Basic AMOS

AMOS is statistical software and it stands for analysis of a moment structures. AMOS is an added SPSS module, and is specially used for Structural Equation Modeling, path analysis, and confirmatory factor analysis. It is also known as analysis of covariance or causal modeling software. AMOS is a visual program for structural equation modeling (SEM). In AMOS, we can draw models graphically using simple drawing tools. AMOS quickly performs the computations for SEM and displays the results.

In calculation of SEM coefficients, AMOS uses the following methods:

* Maximum likelihood
* Unweighted least squares
* Generalized least squares
* Browne’s asymptotically distribution-free criterion
* Scale-free least squares

Construction of model in AMOS:

First, we have to run AMOS. By clicking the “start” menu and selecting the “AMOS graphic” option from AMOS, we can run AMOS. The moment AMOS starts running, a window appears called the “AMOS graphic.” In this window, we can manually draw our SEM model.

Attaching data in AMOS: By selecting a file name from the data file option, we can attach data in AMOS for SEM analysis. This option also appears if we will click on the “select data” icon.

Observed variable: In AMOS, a rectangle icon is used to draw the observed variable.

Unobserved variable: In AMOS, a circle icon is used to draw the unobserved variable.

Cause effect relationdship: A single headed arrow in AMOS is used to draw the cause effect relationship between the observed and unobserved variables.

Covariance: A double headed arrow is used in AMOS to draw the covariance between variables.

Error term: In AMOS, the error term icon is next to the unobserved variable icon, and it is used to draw the latent variable.

Naming the Variable: When we right click on a variable in a graphical window, the first option, “object properties,” is used to give the name of the varible in AMOS.

There are other icons as well, and these icons help in drawing the SEM model graphically. Icons such as erase icon, moving icon, caculate icon, view text, analysis properties, etc., help in drawing the SEM model graphically.

Understanding the text output in AMOS: After running the analysis, we can see the results on the graphic window. We can also see the text output. The graphic window will only show the standardized and unstandrized regressions and error term weights. All results will be shown in the text output.
AMOS will produce the following important output:

Variable Summary: In AMOS and its text output variable summery, we can see how many variables and which variables are used for SEM analysis. We can see how many observerd variables and how many unobserved variables were in the model.

Accessing the normality: In SEM model, data should be normaly distributed. AMOS will give the text output, and Skewness, Kurtosis and Mahalanobis d-squared test will tell us about the normality of the data.

Estimates: In AMOS text output, the estimate option will give the result for regression weight, standardized loading for factor, residual, correlation, covariance, direct effect, indirect effect, total effect, etc.

Modification index: In AMOS text output, the modification index result shows the reliability of the path drawn in the SEM model. If MI index value is large, then we can add more paths to the SEM model.

Model fit: In AMOS text output, Model fit option will give the result for goodness of fit model statistics. It will show all the goodness of fit indexes, such as GFI, RMR, TLI, BIC, RMSER, etc.

Error message: If there is any problem, during the process of drawing the model (for example, if we forget to draw the error term or if we draw the covariance between two variables, or if missing data is present), then AMOS will either not calculate the result or it will give an error message.

Source: http://www.statisticssolutions.com/statistics-software/amos/


Structural equation modeling

Structural equation modeling is a multivariate statistical analysis technique that is used to analyze structural relationships. Structural equation modeling technique is the combination of factor analysis and multiple regression analysis, and it is used to analyze the structural relationship between measured variables and latent constructs. Structural equation modeling is preferred by the researcher because it estimates the multiple and interrelated dependence in a single analysis.

In structural equation modeling, two types of variables are used endogenous variables and exogenous variables. In structural equation modeling, endogenous variables are equivalent to dependent variables. In structural equation modeling, exogenous variables are equal to the independent variable.
Structural equation modeling and theory:

In structural equation modeling, theory can be thought of as a set of relationships providing consistency and comprehensive explanations of the actual phenomena. Structural equation modeling consists of two types of models:

1. Measurement model: In structural equation modeling, the measurement model represents the theory that specifies how measured variables come together to represent the theory.

