The mice package implements a method to deal with missing data. The package creates multiple imputations (replacement values) for multivariate missing data. The method is based on Fully Conditional Specification, where each incomplete variable is imputed by a separate model. The MICE algorithm can impute mixes of continuous, binary, unordered categorical and ordered categorical data. In addition, MICE can impute continuous two-level data, and maintain consistency between imputations by means of passive imputation. Many diagnostic plots are implemented to inspect the quality of the imputations.

Generates Multivariate Imputations by Chained Equations (MICE)

## Usage

```
mice(
data,
m = 5,
method = NULL,
predictorMatrix,
ignore = NULL,
where = NULL,
blocks,
visitSequence = NULL,
formulas,
blots = NULL,
post = NULL,
defaultMethod = c("pmm", "logreg", "polyreg", "polr"),
maxit = 5,
printFlag = TRUE,
seed = NA,
data.init = NULL,
...
)
```

## Arguments

- data
A data frame or a matrix containing the incomplete data. Missing values are coded as

`NA`

.- m
Number of multiple imputations. The default is

`m=5`

.- method
Can be either a single string, or a vector of strings with length

`length(blocks)`

, specifying the imputation method to be used for each column in data. If specified as a single string, the same method will be used for all blocks. The default imputation method (when no argument is specified) depends on the measurement level of the target column, as regulated by the`defaultMethod`

argument. Columns that need not be imputed have the empty method`""`

. See details.- predictorMatrix
A numeric matrix of

`length(blocks)`

rows and`ncol(data)`

columns, containing 0/1 data specifying the set of predictors to be used for each target column. Each row corresponds to a variable block, i.e., a set of variables to be imputed. A value of`1`

means that the column variable is used as a predictor for the target block (in the rows). By default, the`predictorMatrix`

is a square matrix of`ncol(data)`

rows and columns with all 1's, except for the diagonal. Note: For two-level imputation models (which have`"2l"`

in their names) other codes (e.g,`2`

or`-2`

) are also allowed.- ignore
A logical vector of

`nrow(data)`

elements indicating which rows are ignored when creating the imputation model. The default`NULL`

includes all rows that have an observed value of the variable to imputed. Rows with`ignore`

set to`TRUE`

do not influence the parameters of the imputation model, but are still imputed. We may use the`ignore`

argument to split`data`

into a training set (on which the imputation model is built) and a test set (that does not influence the imputation model estimates). Note: Multivariate imputation methods, like`mice.impute.jomoImpute()`

or`mice.impute.panImpute()`

, do not honour the`ignore`

argument.- where
A data frame or matrix with logicals of the same dimensions as

`data`

indicating where in the data the imputations should be created. The default,`where = is.na(data)`

, specifies that the missing data should be imputed. The`where`

argument may be used to overimpute observed data, or to skip imputations for selected missing values. Note: Imputation methods that generate imptutations outside of`mice`

, like`mice.impute.panImpute()`

may depend on a complete predictor space. In that case, a custom`where`

matrix can not be specified.- blocks
List of vectors with variable names per block. List elements may be named to identify blocks. Variables within a block are imputed by a multivariate imputation method (see

`method`

argument). By default each variable is placed into its own block, which is effectively fully conditional specification (FCS) by univariate models (variable-by-variable imputation). Only variables whose names appear in`blocks`

are imputed. The relevant columns in the`where`

matrix are set to`FALSE`

of variables that are not block members. A variable may appear in multiple blocks. In that case, it is effectively re-imputed each time that it is visited.- visitSequence
A vector of block names of arbitrary length, specifying the sequence of blocks that are imputed during one iteration of the Gibbs sampler. A block is a collection of variables. All variables that are members of the same block are imputed when the block is visited. A variable that is a member of multiple blocks is re-imputed within the same iteration. The default

`visitSequence = "roman"`

visits the blocks (left to right) in the order in which they appear in`blocks`

. One may also use one of the following keywords:`"arabic"`

(right to left),`"monotone"`

(ordered low to high proportion of missing data) and`"revmonotone"`

(reverse of monotone).*Special case*: If you specify both`visitSequence = "monotone"`

and`maxit = 1`

, then the procedure will edit the`predictorMatrix`

to conform to the monotone pattern. Realize that convergence in one iteration is only guaranteed if the missing data pattern is actually monotone. The procedure does not check this.- formulas
A named list of formula's, or expressions that can be converted into formula's by

