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The pool() function combines the estimates from m repeated complete data analyses. The typical sequence of steps to perform a multiple imputation analysis is:

  1. Impute the missing data by the mice() function, resulting in a multiple imputed data set (class mids);

  2. Fit the model of interest (scientific model) on each imputed data set by the with() function, resulting an object of class mira;

  3. Pool the estimates from each model into a single set of estimates and standard errors, resulting in an object of class mipo;

  4. Optionally, compare pooled estimates from different scientific models by the D1() or D3() functions.

A common error is to reverse steps 2 and 3, i.e., to pool the multiply-imputed data instead of the estimates. Doing so may severely bias the estimates of scientific interest and yield incorrect statistical intervals and p-values. The pool() function will detect this case.


pool(object, dfcom = NULL, rule = NULL, custom.t = NULL)

pool.syn(object, dfcom = NULL, rule = "reiter2003")



An object of class mira (produced by with.mids() or as.mira()), or a list with model fits.


A positive number representing the degrees of freedom in the complete-data analysis. Normally, this would be the number of independent observation minus the number of fitted parameters. The default (dfcom = NULL) extract this information in the following order: 1) the component residual.df returned by glance() if a glance() function is found, 2) the result of df.residual( applied to the first fitted model, and 3) as 999999. In the last case, the warning "Large sample assumed" is printed. If the degrees of freedom is incorrect, specify the appropriate value manually.


A string indicating the pooling rule. Currently supported are "rubin1987" (default, for missing data) and "reiter2003" (for synthetic data created from a complete data set).


A custom character string to be parsed as a calculation rule for the total variance t. The custom rule can use the other calculated pooling statistics where the dimensions must come from .data$. The default t calculation would have the form ".data$ubar + (1 + 1 / .data$m) * .data$b". See examples for an example.


An object of class mipo, which stands for 'multiple imputation pooled outcome'. For rule "reiter2003" values for lambda and fmi are set to `NA`, as these statistics do not apply for data synthesised from fully observed data.


The pool() function averages the estimates of the complete data model, computes the total variance over the repeated analyses by Rubin's rules (Rubin, 1987, p. 76), and computes the following diagnostic statistics per estimate:

  1. Relative increase in variance due to nonresponse r;

  2. Residual degrees of freedom for hypothesis testing df;

  3. Proportion of total variance due to missingness lambda;

  4. Fraction of missing information fmi.

The degrees of freedom calculation for the pooled estimates uses the Barnard-Rubin adjustment for small samples (Barnard and Rubin, 1999).

The pool.syn() function combines estimates by Reiter's partially synthetic data pooling rules (Reiter, 2003). This combination rule assumes that the data that is synthesised is completely observed. Pooling differs from Rubin's method in the calculation of the total variance and the degrees of freedom.

Pooling requires the following input from each fitted model:

  1. the estimates of the model;

  2. the standard error of each estimate;

  3. the residual degrees of freedom of the model.

The pool() and pool.syn() functions rely on the broom::tidy and broom::glance for extracting these parameters.

Since mice 3.0+, the broom package takes care of filtering out the relevant parts of the complete-data analysis. It may happen that you'll see the messages like Error: No tidy method for objects of class ... or Error: No glance method for objects of class .... The message means that your complete-data method used in with(imp, ...) has no tidy or glance method defined in the broom package.

The broom.mixed package contains tidy and glance methods for mixed models. If you are using a mixed model, first run library(broom.mixed) before calling pool().

If no tidy or glance methods are defined for your analysis tabulate the m parameter estimates and their variance estimates (the square of the standard errors) from the m fitted models stored in fit$analyses. For each parameter, run pool.scalar to obtain the pooled parameters estimate, its variance, the degrees of freedom, the relative increase in variance and the fraction of missing information.

An alternative is to write your own glance() and tidy() methods and add these to broom according to the specifications given in In versions prior to mice 3.0 pooling required that coef() and vcov() methods were available for fitted objects. This feature is no longer supported. The reason is that vcov() methods are inconsistent across packages, leading to buggy behaviour of the pool() function.

Since mice 3.13.2 function pool() uses the robust the standard error estimate for pooling when it can extract from the tidy() object.


Barnard, J. and Rubin, D.B. (1999). Small sample degrees of freedom with multiple imputation. Biometrika, 86, 948-955.

Rubin, D.B. (1987). Multiple Imputation for Nonresponse in Surveys. New York: John Wiley and Sons.

Reiter, J.P. (2003). Inference for Partially Synthetic, Public Use Microdata Sets. Survey Methodology, 29, 181-189.

van Buuren S and 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


# impute missing data, analyse and pool using the classic MICE workflow
imp <- mice(nhanes, maxit = 2, m = 2)
#>  iter imp variable
#>   1   1  bmi  hyp  chl
#>   1   2  bmi  hyp  chl
#>   2   1  bmi  hyp  chl
#>   2   2  bmi  hyp  chl
fit <- with(data = imp, exp = lm(bmi ~ hyp + chl))
#>          term    estimate  std.error statistic       df     p.value
#> 1 (Intercept) 20.60793363 4.64881537 4.4329430 10.12935 0.001229288
#> 2         hyp  0.21992681 1.99646262 0.1101582 19.71919 0.913397289
#> 3         chl  0.02823202 0.02531475 1.1152403 11.47827 0.287552697

# generate fully synthetic data, analyse and pool
imp <- mice(cars,
  maxit = 2, m = 2,
  where = matrix(TRUE, nrow(cars), ncol(cars))
#>  iter imp variable
#>   1   1  speed  dist
#>   1   2  speed  dist
#>   2   1  speed  dist
#>   2   2  speed  dist
fit <- with(data = imp, exp = lm(speed ~ dist))
#>          term    estimate  std.error statistic       df      p.value
#> 1 (Intercept) 11.45981833 1.35768034  8.440734 9.843846 8.128227e-06
#> 2        dist  0.08662892 0.03546026  2.442986 2.896935 9.529340e-02

# use a custom pooling rule for the total variance about the estimate
# e.g. use t = b + b/m instead of t = ubar + b + b/m
imp <- mice(nhanes, maxit = 2, m = 2)
#>  iter imp variable
#>   1   1  bmi  hyp  chl
#>   1   2  bmi  hyp  chl
#>   2   1  bmi  hyp  chl
#>   2   2  bmi  hyp  chl
fit <- with(data = imp, exp = lm(bmi ~ hyp + chl))
pool(fit, custom.t = ".data$b + .data$b / .data$m")
#> Class: mipo    m = 2 
#>          term m    estimate         ubar            b            t dfcom df
#> 1 (Intercept) 2 24.13311859 2.244648e+01 6.5406334371 9.8109501556    22  0
#> 2         hyp 2 -2.26888722 4.854083e+00 0.2991634667 0.4487452001    22  0
#> 3         chl 2  0.02693878 6.328178e-04 0.0001986903 0.0002980354    22  0
#>          riv lambda       fmi
#> 1 0.43708196      1 0.7680485
#> 2 0.09244696      1 0.6948746
#> 3 0.47096565      1 0.7733915