university of tennessee at chattanooga · web viewstructural equation models: models that combine...

Measurement and Structural Models 1

Measurement Models and Structural ModelsStructural Equation Models: Models that combine factor analyses with regression analyses. The study of relations between dependent and independent variables in a factor analytic context.They consist of two parts – measurement of constructs and tests of relations among constructs.

Possibilities for SEMs

Latent Variable to Latent Variable

Latent Variable to Observed Variable

Observed Variable to Latent Variable

P1P1P1P1C1C1C1C1

Employee Performance

Conscientiousness

Performance Rating

C1C1C1C1

Conscientiousness

Conscientiousness

Scale score

P1 P1 P1 P1

Employee Performance


Measurement models. Measurement of constructs is done with a measurement model. It is the part that specifies how the independent and dependent latent variables in the model are to be measured – their indicators, and the relationships between the latent variables and the indicators.

The measurement model has traditionally been a confirmatory factor analysis model with 1 or more latent variable(s) and multiple indicators for each latent variable. Recent advances in software may now make it practicable to use exploratory factor analysis models as measurement models in some cases although we will not be using such software (Mplus) in this course.

Example – Does Organization predict GPA over and above Conscientiousness?

Bi-factor EFA as measurement model for Conscientiousness constructs

Bi-factor CFA as measurement model for Conscientiousness constructs

E

GPA

E

GPA

This SEM would be preferred because most researchers would view G and F1 (Org) as being measured more validly than in the CFA measurement model below.

This is the type of SEM we’ll have to use in this course because our software (lavaan) does not yet have the ability to create an EFA bifactor measurement model.


More on Measurement and Structural Models

Measurement Models

The measurement model specifies how each construct represented by a latent variable is indicated by observed variables. It connects the latent variables in the model to the real world.

A measurement model can be both an end and a means to an end depending on the researcher.

It’s an end if it specifies a way to measure a construct that has not been specified previously and that way of measurement is what is important to the researcher. Some researchers, e.g., Biderman, are mainly interested in doing this. We believe that summated scales scores of Big Five domains are contaminated and that factor scores from bifactor models are more appropriate.

It’s a means to an end if it is introduced mainly to provide a solid foundation for the measurement of constructs whose relations are the primary focus. As mentioned before, latent variables are “purer” measures of constructs than summated scores. So, in many instances, researchers use latent variable techniques to measure constructs just so they’ll have good measures of those constructs.

Structural model.

The relations involving factors are called the structural models of the SEM. Many researchers are mainly interested in these relations. For them, the measurement model is simply a means to better assess the constructs involved in those relations.

Structural models are typically regression analyses involving the latent variables within a SEM.

As shown above, the regression analyses can be between latent variables and other latent variables or between latent variables and observed variables. They are called structural regressions by some persons.

You might say, “Wait, I don’t want to conduct a regression analysis, I just want to compare means.” Remember, that with group coding variables, you CAN compare means using regression techniques. So, everything is possible.


Example 1A. Predicting GPA by Conscientiousness Correlated Factor Model factors.

The lavaan commands . . .

> HEXCon <- + read.spss("G:/MDBR/1Nhung/Towson Mplus Presentation/HEXCon n1195 181112.sav",+ use.value.labels=TRUE, max.value.labels=Inf, to.data.frame=TRUE)

> colnames(HEXCon) <- tolower(colnames(HEXCon))

> hexcon.lavaan <- '+ f1 =~ hc1+hc5+hc9+hc13+ f2 =~ hc2+hc6+hc10+hc14+ f3 =~ hc3+hc7+hc11+hc15+ f4 =~ hc4+hc8+hc12+hc16+ eosgpa1 ~ f1+f2+f3+f4'

> hexcon.fit <- sem(hexcon.lavaan,data=HEXCon)

> summary(hexcon.fit,fit.measures=T,standardized=T)

Meta-analyses of the correlation of GPA with Conscientiusness have found that the raw correlation is about 0.2.

The number of studies investigating the relation of GPA to facets can probably be counted on the thumb of one hand.

