# Multiple imputation inference

### Description

Combines estimates and standard errors from m complete-data analyses performed on m imputed datasets to produce a single inference. Uses the technique described by Rubin (1987) for multiple imputation inference for a scalar estimand.

### Usage

1 | ```
mi.inference(est, std.err, confidence=0.95)
``` |

### Arguments

`est` |
a list of $m$ (at least 2) vectors representing estimates (e.g., vectors of estimated regression coefficients) from complete-data analyses performed on $m$ imputed datasets. |

`std.err` |
a list of $m$ vectors containing standard errors from the
complete-data analyses corresponding to the estimates in |

`confidence` |
desired coverage of interval estimates. |

### Value

a list with the following components, each of which is a vector of the
same length as the components of `est`

and `std.err`

:

`est` |
the average of the complete-data estimates. |

`std.err` |
standard errors incorporating both the between and the within-imputation uncertainty (the square root of the "total variance"). |

`df` |
degrees of freedom associated with the t reference distribution used for interval estimates. |

`signif` |
P-values for the two-tailed hypothesis tests that the estimated quantities are equal to zero. |

`lower` |
lower limits of the (100*confidence)% interval estimates. |

`upper` |
upper limits of the (100*confidence)% interval estimates. |

`r` |
estimated relative increases in variance due to nonresponse. |

`fminf` |
estimated fractions of missing information. |

### METHOD

Uses the method described on pp. 76-77 of Rubin (1987) for combining the complete-data estimates from $m$ imputed datasets for a scalar estimand. Significance levels and interval estimates are approximately valid for each one-dimensional estimand, not for all of them jointly.

### References

See Rubin (1987) or Schafer (1996), Chapter 4.

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