈ Conceptualization of the Latent Trait (Θ) ◈ Percentile Rank interpretation ◈ Test Response Function (TRF) ◈ Assumption of Conditional Independence ◈ Parameter Invariance across populations ⦿ 4. Model Validation & Testing ◈ Unidimensionality assessment ◈ Root Mean Square Error (RMSE) and Root Mean Squared Difference (RMSD) for model fit ◈ Binomial Tests for guessing and carelessness detection ◈ Differential Item Functioning (DIF) Analysis using Logistic Regression and Mantel-Haenszel Tests ⦿ 5. Practical Applications & Robustness ◈ Ensuring fairness and accuracy in educational and psychological assessments This structured approach ensures a deep and coherent understanding of IRT, from its theoretical foundations to practical validation, comparison, and fairness testing. Each phase builds logically on prior knowledge, culminating in a robust framework for advanced psychometric analysis.

This work formalizes a framework for handling missing data, progressing from mechanism identification (via Little's MCAR test) to optimized treatment strategies, including Multiple Imputation for uncertainty quantification and EM algorithms for multivariate missingness. The structured approach balances theoretical rigor with practical implementation across data patterns. ⦿ 1. Problem Identification ◈ The Impact of Missing Data  ▣ Biased parameter estimates  ▣ Reduced statistical power  ▣ Compromised generalizability ◈ Limitations of Traditional Methods  ▣ Listwise deletion: Inefficient and often invalid  ▣ Ad hoc fixes: May introduce new biases ⦿ 2. Theoretical Foundations ◈ Missing Data Patterns  ▣ Monotone vs. arbitrary patterns ◈ Missing Data Mechanisms  ▣ MCAR (Missing Completely at Random)  ▣ MAR (Missing at Random)  ▣ MNAR (Missing Not at Random) ◈ Diagnostic Tools  ▣ Little's MCAR Test (formal hypothesis testing) ⦿ 3. Basic Solutions ◈ Single Imputation Techniques  ▣ Mean/median imputation (with caveats)  ▣ Random hot-deck imputation ◈ Preliminary Analysis  ▣ Bootstrapping for robust parameter estimation ⦿ 4. Advanced Methods ◈ Multiple Imputation  ▣ Creates multiply-imputed datasets  ▣ Accounts for imputation uncertainty ◈ Model-Based Approaches  ▣ Maximum Likelihood Estimation (MLE)  ▣ Expectation-Maximization (EM) Algorithm   ◆ Bivariate EM applications   ◆ Extension to multivariate data

This project follows a systematic, step-by-step approach to data analysis, ensuring rigor, reproducibility, and transparency. The progression is divided into two core phases: ▣ 1. Data Preparation & Screening ⦿ Data Cleaning & Screening ⦿ Missing Data Handling ⦿ Missing Completely at Random (MCAR) Test ⦿ Outlier Detection & Treatment  ◈ Winsorizing (capping extreme values)  ◈ Mahalanobis Distance (multivariate outliers)  ◈ Cook’s D (influence in regression) ⦿ Likert Scale Data Validation  ◈ Cronbach’s Alpha (reliability testing) ▣ 2. Statistical Analysis ⦿ Descriptive Statistics (summarizing data) ⦿ Correlation & Regression  ◈ Pearson Product-Moment Correlation  ◈ Bivariate Linear Regression ⦿ Chi-Square Tests  ◈ Goodness-of-fit tests  ◈ Test of independence (cross-tabulation) ⦿ T-Tests & Non-Parametric Alternatives  ◈ One-sample t-test  ◈ Paired t-test  ◈ Independent t-test  ◈ Mann-Whitney U test (non-parametric alternative)  ◈ Cohen’s d (effect size) ⦿ ANOVA & Post-Hoc Tests  ◈ One-way ANOVA  ◈ Bartlett’s & Levene’s tests (homogeneity of variance)  ◈ Tukey’s HSD (post-hoc comparisons)  ◈ Kruskal-Wallis test (non-parametric alternative) ⦿ Homoscedasticity Checks (variance stability) ⦿ Effect Size Calculations in R (e.g., Cohen’s d, η²) This structured approach ensures rigorous, replicable, and well-documented statistical analysis.

