Say all the variables with 5-points Likert scale or with 7-points Likert scale; or could we use a combination of both types (if we intend to perform Confirmatory factor analysis for all the individual constructs as well as measurement model)?
It is not necessary that all measures have the same response scale, or the same mean or median, or the same variance or range of scores.
If your measures are based on Likert-type response scales, and you're not aggregating individual item/question/stimulus responses into some sort of overall or composite score, then you may wish to consider the use of polychoric correlations for the CFA, or to use CFA software that allows for ordinal manifest variables (such as Mplus or lavaan).
Taking this a step forward where you may eventually consider looking at relationships between factors in a SEM-framework. In this regard, it is easier to think about it in terms that you are looking at relationships between latent constructs, where each construct is measured by some set of reflective measures. Those reflective measures can be in any form such as a 5 point or 7 point likert scale etc.
Thank you very much David Morse Kujtim Hameli Paul F Burke for your replies. Your responses clarify my doubt. However, just a related small query I have is whether it makes any difference when we go for overall/composite score vis-a-vis mean of the total score of the variables?
No it shouldn't matter as the standard error will be adjusted in the same manner the mean is adjusted. Hence the est./se to denote significance is unit free.
It is like getting the same result in regression whether you enter income in dollars or thousands of dollars.