structural equations modeling – part 1: conﬁrmatory factor ... · structural equations modeling...

Structural Equations Modeling – Part 1:Confirmatory Factor Analysis

Pekka Malo 30E00500 – Quantitative Empirical Research Spring 2016

Agenda

•  Basic concepts

•  Confirmatory Factor Analysis (CFA)

•  Practical guidelines

•  Tutorial on SPSS Amos and CFA

25.01.16 Confirmatory Factor Analysis

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Section 1: What is Structural Equations Modeling?

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What is SEM?

Structural equation modeling (SEM) is a collection of statistical techniques that allow a set of

relationships between one or more independent variables (IV’s), either continuous or discrete, and

one or more dependent variables (DV’s), either continuous or discrete, to be examined.

(~ Series of multiple regression equations)


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Why use SEM?

1.  Estimation of several interrelated relationships

2.  Ability to represent unobserved (latent) concepts and correct for measurement error

3.  Defines a model to explain an entire set of relationships


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What is a latent construct?

•  Represents theoretical concepts, which cannot be observed directly

•  Similar to factors discussed in Exploratory Factor Analysis

•  Needs to be measured indirectly using multiple measured variables (a.k.a. indicator or manifest variables)


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Exogenous vs. Endogenous Constructs •  Exogenous construct ~ latent, multi-item equivalent of an

independent variable –  Variate (linear combination) of measures is used to represent a

construct

–  Multiple measured variables represent the exogenous constructs

•  Endogenous construct ~ latent, multi-item equivalent to a dependent variable

–  Theoretically determined by factors within the model

–  Multiple measured variables represent the endogenous constructs


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Example: Two latent constructs

•  Loadings represent the relationships from constructs to variables as in factor analysis.

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

Source: Hair et al. (2010)

EndogenousConstruct

Y1 Y2 Y3 Y4

Exogenous Construct

X1 X2 X3 X4


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Visual modeling: Path diagrams •  SEM models are commonly described in visual form using

“path diagrams”, which present relations between constructs and measured variables

•  Path diagrams generally consist of two parts:

Measurement model •  How are the constructs related to measured variables?

Structural model •  What are the relationships between the constructs?


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Types of relationships in SEM 1.  Relationship between a Construct and a Measured

Variable

Endogenous

Exogenous X

Y


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Types of relationships in SEM 2.  Relationship between a Construct and multiple Measured

Variables

Exogenous

X1

X2

X3


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Types of relationships in SEM 3.  Dependence relationship between two constructs

(Structural relationship)

Exogenous Endogenous


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Types of relationships in SEM 4.  Correlational relationship between constructs

Construct 1

Construct 2


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Measurement and structural model

Measurement model

Structural model


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“Cause-and-effect” relationships

Substantial evidence required:

1.  Covariation

2.  Sequence

3.  Non-spurious covariance

4.  Theoretical support


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Causal modeling in SEM?


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Non-spurious relationships •  Original relationship:

•  Testing for alternate cause:

Supervisor Job satisfaction

0.50

Supervisor Job satisfaction

Working conditions

0.50

0.00 0.30


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Reliability and measurement error

•  A certain degree of measurement error is practically always present

•  Reliability = measure for the degree to which a set of indicators of a latent construct are internally consistent (i.e. the extent to which they measure the same thing)

–  Reliability is generally inversely related to measurement error


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Improving statistical estimation •  In the previous multivariate techniques, we have assumed

that we can overlook the measurement error in the variables

•  SEM automatically applies a “correction” for the amount of measurement error and estimates the correct structural coefficient (i.e. the relationships between constructs)

•  Relationship coefficients estimated by SEM tend to be larger than coefficients obtained from multiple regression


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Strong theoretical basis needed

•  No SEM model should be considered without an underlying theory

•  Theory is needed for specifying the path diagram: –  Measurement model

–  Structural model


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Modeling strategies •  Confirmatory modeling strategy

–  Specify a single model

–  “It either works or it doesn’t”

