aregImpute {Hmisc}  R Documentation 
The transcan
function creates flexible additive imputation models
but provides only an approximation to true multiple imputation as the
imputation models are fixed before all multiple imputations are
drawn. This ignores variability caused by having to fit the
imputation models. aregImpute
takes all aspects of uncertainty in
the imputations into account by using the bootstrap to approximate the
process of drawing predicted values from a full Bayesian predictive
distribution. Different bootstrap resamples are used for each of the
multiple imputations, i.e., for the i
th imputation of a sometimes
missing variable, i=1,2,... n.impute
, a flexible additive
model is fitted on a sample with replacement from the original data and
this model is used to predict all of the original missing and
nonmissing values for the target variable.
areg
is used to fit the imputation models. By default, linearity
is assumed for target variables (variables being imputed) and
nk=3
knots are assumed for continuous predictors transformed
using restricted cubic splines. If nk
is three or greater and
tlinear
is set to FALSE
, areg
simultaneously find transformations of the target variable and of all of
the predictors, to get a good fit assuming additivity, maximizing
R^2, using the same canonical correlation method as
transcan
. Flexible transformations may be overridden for
specific variables by specifying the identity transformation for them.
When a categorical variable is being predicted, the flexible
transformation is Fisher's optimum scoring method. Nonlinear transformations for continuous variables may be nonmonotonic. If
nk
is a vector, areg
's bootstrap and crossval=10
options will be used to help find the optimum validating value of
nk
over values of that vector, at the last imputation iteration.
For the imputations, the minimum value of nk
is used.
Instead of defaulting to taking random draws from fitted imputation
models using random residuals as is done by transcan
,
aregImpute
by default uses predictive mean matching with optional
weighted probability sampling of donors rather than using only the
closest match. Predictive mean matching works for binary, categorical,
and continuous variables without the need for iterative maximum
likelihood fitting for binary and categorical variables, and without the
need for computing residuals or for curtailing imputed values to be in
the range of actual data. Predictive mean matching is especially
attractive when the variable being imputed is also being transformed
automatically. See Details below for more information about the
algorithm. A "regression"
method is also available that is
similar to that used in transcan
. This option should be used
when mechanistic missingness requires the use of extrapolation during
imputation.
A print
method summarizes the results, and a plot
method plots
distributions of imputed values. Typically, fit.mult.impute
will
be called after aregImpute
.
If a target variable is transformed nonlinearly (i.e., if nk
is
greater than zero and tlinear
is set to FALSE
) and the
estimated target variable transformation is nonmonotonic, imputed
values are not unique. When type='regression'
, a random choice
of possible inverse values is made.
aregImpute(formula, data, subset, n.impute=5, group=NULL, nk=3, tlinear=TRUE, type=c('pmm','regression'), match=c('weighted','closest'), fweighted=0.2, curtail=TRUE, boot.method=c('simple', 'approximate bayesian'), burnin=3, x=FALSE, pr=TRUE, plotTrans=FALSE, tolerance=NULL, B=75) ## S3 method for class 'aregImpute': print(x, digits=3, ...) ## S3 method for class 'aregImpute': plot(x, nclass=NULL, type=c('ecdf','hist'), datadensity=c("hist", "none", "rug", "density"), diagnostics=FALSE, maxn=10, ...)
formula 
an S model formula. You can specify restrictions for transformations
of variables. The function automatically determines which variables
are categorical (i.e., factor , category , or character vectors).
Binary variables are automatically restricted to be linear. Force
linear transformations of continuous variables by enclosing variables
by the identify function (I() ). It is recommended that
factor() or as.factor() do not appear in the formula but
instead variables be converted to factors as needed and stored in the
data frame. That way imputations for factor variables (done using
impute.transcan for example) will be correct.

x 
an object created by aregImpute . For aregImpute , set
x to TRUE to save the data matrix containing the final (number
n.impute ) imputations in the result. This
is needed if you want to later do outofsample imputation.
Categorical variables are coded as integers in this matrix.