GPA


lavaan abbreviated output

lavaan (0.5-23.1097) converged normally after 46 iterations

Number of observations 1195

Estimator ML Minimum Function Test Statistic 813.857 Degrees of freedom 110 P-value (Chi-square) 0.000

Comparative Fit Index (CFI) 0.876

RMSEA 0.073 90 Percent Confidence Interval 0.069 0.078 P-value RMSEA <= 0.05 0.000

SRMR 0.058

Parameter Estimates:

Information Expected Standard Errors Standard

Latent Variables: Estimate Std.Err z-value P(>|z|) Std.lv Std.all f1 =~ hc1 1.000 0.926 0.603 hc5 1.042 0.064 16.215 0.000 0.965 0.620 hc9 1.330 0.076 17.432 0.000 1.232 0.694 hc13 1.193 0.066 17.944 0.000 1.105 0.734 f2 =~ hc2 1.000 0.780 0.658 hc6 1.207 0.060 20.194 0.000 0.942 0.743 hc10 1.062 0.060 17.583 0.000 0.829 0.616 hc14 1.055 0.059 17.754 0.000 0.824 0.623 f3 =~ hc3 1.000 0.737 0.531 hc7 1.091 0.076 14.439 0.000 0.804 0.599 hc11 1.094 0.073 15.029 0.000 0.806 0.644 hc15 1.287 0.093 13.814 0.000 0.948 0.557 f4 =~ hc4 1.000 0.912 0.705 hc8 1.152 0.062 18.526 0.000 1.051 0.749 hc12 0.668 0.047 14.272 0.000 0.609 0.499 hc16 0.639 0.049 13.039 0.000 0.582 0.451

Correlated factors CFA does not fit the data well.

All of the loadings are large, leaving the false impression that things are going well for this analysis.


Covariances: Estimate Std.Err z-value P(>|z|) Std.lv Std.all f1 ~~ f2 0.439 0.038 11.557 0.000 0.608 0.608 f3 0.427 0.040 10.605 0.000 0.625 0.625 f4 0.471 0.043 11.025 0.000 0.558 0.558 f2 ~~ f3 0.494 0.040 12.471 0.000 0.860 0.860 f4 0.312 0.032 9.680 0.000 0.438 0.438 f3 ~~ f4 0.379 0.037 10.335 0.000 0.564 0.564

The factors are all highly correlated with each other, as we would expect, since they’re all facets of conscientiousness.

But, recall the problems associated with using highly correlated predictors in a multiple regression analysis.

Regressions: Estimate Std.Err z-value P(>|z|) Std.lv Std.all eosgpa1 ~ f1 -0.009 0.036 -0.246 0.805 -0.008 -0.013 f2 0.158 0.089 1.772 0.076 0.124 0.190 f3 0.087 0.106 0.821 0.411 0.064 0.099 f4 0.048 0.036 1.338 0.181 0.044 0.068

This is an example of how you can do analyses that seem to be OK but possibly lead to stupid conclusions.

The measurement model of the predictors – the four factors from the correlated factors solution - is NOT CORRECT.

Because the measurement of the predictors was screwed up, the results also make no sense.

The reason that none of the predictors was significant is because they are all highly correlated with each other. Remember the highly correlated predictors example from PSY 5130. This is another example of a situation in which the wrong conclusion would be reached.

lavaan’s strange choice to not print the overall R or R-squared prevented us from seeing that, in fact, the conscientiousness facets, AS A GROUP, did predict GPA.

I replicated the analysis using Mplus so that I could get the overall R2.

Hello!! NONE of the predictors is significant in this multiple regression.


Example 1B. Mplus analysis of the same data and diagram.

Mplus commands . . .

title: Spring 2014 HEXACO Data; data: FILE IS 'G:\MDBR\1Nhung\Towson Mplus Presentation\HEX GPA n1195 170409.csv'; variable: names are bidid hh1 - hh16 hs1 - hs16 hx1 - hx16 ha1 - ha16 hc1 - hc16 ho1 - ho16 eosgpa ; usevariables are hc1 - hc16 eosgpa; analysis: Type = general; model: f1 by hc1 hc5 hc9 hc13; f2 by hc2 hc6 hc10 hc14; f3 by hc3 hc7 hc11 hc15; f4 by hc4 hc8 hc12 hc16; eosgpa on f1 f2 f3 f4; !THIS IS THE STRUCTURAL MODEL; output: STDYX;

