Strictly speaking, it depends on what exactly you mean by "equivalent" because there are different levels and definitions of "equivalence" of latent variables in measurement theory, for example, tau equivalence versus essential tau equivalence versus tau congenericity in classical test theory.

"Less strictly speaking" (which may be enough for the purposes of your study), if you can demonstrate that the observed variables in question load highly on the same factor in a CFA or EFA and/or that two factors for these item sets are perfectly or close to perfectly correlated (like you have already done), then I would say it's fairly plausible that these variables measure very similar attributes or even the exact same attribute.

In terms of Campbell and Fiske's (1959) multitrait-multimethod framework for convergent and discriminant validity, a (near) perfect correlation of the factors would indicate (near) perfect convergent validity of the scales/measures. Similarly, high loadings of the measured variables/items on a single factor would also indicate strong convergent validity.

Campbell, D. T., & Fiske, D. W. (1959). Convergent and discriminant validation by the multitrait-multimethod matrix. Psychological Bulletin, 56(2), 81–105. https://doi.org/10.1037/h0046016

One should go multicollinearity assessment for EFA for collinear factor and Discriminant validity assessment for distinguishing different construct.

About your question, 'How do I prove or test whether two latent variables are equivalent and measuring the same thing?', It appears your preamble answers the question. To prove that variables are measuring the same thing, EFA may lead to the variables loading on the same factor. Again, CFA or correlation analysis between the two scales/variables may be very high (i.e., from 0.9 and above). Implications of this could be (1) You may have to merge the two scales or (2) Drop one of them.

I would like to thank you all for you answers. It is very kind of you and it helped a lot.

@Sebastian Gyamfi You said that if in the CFA the correlation is 0,9 and above I could eplace one scale with the other. Is there any reference for that implication?? Thank you !

You could check these up:

Knekta, E., Runyon, C., & Eddy, S. (2019). One size doesn’t fit all: Using factor analysis to gather validity evidence when using surveys in your research. CBE—Life Sciences Education, 18(1), rm1.

Watkins, M. W. (2018). Exploratory factor analysis: A guide to best practice. Journal of Black Psychology, 44(3), 219-246.

Sebastian Gyamfi Thank you sooo much for the reference !

Hello,

It is surprising to see a correlation of 1, even if it assumes that these measures of the same construct, the items, the sample, etc. are different.

As for equivalence, it can relate to different attributes including:

- The definition of the construct(s)

- The dimensionality of the construct(s)

-The indicators

- Scale format etc.

To better appreciate the equivalence, I suggest you read chapter 2 of the following book, which exists in two versions (English and French)

· Frikha Azza (2019), La mesure en marketing : opérationnalisation des construits latents, édition ISTE, London UK.

· Frikha Azza (2019), Measurement in marketing: operationalization of latent constructs, co-edition ISTE-Wiley, London UK-Hoboken USA.

You can also find the different chapters of the book from Wiley publisher.

Here are some links:

https://onlinelibrary.wiley.com/action/doSearch?AllField=azza+frikha

https://iste-editions.fr/products/la-mesure-en-marketing

https://www.amazon.fr/mesure-en-marketing-Azza-Frikha/dp/1784055697

https://www.amazon.fr/Measurement-Marketing-Operationalization-Latent-Constructs/dp/1786304627/ref=sr_1_1?keywords=9781786304629&linkCode=qs&qid=1570923296&s=books&sr=8-1

Azza Frikha The sample is the same. I had one shot for the survey so I included both scales in the questionnaire. I know it is not very usual, but since it was my first research as a PhD candidate I was afraid to mess the whole research. So I decided to have two groups of questions, one group which was one latent variable, (this was a widely used scale) and the other one was the scale I developed, the second latent variable. The correlation between these two latent variables was 1, all participants included without any exception. In addition the correlations between these two scales and two other variables was almost the same. I downloaded your chapter and I am going to read it after sending you this message. Merci beaucoup !

when indicators that are probably similar are used in the same questionnaire administered to the same individuals, the respondents will opt for similar answers. Thus, changing the survey methodology (e.g. different sample, deferred administration for example two weeks apart from the two-scale surveys, etc.) could enlighten you.