This function performs multiple imputation (MI) with the Approximate Bayesian Bootstrap (ABB) of Rubin and Schenker (1986).

`ABB(X, K=1)`

X

This is a vector or matrix of data that must include both
observed and missing values. When `X`

is a matrix, missing
values must occur somewhere in the set, but are not required to
occur in each variable.

K

This is the number of imputations.

This function returns a list with \(K\) components, one for each set of imputations. Each component contains a vector of imputations equal in length to the number of missing values in the data.

`ABB`

does not currently return the mean of the imputations, or
the between-imputation variance or within-imputation variance.

The Approximate Bayesian Bootstrap (ABB) is a modified form of the
`BayesianBootstrap`

(Rubin, 1981) that is used for
multiple imputation (MI). Imputation is a family of statistical
methods for replacing missing values with estimates. Introduced by
Rubin and Schenker (1986) and Rubin (1987), MI is a family of
imputation methods that includes multiple estimates, and therefore
includes variability of the estimates.

The data, \(\textbf{X}\), are assumed to be independent and
identically distributed (IID), contain both observed and missing
values, and its missing values are assumed to be ignorable (meaning
enough information is available in the data that the missingness
mechanism can be ignored, if the information is used properly) and
Missing Completely At Random (MCAR). When `ABB`

is used in
conjunction with a propensity score (described below), missing values
may be Missing At Random (MAR).

`ABB`

does not add auxiliary information, but performs imputation
with two sampling (with replacement) steps. First,
\(\textbf{X}^\star_{obs}\) is sampled from
\(\textbf{X}_{obs}\). Then,
\(\textbf{X}^\star_{mis}\) is sampled from
\(\textbf{X}^\star_{obs}\). The result is a sample of
the posterior predictive distribution of
\((\textbf{X}_{mis}|\textbf{X}_{obs})\). The first
sampling step is also known as hotdeck imputation, and the second
sampling step changes the variance. Since auxiliary information is not
included, `ABB`

is appropriate for missing values that are
ignorable and MCAR.

Auxiliary information may be included in the process of imputation by
introducing a propensity score (Rosenbaum and Rubin, 1983; Rosenbaum
and Rubin, 1984), which is an estimate of the probability of
missingness. The propensity score is often the result of a binary
logit model, where missingness is predicted as a function of other
variables. The propensity scores are discretized into quantile-based
groups, usually quintiles. Each quintile must have both observed and
missing values. `ABB`

is applied to each quintile. This is called
within-class imputation. It is assumed that the missing mechanism
depends only on the variables used to estimate the propensity score.

With \(K=1\), `ABB`

may be used in MCMC, such as in
`LaplacesDemon`

, more commonly along with a propensity
score for missingness. MI is performed, despite \(K=1\), because
imputation occurs at each MCMC iteration. The practical advantage of
this form of imputation is the ease with which it may be
implemented. For example, full-likelihood imputation should perform
better, but requires a chain to be updated for each missing value.

An example of a limitation of `ABB`

with propensity scores is to
consider imputing missing values of income from age in a context where
age and income have a positive relationship, and where the highest
incomes are missing systematically. `ABB`

with propensity scores
should impute these highest missing incomes given the highest observed
ages, but is unable to infer beyond the observed data.

ABB has been extended (Parzen et al., 2005) to reduce bias, by
introducing a correction factor that is applied to the MI variance
estimate. This correction may be applied to output from `ABB`

.

Parzen, M., Lipsitz, S.R., and Fitzmaurice, G.M. (2005). "A Note on
Reducing the Bias of the Approximate Bayesian Bootstrap Imputation
Variance Estimator". *Biometrika*, 92, 4, p. 971--974.

Rosenbaum, P.R. and Rubin, D.B. (1983). "The Central Role of the
Propensity Score in Observational Studies for Causal
Effects". *Biometrika*, 70, p. 41--55.

Rosenbaum, P.R. and Rubin, D.B. (1984). "Reducing Bias in
Observational Studies Using Subclassification in the Propensity
Score". *Journal of the American Statistical Association*, 79,
p. 516--524.

Rubin, D.B. (1981). "The Bayesian Bootstrap". *Annals of
Statistics*, 9, p. 130--134.

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

Rubin, D.B. and Schenker, N. (1986). "Multiple Imputation for Interval
Estimation from Simple Random Samples with Ignorable
Nonresponse". *Journal of the American Statistical Association*,
81, p. 366--374.

`BayesianBootstrap`

,
`LaplacesDemon`

, and
`MISS`

.

```
# NOT RUN {
library(LaplacesDemon)
### Create Data
J <- 10 #Number of variables
m <- 20 #Number of missings
N <- 50 #Number of records
mu <- runif(J, 0, 100)
sigma <- runif(J, 0, 100)
X <- matrix(0, N, J)
for (j in 1:J) X[,j] <- rnorm(N, mu[j], sigma[j])
### Create Missing Values
M1 <- rep(0, N*J)
M2 <- sample(N*J, m)
M1[M2] <- 1
M <- matrix(M1, N, J)
X <- ifelse(M == 1, NA, X)
### Approximate Bayesian Bootstrap
imp <- ABB(X, K=1)
### Replace Missing Values in X (when K=1)
X.imp <- X
X.imp[which(is.na(X.imp))] <- unlist(imp)
X.imp
# }
```

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