data 

subset 
These may be also be specified. You may not specify na.action as
na.retain is always used.

n.impute 
number of multiple imputations. n.impute=5 is frequently
recommended but 10 or more doesn't hurt.

group 
a character or factor variable the same length as the
number of observations in data and containing no NA s.
When group is present, causes a bootstrap sample of the
observations corresponding to nonNA s of a target variable to
have the same frequency distribution of group as the
that in the nonNA s of the original sample. This can handle
ksample problems as well as lower the chance that a bootstrap sample
will have a missing cell when the original cell frequency was low.

nk 
number of knots to use for continuous variables. When both
the target variable and the predictors are having optimum
transformations estimated, there is more instability than with normal
regression so the complexity of the model should decrease more sharply
as the sample size decreases. Hence set nk to 0 (to force
linearity for noncategorical variables) or 3 (minimum number of knots
possible with a linear tailrestricted cubic spline) for small sample
sizes. Simulated problems as in the examples section can assist in
choosing nk . See nk to a vector to get bootstrapvalidated
and 10fold crossvalidated R^2 and mean and median absolute
prediction errors for imputing each sometimesmissing variable, with
nk ranging over the given vector. The errors are on the
original untransformed scale. The mean absolute error is the
recommended basis for choosing the number of knots (or linearity).

tlinear 
set to FALSE to allow a target variable (variable
being imputed) to have a nonlinear lefthandside transformation when
nk is 3 or greater 
type 
The default is "pmn" for predictive mean matching,
which is a more nonparametric approach that will work for categorical
as well as continuous predictors. Alternatively, use
"regression" when all variables that are sometimes missing are
continuous and the missingness mechanism is such that entire intervals
of population values are unobserved. See the Details section for more
information. For the plot method, specify type="hist"
to draw histograms of imputed values with rug plots at the top, or
type="ecdf" (the default) to draw empirical CDFs with spike
histograms at the bottom.

match 
Defaults to match="weighted" to do weighted multinomial
probability sampling using the tricube function (similar to lowess)
as the weights. The argument of the tricube function is the absolute
difference in transformed predicted values of all the donors and of
the target predicted value, divided by a scaling factor.
The scaling factor in the tricube function is fweighted times
the mean absolute difference between the target predicted value and
all the possible donor predicted values. Set match="closest"
to find as the donor the observation having the closest predicted
transformed value, even if that same donor is found repeatedly. 
fweighted 
Smoothing parameter (multiple of mean absolute difference) used when
match="weighted" , with a default value of 0.2. Set
fweighted to a number between 0.02 and 0.2 to force the donor
to have a predicted value closer to the target, and set
fweighted to larger values (but seldom larger than 1.0) to allow
donor values to be less tightly matched. See the examples below to
learn how to study the relationship between fweighted and the
standard deviation of multiple imputations within individuals. 
curtail 
applies if type='regression' , causing imputed
values to be curtailed at the observed range of the target variable.
Set to FALSE to allow extrapolation outside the data range. 
boot.method 
By default, simple boostrapping is used in which the
target variable is predicted using a sample with replacement from the
observations with nonmissing target variable. Specify
boot.method='approximate bayesian' to build the imputation
models from a sample with replacement from a sample with replacement
of the observations with nonmissing targets. Preliminary simulations
have shown this results in good confidence coverage of the final model
parameters when type='regression' is used. Not implemented
when group is used. 
burnin 
aregImpute does burnin + n.impute iterations of the
entire modeling process. The first burnin imputations are
discarded. More burnin iteractions may be requied when multiple
variables are missing on the same observations. 
pr 
set to FALSE to suppress printing of iteration messages

plotTrans 
set to TRUE to plot ace or avas transformations
for each variable for each of the multiple imputations. This is
useful for determining whether transformations are reasonable. If
transformations are too noisy or have long flat sections (resulting in
"lumps" in the distribution of imputed values), it may be advisable to
place restrictions on the transformations (monotonicity or linearity).

tolerance 
singularity criterion; list the source code in the
lm.fit.qr.bare function for details 
B 
number of bootstrap resamples to use if nk is a vector 
digits 
number of digits for printing 
nclass 
number of bins to use in drawing histogram 
datadensity 
see Ecdf 
diagnostics 
Specify diagnostics=TRUE to draw plots of imputed values against
sequential imputation numbers, separately for each missing
observations and variable.

maxn 
Maximum number of observations shown for diagnostics. Default is
maxn=10 , which limits the number of observations plotted to at most
the first 10.

