Title | Accounting for clustering with mi impute | |

Authors | Wesley Eddings and Yulia Marchenko, StataCorp |

Note 1: This frequently asked question (FAQ)
assumes familiarity with multiple imputation. Please see
the documentation entries [MI] **intro substantive** and
[MI] **intro** if you are unfamiliar with the method. Also, if your data have
already been imputed, see the documentation entry [MI] **mi import** on
how to import your data to
**mi** and see
[MI] **mi estimate** on how to analyze your multiply imputed data.

Note 2: Because the **mi impute**
command is based on random draws, results may differ on previous
versions as a consequence of the 64-bit Mersenne Twister pseudorandom numbers,
which was added to Stata in version 14.

As of Stata 11.1, the
**mi estimate**
command can be used to analyze multiply imputed clustered (panel or longitudinal) data by fitting
several clustered-data models, such as
xtreg,
xtlogit, and
**mixed**; see
mi estimation
for the full list.
However, we must also account for clustering when creating
multiply imputed data; this FAQ will show how.

We can create multiply imputed data with mi impute, Stata’s official command for imputing missing values. There is no definitive recommendation in the literature on the best way to impute clustered data, but three strategies have been suggested:

- Include indicator variables for clusters in the imputation model.
- Impute data separately for each cluster.
- Use a multivariate normal model to impute all clusters simultaneously.

We will explain how to carry out each strategy with **mi impute**.
We will assume for now that we have data in long form and
that only one variable has missing values; extensions to more than one
imputed variable will be described later.

If there are not too many clusters, we can account for clustering by
including cluster indicators in our imputation model. The
factor-variable syntax of Stata makes it easy to include the indicators
with **mi impute**: we do not even have to
generate
any new variables.

Our first example dataset, **data1.dta**, has 40 observations within each of 10
clusters; the variable **id** indexes observations within clusters. Ten
percent of the observations have missing values for the observation-level
predictor **x**; no values of the response **y** are missing. We want
to study the association between **y** and the partially observed
predictor **x** while accounting for the association within clusters.

.use http://www.stata.com/support/faqs/data1.describeContains data from http://www.stata.com/support/faqs/data1.dta Observations: 400 Variables: 4 29 Jul 2010 14:56

Variable Storage Display Value |

name type format label Variable label |

cluster float %9.0g id float %9.0g y double %10.0g x double %10.0g |

-> cluster = 1 |

Variable | Obs Mean Std. dev. Min Max | |

y | 40 97.49968 33.62784 23.14309 160.8217 | |

x | 38 30.00173 7.943642 12.95944 42.72091 |

-> cluster = 2 |

Variable | Obs Mean Std. dev. Min Max | |

y | 40 100.2756 31.70555 20.78498 151.5145 | |

x | 39 30.77486 8.020621 5.549631 44.48839 |

-> cluster = 3 |

Variable | Obs Mean Std. dev. Min Max | |

y | 40 147.5954 37.44895 71.18038 217.404 | |

x | 38 31.85539 8.794805 16.45632 49.47706 |

We impute the missing values of **x** with **mi impute regress**, a
Gaussian regression imputation method. We account for clustering by
including in our imputation model the factor variable **i.cluster**. The
response **y** should also be included as a predictor:

.mi set wide.mi register imputed x.mi impute regress x y i.cluster, add(5) noisily rseed(123)Running regress on observed data:

Source | SS df MS | Number of obs = 360 | |

F(10, 349) = 32.74 | |||

Model | 11088.9434 10 1108.89434 | Prob > F = 0.0000 | |

Residual | 11821.207 349 33.8716533 | R-squared = 0.4840 | |

Adj R-squared = 0.4692 | |||

Total | 22910.1504 359 63.8165749 | Root MSE = 5.8199 |

x | Coefficient Std. err. t P>|t| [95% conf. interval] | |

y | .1572187 .0088668 17.73 0.000 .1397797 .1746578 | |

cluster | ||

2 | .5249299 1.326672 0.40 0.693 -2.084348 3.134208 | |

3 | -6.21639 1.410625 -4.41 0.000 -8.990786 -3.441994 | |

4 | -1.153281 1.364677 -0.85 0.399 -3.837306 1.530743 | |

5 | .6848743 1.387169 0.49 0.622 -2.043388 3.413136 | |

6 | -4.79826 1.409348 -3.40 0.001 -7.570143 -2.026376 | |

7 | -1.828347 1.34203 -1.36 0.174 -4.46783 .8111363 | |

8 | -1.427531 1.349231 -1.06 0.291 -4.081178 1.226117 | |

9 | 1.565089 1.353659 1.16 0.248 -1.097267 4.227444 | |

10 | -2.067867 1.384883 -1.49 0.136 -4.791633 .6558993 | |

_cons | 14.49285 1.287011 11.26 0.000 11.96157 17.02412 | |

Observations per m | ||||

Variable | Complete Incomplete Imputed | Total | ||

x | 360 40 40 | 400 | ||

We used the **noisily** option of **mi impute** to display the
intermediate regression output, which shows that nine dummy variables were
properly included for the ten clusters. We now fit our analysis model by
using, for example, **mixed** with the **mi estimate:** prefix:

