I do not use SAS, but generally you could use a LOESS/LOWESS approach to visualize how "linear" it is. But I would be concerned with heterogeneity for some variables.

Dear David E Jones As a senior data analyst, when dealing with Likert-scale variables (1-6) treated as continuous predictors in regression-based analyses, such as Hayes conditional process models, it’s common to assess linearity through residual plots and scatterplots.

In your SAS residual plots, focus on patterns—ideally, residuals should appear randomly scattered around zero without clear curvature or systematic trends. If residual plots indicate curvature or distinct patterns, this suggests nonlinearity, and you might consider modeling your Likert predictors categorically or testing polynomial (quadratic) terms.

Although Likert data technically represent ordinal scales, it's widely acceptable to treat them as continuous if distributions appear approximately symmetrical and intervals roughly equal. Ultimately, ensuring your residuals lack obvious patterns strengthens the linearity assumption and enhances your model’s validity.

The regression model treats the Likert variable as continuous, but the vertical stacking of residuals suggests that responses are grouped at specific levels (1 to 6). This does not necessarily indicate a violation of linearity, but it does suggest the need for further checks. The dots within each vertical stack looks randomly distributed above and below the zero line, which means linearity is still reasonable.