Summing up ordered items to a sum-score: implications? It is common practice in psychology to add together ordered items to sum-scores, and then use these sum-scores for subsequent statistics. This is usually done without explicitly mentioning measurement level, and I guess people simply assume items to be interval or ratio. 
I wonder about this practice, and the implications. An example are screening instruments for mental disorders. Items are mostly ordered (and not interval or ratio), and a sum-score is used as threshold to determine whether someone has a disease or not. 
Isn't it correct that for a sum to be meaningful, equidistant points between each of the scale elements would be needed, allowing for an interpretation of differences in the degree of order? Adding up the numbers 2, 2, and 3 to a sum-score only makes sense if the first 2 is equal to the second 2, and if the 3 is 1.5 times larger than each of the 2s (which is highly problematic with ordered items, obviously). 
Is there (critical) statistical literature regarding that topic? What would be your opinion? I have looked for a while now but probably wouldn't even know the technical terms to look properly. 
 A: There is a critical literature all right, but it is rather full of criticism of the ordinal/interval/ratio level of measurement framework. Interestingly, even the originator of the whole thing was less than categorical about this particular issue:

As a matter of fact, most of the scales used widely and effectively by psychologists are ordinal scales. In the strictest propriety the ordinary statistics involving means and standard deviations ought not to be used with these scales, for these statistics imply a knowledge of something more than the relative rank-order of data. On the other hand, for this 'illegal' statisticizing there can be invoked a kind of pragmatic sanction: In numerous instances it leads to fruitful results. While the outlawing of this procedure would probably serve no good purpose, it is proper to point out that means and standard deviations computed on an ordinal scale are in error to the extent that the successive intervals on the scale are unequal in size. When only the rank-order of data is known, we should proceed cautiously with our statistics, and especially with the conclusions we draw from them.

Stevens, S.S. (1946). On the Theory of Scales of Measurement, Science, 103 (2684), June 7, 1946, 677-680.
Now, addressing your question directly:

Isn't it correct that for a sum to be meaningful, equidistant points between each of the scale elements would be needed, allowing for an interpretation of differences in the degree of order? 

No, I don't think so. It is true that strictly speaking you are making such an assumption but it does not follow that the result is completely meaningless if the assumption is not fully satisfied or that the response format automatically determines whether it is reasonable to sum several items or not. Many such scales certainly seem meaningful, as evidenced by many empirical results, correlations with other variables, theories built upon them, etc. One interpretation of all this is that people just do their best to interpret the silly tasks we give them as reasonably as possible and Likert-type items are not merely ordinal.
Now, there might be some questionable inferences along the way and many disadvantages to the poor state of measurement in many parts of psychology but it does not seem reasonable to consider that all of this is just pure noise or that the sum of several closely related items is meaningless merely because these items are measured in a 1-to-4 format. Where that does leave us, I am not sure but talks about ordinal and interval scales apparently have not been much help.
Incidentally, if you want some reasons to doubt the whole enterprise, forget about the response format and read Joel Michell, Walter Mischel on personality and Cosma Shalizi on intelligence. The main take-away would be that assuming there is some stable quantitative attribute like intelligence or (some dimension of) personality out there to be measured is in itself highly problematic, long before any concern about the distance between different points on your scale.

Beside “level of measurement”, another relevant key word would be “scaling”. Note that scale building involve other issues than level of measurement and some of these issues could be more serious. Therefore summing a few items and comparing groups in an experiment is completely reasonable and safe in my opinion but making individual decisions or comparing different cultures is a lot more difficult. 
PS: Do check chl's impressive posts on the topic in any case.