... 
other arguments that are ignored 
The sequence of steps used by the aregImpute
algorithm is the
following.
(1) For each variable containing m NA
s where m > 0, initialize the
NA
s to values from a random sample (without replacement if
a sufficient number of nonmissing values exist) of size m from the
nonmissing values.
(2) For burnin+n.impute
iterations do the following steps. The
first burnin
iterations provide a burnin, and imputations are
saved only from the last n.impute
iterations.
(3) For each variable containing any NA
s, draw a sample with
replacement from the observations in the entire dataset in which the
current variable being imputed is nonmissing. Fit a flexible
additive model to predict this target variable while finding the
optimum transformation of it (unless the identity
transformation is forced). Use this fitted flexible model to
predict the target variable in all of the original observations.
Impute each missing value of the target variable with the observed
value whose predicted transformed value is closest to the predicted
transformed value of the missing value (if match="closest"
and
type="pmm"
),
or use a draw from a multinomial distribution with probabilities derived
from distance weights, if match="weighted"
(the default).
(4) After these imputations are computed, use these random draw
imputations the next time the curent target variable is used as a
predictor of other sometimesmissing variables.
When match="closest"
, predictive mean matching does not work well
when fewer than 3 variables are used to predict the target variable,
because many of the multiple imputations for an observation will be
identical. In the extreme case of one righthandside variable and
assuming that only monotonic transformations of left and rightside
variables are allowed, every bootstrap resample will give predicted
values of the target variable that are monotonically related to
predicted values from every other bootstrap resample. The same is true
for Bayesian predicted values. This causes predictive mean matching to
always match on the same donor observation.
When the missingness mechanism for a variable is so systematic that the
distribution of observed values is truncated, predictive mean matching
does not work. It will only yield imputed values that are near observed
values, so intervals in which no values are observed will not be
populated by imputed values. For this case, the only hope is to make
regression assumptions and use extrapolation. With
type="regression"
, aregImpute
will use linear
extrapolation to obtain a (hopefully) reasonable distribution of imputed
values. The "regression"
option causes aregImpute
to
impute missing values by adding a random sample of residuals (with
replacement if there are more NA
s than measured values) on the
transformed scale of the target variable. After random residuals are
added, predicted random draws are obtained on the original untransformed
scale using reverse linear interpolation on the table of original and
transformed target values (linear extrapolation when a random residual
is large enough to put the random draw prediction outside the range of
observed values). The bootstrap is used as with type="pmm"
to
factor in the uncertainty of the imputation model.
As model uncertainty is high when the transformation of a target
variable is unknown, tlinear
defaults to TRUE
to limit the
variance in predicted values when nk
is positive.
a list of class "aregImpute"
containing the following elements:
call 
the function call expression 
formula 
the formula specified to aregImpute

match 
the match argument

fweighted 
the fweighted argument

n 
total number of observations in input dataset 
p 
number of variables 
na 
list of subscripts of observations for which values were originally missing 
nna 
named vector containing the numbers of missing values in the data 
type 
vector of types of transformations used for each variable
("s","l","c" for smooth spline, linear, or categorical with dummy
variables)

tlinear 
value of tlinear parameter 
nk 
number of knots used for smooth transformations 
cat.levels 
list containing character vectors specifying the levels of
categorical variables

df 
degrees of freedom (number of parameters estimated) for each variable 
n.impute 
number of multiple imputations per missing value 
imputed 
a list containing matrices of imputed values in the same format as
those created by transcan . Categorical variables are coded using
their integer codes. Variables having no missing values will have
NULL matrices in the list.

x 
if x is TRUE , the original data matrix with
integer codes for categorical variables 
rsq 
for the last round of imputations, a vector containing the Rsquares
with which each sometimesmissing variable could be predicted from the
others by ace or avas .

