Montalto, C. P. & Sung, J. (1996). Multiple Imputation in the 1992 Survey of Consumer Finances, Financial Counseling and Planning, 7, 133-146
Copyright 1996 by AFCPE
Multiple Imputation In The 1992 Survey Of Consumer Finances
Catherine Phillips Montalto,(1) The Ohio State UniversityJaimie Sung,(2) The Ohio State University
The 1992 Survey of Consumer Finances consists of five complete data sets because missing data are
multiply imputed. The incidence of missing data in the 1992 SCF is addressed and illustrates the
difficulty of obtaining financial information from individuals. The value of using all five data sets and
the risk of using only a single data set in empirical research are explained. Estimates derived
separately from each data set are compared to results using all five data sets to illustrate the extra
variability in the data due to imputation. Researchers are encouraged to use information from all five
data sets in order to make valid inferences.
KEY WORDS: inference, missing data, multiple imputation, repeated-imputation inference (RII),
Survey of Consumer Finances
The 1992 Survey of Consumer Finances (SCF) is a rich
source of information on assets and liabilities of U.S.
households. The wealth information in the SCF is
exceptionally good because the survey uses a sample
designed specifically to support wealth estimation. An
interesting characteristic of the 1992 SCF is that the
public use data tape contains five complete data sets,
instead of the more common single data set. Persons new
to the SCF, who are either interested in practical
implications from SCF research or in using the SCF in
applied research, may wonder why five complete data
sets are provided in the 1992 SCF and what techniques
need to be used to analyze data appropriately in the
presence of five data sets. Previous research using the
SCF has been inconsistent in the treatment of the
multiple data sets and in reporting this information.a
The 1992 SCF consists of five complete data sets as a
result of the procedure used to handle missing data.
Missing or incomplete information is common in all
survey data. Data can be missing because respondents
are unable or unwilling to provide information, or due to
errors in data recording and processing which render data
unuseable. For the final release of the 1992 SCF public
use tapes, missing and incomplete data were imputed,
using the multiple imputation technique developed for
the SCF (Kennickell, 1991). The multiple imputation
technique produces five complete data sets, referred to as
“implicates” (Board of Governors of the Federal Reserve
System, 1996).
This paper provides information on the incidence of
missing data in the 1992 SCF, summarizes alternative
methods of dealing with missing data, and describes
strengths of multiple imputation techniques for imputing
missing data. The value of incorporating information
from all implicates of the 1992 SCF in empirical analysis
is addressed. The “repeated-imputation inference” (RII)
approach, a technique which uses information from all
five implicates, is described and used to present point
estimates, variance estimates and test statistics for
selected variables in the 1992 SCF.
The SCF is conducted by the Board of Governors of the
Federal Reserve System in cooperation with the Statistics
of Income Division (SOI) of the Internal Revenue
Service. Since 1983 the survey has been conducted
every three years. The SCF is intended to provide
information on the financial characteristics of U.S.
households. Detailed information is collected on
household assets and liabilities. Information is also
collected on current and past employment, pension rights,
inheritances, income, marital history, household
demographic characteristics, attitudes, and numerous
other items. A more complete description of the data is
given in Kennickell and Starr-McCluer (1994).
The 1992 SCF employs a dual-frame sample
incorporating both an area-probability (AP) sample and
a special list sample developed from a sample of tax
records. The area-probability sample provides good
information on financial variables which are broadly-distributed in the population, such as automobile
ownership, home mortgages, and credit card debt. The
special list sample oversamples households which are
more likely to be wealthy, and provides good information
on financial variables which are highly correlated with
wealth, such as ownership of stocks, real estate
investments, and business assets. Of the 3,906
completed cases in the 1992 survey, 2,456 households
were part of the area-probability sample, and the
remaining 1,450 were part of the list sample.
Reporting rates for selected financial items for the
weighted full sample of the 1992 SCF are provided in
Table 1. (Kennickell, 1996, summarizes response rates
for the 1983, 1986, 1989, 1992 and 1995 Surveys of
Consumer Finances for the unweighted full sample, the
unweighted area-probability sample, and the weighted
full sample). Focusing on asset information, households
were much more likely to have checking accounts (83%)
and to own their homes (59%) than to have stocks (17%)
or business assets (13%). Households which reported
that they did have a financial item were then asked to
report the value of the item. In most cases, respondents
provided a dollar value. Respondents who were reluctant
to provide dollar values directly were offered a card
containing dollar ranges and asked to select the range
encompassing the dollar value. In some cases
respondents refused to answer or indicated they didn’t
know. The unwillingness or inability of respondents to
provide information, as well as errors in data recording
and processing which render data unusable, result in
missing data.
The incidence of missing data in the SCF varies widely,
but is generally within a range typical of other economic
surveys. In the 1992 SCF, the incidence of missing data
ranges from very low for data on monthly mortgage
payments, rent payments, and credit card balances, to
much higher for data on business assets and stocks. Less
than 5% of households in the 1992 SCF that had
mortgage payments, rent payments, or balances on credit
cards were unwilling or unable to provide information on
the dollar amount of the item. In contrast, over one-quarter of households with business assets and nearly
one-fifth of households with stocks did not know the
value of the assets, resulting in much higher rates of
missing data.
The response rates in the 1992 SCF illustrate the
difficulty which can be encountered in obtaining
information, particularly financial information, from
individuals. In general, individuals are more willing and
able to provide information on financial variables which
are more easily quantified and considered less private
(e.g. monthly housing payments in contrast to stocks).
The problem created by missing survey data is that data
intended by the survey design to be observed are in fact
missing. Missing information raises issues of both
efficiency and bias for users of the data. Nonresponse to
selected survey components means less efficient
estimates due to the reduced size of the useable data
base. In addition, the useable data base is vulnerable to
possible bias since nonrespondents are often
systematically different from respondents.
