A previous article
discussed some of the causes of missing data and some of the
consequences of analyzing only complete cases. This newsletter will
discuss some other common ways of dealing with missing data, with a
discussion of their advantages and disadvantages.
Available case analysis
(pairwise deletion) calculates each step of the analysis separately
using the cases that have data available for that step. Therefore, a
case with data missing on one variable will be used only in steps that
do not involve that variable. The advantage is that the sample size for
each individual analysis is generally higher than with complete case
analysis, but the results are unbiased only if the data are MCAR. It
can also lead to mathematical problems in computing estimates of some
parameters, and is not recommended.
Most other methods involve
imputation-replacing the missing values with an estimate, then
analyzing the full data set as if the imputed values were actual
observed values. There are many ways to choose an estimate. The
following are common methods:
Mean: the mean of the
observed values for that variable
Substitution: the value from
a new individual who was not selected to be in the sample
Hot deck: a randomly chosen
value from an individual who has similar values on other variables
Cold deck: a systematically
chosen value from an individual who has similar values on other
variables
Regression: the predicted
value obtained by regressing the missing variable on other variables
Stochastic regression: the
predicted value from a regression plus a random residual value.
Interpolation and
extrapolation: an estimated value from other observations from the same
individual.
Imputation is popular because
it is conceptually simple and because the resulting sample has the same
number of observations as the full data set. It can be very tempting
when complete-case analysis eliminates a large proportion of the data
set. But it has limitations. Some imputation methods result in biased
parameter estimates, such as means and correlations, unless the data
are MCAR. The bias is often worse than with complete-case analysis,
especially for mean imputation. The extent of the bias depends on many
factors, including the missing data mechanism, the proportion of the
data that is missing, and the information available in the data set.
Moreover, all of these
imputation methods underestimate standard errors. Since the imputed
observations are themselves estimates, their values have corresponding
random error. Despite this, imputed values are treated as actual
observations in analyses. The extra source of error is ignored,
resulting in too-small standard errors and too-small p-values.
Furthermore, although imputation is conceptually simple, it is usually
difficult to do well in practice. Therefore, these imputation methods
are not satisfactory in most circumstances.
Two alternate methods maintain
the full sample size and can result in unbiased estimates of parameters
and standard errors for ignorable missing data: multiple imputation and
maximum likelihood estimation. These techniques are now available in
common statistical software. Subsequent newsletters will describe these
methods and discuss their availability in software packages.
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