class: title-slide, middle <div style = "position:fixed; visibility: hidden"> `$$\require{color}\definecolor{red}{rgb}{0.698039215686274, 0.133333333333333, 0.133333333333333}$$` `$$\require{color}\definecolor{green}{rgb}{0.125490196078431, 0.698039215686274, 0.666666666666667}$$` `$$\require{color}\definecolor{blue}{rgb}{0.274509803921569, 0.509803921568627, 0.705882352941177}$$` `$$\require{color}\definecolor{yellow}{rgb}{0.823529411764706, 0.411764705882353, 0.117647058823529}$$` `$$\require{color}\definecolor{purple}{rgb}{0.866666666666667, 0.627450980392157, 0.866666666666667}$$` </div> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ TeX: { Macros: { red: ["{\\color{red}{#1}}", 1], green: ["{\\color{green}{#1}}", 1], blue: ["{\\color{blue}{#1}}", 1], yellow: ["{\\color{yellow}{#1}}", 1], purple: ["{\\color{purple}{#1}}", 1] }, loader: {load: ['[tex]/color']}, tex: {packages: {'[+]': ['color']}} } }); </script> <style> .red {color: #B22222;} .green {color: #20B2AA;} .blue {color: #4682B4;} .yellow {color: #D2691E;} .purple {color: #DDA0DD;} </style> ### Statistical Modeling in Experimental Psychology # W09 Structural Equation Models ## CFA + Path Models as one framework #### Han Hao @ Tarleton State University --- ## Roadmap 1. SEM as a workflow for theory testing 2. Structural Regression (SR) modeling strategies: 1-step vs 2-step vs 4-step 3. Full-latent vs hybrid/partial SEM models 4. Key fitting choices: estimator, missing data, scaling/identification, correlated residuals, MI 5. Assumptions + limitations of (linear) LVMs 6. Short previews: IRT and network psychometrics --- ## SEM = two "layers" in one set **Measurement model (CFA layer)** - indicators `\(\rightarrow\)` latent constructs - loadings and unique variances define measurement quality **Structural model (SR layer)** - regressions (and sometimes covariates) among constructs - addresses the theoretical relations among constructs **.red[Structural claims are different from measurement problems.]** --- ## SEM research workflow A defensible SEM paper usually answers: 1. What is the construct theory and what is measured? 2. What is the structural theory among constructs? 3. Are measurement and structure identified and estimable? 4. How sensitive are conclusions to reasonable modeling choices? 5. What limitations remain (design, invariance, generalizability)? --- class: inverse, middle ## Structural Regression (SR) modeling strategies ### 1-step vs 2-step vs 4-step --- ## 1-step SR modeling - Fit measurement + structural relations in a single SEM run - OK when measurement is already well-established and stable **Why people do it** - Simple and fast; “one model object” - More likely to get away with measurement issues **Common failures** - Global misfit appears, **.red[but is it due to measurement, structure, or both]**? - Chasing **.red[only good global fit and purly relying on MIs]**, and accidentally change the meaning of constructs --- ## 2-step SR modeling Classic recommendation ([Anderson & Gerbing, 1988](https://arifkamarbafadal.wordpress.com/wp-content/uploads/2011/08/jurnal-anderson-and-gerbing-1988.pdf)): - **Step 1:** test and validate the .red[measurement model] - **Step 2:** test the .red[structural model] based on acceptable measurement **Why it helps** - Measurement issues are handled before structural claims - Structural comparisons are cleaner - We “earn” the right to safely interpret paths among latent variables) --- ## Step 1 Checklist Typical measurement considerations: - Theoretical factor structure - Factor loadings (size, sign, or even generalizability/stability) - Factor correlations (criterion validity) - Local dependence issue "addressed" by the model (e.g., correlating certain residuals) - Decide what is **fixed and confirmed** in a baseline measurement model, before going with structural estimates and constraints --- ## Step 2 Checklist Typical structural considerations: - Most measurement model properties should be consistent and stable - Same indicators, similar loadings (also same estimators and other settings for potential comparison) - Some latent correlations may become directional paths; others may be constrained to 0 (omitted in the structural model) - Include covariates thoughtfully (theory-driven rather than "why not") - Interpret **structural coefficients as construct-level relations** --- ## 4-step SR modeling #### Kline-style diagnostic workflow 1. Start with a broad, plausible measurement model (exploratory and/or less restrictive methods such as **EFA**) 2. Move to a clearer confirmatory measurement model (preferably with simple and clear structure, **CFA**) 3. Fit a more complete structural model (theory-relevant relations included in **SEM**) 4. Test specific hypotheses: e.g., compare alternative models as competing theories (**Model comparisons**) --- ## When to choose 2-step vs 4-step Choose **2-step** when: - Constructs and indicators are conventional - Limited time/resource and prefer fewer moving parts - Sample size considerations Choose **4-step** when: - Measurement is in some level of uncertainty (new scale, new population, etc.) - The structural claims are high-stakes or situation-dependent (e.g., clinical/selection decisions) --- class: inverse, middle ## Full-latent vs Partial SEM ### When your world is not perfect --- ## Full-latent SEM **All major constructs are latent, multi-indicator:** - Predictors and outcomes are latent variables so measurement errors are explicitly modeled for all - How many manifests?: **.red["Two might be fine, three is better, four is best, and anything more is gravy."]