Week 12 S.E.M. Structural Equation Modelling

Question 1

What is S.E.M. (Structural Equation Modelling)?

Answer 1

S.E.M. is a combination of Regression and PCA (Principle Component Analysis)

Question 2

S.E.M. can be defined as…..

Answer 2

a family of statistical models that seek to explain the relationships among multiple variables.
*by examining the structure of interrelationships expressed in a series of equations, similar to a series of multiple regression equations

Question 3

What do the equations produced via S.E.M. depict?

Answer 3

The equations depict all of the relationships among constructs (DV’s & IV’s) involved in the analysis

Question 4

How does S.E.M. define a construct?

Answer 4

Constructs are unobservable (aka ‘latent’) factors that are represented by multiple variables
e.g. DASS variables represent the unobservable constructs of Depression, Anxiety, & Stress

Question 5

What are the 4 distinguishable features of SEM?

Answer 5

• it provides an estimation of multiple and interrelated relationships
• it represents unobservable (AKA latent) concepts & corrects for measurement error
• It produces a model to explain an entire set of relationships
• it defines error terms for each parameter

Question 6

What exactly is SEM?

Answer 6

• SEM is a stats package the presents models detailing structure and measurement
• It is included in AMOS, LISREL, SAS & MPlus
• SEM allows testing of a theoretical combination of variables that can represent a cause and effect relationship

Question 7

Barbara Byrne is a bit of a SEM expert, I heard she conceptualises SEM as Confirmatory Factor Analysis, tell me more?

Answer 7

• Byrne (2010) suggests SEM takes a confirmatory rather than exploratory approach to analyse the data for inferential aims.
• Other multivariate procedures can not assess or correct for measurement error. SEM provides estimates of error variance parameters at every stage of the model.
• SEM can incorporate both latent, observed variables, covariances and error terms.

Question 8

What does Byrne say about analysis that does not utilise SEM?

Answer 8

*Other methods ignore the errors in explanatory variables and therefore can lead to serious errors in interpretation. *Analyses using other methods are based on observed measurements only and don’t incorporate measurement and structure simultaneously.

Question 9

What does the ‘full latent variable model’ allow in terms of the regression structure among latent variables?

Answer 9

The full latent model allows specification of the regression structure among latent variables
*This allows the researcher to hypothesis the impact of one latent construct on another in the modelling of causal direction

Question 10

What does a full or complete model depict?

Answer 10

A full or complete model’s measurement depicts the links between the latent variables and their observed measures and the structural model depicts the links among the latent variables.

Question 11

What is a recursive model?

Answer 11

a recursive model is one in which I specify the direction of causality from only one direction.
i.e. single direction arrow

Question 12

What is a non-recursive model?

Answer 12

A model that allows reciprocal or feedback effects

i.e. double headed arrow

Question 13

What information does a researcher get from SEM to test the plausibility of the model?

Answer 13

• The plausibility of the model is based on sample data that comprises all observed variables in the model
• The goal is to determine the goodness of fit (GoF) between the hypothesised model & sample data to estimate the difference between the 2.

Question 14

What is the residual?

Answer 14

The residual is the difference between the observed and hypothesised model

Question 15

How can the SEM model be summarised?

Answer 15

Data = Model + Residual

Question 16

What are Exogenous Constructs?

Answer 16

*IV
Exogenous constructs are the latent, multi-item equivalent to independent variables.
*They use a variate (linear combination) of measures to represent the construct, which acts as an IV in the model.
*Multiple measured variables (x) represent the exogenous constructs.

Question 17

What are Exogenous variables synonymous with?

Answer 17

Exogenous variables are synonymous with IVs, may cause fluctuations in the values of other latent variables. Gender, age and SES are possible influences external to the model that cause change in the exogenous variables.

Question 18

What are Endogenous Constructs?

Answer 18

*DV
Endogenous constructs are the latent, multi-item equivalent to DVs.
(enDogenous = D for DV)
*These constructs are theoretically determined by factors within the model.
*Multiple measured variables (y) represent the endogenous constructs.

Question 19

What are Endogenous Constructs synonymous with?

Answer 19

Endogenous latent variables are synonymous with DVs, which are influenced by exogenous variables and explained by the model because all latent variables that influence them are included.

