IBM SPSS Amos 21 - Manual

vii Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Results for Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Testing Model B against Model A. . . . . . . . . . . . . . . . . . . . . . . 96 Modeling in VB.NET. . . . . . . . . . . . . . ...

viii Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Model C . . . . . . . . . . . . . . . . . . . . . . . . . ...

ix 9 An Alternative to Analysis of Covariance 145 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Analysis of Covariance and Its Alternative . . . . . . . . . . . . . . . . 145 About the Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 An...

x Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Graphics Output. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . ...

xii Multiple Model Input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Mean Structure Modeling in VB.NET . . . . . . . . . . . . . . . . . . . 217 Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Model B . . . . . . . . . . . . . . . . . . . . . . . . . . ...

xiv Results for Model Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Modeling in VB.NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 Model B . . . . . . . . . . . . . . . . . . . . . . . ....

xv Graphics Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 Output from Models A and B. . . . . . . . . . . . . . ....

xvi About the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 About the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 Text Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 Modeling in VB.NET . . . . . . . . . . . . . . . . . ...

xvii Performing the Specification Search . . . . . . . . . . . . . . . . . 346 Using BIC to Compare Models . . . . . . . . . . . . . . . . . . . . . 347 Viewing the Scree Plot . . . . . . . . . . . . . . . . . . . . . . . . . 348 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

xviii Customizing the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 369 Model 24b: Comparing Factor Means . . . . . . . . . . . . . . . . . . . 370 Specifying the Model. . . . . . . . . . . . . . . . . . . . . . . . . . 370 Removing Constraints . . . . . . . . . . . . . . . . . . . . ...

xx 29 Estimating a User-Defined Quantityin Bayesian SEM 437 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 About the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 The Stability of Alienation Model . . . . . . . . . . . . . . . . . . . ....

xxi Posterior Predictive Distributions . . . . . . . . . . . . . . . . . . . . . . 481 Imputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 General Inequality Constraints on Data Values . . . . . . . . . . . . . . 488 33 Ordered-Categorical Data 489 Introduction . . ....

xxii Specifying the Data File . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 Specifying the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 Constraining the Parameters . . . . . . . . . . . . . . . . . . . . . 546 Fitting the Model . . . . . . . . . . . . . . . . . . . ....

xxiii Other Aspects of the Analysis in Addition to Model Specification . . . 588 Defining Program Variables that Correspond to Model Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588 Part III: Appendices A Notation 591 B Discrepancy Functions 593 C Measures of Fit 597 Mea...

xxiv Comparisons to a Baseline Model . . . . . . . . . . . . . . . . . . . . . 608 NFI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 RFI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 IFI . . . . . . . . . . . . . . . . . . . . . . . . . . ....

1 C h a p t e r 1 Introduction IBM SPSS Amos implements the general approach to data analysis known as structural equation modeling (SEM), also known as analysis of covariance structures , or causal modeling . This approach includes, as special cases, many well- known conventional techniques, includ...

2 C h a p t e r 1 Structural equation modeling (SEM) is sometimes thought of as esoteric and difficult to learn and use. This is incorrect. Indeed, the growing importance of SEM in data analysis is largely due to its ease of use. SEM opens the door for nonstatisticians to solve estimation and hypoth...

3 I n t r o d u c t i o n models. It provides a test of univariate normality for each observed variable as well as a test of multivariate normality and attempts to detect outliers. IBM SPSS Amos accepts a path diagram as a model specification and displays parameter estimates graphically on a path di...

4 C h a p t e r 1 Example 9 and those that follow demonstrate advanced techniques that have so far not been used as much as they deserve. These techniques include: „ Simultaneous analysis of data from several different populations. „ Estimation of means and intercepts in regression equations. „ Maxi...

5 I n t r o d u c t i o n „ Structural Equation Modeling: A Multidisciplinary Journal contains methodological articles as well as applications of structural modeling. It is published by Taylor and Francis ( http://www.tandf.co.uk ). „ Carl Ferguson and Edward Rigdon established an electronic mailing...

8 C h a p t e r 2 demonstrates the menu path. For information about the toolbar buttons and keyboard shortcuts, see the online help. About the Data Hamilton (1990) provided several measurements on each of 21 states. Three of the measurements will be used in this tutorial: „ Average SAT score „ Per c...

9 T u t o r i a l : G e t t i n g S t a r t e d w i t h A m o s G r a p h i c s The following path diagram shows a model for these data: This is a simple regression model where one observed variable, SAT , is predicted as a linear combination of the other two observed variables, Education and Income...

10 C h a p t e r 2 Creating a New Model E From the menus, choose File > New . Your work area appears. The large area on the right is where you draw path diagrams. The toolbar on the left provides one-click access to the most frequently used buttons. You can use either the toolbar or menu commands...

11 T u t o r i a l : G e t t i n g S t a r t e d w i t h A m o s G r a p h i c s Specifying the Data File The next step is to specify the file that contains the Hamilton data. This tutorial uses a Microsoft Excel 8.0 ( *.xls ) file, but Amos supports several common database formats, including SPSS S...

12 C h a p t e r 2 E In the drawing area, move your mouse pointer to the right of the three rectangles and click and drag to draw the ellipse. The model in your drawing area should now look similar to the following: Naming the Variables E In the drawing area, right-click the top left rectangle and c...

13 T u t o r i a l : G e t t i n g S t a r t e d w i t h A m o s G r a p h i c s Your path diagram should now look like this: Drawing Arrows Now you will add arrows to the path diagram, using the following model as your guide: E From the menus, choose Diagram > Draw Path . E Click and drag to dra...

14 C h a p t e r 2 Constraining a Parameter To identify the regression model, you must define the scale of the latent variable Other . You can do this by fixing either the variance of Other or the path coefficient from Other to SAT at some positive value. The following shows you how to fix the path ...

15 T u t o r i a l : G e t t i n g S t a r t e d w i t h A m o s G r a p h i c s Altering the Appearance of a Path Diagram You can change the appearance of your path diagram by moving and resizing objects. These changes are visual only; they do not affect the model specification. To Move an Object E...

16 C h a p t e r 2 To Undo an Action E From the menus, choose Edit > Undo . To Redo an Action E From the menus, choose Edit > Redo . Setting Up Optional Output Some of the output in Amos is optional. In this step, you will choose which portions of the optional output you want Amos to display a...

17 T u t o r i a l : G e t t i n g S t a r t e d w i t h A m o s G r a p h i c s E Close the Analysis Properties dialog box.

18 C h a p t e r 2 Performing the Analysis The only thing left to do is perform the calculations for fitting the model. Note that in order to keep the parameter estimates up to date, you must do this every time you change the model, the data, or the options in the Analysis Properties dialog box. E F...

19 T u t o r i a l : G e t t i n g S t a r t e d w i t h A m o s G r a p h i c s To View Graphics Output E Click the Show the output path diagram button . E In the Parameter Formats pane to the left of the drawing area, click Standardized estimates .

20 C h a p t e r 2 Your path diagram now looks like this: The value 0.49 is the correlation between Education and Income . The values 0.72 and 0.11 are standardized regression weights. The value 0.60 is the squared multiple correlation of SAT with Education and Income . E In the Parameter Formats pa...

21 T u t o r i a l : G e t t i n g S t a r t e d w i t h A m o s G r a p h i c s E Click Print . Copying the Path Diagram Amos Graphics lets you easily export your path diagram to other applications such as Microsoft Word. E From the menus, choose Edit > Copy (to Clipboard) . E Switch to the othe...

23 E x a m p l e 1 Estimating Variances and Covariances Introduction This example shows you how to estimate population variances and covariances. It also discusses the general format of Amos input and output. About the Data Attig (1983) showed 40 subjects a booklet containing several pages of advert...

24 E x a m p l e 1 Bringing In the Data E From the menus, choose File > New . E From the menus, choose File > Data Files . E In the Data Files dialog box, click File Name . E Browse to the Examples folder. If you performed a typical installation, the path is C:\Program Files\IBM\SPSS\Amos\21\E...

25 E s t i m a t i n g V a r i a n c e s a n d C o v a r i a n c e s As you scroll across the worksheet, you will see all of the test variables from the Attig study. This example uses only the following variables: recall1 (recall pretest), recall2 (recall posttest), place1 (place recall pretest), an...

27 E s t i m a t i n g V a r i a n c e s a n d C o v a r i a n c e s Changing the Font E Right-click a variable and choose Object Properties from the pop-up menu. The Object Properties dialog box appears. E Click the Text tab and adjust the font attributes as desired. Establishing Covariances If you...

28 E x a m p l e 1 Performing the Analysis E From the menus, choose Analyze > Calculate Estimates . Because you have not yet saved the file, the Save As dialog box appears. E Enter a name for the file and click Save . Viewing Graphics Output E Click the Show the output path diagram button . Amos ...

29 E s t i m a t i n g V a r i a n c e s a n d C o v a r i a n c e s In the output path diagram, the numbers displayed next to the boxes are estimated variances, and the numbers displayed next to the double-headed arrows are estimated covariances. For example, the variance of recall1 is estimated at...

30 E x a m p l e 1 observation on an approximately normally distributed random variable centered around the population covariance with a standard deviation of about 1.16, that is, if the assumptions in the section “Distribution Assumptions for Amos Models” on p. 35 are met. For example, you can use ...

31 E s t i m a t i n g V a r i a n c e s a n d C o v a r i a n c e s example, Runyon and Haber, 1980, p. 226) is 2.509 with 38 degrees of freedom . In this example, both p values are less than 0.05, so both tests agree in rejecting the null hypothesis at the 0.05 level. However, in other situations,...

33 E s t i m a t i n g V a r i a n c e s a n d C o v a r i a n c e s Optional Output So far, we have discussed output that Amos generates by default. You can also request additional output. Calculating Standardized Estimates You may be surprised to learn that Amos displays estimates of covariances r...

34 E x a m p l e 1 Rerunning the Analysis Because you have changed the options in the Analysis Properties dialog box, you must rerun the analysis. E From the menus, choose Analyze > Calculate Estimates . E Click the Show the output path diagram button. E In the Parameter Formats pane to the left ...

35 E s t i m a t i n g V a r i a n c e s a n d C o v a r i a n c e s E In the tree diagram in the upper left pane of the Amos Output window, expand Estimates , Scalars, and then click Correlations . Distribution Assumptions for Amos Models Hypothesis testing procedures, confidence intervals, and cla...

36 E x a m p l e 1 „ The (conditional) expected values of the random variables depend linearly on the values of the fixed variables. A typical example of a fixed variable would be an experimental treatment, classifying respondents into a study group and a control group, respectively. It is all right...

37 E s t i m a t i n g V a r i a n c e s a n d C o v a r i a n c e s E Enter the VB.NET code for specifying and fitting the model in place of the ‘Your code goes here comment. The following figure shows the program editor after the complete program has been entered. Note: The Examples directory cont...

38 E x a m p l e 1 To open the VB.NET file for the present example: E From the Program Editor menus, choose File > Open . E Select the file Ex01.vb in the \Amos\21\Examples\<language> directory. The following table gives a line-by-line explanation of the program. E To perform the analysis, ...

39 E s t i m a t i n g V a r i a n c e s a n d C o v a r i a n c e s Generating Additional Output Some AmosEngine methods generate additional output. For example, the Standardized method displays standardized estimates. The following figure shows the use of the Standardized method: Modeling in C# Wr...

40 E x a m p l e 1 Other Program Development Tools The built-in program editor in Amos is used throughout this user’s guide for writing and executing Amos programs. However, you can use the development tool of your choice. The Examples folder contains a VisualStudio subfolder where you can find Visu...

41 E x a m p l e 2 Testing Hypotheses Introduction This example demonstrates how you can use Amos to test simple hypotheses about variances and covariances. It also introduces the chi-square test for goodness of fit and elaborates on the concept of degrees of freedom. About the Data We will use Atti...

42 E x a m p l e 2 You can fill these boxes yourself instead of letting Amos fill them. Constraining Variances Suppose you want to set the variance of recall1 to 6 and the variance of recall2 to 8. E In the drawing area, right-click recall1 and choose Object Properties from the pop-up menu. E Click ...

43 T e s t i n g H y p o t h e s e s E Close the dialog box. The path diagram displays the parameter values you just specified. This is not a very realistic example because the numbers 6 and 8 were just picked out of the air. Meaningful parameter constraints must have some underlying rationale, perh...

44 E x a m p l e 2 require both of the variances to have the same value without specifying ahead of time what that value is. Benefits of Specifying Equal Parameters Before adding any further constraints on the model parameters, let’s examine why we might want to specify that two parameters, like the...

45 T e s t i n g H y p o t h e s e s Moving and Formatting Objects While a horizontal layout is fine for small examples, it is not practical for analyses that are more complex. The following is a different layout of the path diagram on which we’ve been working:

46 E x a m p l e 2 You can use the following tools to rearrange your path diagram until it looks like the one above: „ To move objects, choose Edit > Move from the menus, and then drag the object to its new location. You can also use the Move button to drag the endpoints of arrows. „ To copy form...

49 T e s t i n g H y p o t h e s e s E From the menus, choose Analyze > Calculate Estimates . Amos recalculates the model estimates. Covariance Matrix Estimates E To see the sample variances and covariances collected into a matrix, choose View > Text Output from the menus. E Click Sample Momen...

51 T e s t i n g H y p o t h e s e s Displaying Covariance and Variance Estimates on the Path Diagram As in Example 1, you can display the covariance and variance estimates on the path diagram. E Click the Show the output path diagram button. E In the Parameter Formats pane to the left of the drawin...

52 E x a m p l e 2 Notice the word \format in the bottom line of the figure caption. Words that begin with a backward slash, like \forma t , are called text macros . Amos replaces text macros with information about the currently displayed model. The text macro \format will be replaced by the heading...

53 T e s t i n g H y p o t h e s e s maximum likelihood estimates, and there is no reason to expect them to resemble the implied covariances. The chi-square statistic is an overall measure of how much the implied covariances differ from the sample covariances. In general, the more the implied covari...