2. Structural model: In structural equation modeling, the structural model represents the theory that shows how constructs are related to other constructs.

Structural equation modeling is also called casual modeling because structural equation modeling tests the proposed casual relationships. The following assumptions are assumed in structural equation modeling:

1. Multivariate normal distribution: In structural equation modeling, the maximum likelihood method is used and assumed for multivariate normal distribution. Small changes in multivariate normality can lead to a large difference in the chi-square test.

2. Linearity: In structural equation modeling, a linear relationship is assumed between endogenous and exogenous variables.

3. Outlier: In structural equation modeling, data should be free of outliers. Outliers affect the model significance.

4. Sequence: In structural equation modeling, there should be a cause and effect relationship between endogenous and exogenous variables, and a cause has to occur before the event.

5. Non-spurious relationship: In structural equation modeling, observed covariance must be true.

6. Model identification: In structural equation modeling, equations must be greater than the estimated parameters or models should be over identified or exact identified. Under identified models are not considered in structural equation modeling.

7. Sample size: In structural equation modeling, most of the researchers prefer a 200 to 400 sample size with 10 to 15 indicators. As a rule of thumb, that is 10 to 20 times as many cases as variables.

8. Uncorrelated error terms: In structural equation modeling, error terms are assumed uncorrelated with other variable error terms.

9. Data: Interval data is used in structural equation modeling.

The following steps are involved in structural equation modeling:

1. Defining individual constructs: The first step in structural equation modeling is to define the constructs theoretically. Conduct a pretest to evaluate the item. A confirmatory test of the measurement model is conducted using CFA.

2. Developing the overall measurement model: In structural equation modeling, the measurement model is also known as path analysis. Path analysis is a set of relationships between exogenous and endogens variables. This is shown by the use of an arrow. The measurement model follows the assumption of unidimensionality. Measurement theory is based on the idea that latent constructs cause the measured variable and that the error term is uncorrelated within measured variables. In a measurement model, an arrow is drawn from the measured variable to the constructs.

3. Design the study to produce the empirical results: In this step of structural equation modeling, the researcher must specify the model. The researcher should design the study to minimize the likelihood of an identification problem. In structural equation modeling, order condition and rank condition methods are used to minimize the identification problem.

4. Assessing the measurement model validity: In structural equation modeling, assessing the measurement model is also called CFA. In CFA, a researcher compares the theoretical measurement against the reality model. In structural equation modeling, the result of the CFA must be associated with the constructs’ validity.

5. Specifying the structural model: In this step of structural equation modeling, structural paths are drawn between constructs. In the structural model, no arrow can enter an exogenous construct. A single-headed arrow is used to represent a hypothesized structural relationship between one construct and another. This shows the cause and effect relationship. In structural equation modeling, each hypothesized relationship uses one degree of freedom. In structural equation modeling, the model can be recursive or nonrecursive.

6. Examine the structural model validity: In the last step of structural equation modeling, a researcher examines the structural model validity. In structural equation modeling, a model is considered a good fit if the value of the chi-square test is insignificant, and at least one incremental fit index (like CFI, GFI, TLI, AGFI, etc.) and one badness of fit index (like RMR, RMSEA, SRMR, etc.) meet the predetermined criteria.

Source: http://www.statisticssolutions.com/methods-chapter/statistical-tests/structural-equation-modeling/

Path Analysis

In statistics analysis, path analysis is considered an extension of the regression model. In a path analysis model from the correlation matrix, two or more casual models are compared. In path analysis, the path of the model is shown by a circle and an arrow, which shows the causation. In path analysis, regression weight is predicated by the model, and is compared by the observed correlation matrix. Then the goodness of fit statistic is calculated in order to see the fitting of the model.
Key concepts and terms in path analysis:

Estimation method in Path analysis: In path analysis, simple OLS and maximum likelihood methods are used to predict the path.