`as.formula`

. List elements correspond to blocks. The block to which the list element applies is identified by its name, so list names must correspond to block names. The`formulas`

argument is an alternative to the`predictorMatrix`

argument that allows for more flexibility in specifying imputation models, e.g., for specifying interaction terms.- blots
A named

`list`

of`alist`

's that can be used to pass down arguments to lower level imputation function. The entries of element`blots[[blockname]]`

are passed down to the function called for block`blockname`

.- post
A vector of strings with length

`ncol(data)`

specifying expressions as strings. Each string is parsed and executed within the`sampler()`

function to post-process imputed values during the iterations. The default is a vector of empty strings, indicating no post-processing. Multivariate (block) imputation methods ignore the`post`

parameter.- defaultMethod
A vector of length 4 containing the default imputation methods for 1) numeric data, 2) factor data with 2 levels, 3) factor data with > 2 unordered levels, and 4) factor data with > 2 ordered levels. By default, the method uses

`pmm`

, predictive mean matching (numeric data)`logreg`

, logistic regression imputation (binary data, factor with 2 levels)`polyreg`

, polytomous regression imputation for unordered categorical data (factor > 2 levels)`polr`

, proportional odds model for (ordered, > 2 levels).- maxit
A scalar giving the number of iterations. The default is 5.

- printFlag
If

`TRUE`

,`mice`

will print history on console. Use`print=FALSE`

for silent computation.- seed
An integer that is used as argument by the

`set.seed()`

for offsetting the random number generator. Default is to leave the random number generator alone.- data.init
A data frame of the same size and type as

`data`

, without missing data, used to initialize imputations before the start of the iterative process. The default`NULL`

implies that starting imputation are created by a simple random draw from the data. Note that specification of`data.init`

will start all`m`

Gibbs sampling streams from the same imputation.- ...
Named arguments that are passed down to the univariate imputation functions.

## Details

The mice package contains functions to

Inspect the missing data pattern

Impute the missing data

*m*times, resulting in*m*completed data setsDiagnose the quality of the imputed values

Analyze each completed data set

Pool the results of the repeated analyses

Store and export the imputed data in various formats

Generate simulated incomplete data

Incorporate custom imputation methods

Generates multiple imputations for incomplete multivariate data by Gibbs sampling. Missing data can occur anywhere in the data. The algorithm imputes an incomplete column (the target column) by generating 'plausible' synthetic values given other columns in the data. Each incomplete column must act as a target column, and has its own specific set of predictors. The default set of predictors for a given target consists of all other columns in the data. For predictors that are incomplete themselves, the most recently generated imputations are used to complete the predictors prior to imputation of the target column.

A separate univariate imputation model can be specified for each column. The default imputation method depends on the measurement level of the target column. In addition to these, several other methods are provided. You can also write their own imputation functions, and call these from within the algorithm.

The data may contain categorical variables that are used in a regressions on other variables. The algorithm creates dummy variables for the categories of these variables, and imputes these from the corresponding categorical variable.

Built-in univariate imputation methods are:

`pmm` | any | Predictive mean matching |

`midastouch` | any | Weighted predictive mean matching |

`sample` | any | Random sample from observed values |

`cart` | any | Classification and regression trees |

`rf` | any | Random forest imputations |

`mean` | numeric | Unconditional mean imputation |

`norm` | numeric | Bayesian linear regression |

`norm.nob` | numeric | Linear regression ignoring model error |

`norm.boot` | numeric | Linear regression using bootstrap |

`norm.predict` | numeric | Linear regression, predicted values |

`lasso.norm` | numeric | Lasso linear regression |

`lasso.select.norm` | numeric | Lasso select + linear regression |

`quadratic` | numeric | Imputation of quadratic terms |

`ri` | numeric | Random indicator for nonignorable data |

`logreg` | binary | Logistic regression |

`logreg.boot` | binary | Logistic regression with bootstrap |

`lasso.logreg` | binary | Lasso logistic regression |

`lasso.select.logreg` | binary | Lasso select + logistic regression |

`polr` | ordered | Proportional odds model |

`polyreg` | unordered | Polytomous logistic regression |

`lda` | unordered | Linear discriminant analysis |

`2l.norm` | numeric | Level-1 normal heteroscedastic |

`2l.lmer` | numeric | Level-1 normal homoscedastic, lmer |

`2l.pan` | numeric | Level-1 normal homoscedastic, pan |

`2l.bin` | binary | Level-1 logistic, glmer |

`2lonly.mean` | numeric | Level-2 class mean |

`2lonly.norm` | numeric | Level-2 class normal |

`2lonly.pmm` | any | Level-2 class predictive mean matching |

These corresponding functions are coded in the `mice`

library under
names `mice.impute.method`

, where `method`

is a string with the
name of the univariate imputation method name, for example `norm`