54 E x a m p l e 2 E In the Figure Caption dialog box, enter a caption that includes the \cmin , \df , and \p text macros, as follows: When Amos displays the path diagram containing this caption, it appears as follows:

55 T e s t i n g H y p o t h e s e s Modeling in VB.NET The following program fits the constrained model of Example 2:

57 T e s t i n g H y p o t h e s e s E To perform the analysis, from the menus, choose File > Run . Timing Is Everything The AStructure lines must appear after BeginGroup ; otherwise, Amos will not recognize that the variables named in the AStructure lines are observed variables in the attg_yng.s...

59 E x a m p l e 3 More Hypothesis Testing Introduction This example demonstrates how to test the null hypothesis that two variables are uncorrelated, reinforces the concept of degrees of freedom, and demonstrates, in a concrete way, what is meant by an asymptotically correct test. About the Data Fo...

60 E x a m p l e 3 E In the Files of type list, select Text (*.txt) , select Attg_old.txt , and then click Open . E In the Data Files dialog box, click OK . Testing a Hypothesis That Two Variables Are Uncorrelated Among Attig’s 40 old subjects, the sample correlation between age and vocabulary is –0...

61 M o r e H y p o t h e s i s T e s t i n g model specified by the simple path diagram above specifies that the covariance (and thus the correlation) between age and vocabulary is 0. The second method of constraining a covariance parameter is the more general procedure introduced in Example 1 and E...

64 E x a m p l e 3 The usual t statistic for testing this null hypothesis is 0.59 ( , two-sided). The probability level associated with the t statistic is exact. The probability level of 0.555 of the chi-square statistic is off , owing to the fact that it does not have an exact chi-square distributi...

65 M o r e H y p o t h e s i s T e s t i n g Modeling in VB.NET Here is a program for performing the analysis of this example: The AStructure method constrains the covariance, fixing it at a constant 0. The program does not refer explicitly to the variances of age and vocabulary . The default behavi...

67 E x a m p l e 4 Conventional Linear Regression Introduction This example demonstrates a conventional regression analysis, predicting a single observed variable as a linear combination of three other observed variables. It also introduces the concept of identifiability . About the Data Warren, Whi...

68 E x a m p l e 4 Here are the sample variances and covariances: Warren5v also contains the sample means. Raw data are not available, but they are not needed by Amos for most analyses, as long as the sample moments (that is, means, variances, and covariances) are provided. In fact, only sample vari...

69 C o n v e n t i o n a l L i n e a r R e g r e s s i o n The single-headed arrows represent linear dependencies. For example, the arrow leading from knowledge to performance indicates that performance scores depend, in part, on knowledge. The variable error is enclosed in a circle because it is no...

70 E x a m p l e 4 E Draw three double-headed arrows that connect the observed exogenous variables ( knowledge , satisfaction , and value ). Your path diagram should look like this: Identification In this example, it is impossible to estimate the regression weight for the regression of performance o...

71 C o n v e n t i o n a l L i n e a r R e g r e s s i o n Setting a regression weight equal to 1 for every error variable can be tedious. Fortunately, Amos Graphics provides a default solution that works well in most cases. E Click the Add a unique variable to an existing variable button. E Click a...

72 E x a m p l e 4 Viewing the Text Output Here are the maximum likelihood estimates: Amos does not display the path performance <— error because its value is fixed at the default value of 1. You may wonder how much the other estimates would be affected if a different constant had been chosen. It...

73 C o n v e n t i o n a l L i n e a r R e g r e s s i o n of the same factor. Extending this, the product of the squared regression weight and the error variance is always a constant. This is what we mean when we say the regression weight (together with the error variance) is unidentified . If you ...

74 E x a m p l e 4 The standardized regression weights and the correlations are independent of the units in which all variables are measured; therefore, they are not affected by the choice of identification constraints. Squared multiple correlations are also independent of units of measurement. Amos...

75 C o n v e n t i o n a l L i n e a r R e g r e s s i o n Here is the standardized solution: Viewing Additional Text Output E In the tree diagram in the upper left pane of the Amos Output window, click Variable Summary .

76 E x a m p l e 4 Endogenous variables are those that have single-headed arrows pointing to them; they depend on other variables. Exogenous variables are those that do not have single- headed arrows pointing to them; they do not depend on other variables. Inspecting the preceding list will help you...

77 C o n v e n t i o n a l L i n e a r R e g r e s s i o n Modeling in VB.NET The model in this example consists of a single regression equation. Each single-headed arrow in the path diagram represents a regression weight. Here is a program for estimating those regression weights: The four lines tha...

78 E x a m p l e 4 the specification of many models, especially models that have parameters. The differences between specifying a model in Amos Graphics and specifying one programmatically are as follows: „ Amos Graphics is entirely WYSIWYG (What You See Is What You Get). If you draw a two-headed ar...

79 C o n v e n t i o n a l L i n e a r R e g r e s s i o n Note that in the AStructure line above, each predictor variable (on the right side of the equation) is associated with a regression weight to be estimated. We could make these regression weights explicit through the use of empty parentheses ...

82 E x a m p l e 5 Here is a list of the input variables: For this example, we will use a Lotus data file, Warren9v.wk1 , to obtain the sample variances and covariances of these subtests. The sample means that appear in the file will not be used in this example. Statistics on formal education ( past...

83 U n o b s e r v e d V a r i a b l e s Model A The following path diagram presents a model for the eight subtests: Four ellipses in the figure are labeled knowledge , value , satisfaction , and performance . They represent unobserved variables that are indirectly measured by the eight split-half t...

84 E x a m p l e 5 Consider, for instance, the knowledge submodel: The scores of the two split-half subtests, 1knowledge and 2knowledge , are hypothesized to depend on the single underlying, but not directly observed variable, knowledge . According to the model, scores on the two subtests may still ...

86 E x a m p l e 5 Changing the Orientation of the Drawing Area E From the menus, choose View > Interface Properties . E In the Interface Properties dialog box, click the Page Layout tab. E Set Paper Size to one of the “Landscape” paper sizes, such as Landscape - A4 . E Click Apply .

87 U n o b s e r v e d V a r i a b l e s Creating the Path Diagram Now you are ready to draw the model as shown in the path diagram on page 83. There are a number of ways to do this. One is to start by drawing the measurement model first. Here, we draw the measurement model for one of the latent var...

88 E x a m p l e 5 Rotating Indicators The indicators appear by default above the knowledge ellipse, but you can change their location. E From the menus, choose Edit > Rotate . E Click the knowledge ellipse. Each time you click the knowledge ellipse, its indicators rotate 90° clockwise. If you cl...

89 U n o b s e r v e d V a r i a b l e s Your path diagram should now look like this: E Create a fourth copy for performance , and position it to the right of the original. E From the menus, choose Edit > Reflect . This repositions the two indicators of performance as follows:

90 E x a m p l e 5 Entering Variable Names E Right-click each object and select Object Properties from the pop-up menu E In the Object Properties dialog box, click the Text tab, and enter a name into the Variable Name text box. Alternatively, you can choose View > Variables in Dataset from the me...

91 U n o b s e r v e d V a r i a b l e s The hypothesis that Model A is correct is accepted. The parameter estimates are affected by the identification constraints. Chi-square = 10.335Degrees of freedom = 14Probability level = 0.737

92 E x a m p l e 5 Standardized estimates, on the other hand, are not affected by the identification constraints. To calculate standardized estimates: E From the menus, choose View > Analysis Properties . E In the Analysis Properties dialog box, click the Output tab. E Enable the Standardized est...

93 U n o b s e r v e d V a r i a b l e s Viewing the Graphics Output The path diagram with standardized parameter estimates displayed is as follows: The value above performance indicates that pure knowledge , value , and satisfaction account for 66% of the variance of performance . The values displa...

94 E x a m p l e 5 split exactly in half. As a result, 2satisfaction is 20% longer than 1satisfaction . Assuming that the tests differ only in length leads to the following conclusions: „ The regression weight for regressing 2satisfaction on satisfaction should be 1.2 times the weight for regressing...

96 E x a m p l e 5 Here are the standardized estimates and squared multiple correlations displayed on the path diagram: Testing Model B against Model A Sometimes you may have two alternative models for the same set of data, and you would like to know which model fits the data better. You can perform...

97 U n o b s e r v e d V a r i a b l e s distribution with degrees of freedom equal to the difference between the degrees of freedom of the competing models. In this example, the difference in degrees of freedom is 8 (that is, ). Model B imposes all of the parameter constraints of Model A, plus an a...

98 E x a m p l e 5 Modeling in VB.NET Model A The following program fits Model A: Because of the assumptions that Amos makes about correlations among exogenous variables (discussed in Example 4), the program does not need to indicate that knowledge , value , and satisfaction are allowed to be correl...

99 U n o b s e r v e d V a r i a b l e s Model B The following program fits Model B: Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Standardized() Sem.Smc() Sem.BeginGroup(Sem.AmosDir & "Examples\Warren9v.wk1") Sem.AStructure("1performance <--- performance (1)")...

101 E x a m p l e 6 Exploratory Analysis Introduction This example demonstrates structural modeling with time-related latent variables, the use of modification indices and critical ratios in exploratory analyses, how to compare multiple models in a single analysis, and computation of implied moments...

102 E x a m p l e 6 variances and covariances, as needed for the analysis. We will not use the sample means in the analysis. Model A for the Wheaton Data Jöreskog and Sörbom (1984) proposed the model shown on p. 103 for the Wheaton data, referring to it as their Model A. The model asserts that all o...

104 E x a m p l e 6 The Wheaton data depart significantly from Model A. Dealing with Rejection You have several options when a proposed model has to be rejected on statistical grounds: „ You can point out that statistical hypothesis testing can be a poor tool for choosing a model. Jöreskog (1967) di...

105 E x p l o r a t o r y A n a l y s i s Modification Indices You can test various modifications of a model by carrying out a separate analysis for each potential modification, but this approach is time-consuming. Modification indices allow you to evaluate many potential modifications in a single a...

106 E x a m p l e 6 The column heading M.I. in this table is short for Modification Index . The modification indices produced are those described by Jöreskog and Sörbom (1984). The first modification index listed ( 5.905 ) is a conservative estimate of the decrease in chi-square that will occur if e...

107 E x p l o r a t o r y A n a l y s i s The theoretical reasons for suspecting that eps1 and eps3 might be correlated apply to eps2 and eps4 as well. The modification indices also suggest allowing eps2 and eps4 to be correlated. However, we will ignore this potential modification and proceed immed...

108 E x a m p l e 6 Text Output The added covariance between eps1 and eps3 decreases the degrees of freedom by 1. The chi-square statistic is reduced by substantially more than the promised 40.911. Model B cannot be rejected. Since the fit of Model B is so good, we will not pursue the possibility, m...

109 E x p l o r a t o r y A n a l y s i s Note the large critical ratio associated with the new covariance path. The covariance between eps1 and eps3 is clearly different from 0. This explains the poor fit of Model A, in which that covariance was fixed at 0. Graphics Output for Model B The following...

110 E x a m p l e 6 Misuse of Modification Indices In trying to improve upon a model, you should not be guided exclusively by modification indices. A modification should be considered only if it makes theoretical or common sense. A slavish reliance on modification indices without such a limitation a...

111 E x p l o r a t o r y A n a l y s i s E From the menus, choose View > Analysis Properties . E In the Analysis Properties dialog box, click the Output tab. E Enable the Critical ratios for differences check box. When Amos calculates critical ratios for parameter differences, it generates names...

112 E x a m p l e 6 The parameter names are needed for interpreting the critical ratios in the following table: Ignoring the 0’s down the main diagonal, the table of critical ratios contains 120 entries, one for each pair of parameters. Take the number 0.877 near the upper left corner of the table. ...

113 E x p l o r a t o r y A n a l y s i s par_1 and par_2 divided by the estimated standard error of this difference. These two parameters are the regression weights for powles71 <– 71_alienation and powles67 <– 67_alienation . Under the distribution assumptions stated on p. 35, the critical r...

114 E x a m p l e 6 Model C for the Wheaton Data Here is the path diagram for Model C from the file Ex06–c.amw : The label path_p requires the regression weight for predicting powerlessness from alienation to be the same in 1971 as it is in 1967. The label var_a is used to specify that eps1 and eps3...

115 E x p l o r a t o r y A n a l y s i s Testing Model C As expected, Model C has an acceptable fit, with a higher probability level than Model B: You can test Model C against Model B by examining the difference in chi-square values ( ) and the difference in degrees of freedom ( ). A chi-square val...

116 E x a m p l e 6 Multiple Models in a Single Analysis Amos allows for the fitting of multiple models in a single analysis. This allows Amos to summarize the results for all models in a single table. It also allows Amos to perform a chi-square test for nested model comparisons. In this example, Mo...

119 E x p l o r a t o r y A n a l y s i s E In the Parameter Constraints box, type: Model A: No AutocorrelationModel C: Time-Invariance These lines tell Amos that Model D incorporates the constraints of both Model A and Model C. Now that we have set up the parameter constraints for all four models, ...

120 E x a m p l e 6 The CMIN column contains the minimum discrepancy for each model. In the case of maximum likelihood estimation (the default), the CMIN column contains the chi-square statistic. The p column contains the corresponding upper-tail probability for testing each model. For nested pairs ...

121 E x p l o r a t o r y A n a l y s i s Obtaining Optional Output The variances and covariances among the observed variables can be estimated under the assumption that Model C is correct. E From the menus, choose View > Analysis Properties . E In the Analysis Properties dialog box, click the Ou...

122 E x a m p l e 6 The matrix of implied covariances for all variables in the model can be used to carry out a regression of the unobserved variables on the observed variables. The resulting regression weight estimates can be obtained from Amos by enabling the Factor score weights check box. Here a...

124 E x a m p l e 6 Model B The following program fits Model B. It is saved as Ex06–b.vb . Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Standardized() Sem.Smc() Sem.Crdiff() Sem.BeginGroup(Sem.AmosDir & "Examples\Wheaton.sav") Sem.AStructure("anomia67 <--- 67_alie...

