How can I combine results other than coefficients in e(b) with multiply imputed data?
| Title |
|
Combining results other than coefficients in e(b) with multiply imputed data |
| Authors |
Isabel Cañette and Yulia Marchenko, StataCorp |
| Date |
October 2010; minor revisions July 2011 |
mi
estimate estimates parameters from multiply imputed data and adjusts
these estimates and their respective standard errors for the imputation
uncertainty using Rubin’s combination rules. mi estimate is designed
to work with Stata estimation commands. As such, it combines the estimates of
coefficients, which are stored in matrix e(b), and their respective
variance–covariance estimates (VCE), stored in matrix e(V). You may
also want to combine results other than e(b). The aim of this
tutorial is to show you how to do that.
This entry is organized as follows:
Finding out where your results are saved
First, you need to know where your results are saved. Most Stata commands
save results either in e() or r(). You can type
. ereturn list
or
. return list
to find out where your value of interest is saved after running a command.
Alternatively, you can look at the Saved results section in the help file or
documentation for the command.
Combining point estimates only
In some situations, we are interested in a point estimate, such as the R-squared
(R2) value from a regression, not in its variance. According to
Rubin's rules, the estimate of the value of interest should be computed for
each imputation, and the overall value will be the mean of these estimates.
We can do this manually, taking advantage of
mi xeq,
which allows you to run sequences of commands of interest on each
individual imputation.
In the following example, results are combined for the
R2 value from a linear regression. We use this example
strictly for the purpose of illustration. See the unofficial command
mibeta (type . findit mibeta to locate and install this
command), which automatically provides R2, adjusted R2, and
standardized coefficients after regression for imputed data.
We will use the dataset from the example of house resale prices in [MI]
mi estimate. Our first step is to identify where our result of interest
is stored. The regress command is an e-class command and it
stores R2 in e(r2):
. webuse mhouses1993s30
(Albuquerque Home Prices Feb15-Apr30, 1993)
. mi xeq 0: regress price tax sqft age nfeatures ne custom corner
m=0 data:
-> regress price tax sqft age nfeatures ne custom corner
Source | SS df MS Number of obs = 66
-------------+------------------------------ F( 7, 58) = 51.86
Model | 9164658.61 7 1309236.94 Prob > F = 0.0000
Residual | 1464105.15 58 25243.1922 R-squared = 0.8623
-------------+------------------------------ Adj R-squared = 0.8456
Total | 10628763.8 65 163519.442 Root MSE = 158.88
------------------------------------------------------------------------------
price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
tax | .4988701 .1584853 3.15 0.003 .1816273 .8161128
sqft | .3522184 .0957476 3.68 0.001 .1605588 .5438779
age | -.5650817 2.002529 -0.28 0.779 -4.57358 3.443416
nfeatures | 4.389607 18.55499 0.24 0.814 -32.75223 41.53145
ne | -17.38534 47.27462 -0.37 0.714 -112.0158 77.2451
custom | 174.9411 53.72371 3.26 0.002 67.40139 282.4808
corner | -73.58234 49.13007 -1.50 0.140 -171.9269 24.76218
_cons | 92.7448 101.607 0.91 0.365 -110.6438 296.1334
------------------------------------------------------------------------------
. ereturn list
scalars:
e(N) = 66
e(df_m) = 7
e(df_r) = 58
e(F) = 51.86495180785406
e(r2) = .8622506644760125
(output omitted)
We used mi xeq 0: in the command line above to execute the regress command
on the original data m=0.
