# Lower Denomination Imputation

Before I start, I will point out that I am very new to imputing data and so any advice would be greatly appreciated. Apologies if there is an obvious answer that I am overlooking.

I have a data set for which I am going to be performing multiple imputation to analyse (using the MICE package in R). The main values that are missing are height values. I have enough other variables to work with. They were recorded in either metric or imperial (to be converted to metric). However, for some of the imperial values, the lower denomination is missing. i.e. feet present but no inches.

My first idea was to assume that all of these missing values were 0. However, when I do this and plot a histogram of the heights, there is a huge spike at 5'0". Clearly this original assumption was wrong as this would massively skew my results. Obviously having someone be somewhere between 5'0" and 5'11" is quite a large range which contains most of the population (all female).

It appears that I will have to impute only the lower denomination values for these entries (i.e. inches). Does anybody know of any documentation/articles available which indicate the best way to go about this? Would I impute 4', 5' & 6' separately as if they were different data sets? I feel that this would cause more bias. I'm unsure whether the multiple imputation methods I've read about so far would be able to account for this specific kind of imputation and so I'm hoping for a bit of advice on how to proceed.

• For readers in this thoroughly international forum puzzled by these quaint units: 1 inch, written sometimes 1", is 25.4 mm; 1 foot, written sometimes 1', is 12 inches. From other data you should able to work out the means of those 4 foot and up, 5 foot and up, 6 foot and up. In other situations using a midpoint would be a much better stab than using zero (which is indeed absurd), but you should be able to do better than 4'6" (I guess much too low) or 6'6" (too high). – Nick Cox Jan 19 '16 at 13:38
• Just to point out that in statistical circles "skew" is not synonymous with "bias". – Nick Cox Jan 19 '16 at 13:46
• Hi Nick, thanks for your input in regards to the units. I had assumed, rather naively, that people knew of feet and inches even if they didn't use them. In your suggestion, do you mean to use the same value for everyone with NAs in the 5' bracket? I don't think this would be the right way to go as it would essentially be performing an overall mean imputation on each bracket which doesn't tend to give good results (plus, as far as I'm aware, wouldn't work under a multiple imputation model). – Michael Barrowman Jan 19 '16 at 15:18
• You seemed to be suggesting zero inches as the replacement input for missing; I am suggesting only that an interval midpoint or an empirical mean would work better. I am happy to agree that you could do better still if you could use other variables within multiple imputation. Although you mention that context clearly, your question didn't seem to be about it. – Nick Cox Jan 19 '16 at 15:22
• I don't know specific literature, which means no more than it says. My impression in general is that data management of this kind is often poorly documented. No one wants to write about it: it can be embarrassing to yourself or data providers if datasets are publicly shown to be lousy, as they usually are; the details often seem too mundane or too localised to deserve writing up; and most of the rewards are elsewhere, for interesting results or a new method or model for analysis. – Nick Cox Jan 19 '16 at 15:42

The problem breaks down to trying to find b such that x % 12 = b where x is height in some continuous units- say cm or inches. It would be simpler if we calculate x for rows where feet and inches are both present.
Then I suggest using knnImpute method from the caret package to preprocess and impute your x values based on clean neighbor columns. After this you could convert back to the form- a = floor(x / 12) and b = x % 12