Several standard procedures to address missing data are
used in empirical research (Little & Rubin, 1987). A
brief description of the most common of these
procedures and the limitations follows. Observations
with missing data can be eliminated from the sample; this
results in less efficient estimates due to the reduced
sample size, and assumes no nonresponse bias.
Observations with missing data can be matched to
observations with complete data on a set of background
characteristics, and then the observations with complete
information can be weighted to compensate for
observations with missing data; this method assumes no
nonresponse bias beyond that explained by the measured
background characteristics. Missing values can be
replaced with the sample mean value of the variable.
This too assumes no nonresponse bias, can distort
correlations among variables, and understates the
variance because all missing values are replaced with the
sample mean. A “hot-deck” approach can be used
whereby each observation with missing data is matched
on a set of defining characteristics to a similar complete
observation within the sample, and the variable value
from the complete observation is substituted in place of
the missing data. In addition to assuming no
nonresponse bias after accounting for the variables used
as defining characteristics, this approach can result in
underestimation of the true variability in the sample.
Multivariate methods can be used to fill-in missing data
by single imputation. Single imputation commonly uses
regression equations to generate estimates of the values
of missing data utilizing information available in the data
set.
With respect to efficiency, methods of handling missing
data which produce data sets with no missing data (i.e.
complete data sets) increase the efficiency of estimation
by allowing the researcher to use all available data. With
respect to bias, methods which incorporate multivariate
techniques can incorporate information to adjust for
observed differences between respondents and
nonrespondents in an effort to reduce nonresponse bias
(Little, 1983). However, all methods which replace each
missing data point with one imputed value share a
common problem — they ignore the extra variability due
to the unknown missing values. Empirical analyses
based on data sets with single imputed values
systematically underestimate variability because the
imputed values are treated as if they were known with
certainty.
Multiple imputation is a procedure for handling missing
data which provides information that can be used to
estimate the extra variability due to the unknown missing
values. Multiple imputation uses stochastic multivariate
methods to replace each missing value with two or more
values generated to simulate the sampling distribution of
the missing values.b The goal of the imputation process
is to obtain the best possible estimates of the true but
unobserved values of data which are missing. As more
imputed values are generated, the approximation to the
true sampling distribution improves. In analysis, the
multiply imputed values are averaged to produce the best
estimate of what the results would have been if the
missing data had been observed, and the variance
estimates are corrected for the uncertainty due to missing
values. Rubin (1987) provides an extensive discussion
of both the theory and practice of multiple imputation.
Since the 1989 SCF, multiple imputation has been used
to replace each missing value with five values. The end
result is five complete data sets, referred to as
“implicates”. For a description of the imputation
procedure used since the 1989 SCF see Kennickell
(1991). Kennickell and McManus (1994) discuss
relevant issues associated with the multiple imputation of
the 1983-89 SCF Panel.
Multiple imputation, like several of the previously
mentioned techniques, offers the advantage of increased
efficiency in estimation and the ability to incorporate
information in an effort to reduce nonresponse bias. The
ability to address nonresponse bias is actually enhanced
because multiple imputation can incorporate information
about nonresponse by modeling either known reasons for
nonresponse or uncertainty about the reasons for
nonresponse.
The distinct advantage of multiple imputation is that it
provides information which can be used to estimate the
uncertainty in estimates due to missing values. As a
result, multiply imputed data sets provide a basis for
more valid inference and tests of significance. As an
illustration, consider a representative sample of
households and no missing data on family income. The
best estimate of mean family income for the population
of households is mean family income for the sample.
Similarly, the best estimate of the variance of mean
family income is the variance of the sample mean.
However, in the presence of missing data and the use of
multiple imputation techniques to fill in the missing data,
the best estimates of mean family income and the
variance of mean family income need to average over all
the imputed values. In addition, the best estimate of
variance needs to incorporate information on the amount
of uncertainty in the estimate due to missing data. When
the uncertainty due to missing values is ignored, as when
missing data are filled in by single imputation or when
the additional information available in multiple
implicates of a data set is not used, the variance estimate
will underestimate the true variance. All inference based
on this biased variance estimate risk misrepresenting the
precision of the estimates and the significance of
relationships. In general, as the proportion of values
which are missing and therefore imputed increases, the
downward bias in the uncorrected variance estimate
increases.
Table 1.
Reporting Rates for Various Items; 1992 SCF, Full Sample, Weighted
|
Item |
Percent of Households That Have
the Item |
Of Households That Have the Item, the Percent Providing Each Type
of Response When Asked to Report the Value of the Item |
||||
|
Yes |
Unknown |
Dollar Value |
Range Card
|
Missing Because: | ||
| Respondent
Didn’t Know |
Other
Reasons |
|||||
| Principal residence
Borrowed on mortgage
Owe on mortgage
Mortgage payment
Rent payment
Other real estate
Business assets
Car loan payment
Credit card balance
Checking account
Money market account
Savings account
Certificates of deposit
IRA/Keogh account
Savings bonds
Municipal bonds
Tax-free mutual funds
Stock
Wage income
Business income
Non-tax. interest income
Taxable interest income
Dividend income
Capital gains and losses
Rent and royalties
Unemployment comp.