Selected Mplus output

STANDARDIZED MODEL RESULTS

STDYX Standardization Two-Tailed Estimate S.E. Est./S.E. P-Value

EOSGPA ON F1 -0.013 0.051 -0.244 0.807 F2 0.190 0.108 1.757 0.079 F3 0.099 0.122 0.812 0.417 F4 0.068 0.051 1.334 0.182

R-SQUARE

Observed Two-Tailed Variable Estimate S.E. Est./S.E. P-Value

EOSGPA 0.097 0.018 5.416 0.000

GPA R2 = .10 R = .31


-.01 .19 .10 .07

So, the four correlated factor model factors did not predict well and gave a very confusing picture of the relation of GPA to Conscientiousness and its facets – a classic highly correlated predictors picture.

But a complete understanding of the relation of GPA to Conscientiousness and all its facets must rely on a bifactor measurement model, in my view.


Example 1C. GPA predicted by HEXACO Conscientiousness Bifactor Model factors.This is from the EAOM Mplus Workshop.

Note that all residual (error) latent variables including the residual latent variable associated with EOSGPA are not included in the path diagram in order to reduce clutter.

Lavaan commands . . .

> HEXCon <- + read.spss("G:/MDBR/1Nhung/Towson Mplus Presentation/HEXCon n1195 181112.sav",+ use.value.labels=TRUE, max.value.labels=Inf, to.data.frame=TRUE)

> colnames(HEXCon) <- tolower(colnames(HEXCon))

> hexcon.lavaan <- '+ f1 =~ hc1+hc5+hc9+hc13+ f2 =~ hc2+hc6+hc10+hc14+ f3 =~ hc3+hc7+hc11+hc15+ f4 =~ hc4+hc8+hc12+hc16+ g =~ hc1+hc5+hc9+hc13+hc2+hc6+hc10+hc14+hc3+hc7+hc11+hc15+hc4+hc8+hc12+hc16+ g ~~ 0*f1+ g ~~ 0*f2+ g ~~ 0*f3+ g ~~ 0*f4+ eosgpa1 ~ f1+f2+f3+f4+g'

> hexcon.fit <- sem(hexcon.lavaan,data=HEXCon)

> summary(hexcon.fit,fit.measures=T,standardized=T)

F1= OrgF2= DilF3= PerF4= Pru


lavaan abbreviated output

lavaan (0.5-23.1097) converged normally after 90 iterations Whew! 90!!!???



Model test baseline model:

Minimum Function Test Statistic 5792.107 Degrees of freedom 136 P-value 0.000

User model versus baseline model:

Comparative Fit Index (CFI) 0.942 re 0.95 Tucker-Lewis Index (TLI) 0.916

Loglikelihood and Information Criteria:

Loglikelihood user model (H0) -31911.670 Loglikelihood unrestricted model (H1) -31702.556

Number of free parameters 60 Akaike (AIC) 63943.341 Bayesian (BIC) 64248.495 Sample-size adjusted Bayesian (BIC) 64057.912

Root Mean Square Error of Approximation:

RMSEA 0.054 re 0.05 90 Percent Confidence Interval 0.049 0.059 P-value RMSEA <= 0.05 0.097

Standardized Root Mean Square Residual:

SRMR 0.034 re 0.08Parameter Estimates:


Latent Variables: Estimate Std.Err z-value P(>|z|) Std.lv Std.all f1 =~ Org hc1 1.000 0.652 0.425 hc5 0.146 0.086 1.693 0.091 0.095 0.061 hc9 1.991 0.232 8.576 0.000 1.298 0.732 hc13 0.740 0.080 9.287 0.000 0.482 0.320 f2 =~ Dil hc2 1.000 0.517 0.436 hc6 1.713 0.194 8.828 0.000 0.886 0.699 hc10 0.697 0.090 7.713 0.000 0.361 0.268 hc14 0.405 0.087 4.641 0.000 0.209 0.158 f3 =~ Per hc3 1.000 0.465 0.335 hc7 0.544 0.124 4.376 0.000 0.253 0.189 hc11 1.220 0.178 6.839 0.000 0.568 0.453 hc15 0.428 0.153 2.791 0.005 0.199 0.117 cont’d below

The CFA bifactor model is almost acceptable.


