Midterm Exam

Question 1

Factor analytic (measurement) model

Answer 1

• Is a specified model relating constructs (i.e., factors or latent variables) to measures consistent with the observed data?
• Measurement model relating factors to measures; no directional effects between factors (ex: EFA, CFA, bi-factor models, multitrait, multimethod (MTMM) models)

Question 2

Path analytic model

Answer 2

• Are the hypothesized directional relations between measured variables consistent with the observed data?
• Directional effects among measures; no latent variables are included

Question 3

Full structural model

Answer 3

• Are the hypothesized directional relations between constructs consistent with the observed data?
• Includes measurement model and structural effects between factors
• This is a combination of characteristics of path analytic models (directional relations between constructs) and factor analytic/measurement models (relations of constructs to measures).

Question 4

Model specification

Answer 4

Specification of models and hypotheses using words, model diagrams with appropriate symbols from the Bentler-Weeks notation system (e.g., F1, g1, V1, E1) and equations

Question 5

Advantage of SEM over regression/ANOVA models

Answer 5

SEM includes measurement error of observed vars in the model, whereas regression/ANOVA assumes observed vars are measured without error (unrealistic)

Question 6

Variance, covariance, correlation

Answer 6

Variance: statistical test of individual differences
Covariance: statistical test of how 2 vars covary
Correlation: standardized covariance between 2 vars (rXY= covXY/sXsY)

Question 7

What is the null hypothesis in SEM?

Answer 7

Null hypothesis suggests perfect fit of specified model to data. Therefore, we DO NOT want to reject the null hypothesis.
Fit is tested through chi-square
- p-value < .05 = We reject our null hypothesis of perfect fit.
- p-value > .05 = fail to reject the null hypothesis

Question 8

Compare/contrast

JKW/LISREL and Bentler-Weeks notation models

Answer 8

JKW/LISREL:

• characterized by 8 basic matrices containing model parameters symbolized using Greek letters
• comprises structural models and measurement models
• distinguishes between exogenous and endogenous latent vars

Bentler-Weeks:

• parameters found in only 3 matrices
• uses VFED system (Variable, Factor, Error, Disturbance) to name vars and residuals; expressed by equation for each DV and covariance matrix among IVs
• measure vars (Vs) and latent vars (Fs) are handled similarly

Question 9

Mathematical equations for DVs in a CFA model (containing 2 uncorrelated factors, 6 vars)

Answer 9

n = Y E (Greek symbols)

V1 = g1F1 + E1
V2 = g2F1 + E2
V3 = g3F1 + E3
V4 = g4F2 + E4
V5 = g5F2 + E5
V6 = g6F2 + E6 + 0F1 + 0E1 +0E2 + 0E3 + 0E4 + 0E5 (this last one is expanded form)

Question 10

n = Y E (Greek symbols)

Answer 10

n (eta) = matrix containing DVs (V’s)
Y (gamma) = matrix containing weight parameters (g’s, 1’s, and 0’s)
E (epsilon) = matrix containing IVs (F’s and E’s)

Question 11

What are the g’s in the Y(gamma) matrix?

Answer 11

g’s are the weights applied to the factors to produce measured vars

Question 12

What should be noted about the last columns of a weight matrix?

Answer 12

They are generally the weights applied to the errors to produce measured vars and produce an identity matrix (i.e., will have 1’s down the diagonal of the matrix with 0’s in the off-diagonal space)

Question 13

What Greek symbol represent the covariance matrix for IVs?

Answer 13

Phi (circle with line vertically in middle)

Question 14

What are parameters of a SEM?

Answer 14

• Variances of exogenous vars (i.e., F’s and E’s)
• Covariances of exogenous vars (i.e., F’s and E’s)
• Weights representing directional effects specified in the model (i.e., g’s, 1’s, 0’s)

Question 15

What is NOT considered a model parameter?

Answer 15

Variances and covariance of measured vars

Question 16

Define ULI and UVI and what they do

Answer 16

ULI and UVI are constraints that scale the factors (i.e., methods of setting metric of latent vars)
ULI constraint: metric of factor is set by fixing the first loading to 1
UVI constraint: metric of factor is set by fixing the variance of the factor to 1 (thus standardizing factor)

Question 17

Equation for model df

What does it consist of?

