# Dependent Variable takes on the values 0, 1, 2, 3 - What is the right (logistic) regression model to use?

I am looking for help to analyze the data from my online experiment.

For my master thesis I conducted an online experiment where participants had to conduct a shopping task where they were provided with a local and a non-local product three times in a row. So for three times, they had to choose between either the local or the non-local product. The result of the shopping task is my dependent variable "Green Shopping Behavior". Every time someone chose the local product, they got a "1" and if they chose the non-local product they got a "0". In the end i added everything up, so for each observation the dependent variable can take the values 0, 1, 2 and 3. So from my understanding i have a metrically scaled variable that can be considered count data. I have two metric/continuous independent variables I want to use to predict if participants chose 0/1/2/3 local products, as well as control variables.

Since the structure of my dependent variable is not fitted for a linear regression I looked into other regression models and landed on different logistic models but am heavily confused about what the right approach is. I understand that a classic logistic regression is not suitable, because my dependent variable ist not binary, I further looked into multinomial and ordered logistic regression analysis from which I think that the ordered regression is most suitable given the structure of my dependent variable. Could you give me some insights into what is the right analysis, if i am on the right track or if i am heavily mistaken and if you would propose a completely different strategy?

Your help is highly appreciated!

edit: I use STATA

• "Since the structure of my dependent variable is not fitted for a linear regression" How did you determine this?
– Stef
Commented Sep 21, 2022 at 14:42

If you stick really close to the data generating process, these are repeated binary decisions. I.e. each participants makes three decisions of choosing the local product or not (each time a 1/0 outcome).

It's not count data, because the counts cannot reach any arbitrary number. Arguably, it could be truncated (at 3) count data or ordinal data, but that's not exactly how the data arose. That does not mean that you could not create a useful model by considering it that way, especially an ordinal model may indeed be useful. However, an ordinal model ignores that participants are asked to make a very similar choices - possibly with some known differences between questions - each time.

If one goes down the repeated binary route, then it matters what we wish to assume/what detailed data we have on each decision. E.g. when asked about the products, the product might sometimes have been a vegetables, exotic fruit, or clothes. People might tend to prefer buying vegetables locally, think that exotic fruits are so much better from further away and be more mixed about clothes. Or perhaps you think people might change their answer based on something else (e.g. is it the first question they get asked, participant's age, favorability rating 0-100 for globalization etc.). In that case modeling each binary choice (with explanatory variables like type of choice, participant age etc. as a fixed effect) and accounting for decisions being by the same participant (e.g. participant random effect) would be an option. Or, perhaps the choices were really all the same, in which one can reduce this to a binomial outcome (i.e. number of yes answers out of 3 with the same probability applying to all of them). Thus, I'm arguing for the use of (random effects) logistic regression (which is well supported e.g. in R using the lme4 or brms packages).

Although ordinal regression is a generally useful choice for ordered outcomes, in this particular case a binary regression could also be considered.

In each of the 3 trials per individual there is a binary choice: local versus non-local. In R you can model this situation with binary regression, coding the outcomes in a two-column integer matrix of c(successes, failures) with a row for each individual. That would also allow for individuals who didn't finish all 3 trials. See the Details of the R help page for family. I suspect that STATA allows for similar coding.

My impression is that would provide the same result as ordinal logistic regression here, but I haven't thought that through carefully. A potential advantage of using binary regression is that it's (perhaps unfortunately) more likely to be familiar to your audience than ordinal regression and thus easier to explain.

A binary regression could also be used to evaluate changes in outcome probabilities as a function of trial number, although it might be unlikely in this study. You could separate out each trial with a 0/1 outcome and annotation of the individual and trial number, treat the individuals as random effects in a mixed model, and include trial number as a covariate.

Yes, ordinal logistic regression (also referred to as the proportional odds model) is your best choice. If your outcome were to have 7+ ordered categories, linear regression may also be used, though because you have few ordered categories (i.e., 4) I would use ordinal logistic regression as suggested in the comments by @mkt. Bauer & Sterba (2011) and Bürkner & Vuorre (2019) are good introductions ordinal regression.

References

Bauer, D. J., & Sterba, S. K. (2011). Fitting multilevel models with ordinal outcomes: performance of alternative specifications and methods of estimation. Psychological methods, 16(4), 373.

Bürkner, P. C., & Vuorre, M. (2019). Ordinal regression models in psychology: A tutorial. Advances in Methods and Practices in Psychological Science, 2(1), 77-101.

• That 7+ categories claim seems surprisingly specific - could you explain the justification?
– mkt
Commented Sep 20, 2022 at 15:10
• Yes, you are correct, it is fairly specific. Most simulation research concerning ordinal logistic regression is influenced by research scenarios where likert scales are commonly used, and most Likert scales do not contain > 7 categories. Simulation studies have found that linear models work adequately well when the 7-point Likert scale is used. Of course, if the sample size is adequately large, it is probably best to always stick with ordinal regression. Commented Sep 20, 2022 at 15:30
• Could you give some literature? Commented 2 days ago

## Beware of adding difficult-to-compare indicators

Treating your variable as an ordinal assumes both that each decision has equal weight (local apple is equivalent to local grape even if the latter is 4 times the price and the former has 2 times the reduction in environmental impact by purchasing locally) and yet that the difference between 3 local choices and 2 is not comparable to the difference between 2 local choices and 1. This is a strange combination.

## Use metrics and weight your data

I would suggest determining variables that matches what you really want to study - some function of (differences in) price or environmental impact for instance - and making use of those variables. This will give the decisions made by your participants the appropriate weight and scale appropriately to future tasks. If you have very few items and they are offered to all participants then modelling them separately as suggested by Björn will make the most of your data.

## Alternative approach which may be more informative to your research question - pivot to decision-level rather than participant-level modelling

I guess your research question is about whether people choose to purchase greener versions of products. You're looking at this from a 'person' perspective - would you be better served by analysing from a 'decision' perspective?

I suggest a more informative model here might be price of local item, environmental impact of local item, price of remote item, environmental impact of remote item (other item characteristics as you present to the participant) and participant characteristics predicting whether the choice is made to take a local item or not. That gives you a binary variable for which you could use logistic regression.

The resulting model has much higher generalisability to other items, which you could test through offering other people other shopping tasks.