f4 =~ Pru hc4 1.000 0.792 0.613 hc8 0.859 0.090 9.540 0.000 0.680 0.485 hc12 0.609 0.069 8.850 0.000 0.482 0.396 hc16 0.333 0.061 5.466 0.000 0.264 0.204 g =~ hc1 1.000 0.671 0.437 hc5 1.486 0.122 12.195 0.000 0.996 0.640 hc9 1.328 0.092 14.498 0.000 0.891 0.502 hc13 1.418 0.102 13.909 0.000 0.951 0.632 hc2 0.870 0.091 9.551 0.000 0.583 0.492 hc6 0.990 0.108 9.200 0.000 0.664 0.523 hc10 1.063 0.103 10.272 0.000 0.713 0.530 hc14 1.177 0.108 10.932 0.000 0.789 0.597 hc3 0.912 0.101 9.010 0.000 0.612 0.441 hc7 1.089 0.104 10.421 0.000 0.730 0.544 hc11 0.971 0.103 9.391 0.000 0.651 0.520 hc15 1.359 0.130 10.490 0.000 0.911 0.536 hc4 0.864 0.088 9.799 0.000 0.580 0.448 hc8 1.111 0.103 10.837 0.000 0.745 0.531 hc12 0.579 0.071 8.119 0.000 0.389 0.319 hc16 0.796 0.082 9.667 0.000 0.534 0.414Regressions: Estimate Std.Err z-value P(>|z|) Std.lv Std.all eosgpa1 ~ f1 -0.230 0.076 -3.043 0.002 -0.150 -0.231 f2 0.084 0.090 0.934 0.350 0.044 0.067 f3 -0.305 0.166 -1.843 0.065 -0.142 -0.219 f4 -0.086 0.049 -1.753 0.080 -0.068 -0.105 g 0.365 0.045 8.138 0.000 0.245 0.378

Alas, lavaan does not print multiple R or R-squared values.I cannot find a reference for why this is the case.Below are the results from the Mplus analysis illustrated in the EAOM workshop

So, what have we learned from this is that whenever analyzing the results of a multidimensional questionnaire, such as the Big Five or HEXACO questionnaires, you should always consider the applicability of a bifactor measurement model for the predictors and for the dependent variable.

By the way, the strength of the relation shown here is about twice that typically reported.

R = .46.


What is Conscientiousness?

Some questions that remain to be answered

• Organization• 1 I clean my office or home quite frequently.• 5 I plan ahead and organize things, to avoid scrambling at the last minute.• 9 People often joke with me about the messiness of my room or desk.• 13 When working, I sometimes have difficulties due to being disorganized.• Diligence• 2 When working, I often set ambitious goals for myself.• 6 I often push myself very hard when trying to achieve a goal.• 10 Often when I set a goal, I end up quitting without having reached it.• 14 I do only the minimum amount of work needed to get by. • Perfectionism• 3 I often check my work over repeatedly to find any mistakes.• 7 When working on something, I don't pay much attention to small details.• 11 I always try to be accurate in my work, even at the expense of time.• 15 People often call me a perfectionist.• Prudence• 4 I make decisions based on the feeling of the moment rather than on careful thought.• 8 I make a lot of mistakes because I don't think before I act.• 12 I don’t allow my impulses to govern my behavior.• 16 I prefer to do whatever comes to mind, rather than stick to a plan.

The bifactor model assumes that conscientiousness is a characteristic that is independent of organization, diligence, perfectionism, and prudence. So, then, what is it?

1. What is Conscientiousness apart from its facets?Are there “facet free” indicators of Conscientiousness

What kind of questionnaire item will access “pure” conscientiousness without tainting it with organization, diligence, perfectionism, prudence, or any of the many other overt manifestations of conscientious behavior?

The bifactor model assumes that the facets are characteristics that are independent of Conscientiousness. So, then, what are they?

2. What is Organization after the contamination by conscientiousness is removed.Are there “Conscientiousness free” indicators of Organization?

What is organization without Conscientiousness? Is it interest in being organized? Is it enjoyment in being organized? Is it some aesthetic aspect of messiness that I don’t appreciate? How do I describe organization without including mention of Conscientiousness or Conscientiousness-like behaviors?