Answer 17

df = v(v + 1) / 2 - (# of free parameters)

v(v + 1) / 2 = variances and covariances

Question 18

What are estimated (free) parameters?

Answer 18

• factor variances
• factor covariances
• factor loadings
• error variances

Question 19

What are free parameters?

Answer 19

Parameters we wish to estimate and are NOT constrained

Question 20

Fixed parameters/constraints

What is the issue with too many constraints?

Answer 20

Parameters constrained to equal

• 0 (as in a path excluded from the model)
• 1 (as in a factor variance or factor loading used to set the metric)
• X value (do this comparing a model across groups)

Constraints produce some lack of fit

Question 21

Independent (exogenous) vars

Dependent (endogenous) vars

Answer 21

• Exogenous latent variables are independent variables
• Endogenous latent variables are dependent variables in that they are predicted by exogenous latent variables and/or other endogenous latent variables

Question 22

what are the model parameters in a CFA and in which matrices do we find them?

Answer 22

• Variances of exogenous vars (i.e., F’s and E’s)
• Covariances of exogenous vars (i.e., F’s and E’s)
• Weights representing directional effects specified in the model (i.e., g’s, 1’s, 0’s)
• Covariance matrices

Question 23

What is the equation for the t rule?

Answer 23

t < or = p (p + 1) / 2

t = freely estimated parameters
p = measured vars (v’s)
p (p + 1) / 2 = unique variances and covariances among measured vars

same components as df equation

t-rule is necessary but not sufficient identification condition

Question 24

What is identification?

Answer 24

Mathematically identifying a unique solution for the model parameters. Starts with scaling factors (ULI or UVI); check t-rule; then mathematically test for unique solution for parameters (2-indicator and 3-indicator rule).

Question 25

2-indicator rule (for CFA)

Answer 25

For a multifactor, standard factor analytic model, if the following conditions are met, they are sufficient to identify a model: (a) at least two variables are a function of each factor, (b) each variable is a function of one and only one factor, and (c) each factor has at least one non-zero covariance with another factor.

Question 26

3-indicator rule (for CFA)

Answer 26

Three-indicator Rule: For a multifactor, standard factor analytic model, if the following conditions are met, they are sufficient to identify a model: (a) at least three measured variables are a function of each factor, and (b) each measured variable is a function of one and only one factor.