This project follows a systematic progression through the development, analysis, and validation of psychological and educational assessments, with a particular emphasis on measuring math anxiety and evaluating scale reliability and validity. The workflow is organized into two major phases: ⦿ Phase 1: Questionnaire Development & Validation ▣ Scale Design ◈ Likert scale optimization (5 vs 7-point scales) ◈ Item pool generation with expert review ◈ Cognitive pretesting of items ▣ Reliability Assessment ◈ Cronbach's α (internal consistency) ◈ Test-retest reliability (temporal stability) ◈ Inter-rater reliability (observational measures) ▣ Factor Structure Analysis ◈ Exploratory Factor Analysis (EFA) ◈ Confirmatory Factor Analysis (CFA) ◈ Model fit evaluation (CFI, RMSEA) ▣ Validity Evidence ◈ Construct validity (HTMT ratios) ◈ Content validity index (CVI) ◈ Known-groups validation This phase ensures the psychometric robustness of the instrument before further statistical analysis. ⦿ Phase 2: Classical Test Theory Analysis ▣ Item Analysis ◈ Difficulty indices (p-values) ◈ Discrimination parameters ◈ Distractor analysis ▣ Reliability Metrics ◈ KR-20 (binary items) ◈ Stratified alpha (multidimensional scales) ◈ Conditional reliability ▣ Score Interpretation ◈ Standard Error of Measurement (SEM) ◈ True score confidence bands ◈ Precision across ability levels This phase ensures that the scale is statistically sound and accurately measures the intended construct. This structured approach supports rigorous scale development, validation, and analysis, with broad applicability in psychology, education, and the social sciences.

This project systematically builds from foundational matrix operations to advanced statistical modeling techniques, emphasizing the role of linear algebra in statistical computations. The progression is structured as follows: ◈ 1. Core Matrix Operations ◇ Deviation Scores Matrix ◇ Sums of Squares and Cross Products (SSCP) ◇ Corrected SSCP Matrix (CSSCP) ◇ Matrix Inversion Techniques ▹ LU decomposition ▹ Cholesky decomposition ◇ Rank Determination ◈ 2. Advanced Decompositions ◇ SWEEP Operator Applications ◇ Cholesky Factorization ▹ Forward substitution ▹ Backward substitution ◇ Eigenvalue Decomposition ◈ 3. Regression Modeling ◇ Hat Matrix (Projection Matrix) ▹ Leverage diagnostics ▹ Cook's distance ◇ Residual Analysis ▹ Covariance structure ▹ Heteroskedasticity checks ◈ 4. ANOVA Framework ◇ Variance Partitioning ▹ Between-group (Model SS) ▹ Within-group (Error SS) ◇ F-Statistic Derivation ▹ Expected mean squares ▹ F-distribution properties ◇ Effect Size Metrics ▹ R² interpretation ▹ Adjusted R² computation

This project systematically explores numerical methods for solving mathematical problems computationally, progressing from foundational concepts to advanced techniques. Here’s the structured narrative of its development: ● 1. Root-Finding Methods ◇ Bracketing Approaches ▹ Bisection Method ▹ Regula Falsi (False Position) ◇ Open Methods ▹ Newton-Raphson ▹ Secant Method ▹ Fixed-Point Iteration ◇ Convergence Analysis ▹ Linear vs Quadratic Convergence ▹ Error Bounds ● 2. Numerical Integration ◇ Basic Quadrature ▹ Riemann Sums ▹ Trapezoidal Rule ◇ Advanced Techniques ▹ Romberg Integration ▹ Simpson's Rule (1/3, 3/8) ◇ Adaptive Methods ▹ Error Estimation ● 3. Linear Systems Solvers ◇ Elimination Methods ▹ Naive Gaussian Elimination ▹ Scaled Partial Pivoting ◇ Matrix Decompositions ▹ LU Factorization ▹ Cholesky Method ● 4. Computational Optimization ◇ Stability Analysis ◇ Floating-Point Considerations ◇ Algorithmic Complexity Each stage builds on prior knowledge, moving from scalar equations to multidimensional systems. The project equips learners with tools to handle real-world problems where analytical solutions are impractical, emphasizing algorithmic thinking, error analysis, and computational efficiency. This structured approach ensures a deep, practical understanding of numerical computing.

This project begins with foundational concepts in Statistical Computing, integrating practical applications with analytical progression to build a robust theoretical and computational foundation: ▣ 1. Descriptive Statistics ⦿ Central Tendency  ◈ Mean, Median, Mode ⦿ Variability Measures  ◈ Variance, MAD, IQR  ◈ Skewness & Kurtosis ⦿ Dependence Structures  ◈ Pearson/Spearman/Kendall Correlations Establishes comprehensive dataset characterization before advanced analysis. ▣ 2. Hypothesis Testing ⦿ Core Concepts  ◈ p-values & Significance Levels  ◈ Type I/II Errors ⦿ Power Analysis  ◈ Statistical Power  ◈ Confidence Intervals ⦿ Bayesian Methods  ◈ Prior/Posterior Distributions Systematically bridges descriptive statistics with inferential testing. ▣ 3. Probability & Sampling ⦿ Distributions  ◈ Continuous (Normal, t, χ²)  ◈ Discrete (Bernoulli, Poisson) ⦿ Sampling Techniques  ◈ Inverse Transform  ◈ Rejection Sampling ▣ 4. Computational Methods ⦿ Monte Carlo  ◈ Classical Integration  ◈ Importance Sampling ⦿ MCMC  ◈ Metropolis-Hastings  ◈ Gibbs Sampling ▣ 5. Parameter Estimation ⦿ Maximum Likelihood (MLE) ⦿ Expectation-Maximization (EM)  ◈ Gaussian Mixtures  ◈ Multivariate Normals