. The
`method`

argument specifies the methods to be used. For the `j`

'th
column, `mice()`

calls the first occurrence of
`paste('mice.impute.', method[j], sep = '')`

in the search path. The
mechanism allows uses to write customized imputation function,
`mice.impute.myfunc`

. To call it for all columns specify
`method='myfunc'`

. To call it only for, say, column 2 specify
`method=c('norm','myfunc','logreg',...{})`

.

*Skipping imputation:* The user may skip imputation of a column by
setting its entry to the empty method: `""`

. For complete columns without
missing data `mice`

will automatically set the empty method. Setting t
he empty method does not produce imputations for the column, so any missing
cells remain `NA`

. If column A contains `NA`

's and is used as
predictor in the imputation model for column B, then `mice`

produces no
imputations for the rows in B where A is missing. The imputed data
for B may thus contain `NA`

's. The remedy is to remove column A from
the imputation model for the other columns in the data. This can be done
by setting the entire column for variable A in the `predictorMatrix`

equal to zero.

*Passive imputation:* `mice()`

supports a special built-in method,
called passive imputation. This method can be used to ensure that a data
transform always depends on the most recently generated imputations. In some
cases, an imputation model may need transformed data in addition to the
original data (e.g. log, quadratic, recodes, interaction, sum scores, and so
on).

Passive imputation maintains consistency among different transformations of
the same data. Passive imputation is invoked if `~`

is specified as the
first character of the string that specifies the univariate method.
`mice()`

interprets the entire string, including the `~`

character,
as the formula argument in a call to ```
model.frame(formula,
data[!r[,j],])
```

. This provides a simple mechanism for specifying deterministic
dependencies among the columns. For example, suppose that the missing entries
in variables `data$height`

and `data$weight`

are imputed. The body
mass index (BMI) can be calculated within `mice`

by specifying the
string `'~I(weight/height^2)'`

as the univariate imputation method for
the target column `data$bmi`

. Note that the `~`

mechanism works
only on those entries which have missing values in the target column. You
should make sure that the combined observed and imputed parts of the target
column make sense. An easy way to create consistency is by coding all entries
in the target as `NA`

, but for large data sets, this could be
inefficient. Note that you may also need to adapt the default
`predictorMatrix`

to evade linear dependencies among the predictors that
could cause errors like `Error in solve.default()`

or ```
Error:
system is exactly singular
```

. Though not strictly needed, it is often useful
to specify `visitSequence`

such that the column that is imputed by the
`~`

mechanism is visited each time after one of its predictors was
visited. In that way, deterministic relation between columns will always be
synchronized.

#'A new argument `ls.meth`

can be parsed to the lower level
`.norm.draw`

to specify the method for generating the least squares
estimates and any subsequently derived estimates. Argument `ls.meth`

takes one of three inputs: `"qr"`

for QR-decomposition, `"svd"`

for
singular value decomposition and `"ridge"`

for ridge regression.
`ls.meth`

defaults to `ls.meth = "qr"`

.

*Auxiliary predictors in formulas specification: *
For a given block, the `formulas`

specification takes precedence over
the corresponding row in the `predictMatrix`

argument. This
precedence is, however, restricted to the subset of variables
specified in the terms of the block formula. Any
variables not specified by `formulas`

are imputed
according to the `predictMatrix`

specification. Variables with
non-zero `type`

values in the `predictMatrix`

will
be added as main effects to the `formulas`

, which will
act as supplementary covariates in the imputation model. It is possible
to turn off this behavior by specifying the
argument `auxiliary = FALSE`

.

## Functions

The main functions are:

`mice()` | Impute the missing data *m* times |

`with()` | Analyze completed data sets |

`pool()` | Combine parameter estimates |

`complete()` | Export imputed data |

`ampute()` | Generate missing data |

## Vignettes

There is a detailed series of six online vignettes that walk you through solving realistic inference problems with mice.

We suggest going through these vignettes in the following order

#'Van Buuren, S. (2018).
Boca Raton, FL.: Chapman & Hall/CRC Press.
The book
*Flexible Imputation of Missing Data. Second Edition.*
contains a lot of example code.

