The pool() function combines the estimates from m repeated complete data analyses. The typical sequence of steps to do 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 is 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)

## Arguments

object 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.

## Value

An object of class mipo, which stands for 'multiple imputation pooled outcome'.

## Details

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 function requires the following input from each fitted model:

1. the estimates of the model, usually obtainable by coef()

2. the standard error of each estimate;

3. the residual degrees of freedom of the model.

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

The pool() function relies on the broom::tidy for extracting the parameters. Versions before mice 3.8.5 failed when no broom::glance() function was found for extracting the residual degrees of freedom. The pool() function is now more forgiving.

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

In versions prior to mice 3.0 pooling required only 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 3.0+, 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 https://broom.tidymodels.org.

## References

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.

van Buuren S and Groothuis-Oudshoorn K (2011). mice: Multivariate Imputation by Chained Equations in R. Journal of Statistical Software, 45(3), 1-67. https://www.jstatsoft.org/v45/i03/

with.mids, as.mira, pool.scalar, glance, tidy https://github.com/amices/mice/issues/142, https://github.com/amices/mice/issues/274

## Examples

# 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  chlfit <- with(data = imp, exp = lm(bmi ~ hyp + chl))
summary(pool(fit))
#>          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