Example 2A: A measurement model for Big Five personality characteristics and a faking characteristic. Below is an example of a measurement model from Wrensen & Biderman (2005). Participants responded to a Big Five questionnaire twice – once under instructions to respond honestly, again under instructions to “fake good”. The model specifies how the Big Five latent variables and a faking latent variable are to be measured. Mike – Discuss it in detail, including the two types of correlated residuals.

.77

HETL1.67

HETL2.77

HETL3.40

FETL1.16

FETL2.63

FETL3

.26

HATL1.86

HATL2.08

HATL3.21

FATL1.08

FATL2.02

FATL3

.77

HCTL1.63

HCTL2.53

HCTL3.66

FCTL1.67

FCTL2.62

FCTL3.71

HSTL1.79

HSTL2.55

HSTL3.63

FSTL1.64

FSTL2.53

FSTL3.71

HOTL1.62

HOTL2.33

HOTL3.51

FOTL1.43

FOTL2.43

FOTL3

e1

e2

e3

e4

e5

e6

e7

e8e9

e10

e11

e12

e13

e14

e15

e16

e17

e18

e19

e20

e21

e22

e23

e24

e25

e26e27

e28

e29

e30

E

.88

.82

.87

A

C

S

O

.51

.93

.28

.88.79

.73

.84

.89

.74

.84

.79

.57

.18

.04

.00

.29

.20

-.08

.12 .27

.09

.33

Measurement ModelChi-square = 564.595df = 350p = .000CFI = .938GFI = .838RMSEA = .058

.27

.22.24

.46.18

.12

.76.17

.17

.72

.26.31

.66.27

.19

.17

.17

.78

.40

.69

.17

.07

.66

.15

.08

.71

.16

.21

FA

.07.22.10

.28

.77 .33.75

.65

.27.60

.63

.57.34

.76

.80

.19

.55

.42.57

.10

.35

.11

-.05

.39

.12.03

.38

.09

.15

.22

.09

-.03

FA represents variance in the F testlets that is a) common across all 5 dimensions and b) not attributable to any of the Big Five characteristics.It’s a Faking Ability factor here because participants were instructed to fake good.

These covariances are estimated separately because they’re assumed to reflect similarities in responses due to similarities in item wording.

These covariances are estimated separately because they’re assumed to reflect slight differences in amount of faking specific to the Big Five dimensions.We included them to improve goodness-of-fit values.We’ve since quit estimating them, although the issue needs to be revisited.


The above model applied using lavaan

wrensen1 <- read.spss("G:/MDBR/0DataFiles/Wrensen_070114.sav", use.value.labels=TRUE, max.value.labels=Inf, to.data.frame=TRUE)colnames(wrensen1) <- tolower(colnames(wrensen1))

library(lavaan)wrensen1.model3 <- 'E =~ hetl1+hetl2+hetl3+fetl1+fetl2+fetl3A =~ hatl1+hatl2+hatl3+fatl1+fatl2+fatl3C =~ hctl1+hctl2+hctl3+fctl1+fctl2+fctl3S =~ hstl1+hstl2+hstl3+fstl1+fstl2+fstl3O =~ hotl1+hotl2+hotl3+fotl1+fotl2+fotl3FA =~ fetl1+fetl2+fetl3 + fatl1+fatl2+fatl3 + fctl1+fctl2+fctl3 + fstl1+fstl2+fstl3 + fotl1+fotl2+fotl3hetl1~~fetl1 hetl2~~fetl2 hetl3~~fetl3hatl1~~fatl1 hatl2~~fatl2 hatl3~~fatl3hctl1~~fctl1 hctl2~~fctl2 hctl3~~fctl3hstl1~~fstl1 hstl2~~fstl2 hstl3~~fstl3hotl1~~fotl1 hotl2~~fotl2 hotl3~~fotl3fetl1~~fetl2fetl1~~fetl3fetl2~~fetl3fatl1~~fatl2fatl1~~fatl3fatl2~~fatl3fctl1~~fctl2fctl1~~fctl3fctl2~~fctl3fstl1~~fstl2fstl1~~fstl3fstl2~~fstl3fotl1~~fotl2fotl1~~fotl3fotl2~~fotl3'wrensen1.fit3 <- cfa(wrensen1.model3,data=wrensen1)summary(wrensen1.fit3,fit.measures=T,standardized=T)

Note on lavaan:

There is a website of information and examples using lavaan. It’s

http://lavaan.ugent.be/tutorial/sem.html

Click on the Tutorial link for even more info.

