Haris, maybe I can add some thoughts to the great replies you have already received. 

I always try to make myself think about correlated errors as latent variables as they are essentially the same thing. In your model the covariation among 2 items is not being adequately explained by the latent variable(s) in the model so you have introduced a correlated error parameter. This would be same as adding another latent variable for these two items (albeit with equal factor loadings). The only difference is that correlated errors are somehow seen as benign parameters, simply mopping up some unwanted residual variance whereas adding latent variables tends to ruin people's models as, in this case, a 3 factor model has to become a 4 factor model. I've seen many cases of claims of unidimensionality based on a one factor model with a handful of correlated errors - that's not evidence of unidimensionality, but multidimensionality.

This is a long winded way of saying that correlated errors are indicative of systematic error variance, which in most people's book is another way of thinking about latent variables. Therefore you could describe the correlated errors in terms of another latent variable, or maybe a better approach is not to include correlated errors. Instead of just testing 1 model why not test a few alternative, or competing models, based on the same variables. there's usually at least 2 or 3 ways that even a small set of indicators can be modelled in a plausible way. This approach increases the likelihood that you'll get at least one model that fits the data well and doesn't need any correlated errors.

Best wishes

Mark

It means that AMOS cannot fit a set of loadings that will reproduce the observed pattern of covariation among the variables you have assigned to each of those factors.

With 8 variables and 3 latent factors, you must have 3 variables on each of 2 factors and just 2 variables on the other one. SEM will fit the 2-variable factor automatically, because there is only on covariance between those 2. So not being able to fit the other factors without correlated errors is general indication that your theoretical model is a poor fit to your data.

I suggest you go back and examine the correlation matrix among the original 8 variables, and order the variables so that they correspond to your proposed factors (i.e., the first three variables correspond to factor #1, then the variables for #2, and finally the set of variables associated with #3). That simplified format should allow you examine the matrix in its "raw" form.

The covariance between residuals are interpreted as an amount of variance non relevant to construct. For example, if the e1 y e2 are correlated, is because shared "any" which can´t be related with construct (item wording, format, etc.). Exactly, this items (1 y 2) could share any word, or focus in any thing, but regardless the construct.

I hope to help.

Thank you for your answers. Now I wonder about the possible write-up of the results. So, the latent construct is wellbeing and is composed of emotional, physical and material wellbeing (theory). Data do not fit this model. When I covariate residuals of emotional and physical wellbeing, than the data fit is OK. Does it mean that, maybe, emotional and physical wellbeing could be seen as one separate construct (health?- or whatever). Any ideas on the possible write-up of this are welcome.  

The more I think about, the more problematic your results become. I'm a little rusty on these calculations, but If you are estimating a factor from three variables, then there are just three covariances involved, and I  believe that using two loadings and a correlated error means that the model has to fit the data because you have used all your degrees of freedom.

Have you run an Exploratory Factor Analysis on the data? Like my earlier suggestion that you examine the correlation matrix, the EFA would help you understand the observed structure in your data, now that it is clear that the theorized pattern in not there.

FYI, I am assuming that you allowed the three factors in your CFA to be correlated, but your suggestion about separating two of them (emotional and physical) makes me wonder. The only reason to separate them would be if they were not strongly correlated in the CFA results.

if the modification in residuals or error terms of same construct is done the there is no issue but as you say that the covariance has been applied on residuals of emotional and physical wellbeing (different constructs). t you have search for possible link between the two and justify that too.

Well, actually when I do factor analysis (principal components) i get a two factor solution with all scales loading highly on first factor (all higher than 0.57), and only two scales loading highly on second factor (emotional and physical wellbeing!?). Varimax solution is almost the same but this time emotional and physical wellbeing are loading highly on second factor only. 

Jeevan, covariance has been applied on the same factor- for scales physical and emotional (the factor consists of 3 scales: emotional, physical and material).

Obviously this is not the good model fit, but i wonder of the write-up when I apply covariance of scales on the same factor. How would it differ from the one when you do not apply the covariance (in statistical and practical terms).