Question 27

Identified vs. unidentified model

Answer 27

``` Identified = we can uniquely solve for all model parameters Unidentified = we cannot solve uniquely for all the model parameters, because multiple solutions exist. ```

Question 28

Empirical underindentification

Answer 28

A model is mathematically identified, but characteristics in the observed data prevent us from obtaining unique values for model parameters. Ex: What if V3 is not correlated with V1 and V2 in our data? The covariances between V3 and the other two variables are not useful in solving for model parameters. The model is empirically underidentified.

Question 29

Null β Rule

Answer 29

- Way to mathematically identify path analytic (or full structural) models - If all elements in the β matrix are zeros, the model is identified. The β matrix contains all zeros if no dependent variable is a function of any other dependent variable (e.g., a multiple regression model). The errors for these models may or may not be correlated.

Question 30

Recursive Rule

Answer 30

- Way to mathematically identify path analytic (or full structural) models - If a model is recursive, it is identified. A recursive model has no reciprocal effects (direct feedback), feedback loops (indirect feedback), or error covariances. A model is recursive if the upper-right triangle of the beta matrix contains all zero values for variables that have been ordered in the standard manner and the covariances among the errors are zero.

Question 31

Purpose of estimation

Answer 31

Calculate the best estimates for the model parameters such that values in the residual matrix are minimized

Question 32

Goal of Maximum likelihood estimation (F.ML)

Answer 32

Find parameter values that have the greatest probability of producing the sample data.

Question 33

Assumptions of ML estimation

Answer 33

- independent observations - large sample size - correctly specified model - multivariate normal data

Question 34

Iterative procedures in ML estimation

Answer 34

* Start values provide first guesses about model parameters; can be generated by the software or provided by the user. * Log-likelihood is computed using these starting values in the density function. * Parameter values are then adjusted such that the log-likelihood value is increased. * The process continues until the increases in log-likelihood value cease (i.e., they do not increase beyond the set convergence criterion), yielding the maximum likelihood parameter estimates.

Question 35

Reproduced covariance matrix

Answer 35

- Used in ML function to approximate the sample covariance matrix based on the factor model and the estimated parameter

Question 36

Residual matrix

Answer 36

= the difference between the sample covariance matrix and the reproduced covariance matrix based on the model parameters - We want to minimize values in the residual matrix in ML estimation (as close to 0.0 as possible)

Question 37

under what conditions might our measures have nonnormal distributions?

Answer 37

- nonnormality of continuous vars (i.e., skewness, kurtosis) - coarsely categorized (discrete) data (ex: dichotomous data [yes/no] and Likert-type data) - all of the above

Question 38

what are the implications of nonnormality for our choice of estimation procedure?

Answer 38

- nonnormality can alter the estimation of standard errors and chi-square statistic - we would have to use MLM (Satorra-Bentler chi-square), MLMV, or MLR (only one to deal with missing data)

Question 39

Evaluation of fit includes...

Answer 39

Global fit: the model perfectly reproduces the variance-covariance matrix among the measures (e.g., model test statistic [chi-square]) Local fit: assessing the individual parameter estimates to see if they are in the expected magnitude and direction (e.g., approximate fit indexes)

Question 40

Model chi-square test

Answer 40

test of exact fit; likelihood ratio test

Question 41

RMSEA

Answer 41

- root mean square error of approximation - a parsimony adjusted index that assesses fit as a fx of degrees of freedom - the larger the df, the smaller the RMSEA; thus, parsimony correction is reduced in strength with larger sample sizes

Question 42

RMSEA thresholds

Answer 42

< .05 = close fit of model to data .05 - .08 = fair fit .08 - .10 = mediocre fit > .10 = poor fit

Question 43

CFI

Answer 43

- comparative fit index - an incremental fit index that estimates the relative improvement in fit of model over a baseline model (i.e., the null model which assumes zero covariances among measures) - CFI > or = .95

Question 44

SRMR

Answer 44

- standardized root mean square residual - the mean absolute correlation residual - SRMR < or = .08

Question 45

AIC

Answer 45

- Akaike information criterion - predictive fit index that assesses fit in hypothetical replications of the same size randomly selected from the same population - model with the smallest AIC is preferred as the most likely to replicate

Question 46

Parsimony

Answer 46

Idea of the simplest model that is able to best explain the data using the least number of parameters

Question 47

Implications of Type I and Type II errors and power in evaluating fit

Answer 47

In SEM, the framework of hypothesis testing does a better job guarding against Type I errors than it does guarding against Type II errors - more power = greater precision in model estimates

Question 48

Nested vs. non-nested models

Answer 48

- Nested can be evaluated using chi-square difference test, LM tests, and Wald tests - Non-nested can be compared with information criteria (AIC and BIC)

Question 49

chi-square difference test

Answer 49

- If one model (more constrained or restricted model) can be derived by imposing one or more constraints on a second model (less constrained or full model), then more constrained model is nested within the less constrained model. - A type of specification search - Assumptions: multivariate normality of vars, large sample size

Question 50

robust chi-square difference tests

Answer 50

MLM, MLR, MLMV

Question 51

Why are specification searches conducted?

Answer 51

- to add parameters to improve model fit | - to delete parameters to produce more parsimonious models

Question 52

Wald tests

Answer 52

- a specification search used to delete parameters to produce more parsimonious models - evaluates whether a model would produce poorer fit in the population if one or more of the free parameters were fixed (non-sig Wald test = no significant loss of fit when parameter is deleted from model, which indicates the parameter can be constrained)

Question 53

Modification indices/LM tests

Answer 53

- Lagrange Multiplier - a specification search - evaluates whether a model would show improved fit in the population if one or more parameters were freed (non-sig LM test = no significant improvement in fit when parameters are added to the model)