Send them an email asking them to incorporate multiple R in the output of their regression analyses.

http://lavaan.ugent.be/tutorial/sem.html


Example 1B. A Structural model of faking ability added to the above measurement model.The following schematic path diagram shows the measurement model above along with a structural model relating faking ability to a collection of observed variables.. The structural model is in red. In this example, the structural model is between exogenous observed variables and the Faking Ability latent variable. Because Faking Ability is being predicted it’s endogenous in this model.

CA: Cognitive ability; EI: Emotional Intelligence; INT: Integrity; SD: Social Desirability; SM: Self Monitoring; Order: Order of presentation of H and F conditions. Notice that all of the regression relationships are partial, since we performed a multiple regression. The CA->FA relationship has been replicated a couple of times – Smarter people are better able to fake when told to do so.

.19*

.16*

.18*

-.24*

-.02

-.13

Order

SM

SD

INT

EI

CA

.37

.30

.30

.39

.18

.21

.06

.19

.07

.51

.49

.56

.71

.14

.55

FA

.37

.74

.43

.83

.37

.80

.26

.55

.25

.85

F-O

H-O

F-S

H-S

F-C

H-C

F-A

H-A

F-E

H-E

O

S

C

A

E

Mean of 3 standardized loadings: .88, .82, .87 (see above).

Mean of 3 residual correlations.















































Some oddities in these models.

1. The standardized loadings of all faking testlets on the trait factors are smaller than the loadings of honest testlets on the same factors!!!???

2. The correlated residuals of taking testlets are smaller than those of honest testlets. Values not shown in this figure.


The above model using lavaan. Same as the above model with the except of line in red.

library(foreign, pos=15)wrensen1 <- read.spss("G:/MDBR/0DataFiles/Wrensen_070114.sav", use.value.labels=TRUE, max.value.labels=Inf, to.data.frame=TRUE)colnames(wrensen1) <- tolower(colnames(wrensen1))

wrensen1.model4 <- 'E =~ hetl1+hetl2+hetl3+fetl1+fetl2+fetl3A =~ hatl1+hatl2+hatl3+fatl1+fatl2+fatl3C =~ hctl1+hctl2+hctl3+fctl1+fctl2+fctl3S =~ hstl1+hstl2+hstl3+fstl1+fstl2+fstl3O =~ hotl1+hotl2+hotl3+fotl1+fotl2+fotl3FA =~ fetl1+fetl2+fetl3 + fatl1+fatl2+fatl3 + fctl1+fctl2+fctl3 + fstl1+fstl2+fstl3 + fotl1+fotl2+fotl3hetl1~~fetl1 hetl2~~fetl2 hetl3~~fetl3hatl1~~fatl1 hatl2~~fatl2 hatl3~~fatl3hctl1~~fctl1 hctl2~~fctl2 hctl3~~fctl3hstl1~~fstl1 hstl2~~fstl2 hstl3~~fstl3hotl1~~fotl1 hotl2~~fotl2 hotl3~~fotl3fetl1~~fetl2fetl1~~fetl3fetl2~~fetl3fatl1~~fatl2fatl1~~fatl3fatl2~~fatl3fctl1~~fctl2fctl1~~fctl3fctl2~~fctl3fstl1~~fstl2fstl1~~fstl3fstl2~~fstl3fotl1~~fotl2fotl1~~fotl3fotl2~~fotl3FA ~ wondlic+ei+sd+selfmon+order'wrensen1.fit4 <- sem(wrensen1.model4,data=wrensen1)summary(wrensen1.fit4,fit.measures=T,standardized=T)


Output of lavaan STRUCTURAL model.

