Many researchers like to do their anova using regression with dummy coding but find
it confusing when they don’t get the same main-effects as in anova. This FAQ
will show you how to get those main-effects.

Example 1

Let’s begin by showing the normal anova using a dataset called crf24 to use
as a comparison.

use https://stats.idre.ucla.edu/stat/stata/faq/crf24, clear

anova y a##b

                         Number of obs =         32    R-squared     =  0.9214
                         Root MSE      =    .877971    Adj R-squared =  0.8985

                  Source | Partial SS         df         MS        F    Prob>F
              -----------+----------------------------------------------------
                   Model |        217          7          31     40.22  0.0000
                         |
                       a |      3.125          1       3.125      4.05  0.0554
                       b |      194.5          3   64.833333     84.11  0.0000
                     a#b |     19.375          3   6.4583333      8.38  0.0006
                         |
                Residual |       18.5         24   .77083333  
              -----------+----------------------------------------------------
                   Total |      235.5         31   7.5967742  

Here is how the above analyses would look using Stata 15’s factor variables with the regress command. The regression model will be followed by a test of the interaction, the margins command and the test of the two main effects using the testparm command.

 

regress y a##b

      Source |       SS       df       MS              Number of obs =      32
-------------+------------------------------           F(  7,    24) =   40.22
       Model |         217     7          31           Prob > F      =  0.0000
    Residual |        18.5    24  .770833333           R-squared     =  0.9214
-------------+------------------------------           Adj R-squared =  0.8985
       Total |       235.5    31  7.59677419           Root MSE      =  .87797

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         2.a |         -2   .6208194    -3.22   0.004    -3.281308   -.7186918
             |
           b |
          2  |        .25   .6208194     0.40   0.691    -1.031308    1.531308
          3  |       3.25   .6208194     5.24   0.000     1.968692    4.531308
          4  |       4.25   .6208194     6.85   0.000     2.968692    5.531308
             |
         a#b |
        2 2  |          1   .8779711     1.14   0.266    -.8120434    2.812043
        2 3  |         .5   .8779711     0.57   0.574    -1.312043    2.312043
        2 4  |          4   .8779711     4.56   0.000     2.187957    5.812043
             |
       _cons |       3.75   .4389856     8.54   0.000     2.843978    4.656022
------------------------------------------------------------------------------

testparm a#b   /* test of a#b interaction */

 ( 1)  2.a#2.b = 0
 ( 2)  2.a#3.b = 0
 ( 3)  2.a#4.b = 0

       F(  3,    24) =    8.38
            Prob > F =    0.0006

Even though the interaction is statistically significant we will go ahead and check out the
main effects. We will demonstrate two methods for computing the main effects for this
example. We need to make clear that there are more than two methods of obtaining the
main effects using the margins command. These are just two of the easier methods.

The first method uses testparm with the equal option.

estimates store m1   /* store regression results for later computations */

margins a b, asbalanced post  /* margins command for main effects: method 1 */


Adjusted predictions                            Number of obs     =         32
Model VCE    : OLS

Expression   : Linear prediction, predict()
at           : a                (asbalanced)
               b                (asbalanced)

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           a |
          1  |     5.6875   .2194928    25.91   0.000     5.234489    6.140511
          2  |     5.0625   .2194928    23.06   0.000     4.609489    5.515511
             |
           b |
          1  |       2.75   .3104097     8.86   0.000     2.109346    3.390654
          2  |        3.5   .3104097    11.28   0.000     2.859346    4.140654
          3  |       6.25   .3104097    20.13   0.000     5.609346    6.890654
          4  |          9   .3104097    28.99   0.000     8.359346    9.640654
------------------------------------------------------------------------------


testparm i.a, equal   /* a main effect */
 ( 1)  - 1bn.a + 2.a = 0

       F(  1,    24) =    4.05
            Prob > F =    0.0554

testparm i.b, equal   /* b main effect */

 ( 1)  - 1bn.b + 2.b = 0
 ( 2)  - 1bn.b + 3.b = 0
 ( 3)  - 1bn.b + 4.b = 0

       F(  3,    24) =   84.11
            Prob > F =    0.0000

Next, we demonstrate the second method for main effects using margins with the
dydx option.

estimates restore m1   /* restore regression results */

margins, dydx(a b) asbalanced post  /* margins command for main effects: method 2 */


Conditional marginal effects                    Number of obs     =         32
Model VCE    : OLS

Expression   : Linear prediction, predict()
dy/dx w.r.t. : 2.a 2.b 3.b 4.b
at           : a                (asbalanced)
               b                (asbalanced)

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         2.a |      -.625   .3104097    -2.01   0.055    -1.265654    .0156541
             |
           b |
          2  |        .75   .4389856     1.71   0.100    -.1560217    1.656022
          3  |        3.5   .4389856     7.97   0.000     2.593978    4.406022
          4  |       6.25   .4389856    14.24   0.000     5.343978    7.156022
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.

testparm i.a   /* a main effect */

  ( 1)  2.a = 0

       F(  1,    24) =    4.05
            Prob > F =    0.0554

testparm i.b   /* b main effect */

 ( 1)  2.b = 0
 ( 2)  3.b = 0
 ( 3)  4.b = 0

       F(  3,    24) =   84.11
            Prob > F =    0.0000

Example 2

This method generalizes to more complex designs with multiple factors, so let’s consider a 3-factor completely crossed design.

