Why is collinearity a problem when imputing missing values?

Question

I'm imputing missing values using R's mice package. My data has three numeric variables and a class variable so I am using a multilevel imputation with the 2l.norm method on the variables. However, like in these two questions, I'm getting an error that the leading minor of order 1 is not positively definite. If I understand it correctly, this error is usually due to collinearity of the variables.

Why is collinearity a bad thing? Consider a set of data where one variable is a measurement of length in cm, and the other one is in inch, and both columns have quite some missing values. We'd have collinearity for the complete cases, and they could be used to predict each other in the non-complete cases. Why does this not work this way?

Am I misunderstanding something here?

Answer 1

I don't think it is just the colinearity, but the structure of the colinearity is probably the problem. To expand your example with length in inches and length in cm, suppose that you have a third variable that is width and width is positively correlated with length in inches (for where you have data on both) and is negatively correlated with length in cm (for where you have data on both). Now how do you make the predictions on length in inches when one of the other variables suggests it should be a high value and the other suggests it should be a low value. This can happen with some missing value patterns.