Now we want to compute the average of the R2 values. To do so, we
can use mi xeq: to run regressions on each imputed dataset and
collect individual R2 values:
. mi query
data mi set mlong, M = 30
. local M = r(M)
. scalar r2 = 0
. mi xeq 1/`M': regress price tax sqft age nfeatures ne custom corner; scalar r2 = r2 + e(r2)
m=1 data:
-> regress price tax sqft age nfeatures ne custom corner
Source | SS df MS Number of obs = 117
-------------+------------------------------ F( 7, 109) = 74.79
Model | 13895850.5 7 1985121.5 Prob > F = 0.0000
Residual | 2893096.28 109 26542.1677 R-squared = 0.8277
-------------+------------------------------ Adj R-squared = 0.8166
Total | 16788946.8 116 144732.3 Root MSE = 162.92
------------------------------------------------------------------------------
price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
tax | .676841 .1140902 5.93 0.000 .450718 .9029641
sqft | .2091065 .065971 3.17 0.002 .0783542 .3398589
age | -.2350972 1.472105 -0.16 0.873 -3.152762 2.682568
nfeatures | 4.17071 12.572 0.33 0.741 -20.74658 29.088
ne | 14.4082 33.3729 0.43 0.667 -51.73581 80.55221
custom | 137.1073 41.98151 3.27 0.001 53.90127 220.3132
corner | -75.24024 39.1843 -1.92 0.057 -152.9023 2.421776
_cons | 149.6721 66.01572 2.27 0.025 18.83105 280.5131
------------------------------------------------------------------------------
-> scalar r2 = r2 + e(r2)
...
(output omitted)
m=30 data:
-> regress price tax sqft age nfeatures ne custom corner
Source | SS df MS Number of obs = 117
-------------+------------------------------ F( 7, 109) = 71.60
Model | 13790079.1 7 1970011.3 Prob > F = 0.0000
Residual | 2998867.71 109 27512.5478 R-squared = 0.8214
-------------+------------------------------ Adj R-squared = 0.8099
Total | 16788946.8 116 144732.3 Root MSE = 165.87
------------------------------------------------------------------------------
price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
tax | .6938792 .1261562 5.50 0.000 .4438417 .9439168
sqft | .2044596 .0710267 2.88 0.005 .0636869 .3452322
age | .3372971 1.591275 0.21 0.833 -2.816558 3.491152
nfeatures | 6.940631 13.04224 0.53 0.596 -18.90865 32.78992
ne | 7.280361 36.79407 0.20 0.844 -65.6443 80.20502
custom | 132.975 43.24393 3.07 0.003 47.26696 218.6831
corner | -70.26249 39.74756 -1.77 0.080 -149.0409 8.515876
_cons | 128.2627 67.20399 1.91 0.059 -4.933393 261.4588
------------------------------------------------------------------------------
-> scalar r2 = r2 + e(r2)
m=30 data:
. scalar r2 = r2/`M'
. di as txt "R2 over imputed data = " as res r2
R2 over imputed data = .82300853
We used mi query above to identify the number of imputations and to store it
in the local macro M. Then we used mi xeq 1/`M': to run
regressions on the imputed data and accumulate the R2 values in the
r2 scalar. We used a semicolon to separate the two commands
we wished to execute on each imputed datum; see [MI] mi xeq for
details. Finally, to obtain the average, we divided r2 by the number
of imputations.
We recommend applying Rubin’s combination rules to
parameters in a metric for which the asymptotic normal approximation works
well. Fisher’s z, or inverse hyperbolic tangent [atanh()],
transformation is often recommended for the correlation coefficient R to
improve its asymptotic normality. Thus we can combine the R2
values in the transformed metric and then use the inverse transformation,
tanh(), to switch back to the original metric.
. local M = 30
. scalar r2 = 0
. qui mi xeq 1/`M': regress price tax sqft age nfeatures ne custom corner; scalar r2 = r2 + atanh(sqrt(e(r2)))
. scalar r2 = tanh(r2/`M')^2
. di as txt "R2 using Fisher's z over imputed data = " as res r2
R2 using Fisher's z over imputed data = .82306058
The two estimates of the R2 are very similar.
Combining point estimates and their variances
If we want to combine point estimates and their variances, we can
use mi estimate. However, we need to create an e-class program
(see program)
that saves the necessary results where mi estimate expects to see
them. The general guidelines are described in
program properties.