Transfers |
58.9
38.2
38.2
37.9
31.3
17.9
13.2
24.6
62.2
83.2
11.1
43.7
16.6
23.0
22.1
2.1
2.7
16.8
71.0
11.1
5.1
38.0
16.4
7.7
8.9
6.0
3.6 |
0.0
0.2
0.2
0.2
0.0
0.2
0.0
0.2
0.1
0.3
0.4
0.5
0.5
0.4
0.5
0.6
0.9
0.5
2.3
2.2
2.5
2.4
2.7
2.7
2.7
2.7
2.7 |
93.7
91.1
85.8
95.7
96.9
89.7
70.0
90.9
95.7
87.2
84.8
84.3
73.1
79.5
84.8
79.7
67.4
73.8
85.6
78.3
72.5
73.0
70.9
73.1
83.0
87.3
87.0 |
0.7
1.0
0.0
0.5
0.3
0.2
0.9
0.4
0.8
1.7
0.9
1.6
1.7
2.4
1.7
1.7
1.2
1.9
3.6
1.8
2.0
2.9
1.6
1.7
1.4
0.7
1.1 |
4.8
4.5
11.1
1.5
0.5
8.4
27.2
3.9
2.0
3.7
4.5
4.1
7.9
8.1
9.8
12.4
15.3
17.9
4.1
7.9
14.1
12.3
12.7
12.3
5.2
3.8
3.1 |
0.8
3.4
3.1
2.3
2.2
1.7
1.9
4.8
1.6
7.4
9.8
10.1
17.3
10.0
3.6
6.3
16.1
6.5
6.7
10.0
11.4
11.8
14.8
11.9
10.4
8.2
8.8 |
SOURCE: Kennickell (1996). Table 4.
Repeated-Imputation Inference
The relevant question for the empirical researcher using
any SCF since the 1989 survey is how to use the
information from all five implicates to generate the best
point estimates and estimates of variance for parameters
of interest. In general, this is achieved by simply
combining results across the five complete data sets (i.e.
implicates). Combining the results often requires only
the calculation of the means and variances of the results
from the five separate implicates. This method of
inference, based on multiple complete data sets, is
referred to as “repeated-imputation inference” (RII)
(Rubin, 1987). RII is based on Bayesian theory, and is
applicable to linear and nonlinear models, and to models
estimated by both least squares and maximum
likelihood. A brief description of RII follows; more
technical information is provided in Appendix A.
Point Estimate
The best point estimate of a parameter of interest is the
average of the point estimates derived independently
from each of the five implicates.
Variance Estimate
The best estimate of variance is the average of the
variance estimates derived independently from each of
the five implicates (“within” imputation variance), plus
an estimate of the “between” imputation variance, with
an adjustment factor for using a finite number of
imputations. The “between” imputation variance is the
sum of the squared deviations of the point estimates in
each implicate from the overall average point estimate (as
described above), divided by the number of implicates
minus one.
Significance Tests
The point estimates and variance estimates derived by
RII techniques can be used to construct confidence
intervals and conduct significance tests. The
adjustments to the test statistics due to the multiple
implicates are described in Appendix A.
Example: Using Repeated-Imputation
Inference (RII) Techniques
To illustrate use of the five implicates in estimating the
amount of uncertainty in estimates due to missing
values, examples using the household liquid assets, total
household income, age of the respondent, and
household size variables in the 1992 SCF are presented.
Estimates of the mean and the variance of the mean for
each variable are computed using the final non-response
adjusted weight provided in the 1992 SCF.c The
multivariate analysis is conducted using ordinary least
squares on the unweighted data.d Results derived
separately from each implicate are compared with
results derived using RII techniques.
Household liquid assets represent the total amount in
checking, savings and money market accounts. The
household liquid asset variable was chosen as the
dependent variable for the multivariate analysis because
it is an interesting financial variable which has not been
extensively studied, and because the percentage of
missing values, and therefore values which are imputed
in the final data set, is moderate. Of households that
had the item, the percentage of cases for which the
amount in the account was missing, either because the
respondent didn’t know or for some other reason, was
11% for checking accounts, 14% for money market
accounts, and 14% for savings accounts (Table 1).
Since we want to illustrate use of RII techniques and to
compare results of RII techniques with results which
ignore variability due to missing values, choice of a
variable with a moderate level of missing values
provides a nice illustration. As the proportion of
missing values increases (decreases), correcting the
variance estimate for variability due to missing values
will produce relatively larger (smaller) increases in the
magnitude of the estimated variance.
Example 1. Estimates of Means and Variances
In the 1992 SCF, 91% of households had liquid assets
and over 99% of households reported a positive value for
total household income (Table 2). A comparison of the
estimates of mean household liquid assets and the
variance of mean household liquid assets derived
separately for each implicate illustrates the variability in
the data due to imputation of missing data. The first
implicate produces relatively large estimates of the mean
($12,089) and standard error of the mean ($1,456)
compared to the other four implicates. When RII
techniques are applied, the best estimate of mean
household liquid assets is $11,898, with a standard error
of $1,344.
Similarly there is variability in the data due to imputation
of missing values for components of total household
income. The fifth implicate produces relatively large
estimates of the mean ($39,221) and standard error of the
mean ($1,316) compared to the other four implicates.
When RII techniques are applied, the best estimate of
mean total household income is $38,914, with a standard
error of $1,294. In contrast, the estimates of the mean
and variance of age of the respondent and household size
are similar across the five separate implicates since little
information on these variables is missing.
Example 2. Regression Model
A linear regression model is used to illustrate the use of
RII techniques in a multivariate framework. The
specification of the model and the estimation procedure
are purposely kept simple in order to focus on the RII
methodology. The dependent variable is the dollar value
of the household’s liquid assets. The independent
variables are selected to analyze the effects of income,
age of the respondent, and household size on household
liquid assets.