Question 20

What is the difference between Exploratory Factor Analysis (EFA) and Confirmatory Factor Analysis (CFA)?

Answer 20

• In EFA the extent to which a group of items measure a given construct is evaluated through factor loadings.
• In contrast CFA is driven by knowledge of the underlying latent construct structure, which tests the structure of the hypothesised model and the strength of the regression paths.
• The CFA model focuses on the link between factors and their measured variables, within the framework of SEM, it represents what has been termed the measurement model.

Question 21

What are the 6 stages of SEM?

Answer 21

Stage 1: Defining Individual Constructs
Stage 2: Developing the Overall Measurement Model
Stage 3: Designing a Study that’s Empirically driven
Stage 4: Assessing the Measurement Model Validity
Stage 5: Specifying the Structural Model
Stage 6: Assessing Structural Model Validity

Question 22

How can SEM Models be represented?

Answer 22

SEM Models can be represented visually in a path diagram.

• Dependence relationships are represented with single headed directional arrows.
• Correlational (covariance) relationships are represented with two-headed arrows.

Question 23

When is SEM not allowed?

Answer 23

When there is no underlying theory

*The theory is required for the measurement model specification and the Structural model specification

Question 24

What conditions need to be present for SEM to be possible?

Answer 24

There must be a genuine (i.e. non-spurious), unique covariance relationship between cause & effect variables which does not include other constructs
There must be theoretical support for the cause & effect relationship

Question 25

What other conditions are necessary for using SEM?

Answer 25

Models developed with a developmental strategy should be cross-validated with an independent sample.

Question 26

Other than using a Path Diagram to represent SEM, what other way can SEM appear?

Answer 26

Structural equations can be schematically presented or through a series of equations.

Question 27

What function does the measurement model perform?

Answer 27

The measurement model defines relations between the observed and unobserved variables.
*It provides the link between scores on a measuring instrument and underlying constructs, which they are designed to measure.

Question 28

What function does the structural model perform?

Answer 28

The structural model defines relations among the unobserved variables.

• It specifies the manner by which particular latent variables directly or indirectly influence changes in the values of certain other latent variables in the model.
• CFA models can be analysed using AMOS with first and second order hierarchical models.

Question 29

What do loadings or Lambda weights represent?

Answer 29

Loadings or Lambda Weights represent the relationships from constructs to variables as in factor analysis

Question 30

What do path estimates represent?

Answer 30

Path estimates represent the relationships between constructs as does β in regression analysis

Question 31

Which 3 types of relationships can a Measurement Model represent?

Answer 31

A. a relationship between a Construct and a Measured Variable
B. a Relationship between a Construct and Multiple Measured Variables
D. Correlational Relationships between Constructs

Question 32

Which type of relationships can a Measurement Model not represent?

Answer 32

C. a Dependence relationship between two Constructs (A Structural Relationship)

Question 33

Which type of relationship can the Structural Model include?

Answer 33

*all of them:
A. a relationship between a Construct and a Measured Variable
B. a Relationship between a Construct and Multiple Measured Variables
C. a Dependence relationship between two Constructs (A Structural Relationship)
D. Correlational Relationships between Constructs

Question 34

In SEM what is the Model?

Answer 34

the hypothesized structure linking the observed variables to the latent variable

Question 35

In SEM what is the Residual?

Answer 35

The Residual represents the discrepancy between the hypothesized model & observed data

Question 36

What 4 geometrical shapes schematically represent structural models?

Answer 36

• A circle or ellipse represents observed latent variables.
• A rectangle represents directly observed variables.
• Single-headed arrows rep the impact of one variable on another.
• Double-headed arrows representing covariances or correlations between pairs of variables

Question 37

How is a Path coefficient for regression of an observed variable onto an unobserved latent variable (or factor) represented?

Answer 37

by a rectangle joined by a single arrow to an ellipse

i.e. rectangle, single arrow, ellipse

Question 38

How is a Path coefficient for regression of one factor onto another factor represented?

Answer 38

by an ellipse, single arrow, joining second ellipse

Question 39

How is the Residual error in the prediction of an unobserved factor represented?

Answer 39

by a circle

Question 40

Structural Models can be just-identified, Over-identified, or Under-Identified. Which is ideal?