•  Competing models strategy –  Multiple alternative specifications

–  Strongest test is to compare models representing different but plausible hypothesized relationships

•  Model development strategy –  Basic model proposed as a starting point

–  SEM used to get insights for re-specification

–  Model needs to be verified with an independent sample


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SEM and other multivariate techniques

•  SEM is most appropriate when researcher has multiple constructs, each represented by several measurement variables

•  SEM ~ hybrid of multiple regression, MANOVA and factor analysis

•  Opposite of exploratory techniques; everything is theory driven


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Example: multiple regression


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Section 2: Confirmatory Factor Analysis

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Confirmatory Factor Analysis

Similar to EFA in many respects, but with a completely different philosophy. With CFA,

researcher needs to specify both number of factors as well as what variables define the factors.


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CFA as a tool for evaluating measurement model

•  Specification of the measurement model is a crucial step in SEM (!)

•  Commonly CFA is used as a tool to validate the measurement model before specifying and estimating the structural model:

–  Are the constructs unidimensional and valid?

–  How many indicators should be used for each construct?

–  Are the measures able to portray the construct or explain it?


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Steps in CFA

Define constructs

Define measurement model

Design the empirical study

Estimate and assess validity


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Step 1: Defining the constructs

•  Operationalization

•  Scales from prior research

•  Development of new scales

•  Pretesting


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Step 2: Defining the measurement model

•  Are the constructs unidimensional (i.e. no cross-loadings)?

•  Is the measurement model congeneric (i.e. no covariance between or within construct error variances)?

•  Is there a sufficient number of indicators per construct (i.e. ensure identification)?


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Compensation

X1 X2 X3 X4

e1 e2 e3 e4

Lx1

Lx 2 Lx 3

Lx 4

Teamwork

X5 X6 X7 X8

e5 e6 e7 e8

Lx 5 L 6 Lx 7

Lx 8


Example: Congeneric model

Each measured variable is related to exactly one construct.


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X5 X6 X7 X8

δ5 δ6 δ7 δ8

λx5,2 λx6,2 λx7,2 λx8,2

X1 X2 X3 X4

λx1,1 λx2,1 λx3,1 λx4,1

λx3,2 λx5,1

δ1 δ2 δ3 δ4

Ф21

θδ 2,1 θδ 7,4

Figure 11.2 A Measurement Model with Hypothesized Cross-Loadings and Correlated Error Variance

Compensation Teamwork


Example: Non-Congeneric model

Each measured variable is not related to exactly one construct – errors are not independent.


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Items per construct •  Good practice dictates a minimum of 3 indicator variables

per construct (4 is preferred)

•  Assessment of single-item constructs is problematic (if included, they don’t generally stand for latent constructs)

•  Rationale for requirement of 3 indicators: –  Measurement model with a single constructs and only 2

indicators is under-identified (= there are more parameters than unique covariances)

–  Remember: the number of unique variances and covariances in the observed covariance matrix = degrees of freedom

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Measured Items X1=Cheerful X2=Stimulated X3=Lively X4=Bright

Loading Estimates λx 1,1=0.78 λx 2,1=0.89 λx 3,1=0.83 λx 4,1=0.87

Error Variance Estimates θδ1,1=0.39 θδ2,2=0.21 θδ3,3=0.31 θδ4,4=0.24

Eight paths to estimate

10 unique variance-covariance terms

ξ1

X1 X2 X3 X4

δ1 δ2 δ3 δ4

λx 1,1 λx 2,1 λx

3,1 λx 4,1

θδ 1,1 θδ 2,2 θδ 3,3 θδ 4,4

Symmetric Covariance Matrix: | X1 X2 X3 X4 --------------------------------------- X1 | 2.01 X2 | 1.43 2.01 X3 | 1.31 1.56 2.24 X4 | 1.36 1.54 1.57 2.00 Model Fit: χ2 = 14.9 df = 2 p = .001 CFI = .99


Example: Over-identified construct


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Formative vs. reflective constructs

•  Reflective measurement theory: –  Latent constructs cause the measured variables –  CFA is based on the reflective approach –  Errors occur due to inability to fully explain variables

•  Formative measurement theory: –  Measured variables “cause” the construct

–  Error term is an inability of measured variables to fully explain the construct

–  Formative constructs are not latent –  Formative constructs are interpreted as indices where each indicator

is a cause of the construct –  Have problems in statistical identification?


