deletion reduced the sample size by 32%. Next, we’ll consider an approach that employs the entire dataset (including cases with missing data).

18.7 Multiple imputation

Multiple imputation (MI) provides an approach to missing values that’s based on repeated simulations. MI is frequently the method of choice for complex missing-val- ues problems. In MI, a set of complete datasets (typically 3 to 10) is generated from an existing dataset that’s missing values. Monte Carlo methods are used to fill in the missing data in each of the simulated datasets. Standard statistical methods are applied to each of the simulated datasets, and the outcomes are combined to provide estimated results and confidence intervals that take into account the uncertainty introduced by the missing values. Good implementations are available in R through the Amelia, mice, and mi packages.

complete datasets (the default is five). Each complete dataset is created by imputing values for the missing data in the original data frame. There’s a random component to the imputations, so each complete dataset is slightly different. The with() function is then used to apply a statistical model (for example, a linear or generalized linear model) to each complete dataset in turn. Finally, the pool() function combines the results of these separate analyses into a single set of results. The standard errors and p-values in this final model correctly reflect the uncertainty produced by both the missing values and the multiple imputations.

How does the mice() function impute missing values?

Missing values are imputed by Gibbs sampling. By default, each variable with missing values is predicted from all other variables in the dataset. These prediction equations are used to impute plausible values for the missing data. The process iterates until convergence over the missing values is achieved. For each variable, you can choose the form of the prediction model (called an elementary imputation method) and the variables entered into it.

By default, predictive mean matching is used to replace missing data on continuous variables, whereas logistic or polytomous logistic regression is used for target variables that are dichotomous (factors with two levels) or polytomous (factors with more than two levels), respectively. Other elementary imputation methods include Bayesian linear regression, discriminant function analysis, two-level normal imputation, and random sampling from observed values. You can supply your own methods as well.

An analysis based on the mice package typically conforms to the following structure

library(mice)

imp <- mice(data, m)

fit <- with(imp, analysis) pooled <- pool(fit) summary(pooled)

where

■data is a matrix or data frame containing missing values.

■imp is a list object containing the m imputed datasets, along with information on how the imputations were accomplished. By default, m = 5.

■analysis is a formula object specifying the statistical analysis to be applied to each of the m imputed datasets. Examples include lm() for linear regression models, glm() for generalized linear models, gam() for generalized additive models, and nbrm() for negative binomial models. Formulas within the parentheses give the response variables on the left of the ~ and the predictor variables (separated by + signs) on the right.

■fit is a list object containing the results of the m separate statistical analyses.

■pooled is a list object containing the averaged results of these m statistical analyses.

Let’s apply multiple imputation to the sleep dataset. You’ll repeat the analysis from section 18.6, but this time use all 62 mammals. Set the seed value for the random number generator to 1,234 so that your results will match the following:

>library(mice)

>data(sleep, package="VIM")

>imp <- mice(sleep, seed=1234)

[...output deleted to save space...]

>fit <- with(imp, lm(Dream ~ Span + Gest))

>pooled <- pool(fit)

>summary(pooled)

Here, you see that the regression coefficient for Span isn’t significant (p 0.08), and the coefficient for Gest is significant at the p < 0.01 level. If you compare these results with those produced by a complete case analysis (section 18.6), you see that you’d come to the same conclusions in this instance. Length of gestation has a (statistically) significant, negative relationship with amount of dream sleep, controlling for life span. Although the complete-case analysis was based on the 42 mammals with complete data, the current analysis is based on information gathered from the full set of 62 mammals. By the way, the fmi column reports the fraction of missing information (that is, the proportion of variability that is attributable to the uncertainty introduced by the missing data).

You can access more information about the imputation by examining the objects created in the analysis. For example, let’s view a summary of the imp object:

From the resulting output, you can see that five synthetic datasets were created and that the predictive mean matching (pmm) method was used for each variable with missing data. No imputation ("") was needed for BodyWgt, BrainWgt, Pred, Exp, or Danger, because they had no missing values. The visit sequence tells you that variables

were imputed from right to left, starting with NonD and ending with Gest. Finally, the predictor matrix indicates that each variable with missing data was imputed using all the other variables in the dataset. (In this matrix, the rows represent the variables being imputed, the columns represent the variables used for the imputation, and 1s/0s indicate used/not used).

You can view the imputations by looking at subcomponents of the imp object. For example,

32.3 2.4 1.9 1.5 2.4

41.2 1.3 5.6 2.3 1.3

140.6 1.0 0.0 0.3 0.5

241.2 1.0 5.6 1.0 6.6

261.9 6.6 0.9 2.2 2.0

301.0 1.2 2.6 2.3 1.4

315.6 0.5 1.2 0.5 1.4

470.7 0.6 1.4 1.8 3.6

530.7 0.5 0.7 0.5 0.5

550.5 2.4 0.7 2.6 2.6

621.9 1.4 3.6 5.6 6.6

displays the 5 imputed values for each of the 12 mammals with missing data on the Dream variable. A review of these matrices helps you determine whether the imputed values are reasonable. A negative value for length of sleep might give you pause (or nightmares).

You can view each of the m imputed datasets via the complete() function. The format is

complete(imp, action=#)

where # specifies one of the m synthetically complete datasets. For example,

>dataset3 <- complete(imp, action=3)

>dataset3

displays the third (out of five) complete dataset created by the multiple imputation process.

Due to space limitations, we’ve only briefly considered the MI implementation provided in the mice package. The mi and Amelia packages also contain valuable