Structural Equation Modeling with AMOS 5 - Denver, Colorado · AMOS automatically compares your model to a “saturated model” (all variables correlated with all others) and to

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Structural Equation Modeling with AMOS 5.0

1. Raw data should be saved in SPSS (.sav) format.

2. Open AMOS Graphics from the Start Menu.

3. Select “New” from the File Menu.

4. Select “Data Files” from the File Menu. Then click on the “File Name” button,

and use the browser window to locate your SPSS data file.

5. You can use the “View Data” button to check your data (it will appear in SPSS).

6. Click “OK” to go on.



7. Use the “List Variables in Dataset” button on the left-hand side to see the

variables in your dataset:

8. Drag and drop the observed variables that you want to include in your model onto

the gray workspace:



9. Use the oval button on the left to draw in any unobserved (latent) variables that

you want to include in your model. Double click on the new variable (the oval

you drew) in order to give it a name. Each variable must be named.

10. Use the arrow tool on the left to draw paths between variables. The path always

goes in the direction of latent variable (oval) to observed variable (rectangle).



11. Use the double-sided arrow to draw covariances:

12. Add error variances (residuals) for each observed variable. These are unique,

unobserved variables for each observed variable. This is an important step, and

AMOS will give you a warning message and be unable to proceed if you omit it.



13. Enter regression weights for each of the relationships between observed and

unobserved variables. (Unless you have reason to believe that some associations

will be stronger than others, use 1 for all regression weights. If you believe, e.g.,

that one association is 2x as strong as another, use 2 for the weight that is twice as

strong, and 1 for the other). Note that it is not necessary to specify regression

weights for every relationship. Specify them for each error term, and specify the

first regression weight for each unobserved to observed variable, as shown below:



14. In the “View/Set” menu, select “Analysis Properties.” Go to the “Output” tab.

Select “Standardized Estimates.”

15. Run the model using the “calculate estimates” button on the left-side control

panel:

AMOS stands for “Analysis of MOment Structures.” AMOS calculates estimates for

the parameters and fits the model based on mean and covariance structures, to make

these fit the data as closely as possible.



16. View the results by clicking on the “View Output” button. This will show you the

solution for the model. Select “standardized” or “unstandardized” estimates.

17. Test the model against other plausible models by clicking on the “view text”

button to see all statistical outputs from the procedure:



18. “Notes for Model” shows the chi-square statistic that tests for significant

deviations between the model and the data. In this case, the p-value is significant,

meaning that the model is not a good fit for the data (chi-square is used as a

“badness of fit” statistic in SEM).

Notes for Model (Default model)

Computation of degrees of freedom (Default model)

Number of distinct sample

moments: 55

Number of distinct parameters

to be estimated: 21

Degrees of freedom (55 - 21): 34

Result (Default model)

Minimum was achieved

Chi-square = 52.738

Degrees of freedom = 34

Probability level = .021

19. If adequate fit was not achieved, go back and change the model to improve it.

There are some theoretical reasons (from earlier exploratory factor analysis work)

to expect that the “Coding” subtest was not strongly related to the underlying

constructs of Verbal and Performance IQ. Therefore, this subtest was deleted

from the model.

20. In the “analysis properties” tab, you can select “modification indices.” This will

print suggestions in the output for how to improve your model. However, be

careful to take suggestions only if they make theoretical sense.

21. This time, the model fits much better (chi square = 38.1, p = .058). Interestingly,

the paths showing the strongest correlations with the underlying factors

(Vocabulary with VIQ and Block Design with PIQ) are the same as those

suggested in published results for this test and found with exploratory factor

analysis in other samples. Overall, the results are consistent with the predicted

two-factor model of intelligence.



22. The chi-square does not have to indicate a complete fit in order to accept the model –

with a large sample, the chi-square’s p-value is always likely to be small. The

following alternative fit indices are provided in the text output. AMOS automatically

compares your model to a “saturated model” (all variables correlated with all others)

and to an “independence model” (all variables uncorrelated with all others), to ensure

that your model is a better fit. The various fit indices are described below:

