# Multiple Imputation by Chained Equations (MICE) Explained

I have seen Multiple Imputation by Chained Equations (MICE) used as a missing data handling method. Is anyone able to provide a simple explanation of how MICE works?

MICE is a multiple imputation method used to replace missing data values in a data set under certain assumptions about the data missingness mechanism (e.g., the data are missing at random, the data are missing completely at random).

If you start out with a data set which includes missing values in one or more of its variables, you can create multiple copies of this data set - for example, you can create 5 copies of the original data set - and replace the missing data values in each copy using the MICE procedure. You can then:

• Analyze the 5 complete data set copies using your intended statistical analysis;
• Combine (or pool) the results of these complete data analyses;
• Report the combined result.

The rules for combining (or pooling) results are specific to the results being combined and were initially developed by Rubin.

Figure 1 in the article Multiple Imputation by Chained Equations in Praxis: Guidelines and Review by Jesper N. Wulff and Linda Ejlskov is visually summarizes the process described above: https://vbn.aau.dk/en/publications/multiple-imputation-by-chained-equations-in-praxis-guidelines-and.

How does MICE replace the missing data values in each copy of the original data set?

The article Multiple Imputation by Chained Equations: What is it and how does it work? by Azur et al. explains what happens underneath the MICE hood with a nice example: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3074241/

In the example, the author articles start out with a simple data set which includes only 3 variables: age, income, and gender. All 3 have at least some missing values.

To apply MICE, create 5 copies (say) of this simple data set and cycle multiple times through the steps below for each copy:

Step 1: Replace (or impute) the missing values in each variable with temporary "place holder" values derived solely from the non-missing values available for that variable. For example, replace the missing age value with the mean age value observed in the data, replace the missing income values with the mean income value observed in the data, etc.

Step 2 Set back to missing the “place holder” imputations for the age variable only. This way, the current data copy contains missing values for age, but not for income and gender.

Step 3: Regress age on income and gender via a linear regression model (though it is possible to also regress age on only one of these variables); to be able to fit the model to the current data copy, drop all the records where age is missing during the model fitting process. In this model, age is the dependent variable and income and gender are the independent variables.

Step 4 Use the fitted regression model in the previous step to predict the missing age values. (When age will be subsequently used as an independent variable in the regression models for other variables, both the observed values of age and these predicted values will be used.) The article doesn't make it clear that a random component should be added to these predictions.

Step 5: Repeat Steps 2–4 separately for each variable that has missing data, namely income and gender.

Cycling through Steps 1–5 once for each of the variables age, income and gender constitutes one cycle. At the end of this cycle, all of the missing values in age, income an gender will have been replaced with predictions from regression models that reflect the relationships observed in the data between these variables.

As stated earlier, MICE requires that we cycle through Steps 1–5 for a number of cycles, with the imputations of the missing values of age, income and gender being updated at each subsequent cycle.

We can specify in advance the number of cycles to be performed (e.g., 10 cycles) – once we reach the final cycle, we retain the imputed values corresponding to that final cycle, obtaining an imputed data set (i.e., a data set where all missing values in age, gender and income were replaced with imputed data values obtained via an iterative procedure).

To sum up, MICE imputes missing values in the variables of a data set by using a divide and conquer approach – in other words, by focusing on one variable at a time. Once the focus is placed on one variable, MICE uses all the other variables in the data set (or a sensibly chosen subset of these variables) to predict missingness in that variable. The prediction is based on a regression model, with the form of the model depending on the nature of the focus variable (e.g., age and income will require linear regression models for prediction of their missing values, but gender will require a logistic regression model).

• Hey Isabella. Thanks for a really wonderful explanation! Really well written. Just to be clear, the only source of variation is the noise component added to the predictions in step 4? Commented Aug 13, 2021 at 16:26
• Thanks @isabella-ghement, your comment in Step 4 about the Azur et al article not making clear that a random component needs to be added was the required lightbulb appearing above my head. Commented Mar 10, 2023 at 16:26
• This is a great answer, +1 Commented Mar 15, 2023 at 16:38