125 E x p l o r a t o r y A n a l y s i s Model C The following program fits Model C. It is saved as Ex06–c.vb . Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Standardized() Sem.Smc() Sem.AllImpliedMoments() Sem.FactorScoreWeights() Sem.TotalEffects() Sem.BeginGroup(Sem.AmosDir &...

126 E x a m p l e 6 Fitting Multiple Models To fit all three models, A, B, and C in a single analysis, start with the following program, which assigns unique names to some parameters: Since the parameter names are unique, naming the parameters does not constrain them. However, naming the parameters ...

130 E x a m p l e 7 Sample correlations, means, and standard deviations for these six variables are contained in the SPSS Statistics file, Fels_fem.sav . Here is the data file as it appears in the SPSS Statistics Data Editor: The sample means are not used in this example. Felson and Bohrnstedt’s Mod...

131 A N o n r e c u r s i v e M o d e l Perceived academic performance is modeled as a function of GPA and perceived attractiveness ( attract ). Perceived attractiveness, in turn, is modeled as a function of perceived academic performance, height , weight , and the rating of attractiveness by childr...

132 E x a m p l e 7 Judging by the critical ratios, you see that each of these three null hypotheses would be accepted at conventional significance levels: „ Perceived attractiveness does not depend on height (critical ratio = 0.050). „ Perceived academic ability does not depend on perceived attract...

133 A N o n r e c u r s i v e M o d e l Obtaining Standardized Estimates Before you perform the analysis, do the following: E From the menus, choose View > Analysis Properties . E In the Analysis Properties dialog box, click the Output tab. E Select Standardized estimates (a check mark appears ne...

134 E x a m p l e 7 E In the Analysis Properties dialog box, click the Output tab. E Select Squared multiple correlations (a check mark appears next to it). E Close the dialog box. The squared multiple correlations show that the two endogenous variables in this model are not predicted very accuratel...

135 A N o n r e c u r s i v e M o d e l Stability Index The existence of feedback loops in a nonrecursive model permits certain problems to arise that cannot occur in recursive models. In the present model, attractiveness depends on perceived academic ability, which in turn depends on attractiveness...

136 E x a m p l e 7 Modeling in VB.NET The following program fits the model of this example. It is saved in the file Ex07.vb . The final AStructure line is essential to Felson and Bohrnstedt’s model. Without it, Amos would assume that error1 and error2 are uncorrelated. You can specify the same mode...

137 E x a m p l e 8 Factor Analysis Introduction This example demonstrates confirmatory common factor analysis. About the Data Holzinger and Swineford (1939) administered 26 psychological tests to 301 seventh- and eighth-grade students in two Chicago schools. In the present example, we use scores ob...

138 E x a m p l e 8 The file Grnt_fem.sav contains the test scores: A Common Factor Model Consider the following model for the six tests: spatial visperc cubes lozenges wordmean paragrap sentence err_v err_c err_l err_p err_s err_w verbal 1 1 1 1 1 1 1 1 Example 8 Factor analysis: Girls' sample Holz...

139 F a c t o r A n a l y s i s This model asserts that the first three tests depend on an unobserved variable called spatial . Spatial can be interpreted as an underlying ability (spatial ability) that is not directly observed. According to the model, performance on the first three tests depends on...

140 E x a m p l e 8 is true that the lack of a unit of measurement for unobserved variables is an ever-present cause of non-identification. Fortunately, it is one that is easy to cure, as we have done repeatedly. But other kinds of under-identification can occur for which there is no simple remedy. ...

141 F a c t o r A n a l y s i s Results of the Analysis Here are the unstandardized results of the analysis. As shown at the upper right corner of the figure, the model fits the data quite well. As an exercise, you may wish to confirm the computation of degrees of freedom. The parameter estimates, b...

143 F a c t o r A n a l y s i s E Also select Squared multiple correlations if you want squared multiple correlation for each endogenous variable, as shown in the next graphic. E Close the dialog box. Viewing Standardized Estimates E In the Amos Graphics window, click the Show the output path diagra...

144 E x a m p l e 8 The squared multiple correlations can be interpreted as follows: To take wordmean as an example, 71% of its variance is accounted for by verbal ability. The remaining 29% of its variance is accounted for by the unique factor err_w . If err_w represented measurement error only, we...

146 E x a m p l e 9 here has been employed by Bentler and Woodward (1979) and others. Another approach, by Sörbom (1978), is demonstrated in Example 16. The Sörbom method is more general. It allows testing other assumptions of analysis of covariance and permits relaxing some of them as well. The Sör...

147 A n A l t e r n a t i v e t o A n a l y s i s o f C o v a r i a n c e Correlations and standard deviations for the five measures are contained in the Microsoft Excel workbook UserGuide.xls , in the Olss_all worksheet. Here is the dataset: There are positive correlations between treatment and eac...

149 A n A l t e r n a t i v e t o A n a l y s i s o f C o v a r i a n c e Specifying Model A To specify Model A, draw a path diagram similar to the one on p. 148. The path diagram is saved as the file Ex09-a.amw . Results for Model A There is considerable empirical evidence against Model A: This is ...

150 E x a m p l e 9 Requesting modification indices with a threshold of 4 produces the following additional output: According to the first modification index in the M.I. column, the chi - square statistic will decrease by at least 13.161 if the unique variables eps2 and eps4 are allowed to be correl...

152 E x a m p l e 9 chi-square statistic that will occur if the corresponding constraint—and only that constraint—is removed. The following raw parameter estimates are difficult to interpret because they would have been different if the identification constraints had been different: As expected, the...

153 A n A l t e r n a t i v e t o A n a l y s i s o f C o v a r i a n c e In this example, we are primarily concerned with testing a particular hypothesis and not so much with parameter estimation. However, even when the parameter estimates themselves are not of primary interest, it is a good idea t...

154 E x a m p l e 9 Drawing a Path Diagram for Model C To draw the path diagram for Model C: E Start with the path diagram for Model B. E Right-click the arrow that points from treatment to post_verbal and choose Object Properties from the pop-up menu. E In the Object Properties dialog box, click th...

155 A n A l t e r n a t i v e t o A n a l y s i s o f C o v a r i a n c e Modeling in VB.NET Model A This program fits Model A. It is saved in the file Ex09–a.vb . Model B This program fits Model B. It is saved in the file Ex09–b.vb . Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Mod...

156 E x a m p l e 9 Model C This program fits Model C. It is saved in the file Ex09–c.vb . Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Mods(4) Sem.Standardized() Sem.Smc() Sem.BeginGroup(Sem.AmosDir & "Examples\UserGuide.xls", "Olss_all") Sem.AStructure("pre...

157 A n A l t e r n a t i v e t o A n a l y s i s o f C o v a r i a n c e Fitting Multiple Models This program (Ex09-all.vb ) fits all three models (A through C). Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Mods(4) Sem.Standardized() Sem.Smc() Sem.BeginGroup(Sem.AmosDir & "...

161 S i m u l t a n e o u s A n a l y s i s o f S e v e r a l G r o u p s Conventions for Specifying Group Differences The main purpose of a multigroup analysis is to find out the extent to which groups differ. Do the groups all have the same path diagram with the same parameter values? Do the group...

163 S i m u l t a n e o u s A n a l y s i s o f S e v e r a l G r o u p s E Connect recall1 and cued1 with a double - headed arrow. E To add a caption to the path diagram, from the menus, choose Diagram > Figure Caption and then click the path diagram at the spot where you want the caption to app...

165 S i m u l t a n e o u s A n a l y s i s o f S e v e r a l G r o u p s E Click Close . E From the menus, choose File > Data Files . The Data Files dialog box shows that there are two groups labeled young subjects and old subjects . E To specify the dataset for the old subjects, in the Data Fil...

166 E x a m p l e 1 0 E Click OK . Text Output Model A has zero degrees of freedom. Amos computed the number of distinct sample moments this way: The young subjects have two sample variances and one sample covariance, which makes three sample moments. The old subjects also have three sample moments,...

169 S i m u l t a n e o u s A n a l y s i s o f S e v e r a l G r o u p s E In the Variance text box, enter a name for the variance of recall1 ; for example, type var_rec . E Select All groups (a check mark will appear next to it). The effect of the check mark is to assign the name var_rec to the va...

170 E x a m p l e 1 0 Text Output Because of the constraints imposed in Model B, only three distinct parameters are estimated instead of six. As a result, the number of degrees of freedom has increased from 0 to 3. Model B is acceptable at any conventional significance level. The following are the p...

171 S i m u l t a n e o u s A n a l y s i s o f S e v e r a l G r o u p s Graphics Output For Model B, the output path diagram is the same for both groups. Modeling in VB.NET Model A Here is a program ( Ex10-a.vb ) for fitting Model A: Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Be...

173 S i m u l t a n e o u s A n a l y s i s o f S e v e r a l G r o u p s Multiple Model Input Here is a program ( Ex10-all.vb ) for fitting both Models A and B. 1 The Sem.Model statements should appear immediately after the AStructure specifications for the last group. It does not matter which Mode...

176 E x a m p l e 1 1 Notice that there are eight variables in the boys’ data file but only seven in the girls’ data file. The extra variable skills is not used in any model of this example, so its presence in the data file is ignored. Specifying Model A for Girls and Boys Consider extending the Fel...

177 F e l s o n a n d B o h r n s t e d t ’ s G i r l s a n d B o y s E In the Figure Caption dialog box, enter a title that contains the text macro \group . For example: In Example 7, where there was only one group, the group’s name didn’t matter. Accepting the default name Group number 1 was good ...

179 F e l s o n a n d B o h r n s t e d t ’ s G i r l s a n d B o y s Text Output for Model A With two groups instead of one (as in Example 7), there are twice as many sample moments and twice as many parameters to estimate. Therefore, you have twice as many degrees of freedom as there were in Examp...

181 F e l s o n a n d B o h r n s t e d t ’ s G i r l s a n d B o y s Graphics Output for Model A For girls, this is the path diagram with unstandardized estimates displayed: The following is the path diagram with the estimates for the boys: You can visually inspect the girls’ and boys’ estimates in...

182 E x a m p l e 1 1 Obtaining Critical Ratios for Parameter Differences E From the menus, choose View > Analysis Properties . E In the Analysis Properties dialog box, click the Output tab. E Select Critical ratios for differences . In this example, however, we will not use critical ratios for d...

184 E x a m p l e 1 1 E Repeat this until you have named every regression weight. Always make sure to select (put a check mark next to) All groups . After you have named all of the regression weights, the path diagram for each sample should look something like this: Results for Model B Text Output M...

185 F e l s o n a n d B o h r n s t e d t ’ s G i r l s a n d B o y s Comparing Model B against Model A gives a nonsignificant chi-square of with degrees of freedom. Assuming that Model B is indeed correct, the Model B estimates are preferable over the Model A estimates. The unstandardized parameter...

186 E x a m p l e 1 1 The unstandardized parameter estimates for the boys are: As Model B requires, the estimated regression weights for the boys are the same as those for the girls. Regression Weights: (boys - Default model) Estimate S.E. C.R. P Label academic <--- GPA .022 .002 9.475 *** p1 att...

187 F e l s o n a n d B o h r n s t e d t ’ s G i r l s a n d B o y s Graphics Output The output path diagram for the girls is: And the output for the boys is: 12.12 GPA 8.43 height 1.02 rating 371.48 weight academic attract .02 .0 1 .00 .1 5 .02 error1 .14 error2 1 1 .5 3 -.4 7 -6 .7 1 1 .8 2 1 9 ....

188 E x a m p l e 1 1 Fitting Models A and B in a Single Analysis It is possible to fit both Model A and Model B in the same analysis. The file Ex11-ab.amw in the Amos Examples directory shows how to do this. Model C for Girls and Boys You might consider adding additional constraints to Model B, suc...

190 E x a m p l e 1 1 E To even out the spacing between the rectangles, from the menus, choose Edit > Select All . E Then choose Edit > Space Vertically . There is a special button for drawing large numbers of double-headed arrows at once. With all six variables still selected from the previou...

192 E x a m p l e 1 1 Modeling in VB.NET Model A The following program fits Model A. It is saved as Ex11-a.vb . Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.BeginGroup(Sem.AmosDir & "Examples\Fels_fem.sav") Sem.GroupName("girls") Sem.AStructure("academic = GP...

193 F e l s o n a n d B o h r n s t e d t ’ s G i r l s a n d B o y s Model B The following program fits Model B, in which parameter labels p1 through p6 are used to impose equality constraints across groups. The program is saved in Ex11-b.vb . Model C The VB.NET program for Model C is not displayed...

195 E x a m p l e 12 Simultaneous Factor Analysis for Several Groups Introduction This example demonstrates how to test whether the same factor analysis model holds for each of several populations, possibly with different parameter values for different populations (Jöreskog, 1971). About the Data We...

196 E x a m p l e 1 2 Model A for the Holzinger and Swineford Boys and Girls Consider the hypothesis that the common factor analysis model of Example 8 holds for boys as well as for girls. The path diagram from Example 8 can be used as a starting point for this two-group model. By default, Amos Grap...

197 S i m u l t a n e o u s F a c t o r A n a l y s i s f o r S e v e r a l G r o u p s E While the Manage Groups dialog box is open, create another group by clicking New . E Then, type Boys in the Group Name text box. E Click Close to close the Manage Groups dialog box. Specifying the Data E From t...

200 E x a m p l e 1 2 The corresponding output path diagram for the 72 boys is: Notice that the estimated regression weights vary little across groups. It seems plausible that the two populations have the same regression weights—a hypothesis that we will test in Model B. Model B for the Holzinger an...

202 E x a m p l e 1 2 The path diagram for either of the two samples should now look something like this: Results for Model B Text Output Because of the additional constraints in Model B, four fewer parameters have to be estimated from the data, increasing the number of degrees of freedom by 4. The ...

