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Stat 415a: Structural equation Modeling

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Stat 415a: Structural equation Modeling

Fall 2013

Instructor: Fred Lorenz

Department of Statistics

Department of Psychology

Outline

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

Part 1: Introduction

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

- 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

Why structural equation modeling?

*Explanation rather than prediction

*Conceptual simplification (parsimony)

*Reliability (random measurement error)

*Validity (systematic measurement error)

- 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?

- 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

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.

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.

Part 1.3: Historic overview

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).

Part 1.4:Data set used in course

The Iowa Family Transition Project (FTP)

550 rural Iowa families (1989 – 2009)

Interview: mother, father, “target child”

And later, target’s romantic partner

Part 2: Classical path analysis

Some themes. . .

Model specification

Model estimation with PROC REG and PROC CALIS

Model evaluation & model comparisons

The decomposition of effects

- 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?

Part 2.2: Model estimation

Ordinary least squares regression (use PROC REG is SAS)

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

Standardized vsunstandardized coefficients

Part 2.3: Model evaluation:

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?

Part 2.3: Model comparison

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.

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

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

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

The statistic:

T = (N – 1)F

F = min(obs – expected)

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

t = # parameters being estimated

T ~ 2 (df)

Part 2.4, cont.: Two types of model comparison

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

Part 2.3, cont.: Some absolute fit indices

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

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

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

Part 2.4: Decomposition of effects

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

* direct effects

*indirect effects

*spurious effects

*associational effects

Part 2.4, cont. : Decomposition of 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

Part 3: Non-recursive models

Reciprocity & causal order in survey data. . .

Model specification: writing the equations

Model identification

Interpreting model results: reciprocity and causal order

Model comparison and evaluation

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

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.

Part 3, cont., Insuring identification

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

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

Rules for identification

*The usual (necessary) t-rule

*The three indicator rule

*The two indicator rule

Comments on identification

Part 4, cont.: CFA model comparison

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

Part 5: Integrated model

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?

Part 5, cont.: Integrated model

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

Part 6: Concluding comments

Concluding comments

Complete homework projects