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Formative vs. reflective constructs (cont.)

•  Practical implications: –  Use of formative constructs require additional variables or

constructs to ensure an over-identified model

–  Formative should represent all items for it: dropping items because of low loadings should not be done (internal consistency and reliability are not so important)

–  In reflective approach, indicators which have low correlations with the other indicators of the same construct, should be removed


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Step 3: Design the empirical study

•  Choice of measurement scales

•  Sampling issues

•  Model specification and identification issues

•  Countering potential estimation problems


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Setting the scales for constructs •  All indicator variables for a construct don’t have to be of

the same scale

•  However, normalization can make interpretation easier

•  Before estimation of the model, you need to ensure that the scale of each construct is defined:

–  Fix one loading and set its value to 1 (i.e. don’t estimate loading parameter); or

–  Fix the construct variance and set its value to 1

•  Check that multiple values are not constrained to 1 for the purpose of defining the scale


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Identification of the model •  Degrees of freedom gives the amount of mathematical

information available to estimate model parameters

•  In the case of SEM, this is given by the number of unique variances and covariances minus number of parameters

•  Where p = number of variables and k=number of parameters


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Identification of the model (cont.)

Order condition: Net degrees of freedom must be > 0

•  Under-identified ~ more parameters than unique covariance and variance terms

•  Just identified ~ df = 0

•  Over-identified ~ df > 0

Rank condition: Each parameter is uniquely defined


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Recognizing identification problems

•  Incorrect indicator specification –  Not linking an indicator to any construct

–  Linking an indicator to two or more constructs

–  Not creating and linking an error term for each indicator

•  Setting the scale of a construct –  Forgetting to set the scale (either loading of an indicator or the

construct variance)

•  Insufficient degree of freedom –  Violation of 3-indicator rule (in particular when sample < 200)

–  More indicators needed or add constraints to free up degrees of freedom


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Recognizing identification problems (cont.)

•  Very large standard errors

•  Inability to invert the information matrix (no solution found)

•  Wildly unreasonable estimates, including negative error variances

•  Unstable parameter values


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Sample size issues •  Multivariate distribution of data

–  Should have 15 observations for each parameter estimated

•  Estimation technique –  If all assumptions OK, ML works already with sample of 50 –  In less than ideal conditions, sample should be at least 200 –  Sample sizes in range of 100-400 are recommended

•  Model complexity (# of constructs, parameters, groups)

•  Amount of missing data

•  Amount of average error variance among the reflective indicators

–  With communalities less than 0.5 (i.e. standardized loadings less than 0.7), large samples required for stable solution


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Thumb rules on sample size •  Minimum sample of 100:

–  5 or less constructs, each with more than 3 indicator variables, and high communalities 0.6 or higher

•  Minimum sample of 150: –  7 or less constructs, modest communalities 0.5, and no under-

identified constructs (i.e. fewer than 3 indicators)

•  Minimum sample of 300: –  7 or fewer constructs, low communalities (below 0.45), and

multiple under-identified constructs

•  Minimum sample of 500: –  Models with large number of constructs, some with lower

communalities, and/or having fewer than 3 indicators


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Step 4: Examination of model validity

•  Are the constructs valid?

•  Is the model fit acceptable?

•  Diagnostics?


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Construct validity •  SEM can be used to evaluate the validity of constructs (i.e.

to what extent do the measured items reflect the theoretical latent construct?)

•  Aspects of construct validity: –  Convergent validity: loadings, variance extracted, reliability

–  Discriminant validity

–  Nomological validity

–  Face validity


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Convergent validity •  Indicators of a specific construct should “converge” or

share a high proportion of variance in common

•  Statistics for convergent validity –  Loadings

–  Average variance extracted

–  Reliability


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Statistics for convergent validity •  Standardized factor loadings and squared factor loadings

–  High loadings indicate convergence

–  Should be statistically significant

•  AVE = average variance extracted

–  where squared standardized factor loadings indicate the amount

of variation in the indicator that can be explained by the factor

–  AVE > 0.5 => adequate convergence

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Statistics for convergent validity(cont.)

•  Construct reliability

–  Where V(ei) = error variance in variable i

–  Should be > 0.7 to warrant good reliability

–  High construct reliability indicates internal consistency, i.e. all measures represent the same construct


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Guidelines for evaluating convergent and discriminant validity

•  Estimated loadings should be 0.5 or higher

•  AVE should be 0.5 or higher to support convergent validity

•  AVE estimates for two factors should be greater than the square of the correlation between two factors to provide evidence of discriminant validity

•  Construct reliability should be 0.7 or higher to suggest convergence and internal consistency


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Discriminant validity •  Is the construct unique?

•  Does it differ from other constructs?

•  Do the individual indicator variables represent only one latent construct?

•  Examine correlations between constructs

•  Presence of cross-loadings is an indicator of discriminant validity problems


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Nomological and face validity •  Face validity ~ “looks like it will work”

–  Needs to be established before experiment

–  Ensure understanding of every indicators content and meaning

•  Nomological validity ~ “does the construct behave as it should with respect to other constructs”

–  Theoretical propositions, e.g. “as age increases, memory loss increases”

–  Check whether the correlations between constructs make sense!


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Assessment of model validity

•  Goodness-of-fit: Does the estimated implied covariance matrix match the observed covariance structure?

–  Absolute goodness-of-fit

–  Incremental goodness-of-fit

–  Parsimonious fit measures

•  Construct validity


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Chi-square test

•  The null hypothesis tests whether the difference between the sample and the estimated covariance matrix is a null or zero matrix

•  Concluding that the null hypothesis holds indicates that the model fits the data


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Problems with Chi-square test •  Chi-square statistic is a function of the sample size N and

the difference between observed and estimated covariance matrices

•  As N increases, so does the test-statistic even when differences between matrices don’t change

•  Chi-square statistic also increases when adding number of observed variables, which makes it more difficult to achieve a fit

Need for complementary statistics!!


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Comparative fit indices •  Based on idea of comparing nested models on

continuum: saturated --- estimated --- independence

•  Bentler-Bonett normed fit index (NFI): compares estimated model to independence model

–  High values (> 0.95) indicate good-fit

•  Bentler’s comparative fit index (CFI): –  High values (> 0.95) indicate good-fit


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Comparative fit indices (cont.) •  Tucker-Lewis Index (TLI):

–  Conceptually similar to NFI

–  Takes model complexity into account

–  Not normalized, but generally models with good fit have values close to 1

•  Relative non-centrality index (RNI) –  Compares observed fit to that of a null model

–  Higher values represent better model (> 0.9)


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Parsimony fit indices

•  Improved either by a better fit or a simpler model

•  Conceptually similar to adjusted R2

•  Examples: –  Adjusted Goodness-of-fit (AGFI)

–  Parsimony normed fit index (PNFI)


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Badness-of-fit indices •  Root mean square error of approximation (RMSEA)

–  Quite broadly used

–  Attempts to correct for tendency of chi-square to reject models with large sample or large number of observed variables

–  Lower values imply better fit (< 0.08)

•  Root mean square residual (RMR) or standardized RMR –  Generally standardized residuals exceeding |4.0| should be

scrutinized

–  SRMR > 0.1 indicates a problem with fit


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Thank you!

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Very Useful Materials •  http://statwiki.kolobkreations.com/

•  Download the helpful Excel-tool by J. Gaskin! It is useful during the course!

•  There are additionally two plugins, which you can install when using your laptops / home computers (requires admin rights).

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structural equations modeling – part 1: conﬁrmatory factor ... · structural equations modeling...

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