In order to reduce heteroskedasticity (unequal variance of
the disturbances), the natural logarithm of annual total
household income (before taxes) is used instead of the
dollar amount.e Linear and quadratic terms for age of the
respondent are included to allow for nonlinear age
effects. Household size is captured with four dummy
variables, one each for household size one, two and
three, and one variable for households with four or more
persons. Interactions terms between income and each of
the age and household size variables are used to allow the
effect of income to vary with age of the respondent and
across households of different sizes. The equation is
estimated by ordinary least squares on the sample of
households with total household income less than or
equal to $100,000 f. Ordinary least squares is chosen
due to the ease of interpretation of the estimated
coefficients.g
There are several consistent results across the five
separate implicates but also some differences (Table 3).
The estimated coefficient on the income term is positive
and statistically significant (p<.05) in all five implicates.
The linear and quadratic age terms are positive and
negative respectively, and statistically significant in all
five implicates. All estimated coefficients on the
household size dummy variables are positive. The
coefficients on the variables for two person households
and four person households are statistically significant
in all but the second implicate. The coefficient on the
variable for three person households is statistically
significant only in the fifth implicate. The interaction
term between income and age has a negative and
statistically significant coefficient in all five implicates;
and the interaction term between income and age
squared has a positive and statistically significant
coefficient in all five implicates. The interaction terms
between income and the three household size variables
all have negative coefficients. The coefficients on the
interaction terms for two person households and four
person households are statistically significant in all but
the second implicate. The coefficient on the interaction
term for three persons households is statistically
significant only in the third, fourth and fifth implicates.
There is also variability across the five separate
implicates in the magnitude of the estimated coefficients
and standard errors. For example, the estimated
coefficient for the income term ranges from 41,174 in
the fourth implicate to 56,771 in the third implicate.
The estimated standard error of the income coefficient
ranges from 8,296 in the fourth implicate to 18,900 in
the second implicate.
Estimates derived using RII techniques use the
information from all five implicates and incorporate
imputation variability. Results which were consistent
across the five separate implicates are confirmed in the
RII results, although the levels of significance are
generally more stringent. For example, the coefficient
on the income term which is positive and significant in
each of the five implicates, is also positive and
significant (p<.01) in the RII results. Similar RII results
are obtained for age, age squared, and the interaction of
the income and age terms. It is informative to examine
variables for which results from the five separate
implicates were not consistent. For example, the
coefficient on the dummy variable for two person
households is positive and significant in four of the five
implicates, and is positive and significant (p=.04) in the
RII results. Similar RII results are obtained for four
person households and the interaction terms between
income and household size two and income and
household size four. The coefficient on the dummy
variable for three person households is positive, but
significant only in one implicate, and is not significant in
the RII results (p=.24). The interaction term between
income and household size three is negative and is
significant in three implicates, but is not significant in the
RII results (p=.18). These results illustrate the risk of
basing inference on results of a single implicate. Based
on the fifth implicate alone, all eleven independent
variables are statistically significant. Based on the
second implicate alone, only six of the eleven
independent variables are statistically significant. The
best estimates are generated through RII techniques
which result in nine of the eleven independent variables
being significant predictors of household liquid assets.
These examples illustrate the quantitative differences in
estimates derived independently from each of the five
implicates in the 1992 SCF. Due to these differences,
inference based on analysis of only a single implicate of
the 1992 SCF risk misrepresenting the magnitude,
variability and statistical significance of parameters of
interest. Further, the best point estimates and estimates
of variance for parameters of interest are generated
through RII techniques which use information from all
five implicates and incorporate information on the
variability due to missing values.
As an example of the combined impact of the differences
in estimates, Figure 1 shows the predicted level of liquid
assets for one person households with a mean income
level ($38,914) for ages ranging from 20 to 80. (The
predicted levels are high because this analysis is
unweighted, so they are not representative of the U.S.
population.) The difference between Implicate 1 and
Implicate 2 in predicted levels of liquid assets is $7,797
at age 20 and $13,135 at age 80. The RII estimates are
generally between the extreme values predicted from the
Implicate estimates. Clearly, when predictions are
desired, the RII method should be used.
Table 2. Estimates of the Mean and the Variance of the Mean Derived Separately for Each Implicate and Using RII
Techniques.
1992 SCF, Full Sample, Weighted.
|
Descriptive Statistic |
Implicate |
RII
|
||||
| First | Second | Third | Fourth | Fifth | ||
| Household liquid assets
Mean
Std. Error of the Mean
Variance of the Mean
Number of nonzero observations
Percent of nonzero observations
Total household income Mean
Std. Error of the Mean
Variance of the Mean
Number of nonzero observations
Percent of nonzero observations
Age of the respondent
Mean
Std. Error of the Mean
Variance of the Mean
Number of nonzero observations
Percent of nonzero observations
Household size
Mean
Std. Error of the Mean
Variance of the Mean
Number of nonzero observations
Percent of nonzero observations |
12,088.57
1,456.23
2.121E+6
3,539
90.6
38,827.96
1,268.74
1.610E+6
3,889
99.6
48.4642
0.2785
0.0776
3900
99.8
2.6112
0.0239
0.0006
3,906
100.0 |
11,921.83
1,419.11
2.014E+6
3,536
90.5
38,836.12
1,284.32
1.649E+6
3,889
99.6
48.4650
0.2785
0.0776
3900
99.8
2.6163
0.0240
0.0006
3,906
100.0 |
11,929.12
1,330.72
1.771E+6
3,538
90.6
38,803.55
1,281.30
1.642E+6
3,889
99.6
48.4629
0.2783
0.0775
3900
99.8
2.6134
0.0240
0.0006
3,906
100.0 |
11,519.49
1,226.16
1.503E+6
3,538
90.6
38,883.52
1.247.48
1.556E+6
3,889
99.6
48.4577
0.2784
0.0775
3900
99.8
2.6142
0.0240
0.0006
3,906
100.0 |
12,030.49
1,153.47
1.330E+6
3,539
90.6
39,220.57
1,316.05
1.732E+6
3,889
99.6
48.4589
0.2784
0.0775
3900
99.8
2.6154
0.0240
0.0006
3,906
100.0 |
11,897.90
1,344.42
1.807E+6
38,914.35
1,293.83
1.674E+6
48.4618
0.2784
0.0775
2.6141
0.0241
0.0006 |
Seventeen households in each implicate had negative values for total household income. These negative values were set equal to zero in the calculation
of the mean and variance.
