# pool: Combine estimates by pooling rules In mice: Multivariate Imputation by Chained Equations

## Description

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.

## Usage

 ```1 2 3``` ```pool(object, dfcom = NULL, rule = NULL) pool.syn(object, dfcom = NULL, rule = "reiter2003") ```

## Arguments

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

## 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 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 https://broom.tidymodels.org. 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 `robust.se` from the `tidy()` object.

## Value

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.

## 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.

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

`with.mids`, `as.mira`, `pool.scalar`, `glance`, `tidy` https://github.com/amices/mice/issues/142, https://github.com/amices/mice/issues/274
 ``` 1 2 3 4 5 6 7 8 9 10``` ```# impute missing data, analyse and pool using the classic MICE workflow imp <- mice(nhanes, maxit = 2, m = 2) fit <- with(data = imp, exp = lm(bmi ~ hyp + chl)) summary(pool(fit)) # generate fully synthetic data, analyse and pool imp <- mice(cars, maxit = 2, m = 2, where = matrix(TRUE, nrow(cars), ncol(cars))) fit <- with(data = imp, exp = lm(speed ~ dist)) summary(pool.syn(fit)) ```