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/