Comparing variables on a different scale

[Laura Freds]

I am using data that measured a person's happiness on a 1 to 7 scale and then the data coders switched to using a 0 or 1 scale. I want to compare how happiness as a DV is related to some independent variable. What is the correct way to transform this dependent variable to be compared? I assumed standardization (z-score) but wasnt sure if there are more options.

[Hemanshu Kumar]

I'm not sure I understand. A z-score would give you a continuous variable, where both versions of your variable are integer-valued categorical variables. Wouldn't you want to look at the scales and transform the initial 1-7 valued variable to its logically equivalent 0/1 counterpart?

I should add: standardisation is not strictly even a valid concept for an ordinal variable (the "mean" and "standard deviation" have no really sensible meaning).

[Laura Freds]

I am equally unsure about the z score so opinions are useful. I like the idea of finding the 0/1 equivalent but do not have a set cutoff to transform.

I do not follow the mean having no meaning. In the ordinal variable, a value of 2 is seen as twice the value of 1, and the value of 4 is seen as twice the value of 2. In this case would not the mean have meaning?

[Hemanshu Kumar]

I do not follow the mean having no meaning. In the ordinal variable, a value of 2 is seen as twice the value of 1, and the value of 4 is seen as twice the value of 2. In this case would not the mean have meaning?

Actually, that's not true. An ordinal variable only defines a ranking, or order. There is no sense is which something like "Extremely happy" coded as say 2, is literally twice as much as "Somewhat happy" which may be coded as a 1. All you can say is that a person who is "extremely happy" is probably happier than when she says she is "somewhat happy". Therefore, it strictly speaking makes no sense to compute means; medians are however okay.

[Mike Lacy]

I have a more pessimistic perspective here: It's a very standard view that there is no generally acceptable way to equate responses to the same question using different response formats. Among other issues, different response formats elicit different cognitive framings. So, for example, suppose you had a four point ordinal response format like "Strongly Agree, Agree, Disagree, and Strongly Disagree," and another with "Agree, Disagree." I would not accept that the first two responses of the four point variable could be collapsed and equated to the first response of the two point format. I believe (but don't know) that there is a serious cognitive psychology psych literature on response formats in questionnaires that would support my view.

The only way I could imagine equating responses would be if one had other studies in a similar population in which persons responded to the same questions once with the 1/7 format and once with the 1/2 format. With that, one could presumably establish some (imperfect) regression function mapping 1/7 responses to the 1/2 responses. What I would expect to find here is that, not only is such a mapping imperfect for any one question (given variations across persons), but worse yet, the parameters of such a mapping would vary across different questions.

I think you could do two different analyses here of the relationship of your explanatory variable and your response, one with the persons responding 1/7, and one with the people responding 1/2, but I don't see any way to included both versions of the response in one analysis.

[Laura Freds]

Actually, that's not true. An ordinal variable only defines a ranking, or order. There is no sense is which something like "Extremely happy" coded as say 2, is literally twice as much as "Somewhat happy" which may be coded as a 1. All you can say is that a person who is "extremely happy" is probably happier than when she says she is "somewhat happy". Therefore, it strictly speaking makes no sense to compute means; medians are however okay.

I should have been more clear. The question is about happiness and people's labour. Respondents were told that the 1-7 scale should be interpreted in this way (e.g., 2 is double 1). The question under the new scale was the same but it was responded to as a 0 or 1. Many of the points I agree. This is more of a measurement point I want to make. I want to forget whether we agree about how the scale was explained.

As Mike said there is work in this area.

I have a more pessimistic perspective here: It's a very standard view that there is no generally acceptable way to equate responses to the same question using different response formats. Among other issues, different response formats elicit different cognitive framings. So, for example, suppose you had a four point ordinal response format like "Strongly Agree, Agree, Disagree, and Strongly Disagree," and another with "Agree, Disagree." I would not accept that the first two responses of the four point variable could be collapsed and equated to the first response of the two point format. I believe (but don't know) that there is a serious cognitive psychology psych literature on response formats in questionnaires that would support my view.

The only way I could imagine equating responses would be if one had other studies in a similar population in which persons responded to the same questions once with the 1/7 format and once with the 1/2 format. With that, one could presumably establish some (imperfect) regression function mapping 1/7 responses to the 1/2 responses. What I would expect to find here is that, not only is such a mapping imperfect for any one question (given variations across persons), but worse yet, the parameters of such a mapping would vary across different questions.

I think you could do two different analyses here of the relationship of your explanatory variable and your response, one with the persons responding 1/7, and one with the people responding 1/2, but I don't see any way to included both versions of the response in one analysis.

The analysis I want to run is on the DV of happy and the IV of years of work experience. I want to interact experience and whether the new scale was used. I want to show that scale matters. This way I need to use a common DV.

[Nick Cox]

I agree with Mike Lacy. It seems to me that you can try to compare one model with an outcome of 0, 1 with another model with an outcome on the 7-point scale -- but the models would necessarily be quite different. The intended conclusion that the scale matters seems inevitable given that the scales are so different. I can't see that there is a solution based on transforming one to the other.