Table 3. Regression Model Separately for Each Implicate and Derived Using RII Techniques. Dependent Variable:
Household Liquid Assets. 1992 SCF, Sample of Households with Total Household Income Less Than or Equal to
$100,000, Unweighted.
|
Coefficient |
Std. Error |
Variance |
t-statistic |
p-value |
R2
Model F
p-value |
|
| First implicate
Intercept
Ln(Income)
Age
Age squared
Household size 2
Household size 3
Household size 4
Ln(inc)*age
Ln(inc)*age sq
Ln(inc)*size 2
Ln(inc)*size 3
Ln(inc)*size 4 |
-533280
54305
25285
-297.72
99998
51659
150020
-2546.4
30.65
-10807
-6065.7
-14918 |
99650
10220
4206
42.75
37630
39450
41120
428.4
4.35
3786
3952
4090 |
9.930E+9
1.045E+8
1.769E+7
1827.37
1.416E+9
1.557E+9
1.691E+9
183517
18.90
1.434E+7
1.561E+7
1.672E+7 |
5.351
5.313
6.011
6.965
2.658
1.309
3.648
5.944
7.050
2.854
1.535
3.648 |
0.0000
0.0000
0.0000
0.0000
0.0079
0.1904
0.0003
0.0000
0.0000
0.0043
0.1248
0.0003 |
0.0510
13.9985
0.0000 |
|
Coefficient |
Std. Error |
Variance |
t-statistic |
p-value |
R2 Model F
p-value |
|
| Second implicate
Intercept
Ln(Income)
Age
Age squared
Household size 2
Household size 3
Household size 4
Ln(inc)*age
Ln(inc)*age sq
Ln(inc)*size 2
Ln(inc)*size 3
Ln(inc)*size 4 |
-437180
43413
21094
-249.09
78733
37950
92180
-2081.4
25.28
-8263.7
-4413.5
-8311.1 |
183900
18900
7773
78.97
69400
72670
75520
793.7
8.05
6981
7279
7516 |
3.38E+10
3.572E+8
6.041E+7
6235.67
4.816E+9
5.281E+9
5.703E+9
629904
64.84
4.874E+7
5.298E+7
5.649E+7 |
2.377
2.297
2.714
3.154
1.135
0.522
1.221
2.622
3.140
1.184
0.606
1.106 |
0.0175
0.0216
0.0067
0.0016
0.2566
0.6015
0.2222
0.0087
0.0017
0.2365
0.5443
0.2688 |
0.0113
2.9671
0.0007 |
| Third implicate
Intercept
Ln(Income)
Age
Age squared
Household size 2
Household size 3
Household size 4
Ln(inc)*age
Ln(inc)*age sq
Ln(inc)*size 2
Ln(inc)*size 3
Ln(inc)*size 4 |
-545390
56771
25398
-296.10
94017
62545
167650
-2613.2
31.02
-10120
-7205.7
-16911 |
92380
9489
3913
39.94
34950
36710
38100
399.1
4.07
3519
3676
3792 |
8.535E+9
9.004E+7
1.531E+7
1594.88
1.222E+9
1.347E+9
1.452E+9
159286
16.55
1.238E+7
1.351E+7
1.438E+7 |
5.904
5.983
6.490
7.414
2.690
1.704
4.400
6.547
7.626
2.876
1.960
4.459 |
0.0000
0.0000
0.0000
0.0000
0.0072
0.0884
0.0000
0.0000
0.0000
0.0040
0.0500
0.0000 |
0.0593
16.4134
0.0000 |
| Fourth implicate
Intercept
Ln(Income)
Age
Age squared
Household size 2
Household size 3
Household size 4
Ln(inc)*age
Ln(inc)*age sq
Ln(inc)*size 2
Ln(inc)*size 3
Ln(inc)*size 4 |
-411510
41176
18318
-210.70
94689
60333
151770
-1801.3
21.27
-9878.0
-6853.0
-15248 |
80810
8296
3405
34.58
30410
31900
33190
347.2
3.52
3060
3195
3304 |
6.531E+9
6.882E+7
1.159E+7
1195.85
9.247E+8
1.018E+9
1.102E+9
120567
12.40
9.361E+6
1.021E+7
1.091E+7 |
5.092
4.963
5.379
6.093
3.114
1.891
4.572
5.188
6.041
3.229
2.145
4.615 |
0.0000
0.0000
0.0000
0.0000
0.0018
0.0586
0.0000
0.0000
0.0000
0.0012
0.0319
0.0000 |
0.0442
12.0380
0.0000 |
| Fifth implicate
Intercept
Ln(Income)
Age
Age squared
Household size 2
Household size 3
Household size 4
Ln(inc)*age
Ln(inc)*age sq
Ln(inc)*size 2
Ln(inc)*size 3
Ln(inc)*size 4 |
-495040
51261
21488
-252.81
133240
92993
189150
-2186.2
26.33
-14493
-10550
-19416 |
102400
10500
4355
44.55
38730
40560
42110
443.8
4.54
3895
4064
4190 |
1.05E+10
1.102E+8
1.896E+7
1985.06
1.500E+9
1.645E+9
1.773E+9
196990
20.57
1.517E+7
1.651E+7
1.756E+7 |
4.836
4.882
4.934
5.674
3.440
2.293
4.492
4.926
5.805
3.721
2.596
4.634 |
0.0000
0.0000
0.0000
0.0000
0.0006
0.0218
0.0000
0.0000
0.0000
0.0002
0.0094
0.0000 |
0.0461
12.5912
0.0000 |
|
Coefficient |
Std. Error |
Variance |
t-statistic |
p-value |
R2 Model F
p-value |
|
| RII techniques
Intercept
Ln(Income)
Age
Age squared
Household size 2
Household size 3
Household size 4
Ln(inc)*age
Ln(inc)*age sq
Ln(inc)*size 2
Ln(inc)*size 3
Ln(inc)*size 4 |
-484479
49385.2
22316.4
-261.28
100136
61096.1
150155
-2245.68
26.91
-10712.4
-7017.55
-14960.9 |
134129
14204.3
5978.19
64.52
49618.6
51605.2
62460.6
627.78
6.81
5138.29
5275.75
6603.03 |
1.80E+10
2.018E+8
3.574E+7
4163.40
2.462E+9
2.663E+9
3.901E+9
394107
46.34
2.640E+7
2.783E+7
4.360E+7 |
3.612
3.477
3.733
4.049
2.018
1.184
2.404
3.577
3.953
2.085
1.330
2.266 |
0.0005
0.0010
0.0006
0.0004
0.0462
0.2389
0.0239
0.0011
0.0007
0.0408
0.1871
0.0358 |
—-
5.5717
0.0000 |
N=2875
Figure 1
Illustration of Difference Between Implicates 1 and 2,
and RII Method
Assuming household size=1. Based on regression estimates shown in
Table 3.