Answer 40

Over-identified is the ideal model: an Over-identified model is one where the number of estimable parameters < the number of data points. Allows for rejection of the model & results in a positive number of degrees of freedom – positive # of dfs.

Question 41

What’s the problem with a just-identified model?

Answer 41

Just-identified – one to one correspondence between the data & structural parameters. No df left so cannot be rejected!!

Question 42

What is the problem with an under-identified model?

Answer 42

Under-identified – one where the number of parameters to be estimated exceeds the number variances and covariances to be identified. Therefore, the model contains insufficient information for the purpose of attaining a determinate solution of parameter estimation – negative # of dfs.

Question 43

How do I work out whether I have sufficient degrees of freedom (df’s) for my model?

Answer 43

This is based on the # of Parameter Estimates:
With a model with 4 factors and 3 items loading on each (4 X 3 = 12), we can ascertain whether the model is identified or not.
How many parameters can you identify? Formula: p(p+1)/2, 12(12+1)/2=78

Question 44

There are 4 requirements for causal inference, what are they?

Answer 44

1. Covariation – Statistically significant estimated paths
2. Sequence – Change in one variable occurs in one direction.
3. Non-spurious Covariance (i.e. genuine covariance) – the size and nature of the relationship between a cause and effect inference should not include other constructs. Should not be false or misleading.
4. Theoretical Support – Requires a compelling rationale to support the cause & effect relationship.

Question 45

The first stage of SEM involves Defining Individual Constructs. What does this involve?

Answer 45

• Operationalizing the Constructs
• Scales from Prior Research
• New Scale Development
• Pretesting

Question 46

The second stage of SEM involves Developing the Overall Measurement Model. What do I need to consider to address this?

Answer 46

• Can the validity and uni-dimensionality of the constructs be supported?
• How many indicators for each construct?
• Is the measurement model reflective or formative? (asking what causes what?)

Question 47

Part of the second stage involves determining whether the Measurement model is Reflective or Formative. What is important to remember?

Answer 47

Kate told us to learn that the Reflective Model Theory and the Formative Measurements Theory are opposites and to learn this for the exam :-)

Question 48

Tell me about The Reflective Model Theory

Answer 48

Reflective Model Theory: based on assumptions that

(1) the latent constructs cause measured variables.
(2) The measurement error results in an inability to fully explain these measures.
* It is the typical representation of a latent construct.

Question 49

Tell me about The Formative Measurements Theory

Answer 49

Formative Measurement Theory: based on assumptions that

(1) the measured variables cause the construct.
(2) The error in measurement is an inability to fully explain the construct.
* The construct is not latent in this case.

Question 50

In the second stage of SEM once I have determined whether the measurement is reflective or formative, what else do I need to do when Developing the Overall Measurement Model?

Answer 50

• Make constructs from measured variables.

* Draw a path diagram for the measurement model

Question 51

What are the 3 questions that arise when specifying a measurement model?

Answer 51

1. Can we empirically support the validity and uni-dimensionality of the constructs?
2. How many indicators should we use and what’s the trade off for increasing or decreasing the number of indicators?
3. Is it a descriptive or explanatory model being depicted? Interpretations will depend on answering this issue, exploratory or confirmatory?

Question 52

Stage 3 involves Designing a Study to Produce Empirical Results – how do I check assumptions?

Answer 52

• Assess the adequacy of the sample size.

* Select the estimation method and missing data approach.

Question 53

What do I do if I find I have missing data?

Answer 53

Missing Data Options:

• Complete Case or List-Wise Deletion.
• All-Available or Pair-Wise Deletion.
• Model-Based Deletion.

Question 54

What are the 6 issues related to Stage 3 (Designing a Study to Produce Empirical Results)?

Answer 54

Research Design:

1. Type of data analysed: covariances or correlations.
2. Missing data.
3. Sample size.

Model Estimation:

4. Model structure.
5. Estimation techniques.
6. Computer software used.

Question 55

What are the 5 sample size issues in SEM?

Answer 55

1. multivariate distribution of the data
2. estimation technique
3. model complexity
4. amount of missing data
5. amount of average error variance among the reflective indicators.

Question 56

During stages 1-3 why might we use a pre-test or pilot study?

Answer 56

A pretest using respondents similar to those from the population to be studied is recommended to screen items for appropriateness is useful when a model has scales borrowed from various sources reporting other research

Question 57

What is the recommended method of handling missing data during stages 1-3?