Below we demonstrate how to use mi estimate to combine results from
r-class command
roctab using
the heart attack example from [MI] mi estimate. The roctab
command posts the estimate of the area under the curve (AUC) in
r(area) and its standard error in r(se):
. webuse mheart1s20
(Fictional heart attack data; bmi missing)
. qui mi passive: generate high_bmi = (bmi>30) if bmi<.
. mi xeq 0: roctab attack high_bmi
-> roctab attack high_bmi
ROC -Asymptotic Normal--
Obs Area Std. Err. [95% Conf. Interval]
--------------------------------------------------------
132 0.5452 0.0322 0.48208 0.60833
. return list
scalars:
r(N) = 132
r(area) = .5452035889029503
r(se) = .0322055279124565
r(lb) = .4820819140914361
r(ub) = .6083252637144645
In the above series of commands, we created a new variable indicating high
body mass index (BMI) values,
high_bmi, which we use as a classification variable in the
receiver operating characteristic (ROC)
analysis of heart attacks. Because bmi is the imputed variable,
high_bmi (being a function of bmi) is a passive variable. As
such, we used mi passive: generate to create the passive variable
high_bmi; see
mi register
and mi passive
for details.
Following the guidelines in
program properties,
we need to create an e-class program that stores the AUC estimate and
its standard error in e(b) and e(V). We also need to store
other results, such as the name of the command in e(cmd), the number
of observations in e(N), and the title of the command in
e(title).
In the following code, the e-class program eroctab is defined.
eroctab calls roctab and posts necessary results to
e(); see ereturn
for details. This program accepts two arguments: the name of the reference
variable (stored in the local macro refvar) and the name of the
classification variable (stored in the local macro classvar). In step
1, we perform ROC analysis of variables in refvar and classvar
using roctab. In step 2, we save the estimates of the area and its
variance in temporary matrices b and V, respectively. Then
in step 3, we label columns of coefficient matrix `b' and rows and
columns of the VCE matrix `V' consistently, as required by ereturn
post. Finally, in step 4, we post coefficient and VCE matrices as well
as other results to e().
cap program drop eroctab
program eroctab, eclass
version 12.0
/* Step 1: perform ROC analysis */
args refvar classvar
roctab `refvar' `classvar'
/* Step 2: save estimate and its variance in temporary matrices*/
tempname b V
mat `b' = r(area)
mat `V' = r(se)^2
local N = r(N)
/* Step 3: make column names and row names consistent*/
mat colnames `b' = AUC
mat colnames `V' = AUC
mat rownames `V' = AUC
/*Step 4: post results to e()*/
ereturn post `b' `V', obs(`N')
ereturn local cmd "eroctab"
ereturn local title "ROC area"
end
Inference about the AUC estimate in roctab is based on the
large-sample normal approximation. Sometimes commands, for example regress,
adjust for small samples and use Student’s t distribution for inference. In
such cases, the corresponding (residual or denominator) degrees of freedom
must be posted to e(df_r). This can be done by specifying the
dof() option with ereturn post.
Now we can use eroctab with the mi estimate: prefix to obtain
multiple-imputation estimates of the ROC area:
. mi estimate, cmdok: eroctab attack high_bmi
Multiple-imputation estimates Imputations = 20
ROC area Number of obs = 154
Average RVI = 0.0753
Largest FMI = 0.0705
DF adjustment: Large sample DF: min = 3873.11
avg = 3873.11
max = 3873.11
------------------------------------------------------------------------------
| Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
AUC | .5450938 .0312929 17.42 0.000 .4837416 .6064459
------------------------------------------------------------------------------
In the command line above, we used the cmdok option with mi estimate because
eroctab is not one of the
estimation
commands officially supported by mi estimate. Alternatively, we
could have included the mi property in the definition of the program
eroctab to use it with mi estimate directly:
cap program drop eroctab
program eroctab, eclass properties(mi)
...
end
. mi estimate: eroctab attack high_bmi
An example using suest
Another situation when you need to define your own program is when your
results are not obtained via a single command, but by using a sequence of
commands such as, for example, with
suest.
