Thank you a lot, Jeffrey, for these two links.

Jeffrey's answer is correct.  Equating is the psychometric methodology for ensuring that test scores for two parallel tests are comparable. It is a large and varied area of practice.

I'd like to add that equating (despite the name) does not ensure parallelism, but rather parallelism is an assumption. That is, you can link (link is the term for equating unlike things) height to intelligence, but that doesn't make them comparable.

So, textbooks on equating and practitioners often emphasize the critical importance of using sound psychometric practices to try to create parallel measures.  For example, the test content should be planned and all tests that will be equated should be composed of the same proportions of the same kinds of items.

Finally, the other methodology for establishing equivalence statistically is using multigroup confirmatory factor analysis and this method is more flexible in establishing the aspects of equivalence and non-equivalence. This methodology does purport to establish the parallelism as well as the metric equivalence. The canonical citation is:

Vandenberg, R. J., & Lance, C. E. (2000). A review and synthesis of the measurement invariance literature: Suggestions, practices, and recommendations for organizational research, Organizational Research Methods, 3, 4-70.

• Thank you, Alan, for your reply and for the reference. I agree that before any statistical method is applied, the test versions should be equivalent in content and in construct and many other conditions have to be fulfilled. Otherwise, it does not make sense to start using psychometrics.
• You mentioned various analysis. Do you know a method called weighted multidimensional scaling?

Scaling is a methodology for affixing numeric values (from an unseen continuum) to stimuli or people.  For example IRT is a methodology for scaling item difficulty and examinee ability on the same scale.  Multidimensional scaling (MDS) refers to scaling that is along more than one dimension.  There are many forms of MDS and I guess "weighted" is one of them.  I'm not an expert on MDS.

To give you an idea how equating and scaling might be related, using IRT (or any other scaling method) you could scale the difficulty of the items of two exams and this might, for example, help you understand that one exam is easier or has more variability in difficulty.  But equating is the procedure that actually adjusts the scores on these exams for whatever differences there are in difficulty to achieve equated scores that are comparable.

If you were asking about how to establish parallelism, probably the simplest analysis you could do would be to correlate the scores of the two exams.  Ideally you would also have some estimate, such as coefficient alpha, of the reliability of the two exams.  Then classical test theory (CTT) has a well-known relation that you can use to estimate the true-score correlation:

true-score correlation = obs. correlation / sqrt( reliability test 1 * reliability test 2)

where "obs. correlation" is the observed correlation.  If the two exams are parallel, then the true-score correlation should be about 1.0. To the degree that it's less than 1.0, you have degrees of less parallelism. For example, I'd say a true-score correlation of 0.90 is still pretty good.  Also, since the true-score correlation is being estimated and you hope the true population value is 1.0, you shouldn't be surprised or disturbed to get about half your estimates > 1.0, although estimates way over 1.0 probably indicate a poorly estimated reliability (or that alpha is not a good estimate for that exam).

I hope this helps.