203 S i m u l t a n e o u s F a c t o r A n a l y s i s f o r S e v e r a l G r o u p s Graphics Output Here are the parameter estimates for the 73 girls: 22.00 spatial visperc cubes lozenges wordmean paragrap sentence 25.08 err_v 12.38 err_c 25.24 err_l 2.83 err_p 8.12 err_s 19.55 err_w 9.72 verbal...

204 E x a m p l e 1 2 Here are the parameter estimates for the 72 boys: Not surprisingly, the Model B parameter estimates are different from the Model A estimates. The following table shows estimates and standard errors for the two models side by side: Parameters Model A Model B Girls’ sample Estima...

206 E x a m p l e 1 2 Modeling in VB.NET Model A The following program ( Ex12-a.vb ) fits Model A for boys and girls: The same model is specified for boys as for girls. However, the boys’ parameter values can be different from the corresponding girls’ parameters. Sub Main() Dim Sem As New AmosEngine...

210 E x a m p l e 1 3 About the Data For this example, we will be using Attig’s (1983) memory data, which was described in Example 1. We will use data from both young and old subjects. The raw data for the two groups are contained in the Microsoft Excel workbook UserGuide.xls , in the Attg_yng and A...

211 E s t i m a t i n g a n d T e s t i n g H y p o t h e s e s a b o u t M e a n s E Select Estimate means and intercepts . Now the path diagram looks like this (the same path diagram for each group): The path diagram now shows a mean, variance pair of parameters for each exogenous variable. There ...

212 E x a m p l e 1 3 The behavior of Amos Graphics changes in several ways when you select (put a check mark next to) Estimate means and intercepts : „ Mean and intercept fields appear on the Parameters tab in the Object Properties dialog box. „ Constraints can be applied to means and intercepts as...

214 E x a m p l e 1 3 Except for the means, these estimates are the same as those obtained in Example 10, Model B. The estimated standard errors and critical ratios are also the same. This demonstrates that merely estimating means, without placing any constraints on them, has no effect on the estima...

216 E x a m p l e 1 3 Results for Model B With the new constraints on the means, Model B has five degrees of freedom. Model B has to be rejected at any conventional significance level. Comparison of Model B with Model A If Model A is correct and Model B is wrong (which is plausible, since Model A wa...

217 E s t i m a t i n g a n d T e s t i n g H y p o t h e s e s a b o u t M e a n s automatically compute the difference in chi-square values as well as the p value for testing Model B against Model A. Mean Structure Modeling in VB.NET Model A Here is a program ( Ex13-a.vb ) for fitting Model A. The...

221 E x a m p l e 14 Regression with an Explicit Intercept Introduction This example shows how to estimate the intercept in an ordinary regression analysis. Assumptions Made by Amos Ordinarily, when you specify that some variable depends linearly on some others, Amos assumes that the linear equation...

222 E x a m p l e 1 4 About the Data We will once again use the data of Warren, White, and Fuller (1974), first used in Example 4. We will use the Excel worksheet Warren5v in UserGuide.xls found in the Examples directory. Here are the sample moments (means, variances, and covariances): Specifying th...

224 E x a m p l e 1 4 With 0 degrees of freedom, there is no hypothesis to be tested. The estimates for regression weights, variances, and covariances are the same as in Example 4, and so are the associated standard error estimates, critical ratios, and p values. Chi-square = 0.000Degrees of freedom...

226 E x a m p l e 1 4 The following program for the model of Example 14 gives all the same results, plus mean and intercept estimates. This program is saved as Ex14.vb . Note the Sem.ModelMeansAndIntercepts statement that causes Amos to treat means and intercepts as explicit model parameters. Anothe...

229 E x a m p l e 15 Factor Analysis with Structured Means Introduction This example demonstrates how to estimate factor means in a common factor analysis of data from several populations. Factor Means Conventionally, the common factor analysis model does not make any assumptions about the means of ...

230 E x a m p l e 1 5 The identification status of the factor analysis model is a difficult subject when estimating factor means. In fact, Sörbom’s accomplishment was to show how to constrain parameters so that the factor analysis model is identified and so that differences in factor means can be es...

232 E x a m p l e 1 5 The boys’ path diagram should look like this: Understanding the Cross-Group Constraints The cross-group constraints on intercepts and regression weights may or may not be satisfied in the populations. One result of fitting the model will be a test of whether these constraints h...

235 F a c t o r A n a l y s i s w i t h S t r u c t u r e d M e a n s Here are the girls’ factor mean estimates from the text output: The girls’ mean spatial ability has a critical ratio of –1.209 and is not significantly different from 0 ( ). In other words, it is not significantly different from t...

237 F a c t o r A n a l y s i s w i t h S t r u c t u r e d M e a n s Results for Model B If we did not have Model A as a basis for comparison, we would now accept Model B, using any conventional significance level. Comparing Models A and B An alternative test of Model B can be obtained by assuming ...

241 E x a m p l e 16 Sörbom’s Alternative to Analysis of Covariance Introduction This example demonstrates latent structural equation modeling with longitudinal observations in two or more groups, models that generalize traditional analysis of covariance techniques by incorporating latent variables ...

242 E x a m p l e 1 6 About the Data We will again use the Olsson (1973) data introduced in Example 9. The sample means, variances, and covariances from the 108 experimental subjects are in the Microsoft Excel worksheet Olss_exp in the workbook UserGuide.xls . The sample means, variances, and covari...

243 S ö r b o m ’ s A l t e r n a t i v e t o A n a l y s i s o f C o v a r i a n c e Changing the Default Behavior E From the menus, choose View > Analysis Properties. E In the Analysis Properties dialog box, click the Bias tab. The default setting used by Amos yields results that are consistent...

244 E x a m p l e 1 6 The following path diagram is Model A for the experimental group: Means and intercepts are an important part of this model, so be sure that you do the following: E From the menus, choose View > Analysis Properties . E Click the Estimation tab. E Select Estimate means and int...

245 S ö r b o m ’ s A l t e r n a t i v e t o A n a l y s i s o f C o v a r i a n c e as verbal ability at the beginning of the study, and post_verbal is interpreted as verbal ability at the conclusion of the study. This is Sörbom’s measurement model. The structural model specifies that post_verbal ...

246 E x a m p l e 1 6 We also get the following message that provides further evidence that Model A is wrong: Can we modify Model A so that it will fit the data while still permitting a meaningful comparison of the experimental and control groups? It will be helpful here to repeat the analysis and r...

247 S ö r b o m ’ s A l t e r n a t i v e t o A n a l y s i s o f C o v a r i a n c e Model B The largest modification index obtained with Model A suggests adding a covariance between eps2 and eps4 in the experimental group. The modification index indicates that the chi-square statistic will drop by...

248 E x a m p l e 1 6 For Model B, the path diagram for the control group is: 0, pre_verbal a_syn1 pre_syn 0, eps1 1 1 a_opp1 pre_opp 0, eps2 opp_v1 1 0 post_verbal a_syn2 post_syn 0, eps3 a_opp2 post_opp 0, eps4 1 1 opp_v2 1 0, zeta 1 Example 16: Model B An alternative to ANCOVA Olsson (1973): cont...

249 S ö r b o m ’ s A l t e r n a t i v e t o A n a l y s i s o f C o v a r i a n c e For the experimental group, the path diagram is: Results for Model B In moving from Model A to Model B, the chi-square statistic dropped by 17.712 (more than the promised 10.508) while the number of degrees of free...

252 E x a m p l e 1 6 Most of these parameter estimates are not very interesting, although you may want to check and make sure that the estimates are reasonable. We have already noted that the variance estimates are positive. The path coefficients in the measurement model are positive, which is reas...

253 S ö r b o m ’ s A l t e r n a t i v e t o A n a l y s i s o f C o v a r i a n c e Next is the path diagram for Model D for the control group: Results for Model D Model D would be accepted at conventional significance levels. Chi-square = 3.976Degrees of freedom = 5Probability level = 0.553 pre_d...

254 E x a m p l e 1 6 Testing Model D against Model C gives a chi-square value of 1.179 (= 3.976 – 2.797) with 1 (that is, 5 – 4) degree of freedom. Again, you would accept the hypothesis of equal regression weights (Model D). With equal regression weights, the comparison of treated and untreated su...

255 S ö r b o m ’ s A l t e r n a t i v e t o A n a l y s i s o f C o v a r i a n c e intercept for the experimental group is significantly different from the intercept for the control group (which is fixed at 0). Model E Another way of testing the difference in post_verbal intercepts for significan...

256 E x a m p l e 1 6 Comparison of Sörbom’s Method with the Method of Example 9 Sörbom’s alternative to analysis of covariance is more difficult to apply than the method of Example 9. On the other hand, Sörbom’s method is superior to the method of Example 9 because it is more general. That is, you ...

257 S ö r b o m ’ s A l t e r n a t i v e t o A n a l y s i s o f C o v a r i a n c e The following is the path diagram for Model X for the control group: The path diagram for the experimental group is identical. Using the same parameter names for both groups has the effect of requiring the two grou...

259 S ö r b o m ’ s A l t e r n a t i v e t o A n a l y s i s o f C o v a r i a n c e Here is the path diagram for the control group: Results for Model Y We must reject Model Y. This is a good reason for being dissatisfied with the analysis of Example 9, since it depended upon Model Y (which, in Exa...

260 E x a m p l e 1 6 covariances of the exogenous variables) imply the assumptions of Model X (equal covariances for the observed variables). Models X and Y are therefore nested models, and it is possible to carry out a conditional test of Model Y under the assumption that Model X is true. Of cours...

261 S ö r b o m ’ s A l t e r n a t i v e t o A n a l y s i s o f C o v a r i a n c e Here is the path diagram for Model Z for the experimental group: Here is the path diagram for the control group: Results for Model Z This model has to be rejected. Chi-square = 84.280Degrees of freedom = 13Probabil...

262 E x a m p l e 1 6 Model Z also has to be rejected when compared to Model Y ( χ 2 = 84.280 – 31.816 = 52.464, df = 13 – 12 = 1). Within rounding error, this is the same difference in chi-square values and degrees of freedom as in Example 9, when Model C was compared to Model B. Modeling in VB.NET...

263 S ö r b o m ’ s A l t e r n a t i v e t o A n a l y s i s o f C o v a r i a n c e Model B To fit Model B, start with the program for Model A and add the line Sem.AStructure("eps2 <---> eps4") to the model specification for the experimental group. Here is the resulting program for M...

264 E x a m p l e 1 6 Model C The following program fits Model C. The program is saved as Ex16-c.vb . Sub Main() Dim Sem As New AmosEngine Try Dim dataFile As String = Sem.AmosDir & "Examples\UserGuide.xls" Sem.TextOutput() Sem.Mods(4) Sem.Standardized() Sem.Smc() Sem.ModelMeansAndInterc...

265 S ö r b o m ’ s A l t e r n a t i v e t o A n a l y s i s o f C o v a r i a n c e Model D The following program fits Model D. The program is saved as Ex16-d.vb . Sub Main() Dim Sem As New AmosEngine Try Dim dataFile As String = Sem.AmosDir & "Examples\UserGuide.xls" Sem.TextOutput() ...

266 E x a m p l e 1 6 Model E The following program fits Model E. The program is saved as Ex16-e.vb . Sub Main() Dim Sem As New AmosEngine Try Dim dataFile As String = Sem.AmosDir & "Examples\UserGuide.xls" Sem.TextOutput() Sem.Mods(4) Sem.Standardized() Sem.Smc() Sem.ModelMeansAndInterc...

269 E x a m p l e 17 Missing Data Introduction This example demonstrates the analysis of a dataset in which some values are missing. Incomplete Data It often happens that data values that were anticipated in the design of a study fail to materialize. Perhaps a subject failed to participate in part o...

270 E x a m p l e 1 7 exclude only persons whose incomes you do not know. Similarly, in computing the sample covariance between age and income, you would exclude an observation only if age is missing or if income is missing. This approach to missing data is sometimes called pairwise deletion . A thi...

272 E x a m p l e 1 7 After specifying the data file to be Grant_x.sav and drawing the above path diagram: E From the menus, choose View > Analysis Properties . E In the Analysis Properties dialog box, click the Estimation tab. E Select Estimate means and intercepts (a check mark appears next to ...

273 M i s s i n g D a t a large number of parameters. In addition, some missing data value patterns can make it impossible in principle to fit the saturated model even if it is possible to fit your model. With incomplete data, Amos Graphics tries to fit the saturated and independence models in addit...

278 E x a m p l e 1 7 The following program fits the saturated model (Model B). The program is saved as Ex17-b.vb . Following the BeginGroup line, there are six uses of the Mean method, requesting estimates of means for the six variables. When Amos estimates their means, it will automatically estima...

280 E x a m p l e 1 7 The AllImpliedMoments method in the program displays the following table of estimates: These estimates, even the estimated means, are different from the sample values computed using either pairwise or listwise deletion methods. For example, 53 people took the visual perception ...

281 M i s s i n g D a t a Computing the Likelihood Ratio Chi-Square Statistic and P Instead of consulting a chi-square table, you can use the ChiSquareProbability method to find the probability that a chi-square value as large as 11.547 would have occurred with a correct factor model. The following ...

282 E x a m p l e 1 7 The p value is 0.173; therefore, we accept the hypothesis that Model A is correct at the 0.05 level. As the present example illustrates, in order to test a model with incomplete data, you have to compare its fit to that of another, alternative model. In this example, we wanted ...

283 E x a m p l e 18 More about Missing Data Introduction This example demonstrates the analysis of data in which some values are missing by design and then explores the benefits of intentionally collecting incomplete data. Missing Data Researchers do not ordinarily like missing data. They typically...

284 E x a m p l e 1 8 About the Data For this example, the Attig data (introduced in Example 1) was modified by eliminating some of the data values and treating them as missing. A portion of the modified data file for young people, Atty_mis.sav , is shown below as it appears in the SPSS Statistics D...

287 M o r e a b o u t M i s s i n g D a t a Results for Model A Graphics Output Here are the two path diagrams containing means, variances, and covariances for the young and old subjects respectively: Text Output E In the Amos Output window, click Notes for Model in the upper left pane. The text out...