Path model: In path analysis, a path model is a diagram which shows the independent, intermediate, and dependent variable. In path analysis, a single headed arrow shows the cause for the independent, intermediate and dependent variable. A double headed arrow shows the covariance between the two variables.

Exogenous and endogenous variables in path analysis: In path analysis, exogenous variables in a path model are those where no error points towards them, except the measurement error term. If exogenous variables are correlated to each other, then a double headed arrow will connect those variables. Endogenous variables may have both the incoming and outgoing arrows.

Path coefficient: In path analysis, a path coefficient is a standardized regression coefficient (beta), showing the direct effect of an independent variable on a dependent variable in the path model.

Disturbance terms: The residual error terms are also called disturbance terms in path analysis. Disturbance terms reflect the unexplained variance and measurement error.

Direct and indirect effect in path analysis: In path analysis, the path model has two types of effects. The first is the direct effect, and the second is the indirect effect. When the exogenous variable has an arrow directed towards the dependent variable, then it is said to be the direct effect. When an exogenous variable has an effect on the dependent variable, through the other exogenous variable, then it is said to be an indirect effect. To see the total effect of the exogenous variable, we have to add the direct and indirect effect. In path analysis, one variable may not have a direct effect, but it may have an indirect effect.

Significance and goodness of fit in path analysis: OLS and maximum likelihood methods are used to predict the path coefficient in path analysis. These days, statistical software like AMOS, SAS and LISREL, etc. are software that calculates the path coefficient and goodness of fit statistics automatically.
The following statistics are used to test the significance and goodness of fit of path analysis:

Chi-square statistics: Non-significant chi-square value in path analysis shows the goodness of fit model. Sometimes, chi-square statistics is significant. However, we still have to test one absolute fit index and one incremental fit index.

Absolute fit index: RMSEA: An absolute fit index using 90% confidence interval for RMSEA should be less than 0.08 for a goodness of fit model.

Increment fit index: CFI, GFI, NNFI, TLI, RFI and AGFI, etc. are some incremental fit indexes, which should be greater than 0.90 for a goodness of fit model.

Modification indexes: Modification indexes (MI) may be used to add arrows to the model. The larger the MI, the more arrows will be added to the model, which will improve the model fit.

Assumptions in Path analysis:

Linearity: In path analysis, relationships should be linear.

Interval level data: In path analysis, data should be at interval scale.

Uncorrelated residual term: In path analysis, error term should not be correlated to any variable.

Disturbance terms: In path analysis, disturbance terms should not be correlated to endogenous variables.

Multicollinearity: In path analysis, low multicollinearity is assumed. Perfect multicollinearity may cause problems in the path analysis.

Identification: In path analysis, the path model should not be under identified. Exactly identified or over identified models are good for path analysis.

Adequate sample size: Kline (1998) recommends that the sample size should be 10 times (or ideally 20 times) as many cases as parameters.

Source: http://www.statisticssolutions.com/methods-chapter/statistical-tests/path-analysis/

Confirmatory factor analysis

Confirmatory factor analysis (CFA) is a multivariate statistical procedure that is used to test how well the measured variables represent the number of constructs. Confirmatory factor analysis (CFA) and exploratory factor analysis (EFA) are similar techniques, but in exploratory factor analysis (EFA), data is simply explored and provides information about the numbers of factors required to represent the data. In exploratory factor analysis, all measured variables are related to every latent variable. But in confirmatory factor analysis (CFA), researchers can specify the number of factors required in the data and which measured variable is related to which latent variable. Confirmatory factor analysis (CFA) is a tool that is used to confirm or reject the measurement theory.
Terms and concepts in confirmatory factor analysis (CFA):

Theory: In confirmatory factor analysis (CFA), theory is a systematic set of causal relationships that provide the comprehensive explanation of a phenomenon.

