2.2 CFA using Amos
Amos can be launched from any computer running Windows in the UITS Student Technology Centers by going to Start → Programs → Statistical Software → Amos 6 → Amos Graphics. The following screen will display:
On the far left appear the different tools that can be used to create path diagrams. Just to the right of the toolbar buttons is a column that will display information about the model after estimates have been calculated. The remainder of the screen contains the area where the path diagram will be drawn.
To load the data, go to File → Data Files. The Data Files dialog box then opens. Click on File Name and navigate to the location where the data file is stored. By default, Amos looks for an SPSS file.
Choose the data file you wish to open, click Open, then OK.
The next step is to draw the path diagram. A model where a single latent variable is assumed to underlie the six observed variables will be drawn first. To add the observed variables to the diagram, first click on the blue rectangle in the upper left corner of the tool bar
(alternatively, click on Diagram → Draw Observed
). Then, in the empty drawing area, hold down the left mouse button to draw a rectangle. Left-click five more times to create a total of six equally sized boxes. To add the latent variables, click on the blue oval in the tool box
(alternatively, click on Diagram → Draw Unobserved
). There will be a total of seven latent variables in the diagram: one representing the common factor and six additional variables representing measurement error specific to each of the observed indicators. After drawing one large oval on the right, draw another smaller one on the left and single click five times. You can move the objects you have just drawn by clicking on the Move Objects
located on the tool bar and dragging the pieces of the diagram to where you want them. The path diagram will look something like this:
Next, click on the Draw Paths
and click and drag from the common and unique factors to the appropriate observed variable. For SEMs, latent variables are almost always assumed to “cause” the manifest variables, thus the arrows all point towards the rectangles.
The next step is to name each of the variables. The easiest way to name the observed variables is to bring up a list of variable names in the loaded data file. Go to View → Variables in Dataset. The Variables in Dataset window then opens.
It is now possible to click-and-drag each variable to its corresponding rectangle in the path diagram.
To name the common latent variable, right click inside it and choose Object Properties. Then click on the Text tab and enter LEFTRGHT in the Variable name box. Amos applies the change immediately to the path diagram. It is also possible, if desired, to add a label describing the variable.
Follow the same process to name the unique factors representing measurement error. Name these d1 through d6, yielding the following path diagram:
Without introducing some additional constraints the scales of the latent variables are meaningless and the model is not identified. A common procedure for setting the latent variable scale is to constrain a factor loading to equal one. Do this for the arrow pointing to PRIVTOWN by left-clicking directly on the arrow and choosing Object Properties. Then click on the Paramaters tab and enter 1 in the field labeled Regression weight.
Close the box by clicking on the X in the upper right hand corner. Follow the same steps for the arrows connecting each of the error terms with their respective indicators. When finished, the path diagram should look like this:
Note that there are several ways to draw a path diagram in Amos. A more efficient means of creating the same diagram may have been to draw the oval representing the common latent variable, clicking the Draw a latent variable or add an indicator to a latent variable
, placing the curser inside the oval, and clicking six times. Amos adds six rectangles representing observed indicators along with ovals representing measurement error. The scales of the unique factors are automatically set by constraining the regression weights to equal one. The variables could then be named as described above.
Before estimating the model it is possible to choose the information that will be provided in the output by going to View → Analysis Properties. Click on the Output tab and choose the following options: Minimization history, Standardized estimates, Squared multiple correlations, and Modification indices.
Close the box by clicking on the X in the upper right hand corner. To start the estimation, choose Analyze → Calculate Estimates. After the estimation is completed it is possible to view the parameter estimates in the path diagram by clicking the View the output path diagram button.
By default the unstandardized estimates will display. To bring up the standardized estimates, click on the Standardized estimates option in the column between the tools and the drawing space.
The screen should look like this:
The path diagram now displays the standardized regression weights (factor loadings) for the common factor and each of the indicators. The squared multiple correlation coefficients (R2), describing the amount of variance the common factor accounts for in the observed variables, are also displayed. Additionally, a χ2 (chi-square) statistic is listed in the column between the tools and the path diagram.
It is evident that the three items related to economic values load on the common factor while the standardized regression weights for the three morality items are near zero. The PRIVTOWN and COMPETE variables appear to be the best indicators of LEFTRGHT. Their standardized regression weights are, respectively, .59 and .15. This means that left-right explains about 35% of the variance in PRIVTOWN and 50% of the variance in COMPETE. GOVTRESP is the poorest among the economic indicators of LEFTRGHT, with an R2 of .02 and a standardized regression weight of .15. Meanwhile the three moral values do not appear to have any relationship with the hypothesized LEFTRGHT factor. Each morality item has a corresponding R2 of 0, meaning that LEFTRGHT explains practically no variance in these items. The χ2 statistic of 798.6 (df=9) is very large. The null hypothesis that the model is a good fit to the data is easily rejected.