288 E x a m p l e 1 8 The parameter estimates and standard errors for young subjects are: The parameter estimates and standard errors for old subjects are: The estimates for the mean of vocab are 56.891 in the young population and 65.001 in the old population. Notice that these are not the same as t...

291 M o r e a b o u t M i s s i n g D a t a E To specify Model B, click New . E In the Model Name text box, change Model Number 2 to Model B . E Type m1_old = m1_yng in the Parameter Constraints text box. E Click Close . A path diagram that fits both Model A and Model B is saved in the file Ex18-b.a...

292 E x a m p l e 1 8 Modeling in VB.NET Model A The following program fits Model A. It estimates means, variances, and covariances of both vocabulary tests in both groups of subjects, without constraints. The program is saved as Ex18-a.vb . The Crdiff method displays the critical ratios for paramet...

295 E x a m p l e 19 Bootstrapping Introduction This example demonstrates how to obtain robust standard error estimates by the bootstrap method. The Bootstrap Method The bootstrap (Efron, 1982) is a versatile method for estimating the sampling distribution of parameter estimates. In particular, the ...

296 E x a m p l e 1 9 The bootstrap has its own shortcomings, including the fact that it can require fairly large samples. For readers who are new to bootstrapping, we recommend the Scientific American article by Diaconis and Efron (1983). The present example demonstrates the bootstrap with a factor...

297 B o o t s t r a p p i n g E Click the Bootstrap tab. E Select Perform bootstrap . E Type 500 in the Number of bootstrap samples text box. Monitoring the Progress of the Bootstrap You can monitor the progress of the bootstrap algorithm by watching the Computation summary panel at the left of the ...

303 E x a m p l e 20 Bootstrapping for Model Comparison Introduction This example demonstrates the use of the bootstrap for model comparison. Bootstrap Approach to Model Comparison The problem addressed by this method is not that of evaluating an individual model in absolute terms but of choosing am...

307 B o o t s t r a p p i n g f o r M o d e l C o m p a r i s o n You would not ordinarily fit the saturated and independence models separately, since Amos automatically reports fit measures for those two models in the course of every analysis. However, it is necessary to specify explicitly the satu...

309 B o o t s t r a p p i n g f o r M o d e l C o m p a r i s o n implied moments obtained from fitting Model 1 to the b-th bootstrap sample. Thus, is a measure of how much the population moments differ from the moments estimated from the b-th bootstrap sample using Model 1. The average of over 1,00...

310 E x a m p l e 2 0 fail for models that fit poorly. If some way could be found to successfully fit Model 2 to these 19 samples—for example, with hand-picked start values or a superior algorithm—it seems likely that the discrepancies would be large. According to this line of reasoning, discarding ...

314 E x a m p l e 2 1 Selecting Bootstrap ADF , Bootstrap ML , Bootstrap GLS , B ootstrap SLS , and Bootstrap ULS specifies that each of C ADF , C ML , C GLS , and C ULS is to be used to measure the discrepancy between the sample moments in the original sample and the implied moments from each boots...

320 E x a m p l e 2 2 Figure 22-1: Felson and Bohrnstedt’s model for girls Specification Search with Few Optional Arrows Felson and Bohrnstedt were primarily interested in the two single-headed arrows, academic ← attract and attract ← academic . The question was whether one or both, or possibly neit...

322 E x a m p l e 2 2 When you perform the exploratory analysis later on, the program will treat the three colored arrows as optional and will try to fit the model using every possible subset of them. Selecting Program Options E Click the Options button on the Specification Search toolbar. E In the ...

323 S p e c i f i c a t i o n S e a r c h Limiting the number of models reported can speed up a specification search significantly. However, only eight models in total will be encountered during the specification search for this example, and specifying a nonzero value for Retain only the best ___ mo...

324 E x a m p l e 2 2 The following table summarizes fit measures for the eight models and the saturated model: The Model column contains an arbitrary index number from 1 through 8 for each of the models fitted during the specification search. Sat identifies the saturated model. Looking at the first...

325 S p e c i f i c a t i o n S e a r c h Figure 22-2: Path diagram for Model 7 Viewing Parameter Estimates for a Model E Click on the Specification Search toolbar. E In the Specification Search window, double-click the row for Model 7. The drawing area displays the parameter estimates for Model 7. ...

326 E x a m p l e 2 2 Using BCC to Compare Models E In the Specification Search window, click the column heading BCC 0 . The table sorts according to BCC so that the best model according to BCC (that is, the model with the smallest BCC ) is at the top of the list. Based on a suggestion by Burnham an...

327 S p e c i f i c a t i o n S e a r c h Viewing the Akaike Weights E Click the Options button on the Specification Search toolbar. E In the Options dialog box, click the Current results tab. E In the BCC, AIC, BIC group, select Akaike weights / Bayes factors (sum = 1) . In the table of fit measure...

328 E x a m p l e 2 2 The Akaike weight has been interpreted (Akaike, 1978; Bozdogan, 1987; Burnham and Anderson, 1998) as the likelihood of the model given the data. With this interpretation, the estimated K-L best model (Model 7) is only about 2.4 times more likely (0.494 / 0.205 = 2.41) than Mode...

329 S p e c i f i c a t i o n S e a r c h E In the Specification Search window, click the column heading BIC 0 . The table is now sorted according to BIC so that the best model according to BIC (that is, the model with the smallest BIC ) is at the top of the list. Model 7, with the smallest BIC , is...

331 S p e c i f i c a t i o n S e a r c h Madigan and Raftery (1994) suggest that only models in Occam’s window be used for purposes of model averaging (a topic not discussed here). The symmetric Occam’s window is the subset of models obtained by excluding models that are much less probable (Madigan...

332 E x a m p l e 2 2 Examining the Short List of Models E Click on the Specification Search toolbar. This displays a short list of models. In the figure below, the short list shows the best model for each number of parameters. It shows the best 16-parameter model, the best 17-parameter model, and s...

333 S p e c i f i c a t i o n S e a r c h Viewing a Scatterplot of Fit and Complexity E Click on the Specification Search toolbar. This opens the Plot window, which displays the following graph: The graph shows a scatterplot of fit (measured by C) versus complexity (measured by the number of paramet...

334 E x a m p l e 2 2 E Choose one of the models from the pop-up menu to see that model highlighted in the table of model fit statistics and, at the same time, to see the path diagram of that model in the drawing area. In the following figure, the cursor points to two overlapping points that represe...

335 S p e c i f i c a t i o n S e a r c h Adjusting the Line Representing Constant Fit E Move your mouse over the adjustable line. When the pointer changes to a hand, drag the line so that NFI 1 is equal to 0.900. (Keep an eye on NFI 1 in the lower left panel while you reposition the adjustable line...

336 E x a m p l e 2 2 Viewing the Line Representing Constant C – df E In the Plot window, select C – df in the Fit measure group. This displays the following: The scatterplot remains unchanged except for the position of the adjustable line. The adjustable line now contains points for which C – df is...

337 S p e c i f i c a t i o n S e a r c h Appendix G). Initially, both CFI 1 and CFI 2 are equal to 1 for points on the adjustable line. When you move the adjustable line, the fit measures in the lower left panel change to reflect the changing position of the line. Adjusting the Line Representing Co...

338 E x a m p l e 2 2 Viewing Other Lines Representing Constant Fit E Click AIC , BCC , and BIC in turn. Notice that the slope of the adjustable line becomes increasingly negative. This reflects the fact that the five measures ( C , C – df , AIC , BCC , and BIC ) give increasing weight to model comp...

339 S p e c i f i c a t i o n S e a r c h Each point in this graph represents a model for which C is less than or equal to that of any other model that has the same number of parameters. The graph shows that the best 16-parameter model has , the best 17-parameter model has , and so on. While Best fi...

340 E x a m p l e 2 2 BIC is the measure among C , C – df , AIC , BCC , and BIC that imposes the greatest penalty for complexity. The high penalty for complexity is reflected in the steep positive slope of the graph as the number of parameters increases beyond 17. The graph makes it clear that, acco...

341 S p e c i f i c a t i o n S e a r c h E In the Fit measure group, select C . The Plot window displays the following graph: Figure 22-6: Scree plot for C In this scree plot, the point with coordinate 17 on the horizontal axis has coordinate 64.271 on the vertical axis. This represents the fact th...

342 E x a m p l e 2 2 The figure on either p. 338 or p. 341 can be used to support a heuristic point of diminishing returns argument in favor of 17 parameters. There is this difference: In the best-fit graph (p. 338), one looks for an elbow in the graph, or a place where the slope changes from relat...

343 S p e c i f i c a t i o n S e a r c h For C – df , AIC , BCC , and BIC , the units and the origin of the vertical axis are different than for C , but the graphs are otherwise identical. This means that the final model selected by the scree test is independent of which measure of fit is used (unl...

344 E x a m p l e 2 2 Specification Search with Many Optional Arrows The previous specification search was largely confirmatory in that there were only three optional arrows. You can take a much more exploratory approach to constructing a model for the Felson and Bohrnstedt data. Suppose that your o...

345 S p e c i f i c a t i o n S e a r c h Specifying the Model E Open Ex22b.amw . If you performed a typical installation, the path will be C:\Program Files\IBM\SPSS\Amos\21\Examples\<language>\Ex22b.amw . Tip: If the last file you opened was in the Examples folder, you can open the file by do...

348 E x a m p l e 2 2 Viewing the Scree Plot E Click on the Specification Search toolbar. E In the Plot window, select Scree in the Plot type group. The scree plot strongly suggests that models with 15 parameters provide an optimum trade-off of model fit and parsimony. E Click the point with the hor...

350 E x a m p l e 2 3 Figure 23-1: Exploratory factor analysis model with two factors Specifying the Model E Open the file Ex23.amw . If you performed a typical installation, the path will be C:\Program Files\IBM\SPSS\Amos\21\Examples\<language>\Ex23.amw . Initially, the path diagram appears a...

351 E x p l o r a t o r y F a c t o r A n a l y s i s b y S p e c i f i c a t i o n S e a r c h Making All Regression Weights Optional E Click on the Specification Search toolbar, and then click all the single-headed arrows in the path diagram. Figure 23-2: Two-factor model with all regression weigh...

352 E x a m p l e 2 3 E Now click the Next search tab. Notice that the default value for Retain only the best ___ models is 10 .

357 E x p l o r a t o r y F a c t o r A n a l y s i s b y S p e c i f i c a t i o n S e a r c h Viewing the Scree Plot E Click on the Specification Search toolbar. E In the Plot window, select Scree in the Plot type group. The scree plot strongly suggests the use of 13 parameters because of the way ...

358 E x a m p l e 2 3 Heuristic Specification Search The number of models that must be fitted in an exhaustive specification search grows rapidly with the number of optional arrows. There are 12 optional arrows in Figure 23-2 on p. 351 so that an exhaustive specification search requires fitting mode...

359 E x p l o r a t o r y F a c t o r A n a l y s i s b y S p e c i f i c a t i o n S e a r c h and Backward searches are alternated until one Forward or Backward search is completed with no improvement. Performing a Stepwise Search E Click the Options button on the Specification Search toolbar. E I...

361 E x p l o r a t o r y F a c t o r A n a l y s i s b y S p e c i f i c a t i o n S e a r c h Limitations of Heuristic Specification Searches A heuristic specification search can fail to find any of the best models for a given number of parameters. In fact, the stepwise search in the present examp...

364 E x a m p l e 2 4 Figure 24-1: Two-factor model for girls and boys This is the same two-group factor analysis problem that was considered in Example 12. The results obtained in Example 12 will be obtained here automatically. Specifying the Model E From the menus, choose File > Open . E In the...

366 E x a m p l e 2 4 subsets that appear in a black (that is, not gray) font are mutually exclusive and exhaustive, so that column 3 generates a model in which all parameters are constant across groups. In summary, columns 1 through 3 generate a hierarchy of models in which each model contains all ...

367 M u l t i p l e - G r o u p F a c t o r A n a l y s i s Viewing the Generated Models E In the Multiple-Group Analysis dialog box, click OK . The path diagram now shows names for all parameters. In the panel at the left of the path diagram, you can see that the program has generated three new mod...

368 E x a m p l e 2 4 Fitting All the Models and Viewing the Output E From the menus, choose Analyze > Calculate Estimates to fit all models. E From the menus, choose View > Text Output . E In the navigation tree of the output viewer, click the Model Fit node to expand it, and then click CMIN ...

369 M u l t i p l e - G r o u p F a c t o r A n a l y s i s Here is the CMIN table: E In the navigation tree, click AIC under the Model Fit node. AIC and BCC values indicate that the best trade-off of model fit and parsimony is obtained by constraining all parameters to be equal across groups (the M...

370 E x a m p l e 2 4 Model 24b: Comparing Factor Means Introducing explicit means and intercepts into a model raises additional questions about which cross-group parameter constraints should be tested, and in what order. This example shows how Amos constrains means and intercepts while fitting the ...

371 M u l t i p l e - G r o u p F a c t o r A n a l y s i s Removing Constraints Initially, the factor means are fixed at 0 for both boys and girls. It is not possible to estimate factor means for both groups. However, Sörbom (1974) showed that, by fixing the factor means of a single group to consta...

372 E x a m p l e 2 4 Now that the constraints on the girls’ factor means have been removed, the girls’ and boys’ path diagrams look like this: Tip: To switch between path diagrams in the drawing area, click either Boys or Girls in the List of Groups pane to the left. Generating the Cross-Group Cons...

373 M u l t i p l e - G r o u p F a c t o r A n a l y s i s The default settings, as shown above, will generate the following nested hierarchy of five models: E Click OK . Fitting the Models E From the menus, choose Analyze > Calculate Estimates . The panel at the left of the path diagram shows t...

374 E x a m p l e 2 4 cross-group constraints, and the Measurement weights model with factor loadings held equal across groups, are unidentified. Viewing the Output E From the menus, choose View > Text Output . E In the navigation tree of the output viewer, expand the Model Fit node. Some fit mea...

