Jochen, your question is remarkably interesting and timely. I do not know documents on the subject, but it seems to me that the concept of entropy can be used to calculate losses when the number of response categories is decreased. However, I have used dichotomizations to find answers to questions such as "what percentage of employees agree to work more than 8 hours a day to improve their income?" (The original question was "Do you agree to work more than 8 hours a day to improve your income?" (1. Strongly disagree, 2. Disagree, 3. Agree, 4. Strongly agree) For the rest of the analysis I worked with its four categories because they allowed to identify the most radical agreements or disagreements.

Dear Jochen Wilhelm ,

I don't have an answer to your question but I would like to make some (hopefully relevant) observations:

• i don't really understand why you would want to reduce the richness of ordinal data and turn it into binary data. This is against the general principles of data analysis.
• The ideal number of "steps" (value of k) is about 7 based on Miller's famous 7+-2 paper as it balances data richness against working memory limits - anything more than about 7 and respondents tend to simplify the choice.
• There is some literature on the subject of whether k should be odd or even. My personal view is k should be odd as there is a psychological tendency for people to please the question setter so answering a middle value is often a deliberate choice which should be permitted.

Dear Peter Samuels ,

that's all not really my topic and I am asking just out of curiosity. I don't "want to reduce the richness"... but I wonder whare this "richness" is actually and practically used? Example I am aware of stop by claims like "there is a significant preference of blah blah" (what means almost nothing practically useful and does not refer to any information that goes beyond a simple odds ratio (like, for instance, if the variance is affected, if there is bimodality, or if there is a concentration at some extreme etc.).

What is the trade-off between simplicity/clarity and richness/power? How can this be measured? What does it depend on? Have there been smart people thinking about that explained it somewhere that I coud possibly understand it?

Your other points touch the phychology of the people answering Likert-kind multiple-choice questions. That's interesting, but I don't even know if this is linked to my question. I was assuming any given scale (value of k), not asking for psychological reasons to prefer one over the other. If I was asking for a preference between two opposite choises/extremes, I would not think of using some kind of "muliple-chioce question" alltogether. I would rather use some graphical way to get the "position" of the person relative to the possible extremes... but this is an entirely different topic (I think, at least... maybe I'm wrong).

Particularily the last point is something that -as I understand it- cannot really be relevant for my problem. It might be "nice" for a person not to be forced to favour a particular direction when the person really feels "neutral"; but if the person really does feel "neutral" but ther are only the two "extreme" choices (no "neutral"), and one of these must be ticked, then the expectation is that Pr(choice A ticked) = Pr(Choice B ticked) = 0.5, and this is correct. If a "neutral" answer is offered, many people that are "close to neutral" may just tick that neutral answer. This is then actually an information loss! Consider all people habe a very slight favour of A over B, but really not much, and you offer A or 0 or B. This would result in all people choosing 0, and you have no information about the slight preference of A. If, instead, 0 was not offered as a possible choice, most choices will be "random", but - so I suppse! - with a slight statitical advantage for A, so you can expect Pr(A) being a slightly bit larger than Pr(B), It may even be that the odds ratio does not reflect the real balance of preferences, if people tend then to tick A ealmost surely even if they are "almost neutral". In this way you would even get a much more sensitive measure about the preference of A over B (the calibration - what is the practical relevance of the preference I see? - remains a problem).

Hi Jochen Wilhelm (nice see you ASKING a question), are you thinking something like a sign test where the 0s are ignore or randomly assigned (or weighted) a direction. The sign test "works" for the NHST because the 0s don't provide information (usually) with the direction, but the 0s do provide for the effect size if you assume there is some underlying construct. So with

values -1 0 +1

freq1 100 2 150

freq2 100 200 150

you presumably want the effect size to be smaller for freq2 than freq1. Randomly allocating people would require some analysis first to decide the weight (obviously the 50% makes interval/linear assumptions) the 0s, and perhaps doing this a few thousand times could produce a single logit coefficient (again, depending on your weighting assumption), but it doesn't seem, to me at least, intuitively easier, it just splits it into two steps.

My question is what do you mean by advantageous? If it is just simplicity of interpreting one versus two coefficients (or however many depending on the scale), then I'd assume a study of researchers would show one is simpler, but I assume you mean something more than that, and would want to include both steps.

Peter Samuels , the "ideal" number of steps is not determined by memory chunk size/working memory capacity (i.e., Miller, and if it were, see https://www.cambridge.org/core/journals/behavioral-and-brain-sciences/article/magical-number-4-in-shortterm-memory-a-reconsideration-of-mental-storage-capacity/44023F1147D4A1D44BDC0AD226838496 and note that working memory capacity for the different parts is now often treated as variable, though still discussion of this).