Frank Harrell
Department of Biostatistics
Vanderbilt University
f.harrell@vanderbilt.edu
Little R, An H. Robust likelihoodbased analysis of multivariate data with missing values. Statistica Sinica 14:949968, 2004.
van Buuren S, Brand JPL, GroothuisOudshoorn CGM, Rubin DB. Fully conditional specifications in multivariate imputation. J Stat Comp Sim 72:10491064, 2006.
de Groot JAH, Janssen KJM, Zwinderman AH, Moons KGM, Reitsma JB. Multiple imputation to correct for partial verification bias revisited. Stat Med 27:58805889, 2008.
Siddique J. Multiple imputation using an iterative hotdeck with distancebased donor selection. Stat Med 27:83102, 2008.
fit.mult.impute
, transcan
, areg
, naclus
, naplot
, mice
,
dotchart2
, Ecdf
# Check that aregImpute can almost exactly estimate missing values when # there is a perfect nonlinear relationship between two variables # Fit restricted cubic splines with 4 knots for x1 and x2, linear for x3 set.seed(3) x1 < rnorm(200) x2 < x1^2 x3 < runif(200) m < 30 x2[1:m] < NA a < aregImpute(~x1+x2+I(x3), n.impute=5, nk=4, match='closest') a matplot(x1[1:m]^2, a$imputed$x2) abline(a=0, b=1, lty=2) x1[1:m]^2 a$imputed$x2 # Multiple imputation and estimation of variances and covariances of # regression coefficient estimates accounting for imputation # Example 1: large sample size, much missing data, no overlap in # NAs across variables x1 < factor(sample(c('a','b','c'),1000,TRUE)) x2 < (x1=='b') + 3*(x1=='c') + rnorm(1000,0,2) x3 < rnorm(1000) y < x2 + 1*(x1=='c') + .2*x3 + rnorm(1000,0,2) orig.x1 < x1[1:250] orig.x2 < x2[251:350] x1[1:250] < NA x2[251:350] < NA d < data.frame(x1,x2,x3,y) # Find value of nk that yields best validating imputation models # tlinear=FALSE means to not force the target variable to be linear f < aregImpute(~y + x1 + x2 + x3, nk=c(0,3:5), tlinear=FALSE, data=d, B=10) # normally B=75 f # Try forcing target variable (x1, then x2) to be linear while allowing # predictors to be nonlinear (could also say tlinear=TRUE) f < aregImpute(~y + x1 + x2 + x3, nk=c(0,3:5), data=d, B=10) f # Use 100 imputations to better check against individual true values f < aregImpute(~y + x1 + x2 + x3, n.impute=100, data=d) f par(mfrow=c(2,1)) plot(f) modecat < function(u) { tab < table(u) as.numeric(names(tab)[tab==max(tab)][1]) } table(orig.x1,apply(f$imputed$x1, 1, modecat)) par(mfrow=c(1,1)) plot(orig.x2, apply(f$imputed$x2, 1, mean)) fmi < fit.mult.impute(y ~ x1 + x2 + x3, lm, f, data=d) sqrt(diag(vcov(fmi))) fcc < lm(y ~ x1 + x2 + x3) summary(fcc) # SEs are larger than from mult. imputation # Example 2: Very discriminating imputation models, # x1 and x2 have some NAs on the same rows, smaller n set.seed(5) x1 < factor(sample(c('a','b','c'),100,TRUE)) x2 < (x1=='b') + 3*(x1=='c') + rnorm(100,0,.4) x3 < rnorm(100) y < x2 + 1*(x1=='c') + .2*x3 + rnorm(100,0,.4) orig.x1 < x1[1:20] orig.x2 < x2[18:23] x1[1:20] < NA x2[18:23] < NA #x2[21:25] < NA d < data.frame(x1,x2,x3,y) n < naclus(d) plot(n); naplot(n) # Show patterns of NAs # 100 imputations