# Validity of modelling different ordinal variables via linear regression

I am attending a course in Marketing. Our latest assignment was to analyze data from a survey regarding dishwashing detergents. The questions we asked respondents were all of an ordinal kind: they had to give an integer score (between 1 and 5, 1 and 7, or 1 and 10) in order to rate various kinds of behaviors connected to dishwashing. Thus, every answer essentially resulted in an ordinal variable with 5-10 levels.

There are two particular questions that I'm concerned with, these were:

1. On a scale from 1 to 7, how important is it to you that a dishwashing detergent be eco-friendly and environmentally sustainable?
• 1 = not at all important to me
• 4 = it's somewhat important to me
• 7 = it is of utmost importance to me
2. On a scale from 1 to 7, how often do you use your home dishwasher, per week?
1. = never
2. = less than once a week
3. = once a week
4. = 2-3 times a week
5. = 4-6 times a week
6. = 7 times a week
7. = More than 7 times a week

Let's say these are the only two variables. We are then tasked with analyzing the data using a linear regression model, with the exact meaning of the dependent variable, Y, being unimportant for the purposes of this question. It's also usually a similar, subjective, rating-like quantity, such as "how likely are you to buy/recommend this product?".

Now it might seem inappropriate to use linear regression for this, but for pedagogical reasons, hardly any statistical method other than linear regression and Student's T-tests are taught in typical marketing courses (people there generally don't have a strong math background). Therefore, we are told to avert our eyes, pretend that the variables are normally distributed, errors are Gaussian and homoscedastic, and there's no collinearity. We are also usually taught to just treat ordinal variables as if they were numerical on an equally-spaced scale, typically positive integers.

With all these, our group had performed the analysis by simply shoving both aforementioned variables into a linear regression model and called it a day. Today, the teacher evaluated our coursework, and she said she is going to deduct a lot of our points (!) because of the inappropriate use of the second variable (the one related to dishwashing frequency).

I could hardly believe my ears. We are always being told that we can just wing it and treat all ordinals as numeric variables, and now she is complaining that we shouldn't do that. But she is being inconsistent, IMO: she is apparently fine with us using the opinion-based values (about the importance of environmental impact) in the regression, but she insists we should have one-hot encoded the answers about dishwashing frequency.

Naturally, I confronted her and asked why she deems using the second variable inappropriate when she's apparently fine with using the first variable, which is of mathematically identical nature. Her response was, in essence: the problem is that the opinion-based answers are equally-spaced (i.e., linear), while the frequency-based answers aren't, because the gaps between frequency ranges are different (as given above).

I just find this completely unsatisfying, unreasonable, and utterly inconsistent. When one asks about people's opinion in a poll, that's inherently not an equally-spaced scale, since opinion (on importance in our case) doesn't have a well-quantifiable, absolute, numerical value. The fact that the answers are numerical and equally-spaced is merely an artifact of the methodological decision of "we have no idea what the exact differences are, so let's just use integers". Opinion is not any more equally-spaced or more quantitative than objective, physical data that is binned non-uniformly.

In my opinion, it is not consistent to insist that the opinion data can be used directly in a regression model but the binned frequency data isn't. They are the exact same kind of feature, they both severely violate the assumptions of linear regressions, and they are both still useful to be analyzed via linear regression nonetheless.

My question is: am I missing something fundamental? Is the teacher right, or am I?

• You are grasping the issues better than the teacher. Be confident in your approach. Your argument would be even stronger had the frequency bins been equal width. Both variables should really be considered ordinal. Commented Nov 29, 2022 at 13:08
• @FrankHarrell Thank you! I also think that uniform bin widths are not even necessarily a requirement – for example, many econometric phenomena (e.g. prices vs. demand) exhibit an exponential relationship, in which case even explicit, intentional logarithmic binning (e.g. 1-2, 2-4, 4-8, etc.) would also make more sense and could actually result in a more predictive model, if I'm not mistaken. Commented Nov 29, 2022 at 13:15
• But for that case you have the original continuous value, and binning is not needed or warranted. Commented Nov 29, 2022 at 23:01
• @FrankHarrell Indeed. Unfortunately, most of the time, people (teachers included) expect binning, for alleged reasons of "interpretability", however misguided that might be. Surveys/opinion polls are also typically expected to contain multiple-choice questions with a finite number of pre-defined bins, and it's frowned upon to present an open-ended "please insert how many times per month you buy this product" numerical entry field. It's ironic (and sad) how artificial discretization is somehow admired in econometry, yet significantly smaller issues are punished harshly. Commented Nov 30, 2022 at 7:06
• Imaging if an olympic track coach had a binned stopwatch or the speedometer in your car read slow or fast. Binning is bad. Commented Nov 30, 2022 at 12:40