ACSPRI Topic 1 EFA and Regr

ACSPRI Summer 2016Applied SEM

using AMOS

Topic 1: Introduction and Review of Regression/EFA

Structural part of the model

Measurement parts of the model

SEM References• Byrne, B.M. (2010). Structural Equation Modeling

with AMOS: Basic Concepts, Applications, and Programming. Mahwah, N.J.: Lawrence Erlbaum Associates.

• Arbuckle, J. L. (2013). AMOS 21 User’s Guide. Chicago: SPSS Inc.

Exploratory factor analysis: A review• used more for theory-generating than for theory-testing:

– explore data to determine the number or the nature of factors that account for the covariation between variables

– when the researcher does not have, a priori, sufficient evidence to form a hypothesis about the number of factors underlying the data.

– helpful in assessing the nature of relationships among variables and in establishing the construct validity of test scores.

– Highly empirical or “data- (cf. Theory-) driven technique

Data Reduction• Just looking at a questionnaire, can often spot which Qs go

together, sometimes they have been designed to go together, to measure a particular aspect of a construct

• Inspecting a correlation matrix, can also see which variables have high correlations

v1 v2 v3 v4 v5v1 1.00 0.75 0.65 0.22 0.15v2 0.75 1.00 0.55 0.10 0.20v3 0.65 0.55 1.00 0.18 0.25v4 0.22 0.10 0.18 1.00 0.80v5 0.15 0.20 0.25 0.80 1.00

Restrained eating

Item 5Item 4Item 3Item 2

Emotionaleating

Item 9Item 8Item 7Item 6

“Factors”

Questionnaire Items

Item 1 Item 10

Exploratory Factor Analysis

Restrained eating

Item 5Item 4Item 3Item 2

Emotionaleating

Item 9Item 8Item 7Item 6

Latent Variables

Questionnaire Items

Item 1 Item 10

Confirmatory Factor Analysis

Spatial Representation• Correlations between variables can be represented spatially • Variables with a small angle between them have a high correlation

• In fact the projection of one variable on the other is the simple correlation between them

• “Vectors” = arrows

Eat when discouraged

Eat when lonely

Eat less at mealsEat less followingWeight gain

Spatial Representation• So, one may imagine the variables graphed in

a space where highly correlated variables are close together, and uncorrelated ones are not

Factor 2

Factor 1

The factors(the latent variables) will go through the middle of clusters of variables (indicators; questionnaire

items)

Eat when discouraged

Eat when lonely

Eat less at mealsEat less following

Weight gain

Steps in performing PCA/FA

1. EXTRACTION : Determining HOW MANY components best represent the data

2. ROTATION : Making the data interpretable

Exercise 1

• Let’s use the dataset “Ex 1.1_Eating.sav” and work through exercise 1 in the notes

Limitations of EFA• EFA was designed to fit the data

– will find you some “factors” whether you expected these factors or not…

• Interpretation of factors measured by a few variables can be quite complicated – because the researcher lacks prior knowledge and therefore has no

basis on which to make an interpretation. • Mulaik (1972): EFA at best suggests hypotheses for further research; does

not justify knowledge.

Confirmatory factor analysis• A theory-testing model as opposed to a theory-generating

method like EFA:– researcher begins with a hypothesis prior to the analysis. – The model or hypothesis specifies

• number of factors• which variables will be correlated with which factors• which factors are correlated.

– The hypothesis should be based on a strong theoretical and/or empirical foundation (Stevens, 1996).

CFA• Often used for evaluating construct validity

– researcher can explicitly test hypotheses concerning the factor structure of the data due to having the predetermined model specifying the number and composition of the factors.

Test 1

Verbal Ability Math Ability

“Intelligence”

Test 2

Test 3

Test 4

Test 5

Test 6

CFA• Allows researcher to compare or test the fit of different models to

the data

Test 1

“Intelligence”

Test 2

Test 3

Test 4

Test 5

Test 6

Test 1

Verbal Ability Math Ability

“Intelligence”

Test 2

Test 3

Test 4

Test 5

Test 6

Which model will provide better fit to the data?

First order factor model or second-order factor model?

Using AMOS to test a CFA• Use “Ex1.2_CFA IQ test”• Open SPSS, under

analysis you’ll find AMOS(note, sometimes AMOS is

install in standalone mode. If so, just open it as a regular program))

First, run an EFA• Descriptives

– Ask for Coefficients, KMO, Bartlett

• Extraction– Use maximum likelihood– Ask for scree plot

• Rotation– Direct oblim– Rotated solution, loading plots

CFA2 latent factorsi. Verbal IQ

i. Informationii. Comprehensioniii. Vocab

ii. Performance IQi. Picture completionii. Block designiii. Object assembly

See notes for details

Using AMOS to test a CFA• Now try running a CFA on the “Ex1.1_Eating.sav” file

Multiple regression revision

r and SEM• SEM is based on the

analysis of covariance-variance matrices

• AMOS = Analysis Of Moment Structures

• Covariance is the unstandardised correlation

Moments:

MeanVariance

SkewnessKurtosis

Mixed moments:Covariance

Goals of correlation coefficients

1. To obtain an estimate of ρ- Reliability and validity estimates

2. Calculate coefficient of determination- i.e., variance accounted

3. Use in other analyses- Basis of factor analysis, multiple regression, SEM,

mediation, moderation, MLM

• if X and Y are perfectly correlated (r = 1 or -1), then prediction is easy…

• just read off the scatterplot

024681012141618

0 5 10

regression

X

Y

in this case, if a person scores 6 on X, they will have a score of 13 on Y

• but correlations are rarely ever perfect!

0123456789

0 3 6 9 12

However we can calculate a line that best represents the general relationship (or trend) of the data points

The “line of best fit” (aka the regression line) allows us to make predictions that are better than if we knew nothing about the relationship. Can be expressed mathematically (linear regression equation)

regression

0123456789

0 3 6 9 12

least squares criterion

= predicted value of Yb = slope of regression line

(rate at which Y changes with each 1-unit increase in X)a = intercept

(the predicted value of Y when X = 0)

we can define the regression line mathematically using a formula - this is the formula that describes a straight line

where:

regression

ˆ Y

abxY = ˆ

Regression using SPSS

Use “Ex1.3_Simple regression.sav”

SPSS regression output

slope intercept

standardised regression equation

Using AMOS to perform Regression• Use “Ex1.4_multiple_regression.sav”• Run MR with concentration, perseverance,

attentiveness and purposefulness as predictors of reading

Now, let’s do this in AMOS

Note that we need to specify two things that SPSS used to do for us: