for Dummies by Dummies February 22 2013 Indiana University Bloomington Joseph J Sudano Jr PhD Center for Health Care Research and Policy Case Western Reserve University at The MetroHealth System ID: 713629
Download Presentation The PPT/PDF document "Applied Structural Equation Modeling" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Applied Structural Equation Modeling for Dummies, by DummiesFebruary 22, 2013Indiana University, Bloomington
Joseph J. Sudano, Jr., PhD
Center for Health Care Research and Policy
Case Western Reserve University at The MetroHealth System
Adam T.
Perzynski
, PhD
Center for Health Care Research and Policy
Case Western Reserve University at The MetroHealth SystemSlide2
AcknowledgementsThanks Joe.Thanks to Bill Pridemore and all of you here at IU.Thanks to Doug Gunzler.Thanks to Kyle Kercher.Slide3
Rejected Titles for this TalkFebruary 22, 2013Indiana University, BloomingtonJoseph J. Sudano, Jr., PhDCenter for Health Care Research and PolicyCase Western Reserve University at The MetroHealth System
Adam T.
Perzynski
, PhD
Center for Health Care Research and Policy
Case Western Reserve University at The MetroHealth SystemSlide4
Structural Equation Modeling for Fashion WeekSlide5
We have lots of Models!Slide6
Structural Equation Modelin’ fer PiratesSlide7
Structural Equation Modelin’ fer Pirates
SEM
be a statistical technique for
testin
' and
estimatin
' causal relations
usin
' a combination o' statistical data and qualitative causal
assumptions
*From WikipediaSlide8
AssumptionsI do not actually assume you are dummiesFeel free to assume what you want about meI do not assume you will be experts in SEM after this presentationI assume you know something about means and regression (hopefully)Slide9
OutlineImportant SEM ResourcesMeasurement (and measurement error)ExamplesMeasurement InvarianceLatent Class AnalysisLatent Growth Mixture ModelingModel SpecificationSlide10
OutlineImportant SEM ResourcesMeasurement (and measurement error)ExamplesMeasurement InvarianceLatent Class AnalysisLatent Growth Mixture ModelingModel SpecificationSlide11
SEM ResourcesSlide12
SEM ResourcesSlide13
SEM ResourcesSlide14
SEM Resources: Statmodel.comSlide15
SEM ResourcesSlide16
SEM ResourcesSlide17
SEM ResourcesSlide18
SEM ResourcesSlide19
SEM ResourcesSlide20
SEM ResourcesSlide21
SEM ResourcesSlide22
OutlineImportant SEM ResourcesMeasurement (and measurement error)ExamplesMeasurement InvarianceLatent Class AnalysisLatent Growth Mixture ModelingModel SpecificationSlide23
Measurement ModelsA special type of causal modelsSurvey items are assumed to have measurement errorEach question has its own amount of errorYour answer to a survey question is causally related to a latent, unobserved variable. Slide24
Perfect MeasurementSelf-rated healthhealth
1.0?Slide25
Causality and the Latent Concept of HealthIn general, how would you describe your health?We assume that every individual varies along an infinite continuum from best possible health to worst possible health. When any given individual answers this question, they are approximating their position on this latent continuum. Slide26
Imperfect MeasurementSelf-rated healthhealth
e4
< 1.0
1.0
Variance > 0Slide27
Measurement Models using Multiple IndicatorsSingle items are unreliable Single cases prevent generalizabilityUse multiple indicators and large samples to estimate the values of the latent, unobservered variables or factorsThe SF36 uses multiple indicators describing multiple factors in order to measure health more reliably. Slide28
OutlineImportant SEM ResourcesMeasurement (and measurement error)ExamplesMeasurement InvarianceLatent Class AnalysisLatent Growth Mixture ModelingModel SpecificationSlide29
Acknowledgement: This study was funded by Grant number R01-AG022459 from the NIH National Institute on Aging. Measuring Disparities: Bias in Self-reported Health Among Spanish-speaking PatientsJ.J. Sudano1,2, A.T. Perzynski
1,2
, T.E. Love
2
, S.A. Lewis
1
,B. Ruo
3
, D.W. Baker
3
1
The MetroHealth System, Cleveland, OH;
2
Case Western Reserve University School of Medicine, Cleveland, OH;
3
Northwestern University Feinberg School of MedicineSlide30
Measurement Model of the SF36Slide31
Objective & SignificanceDo observed differences in SRH reflect true differences in health? Cultural and language differences may create measurement biasIf outcomes aren’t measuring the same thing in different groups, we have a problemSlide32
Measurement Equivalence &Factorial InvarianceIt is only possible to properly interpret group differences after measurement equivalence has been established (Horn & McArdle, 1992; Steenkamp & Baumgartner, 1998). “It may be the case that the groups differ … but it also may be the case that extraneous influences are giving rise to the observed difference.” Meredith & Teresi (2006 p. S69)The external validity of any conclusion regarding group differences rests securely on whether the measurement equivalence of the scale has been established (Borsboom, 2006).Slide33
Cross-sectional StudyN= 1281Medical patients categorized into four groups:White, Black, English-speaking Hispanic and Spanish-speaking Hispanic. Multigroup Confirmatory Factor Analysis (MGCFA)Slide34Slide35
Two Types of InvarianceMetric (Weak) InvarianceAre the item factor loadings equivalent across groups?Is a one unit change in the item equal to a one unit change in the factor score for all groups?Scalar (Strong) InvarianceAre the item intercepts equivalent across groups? Unequal intercepts results in unequal scaling of factor scoresSlide36Slide37Slide38Slide39
Self-rated healthhealth
e4
What happens to the model fit when we constrain all of these paths (loadings) to be equal across groups?
Weak invarianceSlide40Slide41
Table 1: Goodness of Fit for SF36 Multigroup Factorial Invariance Testing (N = 1281)
Model
Description
RMSEA (95% CI)
CFI
B-S
χ
2
*
df
Ref
∆RMSEA
∆CFI
B-S ∆
χ
2
∆df
1
Unconstrained Model
0.028
(.017 - .030)
0.936
3001
2172
2
Metric Invariance (Factor Weights)
0.029
(.028 - .030)
0.931
3110
2253
1
0.001
-0.005
109
81
3
Scalar Invariance (Intercepts)
0.033
(.032 - .034)
0.907
3215
2358
2
0.004
-0.024
105
105
4
Partial Scalar Invariance (B=W=HS not HE)
0.033
(.032 - .034)
0.909
3179
2323
2
0.004
-0.022
69
70
5
Partial Scalar Invariance (B=W=HE not HS)
0.030
(.029 - .032)
0.921
3180
2323
2
0.001
-0.010
70
70
6
2nd Order Structural Invariance**
0.030
(.029 - .032)
0.921
3187
2333
2
0.001
-0.010
77
80
7
2nd & 3rd Order Structural Invariance**
0.030
(.029 - .032)
0.921
3196
2339
2
0.001
-0.010
86
86
* The bootstrapped Bollen - Stine χ2 value is reported because of significant (p<.01) multivariate non-normality. ** Structural factor weights are constrained equal for Blacks, Whites and Hispanic English (Hispanic Spanish are unconstrained). Slide42Slide43
Table 1: Goodness of Fit for SF36 Multigroup Factorial Invariance Testing (N = 1281)
Model
Description
RMSEA (95% CI)
CFI
B-S
χ
2
*
df
Ref
∆RMSEA
∆CFI
B-S ∆
χ
2
∆df
1
Unconstrained Model
0.028
(.017 - .030)
0.936
3001
2172
2
Metric Invariance (Factor Weights)
0.029
(.028 - .030)
0.931
3110
2253
1
0.001
-0.005
109
81
3
Scalar Invariance (Intercepts)
0.033
(.032 - .034)
0.907
3215
2358
2
0.004
-0.024
105
105
4
Partial Scalar Invariance (B=W=HS not HE)
0.033
(.032 - .034)
0.909
3179
2323
2
0.004
-0.022
69
70
5
Partial Scalar Invariance (B=W=HE not HS)
0.030
(.029 - .032)
0.921
3180
2323
2
0.001
-0.010
70
70
6
2nd Order Structural Invariance**
0.030
(.029 - .032)
0.921
3187
2333
2
0.001
-0.010
77
80
7
2nd & 3rd Order Structural Invariance**
0.030
(.029 - .032)
0.921
3196
2339
2
0.001
-0.010
86
86
* The bootstrapped Bollen - Stine χ2 value is reported because of significant (p<.01) multivariate non-normality. ** Structural factor weights are constrained equal for Blacks, Whites and Hispanic English (Hispanic Spanish are unconstrained).
The Unconstrained Model Fits the Data WellSlide44
Table 1: Goodness of Fit for SF36 Multigroup Factorial Invariance Testing (N = 1281)
Model
Description
RMSEA (95% CI)
CFI
B-S
χ
2
*
df
Ref
∆RMSEA
∆CFI
B-S ∆
χ
2
∆df
1
Unconstrained Model
0.028
(.017 - .030)
0.936
3001
2172
2
Metric Invariance (Factor Weights)
0.029
(.028 - .030)
0.931
3110
2253
1
0.001
-0.005
109
81
3
Scalar Invariance (Intercepts)
0.033
(.032 - .034)
0.907
3215
2358
2
0.004
-0.024
105
105
4
Partial Scalar Invariance (B=W=HS not HE)
0.033
(.032 - .034)
0.909
3179
2323
2
0.004
-0.022
69
70
5
Partial Scalar Invariance (B=W=HE not HS)
0.030
(.029 - .032)
0.921
3180
2323
2
0.001
-0.010
70
70
6
2nd Order Structural Invariance**
0.030
(.029 - .032)
0.921
3187
2333
2
0.001
-0.010
77
80
7
2nd & 3rd Order Structural Invariance**
0.030
(.029 - .032)
0.921
3196
2339
2
0.001
-0.010
86
86
* The bootstrapped Bollen - Stine χ2 value is reported because of significant (p<.01) multivariate non-normality. ** Structural factor weights are constrained equal for Blacks, Whites and Hispanic English (Hispanic Spanish are unconstrained).
The Unconstrained Model fits the data well
The model with factor loadings constrained still fits the data well.Slide45
Metric (Weak) Invariance was ConfirmedSlide46
I forget what an intercept isScalar (Strong) InvarianceAre the item intercepts equivalent across groups? Intercept: the intercept in a multiple regression model is the mean for the response when all of the explanatory variables take on the value 0.Could be called the “starting point”Slide47Slide48
Table 1: Goodness of Fit for SF36 Multigroup Factorial Invariance Testing (N = 1281)
Model
Description
RMSEA (95% CI)
CFI
B-S
χ
2
*
df
Ref
∆RMSEA
∆CFI
B-S ∆
χ
2
∆df
1
Unconstrained Model
0.028
(.017 - .030)
0.936
3001
2172
2
Metric Invariance (Factor Weights)
0.029
(.028 - .030)
0.931
3110
2253
1
0.001
-0.005
109
81
3
Scalar Invariance (Intercepts)
0.033
(.032 - .034)
0.907
3215
2358
2
0.004
-0.024
105
105
4
Partial Scalar Invariance (B=W=HS not HE)
0.033
(.032 - .034)
0.909
3179
2323
2
0.004
-0.022
69
70
5
Partial Scalar Invariance (B=W=HE not HS)
0.030
(.029 - .032)
0.921
3180
2323
2
0.001
-0.010
70
70
6
2nd Order Structural Invariance**
0.030
(.029 - .032)
0.921
3187
2333
2
0.001
-0.010
77
80
7
2nd & 3rd Order Structural Invariance**
0.030
(.029 - .032)
0.921
3196
2339
2
0.001
-0.010
86
86
* The bootstrapped Bollen - Stine χ2 value is reported because of significant (p<.01) multivariate non-normality. ** Structural factor weights are constrained equal for Blacks, Whites and Hispanic English (Hispanic Spanish are unconstrained).
The Unconstrained Model fits the data well
The model with factor loadings constrained still fits the data well.
Constraining the intercepts results in a worsening of model fitSlide49
Table 1: Goodness of Fit for SF36 Multigroup Factorial Invariance Testing (N = 1281)
Model
Description
RMSEA (95% CI)
CFI
B-S
χ
2
*
df
Ref
∆RMSEA
∆CFI
B-S ∆
χ
2
∆df
1
Unconstrained Model
0.028
(.017 - .030)
0.936
3001
2172
2
Metric Invariance (Factor Weights)
0.029
(.028 - .030)
0.931
3110
2253
1
0.001
-0.005
109
81
3
Scalar Invariance (Intercepts)
0.033
(.032 - .034)
0.907
3215
2358
2
0.004
-0.024
105
105
4
Partial Scalar Invariance (B=W=HS not HE)
0.033
(.032 - .034)
0.909
3179
2323
2
0.004
-0.022
69
70
5
Partial Scalar Invariance (B=W=HE not HS)
0.030
(.029 - .032)
0.921
3180
2323
2
0.001
-0.010
70
70
6
2nd Order Structural Invariance**
0.030
(.029 - .032)
0.921
3187
2333
2
0.001
-0.010
77
80
7
2nd & 3rd Order Structural Invariance**
0.030
(.029 - .032)
0.921
3196
2339
2
0.001
-0.010
86
86
* The bootstrapped
Bollen
- Stine χ2 value is reported because of significant (p<.01) multivariate non-normality. ** Structural factor weights are constrained equal for Blacks, Whites and Hispanic English (Hispanic Spanish are unconstrained).
The model with factor loadings constrained still fits the data well.
Constraining the intercepts results in a worsening of model fit
The fit is still poor if you allow intercepts for English-speaking Hispanics to varySlide50
Table 1: Goodness of Fit for SF36 Multigroup Factorial Invariance Testing (N = 1281)
Model
Description
RMSEA (95% CI)
CFI
B-S
χ
2
*
df
Ref
∆RMSEA
∆CFI
B-S ∆
χ
2
∆df
1
Unconstrained Model
0.028
(.017 - .030)
0.936
3001
2172
2
Metric Invariance (Factor Weights)
0.029
(.028 - .030)
0.931
3110
2253
1
0.001
-0.005
109
81
3
Scalar Invariance (Intercepts)
0.033
(.032 - .034)
0.907
3215
2358
2
0.004
-0.024
105
105
4
Partial Scalar Invariance (B=W=HS not HE)
0.033
(.032 - .034)
0.909
3179
2323
2
0.004
-0.022
69
70
5
Partial Scalar Invariance (B=W=HE not HS)
0.030
(.029 - .032)
0.921
3180
2323
2
0.001
-0.010
70
70
6
2nd Order Structural Invariance**
0.030
(.029 - .032)
0.921
3187
2333
2
0.001
-0.010
77
80
7
2nd & 3rd Order Structural Invariance**
0.030
(.029 - .032)
0.921
3196
2339
2
0.001
-0.010
86
86
* The bootstrapped
Bollen
- Stine χ2 value is reported because of significant (p<.01) multivariate non-normality. ** Structural factor weights are constrained equal for Blacks, Whites and Hispanic English (Hispanic Spanish are unconstrained).
The model with factor loadings constrained still fits the data well.
The fit is acceptable if you allow intercepts for Spanish speaking Hispanics to varySlide51
Scalar (Strong) Invariance is NOT ConfirmedMeasurement equivalence of the SF36 does not exist for Spanish speaking HispanicsSlide52
Intercepts are lower for Spanish-speaking Hispanics on nearly all itemsMeasurement equivalence of the SF36 does not exist for Spanish speaking HispanicsSlide53
Use of English Rating Categories on TwiterUsing of Spanish Rating Categories on TwitterSlide54
OutlineImportant SEM ResourcesMeasurement (and measurement error)ExamplesMeasurement InvarianceLatent Class AnalysisLatent Growth Mixture ModelingModel SpecificationSlide55
Everywhere and Nowhere: Latent Class Analysis of Knowledge of the Spread of Hepatitis CAdam T. Perzynski, PhD
E-mail: Adam.Perzynski@case.eduSlide56
IntroductionHepatitis C is a widespread and serious disease that affects the liver. 170 million people worldwide are infected.3.9 million Americans infected with HCV. (AHRQ 2003) More Americans die every year from Chronic HCV infection than from HIVSlide57
HCV TransmissionBlood Injection Drug UseBlood TransfusionsNeedle Sticks Shared Household Items (Razor or Toothbrush) Sexual transmission of HCV is recognized but is infrequent. HCV is not transmitted by Coughing, Kissing, Sneezing, Touching, Bathrooms, Fecal Matter, or Contaminated FoodSlide58
SampleBehavior Risk Factor Surveillance System (BRFSS), 2001, ArizonaConducted by the Centers for Disease Control (CDC)The world’s largest telephone surveyNearly 200,000 people participated in 2001Slide59
MeasureDo you think hepatitis C can be spread thru?Sneezing or CoughingKissingUnprotected SexFood or WaterSharing Needles to Inject Street DrugsUsing the Same BathroomContact with the Blood of an Infected PersonSlide60
Methods of AnalysisAnalyzed with Mplus Analysis proceeded in several stagesExploratory Factor Analysis Confirmatory Factor AnalysisCluster Analysis (Not reported)Latent Class AnalysisMixture Modeling
Robust estimation for binary indicators
Missing Values Imputation using Full Information Maximum Likelihood Estimation (FIML)Slide61
Distribution of Outcome VariablesSlide62
Means and Standard DeviationsSlide63
Exploratory Factor AnalysisScree plot, Eigenvalues, and Root Mean Square Residuals more or less supported a two factor solutionSlide64Slide65
What is different about LCA?Instead of assuming that the latent variable is continuous (infinitely poor to infinitely good)We assume the latent variable is categorical. Membership in “hidden” empirical forms determines answers rather than a single latent continuum. Slide66Slide67
Three Latent ClassesThe Two Category and Four Category models do not fit the data as well as as the Three Category model. HCV is NowhereN = 1683 (The largest class!)Full Awareness of how HCV is SpreadN = 930HCV is EverywhereN = 479Slide68
Figure 3: Estimated Probabilities of Knowing How HCV is Spread by Class MembershipSlide69
Additional AnalysesWhat predicts membership in each latent class? Do the relationships between variables vary inside of a particular class? Mixture Modeling Simultaneously test continuous and categorical predictors of class membership. Slide70Slide71
OutlineImportant SEM ResourcesMeasurement (and measurement error)ExamplesMeasurement InvarianceLatent Class AnalysisLatent Growth Mixture ModelingModel SpecificationSlide72
Longitudinal Patterns of Depressive Symptoms in the Health and Retirement Study Adam T. Perzynski, PhD & Joseph S. Sudano, Jr., PhDCenter for Health Care Research and PolicyCase Western Reserve University and MetroHealthPresentation at the Annual Meeting of the Gerontological Society of America on November 22, 2010Slide73
IntroductionThis is another measurement studyExplore the use of Latent Class Growth Analysis to model changes in depressive symptoms over time in the Health and Retirement Study. Most studies compare the change in means scores between two waves.A small number of studies have modeled change as a single growth trajectory Slide74
Change in Means Between WavesOften we simply calculate the mean depressive symptoms at Wave 1 (baseline). Subract it from the mean at Wave 2 (followup). Slide75
What is a trajectory?Regrettably, the term “trajectory” has taken on multiple meanings across disciplines and research studies. A broad, inclusive definition of trajectory modeling is the analysis of patterns of change or stability. Confusion is possible between aggregate trajectories which summarize an overall average pattern of change for a population and disaggregated trajectories which examine multiple potential trajectories of different shapes (George 2006). Slide76
Example of a single growth trajectorySlide77
Continuous Latent Growth Curve AnalysisLGA / LGCAStudies in older adults (ie George and Lynch 2003) typically find that the slope of the latent growth curve for depressive symptoms is small and positive, and that the slope of the curve is steepest in the oldest cohorts. Slide78
Example from George and Lynch (2003)Slide79
Example of an LGA findingSlide80
LGA estimates a single Aggregate trajectoryAssumes that the average population starting point (intercept for the growth curve) and average amount of change (slope) are a sufficient depiction of variation over time in depressive symptoms. If discrete subtypes of depressive symptom trajectories exist, but are ignored (as in single latent growth curve and autoregressive models) the magnitude of associations could be grossly misestimated. Slide81
What is Latent Class Growth Analysis?Latent Class Growth Analysis (LCGA), also referred to as growth mixture modeling, belongs to a family of statistical techniques referred to as general latent variable modeling or GLVM. Slide82
Why would we ever think we should use LCGA?Studying the mean change or using a single trajectory for everyone assumes uniform heterogeneity in the population. Researchers use familiar methods and typically assume that the underlying (latent or real) distribution of variables is continuous. We have theoretical reasons to suspect that underlying distributions could be categorical. Life course theorists (Dannefer) specifically caution that intracohort differentiation is unlikely to be homogeneous. Slide83
Why would we use LCGA?We think individuals and cohorts diverge over timeCumulative change differentiates individuals and cohorts.Slide84
Prior LCGA Models of Depression or Depressive SymptomsLCGA models and closely related Longitudinal Latent Class Analysis (LLCA) have been used to estimate models of depressive symptoms in prior studies of maternity (Campbell et al 2009; Mora et al 2009)childhood and adolescence (Meadows et al 2006)adolescence through young adulthood (Olino et al 2009)response to antidepressants among adults (Muthen et al, 2007; Hunter et al 2009)patients who have had a cardiovascular event (Kaptein et al 2006). Slide85
Methods 5,195 age-eligible respondents from the 1992 Health and Retirement Study cohort, who completed interviews in all seven waves through 2004. Depressive symptoms in HRS are measured using a dichotomous, 8-item version of the CES-D. Analysis begins with Wave 2 data due to a change in response categories from Wave 1.Using MPlus, we compared the fit of LCGA models of two to eight classes while also accounting for the HRS complex sampling design. We then tested the effect of a small number of covariates. This is very similar to a multinomial logistic regression.Slide86
Demographic characteristics Gender60.3% femaleRace/ethnicity76.4% non-Hispanic White 14.4% Black 7.4% Hispanic1.8% other racial/ethnic groupsAgeMedian=55 EducationMean=12.4 years (SD=3.0). Slide87
Rule for Determining the number of Latent Classes“How many trajectories are there?” Measures of model fit including: Lo-Mendell-Rubin Test (LMR test) log-likelihood (LL) Bayesian Information Criteria (BIC) (Vuong, 1989; Muthen, 2004; Muthen, & Muthen, 2005; Nylund et al, 2007). Here we will use the LMR TestWhere k is the number of latent classes, this test gives a p-value for the k-1 versus the k-class model when running the k-class model (Vuong, 1989; Muthen, B. 2005).
The first time p > .05, k-1 is the preferred number of classes. Slide88
ResultsHow many classes are there?What do the classes look like?How is this different from looking at means or single trajectory?Are any demographic variables associated with being in a particular class?Slide89
How many Classes are there?Slide90
How many Classes are there?Slide91
What do the classes look like?Slide92
How is this different from looking at Means or a Single Trajectory?Online at: http://spreadsheets.google.com/pub?key=0ApRkae54BRnudEYyUGdXZWlES3Z4VzZ6akNaOFFiekE&gid=5Slide93
Does anything influence the chances of being in a particular class?Slide94
Does anything influence the chances of being in a particular class?Females, African Americans and those with fewer years of education have a higher probability of being in the Many Symptoms trajectory. Slide95
OutlineImportant SEM ResourcesMeasurement (and measurement error)ExamplesMeasurement InvarianceLatent Class AnalysisLatent Growth Mixture ModelingModel SpecificationSlide96
Model SpecificationChoosing the model that best represents the data structure and addresses the research questions of interest can be a daunting task. Brief overview of model specification tests and procedures. Slide97
Model Specification“First, your return to shore was not part of our negotiations nor our agreement so I must do nothing. And secondly, you must be a pirate for the pirate's code to apply and you're not. And thirdly, the code is more what you'd call ‘guidelines’ than actual rules.”Captain Barbossa from Pirates of the Caribbean: The Curse of the Black Pearl (2003)Slide98
Model SpecificationIn model specification a researcher can use:logic, theory and prior empirical evidence to choose the initial modelmodel comparison testing to compare the initial model to competing modelsa combination of theory, prior evidence, and the results of the model comparison testing to decide upon which model or models are appropriate for a given studySlide99Slide100
Nested or Not Nested?Slide101
Chi Square TestThe Chi-square statistic is computed and used to test whether the model does fit the data well. It is the basis for most other fit tests.Along with other fit tests we use it to evaluate whether to include or exclude model paths relating measures to each other for a given study. Slide102
Chi Square TestAlso called the discrepancy functionIf not significant, the model is regarded as acceptable.*Slide103
Chi Square Test*Some limitations are:Complex models with many parametersWith large samples, models will most often be rejected, sometimes unfairlyWhere multivariate non-normality is present, the chi-square fit index is inaccurate. Modified tests (The Satorra-Bentler scaled chi-square) are available.Slide104
Modification IndicesModification indices can be calculated individually for every path that is fixed to zero, by estimating a chi-square test statistic with one df. The higher the value of the modification index for a causal path, the better the predicted improvement in overall model fit if that path were added to the model. Jöreskog suggested that a modification index should be at least five before the researcher considers adding the causal path and modifying the hypothesized model.Slide105
R-squaredIn linear regression analysis, we interpret the r2 value as the amount of variation in the response that can be explained by the regressors in the model. In SEM, it is pretty much the same**Not exactly, but that is beyond the “for dummies” version of this talkSlide106
AIC, BIC (and BCC)Bayesian Information Criterion (BIC) Akaike Information Criterion (AIC) Based on the chi-squared test statistic While the models under comparison can be nested or non-nested, in both these tests, as with all tests in this section, for a truly direct comparison, we prefer that the same observed measures are used in the models we are comparing. Both BIC and AIC feature the goodness-of-fit term for our model , derived directly from the discrepancy function when applicable, along with a penalty term.