Model: In confirmatory factor analysis (CFA), model is a specified set of dependant relationships that can be used to test the theory.

Path analysis: In confirmatory factor analysis (CFA), path analysis is used to test structural equations.

Path diagram: In confirmatory factor analysis (CFA), the path diagram shows the graphical representation of cause and effect relationships of the theory.

Endogenous variable: In confirmatory factor analysis (CFA), endogenous variables are the resulting variables that are a causal relationship.

Exogenous variable: In confirmatory factor analysis (CFA), exogenous variables are the predictor variables.

Confirmatory analysis: In confirmatory factor analysis (CFA), confirmatory analysis is used to test the pre-specified relationship.

Cronbach’s alpha: In confirmatory factor analysis (CFA), Cronbach’s alpha is used to measure the reliability of two or more construct indicators.

Identification: In confirmatory factor analysis (CFA), identification is used to test whether or not there are a sufficient number of equations to solve the unknown coefficient. In confirmatory factor analysis (CFA) identifications are of three types: (1) underidentified, (2) exact identified, and (3) over-identified.

Goodness of fit: In confirmatory factor analysis (CFA), goodness of fit is the degree to which the observed input matrix is predicted by the estimated model.
The following are the procedures involved in confirmatory factor analysis (CFA):

1. Defining individual construct: In confirmatory factor analysis (CFA), first we have to define the individual constructs. In confirmatory factor analysis (CFA), the first step involves the procedure that defines constructs theoretically. This involves a pretest to evaluate the construct items, and a confirmatory test of the measurement model that is conducted using confirmatory factor analysis (CFA), etc.

2. Developing the overall measurement model theory: In confirmatory factor
analysis (CFA), we should consider the concept of unidimensionality between construct error variance and within construct error variance. At least four constructs and three items per constructs should be present in the research.

3. Designing a study to produce the empirical results: In confirmatory factor analysis (CFA), the measurement model must be specified. In confirmatory factor analysis (CFA), most commonly, the value of one loading estimate should be one per construct. In confirmatory factor analysis (CFA), two methods are available for identification. The first is rank condition, and the second is order condition.

4. Assessing the measurement model validity: In confirmatory factor analysis (CFA), assessing the measurement model validity occurs when the theoretical measurement model is compared with the reality model to see how well the data fits. In confirmatory factor analysis (CFA), to check the measurement model validity, the number of the indicator helps us. For example, in confirmatory factor analysis (CFA), the factor loading latent variable should be greater than 0.7. Chi-square test and other goodness of fit statistics like RMR, GFI, NFI, RMSEA, SIC, BIC, etc., are some key indicators that help in measuring the model validity in confirmatory factor analysis (CFA).

Confirmatory factor analysis (CFA) and statistical software: Usually, statistical software like AMOS, LISREL, EQS and SAS are used for confirmatory factor analysis (CFA). In AMOS, visual paths are manually drawn on the graphic window and analysis is performed. In confirmatory factor analysis (CFA) in LISREL, confirmatory factor analysis (CFA) can be performed graphically as well as from the menu. In SAS, confirmatory factor analysis (CFA) can be performed by using the programming languages.

Source: http://www.statisticssolutions.com/methods-chapter/statistical-tests/confirmatory-factor-analysis/

Is AMOS necessary to get a PhD?

Source: IBM SPSS. Click to enlarge image

This is a big question posted by most participants of OUM's 3rd colloquium held in KL from Aug 7 - 8, 2010.

So should you use AMOS? Ask yourself these questions:

First, do your models have multiple regressions that are embedded within each other? Second, do you have latent variables in your models? If you answered yes to both or either of these questions, you probably need to apply structural equation models. AMOS may be the best option for you.

Generally there are many statistical tools available to assist us in analyzing our data. Among the popular ones are SPSS, LISREL, MPlus and MStat. How about AMOS? Well actually AMOS is an add-on of SPSS, which most of us are familiar with. Before we dwell further, lets look at some of the writings about AMOS.