.mi estimate: mixed y x || cluster:Multiple-imputation estimates Imputations = 5 Mixed-effects ML regression Number of obs = 400 Group variable: cluster Number of groups = 10 Obs per group: min = 40 avg = 40.0 max = 40 Average RVI = 0.0723 Largest FMI = 0.1366 DF adjustment: Large sample DF: min = 238.81 avg = 22,755.03 max = 89,559.86 Model F test: Equal FMI F( 1, 292.0) = 285.80 Prob > F = 0.0000

y | Coefficient Std. err. t P>|t| [95% conf. interval] | |

x | 2.92631 .1730981 16.91 0.000 2.585632 3.266988 | |

_cons | 22.40641 7.02119 3.19 0.001 8.627185 36.18563 | |

Random-effects parameters | Estimate Std. err. [95% conf. interval] | |

cluster: Identity | ||

sd(_cons) | 14.02273 3.406021 8.711221 22.57284 | |

sd(Residual) | 25.46452 .9772013 23.61045 27.46419 | |

The coefficient of **x** is estimated to be about 3 with a standard error of
about 0.2, and the cluster-level intercepts have a mean of about 22 with a
standard deviation of about 14. Had we not included the cluster variable in
our imputation model, we would have obtained a smaller estimate of the
variance component for clusters.

Graham (2009) suggests that cluster indicators can work well for as many as 35 indicator variables. Strategy 1 is best suited for data with few clusters and many observations within each cluster.

By including clusters as indicator variables in our imputation model (strategy 1), we allow the regression function of the
imputed variable to vary by cluster. More generally, we can allow the
distributions of the imputed values to differ among clusters by imputing
each cluster separately (Graham 2009). Since Stata 12, we can use **mi impute**
with the **by()** option.

Our second example dataset, **data2.dta**, like the first, includes a
response variable
with no missing values and a predictor **x** with 10% missing
values. We have 50 observations within each of 20 clusters. We will
impute each cluster separately
and then fit an analysis model with **mixed**.

.use http://www.stata.com/support/faqs/data2.dta.mi set wide.mi register imputed x.mi impute regress x y, add(5) by(cluster, noreport) rseed(123)Univariate imputation Imputations = 5 Linear regression added = 5 Imputed:m=1 throughm=5 updated = 0

Observations per m | |||||

by() | |||||

Variable | Complete Incomplete Imputed | Total | |||

cluster = 1 | |||||

x | 44 6 6 | 50 | |||

cluster = 2 | |||||

x | 47 3 3 | 50 | |||

... | |||||

cluster = 19 | |||||

x | 45 5 5 | 50 | |||

cluster = 20 | |||||

x | 46 4 4 | 50 | |||

Overall | |||||

x | 900 100 100 | 1000 | |||

We now fit **mi estimate: mixed** to our multiply imputed data:

.mi estimate: mixed y x || cluster:Multiple-imputation estimates Imputations = 5 Mixed-effects ML regression Number of obs = 1,000 Group variable: cluster Number of groups = 20 Obs per group: min = 50 avg = 50.0 max = 50 Average RVI = 0.0546 Largest FMI = 0.1551 DF adjustment: Large sample DF: min = 187.43 avg = 156,793.24 max = 422,770.46 Model F test: Equal FMI F( 1, 2471.3) = 1577.92 Prob > F = 0.0000

y | Coefficient Std. err. t P>|t| [95% conf. interval] | |

x | 8.134255 .2047741 39.72 0.000 7.732709 8.535802 | |

_cons | 19.50823 6.343581 3.08 0.002 7.07497 31.9415 | |

Random-effects parameters | Estimate Std. err. [95% conf. interval] | |

cluster: Identity | ||

sd(_cons) | 26.52947 4.312226 19.29167 36.48273 | |

sd(Residual) | 30.48525 .7451697 29.05013 31.99127 | |

The coefficient for **x** is about 8 with a standard error of about 0.2,
and the intraclass correlation is about (27^{2})/(27^{2} +
30^{2}) = 0.45. The intraclass correlation ranges from zero to one,
and larger values mean that the clustering variable is more informative.

Imputing each cluster separately requires a sufficient number of observations in each cluster.

A third way to account for within-cluster correlation is to impute jointly over clusters using a multivariate normal model. Observations within clusters may be viewed as a sample from a multivariate normal distribution with an unrestricted covariance structure. The multivariate normal strategy works well when there are only a few observations in each cluster (Allison 2002). There is a limitation to this strategy: it is best suited to balanced repeated-measures data.