Slide107
AIC, BIC (and BCC)Cannot identify if a model has good fit. Only if one model fits better than another. The lower the value of BIC, AIC and BCC, the better the fit.BCC penalizes for model complexity more than AIC and BIC.BIC penalizes for model complexity more than AIC.Slide108
Specification SearchAllows researchers to choose a model from among a number of candidates.ExploratoryShould be guided by theorySlide109
Specification SearchGiven the model in Figure 1, with 7 unknown paths, the number of models is equivalent to 27=128 possible specifications of the model. 128 different possible models!Slide110Slide111
Specification SearchThe unconstrained model (The one with all seven ambiguous paths in the model) demonstrates satisfactory overall model fit CFI=.95TLI=.92 RMSEA =.07chisq= 476.33, DF=63Slide112
Table. Results of Specification Search in AMOSModelName
Params
df
C
C -
df
AIC
BCC
BIC
C /
df
p
R
2
A
Unconstrained
56
63
476.336
413.336
588.336
589.664
877.039
7.561
0
.11
B
No DIF path to PF1
55
64
476.357
412.357
586.357
587.661
869.904
7.443
0
.11
C
No DIF path to PF1, PF2
54
65
476.396
411.396
584.396
585.677
862.787
7.329
0
.11
D
No DIF paths to PF1, PF2, or PF3
53
66
476.857
410.857
582.857
584.114
856.093
7.225
0
.11
E
No DIF paths, No
educ
to SS
52
67
478.536
411.536
582.536
583.769
850.616
7.142
0
.11
F
No DIF paths, No
educ
to SS, SS to PP
51
68
482.272
414.272
584.272
585.481
847.197
7.092
0
.10
G
No DIF paths, No
educ
to SS, SS to PP/PF
50
69
495.104
426.104
595.104
596.289
852.873
7.175
0
.09
H
Fully Constrained (No DIF or SS paths)
49
70
540.892
470.892
638.892
640.054
891.506
7.727
0
.09
Notes: Reported R
2
values are for the equation in each model with the endogenous PF latent variable with the interpretation of total explained variance in physical functioning given all other paths in the model.
C
is the chi-squared test statistic and
df
are the associated degrees of freedom.Slide113
Specification SearchWhen a number of models are plausible, specification tests can be used as evidence for verification of or improvement over an initial model. Slide114
‘Guidelines’a researcher is ultimately left to decide if the results of the specification tests are unjustly in favor of a certain model due to complexity or sample size, rather than the meaning behind the causal paths. Thus the specification tests act more like guidelines, rather than strict codes dictating the “best” fitting model. Slide115
Selected Strengths & Limitations of SEMStrengthsVery flexibleEstimate and correct for measurement errorLimitationsLarge sample sizesChallenging to learnNeed lots of hands-on experience to learnNeed a strong theoretical basisIt’s easy to mis-specify a model if you have no idea what you are doing.Slide116
Applied Structural Equation Modeling for Everyone!February 22, 2013Indiana University, BloomingtonJoseph J. Sudano, Jr., PhDCenter for Health Care Research and PolicyCase Western Reserve University at The MetroHealth System
Adam T.
Perzynski
, PhD
Center for Health Care Research and Policy
Case Western Reserve University at The MetroHealth System