## Methodology

The mice software was published in the Journal of Statistical Software (Van Buuren and Groothuis-Oudshoorn, 2011). doi:10.18637/jss.v045.i03
The first application of the method
concerned missing blood pressure data (Van Buuren et. al., 1999).
The term *Fully Conditional Specification* was introduced in 2006 to describe a general class of methods that specify imputations model for multivariate data as a set of conditional distributions (Van Buuren et. al., 2006). Further details on mixes of variables and applications can be found in the book
*Flexible Imputation of Missing Data. Second Edition.*
Chapman & Hall/CRC. Boca Raton, FL.

## Enhanced linear algebra

Updating the BLAS can improve speed of R, sometime considerably. The details depend on the operating system. See the discussion in the "R Installation and Administration" guide for further information.

## References

van Buuren, S., Boshuizen, H.C., Knook, D.L. (1999) Multiple
imputation of missing blood pressure covariates in survival analysis.
*Statistics in Medicine*, **18**, 681--694.

van Buuren, S., Brand, J.P.L., Groothuis-Oudshoorn C.G.M., Rubin, D.B. (2006)
Fully conditional specification in multivariate imputation. *Journal of
Statistical Computation and Simulation*, **76**, 12, 1049--1064.

van Buuren, S., Groothuis-Oudshoorn, K. (2011). `mice`

:
Multivariate Imputation by Chained Equations in `R`

. *Journal of
Statistical Software*, **45**(3), 1--67. doi:10.18637/jss.v045.i03

Van Buuren, S. (2018).
*Flexible Imputation of Missing Data. Second Edition.*
Chapman & Hall/CRC. Boca Raton, FL.

Van Buuren, S., Groothuis-Oudshoorn, K. (2011). `mice`

:
Multivariate Imputation by Chained Equations in `R`

. *Journal of
Statistical Software*, **45**(3), 1-67.
doi:10.18637/jss.v045.i03

Van Buuren, S. (2018).
*Flexible Imputation of Missing Data. Second Edition.*
Chapman & Hall/CRC. Boca Raton, FL.

Van Buuren, S., Brand, J.P.L., Groothuis-Oudshoorn C.G.M., Rubin, D.B. (2006)
Fully conditional specification in multivariate imputation. *Journal of
Statistical Computation and Simulation*, **76**, 12, 1049--1064.

Van Buuren, S. (2007) Multiple imputation of discrete and continuous data by
fully conditional specification. *Statistical Methods in Medical
Research*, **16**, 3, 219--242.

Van Buuren, S., Boshuizen, H.C., Knook, D.L. (1999) Multiple imputation of
missing blood pressure covariates in survival analysis. *Statistics in
Medicine*, **18**, 681--694.

Brand, J.P.L. (1999) *Development, implementation and evaluation of
multiple imputation strategies for the statistical analysis of incomplete
data sets.* Dissertation. Rotterdam: Erasmus University.

## Author

Stef van Buuren stef.vanbuuren@tno.nl, Karin Groothuis-Oudshoorn c.g.m.oudshoorn@utwente.nl, 2000-2010, with contributions of Alexander Robitzsch, Gerko Vink, Shahab Jolani, Roel de Jong, Jason Turner, Lisa Doove, John Fox, Frank E. Harrell, and Peter Malewski.

