# Hot-deck methods and reduce models

My database has missing data values at random. I want to impute these missing values. The data values are all categorical. I would like to work with hot-deck techniques and reduced models. What are the most well-known or representative algorithms of such techniques?

A brief explanation in a few words of these techniques would be appreciated.

• Your question isn't enough clear. What are reduced models for you, how they cross with hot-dock imputation? HD imputation is simply the method of borrowing valid values from other cases of the dataset, conditioned on categorical background variables. – ttnphns Aug 10 '16 at 17:23

Hot-deck imputation is one of the multiple methods for imputing missing data (you can also impute missing data using cold-deck i.e. information from external sources, unconditional or conditional mean, predictions from some model, random draws from assumed distribution, and in many other ways as described e.g. by Gelman and Hill, 2007, or Little, 1992 for accessible reviews).

The idea of hot-deck method is simple: if you have missing value in multivariate dataset you replace such value with non-missing one from another case. The donor case may be chosen at random, or by choosing case that is similar to the one with missing data.

If you are dealing with discrete variables, then for calculating similarity between cases you can use affinity scores defined as

$$\alpha_{ij} = \frac{k - q_i - z_{ij}}{k - q_i}$$

where $k$ is number of variables, $q_i$ is number of missing values in $i$-th case, $z_{ij}$ is number of variables for which the potential donor $j$ and the recipient $i$ have different values.

In case of continuous, or discrete with multiple categories variables, you need to decide how to measure distance between vectors of values since you would not be expecting exact matches. The most simple approach is to match with some tolerance $\varepsilon_p$

$$\alpha_{ij} = \frac{k - q_i - \sum_{p=1}^k \left[ |x_{ip} - x_{jp}| > \varepsilon_p \right] }{k - q_i}$$

Among other choices are multiple popular distance metrics (this is problem-specific!) as mentioned briefly by Cranmer and Gill (2013). Moreover, to deal with missing cases when calculating distance you can first use mean imputation to replace the missing cases and then calculate full-data distances.

Based on distance metric, donor can be chosen by choosing the single one within smallest distance, or at random from group of best donors.

In many cases you wouldn't have single best match, or it wouldn't be reasonable to stick to the single best match, so wiser idea is to use multiple hot-deck imputation, i.e. replicate your dataset with several copies and in each of them randomly assign values from different "best" donors. In the end you would calculate your statistic of interest on all of the datasets and take arithmetic mean of those estimates as your final estimate.

Another question that you can ask yourself is if you are going to re-use donors if they match multiple missing-data cases. As described by Joenssen and Bankhofer (2012), p. 75):

Under some situations, donor limitation leads to better parameter estimations. Splitting the data into a low amount of imputation classes leads to better estimation of variance and quartile distance for quantitative and ordinal variables, respectively. For low amounts of objects per imputation class the variance of quantitative variables is estimated better with a donor limitation, while binary variables with many objects per imputation class also profit from a donor limit. This is also the case for data matrixes with high amounts of missingness. Estimation of location, such as mean and median are not influenced by limiting donor usage.

You can check also Reilly (1993) for theoretical discussion of multiple hot-deck imputation.

Cranmer, S. J., & Gill, J. (2013). We have to be discrete about this: A non-parametric imputation technique for missing categorical data. British Journal of Political Science, 43(02), 425-449.

Little, R. J. (1992). Regression with missing X's: a review. Journal of the American Statistical Association, 87(420), 1227-1237.

Gelman, A. and Hill, J. (2006). Data analysis using regression and multilevel/hierarchical models. Cambridge University Press.

Joenssen, D. W., & Bankhofer, U. (2012, July). Hot deck methods for imputing missing data. In International Workshop on Machine Learning and Data Mining in Pattern Recognition (pp. 63-75). Springer Berlin Heidelberg.

Reilly, M. (1993). Data analysis using hot deck multiple imputation. The Statistician, 307-313.