Summary
The 1992 Survey of Consumer Finances contains five
complete data sets, instead of the more common single
data set, due to the use of multiple imputation techniques
to handle missing data. When imputation techniques are
used to fill in missing data, there will inherently be extra
variability in the data due to the missing values. This
variability needs to be incorporated into empirical
estimates. “Repeated-imputation inference” (RII)
techniques can be used to estimate this variability. Point
estimates and estimates of variance derived by RII
techniques provide a basis for more valid inference and
tests of significance. Inference based on results from a
single implicate ignores the extra variability due to
missing values and risks misrepresenting the precision of
estimates and significance of relationships.
In general, adjusting variance estimates for imputation
variability will increase the estimate of variance
compared to estimates which ignore this variability. The
magnitude of this change is an empirical question — it
could make a large difference in some situations; an
unnoticeable change in others. When the proportion of
cases with imputed values is high, as for business assets
and stocks in the 1992 SCF (Table 1), incorporating
information on variability due to imputation is likely to
have large effects relative to when only a moderate
proportion (i.e. liquid assets) or small proportion (i.e.
credit card balance) of cases have imputed values.
The use of RII techniques in analyzing the 1992 SCF is
straightforward, and with the use of computers does little
to complicate the estimation. Researchers should use RII
techniques in order to produce estimates which
incorporate the variability in the data due to missing
values. Only then can the practical significance (i.e. how
much does it matter) of incorporating this imputation
variability be evaluated. Future research should carefully
study and document the practical importance of
imputation variability. The implications of high
proportions of imputed values for variables of interest,
both as dependent and independent variables in
multivariate frameworks, need to be better understood.
a. The public use tapes of the 1992 SCF were released during Spring
1996, so empirical research using this data set is just beginning to
be published. However, numerous studies using the 1989 SCF,
which contains five complete data sets produced by multiple
imputation, have been published. A review of some of this empirical
research revealed two studies which used information from all five
data sets for all analysis (Choi & DeVaney, 1995; Kao, 1995), and
two studies which used information from all five data sets for
descriptive statistics or bivariate analyses, but not in multivariate
analyses (DeVaney, 1995c; Xiao, 1995b). Nine studies clearly
stated that analyses were performed on the first data set only
(DeVaney, 1995a, 1995b; Hong & Swanson, 1995; Kao, 1994;
Malroutu & Xiao, 1995a, 1995b; McGurr, 1995; Xiao, 1995a;
Zhong & Xiao, 1995). In eleven studies, the treatment of the
multiple data sets was not clearly specified (Drollinger & Johnson,
1995; Hatcher, 1995; Hong & Yu, 1995; Kokrda & Cramer, 1995;
Liao, 1994; Steidle, 1994; Xiao, Malroutu & Olson, 1995; Yieh &
Widdows, 1995; Yu & Kao, 1994; Zhong, 1994; Zhou,1995).
b. In a stochastic imputation method the estimating equation includes
a non-zero residual term. This residual term captures the residual
(within-class) variance, thus producing better estimates of the
standard deviation and the distribution of variables of interest.
c. Descriptive statistics are computed using the SCF weight variable
in order to produce unbiased estimates of means and variances
generalizable to the population of U.S. households. Variance
estimates are corrected for imputation variability using the
described RII technique. However, the variance estimates are not
corrected for variability due to the complex sample design (i.e.
sampling variability). Kennickell, McManus & Woodburn (1996)
show that imputation variability is small (though not unimportant)
relative to sampling variability for their estimates of concentration
ratios. The 1992 SCF contains information (i.e. bootstrap
replicates) which can be used to correct variance estimates for
sampling variability. Variance estimates which are corrected for
both imputation variability and sampling variability provide the
best basis for valid inference. Montalto and Sung (1996) discuss
use of the bootstrap replicates to estimate variability due to
sampling and present empirical results for selected variables in the
1992 SCF.
d. The regressions reported in this paper are unweighted and the
standard errors of the estimated coefficients are corrected for
imputation variability using the described RII techniques. This
method produces unbaised parameter estimates and valid estimates
of the standard errors for testing the statistical significance of
individual independent variables. In contrast, when weighted OLS
is used, the parameter estimates will be unbiased, but weighted
OLS will not correct the standard errors for the complex sample
design. Therefore, inference based on weighted OLS results will
not be valid. Statistical packages specifically designed for analysis
of complex survey data are available, for example SUDAAN®.
