Below we show how to perform post estimation hypothesis tests on models based
on multiply imputed data with mi estimate, mi test and mi testtransform.

The example for this faq uses data on high school students. The
variables read, write, and math give the student’s
scores in reading, writing, and math respectively. The variable female
is equal to one if the student is female and zero otherwise. Finally,
prog contains information on the type of program the student is in
either general, academic, and vocational. The multiply imputed datasets are
created using mi impute  and are saved into in a single file which contains all 10 imputations
as well as the original data. The variable _mi_m gives the imputation number,  _mi_m = 0
contains the original data.

use https://stats.idre.ucla.edu/stat/data/hsbmar, clearmi set mlongmi register imputed female math read science socstmi impute chain (logit) female (regress) math science socst read = ///ses write awards, add(10) force

Below we use mi estimate:regress to fit a linear regression model. The mi estimate: 
prefix informs Stata that we want to analyze multiply imputed
datasets, without it, the command would be performed on the dataset as though it
were a single dataset, rather than a series of multiply imputed
datasets.

mi estimate: regress read write i.female math i.prog            

Multiple-imputation estimates                   Imputations       =         10
Linear regression                               Number of obs     =        199
                                                Average RVI       =     0.1481
                                                Largest FMI       =     0.2715
                                                Complete DF       =        193
DF adjustment:   Small sample                   DF:     min       =      69.10
                                                        avg       =     127.30
                                                        max       =     174.50
Model F test:       Equal FMI                   F(   5,  172.3)   =      33.18
Within VCE type:          OLS                   Prob > F          =     0.0000

------------------------------------------------------------------------------
        read |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       write |   .4059465   .0840575     4.83   0.000     .2395667    .5723263
             |
      female |
     female  |   -3.18419   1.296704    -2.46   0.016    -5.763144   -.6052363
        math |   .3980997   .0898392     4.43   0.000       .21888    .5773194
             |
        prog |
   academic  |   1.467474   1.444345     1.02   0.311    -1.385213    4.320161
   vocation  |  -.7782421   1.586677    -0.49   0.624     -3.90979    2.353306
             |
       _cons |   11.04381   3.939979     2.80   0.006     3.260903    18.82671
------------------------------------------------------------------------------

Once the model is estimated the mi test command with the prefix
can be used to perform multiple degree of freedom tests. One common use for this is to
test for an overall effect ofa nominal variable represented by a series of dummy variables.
Below we use mi test: to test for an overall effect of type of program (prog).

mi test 2.prog 3.prog

 ( 1)  2.prog = 0
 ( 2)  3.prog = 0

       F(  2, 181.6) =    1.23
            Prob > F =    0.2952

The mi test command can also be used to test nested models, where the null
hypothesis is that the coefficients on two or more variables are simultaneously equal to zero.

mi test math write

( 1)  math = 0
 ( 2)  write = 0

       F(  2, 132.6) =   52.12
            Prob > F =    0.0000

It is also possible to test linear combinations of variables. Below we test a model
with an interaction between math and female. The variable female
is dummy coded (0=male, 1=female). First we create the interaction as we
normally would, then we use the regress command with the mi estimate: prefix to
fit a regression model.

Then the mi estimate: and mitesttransform command
can be used  to test the null hypothesis that the effect of math on read is zero when
female=1. The coeflegend option specifies the legend of coefficients and
how to specify them in an expression. We will need these coefficient names in order to estimate
the effect of math for female=1.

mi estimate (math_slope_female:_b[math] + _b[1.female#c.math]), coeflegend: /// 
regress read write i.female##c.math i.prog

Transformations                                 Average RVI       =     0.1536
                                                Largest FMI       =     0.1379
                                                Complete DF       =        192
DF adjustment:   Small sample                   DF:     min       =     124.38
                                                        avg       =     124.38
Within VCE type:          OLS                           max       =     124.38

 math_slope~e: _b[math] + _b[1.female#c.math]

-----------------------------------------------------------------------------------
             read |      Coef.  Legend
------------------+----------------------------------------------------------------
math_slope_female |   .4542803  _b[math_slope_female]
-----------------------------------------------------------------------------------

mi testtransform math_slope_female

 math_slope~e: _b[math] + _b[1.female#c.math]

 ( 1)  math_slope_female = 0

       F(  1, 124.4) =   20.67
            Prob > F =    0.0000