377 E x a m p l e 25 Multiple-Group Analysis Introduction This example shows you how to automatically implement Sörbom’s alternative to analysis of covariance. Example 16 demonstrates the benefits of Sörbom’s approach to analysis of covariance with latent variables. Unfortunately, as Example 16 also...

379 M u l t i p l e - G r o u p A n a l y s i s E In the drawing area, right-click pre_verbal and choose Object Properties from the pop- up menu. E In the Object Properties dialog box, click the Parameters tab. E In the Mean text box, type 0 . E With the Object Properties dialog box still open, clic...

380 E x a m p l e 2 5 E Click OK to generate the following nested hierarchy of eight models: Model Constraints Model 1 (column 1) Measurement weights (factor loadings) are constant across groups. Model 2 (column 2) All of the above, and measurement intercepts (intercepts in the equations for predict...

382 E x a m p l e 2 5 There are many chi-square statistics in this table, but only two of them matter. The Sörbom procedure comes down to two basic questions. First, does the Structural weights model fit? This model specifies that the regression weight for predicting post_verbal from pre_verbal be c...

383 M u l t i p l e - G r o u p A n a l y s i s E Now click experimental in the panel on the left. As you can see in the following covariance table for the experimental group, there are four modification indices greater than 4: Of these, only two modifications have an obvious theoretical justificati...

384 E x a m p l e 2 5 Now that the Structural weights model fits the data, it can be asked whether the Structural intercepts model fits significantly worse. Assuming the Structural weights model to be correct: The Structural intercepts model does fit significantly worse than the Structural weights m...

385 E x a m p l e 26 Bayesian Estimation Introduction This example demonstrates Bayesian estimation using Amos. Bayesian Estimation In maximum likelihood estimation and hypothesis testing, the true values of the model parameters are viewed as fixed but unknown , and the estimates of those parameters...

386 E x a m p l e 2 6 interpret. A good way to start is to plot the marginal posterior density for each parameter, one at a time. Often, especially with large data samples, the marginal posterior distributions for parameters tend to resemble normal distributions. The mean of a marginal posterior dis...

387 B a y e s i a n E s t i m a t i o n Selecting Priors A prior distribution quantifies the researcher’s belief concerning where the unknown parameter may lie. Knowledge of how a variable is distributed in the population can sometimes be used to help researchers select reasonable priors for paramet...

388 E x a m p l e 2 6 Performing Bayesian Estimation Using Amos Graphics To illustrate Bayesian estimation using Amos Graphics, we revisit Example 3, which shows how to test the null hypothesis that the covariance between two variables is 0 by fixing the value of the covariance between age and vocab...

389 B a y e s i a n E s t i m a t i o n This is the resulting path diagram (you can also find it in Ex26.amw ): Results of Maximum Likelihood Analysis Before performing a Bayesian analysis of this model, we perform a maximum likelihood analysis for comparison purposes. E From the menus, choose Analy...

390 E x a m p l e 2 6 Bayesian Analysis Bayesian analysis requires estimation of explicit means and intercepts. Before performing any Bayesian analysis in Amos, you must first tell Amos to estimate means and intercepts. E From the menus, choose View > Analysis Properties . E Select Estimate means...

392 E x a m p l e 2 6 Replicating Bayesian Analysis and Data Imputation Results The multiple imputation and Bayesian estimation algorithms implemented in Amos make extensive use of a stream of random numbers that depends on an initial random number seed . The default behavior of Amos is to change th...

393 B a y e s i a n E s t i m a t i o n maintains a log of previous seeds used, so it is possible to match the file creation dates of previously generated analysis results or imputed datasets with the dates reported in the Seed Manager. Changing the Current Seed E Click Change and enter a previously...

394 E x a m p l e 2 6 Record the value of this seed in a safe place so that you can replicate the results of your analysis at a later date. Tip: We use the same seed value of 14942405 for all examples in this guide so that you can reproduce our results. We mentioned earlier that the MCMC algorithm u...

395 B a y e s i a n E s t i m a t i o n The likely distance between the posterior mean and the unknown true parameter is reported in the third column, labeled S.D. , and that number is analogous to the standard error in maximum likelihood estimation. Additional columns contain the convergence statis...

396 E x a m p l e 2 6 You can change the refresh interval to something other than the default of 1,000 observations. Alternatively, you can refresh the display at a regular time interval that you specify. If you select Refresh the display manually , the display will never be updated automatically. R...

397 B a y e s i a n E s t i m a t i o n samples, one may ask whether there are enough of these samples to accurately estimate the summary statistics, such as the posterior mean. That question pertains to the second type of convergence, which we may call convergence of posterior summaries . Convergen...

398 E x a m p l e 2 6 At this point, we have 22,501 analysis samples, although the display was most recently updated at the 22,500 th sample. The largest C.S. is 1.0012, which is below the 1.002 criterion that indicates acceptable convergence. Reflecting the satisfactory convergence, Amos now displa...

401 B a y e s i a n E s t i m a t i o n In this example, the distributions of the first and last thirds of the analysis samples are almost identical, which suggests that Amos has successfully identified the important features of the posterior distribution of the age-vocabulary covariance. Note that ...

402 E x a m p l e 2 6 E To view the trace plot, select Trace . The plot shown here is quite ideal. It exhibits rapid up-and-down variation with no long-term trends or drifts. If we were to mentally break up this plot into a few horizontal sections, the trace within any section would not look much di...

403 B a y e s i a n E s t i m a t i o n E To display this plot, select Autocorrelation . Lag , along the horizontal axis, refers to the spacing at which the correlation is estimated. In ordinary situations, we expect the autocorrelation coefficients to die down and become close to 0, and remain near...

404 E x a m p l e 2 6 when the missing values fall in a peculiar pattern, or in models with some parameters that are poorly estimated. If this should happen, the trace plots for one or more parameters in the model will have long-term drifts or trends that do not diminish as more and more samples are...

405 B a y e s i a n E s t i m a t i o n E Select Histogram to display a similar plot using vertical blocks. E Select Contour to display a two-dimensional plot of the bivariate posterior density.

406 E x a m p l e 2 6 Ranging from dark to light, the three shades of gray represent 50%, 90%, and 95% credible regions , respectively. A credible region is conceptually similar to a bivariate confidence region that is familiar to most data analysts acquainted with classical statistical inference me...

407 B a y e s i a n E s t i m a t i o n Credible Intervals Recall that the summary table in the Bayesian SEM window displays the lower and upper endpoints of a Bayesian credible interval for each estimand. By default, Amos presents a 50% interval, which is similar to a conventional 50% confidence in...

408 E x a m p l e 2 6 Learning More about Bayesian Estimation Gill (2004) provides a readable overview of Bayesian estimation and its advantages in a special issue of Political Analysis . Jackman (2000) offers a more technical treatment of the topic, with examples, in a journal article format. The b...

409 E x a m p l e 27 Bayesian Estimation Using a Non-Diffuse Prior Distribution Introduction This example demonstrates using a non-diffuse prior distribution. About the Example Example 26 showed how to perform Bayesian estimation for a simple model with the uniform prior distribution that Amos uses ...

410 E x a m p l e 2 7 As the sample size increases, the likelihood function becomes more and more tightly concentrated about the ML estimate. In that case, a diffuse prior tends to be nearly flat or constant over the region where the likelihood is high; the shape of the posterior distribution is lar...

411 B a y e s i a n E s t i m a t i o n U s i n g a N o n - D i f f u s e P r i o r D i s t r i b u t i o n experimental condition, measured their levels of depression, treated the experimental group, and then re-measured participants’ depression. The researchers did not rely on a single measure of ...

412 E x a m p l e 2 7 Bayesian Estimation with a Non-Informative (Diffuse) Prior Does a Bayesian analysis with a diffuse prior distribution yield results similar to those of the maximum likelihood solution? To find out, we will do a Bayesian analysis of the same model. First, we will show how to inc...

413 B a y e s i a n E s t i m a t i o n U s i n g a N o n - D i f f u s e P r i o r D i s t r i b u t i o n The summary table should look something like this:

414 E x a m p l e 2 7 In this analysis, we allowed Amos to reach its default limit of 100,000 MCMC samples. When Amos reaches this limit, it begins a process known as thinning . Thinning involves retaining an equally-spaced subset of samples rather than all samples. Amos begins the MCMC sampling pro...

416 E x a m p l e 2 7 E Select the variance of e5 in the Bayesian SEM window to display the default prior distribution for e5 . E Replace the default lower bound of with 0. 3.4 – 10 38 – ×

417 B a y e s i a n E s t i m a t i o n U s i n g a N o n - D i f f u s e P r i o r D i s t r i b u t i o n E Click Apply to save this change. Amos immediately discards the accumulated MCMC samples and begins sampling all over again. After a while, the Bayesian SEM window should look something like ...

418 E x a m p l e 2 7 The posterior mean of the variance of e5 is now positive. Examining its posterior distribution confirms that no sampled values fall below 0.

419 B a y e s i a n E s t i m a t i o n U s i n g a N o n - D i f f u s e P r i o r D i s t r i b u t i o n Is this solution proper? The posterior mean of each variance is positive, but a glance at the Min column shows that some of the sampled values for the variance of e2 and the variance of e3 are...

424 E x a m p l e 2 8 Indirect Effects Suppose we are interested in the indirect effect of ses on 71_alienation through the mediation of 67_alienation . In other words, we suspect that socioeconomic status exerts an impact on alienation in 1967, which in turn influences alienation in 1971.

425 B a y e s i a n E s t i m a t i o n o f V a l u e s O t h e r T h a n M o d e l P a r a m e t e r s Estimating Indirect Effects E Before starting the Bayesian analysis, from the menus in Amos Graphics, choose View > Analysis Properties . E In the Analysis Properties dialog box, click the Outp...

427 B a y e s i a n E s t i m a t i o n o f V a l u e s O t h e r T h a n M o d e l P a r a m e t e r s You do not have to work the standardized indirect effect out by hand. To view all the standardized indirect effects: E From the menus, choose View > Text Output . E In the upper left corner of ...

428 E x a m p l e 2 8 The MCMC algorithm converges quite rapidly within 22,000 MCMC samples. Additional Estimands The summary table displays results for model parameters only. To estimate the posterior of quantities derived from the model parameters, such as indirect effects: E From the menus, choos...

429 B a y e s i a n E s t i m a t i o n o f V a l u e s O t h e r T h a n M o d e l P a r a m e t e r s Estimating the marginal posterior distribution of the additional estimands may take a while. A status window keeps you informed of progress. Results are displayed in the Additional Estimands windo...

430 E x a m p l e 2 8 E To print the results, select the items you want to print. (A check mark will appear next to them). E From the menus, choose File > Print . Be careful because it is possible to generate a lot of printed output. If you put a check mark in every check box in this example, the...

431 B a y e s i a n E s t i m a t i o n o f V a l u e s O t h e r T h a n M o d e l P a r a m e t e r s Inferences about Indirect Effects There are two methods for finding a confidence interval for an indirect effect or for testing an indirect effect for significance . Sobel ( 1982, 1986 ) gives a m...

432 E x a m p l e 2 8 The lower boundary of the 95% credible interval for the indirect effect of socioeconomic status on alienation in 1971 is –0.382. The corresponding upper boundary value is –0.270, as shown below: We are now 95% certain that the true value of this standardized indirect effect lie...

433 B a y e s i a n E s t i m a t i o n o f V a l u e s O t h e r T h a n M o d e l P a r a m e t e r s E Select Mean and Standardized Indirect Effects in the Additional Estimands window.

438 E x a m p l e 2 9 ( “c” ) and the two components of the indirect effect ( “a” and “b” ). Although not required, parameter labels make it easier to specify custom estimands. To begin a Bayesian analysis of this model: E From the menus, choose Analyze > Bayesian Estimation . After a while, the ...

440 E x a m p l e 2 9 E From the menus, choose View > Additional Estimands . E In the Additional Estimands window, select Standardized Direct Effects and Mean . The posterior mean for the direct effect of ses on 71_alienation is –0.195.

442 E x a m p l e 2 9 The posterior distribution of the indirect effect lies entirely to the left of 0, so we are practically certain that the indirect effect is less than 0. You can also display the posterior distribution of the direct effect. The program does not, however, have any built-in way to...

443 E s t i m a t i n g a U s e r - D e f i n e d Q u a n t i t y i n B a y e s i a n S E M Numeric Custom Estimands In this section, we show how to write a Visual Basic program for estimating the numeric difference between a direct effect and an indirect effect. (You can use C# instead of Visual Ba...

446 E x a m p l e 2 9 The placeholder ‘TODO: Your code goes here needs to be replaced with lines for evaluating the estimands called “direct” , “indirect” and “difference” . We start by writing Visual Basic code for computing the direct effect. In the following figure, we have already typed part of ...

447 E s t i m a t i n g a U s e r - D e f i n e d Q u a n t i t y i n B a y e s i a n S E M We need to finish the statement by adding additional code to the right of the equals ( =) sign, describing how to compute the direct effect. The direct effect is to be calculated for a set of parameter values...

448 E x a m p l e 2 9 Tip: When you get the mouse pointer on the right spot, a plus (+) symbol will appear next to the mouse pointer. E Hold down the left mouse button, drag the mouse pointer into the Visual Basic window to the spot where you want the expression for the direct effect to go, and rele...

449 E s t i m a t i n g a U s e r - D e f i n e d Q u a n t i t y i n B a y e s i a n S E M The parameter on the right side of the equation is identified by the label ( “c” ) that was used in the path diagram shown earlier. We next turn our attention to calculating the indirect effect of socioeconom...

451 E s t i m a t i n g a U s e r - D e f i n e d Q u a n t i t y i n B a y e s i a n S E M E Next, drag and drop the direct effect of 1967 alienation on 1971alienation . This second direct effect appears in the Unnamed.vb window as sem.ParameterValue(“b”) .

452 E x a m p l e 2 9 E Finally, use the keyboard to insert an asterisk ( * ) between the two parameter values.

453 E s t i m a t i n g a U s e r - D e f i n e d Q u a n t i t y i n B a y e s i a n S E M Hint: For complicated custom estimands, you can also drag and drop from the Additional Estimands window to the Custom Estimands window. To compute the difference between the direct and indirect effects, add a...

