# Assumptions of linearity and homoskedasticity when the DV is a Likert scale - how to interpret the (optimized) residual plots?

I am currently building mixed models for several analyses where each the DV is ordinal, specifically a 7-point Likert scale. Now, I am fully aware of the controversies of using Likert scale as a DV for a linear model, however after some deliberation, at least for the sake of my current project I have decided to proceed with this.

Of course, this entails problems with interpreting the residual plots, needed for the assumptions of homoskedasticity and linearity, since all the residuals are fitted onto descrete lines, as seen on the plot below: I have found an interesting way to optimize the residual plot in a way that allows for easier interpretation of Likert scale-derived residuals, that someone posted online. Specifically, they recommend introducing some disturbance to the DV before plotting. I have done so, using this code (in R):

Y1 <- df\$Y + runif(N,-.5,.5)


where N is the number of observations.

Having plotted the residuals of the model containing the new Y1, I indeed found the new plots to be much more "typical". Nonetheless, I struggle somewhat with understanding if the assumptions of linearity and homoskedasticity are met, perhaps because the plots are still not as good as the typical plot with a continuous DV, or perhaps simply because I've reached the point where I completely lost any confidence regarding my knowledge in stats. Either way, an input of a another person would be greatly appreciated...

Below I attach the residual plots for two of the 10 models I run. The first one looked the best to my ignorant student I out of all of them, while the second one look the worst, therefore I figured it might be more informative to include both and see what you think.

Plot 1 is below. For this DV, the data was normally distributed, there were no outliers, the mean was pretty much in the middle of the range of values. It seems pretty random to me, however the tilt is still present, and I am simply unsure if this suggests lack of linearity and/or homoskedasticity or not... And here's Plot 2. The data here does have many issues. The data was not normally distributed (kurtosis value over 9), there were multiple outliers, and the mean was close to 0 -> floor effect. How would you interpret this in terms of the two assumptions? And, what exactly here would indicate heterostedasticity and lack of linearity? Would the very strong floor effect itself be the thing that shows that the assumption of linearity is not met? And, while there's no fanning or curve that I know would indicate heteroskedasticity, can I really assume homoskedasticity? • I am not doing anything about the outliers in my dataset, since they did not look erroneous upon inspecton and exist only due to the floor effect observed, unfortunately, for a majority of my DVs
• The sample is very large, there's 5056 observations per each DV (repeated measures, 79 subjects x 64 stimuli)

Thank you in advance for the input!