> summary(wrensen1.fit4,fit.measures=T,standardized=T)lavaan (0.5-23.1097) converged normally after 134 iterations



Model test baseline model:

Minimum Function Test Statistic 3542.818 Degrees of freedom 585 P-value 0.000

User model versus baseline model:

Comparative Fit Index (CFI) 0.892 Tucker-Lewis Index (TLI) 0.872

Loglikelihood and Information Criteria:

Loglikelihood user model (H0) -6337.124 Loglikelihood unrestricted model (H1) -5929.261

Number of free parameters 120 Akaike (AIC) 12914.247 Bayesian (BIC) 13287.686 Sample-size adjusted Bayesian (BIC) 12907.756

Root Mean Square Error of Approximation:

RMSEA 0.062 90 Percent Confidence Interval 0.055 0.070 P-value RMSEA <= 0.05 0.005

Standardized Root Mean Square Residual:

SRMR 0.085

Parameter Estimates:


Latent Variables: Estimate Std.Err z-value P(>|z|) Std.lv Std.all E =~ hetl1 1.000 0.751 0.785 hetl2 1.118 0.178 6.299 0.000 0.839 0.930 hetl3 0.246 0.059 4.179 0.000 0.185 0.339 fetl1 0.362 0.076 4.762 0.000 0.272 0.317 fetl2 0.281 0.096 2.923 0.003 0.211 0.280 fetl3 0.093 0.053 1.763 0.078 0.070 0.129 A =~ hatl1 1.000 0.410 0.553 hatl2 1.534 0.474 3.238 0.001 0.629 0.839 hatl3 0.247 0.089 2.788 0.005 0.101 0.264 fatl1 0.796 0.088 9.023 0.000 0.327 0.493 fatl2 0.366 0.215 1.698 0.090 0.150 0.224 fatl3 0.169 0.090 1.879 0.060 0.069 0.172 C =~ hctl1 1.000 0.643 0.864 hctl2 1.170 0.105 11.176 0.000 0.753 0.843 hctl3 0.810 0.082 9.877 0.000 0.521 0.727 fctl1 0.811 0.042 19.154 0.000 0.522 0.751 fctl2 0.244 0.080 3.043 0.002 0.157 0.176


fctl3 0.181 0.059 3.066 0.002 0.116 0.176 S =~ hstl1 1.000 0.886 0.917 hstl2 0.775 0.068 11.426 0.000 0.687 0.761 hstl3 0.801 0.063 12.735 0.000 0.710 0.830 fstl1 0.346 0.061 5.626 0.000 0.306 0.314 fstl2 0.358 0.060 5.945 0.000 0.317 0.360 fstl3 0.241 0.064 3.746 0.000 0.213 0.223 O =~ hotl1 1.000 0.539 0.819 hotl2 1.021 0.125 8.154 0.000 0.551 0.786 hotl3 0.923 0.130 7.107 0.000 0.498 0.611 fotl1 0.759 0.057 13.240 0.000 0.410 0.641 fotl2 0.350 0.096 3.660 0.000 0.189 0.285 fotl3 0.249 0.098 2.543 0.011 0.135 0.181 FA =~ fetl1 1.000 0.481 0.561 fetl2 0.524 0.094 5.556 0.000 0.252 0.335 fetl3 0.555 0.097 5.713 0.000 0.267 0.493 fatl1 0.095 0.068 1.401 0.161 0.046 0.069 fatl2 0.311 0.110 2.813 0.005 0.150 0.223 fatl3 0.115 0.069 1.668 0.095 0.055 0.137 fctl1 0.375 0.069 5.451 0.000 0.180 0.260 fctl2 1.493 0.194 7.704 0.000 0.719 0.803 fctl3 1.087 0.143 7.615 0.000 0.523 0.794 fstl1 1.572 0.205 7.675 0.000 0.757 0.776 fstl2 1.249 0.174 7.178 0.000 0.601 0.682 fstl3 1.474 0.200 7.366 0.000 0.710 0.743 fotl1 0.334 0.068 4.887 0.000 0.161 0.252 fotl2 0.763 0.124 6.148 0.000 0.368 0.555 fotl3 0.947 0.144 6.563 0.000 0.456 0.615

Regressions: Estimate Std.Err z-value P(>|z|) Std.lv Std.all FA ~ wondlic 0.016 0.006 2.580 0.010 0.033 0.209 ei 0.006 0.003 1.957 0.050 0.013 0.157 sd -0.044 0.016 -2.757 0.006 -0.091 -0.229 selfmon -0.007 0.009 -0.718 0.473 -0.014 -0.057 order -0.153 0.076 -1.998 0.046 -0.317 -0.158