These programs compute robust variance estimates which fully
account for unequal weighting and stratification.
e. Disturbance terms are heteroskedastic when they have different
variances. Heteroskedasticity usually arises in cross-section data
where the scale of the dependent variable and the explanatory
power of the model vary across observations. In our example, there
is likely to be greater variation in the level of household liquid
assets among high-income households than low-income households
due to the greater discretion allowed by higher income. In
response, the income variable is transformed to the natural log
value in an attempt to make the variance more homogeneous on the
transformed scale.
f. Households with total household income greater than $100,000 are
omitted from our multivariate analysis to eliminate the influence of
these households which were over sampled in the SCF.
g. In the 1992 SCF, 9.4% of the households reported a value of zero
for household liquid assets. When the dependent variable is
truncated at zero for some observations, parameters estimated by
ordinary least squares are biased and inconsistent. The bias in the
OLS coefficient estimates can be corrected by dividing each
estimated coefficient by the proportion of the sample observations
which are not truncated at zero (Greene, 1981). Therefore, the
coefficients reported in Table 3 for each of the five implicates
would be divided by 90.6 to correct for the bias. Greene (1981)
also provides the appropriate correction for the other OLS
structural parameters, including the intercept and the standard
errors.
Repeated-Imputation Inference
To analyze any SCF since the 1989 survey taking advantage of the
information provided by the multiple complete data sets, separate
analyses are performed on each of the five complete data sets
(implicates) and the results are combined. Combining the results
requires the calculation of the means of the results from the five
separate implicates, plus the calculation of the “between” imputation
variance, as described below. (Notation closely follows Rubin, 1987.)
The quantity of interest, Q, may be a scalar or a vector, and may
represent simple descriptive statistics, such as means, proportions or
totals, or more complicated estimators, such as regression coefficients.
Let Q1, Q2, Q3, Q4, and Q5 represent the point estimates and U1, U2, U3,
U4, and U5 represent the variance estimates from the first, second, third,
fourth and fifth implicates, respectively. The best point estimate of the
variable of interest is the average of the five separate point estimates
Since there are five implicates in the 1992 SCF, m equals five. For
simple statistics, like means, the average of the mean of each of the five
implicates is the same as the mean calculated over all five implicates
(N=19,530).
To estimate the total variance of the point estimate it is necessary to
calculate both the average of the five separate variance estimates (the
“within” imputation variance), and the variance due to imputation of
missing values (the “between” imputation variance). The average
“within” imputation variance is estimated by
The “between” imputation variance is estimated by
where t indicates the transpose when Q is a vector.
Thus the total variance of the point estimate is given by
and the standard deviation of the point estimate, defined as the square
root of the variance, is given by
The total variance is the sum of the average “within” imputation
variance, and the “between” imputation variance weighted by an
adjustment factor for using a finite number of imputations. The
adjustment factor is inversely related to the number of implicates. As
the number of implicates increases, the adjustment factor decreases in
size, thus reducing the relative importance of the “between” imputation
variance in the estimate of total variance. For example, the adjustment
factor is 1.5 when there are two implicates, 1.2 when there are five
implicates, and 1.01 when there are 100 implicates.
Example A1. Estimates of Descriptive Statistics for Scalar Q
Table A1 summarizes the use of RII techniques to derive estimates of
the mean, variance and standard deviation of the household liquid
assets, total household income, age and household size variables from
the 1992 SCF. The RII techniques average the results from separate
analyses on each of the five implicates, and correct the variance
estimate for the uncertainty due to missing values. (The results of the
separate analysis on each of the five implicates are summarized in
Table 2).
Statistical Inference
When the quantity of interest, Q, represents estimated parameters, such
as regression coefficients, we are often interested in conducting
hypothesis tests and interval estimates of single parameters, as well as
joint hypothesis tests on sets of parameters (i.e. vectors).
Confidence intervals for scalar Q For a single parameter, the point
estimate defined in equation 1 and the estimate of the standard
deviation (i.e. the standard error) defined in equation 5 can be used to
construct the standard confidence interval
where tv(/2) is the upper 100 /2 percentage point of the student t
distribution with v degrees of freedom. In the case of repeated-imputation inferences, the degrees of freedom is given by
where rm is the relative increase in variance due to nonresponse
The statistic, rm, is the ratio of the “between” imputation variance to the
average “within” imputation variance, with an adjustment factor for
using a finite number of imputations. Notice that as the relative
increase in variance due to nonresponse becomes larger, the degrees of
freedom become smaller (i.e. rm and v are inversely related), resulting
in more conservative significance levels. The statistic, rm, is positively
related to the fraction of information about the parameter, Q, which is
missing
Thus, as the fraction of information about a parameter of interest that
is missing increases, the relative increase in variance due to
nonresponse will increase, reducing the degrees of freedom and the
level of significance of tested relationships.