What Is AMOS?

AMOS (Analysis of Moment Structures) is an add-on module for SPSS. It is designed primarily for structural equation modeling, path analysis, and covariance structure modeling, though it may be used to perform linear regression analysis and ANOVA and ANCOVA. It features an intuitive graphical interface that allows the analyst to specify models by drawing them. It also has a built-in bootstrapping routine and superior handling of missing data. It reads data from a number of sources, including MS Excel spreadsheets and SPSS databases.

Source: http://www.utexas.edu/its/help/spss/526

Overview of AMOS:



Build structural equation models with more accuracy than standard multivariate statistics models using intuitive drag-and-drop functionality

IBM® SPSS® Amos (formerly Amos™) gives you the power to easily perform structural equation modeling (SEM). Using SEM, you can quickly create models to test hypotheses and confirm relationships among observed and latent variables – moving beyond regression to gain additional insight.
Structural equation modeling (SEM) can take your research to the next level

When you conduct research, you’re probably already using factor and regression analyses in your work. Structural equation modeling (sometimes called path analysis) can help you gain additional insight into causal models and explore the interaction effects and pathways between variables. SEM lets you more rigorously test whether your data supports your hypothesis. You create more precise models – setting your research apart and increasing your chances of getting published.

IBM SPSS Amos is the perfect modeling tool for a variety of purposes, including:

* Psychology – Develop models to understand how drug, clinical, and art therapies affect mood
* Medical and healthcare research – Confirm which of three variables –confidence, savings, or research – best predicts a doctor’s support for prescribing generic drugs
* Social sciences – Study how socioeconomic status, organizational membership, and other determinants influence differences in voting behavior and political engagement
* Educational research – Evaluate training program outcomes to determine impact on classroom effectiveness
* Market research – Model how customer behavior impacts new product sales or analyze customer satisfaction and brand loyalty
* Institutional research – Study how work-related issues affect job satisfaction
* Business planning – Create econometric and financial models and analyze factors affecting workplace job attainment
* Program evaluation – Evaluate program outcomes or behavioral models using SEM to replace traditional stepwise regression

Features and Benefits:

IBM SPSS Amos makes structural equation modeling (SEM) easy and accessible

IBM SPSS Amos builds models that more realistically reflect complex relationships because any numeric variable, whether observed (such as non-experimental data from a survey) or latent (such as satisfaction and loyalty) can be used to predict any other numeric variable.

Its rich, visual framework lets you to easily compare, confirm and refine models.

Quickly build graphical models using IBM SPSS Amos’ simple drag-and-drop drawing tools. Models that used to take days to create are just minutes away from completion. And once the model is finished, simply click your mouse and assess your model’s fit. Then make any modifications and print a presentation-quality graphic of your final model.

Its approach to multivariate analysis encompasses and extends standard methods – including regression, factor analysis, correlation and analysis of variance. New capabilities include bootstrapping of user-defined functions of the model parameter for increased model stability.

Obtain Bayesian estimates of model parameters and other quantities

Bayesian analysis enables you to apply your subject-area expertise or business insight to improve estimates by specifying an informative prior distribution. Markov chain Monte Carlo (MCMC) is the underlying computational method for Bayesian estimation. The MCMC algorithm is fast and the MCMC tuning parameter can be adjusted automatically.

Perform estimation with ordered categorical and censored data Create a model based on non-numerical data without having to assign numerical scores to the data. Or work with censored data without having to make assumptions other than the assumption of normality. You can also impute numerical values for ordered-categorical and censored data. The resulting dataset can be used as input to programs that require complete numerical data.

Impute missing values or latent variable scores

Choose from three data imputation methods: regression, stochastic regression, or Bayesian. Use regression imputation to create a single completed dataset. Use stochastic regression imputation or Bayesian imputation to create multiple imputed datasets. You can also impute missing values or latent variable scores.

