Likert style? Dependent variable and linear regression model

Question

I’m a beginner to using R and have a project due in a couple of weeks I’ve been working on, basically choose any dataset and perform analysis on it. However you can only use a linear regression model not any others.

I chose my topic on leadership approval and how much they are liked. So my dependent variable is on a scale of 0-100 with 0 being severely disliked and 10 like a lot.. So my plan was to have a few independent variables such as education, age, gender, then several national policies which are all in the dataset and see how much impact they all have

My concern is that as I’m about to start to get to the regression stage (done all the visualisations and statistics) that my whole things is invalid because it is not a continuous variable and only continuous variables can utilise the linear model. However the example I saw in the textbook was similar to mine except it was between 0-10. I have seen many people on the internet argue it is fine as there is 11 options to choose from even if it is a likert scale

Any thoughts would be appreciated as I am freaking out and can’t bear the thought of starting again with it due in two weeks

EDIT: if it makes any difference the mean of the dependent variable is 4.36, median 4, mode 0 and standard deviation 2.89

Answer 1

With discrete/categorical variables like Gender (M, F), Coin tosses (H, T), Die rolls (1,2,3,4,5,6), etc. you need to use Logistic Regression instead of Linear Regression. There are many examples online which explain Logistic Regression. The core idea in Logistic Regression is to use "discrete" distributions like Bernoulli, Binomial, Multinomial, etc. over observed variables, unlike Linear Regression which uses continuous distributions (Normal, Student-T).

Likert scale values are indeed discrete - but they are very different from Gender/Coins in the sense that there is an ordering among values.

a. For Coin toss, we can't claim H > T or H < T.

b. For Likert scale, 7 is better than 3. Also, the difference between 6 and 7 is not the same as the difference between 8 and 9. Most humans don't know how different is 55% from 60%, for example. 60% is better, sure - but is it the same as 60% -> 65% subjectively? Maybe not.

To model these peculiar Likert scale data, we use something called Ordinal Regression. This lecture from Richard McElreath's Statistical Rethinking is a great primer on ordinal regression - here.

Some examples on ordinal regression. Hope this helps :)

Answer 2

@stbv has already pointed out the obvious idea to use ordinal regression (that's assuming a regression model of some kind make sense in the first place, see further down).

Given that this is for some reason not permitted in your case (a restriction nobody ever faces for any real problems), your question seems to be how reasonable a linear model might work on ordinal data (i.e. can we argue that while it's technically clearly not quite right, it's still a useful model that is not going to be too badly wrong - after all, some people actually even use linear regression for binary data). For a start, you would be assuming that a difference of x (e.g. 1) always means the same thing along the whole scale (e.g. 0 vs. 1 is as different as 5 vs. 6 and that's a different as 9 vs. 10), otherwise it just make no sense to use a linear model.

If that's the case, then there's various things that would need thinking about. E.g. if you were mostly in the middle part of a very wide ordinal scale (e.g. 0 to 100 or even a wide range) with very few values coming near the edge and lots of different individual values occurring, then one might feel that the values are almost continuous anyway without any limitations of the range (to e.g. [0, 100]) really coming into play. From what you write that might not be the case, e.g. the mode being 0 (=at the edge of the range) sounds like it could be problematic (e.g. errors cannot really be symmetric and equally variable across the range of the data). There's of course also various things one could look at like QQ-plots for residuals to see whether perhaps it all doesn't look that bad.

Finally, to get back to the start. Make sure that it's clear why you are doing a regression model and whether it will allow the interpretations you want to do. You said very little about what your actual questions are. Is it some causal question like "Does going to a certain university result in better leadership style?" or "Does sending a manager to a leadership course improve their leadership scores?", or a prediction question "If we have a manager with some history, can we predict their leadership score for the next year?", or something else? In this context it will also then matter how the data were generated/collected (e.g. a randomized experiment would make causal conclusions about the randomly assigned interventions easier, non-experimental data makes it harder, selection effects might affect the data etc.).