Model Fit Summary

CMIN

Model NPAR CMIN DF P CMIN/DF

Default model 19 38.177 26 .058 1.468

Saturated model 45 .000 0

Independence model 9 429.884 36 .000 11.941

RMR, GFI

Model RMR GFI AGFI PGFI

Default model .433 .952 .918 .550

Saturated model .000 1.000



Model RMR GFI AGFI PGFI

Independence model 2.521 .532 .415 .425

Baseline Comparisons

Model NFI

Delta1

RFI

rho1

IFI

Delta2

TLI

rho2 CFI

Default model .911 .877 .970 .957 .969

Saturated model 1.000 1.000 1.000

Independence model .000 .000 .000 .000 .000

Parsimony-Adjusted Measures

Model PRATIO PNFI PCFI

Default model .722 .658 .700

Saturated model .000 .000 .000

Independence model 1.000 .000 .000

NCP

Model NCP LO 90 HI 90

Default model 12.177 .000 32.787

Saturated model .000 .000 .000

Independence model 393.884 330.800 464.416

FMIN

Model FMIN F0 LO 90 HI 90

Default model .219 .070 .000 .188

Saturated model .000 .000 .000 .000

Independence model 2.471 2.264 1.901 2.669

RMSEA

Model RMSEA LO 90 HI 90 PCLOSE

Default model .052 .000 .085 .432

Independence model .251 .230 .272 .000

HOELTER

Model HOELTER

.05

HOELTER

.01

Default model 178 209

Independence model 21 24



• CMIN – minimum value of the discrepancy between the model and the data.

This is the same as the chi-square statistic in the “notes for model” section.

• CMIN/DF – the chi-square divided by its degrees of freedom. Acceptable

values are in the 3/1 or 2/1 range. Using this criterion, our earlier model

(without the added path from PIQ to COMP) was also acceptable

(CMIN/DF = 1.65)

• RMR, GFI – The RMR [Root Mean Square Residual] is the square root of

the average squared amount by which your model’s estimated sample

variances and covariances differ from their actual values in the data. The

smaller the RMR the better, with RMR = 0 indicating a perfect fit. The GFI

[Goodness of Fit Index] is similar to the Baseline Comparisons described

below, giving a statistic between 0 and 1, with 1 indicating perfect fit, and is

used with maximum likelihood estimation for missing data. The AGFI

[Adjusted Goodness of Fit Index] takes into account the degrees of freedom

available for testing the model (this statistic can have values below zero). The

PGFI [Parsimony Goodness of Fit Index] is another modification of the GFI

that also takes into account the degrees of freedom available for testing the

model.

• Baseline Comparisons – NFI [Normed Fit Index] shows how far between the

(terribly fitting) independence model and the (perfectly fitting) saturated

model the detaulf model is. In this case, it’s 91% of the way to perfect fit. RFI

[Relative Fit Index] is the NFI standardized based on the df of the models,

with values close to 1 again indicating a very good fit. IFI [Incremental Fit

Index], TLI [Tucker-Lewis Coefficient], and CFI [Comparative Fit Index] are

similar. Note that TLI is usually between 0 and 1, but is not limited to that

range.

• Parsimony-Adjusted Measures – The PRATIO [Parsimony Ratio] is an

overall measure of how parsimonious the model is. It is defined as the df of

the current model divided by the df of the independence model. It can be

interpreted as “the current model is X% as complex as the independence

model.” The difference between this number and 1 is how much more

efficient your model is than the independence model. PRATIO is used to

calculate two other statistics: PNFI [Parsimonious Normed Fit Index] is

another modification of the NFI that takes into account the df (i.e.,

complexity) of the model. Similarly, the PCFI [Parsimonious Comparative Fit

Index] is a df-adjusted modification of the CFI. These two measures are likely

to be lower than the NFI and CFI, because they take model complexity into

account.

• NCP – the noncentrality parameter. The columns labeled “LO 90” and “HI

90” give the 90% confidence interval for this statistic. This statistic can also

be interpreted as a chi-square, with the same degrees of freedom as in CMIN.

• FMIN – F0 is the noncentrality parameter (NCP) divided by its degrees of

freedom. This is similar to the CMIN/DF statistic. The results also give the

lower and upper limits of a 90% confidence interval for this statistic (LO 90

and HI 90 under the FMIN heading).



• RMSEA – F0 tends to favor more complex models. RMSEA is a corrected

statistic that gives a penalty for model complexity, calculated as the square

root of F0 divided by DF (RMSEA stands for “root mean squared error of

approximation”). Again, upper and lower bounds of a 90% confidence interval

are given. RMSEA values of .05 or less are good fit, <.1 to >.05 are moderate,

and .1 or greater are unacceptable. RMSEA = .00 indicates perfect fit. The

“PCLOSE” statistic that goes with this result is the probability of a hypothesis

test that the population RMSEA is no greater than .05 (so, you want this result

to be nonsignificant [p > .05], because you do not want to prove that the

RMSEA is significantly greater than .05).