455 E s t i m a t i n g a U s e r - D e f i n e d Q u a n t i t y i n B a y e s i a n S E M The marginal posterior distributions of the three custom estimands are summarized in the following table: The results for direct can also be found in the Bayesian SEM summary table, and the results for indire...

456 E x a m p l e 2 9 Most of the area lies to the left of 0, meaning that the difference is almost sure to be negative. In other words, it is almost certain that the indirect effect is more negative than the direct effect. Eyeballing the posterior, perhaps 95% or so of the area lies to the left of ...

457 E s t i m a t i n g a U s e r - D e f i n e d Q u a n t i t y i n B a y e s i a n S E M Dichotomous Custom Estimands Visual inspection of the frequency polygon reveals that the majority of difference values are negative, but it does not tell us exactly what proportion of values are negative. Tha...

458 E x a m p l e 2 9 E Add lines to the CalculateEstimates function specifying how to compute them. In this example, the first dichotomous custom estimand is true when the value of the indirect effect is less than 0. The second dichotomous custom estimand is true when the indirect effect is smaller...

461 E x a m p l e 30 Data Imputation Introduction This example demonstrates multiple imputation in a factor analysis model. About the Example Example 17 showed how to fit a model using maximum likelihood when the data contain missing values. Amos can also impute values for those that are missing. In...

462 E x a m p l e 3 0 „ Bayesian imputation is like stochastic regression imputation except that it takes into account the fact that the parameter values are only estimated and not known. Multiple Imputation In multiple imputation (Schafer, 1997), a nondeterministic imputation method (either stochas...

463 D a t a I m p u t a t i o n „ Step 1: Use the Data Imputation feature of Amos to create m complete data files. „ Step 2: Perform an analysis of each of the m completed data files separately. Performing this analysis is up to you. You can perform the analysis in Amos but, typically, you would use...

465 D a t a I m p u t a t i o n SPSS Statistics to analyze the completed datasets, the simplest thing would be to select Single output file . Then, the split file capability of SPSS Statistics could be used in Step 2 to analyze each completed dataset separately. However, to make it easy to replicate...

469 E x a m p l e 31 Analyzing Multiply Imputed Datasets Introduction This example demonstrates the analysis of multiply (pronounced multiplee ) imputed datasets. Analyzing the Imputed Data Files Using SPSS Statistics Ten completed datasets were created in Example 30. That was Step 1 in a three-step...

470 E x a m p l e 3 1 Step 2: Ten Separate Analyses For each of the 10 completed datasets from Example 30, we need to perform a regression analysis in which sentence is used to predict wordmean . We start by opening the first completed dataset, Grant_Imp1.sav , in SPSS Statistics. E From the SPSS St...

471 A n a l y z i n g M u l t i p l y I m p u t e d D a t a s e t s Step 3: Combining Results of Multiply Imputed Data Files The standard errors from an analysis of any single completed dataset are not accurate because they do not take into account the uncertainty arising from imputing missing data ...

473 A n a l y z i n g M u l t i p l y I m p u t e d D a t a s e t s Further Reading Amos provides several advanced methods of handling missing data, including FIML (described in Example 17), multiple imputation, and Bayesian estimation. To learn more about each method, consult Schafer and Graham (20...

475 E x a m p l e 32 Censored Data Introduction This example demonstrates parameter estimation, estimation of posterior predictive distributions, and data imputation with censored data. About the Data For this example, we use the censored data from 103 patients who were accepted into the Stanford He...

476 E x a m p l e 3 2 Reading across the first visible row in the figure above, Patient 17 was accepted into the program in 1968. The patient at that time was 20.33 years old. The patient died 35 days later. The next number, 5.916, is the square root of 35. Amos assumes that censored variables are n...

477 C e n s o r e d D a t a known more precisely than it is. Of course, wherever the data provide an exact numeric value, as in the case of Patient 24 who is known to have survived for 218 days, that exact numeric value is used. Recoding the Data The data file needs to be recoded before Amos reads i...

478 E x a m p l e 3 2 E Then in the Data Files dialog box, click the File Name button. E Select the data file transplant-b.sav . E Select Allow non-numeric data (a check mark appears next to it). Recoding the data as shown above and selecting Allow non-numeric data are the only extra steps that are ...

479 C e n s o r e d D a t a To fit the model: E Click on the toolbar. or E From the menus, choose Analyze > Bayesian Estimation . Note: The button is disabled because, with non-numeric data, you can perform only Bayesian estimation. After the Bayesian SEM window opens, wait until the unhappy face...

481 C e n s o r e d D a t a Posterior Predictive Distributions Recall that the dataset contains some censored values like Patient 25’s survival time. All we really know about Patient 25’s survival time is that it is longer than 1,799 days or, equivalently, that the square root of survival time excee...

486 E x a m p l e 3 2 E Double-click the file name to display the contents of the single completed data file, which contains 10 completed datasets. The file contains 1,030 cases because each of the 10 completed datasets contains 103 cases. The first 103 rows of the new data file contain the first co...

487 C e n s o r e d D a t a The first row of the completed data file contains a timesqr value of 7. Because that was not a censored value, 7 is not an imputed value. It is just an ordinary numeric value that was present in the original data file. On the other hand, Patient 25’s timesqr was censored,...

488 E x a m p l e 3 2 Normally, the next step would be to use the 10 completed datasets in transplant- b_C.sav as input to some other program that cannot accept censored data. You would use that other program to perform 10 separate analyses, using each one of the 10 completed datasets in turn. Then ...

489 E x a m p l e 33 Ordered-Categorical Data Introduction This example shows how to fit a factor analysis model to ordered-categorical data. It also shows how to find the posterior predictive distribution for the numeric variable that underlies a categorical response and how to impute a numeric val...

490 E x a m p l e 3 3 One way to analyze these data is to assign numbers to the four categorical responses; for example, using the assignment 1 = SD , 2 = D , 3 = A , 4 = SA . If you assign numbers to categories in that way, you get the dataset in environment-nl-numeric.sav . In an Amos analysis, it...

492 E x a m p l e 3 3 Recoding the Data within Amos The ordinal properties of the data cannot be inferred from the data file alone. To give Amos the additional information it needs so that it can interpret the data values SD , D , A , and SA : E From the Amos Graphics menus, choose Tools > Data R...

495 O r d e r e d - C a t e g o r i c a l D a t a column, based on the assumption that scores on the underlying numeric variable are normally distributed with a mean of 0 and a standard deviation of 1. The ordering of the categories in the Original Value column needs to be changed. To change the ord...

496 E x a m p l e 3 3 You can rearrange the categories and the boundaries. To do this: E Drag and drop with the mouse. or E Select a category or boundary with the mouse and then click the Up or Down button. After putting the categories and boundaries in the correct order, the Ordered-Categorical Det...

498 E x a m p l e 3 3 distributed with a mean of 0 and a standard deviation of 1. Alternatively, you can assign a value to a boundary instead of letting Amos estimate it. To assign a value: E Select the boundary with the mouse. E Type a numeric value in the text box. The following figure shows the r...

499 O r d e r e d - C a t e g o r i c a l D a t a The changes that were just made to the categories and the interval boundaries are now reflected in the frequency table at the bottom of the Data Recode window. The frequency table shows how the values that appear in the data file will be recoded befo...

500 E x a m p l e 3 3 That takes care of item1 . What was just done for item1 has to be repeated for each of the five remaining observed variables. After specifying the recoding for all six observed variables, you can view the original dataset along with the recoded variables. To do this: E Click th...

501 O r d e r e d - C a t e g o r i c a l D a t a environment. The other three items were designed to be measures of awareness of environmental issues. This design of the questionnaire is reflected in the following factor analysis model, which is saved in the file Ex33-a.amw . The path diagram is dr...

504 E x a m p l e 3 3 MCMC Diagnostics If you know how to interpret the diagnostic output from MCMC algorithms (for example, see Gelman, et al, 2004), you might want to view the Trace plot and the Autocorrelation plot.

506 E x a m p l e 3 3 Posterior Predictive Distributions When you think of estimation, you normally think of estimating model parameters or some function of the model parameters such as a standardized regression weight or an indirect effect. However, there are other unknown quantities in the present...

507 O r d e r e d - C a t e g o r i c a l D a t a We are in an even better position to guess at Person 1’s score on the numeric variable that underlies item1 because Person 1 gave a response to item1 . This person’s response places his or her score in the middle interval, between the two boundaries....

509 O r d e r e d - C a t e g o r i c a l D a t a That is because the program is building up an estimate of the posterior distribution as MCMC sampling proceeds. The longer you wait, the better the estimate of the posterior distribution will be. After a while, the estimate of the posterior distribut...

510 E x a m p l e 3 3 E Next, click the table entry in the first column of the 22 nd row to estimate Person 22’s score on the numeric variable that underlies his or her response to item1 . After you wait a while to get a good estimate of the posterior distribution, you see this: Both Person 1 and Pe...

511 O r d e r e d - C a t e g o r i c a l D a t a The mean of the posterior distribution (0.52) can be taken as an estimate of Person 1’s score on the underlying variable if a point estimate is required. Looking at the plot of the posterior distribution, we can be nearly 100% sure that the score is ...

514 E x a m p l e 3 3 E You can optionally view the recoded dataset that includes the new WILLING variable by clicking the View Data button.

517 O r d e r e d - C a t e g o r i c a l D a t a After drawing the path diagram for the saturated model, you can begin the imputation. E From the Amos Graphics menu, choose Analyze > Data Imputation . item1 item2 item3 item6 item4 item5

519 O r d e r e d - C a t e g o r i c a l D a t a E Click OK in the Data Imputation dialog box. The Summary window shows a list of the completed data files that were created. In this case, only one completed data file was created. E Double-click the file name in the Summary window to display the con...

520 E x a m p l e 3 3 Normally, the next step would be to use the 10 completed datasets in environment-nl- string_C.sav as input to some other program that requires numeric (not ordered- categorical) data. You would use that other program to perform 10 separate analyses using each one of the 10 comp...

521 E x a m p l e 34 Mixture Modeling with Training Data Introduction Mixture modeling is appropriate when you have a model that is incorrect for an entire population, but where the population can be divided into subgroups in such a way that the model is correct in each subgroup. Mixture modeling is...

522 E x a m p l e 3 4 The dataset contains four measurements on flowers from 150 different plants. The first 50 flowers were irises of the species setosa . The next 50 were irises of the species versicolor . The last 50 were of the species virginica . A scatterplot of two of the numeric measurements...

523 M i x t u r e M o d e l i n g w i t h T r a i n i n g D a t a The setosa flowers are all by themselves in the lower left corner of the scatterplot. It should therefore be easy for Amos to use PetalLength and PetalWidth to distinguish the setosa flowers from the others. On the other hand, there i...

527 M i x t u r e M o d e l i n g w i t h T r a i n i n g D a t a E In the Data Files dialog box, click Group Value and then double-click setosa in the Choose Value for Group dialog box.

528 E x a m p l e 3 4 The Data Files dialog box should now look like this:

530 E x a m p l e 3 4 E Click OK to close the Data Files dialog box. Specifying the Model We will use a saturated model for the variables PetalLength and PetalWidth . The scatterplot that was shown earlier suggests that these two variables will allow the program to do a good job of classifying the f...

533 M i x t u r e M o d e l i n g w i t h T r a i n i n g D a t a After the Bayesian SEM window opens, wait until the unhappy face changes into a happy face . The table of estimates in the Bayesian SEM window should look something like this: The Bayesian SEM window displays all of the parameter esti...

535 M i x t u r e M o d e l i n g w i t h T r a i n i n g D a t a The Posterior window shows that the proportion of flowers that belong to the setosa species is almost certainly between 0.25 and 0.45. It looks like there is about a 50–50 chance that the proportion is somewhere between 0.3 and 0.35. ...

537 M i x t u r e M o d e l i n g w i t h T r a i n i n g D a t a Latent Structure Analysis It was mentioned earlier that you are not limited to saturated models when doing mixture modeling. You can use a factor analysis model, a regression model, or any model at all. You may want to become familiar...

539 E x a m p l e 35 Mixture Modeling without Training Data Introduction Mixture modeling is appropriate when you have a model that is incorrect for an entire population, but where the population can be divided into subgroups in such a way that the model is correct in each subgroup. When Amos perfor...

541 M i x t u r e M o d e l i n g w i t h o u t T r a i n i n g D a t a E Click New to create a second group. E Click New once more to create a third group. E Click Close . This example fits a three-group mixture model. When you aren’t sure how many groups there are, you can run the program multiple...

543 M i x t u r e M o d e l i n g w i t h o u t T r a i n i n g D a t a E Repeat the preceding steps for Group number 2 , specifying the same data file ( iris2.sav ) and the same grouping variable ( Species ). E Repeat the preceding steps once more for Group number 3 , specifying the same data file ...

544 E x a m p l e 3 5 E Select Assign cases to groups (a check mark will appear next to it). So far, this has been just like any ordinary multiple-group analysis except for the check mark next to Assign cases to groups . That check mark turns this into a mixture modeling analysis. The check mark tel...

546 E x a m p l e 3 5 Constraining the Parameters In this example, variances and covariances will be required to be invariant across groups. This is the assumption of homogeneity of variances and covariances that is often made in discriminant analysis and some kinds of clustering. In principle, the ...

547 M i x t u r e M o d e l i n g w i t h o u t T r a i n i n g D a t a E Right-click PetalLength in the path diagram, choose Object Properties from the pop-up menu, and enter the parameter name, v1 , in the Variance text box. While the Object Properties dialog is still open, click PetalWidth in the...

548 E x a m p l e 3 5 E In the Object Properties dialog box, enter the parameter name, c12 , in the Covariance text box. The path diagram should now look like the following figure. (This path diagram is saved as Ex35-a.amw .) Fitting the Model E Click on the toolbar. or E From the menus, choose Anal...