See how easy it is to use IBM SPSS Amos:

1. Select a data file:

Click on picture to enlarge image

Input data from a variety of file formats (IBM® SPSS® Statistics, Micosoft® Excel, text files, or many others). Select grouping variables and group values. IBM SPSS Amos also accepts data in a matrix format if you’ve computed a correlation or covariance matrix.

2. Specify your model:

Click to enlarge image

Use drag-and-drop drawing tools to quickly specify your path diagram model. Click on objects in the path diagram to edit values, such as variable names and parameter values. Or simply drag variable names from the variable list to the object in the path diagram to specify variables in your model.

3. Select analysis properties:


Select the analysis properties you wish to examine, such as standardized estimates of parameters or squared multiple correlations. Constrain parameters for more precise models by directly specifying path coefficients

4. View output:

Click to enlarge image

IBM SPSS Amos output provides standardized or un-standardized estimates of covariances and regression weights as well as a variety of model fit measures. Hotlinks in the help system link to explanations of the analysis in plain English.

Click to enlarge image

IBM SPSS Amos output provides standardized or un-standardized estimates of covariances and regression weights as well as a variety of model fit measures. Hotlinks in the help system link to explanations of the analysis in plain English.

5. Assess your model’s fit:

Click to enlarge image

Make any modifications to your model and print publication-quality output.

System Requirements:

Operating system

* Microsoft® Windows® Vista® or Windows XP

Hardware

* Memory: 256 MB RAM minimum
* Minimum free drive space: 125 MB

Software

* Internet Explorer 6
* Microsoft .NET Framework 3.5 SP1 or higher


Source: http://www.spss.com/amos/


Example of an application of AMOS:

Antecedents and consequences of parasocial interaction with sport athletes and identification with sport teams.

RESULTS

Due to the limited sample size, the items of each construct were first summated to create single-item indicators. Structural equation modelling was then used to test the above hypotheses. The parsimony of this partially mediated structural model was appraised with the maximum likelihood method of parameter estimation via AMOS 4.0.

The overall model fit for the proposed model was not satisfactory ([chi square] =90.935, df= 20, p<.00 l; NFI =.984; RFI=.955; IFI=.987; TFI=.965; CFI=.987; RMSEA=. 134), since RMSEA is higher than .08 (Browne & Cudeck, 1993). Thus the original model was rejected and the modification indices were subsequently examined as a way of improving the model fit (Anderson & Gerbing, 1988). The modification indices showed that the model fit could be improved if the error (or unique) terms of parasocial interaction and team identification were allowed to correlate (the chi-square statistics would decrease by at least 53.456). Each error term represents much more than random fluctuations due to measurement error. It stands for anything else on which each variable may depend, but which are not measured in the model (Arbuckle & Wothke, 1999). Since parasocial interaction with favorite athletes and sport fan team identification both cover fans' sport involvement, it is reasonable to correlate their error terms. After the model modification, the goodness of fit statistics demonstrated that the modified model provided a much stronger fit ([chi square] =27.015, df= 19,p=. 104; NFI=.995; RFI=.986; IFI=.999; TFI=.996; CFI=.999; RMSEA=.046). Figure 2 shows the modified model and Table 2 reports the parameter estimates for causal paths.

Further reading: http://www.thefreelibrary.com/Antecedents+and+consequences+of+parasocial+interaction+with+sport...-a0227279262

Click here for the guide on AMOS 16:



http://www.amosdevelopment.com/download/Amos%2016.0%20User%27s%20Guide.pdf

e-Customers Retention Management (eCRM) Model

This is the model developed to understand how online community helps to reduce feeling of isolation among e-learners and increase sense of community. Higher sense of community is perceived to cause higher level of satisfaction, motivation and commitment of students and thus increase retention rate.



This is my AMOS output

Any idea how to interpret this?