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### Stat 415a: Structural equation Modeling

### Outline

### Part 1: Introduction

### Part 1.3: Historic overview

### Part 1.4:Data set used in course

### Part 2: Classical path analysis

### Part 2.2: Model estimation

### Part 2.3: Model evaluation:

### Part 2.3: Model comparison

### Part 2.3, cont: Chi-square goodness-of-fit

### Part 2.3, cont.: The chi-square in SEM

### Part 2.4, cont.: Two types of model comparison

### Part 2.3, cont.: Some absolute fit indices

### Part 2.4: Decomposition of effects

### Part 2.4, cont. : Decomposition of effects

### Part 3: Non-recursive models

### Part 3, cont.: What is “identification?”

### Part 3, cont., Insuring identification

### Part 4, cont.: CFA model comparison

### Part 5: Integrated model

### Part 5, cont.: Integrated model

Fall 2013

Instructor: Fred Lorenz

Department of Statistics

Department of Psychology

Part 1: Introduction

Part 2: Classical path analysis

Part 3: Non-recursive models for reciprocity

Part 4: Measurement models

Part 5: Model integration

Part 6: Concluding comments

The language of structural equations

Motivation: Why structural equation modeling?

Historic overview of SEM

Introduction to the data sets used in class examples and homework assignments

Part 1.1: The language of SEM

- Path diagrams:
- Denote concepts (ellipse) and their measures (box)
- Distinguish uni-directed & bi-directed arrows
- Notation & subscripts
- Distinguish
- explanatory (independent) & response (dependent) variables
- endogenous & exogenous variables

Part 1.2: Motivation

Why structural equation modeling?

* Explanation rather than prediction

* Conceptual simplification (parsimony)

* Reliability (random measurement error)

* Validity (systematic measurement error)

Part 1.2, cont. . . .

- Explanation
- We have causal interests. . . . Does X cause Y?
- We want to know “why?”
- Is there a 3rd variable that mediates between X and Y?
- Or is there a 3rd antecedent variable that explains both?

Part 1.2, cont. . .

- Conceptual simplification

* Ockum’s razor: all things equal, we prefer the simplest (most parsimonious) explanation

* And we avoid the “bulkanization” of concepts

* We distinguish “latent” variable from their observed manifestations

* And we distinguish causative (formative) from effects (reflexive) variables

Part 1.2, cont.

Reliability (random measurement error)

- We do not measure our concepts without error .
- Measurement error “attenuates” (reduces) the strength of correlations, and the strength of regression coefficients.
- Goal : work harder so that you measure your concepts real good!
- Or: obtain auxiliary information to estimate measurement error.

Part 1.2, cont. . .

Validity (systematic measurement error)

- Our observed indicators do not always measure what they are intended to measure.
- Maybe less than they should; sometime more.
- Often validity is threatened by “method variance,” one example of which is “glop.”
- Sometimes the systematic error associated with method variance can be estimated.

Wright traced genetic inheritability

Path analysis (recursive models) in Sociology (Duncan 1966).

Non-recursive (simultaneous equation) models in economics

Measurement models from psychometrics (Lawley 1940)

Integrated recursive & non-recursive measurement error models (Joreskog, Keesing & Wiley notation).

The Iowa Family Transition Project (FTP)

550 rural Iowa families (1989 – 2009)

Interview: mother, father, “target child”

And later, target’s romantic partner

Some themes. . .

Model specification

Model estimation with PROC REG and PROC CALIS

Model evaluation & model comparisons

The decomposition of effects

Part 2.1: Model specification

- Theoretical specification
- What is the order of variables in a model
- And how do you decide?
- On logic
- Or empirical evidence?
- Specifying (operationalizing) the model
- How do you express a theoretical model in paths and equations?

Ordinary least squares regression (use PROC REG is SAS)

Maximum likelihood estimation, or variations on it (PROC CALIS)

Standardized vsunstandardized coefficients

Evaluate specific paths of a model

Is a proposed path significant?

Is an hypothesized null path really not significant?

Is there evidence of mediation? Spuriousness?

Evaluate the overall model

How well does a model fit the data?

The two extremes:

saturated (fully recursive) model

null model of complete independence

Where does your model fit?

Implicit model comparison and the chi-square test

Explicit model comparisons and the change in chi- square.

What does the chi-square statistics test do when comparing model?

Compare the expected distribution under the null hypothesis to the observed distribution

The greater the difference between expected and observed distributions, the larger the chi- square

The statistic:

T = (N – 1)F

F = min(obs – expected)

df = p*(p+1)/2 - t

t = # parameters being estimated

T ~ 2 (df)

Absolute fit: Compare model to the “saturated” model, which fits the data perfectly

Relative fit: Compare model to the model of complete independence (like a model with no predictors in OLS regression

Goodness of fit (GFI) and adjusted goodness of fit (AGFI) indices

Standardized root mean residual (SRMR)

Root mean square error of approximation (RMSEA)

Goal: Get the values as small (close to zero) as possible

Part 2.3, cont.: Some relative fit indices

- Tucker-Lewis non-normed fit index (TLI; 2)
- Normed fit index (NFI; 1)
- Relative fit index (RFI; 1)
- Incremental fit index (IFI; 2 )

Part 2.3, cont. : Some more relative fit indices

- Comparative fit index (CFI)
- Goal: Get the values as close to 1.0 as possible
- Akaike Information Criterion (AIC)

The correlation between two variables can be decomposed into 4 parts:

* direct effects

* indirect effects

* spurious effects

* associational effects

The total effect of one variable on another can be decomposed into 2 parts:

Total effect = Direct effect + Indirect effect

Direct effect

Indirect effect

To calculate, use EFFPART in PROC CALIS

Reciprocity & causal order in survey data. . .

Model specification: writing the equations

Model identification

Interpreting model results: reciprocity and causal order

Model comparison and evaluation

Identification in simple algebra:

* two unknowns at least two equations

* k unknowns at least k equations

In SEM: the necessary condition

df = (p)(p+1)/2 – t

t = number of parameters estimated

p = number of observed variables.

Bollens (1989) rules for identification for non-recursive models

Line up the equations

The necessary order condition

The sufficient rank condition

Some observations on identification

Part 4: Measurement models

Measurement error vs. mistakes in measuring

* A note on classical test theory

Random vs systematic measurement error

* The concept of method variance

* Managing method variance

Confirmatory vs. exploratory factor analysis

* A note on notation

* Write down the (restricted) equations

Confirmatory factor analysis (CFA)

* Model specification & identification

* Model estimation & evaluation

Part 4, cont., CFA identification

Rules for identification

* The usual (necessary) t-rule

* The three indicator rule

* The two indicator rule

Comments on identification

Nested vs. non-nested models

Evaluating specific models using the chi-square

Comparing nested models using the change in chi- square: indices and graphic displays

What do integrated models look like?

Integrating the measurement & structural components

Specifying (writing) the equations

Model identification

Is the measurement model identified?

Is the structural portion of the model identified?

Model estimation using PROC CALIS

Standardized vs non-standardized coefficients

Model evaluation

Significant and non-significant structural paths

Significant and non-significant correlated errors

Model comparison

Chi-square goodness of fit statistic for absolute fit

Relative fit indices for comparing nested models

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