Covariances: Estimate Std.Err z-value P(>|z|) Std.lv Std.all .hetl1 ~~ .fetl1 0.051 0.044 1.163 0.245 0.051 0.131 .hetl2 ~~ .fetl2 0.046 0.043 1.059 0.289 0.046 0.203 .hetl3 ~~ E ~~ A 0.055 0.033 1.658 0.097 0.180 0.180 C -0.011 0.043 -0.265 0.791 -0.023 -0.023 S -0.043 0.058 -0.749 0.454 -0.065 -0.065 O 0.097 0.040 2.426 0.015 0.239 0.239 A ~~ C 0.051 0.029 1.758 0.079 0.192 0.192 S -0.031 0.035 -0.869 0.385 -0.084 -0.084 O 0.028 0.023 1.187 0.235 0.124 0.124 C ~~ S 0.126 0.051 2.449 0.014 0.221 0.221 O 0.031 0.032 0.958 0.338 0.089 0.089 S ~~ O 0.156 0.046 3.380 0.001 0.326 0.326

Variances: Estimate Std.Err z-value P(>|z|) Std.lv Std.all .hetl1 0.352 0.091 3.882 0.000 0.352 0.384 .hetl2 0.110 0.103 1.073 0.283 0.110 0.135 .hetl3 0.262 0.029 8.928 0.000 0.262 0.885 .fetl1 0.432 0.053 8.121 0.000 0.432 0.585

E 0.563 0.124 4.552 0.000 1.000 1.000 A 0.168 0.066 2.561 0.010 1.000 1.000 C 0.414 0.065 6.378 0.000 1.000 1.000 S 0.786 0.109 7.180 0.000 1.000 1.000 O 0.291 0.054 5.425 0.000 1.000 1.000 .FA 0.198 0.050 3.954 0.000 0.853 0.853


Example 3. From Biderman, Nguyen, Mullins, & Luna (2008). Example 3A. The Independent Variable Measurement Model. It’s similar to the above measurement model, with the exception that it’s from only one administration of the Big Five. The model here was applied to data obtained under instructions to respond honestly. So the latent variable that represents variance not attributable to the Big Five is called M, for Method Bias. (We now know that it’s more than method bias, but the letter, M, has stuck.)

M

O

S

C

A

E

O10O9O8O7O6O5O4O3O2O1

S10S9S8S7S6S5S4S3S2S1

C10C9C8C7C6C5C4C3C2C1

A10A9A8A7A6A5A4A3A2A1

E10E9E8E7E6E5E4E3E2E1 The model applied here is

called a bifactor model. It’s called that because each observed variable has two (hence “bi”) latent variables influencing it. The first is the trait latent variable. For example, each Extraversion item is influenced by the trait of Extraversion.

But each item is also influence by a 2nd latent variable – called M in this model. Having each observed variable influenced by two latent variables is what causes a model to be called a bifactor model.

Bifactor models are not well known in the CFA or SEM literature, although they’re gaining in popularity.

The M factor is known as the general factor in bifactor literature. Such factors have also been studied under the name “common method factors”.

See Reise, S. P. (2012). The rediscovery of bifactor measurement models. Multivariate Behavioral Research, 47, 667-696.


Example 3B. The Dependent Variable Measurement Model

Supervisors rated their employees on three characteristics . . .

1. Customer Service2. Ability to make sales.3. Ability to collect delinquent payments.

These three ratings correlated above .5 with each other.

They were treated as indicators of a Performance Latent variable, labeled P in the diagram.

P

Colls

Sales

Custs


Example 3C. The Structural Model. Predicting supervisor ratings of performance.In this diagram, the regression links of the structural model are red. All of the Big Five latent variables and M are part of both the measurement model and the structural model. N=770.

.

M

O

S

C

A

E

O10O9O8O7O6O5O4O3O2O1

S10S9S8S7S6S5S4S3S2S1

C10C9C8C7C6C5C4C3C2C1

A10A9A8A7A6A5A4A3A2A1

E10E9E8E7E6E5E4E3E2E1

.174c

-.011

P

Colls

Sales

Custs

-.104

-.102

-.142a

-.134a

We found M was the best predictor of Supervisor Ratings. We were puzzled about that until recently when we discovered that under instructions to respond honestly, M appears to represent the respondent’s affective state. Thus, persons with positive affective states (happy people) had higher ratings than those with negative affective states

Note: a p < .05b p < .01c p < .001

university of tennessee at chattanooga · web viewstructural equation models: models that combine...

Documents