553 M i x t u r e M o d e l i n g w i t h o u t T r a i n i n g D a t a Latent Structure Analysis There is a variation of mixture modeling called latent structure analysis in which observed variables are required to be independent within each group. E To require that PetalLength and PetalWidth be un...

554 E x a m p l e 3 5 Label Switching If you attempt to replicate the analysis in this example, it is possible that you will get the results that are reported here but with the group names permuted. The results reported here for Group number 1 might correspond to the results you get for Group number...

557 E x a m p l e 36 Mixture Regression Modeling Introduction Mixture regression modeling (Ding, 2006) is appropriate when you have a regression model that is incorrect for an entire population, but where the population can be divided into subgroups in such a way that the regression model is correct...

558 E x a m p l e 3 6 A scatterplot of dosage and performance shows two distinct groups of people in the sample. In one group, performance improves as dosage goes up. In the other group, performance gets worse as dosage goes up. -2.00 0.00 2.00 4.00 6.00 dosage 0.00 10. 00 20. 00 p e rf o rm a n c e...

559 M i x t u r e R e g r e s s i o n M o d e l i n g It would be a mistake to try to fit a single regression line to the whole sample. On the other hand, two straight lines, one for each group, would fit the data well. This is a job for mixture regression modeling. A mixture regression analysis wou...

560 E x a m p l e 3 6 The Group Variable in the Dataset Both of the datasets just described include a string (non-numeric) variable called group that contains no data. In a mixture regression analysis, Amos will use the group variable to classify individual cases. (The fact that the variable is call...

561 M i x t u r e R e g r e s s i o n M o d e l i n g The program will then use the five cases that have been pre-classified to assist in classifying the remaining cases. Pre-assigning selected individual cases to groups is mentioned here only as a possibility. In the present example, no cases will ...

563 M i x t u r e R e g r e s s i o n M o d e l i n g Specifying the Data File E From the menus, choose File > Data Files . E Click Group number 1 to select that row. E Click File Name , select the DosageAndPerformance2.sav file that is in the Amos Examples directory, and click Open . E Click Gro...

573 M i x t u r e R e g r e s s i o n M o d e l i n g Improving Parameter Estimates You can improve the parameter estimates (and also improve Amos’s ability to form clusters) by reducing the number of parameters that need to be estimated. As we have seen, the slope of the regression line is about th...

574 E x a m p l e 3 6 The path diagram should now look like the following figure. (This path diagram is saved as Ex36-b.amw .) After constraining the slope and error variance to be the same for the two groups, you can repeat the mixture modeling analysis by clicking the Bayesian button . The results...

575 M i x t u r e R e g r e s s i o n M o d e l i n g Prior Distribution of Group Proportions For the prior distribution of group proportions, Amos uses a Dirichlet distribution with parameters that you can specify. By default, the Dirichlet parameters are 4, 4, …. E To specify the Dirichlet paramet...

576 E x a m p l e 3 6 Label Switching It is possible that the results reported here for Group number 1 will match the results that you get for Group number 2, and that the results reported here for Group number 2 will match those that you get for Group number 1. In other words, your results may matc...

577 E x a m p l e 37 Using Amos Graphics without Drawing a Path Diagram Introduction People usually specify models in Amos Graphics by drawing path diagrams; however, Amos Graphics also provides a non-graphical method for model specification. If you don't want to draw a path diagram, you can specify...

578 E x a m p l e 3 7 About the Data The Holzinger and Swineford (1939) dataset from Example 8 is used for this example. A Common Factor Model The factor analysis model from Example 8 is used for this example. Whereas the model was specified in Example 8 by drawing its path diagram, the same model w...

579 U s i n g A m o s G r a p h i c s w i t h o u t D r a w i n g a P a t h D i a g r a m The Program Editor window opens. E In the Program Editor window, change the Name and Description functions so that they return meaningful strings. You may find it helpful at this point to refer to the first pat...

580 E x a m p l e 3 7 E In the Program Editor, enter the line pd.Observed("visperc") as the first line in the Mainsub function. If you save the plugin now, you can use it later on to draw a rectangle representing a variable called visperc . The rectangle will be drawn with arbitrary height a...

582 E x a m p l e 3 7 E Enter the following lines so that the plugin will draw the 12 single-headed arrows: pd.Path("visperc", "spatial", 1)pd.Path("cubes", "spatial")pd.Path("lozenges", "spatial")pd.Path("paragrap", "verbal", 1)pd.Path...

583 U s i n g A m o s G r a p h i c s w i t h o u t D r a w i n g a P a t h D i a g r a m E Specify a height, width and location each time you use the Observed, Unobserved and Caption methods of the pd class. (See the online help for the Observed, Unobserved and Caption methods.) or E In your plugin...

584 E x a m p l e 3 7 The Mainsub function now looks like this in the Program Editor:

585 U s i n g A m o s G r a p h i c s w i t h o u t D r a w i n g a P a t h D i a g r a m This completes the plugin for specifying the factor analysis model from Example 8. You can find a pre-written copy of the plugin in a file called Ex37a-plugin.vb located in a subfolder of Amos’s plugins folder....

586 E x a m p l e 3 7 After you have saved your plugin, its name, Example 37a , appears on the list of plugins in the Plugins window. (Recall that Example 37a is the string returned by the plugin’s Name function.) E Close the Plugins window. Using the Plugin E From the menus, choose File > New to...

588 E x a m p l e 3 7 Other Aspects of the Analysis in Addition to Model Specification In Example 8, the data file Grnt_fem.sav was specified interactively (by choosing File > Data Files on the menus). You can do the same thing here as well. As an alternative, you can specify the Grnt_fem.sav dat...

591 A p p e n d i x A Notation q = the number of parameters = the vector of parameters (of order q ) G = the number of groups = the number of observations in group g = the total number of observations in all groups combined = the number of observed variables in group g = the number of sample moments...

592 A p p e n d i x A = the covariance matrix for group g , according to the model = the mean vector for group g , according to the model = the population covariance matrix for group g = the population mean vector for group g = the distinct elements of arranged in a single column vector r = the non-...

593 A p p e n d i x B Discrepancy Functions Amos minimizes discrepancy functions (Browne, 1982, 1984) of the form: (D1) Different discrepancy functions are obtained by changing the way f is defined. If means and intercepts are unconstrained and do not appear as explicit model parameters, and will be...

594 A p p e n d i x B (D2) For generalized least squares estimation ( GLS ), , and are obtained by taking f to be: (D3) For asymptotically distribution-free estimation ( ADF ), , and are obtained by taking f to be: (D4) where the elements of are given by Browne (1984, Equations 3.1–3.4): ( ) ( ) ( )...

595 D i s c r e p a n c y F u n c t i o n s For scale-free least squares estimation ( SLS ), , and are obtained by taking f to be: (D5) where . For unweighted least squares estimation ( ULS ), , and are obtained by taking f to be: (D6) The Emulisrel6 method in Amos can be used to replace (D1) with: ...

596 A p p e n d i x B Suppose you have two independent samples and a model for each. Furthermore, suppose that you analyze the two samples simultaneously, but that, in doing so, you impose no constraints requiring any parameter in one model to equal any parameter in the other model. Then, if you min...

597 A p p e n d i x C Measures of Fit Model evaluation is one of the most unsettled and difficult issues connected with structural modeling. Bollen and Long (1993), MacCallum (1990), Mulaik, et al . (1989), and Steiger (1990) present a variety of viewpoints and recommendations on this topic. Dozens ...

598 A p p e n d i x C Measures of Parsimony Models with relatively few parameters (and relatively many degrees of freedom) are sometimes said to be high in parsimony, or simplicity . Models with many parameters (and few degrees of freedom) are said to be complex , or lacking in parsimony. This use o...

599 M e a s u r e s o f F i t where p is the number of sample moments and q is the number of distinct parameters. Rigdon (1994a) gives a detailed explanation of the calculation and interpretation of degrees of freedom. Note: Use the \df text macro to display the degrees of freedom in the output path...

600 A p p e n d i x C specified model). That is, P is a “ p value” for testing the hypothesis that the model fits perfectly in the population. One approach to model selection employs statistical hypothesis testing to eliminate from consideration those models that are inconsistent with the available ...

601 M e a s u r e s o f F i t Our opinion...is that this null hypothesis [of perfect fit] is implausible and that it does not help much to know whether or not the statistical test has been able to detect that it is false. (Browne and Mels, 1992, p. 78). See also “PCLOSE” on p. 605. Note: Use the \p ...

602 A p p e n d i x C FMIN FMIN is the minimum value, , of the discrepancy, F (see Appendix B). Note: Use the \fmin text macro to display the minimum value of the discrepancy function F in the output path diagram. Measures Based On the Population Discrepancy Steiger and Lind (1980) introduced the us...

603 M e a s u r e s o f F i t for , and is obtained by solving for , where is the distribution function of the noncentral chi-squared distribution with noncentrality parameter and d degrees of freedom. Note: Use the \ncp text macro to display the value of the noncentrality parameter estimate in the ...

604 A p p e n d i x C error of approximation , called RMS by Steiger and Lind, and RMSEA by Browne and Cudeck (1993). The columns labeled LO 90 and HI 90 contain the lower limit and upper limit of a 90% confidence interval on the population value of RMSEA . The limits are given by Rule of Thumb Prac...

605 M e a s u r e s o f F i t PCLOSE is a p value for testing the null hypothesis that the population RMSEA is no greater than 0.05. By contrast, the p value in the P column (see “P” on p. 599) is for testing the hypothesis that the population RMSEA is 0. Based on their experience with RMSEA , Brown...

607 M e a s u r e s o f F i t Note: Use the \bic text macro to display the value of the Bayes information criterion in the output path diagram. CAIC Bozdogan’s (1987) CAIC (consistent AIC ) is given by the formula CAIC assigns a greater penalty to model complexity than either AIC or BCC but not as g...

608 A p p e n d i x C MECVI Except for a scale factor, MECVI is identical to BCC . where if the Emulisrel6 command has been used, or if it has not. See also “BCC” on p. 606. Note: Use the \mecvi text macro to display the modified ECVI statistic in the output path diagram. Comparisons to a Baseline M...

609 M e a s u r e s o f F i t This things-could-be-much-worse philosophy of model evaluation is incorporated into a number of fit measures. All of the measures tend to range between 0 and 1, with values close to 1 indicating a good fit. Only NFI (described below) is guaranteed to be between 0 and 1,...

610 A p p e n d i x C Looked at in this way, the fit of Model A is a lot closer to the fit of the saturated model than it is to the fit of the independence model. In fact, you might say that Model A has a discrepancy that is 96.6% of the way between the (terribly fitting) independence model and the ...

611 M e a s u r e s o f F i t where and d are the discrepancy and the degrees of freedom for the model being evaluated, and and are the discrepancy and the degrees of freedom for the baseline model. The RFI is obtained from the NFI by substituting F / d for F . RFI values close to 1 indicate a very ...

612 A p p e n d i x C CFI The comparative fit index ( CFI ; Bentler, 1990) is given by where , d , and NCP are the discrepancy, the degrees of freedom, and the noncentrality parameter estimate for the model being evaluated, and , , and are the discrepancy, the degrees of freedom, and the noncentrali...

613 M e a s u r e s o f F i t PNFI The PNFI is the result of applying James, et al . ’s (1982) parsimony adjustment to the NFI where d is the degrees of freedom for the model being evaluated, and is the degrees of freedom for the baseline model. Note: Use the \pnfi text macro to display the value of...

614 A p p e n d i x C The GFI is given by where is the minimum value of the discrepancy function defined in Appendix B and is obtained by evaluating F with , g = 1, 2,...,G . An exception has to be made for maximum likelihood estimation, since (D2) in Appendix B is not defined for . For the purpose ...

615 M e a s u r e s o f F i t PGFI The PGFI (parsimony goodness-of-fit index), suggested by Mulaik, et al . (1989), is a modification of the GFI that takes into account the degrees of freedom available for testing the model where d is the degrees of freedom for the model being evaluated, and is the ...

616 A p p e n d i x C Here are the critical N ’s displayed by Amos for each of the models in Example 6: Model A, for instance, would have been accepted at the 0.05 level if the sample moments had been exactly as they were found to be in the Wheaton study but with a sample size of 164. With a sample ...

617 M e a s u r e s o f F i t The smaller the RMR is, the better. An RMR of 0 indicates a perfect fit. The following output from Example 6 shows that, according to the RMR , Model A is the best among the models considered except for the saturated model: Note: Use the \rmr text macro to display the v...

621 A p p e n d i x E Using Fit Measures to Rank Models In general, it is hard to pick a fit measure because there are so many from which to choose. The choice gets easier when the purpose of the fit measure is to compare models to each other rather than to judge the merit of models by an absolute s...

622 A p p e n d i x E (not reported by Amos) The following fit measures depend monotonically on and not at all on d . The specification search procedure reports only as representative of them all. Each of the following fit measures is a weighted sum of and d and can produce a distinct rank order of ...

625 A p p e n d i x F Baseline Models for Descriptive Fit Measures Seven measures of fit ( NFI , RFI , IFI , TLI , CFI , PNFI , and PCFI ) require a null or baseline bad model against which other models can be compared. The specification search procedure offers a choice of four null, or baseline, mo...

627 A p p e n d i x G Rescaling of AIC, BCC, and BIC The fit measures, AIC , BCC , and BIC , are defined in Appendix C. Each measure is of the form , where k takes on the same value for all models. Small values are good, reflecting a combination of good fit to the data (small ) and parsimony (small ...

628 A p p e n d i x G The rescaled values are either 0 or positive. For example, the best model according to AIC has , while inferior models have positive values that reflect how much worse they are than the best model. E To display , , and after a specification search, click on the Specification Se...

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647 I n d e x additive constant (intercept), 221ADF, asymptotically distribution-free, 594admissibility test in Bayesian estimation, 420AGFI, adjusted goodness-of-fit index, 614AIC Akaike information criterion, 309, 605 Burnham and Anderson’s guidelines for, 326 Akaike weights, 628, 629 interpreting...