It is possible to get more information about the model than what appears in the path diagram by going to View → Text output. This opens an output window giving information about the raw data, the model, estimation, model fit, and any additional information requested with the Analysis Properties box utilized earlier. For now consider only the parameter estimates, which are displayed in the output window as follows:
In addition to the information available in the path diagram, the output also displays standard errors, critical ratios (estimate/standard error), and p-values for the regression weights. No p-value is listed for the PRIVTOWN variable because it was constrained to one. Three stars (***) mean that the p-value is less than .001. GOVTRESP and COMPETE have p-values smaller than the conventional .05 significance level while the three p-values for the moral indicators are quite large. In other words, the regression weights associated with the moral values indicators are not significantly different from zero.
It is likely that a two factor model is more appropriate to describe the economic and moral values of Americans. To test this possibility, bring the path diagram back up without the estimates by clicking on View the path diagram:
Click on the Erase objects
and then on the three lines connecting the common factor to the moral values indicators. Next, add a new latent variable as before and name it MORALS. Change the name LEFTRGHT to ECONOMIC to keep the theoretical meaning of the latent variables clear. Draw three single-headed arrows to connect each of the observed moral values items to the MORALS factor and set the scale of the new latent variable by constraining the regression weight for HOMOSEX to equal 1. Finally, allow ECONOMIC and MORALS to covary by connecting them with a two-headed arrow. The new diagram should look like the following:
To estimate this model, go to Analyze → Calculate Estimates. The standardized output can be viewed by clicking on the View the output path diagram button and then on Standardized estimates. This produces:
To get more detailed information about the results, go to View → Text output. Here is a sampling of the output:
The χ2 statistic for model fit is still significant, meaning that the null hypothesis of a good fit to the data can be rejected. The RMSEA likewise suggests that the fit of the model is questionable. The value of .054 exceeds the .05 suggested as a cut-off for accepting the model fit.
Under the Regression Weights heading, the unstandardized loadings appear along with standard errors, a critical ratio, and p-values. All of the unconstrained estimates are significant. When a variable loads on only a single common factor the Standardized Regression Weights can be interpreted as the correlation between the observed variable and that factor. For this model, the regression weights are all significant. In addition, the R2 corresponding to five of the six observed variables indicate that the respective factor explains a respectable portion of the variance (between 35.1% and 62.7%). Only GOVTRESP has a negligible R2 (.022), raising the possibility that this item does not tap the same values dimension as the other two economics questions.
While this model appears to fit the data substantially better than the single factor model, it may still be possible to improve the fit further. Previously when the Analysis Properties dialog box was opened we requested that Amos report modification indices. These indices make suggestions about loosening constraints on certain model parameters in order to improve the overall model fit. As long as any decisions made on the basis of modification indices are theoretically meaningful and do not result in an unidentified model, these suggestions can be helpful in improving model specification. Amos made the following recommendations:
The first box suggests adding either a covariance between the d2 error term and MORALS or a covariance between d4 and d2. The first option violates the assumption that the common and the unique factors are uncorrelated, and the second does not make much sense theoretically. The Regression Weights box, meanwhile, proposes adding single-headed arrows between GOVTRESP and any of several other variables. The first suggestion, to add a path from MORALS to GOVTRESP, seems most plausible. The GOVTRESP variable was only weakly accounted for by the ECONOMIC variable, hinting that the survey item was not tapping the same values dimension as the other two economic values indicators. An alternative possibility is that GOVTRESP is also tied to the morality dimension. According to the modification index, freeing this loading will reduce the χ2 value by about 26.728.
A final model was thus estimated after adding a path from MORALS to GOVTRESP. The standardized output, as displayed in the path diagram, is the following:
Selected output is the following:
The overall model fit appears quite good. The χ2 test yields a value of 7.927 which, evaluated with 7 degrees of freedom, has a corresponding p-value of .339. This p-value is too high to reject the null of a good fit. Additionally the RMSEA is .011, well below the .05 cut-off. Both tests suggest that the model is a good fit to the data.
GOVTRESP has low standardized loadings on both factors (.150 for ECONOMIC and .176 for MORALS), suggesting that it is a rather unreliable indicator of both economic and moral values. However, the other items have moderate to strong standardized loadings. For PRIVTOWN the loading is .615; for COMPETE it is .680; for HOMOSEX it is .638; for ABORTION it is .785, and for EUTHANAS it is .666. The squared multiple correlations provide information on how much variance the common factors account for in the observed variables. Despite receiving a path from both latent variables, GOVTRESP has a low R2 of only .053. The remaining R2 statistics are, in order of increasing magnitude, PRIVTOWN (.378), HOMOSEX (.408), EUTHANAS (.443), COMPETE (.463), and ABORTION (.617). Finally, the correlation between the two common factors is a very small -.011, and the covariance estimate of -.030 is not statistically discernable from zero (p<.808).
In summary, there appear to be two orthogonal dimensions which underlie American attitudes on a number of different issues: one representing economic values and the other representing moral values. It is unclear which dimension the GOVTRESP item was tapping, however, and future surveys should employ a more reliable measure.