If you want your Exploratory Factor Analysis to match what you are doing in AMOS, then you should use Maximum Likelihood as your method of factoring with an oblique rotation to obtain correlated factors.

it is hard to report.. I correlated the two error terms to get model fit and get through the review process.

you may be able to justify it if the variable or question items were the same. Or had similar work or question or statement structure. One question might be reverse scored perhaps..

it is really hard to justify otherwise...

but cbsem justify 3-4 good questions after deletion for model specification.

my .02

Brad

What is the scale of measurement of the items?  If they are Lickert-scale items (ordinal), I would be cautious about maximum-likelihood (ML) as it assumes a degree of multivariate normality.  If they deviate substantially from normality, principal-axis factoring (PAF) is the next option and, all other things being equal, should get you results similar to ML.  HOWEVER, I strongly suspect David is correct here.  A part of your model is not identified and you are being asked/forced by the software to accommodate by adding spurious additional constraints to make the math work.

The main aim of CFA is to validate the measurement model of latent constructs. Fitness Indexes reflects the construct validity which reflect how fit is the model to the data from the field.

If the model is not fit, you need to remove the items having low factor loading and re specify the model.

However, if the remaining items have good factor loading but the model is still not fit, then there are certain items in the model which is redundant of each other. The Modification Indices will identify this particular pair of redundant items. Doing co-variation between the measurement errors of redundant items will constrain the redundancy effects and thus will increase the fitness. 

Haris, maybe I can add some thoughts to the great replies you have already received. 

I always try to make myself think about correlated errors as latent variables as they are essentially the same thing. In your model the covariation among 2 items is not being adequately explained by the latent variable(s) in the model so you have introduced a correlated error parameter. This would be same as adding another latent variable for these two items (albeit with equal factor loadings). The only difference is that correlated errors are somehow seen as benign parameters, simply mopping up some unwanted residual variance whereas adding latent variables tends to ruin people's models as, in this case, a 3 factor model has to become a 4 factor model. I've seen many cases of claims of unidimensionality based on a one factor model with a handful of correlated errors - that's not evidence of unidimensionality, but multidimensionality.

This is a long winded way of saying that correlated errors are indicative of systematic error variance, which in most people's book is another way of thinking about latent variables. Therefore you could describe the correlated errors in terms of another latent variable, or maybe a better approach is not to include correlated errors. Instead of just testing 1 model why not test a few alternative, or competing models, based on the same variables. there's usually at least 2 or 3 ways that even a small set of indicators can be modelled in a plausible way. This approach increases the likelihood that you'll get at least one model that fits the data well and doesn't need any correlated errors.

Best wishes

Mark

Thank you Mark. Very informative and useful comment!

This has been an interesting and useful thread with some very good advice.  I do want to add an additional note for the original petitioner and other readers that may be new to CFA.  Structural Equation Modeling (SEM), Latent Variable Analysis, whatever you chose to call it, is a LOT like modern American politics.  Practitioners run the gamut from very conservative (Les Hayduk at the University of Alberta) to very liberal, and many more that are in between (like myself).  The very conservative view would posit that any deviation from the original model means that it (and, thus, the theory behind the model) is an incorrect representation of the population model. They would discourage the use of modification indices for any purpose but for disconfirmation of the original model.  Very liberal practitioners will tell you that modification hunting is akin to data mining and that there is nothing wrong it.  The state of the art (in PRACTICAL terms) is probably some where in between.  Use modification indices intelligently and sparingly and ALWAYS be cautious in the attribution of causality. And, be honest in your write up about the addition of paths outside of the original model and accept that they may be interpreted as caveats to your theory by other researchers and reviewers.  Be ready to defend them.   

Monroe (2005) states that correlated measurement error occurs in the situation that variables hasn`t been identified clearly or not measured directly, so it can affect the answers to the items (Cicirelli, 1999). A latent variable include of only the true scores of a construct’s indicators measurement error is a problem (Jiang, 2014). Measurement errors may be caused by method effects such as self-reported measurement method. On the other hand, measurement errors can be the result of similar meaning or close to the meanings of words and phrases in both positive and negative statements (Harrington, 2008).

you cannot correlate the indicators in underidentified variables. (Brown,T.A., 2015,p.157) And as far as i understand, your model includes at least two underidentified variables because you have 8 items on 3 factors (latent variables).