Answer 57

• Pair-wise deletion of missing cases (all available approach) is a good alternative for handling missing data when the amount of missing data is less than 10% & the sample size is approx 250+
• As sample sizes become small or when missing data exceeds 10%, one of the replacement (imputation) methods for missing data becomes a good alternative.
• When the amount of missing data becomes very high (15% or more), SEM may not be appropriate.

Question 58

Determining the minimum sample size for a particular SEM model depends on several factors including model complexity and communalities (average variance extracted among items) in each factor. If a SEM models containing 5 or fewer constructs, each with more than 3 items (observed variables), & with high Item communalities (.6 or more) what is considered an adequate sample size?

Answer 58

*SEM models containing 5 or fewer constructs, each with more than 3 items (observed variables), & with high Item communalities (.6 or more), can be adequately estimated with samples as low as 100-150.

Question 59

Determining the minimum sample size for a particular SEM model depends on several factors including model complexity and communalities (average variance extracted among items) in each factor. If a SEM models containing When the number of factors is larger than six, some of which have fewer than 3 measured items as indicators, & multiple low communalities are present, what is considered an adequate sample size?

Answer 59

*When the number of factors is larger than six, some of which have fewer than 3 measured items as indicators, & multiple low communalities are present, sample size requirements may be > 500.

Question 60

Determining the minimum sample size for a particular SEM model depends on several factors including model complexity and communalities (average variance extracted among items) in each factor. What is a general rule of thumb for SEM sample sizes?

Answer 60

• The sample size must be sufficient to allow the model to run, but more important, it must adequately represent the population.
• as with all studies, the greater the complexity of the model, the larger the sample size required

Question 61

What is the biggest limitation in SEM?

Answer 61

Mahalanobis Distance:

Measure the influence of a case by measuring the distance of cases from the mean of the predictor variable

Question 62

Stage 4 involves Assessing Measurement Model Validity, how do we do this?

Answer 62

We assess measurement of the model validity using Goodness of Fit (GoF) which evaluates the construct validity of the measurement model
*The number of parameters +1 & divided by 2 tells us how many ‘lines’ we can have in our matrix

Question 63

There are 3 types of Goodness of Fit, name them?

Answer 63

• Absolute Fit Measures.
• Incremental Fit Measures.
• Parsimonious Fit Measures.

Question 64

SEM has no single statistical test that best describes the “strength” of the model. Instead, researchers have developed 3 different types of measures that in combination assess results from 3 perspectives: overall fit, comparative fit & model parsimony.
Tell me about Absolute Fit Measures.

Answer 64

Absolute (overall) = measures overall goodness-of-fit for both the structural and measurement models collectively. This type of measure does not make any comparison to a specified null model (incremental fit measure) or adjust for the number of parameters in the estimated model (parsimonious fit measure).

Question 65

SEM has no single statistical test that best describes the “strength” of the model. Instead, researchers have developed 3 different types of measures that in combination assess results from 3 perspectives: overall fit, comparative fit & model parsimony.
Tell me about Incremental Fit Measures.

Answer 65

Incremental (comparative) = measures goodness-of-fit that compares the current model to a specified “null” (independence) model to determine degree of improvement over the null model.

Question 66

SEM has no single statistical test that best describes the “strength” of the model. Instead, researchers have developed 3 different types of measures that in combination assess results from 3 perspectives: overall fit, comparative fit & model parsimony.
Tell me about Parsimonious Fit Measures.

Answer 66

Parsimonious = measures goodness-of-fit representing the degree of model fit per estimated coefficient. This measure attempts to correct for any “over-fitting” of the model & evaluates the parsimony of the model compared to the GoF. Selecting a rigid cut-off for fit indices is like selecting a minimum R2 for a regression equation. Almost any value can be challenged. Awareness of the factors affecting the values & good judgement are the best guides to evaluate the size of the GoF indices.

Question 67

Stage five involves specifying the structural model - how do we do this?

Answer 67

• Stage five involves specifying the structural model by assigning relationships from one construct to another based on the proposed theoretical model. That is, the dependence relationships that exist among the constructs representing each of the hypotheses are specified.
• The end result is to convert the measurement model to a structural model

Question 68

As models become more complex, the likelihood of alternative models with equivalent fit increases. To address this, multiple fit indices should be used to assess a model’s goodness of fit. What are the names?