• HOELTER – Hoelter’s “critical N” is the largest sample size for which one

would accept the hypothesis that a model is correct (in other words, the

sample size above which the chi-square goodness of fit test would go from

nonsignificant to significant). AMOS reports a critical N for significance

levels of .05 (which was used by Hoelter) and .01. The result can be

interpreted as “the model would be rejected at the [.05/.01] level with a

sample size of greater than X.” Hoelter suggests that models which would be

rejected only with 200 or more participants (a number of 200 or higher in the

Hoelter section of the output) are an adequate fit for the data. Numbers

smaller than 200 suggest an inadequate fit. Arbuckle disagrees with the use of

this criterion, and it’s not one of the more commonly reported statistics for

SEM, but other experts may use it.

The remaining measures are intended for comparing multiple models, rather than

evaluating goodness of fit for a single model:

AIC

Model AIC BCC BIC CAIC

Default model 76.177 78.494 136.308 155.308

Saturated model 90.000 95.488 232.415 277.415


ECVI

Model ECVI LO 90 HI 90 MECVI

Default model .438 .368 .556 .451

Saturated model .517 .517 .517 .549


• AIC – the Akaike Information Criterion is calculated as the discrepancy (C) +

2q (a complexity statistic).

• BCC – the Browne-Cudeck Criterion imposes a slightly greater penalty for

model complexity than the AIC does. Arbuckle (author of the AMOS

program) recommends this particular fit index.



• BIC – the Bayesian Information Criterion assigns an even greater penalty for

complexity, and therefore has a tendency to choose parsimonious models.

This can only be used in single-group models.

• CAIC – the Consistent AIC has a greater penalty for complexity than AIC or

BCC, but not as much as BIC does. Can only be used in single-group models.

• ECVI – Arbuckle reports that “except for a constant scale factor, ECVI

[expected cross-validation index] is the same as AIC” (it is equal to AIC / n).

Upper and lower 90% confidence interval limits are also given for ECVI.

• MECVI is similar, equal to BCC / n. When maximum likelihood estimation

has been used to compensate for missing data, Arbuckle provides a

recommendation to use MECVI instead of ECVI.

23. Specification Search options – An extra feature in AMOS 5.0 is the ability to do a

specification search. This feature allows you to test various models simultaneously,

by specifying that some relationships between variables are optional. For instance,

what if I wanted to test the different fit obtained by using a model with two correlated

factors of IQ versus two uncorrelated factors? I could use the “Specification Search”

command in the “Model-Fit” menu.

a. To see the optional relationships correctly, first go to the “View-Set”

menu and select “Interface Properties”:



b. On the “Accessibility” tab, check the “alternative to color” checkbox. This

will cause optional relationships between variables to show up as dashed

lines on the path diagram (rather than just showing up in blue, which is

harder to see):

c. Open the Specification Search toolbar using the “binoculars” tool:



d. Use the dashed-arrow tool to make some relationships between variables

optional components of the model. Just click on the tool, then click on the

relationship to be changed. You can reverse this action if necessary by

using the solid-line tool that’s just to the right on the toolbar.

e. Run the specification search using the arrow tool on the toolbar. An output

section will appear in the toolbar itself, showing results for each model:

f. You can interpret models using the chi-square and p-values provided, or

you can see more details. If you double click on one of the lines in this

window, or select it and then use the blue-box tool on the toolbar, you will

see the actual path diagram to know which of the optional components

have been included or omitted. In this case, Model 2 has a higher p-value

(which you want with SEM), and includes the VIQ-PIQ covariance.



g. If you got a lot of options, AMOS has a “view short list” button on the

toolbar to help you sift through them.

h. To see RMSEA and other statistics, use the checkbox icon on the toolbar,

and under the “Current Results” tab, check the box for “Derived fit

indices” in the “Display” section. Also, if you select “Akaike weights” in

the lower section of this menu, you can get a statistic in the “BCC”

column that can be interpreted as the likelihood of the model given the

data. This is a useful way to compare models. The model with the smallest

value here (or the lowest BIC value originally) is the best fit for the data.

i. You can also see comparisons between models graphically, using the

“Plots” button on the toolbar. Select “Best Fit” and “AIC” or “BIC.”

“BCC” is also possible, but imposes a greater penalty for complexity.

This graph shows you fit (on the y axis) vs. complexity (on the x axis), wth

the lowest y value showing you the best fit vs. complexity ratio. Click on

the individual data point, and AMOS will tell you which model number

was selected. Click on that model in the output window to see the model.

Structural Equation Modeling with AMOS 5 - Denver, Colorado · AMOS automatically compares your model to a “saturated model” (all variables correlated with all others) and to

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