Executive Summary | 1. The Core Challenge: Ordinal Data & WLS Estimation | 2. Omnibus Test Statistics: Browne vs. Satorra–Bentler | Raw WLS Objective ($F_{\text{WLS}}$) | Browne (1984) Residual $\chi^2$ | Satorra–Bentler (1994, 2010) Scaled $\chi^2$ | 3. Why Standard WLS $\chi^2$ Fails for CFI / TLI / RMSEA (Savalei, 2021) | 4. The Solution: Savalei (2021) CatML Rescaling | RAM Matrix Rescaling ($A^* = D_s^{-1} A D_s$, $S^* = D_s^{-1} S D_s^{-1}$) | 5. Comparative Logic: umx (Savalei catML) vs. lavaan (WLSMV) | A. Understanding the $\chi^2$ Scaling Difference | B. The Impact of Mixed Categorical Levels (2-, 3-, and 4-Category Indicators) | 6. Practical Guide: Interpreting umxSummary & Writing Your Paper | A. Understanding $c_{\text{model}}$ and $c_{\text{null}}$ | B. Robust Standard Errors (uncertainty = "SE" vs "RobustSE") | C. Recommended Manuscript Text for Publications | 7. Summary Comparison Table | References

Purpose | Data: Psych_LDSC | Fit a common factor with umxGSEM | Decimal match: umxGSEM vs GenomicSEM commonfactor | Factor loadings | Residual variances | Fit indices (do not require decimal match) | Implementation notes (for maintainers) | SNP summary data for multivariate GWAS (pipeline roadmap) | How this maps to our LDSC objects | Toy (subset) SNPs — for smoke tests only | Full-fat SNPs — production | umx API for SNPs (implemented) | Session

Introduction | Data | One-factor model (worst fit → one MI call) | Three-factor alternative (no second MI) | Bifactor / multi-group (optional, local) | Summary | References

Introduction | The Holzinger and Swineford (1939) data | Build a one-factor model with umxRAM() | View results with umxSummary() | Standardized solution | Alternative: lavaan-style syntax | Path diagram as PNG | Ordinal and WLS models | Next steps | References

Introduction | 1. Point Estimates vs. Inference (SEs vs. CIs) | Standard Errors (Wald CIs) | Profile Likelihood Confidence Intervals | 2. ML Standard Errors vs. Robust Standard Errors | Standard ML SEs | Robust Standard Errors (Sandwich Estimator) | When to prefer Robust SEs: | Limitations: | 3. Practical Workflow in umx | Step 1: Fit a Base Model | Step 2: Request Robust Standard Errors | Compute robust standard errors | Compare SEs | Step 3: Profile likelihood CIs (two parameters) | 4. Summary: When to Prefer Which? | Recommendations for Reporting

Introduction | 1. Data: three ordinal visual items | 2. Estimators in umxRAM | Live demo: one-factor DWLS on three binary items | Nested comparison | Optional: full WLS (local only) | 3. Practical notes | Sample size | Convergence | Reporting | References