to study them; normally use 5 or 10 f < aregImpute(~y + x1 + x2 + x3, n.impute=100, nk=0, data=d) par(mfrow=c(2,3)) plot(f, diagnostics=TRUE, maxn=2) # Note: diagnostics=TRUE makes graphs similar to those made by: # r < range(f$imputed$x2, orig.x2) # for(i in 1:6) { # use 1:2 to mimic maxn=2 # plot(1:100, f$imputed$x2[i,], ylim=r, # ylab=paste("Imputations for Obs.",i)) # abline(h=orig.x2[i],lty=2) # } table(orig.x1,apply(f$imputed$x1, 1, modecat)) par(mfrow=c(1,1)) plot(orig.x2, apply(f$imputed$x2, 1, mean)) fmi < fit.mult.impute(y ~ x1 + x2, lm, f, data=d) sqrt(diag(vcov(fmi))) fcc < lm(y ~ x1 + x2) summary(fcc) # SEs are larger than from mult. imputation # Study relationship between smoothing parameter for weighting function # (multiplier of mean absolute distance of transformed predicted # values, used in tricube weighting function) and standard deviation # of multiple imputations. SDs are computed from average variances # across subjects. match="closest" same as match="weighted" with # small value of fweighted. # This example also shows problems with predicted mean # matching almost always giving the same imputed values when there is # only one predictor (regression coefficients change over multiple # imputations but predicted values are virtually 11 functions of each # other) set.seed(23) x < runif(200) y < x + runif(200, .05, .05) r < resid(lsfit(x,y)) rmse < sqrt(sum(r^2)/(2002)) # sqrt of residual MSE y[1:20] < NA d < data.frame(x,y) f < aregImpute(~ x + y, n.impute=10, match='closest', data=d) # As an aside here is how to create a completed dataset for imputation # number 3 as fit.mult.impute would do automatically. In this degenerate # case changing 3 to 12,410 will not alter the results. imputed < impute.transcan(f, imputation=3, data=d, list.out=TRUE, pr=FALSE, check=FALSE) sd < sqrt(mean(apply(f$imputed$y, 1, var))) ss < c(0, .01, .02, seq(.05, 1, length=20)) sds < ss; sds[1] < sd for(i in 2:length(ss)) { f < aregImpute(~ x + y, n.impute=10, fweighted=ss[i]) sds[i] < sqrt(mean(apply(f$imputed$y, 1, var))) } plot(ss, sds, xlab='Smoothing Parameter', ylab='SD of Imputed Values', type='b') abline(v=.2, lty=2) # default value of fweighted abline(h=rmse, lty=2) # root MSE of residuals from linear regression ## Not run: # Do a similar experiment for the Titanic dataset getHdata(titanic3) h < lm(age ~ sex + pclass + survived, data=titanic3) rmse < summary(h)$sigma set.seed(21) f < aregImpute(~ age + sex + pclass + survived, n.impute=10, data=titanic3, match='closest') sd < sqrt(mean(apply(f$imputed$age, 1, var))) ss < c(0, .01, .02, seq(.05, 1, length=20)) sds < ss; sds[1] < sd for(i in 2:length(ss)) { f < aregImpute(~ age + sex + pclass + survived, data=titanic3, n.impute=10, fweighted=ss[i]) sds[i] < sqrt(mean(apply(f$imputed$age, 1, var))) } plot(ss, sds, xlab='Smoothing Parameter', ylab='SD of Imputed Values', type='b') abline(v=.2, lty=2) # default value of fweighted abline(h=rmse, lty=2) # root MSE of residuals from linear regression ## End(Not run)