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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.

It extracts the latent construct from other variables and shares the most variance with related variables

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.

Confirmatory factor analysis estimates latent variables based on the correlated variations of the dataset (e.g., association, causal relationship) and can reduce the data dimensions, standardize the scale of multiple indicators, and account for the correlations inherent in the dataset (Byrne 2013)

Defining individual construct: First, we have to define the individual constructs. 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.

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.

Designing a study to produce the empirical results: The measurement model must be specified.  Most commonly, the value of one loading estimate should be one per construct.  Two methods are available for identification; the first is rank condition, and the second is order condition.

Assessing the measurement model validity: Assessing the measurement model validity occurs when the theoretical measurement model is compared with the reality model to see how well the data fits.

To check the measurement model validity, the number of the indicator helps us.  For example, the factor loading latent variable should be greater than 0.7.

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Defining individual construct: First, we have to define the individual constructs. 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.

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

Theory: A systematic set of causal relationships that provide the comprehensive explanation of a phenomenon.

Usually, statistical software like AMOS, LISREL, EQS and SAS are used for confirmatory factor analysis.  In AMOS, visual paths are manually drawn on the graphic window and analysis is performed.

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You wrote: "the model fit is fine, except for the fact that the factors correlate (see picture 1)." It is not a problem per se to have correlated factors in CFA, so I don't think this necessarily speaks against Model

unless you have very specific theoretical expectations about the correlations that are at odds with the estimated correlations to you obtained in the application).

The second-order model (Model 2) is very restrictive as it implies that the covariances of all seven factors follow a unidimensional structure. This implication may be violated here as shown by the worse fit relative to Model 1.

I do not quite understand Model 3. It appears that this model has an 8th factor (BAS) that has no indicators (no measurement model). Otherwise, it is certainly possible to have a model in which only some first-order factors load onto a second order factor.

Also, the first-order factors that do not load on the second-order factor can be correlated with the residuals of the first-order factors that do load onto the second-order factor. Whether or not this makes sense is a substantive / theoretical question.

My personal preference would be Model 1 with correlated first-order factors and no second-order factor. I find second-order factors to be generally difficult to interpret.

I'll go for the first one. There is no problem with correlation between factors in the measurement model and the model fit is good, so why not.

For the third, I see that there are endogenous factors in the second order with no path to the first order, which is not correct.

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Criteria for determining the number of factors: According to the Kaiser Criterion,

Eigenvalues is a good criteria for determining a factor.

If Eigenvalues is greater than one, we should consider that a factor

if Eigenvalues is less than one, then we should not consider that a factor.

In the structural model, the conceptual variables are linked to each other, which is the equivalent of a system of regression equations or a path model.

In the structural model, the conceptual variables are linked to each other, which is the equivalent of a system of regression equations or a path model.

Maximum likelihood method: This method also works on correlation metric but it uses maximum likelihood method to factor.

Other methods of factor analysis: Alfa factoring outweighs least squares.  Weight square is another regression based method which is used for factoring.

Factor loading: Factor loading is basically the correlation coefficient for the variable and factor.  Factor loading shows the variance explained by the variable on that particular factor. 

In the SEM approach, as a rule of thumb, 0.7 or higher factor loading represents that the factor extracts sufficient variance from that variable

Agree with Varunani Shindey

Eigenvalues: Eigenvalues is also called characteristic roots.  Eigenvalues shows variance explained by that particular factor out of the total variance.  From the commonality column, we can know how much variance is explained by the first factor out of the total variance. 

For example, if our first factor explains 68% variance out of the total, this means that 32% variance will be explained by the other factor.

Factor score: The factor score is also called the component score.  This score is of all row and columns, which can be used as an index of all variables and can be used for further analysis.  We can standardize this score by multiplying a common term.  With this factor score, whatever analysis we will do, we will assume that all variables will behave as factor scores and will move

We can standardize this score by multiplying a common term.  With this factor score, whatever analysis we will do, we will assume that all variables will behave as factor scores and will move

Criteria for determining the number of factors: According to the Kaiser Criterion, Eigenvalues is a good criteria for determining a factor.  If Eigenvalues is greater than one, we should consider that a factor and if Eigenvalues is less than one, then we should not consider that a factor.  According to the variance extraction rule, it should be more than 0.7.  If variance is less than 0.7, then we should not consider that a factor.

If Eigenvalues is greater than one, we should consider that a factor and if Eigenvalues is less than one, then we should not consider that a factor.  According to the variance extraction rule, it should be more than 0.7.  If variance is less than 0.7, then we should not consider that a factor

Rotation method: Rotation method makes it more reliable to understand the output.  Eigenvalues do not affect the rotation method, but the rotation method affects the Eigenvalues or percentage of variance extracted.  .

There are a number of rotation methods available: (1) No rotation method, (2) Varimax rotation method, (3) Quartimax rotation method, (4) Direct oblimin rotation method, and (5) Promax rotation method.  Each of these can be easily selected in SPSS, and we can compare our variance explained by those particular methods

Adequate sample size: The case must be greater than the factor.

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