We will illustrate the multivariate normal strategy with a new balanced
dataset. It has 50 clusters but only 5 observations within each
cluster. (Such data might occur, for example, in a repeated-measures study
of subjects’ test scores.) We would once again like to impute missing
values of **x** and then fit a linear mixed-effects model with
**mixed**.

Before we can fit the multivariate normal imputation model, we will need
to reshape
our data to wide form so that each cluster occupies a single row. The
variable **id** indexes observations within clusters.

.use http://www.stata.com/support/faqs/data3.reshape wide x y, i(cluster) j(id)(j = 1 2 3 4 5)

Data Long -> Wide |

Number of observations 250 -> 50 Number of variables 4 -> 11 j variable (5 values) id -> (dropped) xij variables: x -> x1 x2 ... x5 y -> y1 y2 ... y5 |

We can now impute with **mi impute mvn**, and the
multivariate normal regression model will allow interdependencies
within clusters.

.mi set wide.mi register imputed x1 x2 x3 x4 x5.mi impute mvn x1 x2 x3 x4 x5 = y1 y2 y3 y4 y5, add(5) rseed(123)Performing EM optimization: observed log likelihood = -296.02862 at iteration 16 Performing MCMC data augmentation ... Multivariate imputation Imputations = 5 Multivariate normal regression added = 5 Imputed: m=1 through m=5 updated = 0 Prior: uniform Iterations = 500 burn-in = 100 between = 100

Observations per m | |||||

Variable | Complete Incomplete Imputed | Total | |||

x1 | 42 8 8 | 50 | |||

x2 | 43 7 7 | 50 | |||

x3 | 46 4 4 | 50 | |||

x4 | 47 3 3 | 50 | |||

x5 | 47 3 3 | 50 | |||

To use **mi estimate: mixed**, we need to reshape our data back to long
form. With **mi** data, we need to use the **mi reshape** command to
do this:

.mi reshape long x y, i(cluster) j(id)reshapingm=0 data ... (j = 1 2 3 4 5)

Data Wide -> Long |

Number of observations 50 -> 250 Number of variables 11 -> 4 j variable (5 values) -> id xij variables: x1 x2 ... x5 -> x y1 y2 ... y5 -> y |

We are now ready to use **mi estimate: mixed**:

.mi estimate: mixed y x || cluster:Multiple-imputation estimates Imputations = 5 Mixed-effects ML regression Number of obs = 250 Group variable: cluster Number of groups = 50 Obs per group: min = 5 avg = 5.0 max = 5 Average RVI = 0.0942 Largest FMI = 0.2699 DF adjustment: Large sample DF: min = 65.13 avg = 551,115.87 max = 2201656.39 Model F test: Equal FMI F( 1, 65.1) = 45.75 Prob > F = 0.0000

y | Coefficient Std. err. t P>|t| [95% conf. interval] | |

x | .8791031 .1299657 6.76 0.000 .6195535 1.138653 | |

_cons | 2.570404 2.13238 1.21 0.229 -1.625142 6.76595 | |

Random-effects parameters | Estimate Std. err. [95% conf. interval] | |

cluster: Identity | ||

sd(_cons) | 11.37668 1.183462 9.278313 13.94961 | |

sd(Residual) | 5.042148 .2573897 4.561861 5.573 | |

All three strategies can be modified to impute more than one variable. The
indicator-variable and separate-imputation strategies, strategies 1 and 2,
require a multivariate imputation method such as **mi impute monotone**,
**mi impute chained**, or **mi impute mvn** in place of a univariate
method such as **mi impute
regress**. The multivariate normal strategy, strategy 3, can be extended
by adding extra variables to the left-hand side of the equation in **mi impute
mvn**. If we wanted to impute **x** and another variable **z**, the
commands might look like this:

.reshape wide x y z, i(cluster) j(id).mi set wide.mi register imputed x1 x2 x3 x4 x5 z1 z2 z3 z4 z5.mi impute mvn x1 x2 x3 x4 x5 z1 z2 z3 z4 z5 = y1 y2 y3 y4 y5, add(5).mi reshape long x y z, i(cluster) j(id)

All our examples had the same two-level structure—observations within clusters. More-complex multilevel structures are an active research area; one recent paper describing imputation for multilevel models is Goldstein et al. (2009).

- Allison, P. D. 2002.
*Missing Data*. Thousand Oaks, CA: Sage.

- Goldstein, H., J. R. Carpenter, M. G. Kenward, and K. A. Levin. 2009.
- Multilevel models with multivariate mixed response types.
*Statistical Modelling*9: 173–197.

- Graham, J. W. 2009.
- Missing data analysis: Making it work in the real world.
*Annual Review of Psychology*60: 549–576.