** ([Kenny, 1979](https://www.researchgate.net/publication/237077423_Correlation_and_Causality); p.179) **Pros:** Clearer construct interpretation and better account for measurement error, better psychometrics properties **Cons:** More indicators (tasks/measures) and larger samples are needed; more parameters means more model identification and convergence issues --- ## Partial SEM **Some constructs are latent; others are observed:** - Latent factors for core constructs - Directly observed covariates (age, SES, etc.) - Sometimes observed outcomes/predictors when only one measure exists **Pros:** Easier and simpler? **Cons:** - Observed variables carry un-estimated measurement error - Interpretation may implicitly shift to “construct level” instead of staying at "task level" --- ## Single-indicator latent variables If there is only one indicator for a core construct, **as a compromise**, you may: - Model it as a latent variable with unstandardized loading fixed to 1 - Manually estimate residual variance using reliability information To separate “true score” vs “error” to your best knowledge. **Risk:** - Need a defensible reliability estimate - Results become dependent to that assumption --- class: inverse, middle ## Key choices to justify in formal investigations ### Estimator, missingness, residuals, MI, etc. --- ## Estimator choice Common lavaan-friendly guidance: - **Continuous-ish, near normal**: ML - **Semi-normal continuous**: robust ML variants (e.g., MLR) - **Ordered categorical indicators**: WLSMV / DWLS family (specify the indictors using `ordered=` in "lavaan") > In your write-up, explicitly state estimator and why the choice. --- ## Missing data Default "lavaan" approach is listwise deletion (`missing = "listwise"`), unless you specify otherwise. **For ML-family estimators:** - Common alternative is case-wise full-information ML via `missing = "ML"` **For categorical/WLSMV estimator(s):** - Pairwise handling using `missing = "pairwise"` is possible, but other considerations will be needed (see documentations in "lavOptions") --- ## Factor correlations vs structural paths **Similar to the correlation vs. regression discussion in PSYC5316:** - Latent correlation: “X and Y co-vary” - Structural path: “X predicts Y given the model” Best practice workflow: - Measurement model: allow theoretically plausible factor correlations - structural model: directional paths that reflect **.red[theoretically meaningful interpretations]** > Turning correlations into arrows does not create causality (if anything, conditional "causality", maybe?) --- ## Correlated residuals **Correlated residuals can reflect:** - Shared wording/content overlap - Shared method (same format, same rater, same stimulus set) - Local dependence such as time proximity / carryover in tasks **legitimate, but require attention** - Keep them minimal and theory-defended - If many residual correlations accumulate, think of alternative model structures rather than adding "patches" --- ## Model comparison > A “competing theories” perspective: For confirmatory modeling, please avoid that “try all possible models and report the best one” approach Nested comparisons: - `\(\Delta\chi^2\)` logic (with robust corrections when needed) - constrained vs unconstrained theory variants Non-nested comparisons: - AIC/BIC (when appropriate for estimator) - out-of-sample validation (preferred when feasible) --- ## Common SEM problems - Model nonconvergence - Solutions with "numeric" issues (e.g., Heywood cases with negative variances) - Unstable estimates across small specification changes (e.g., from measurement to structural models) - ... **What to consider before panicking:** - "Wrong" model structure - Model complexity (remove weak indicators; simplify residual structure) - Estimator and scaling (both scaling for model estimation and scaling of variables) - Other data issues (outliers, extreme response distributions, etc.) --- class: inverse, middle ## Assumptions + limitations of linear LVMs ### IRT and networks as natural “next steps” --- ## Assumptions for SEM Common assumptions in common LVMs: - Common cause assumption - Linear relations (loadings and regressions) - Distributional assumptions (e.g., ML-family normality assumptions) - Local independence and independent observations (unless multilevel SEM or growth curve modeling) --- ## Formative models .center[  ] Image from [.purple[Baxter (2009)]](https://www.sciencedirect.com/science/article/abs/pii/S0148296308002816) --- ## Item response models (IRT) .pull-left[  ] .pull-right[  ] Image from my [.purple[Intro to IRT tutorial]](https://hanhao23.github.io/project/irttutorial/) --- ## Network psychometrics .center[  ] Image from [.purple[Borsboom et al. (2021)]](https://www.nature.com/articles/s43586-021-00055-w) --- ## Notes for class - Week 09 Demo: We will do something different - Lab 04: The final one for latent variable modeling (LVM) - Spring break next week & plans after spring break - Final project: Find data for your project - Cognitive modeling topics - Paper Critiques 3 & 4 --- class: inverse # End