Answer 68

• The χ2 value and the associated df
• One absolute fit index (like the GFI, RMSEA or SRMR)
• One incremental fit index (like the CFI or TLI)
• One goodness of fit index (GFI, CFI, TLI, . . . )
• One badness of fit index (RMSEA, SRMR, . . . )

Question 69

There are several requirements for Goodness of Fit indices. What is the RMSEA? and how is it different to the other Goodness of Fit indices?

Answer 69

RMSEA = Root mean square error of approximation <.05
RMSEA is only GoF index that needs to be less than .05, all others need to be between 0-1 with the closer to 0.95 the better

Question 70

There are several requirements for Goodness of Fit indices. What is χ2 ?

Answer 70

Chi-square – Likelihood Ratio Test statistic. Appears as CMIN/DF. Large χ2 relative to df indicate a model that needs modification.

Question 71

There are several requirements for Goodness of Fit indices. What is CMIN?

Answer 71

CMIN – Minimum discrepancy – represents the discrepancy between the unrestricted sample covariance matrix S, and the restricted covariance matrix. Representing the Likelihood Ration Test statistic.

Question 72

There are several requirements for Goodness of Fit indices that need to be between 0-1 with the closer to 0.95 the better. What are they?

Answer 72

CFI – Comparative Fit Indices – 0-1, with .95 indicating good fit
GFI – Goodness of Fit Indices - Index – 0-1, with .95 indicating good fit. Closer to 1 the better
AGFI – Adjusted Goodness of Fit (adjusts for the # of df in the model)
TLI – Tucker-Lewis Index – 0-1, with .95 indicating good fit
AIC – Comparing two competing models.

Question 73

Stage 6 involves assessing the structural model validity & predictive accuracy, how do we do this?

Answer 73

No single “magic” value for the fit indices that separates good from poor models. It’s not practical to apply a single set of cut-off rules that apply for all measurement models.
The quality of fit depends heavily on model characteristics, including sample size and complexity

Question 74

Stage 6 involves assessing the structural model validity & predictive accuracy, how do we do this with simple models?

Answer 74

Simple models with small samples should be held to very strict fit standards. Even an insignificant p-value for a very simple model may not be very meaningful.

Question 75

Stage 6 involves assessing the structural model validity & predictive accuracy, how do we do this with complex models?

Answer 75

More complex models with larger samples should not be held to the same strict standards. Thus, when samples are large and the model contains a large number of measured variables and parameter estimates, cut-off values of .95 on key GoF measures may be unrealistic.

Question 76

How do we assess Normality in SEM?

Answer 76

Go to Analysis Summary - Assessment of Normality and check the Multivariate Normality, according to Bentler (2005, cited Byrne, 2010, p. 104) this figure should be <.5

Question 77

How do we assess for multivariate outliers?

Answer 77

An outlying case will have D2 (Mahalanobis squared) value that stands distinctively apart from all the other D2 values.

Question 78

What are Standardised Residuals and what is the cut off score for Standardised residuals in SEM?

Answer 78

• Any difference between the hypothesized model & sample covariance is captured in the Standardized residuals
• the cut off score for standard residuals in SEM is 2.58

Question 79

What can you tell me about Modification indices (MI)?

Answer 79

• The M.I. value indicates how much the overall χ2 would drop by at least the indicated amount in the table and the Par Change you will find the value the estimated parameter will assume.
• Large MI’s suggest cross loadings. We can modify the model by applying covariances to items that appear to cross-load. However, this must have a conceptual or theoretical basis for application. Look at including covariances to assess whether the model’s estimation is improved.
• Can check the value in M.I. and Par Change.

Question 80

How does “Model Maximum Likelihood (ML)” treat true and estimated values of the model parameters?

Answer 80

*Model maximum likelihood (ML) estimates of the true values of the model parameters are considered to be fixed but unknown, whereas their estimates are considered to be random but known.

Question 81

How does the Bayesian Estimation differ in its treatment of true and estimated model parameter values?

Answer 81

*In contrast, from the Bayesian perspective, true model parameters are unknown and therefore considered to be random as Bayesian estimation considers any unknown quantity as a random variable and therefore assigns it a probability distribution