Below we demonstrate an example in which we want to compare the effects of
smoking on heart attacks from two logistic models: one adjusting for both
age and BMI and the other adjusting only for BMI.
First, we create a new program, mysuest, which fits two models of
interest and then uses suest to combine the two estimation results.
The models of interest are passed as two arguments and are stored in local
macros model1 and model2. To avoid specifying the
cmdok option with mi estimate, we add properties(mi)
to the definition of the mysuest program.
cap program drop mysuest
program mysuest, eclass properties(mi)
version 12.0
args model1 model2
qui `model1'
estimates store est1
qui `model2'
estimates store est2
suest est1 est2
estimates drop est1 est2
ereturn local title "Seemingly unrelated estimation"
end
Because suest is an estimation command and already stores the
appropriate results in e(), we did not need to manually post (or repost)
anything to e() except e(title), which is not stored
by suest.
We can now use mysuest with mi estimate:
. webuse mheart1s20, clear
(Fictional heart attack data; bmi missing)
. mi estimate: mysuest "logit attack smokes age bmi" "logit attack smokes bmi"
Multiple-imputation estimates Imputations = 20
Seemingly unrelated estimation Number of obs = 154
Average RVI = 0.0541
Largest FMI = 0.1568
DF adjustment: Large sample DF: min = 794.17
avg = 234946.42
Within VCE type: Robust max = 667104.68
------------------------------------------------------------------------------
| Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
est1_attack |
smokes | 1.187115 .3574611 3.32 0.001 .486502 1.887728
age | .0352896 .0162749 2.17 0.030 .0033913 .067188
bmi | .102868 .0478936 2.15 0.032 .0089024 .1968336
_cons | -5.312597 1.69795 -3.13 0.002 -8.641634 -1.983561
-------------+----------------------------------------------------------------
est2_attack |
smokes | 1.172281 .3496687 3.35 0.001 .4869414 1.85762
bmi | .0921183 .0470325 1.96 0.051 -.0002044 .1844411
_cons | -3.038422 1.229553 -2.47 0.014 -5.451692 -.6251518
------------------------------------------------------------------------------
Recall that within multiple-imputation framework, to test the equality of
coefficients, we must first estimate their difference and then use
mi
testtransform to test the hypothesis; see [MI] mi test for details.
So, to test the coefficients on smokes from the two logistic models,
we first estimate their difference with mi estimate. To display only
transformed results, we specify the nocoef option:
. mi estimate (diff: [est1_attack]smokes - [est2_attack]smokes), nocoef:
> mysuest "logit attack smokes age bmi" "logit attack smokes bmi"
Multiple-imputation estimates Imputations = 20
Seemingly unrelated estimation Number of obs = 154
Average RVI = 0.0047
Largest FMI = 0.0047
DF adjustment: Large sample DF: min = 866779.11
avg = 866779.11
Within VCE type: Robust max = 866779.11
diff: [est1_attack]smokes - [est2_attack]smokes
------------------------------------------------------------------------------
| Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
diff | .0148342 .079597 0.19 0.852 -.1411732 .1708417
------------------------------------------------------------------------------
Although the t test, which is automatically reported by mi estimate, is
sufficient for testing the hypothesis of no difference between the two
coefficient estimates of smokes, we can also use the following mi
testtransform command to test this hypothesis:
. mi testtransform diff
note: assuming equal fractions of missing information
diff: [est1_attack]smokes - [est2_attack]smokes
( 1) diff = 0
F( 1,866779.1) = 0.03
Prob > F = 0.8522
We do not have sufficient evidence to reject the null hypothesis of the
equality of the smoking effects from the considered logistic models.
We could easily extend our mysuest program to allow more than two
models to be combined by suest.
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