## Examples

```
# do default multiple imputation on a numeric matrix
imp <- mice(nhanes)
#>
#> iter imp variable
#> 1 1 bmi hyp chl
#> 1 2 bmi hyp chl
#> 1 3 bmi hyp chl
#> 1 4 bmi hyp chl
#> 1 5 bmi hyp chl
#> 2 1 bmi hyp chl
#> 2 2 bmi hyp chl
#> 2 3 bmi hyp chl
#> 2 4 bmi hyp chl
#> 2 5 bmi hyp chl
#> 3 1 bmi hyp chl
#> 3 2 bmi hyp chl
#> 3 3 bmi hyp chl
#> 3 4 bmi hyp chl
#> 3 5 bmi hyp chl
#> 4 1 bmi hyp chl
#> 4 2 bmi hyp chl
#> 4 3 bmi hyp chl
#> 4 4 bmi hyp chl
#> 4 5 bmi hyp chl
#> 5 1 bmi hyp chl
#> 5 2 bmi hyp chl
#> 5 3 bmi hyp chl
#> 5 4 bmi hyp chl
#> 5 5 bmi hyp chl
imp
#> Class: mids
#> Number of multiple imputations: 5
#> Imputation methods:
#> age bmi hyp chl
#> "" "pmm" "pmm" "pmm"
#> PredictorMatrix:
#> age bmi hyp chl
#> age 0 1 1 1
#> bmi 1 0 1 1
#> hyp 1 1 0 1
#> chl 1 1 1 0
# list the actual imputations for BMI
imp$imp$bmi
#> 1 2 3 4 5
#> 1 29.6 25.5 22.0 30.1 25.5
#> 3 28.7 27.2 29.6 26.3 28.7
#> 4 22.5 21.7 22.5 20.4 25.5
#> 6 25.5 24.9 25.5 25.5 22.5
#> 10 20.4 22.5 28.7 21.7 30.1
#> 11 27.5 27.2 35.3 30.1 27.4
#> 12 27.5 27.2 27.5 22.5 27.4
#> 16 29.6 33.2 28.7 22.0 28.7
#> 21 20.4 22.7 30.1 30.1 33.2
# first completed data matrix
complete(imp)
#> age bmi hyp chl
#> 1 1 29.6 1 238
#> 2 2 22.7 1 187
#> 3 1 28.7 1 187
#> 4 3 22.5 2 186
#> 5 1 20.4 1 113
#> 6 3 25.5 2 184
#> 7 1 22.5 1 118
#> 8 1 30.1 1 187
#> 9 2 22.0 1 238
#> 10 2 20.4 1 187
#> 11 1 27.5 1 187
#> 12 2 27.5 1 218
#> 13 3 21.7 1 206
#> 14 2 28.7 2 204
#> 15 1 29.6 1 238
#> 16 1 29.6 1 238
#> 17 3 27.2 2 284
#> 18 2 26.3 2 199
#> 19 1 35.3 1 218
#> 20 3 25.5 2 206
#> 21 1 20.4 1 187
#> 22 1 33.2 1 229
#> 23 1 27.5 1 131
#> 24 3 24.9 1 186
#> 25 2 27.4 1 186
# imputation on mixed data with a different method per column
mice(nhanes2, meth = c("sample", "pmm", "logreg", "norm"))
#>
#> iter imp variable
#> 1 1 bmi hyp chl
#> 1 2 bmi hyp chl
#> 1 3 bmi hyp chl
#> 1 4 bmi hyp chl
#> 1 5 bmi hyp chl
#> 2 1 bmi hyp chl
#> 2 2 bmi hyp chl
#> 2 3 bmi hyp chl
#> 2 4 bmi hyp chl
#> 2 5 bmi hyp chl
#> 3 1 bmi hyp chl
#> 3 2 bmi hyp chl
#> 3 3 bmi hyp chl
#> 3 4 bmi hyp chl
#> 3 5 bmi hyp chl
#> 4 1 bmi hyp chl
#> 4 2 bmi hyp chl
#> 4 3 bmi hyp chl
#> 4 4 bmi hyp chl
#> 4 5 bmi hyp chl
#> 5 1 bmi hyp chl
#> 5 2 bmi hyp chl
#> 5 3 bmi hyp chl
#> 5 4 bmi hyp chl
#> 5 5 bmi hyp chl
#> Class: mids
#> Number of multiple imputations: 5
#> Imputation methods:
#> age bmi hyp chl
#> "" "pmm" "logreg" "norm"
#> PredictorMatrix:
#> age bmi hyp chl
#> age 0 1 1 1
#> bmi 1 0 1 1
#> hyp 1 1 0 1
#> chl 1 1 1 0
if (FALSE) {
# example where we fit the imputation model on the train data
# and apply the model to impute the test data
set.seed(123)
ignore <- sample(c(TRUE, FALSE), size = 25, replace = TRUE, prob = c(0.3, 0.7))
# scenario 1: train and test in the same dataset
imp <- mice(nhanes2, m = 2, ignore = ignore, print = FALSE, seed = 22112)
imp.test1 <- filter(imp, ignore)
imp.test1$data
complete(imp.test1, 1)
complete(imp.test1, 2)
# scenario 2: train and test in separate datasets
traindata <- nhanes2[!ignore, ]
testdata <- nhanes2[ignore, ]
imp.train <- mice(traindata, m = 2, print = FALSE, seed = 22112)
imp.test2 <- mice.mids(imp.train, newdata = testdata)
complete(imp.test2, 1)
complete(imp.test2, 2)
}
```