Table A1. Scalar Q: Estimates of the Mean and the Variance of the Mean Derived From “Repeated-Imputation Inference”
Techniques; 1992 SCF, Full Sample, Weighted.
| Descriptive
Statistic |
Explanation | Formula/Notation | Numeric Result | |||
| Household
Liquid Assets |
Total
Household Income |
Household
|
Age of
|
|||
| Point
estimate |
average of the five
separate point estimates |
 |
11,897.90 | 38,914.35 | 2.6141 | 48.4618 |
| Average
“within” imputation variance |
average of the five
separate variance estimates |
 |
1.74785E+6 | 1.63782E+6 | 0.0006 | 0.0775 |
| “Between”
imputation variance |
sample variance in
the estimates from the five implicates |
 |
49,668.50 | 30,144.45 | 3.8805E-6 | 1.05E-5 |
| Descriptive
Statistic |
Explanation | Formula/Notation | Numeric Result | |||
| Household
Liquid Assets |
Total
Household Income |
Household
|
Age of
|
|||
| Total
variance |
sum of the average
“within” and “between” variance weighted by an adjustment factor |
 |
1.80745E+6 | 1.67399E+6 | 0.0006 | 0.0775 |
| Std. error of
the mean |
square root of the
variance |
 |
1,344.41 | 1,293.83 | 0.0241 | 0.2784 |
Testing the null hypothesis for scalar Q The test statistic for the null
hypothesis that the point estimate of the scalar Q equals some value,
Qo, is given by
which has a t distribution with v degrees of freedom, with v defined in
equation 7. When we are interested in whether a parameter, Q, is
significantly different from zero, a common test for estimated
parameters in a regression model, the test statistic simplifies to
Single linear constraints can also be tested with an F statistic given by
which has an F distribution with one and v degrees of freedom, with v
defined in equation 7. The resulting F statistic will be the square of the
t statistic defined in equation 10. This reflects the general result that
the square of a t statistic is an F statistic with degrees of freedom one
and the degrees of freedom for the t test. When we are interested in
testing whether a single coefficient is equal to a specific value, it is
usually easier to perform a t test than an F test.
Testing the null hypothesis for a k-dimensional vector Q In many
situations we are interested in a set of parameters rather than an
individual scalar, and Q becomes a k-dimensional vector. For example,
in a regression model we are interested in the overall significance of the
regression equation. This is a joint test of the hypothesis that all of the
coefficients except the intercept term are zero. The test statistic for a
joint hypothesis when the number of implicates is modest relative to the
number of variables (according to Rubin, 1987, when m < 5k), is given
by
which has an F distribution with k and (k + 1)v/2 degrees of freedom;
v is given in equation 7 with rm generalized to be the average relative
increase in variance due to nonresponse
where Tr(A) is the sum of the diagonal elements in the k x k matrix A.
A test statistic for a joint hypothesis test on a set of parameters can also
be computed from the m complete-data 2 statistics and the value of rm
as defined in equation 14. This test statistic is asymptotically
equivalent to the test statistic in equation 13 (Rubin, 1987, pp. 78-79).
This test statistic is given by
which has an F distribution with k and (k + 1)v/2 degrees of freedom,
with v defined in equation 7. The test statistic in equation 15 depends
on the scalar 2 statistics and the scalar rm and is therefore more easily
computed than the test statistic in equation 13 which depends on k x k
matrices. Nonlinear models often generate 2 statistics for testing the
joint hypothesis that all of the coefficients except the intercept term are
zero.
Example A2. Inference for regression coefficients
A linear regression model is used to illustrate the use of RII techniques
in a multivariate framework. Household liquid assets is regressed on
an intercept term, and eleven independent variables. The regression
model is estimated separately on each of the five implicates. (These
regression results are provided in Table 3).
The quantity of interest, Q, is now a k-dimensional vector (k=12), and
the variance estimates are now kxk variance-covariance matrices. The
relevant statistics for total household income and age of the respondent
are the second and third elements in the vector Q and the second and
third diagonal elements in the variance-covariance matrices. In the
following example, numeric results are shown only for these elements
to simplify the illustration.
Equation 1, with Qi a 1xk vector, is used to calculate the average of the
results from the five implicates which produces the best point estimates
of the structural parameters
Equation 3 is used to calculate the variance-covariance matrix
Equation 2 is used to calculate the average of the variance-covariance
matrices from the five implicates
Equation 4 is used to calculate the total variance
The relative increase in variance due to nonresponse is, from equation
8, 0.38 and 0.44 for the estimated coefficients on total household
income and age of the respondent, respectively, implying degrees of
freedom, from equation 7, of 53 and 43, respectively.
The test statistic in equation 11 can be used to test whether each
estimated coefficient is significantly different from zero. For total
household income, the t-statistic is 3.477; with 53 degrees of freedom,
the p-value is 0.0010. Similarly, the t-statistic for age of the respondent
is 3.733; with 43 degrees of freedom, the p-value is 0.0006.
From equation 9 the fraction of missing information for total household
income is
and for age of the respondent is
Equation 13 can be used to test the overall significance of the
regression equation. Since this is a joint test of the hypothesis that the
coefficients on the eleven independent variables are all equal to zero,
information on the intercept term is irrelevant and Q is an 11-dimensional vector and U is a 11×11 matrix. For this joint hypothesis
test, rm is generalized to be the average relative increase in variance due
to nonresponse which from equation 14 equals 0.4781. The test
statistic equals 5.5717 and has an F distribution with k = 11 and
(k+1)v/2 = 230 degrees of freedom. The test statistic is large, and the
associated p-value is 0.0000.
SAS code for an example of using the techniques described in this
article will be available at the Financial Counseling and Planning
Home Page on the Worldwide Web, at:
http://hec.osu.edu/people/shanna/index.htm
Look for the 1996 issue link.
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1. Catherine Phillips Montalto, Assistant Professor, Consumer Sciences Department, The Ohio State University, Columbus, OH 43210-1295.
Phone: (614) 292-4571. Fax: (614) 292-7536. E-mail: montalto.2@osu.edu.
2. Jaimie Sung, Graduate Student, Consumer Sciences Department, The Ohio State University, Columbus, OH 43210-1295. Phone: (614)
292-4590. Fax: (614) 292-7536. E-mail: sung.13@osu.edu.