Here is one way to test the missingness-at-random assumption.
Suppose the question on participant's income has some missing entries. Run a logistic regression with income as your response and everything else as predictors. Your response would be 1 if it's missing, 0 otherwise. The p-value of the predictors should give you an idea whether this MAR assumption is any good.
Do the same for all other columns with missing data.
There is a huge literature behind this issue. I'm risking possibly misleading simplification here. See Ch 25 of this,
Gelman, A., & Hill, J. (2006). Data analysis using regression and multilevel/hierarchical models. Cambridge University Press.