 

use https://stats.idre.ucla.edu/stat/stata/faq/threeway, clear

anova y a##b##c

                         Number of obs =         24    R-squared     =  0.9689
                         Root MSE      =     1.1547    Adj R-squared =  0.9403

                  Source | Partial SS         df         MS        F    Prob>F
              -----------+----------------------------------------------------
                   Model |  497.83333         11   45.257576     33.94  0.0000
                         |
                       a |        150          1         150    112.50  0.0000
                       b |  .66666667          1   .66666667      0.50  0.4930
                     a#b |  160.16667          1   160.16667    120.13  0.0000
                       c |  127.58333          2   63.791667     47.84  0.0000
                     a#c |      18.25          2       9.125      6.84  0.0104
                     b#c |  22.583333          2   11.291667      8.47  0.0051
                   a#b#c |  18.583333          2   9.2916667      6.97  0.0098
                         |
                Residual |         16         12   1.3333333  
              -----------+----------------------------------------------------
                   Total |  513.83333         23    22.34058  

And here is the same model using the regress command.

regress y a##b##c


      Source |       SS           df       MS      Number of obs   =        24
-------------+----------------------------------   F(11, 12)       =     33.94
       Model |  497.833333        11  45.2575758   Prob > F        =    0.0000
    Residual |          16        12  1.33333333   R-squared       =    0.9689
-------------+----------------------------------   Adj R-squared   =    0.9403
       Total |  513.833333        23  22.3405797   Root MSE        =    1.1547

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         2.a |        -.5   1.154701    -0.43   0.673    -3.015876    2.015876
         2.b |        -.5   1.154701    -0.43   0.673    -3.015876    2.015876
             |
         a#b |
        2 2  |        6.5   1.632993     3.98   0.002     2.942014    10.05799
             |
           c |
          2  |          4   1.154701     3.46   0.005     1.484124    6.515876
          3  |          8   1.154701     6.93   0.000     5.484124    10.51588
             |
         a#c |
        2 2  |          1   1.632993     0.61   0.552    -2.557986    4.557986
        2 3  |   3.09e-14   1.632993     0.00   1.000    -3.557986    3.557986
             |
         b#c |
        2 2  |         -4   1.632993    -2.45   0.031    -7.557986   -.4420135
        2 3  |         -9   1.632993    -5.51   0.000    -12.55799   -5.442014
             |
       a#b#c |
      2 2 2  |          3   2.309401     1.30   0.218    -2.031753    8.031753
      2 2 3  |        8.5   2.309401     3.68   0.003     3.468247    13.53175
             |
       _cons |         11   .8164966    13.47   0.000     9.221007    12.77899
------------------------------------------------------------------------------


testparm a#b#c   /* test of the a#b#c interaction */


 ( 1)  2.a#2.b#2.c = 0
 ( 2)  2.a#2.b#3.c = 0

       F(  2,    12) =    6.97
            Prob > F =    0.0098

Before we get to the main effects, we will test the three two-way interactions.

estimates store m1   /* store regression results for later computations */

margins, dydx(a) over(b) asbal post noatlegend /* margins for a#b interaction */

Conditional marginal effects Number of obs = 24
Model VCE : OLS

Expression : Linear prediction, predict()
dy/dx w.r.t. : 2.a
over : b

——————————————————————————
| Delta-method
| dy/dx Std. Err. t P>|t| [95% Conf. Interval]
————-+—————————————————————-
1.a | (base outcome)
————-+—————————————————————-
2.a |
b |
1 | -.1666667 .6666667 -0.25 0.807 -1.619209 1.285875
2 | 10.16667 .6666667 15.25 0.000 8.714125 11.61921
——————————————————————————
Note: dy/dx for factor levels is the discrete change from the base level.

test [2.a]1.b=[2.a]2.b /* test of a#b interaction */

( 1) [2.a]1bn.b – [2.a]2.b = 0

F( 1, 12) = 120.12
Prob > F = 0.0000

estimates restore m1

margins, dydx(a) over(c) asbal post noatlegend /* margins for a#c interaction */

Conditional marginal effects Number of obs = 24
Model VCE : OLS

Expression : Linear prediction, predict()
dy/dx w.r.t. : 2.a
over : c

——————————————————————————
| Delta-method
| dy/dx Std. Err. t P>|t| [95% Conf. Interval]
————-+—————————————————————-
1.a | (base outcome)
————-+—————————————————————-
2.a |
c |
1 | 2.75 .8164966 3.37 0.006 .9710068 4.528993
2 | 5.25 .8164966 6.43 0.000 3.471007 7.028993
3 | 7 .8164966 8.57 0.000 5.221007 8.778993
——————————————————————————
Note: dy/dx for factor levels is the discrete change from the base level.

test ([2.a]1.c=[2.a]2.c)([2.a]1.c=[2.a]3.c) /* test of a#c interaction */

( 1) [2.a]1bn.c – [2.a]2.c = 0
( 2) [2.a]1bn.c – [2.a]3.c = 0

F( 2, 12) = 6.84
Prob > F = 0.0104

estimates restore m1

margins, dydx(b) over(c) asbal post noatlegend /* margins for b#c interaction */

Conditional marginal effects Number of obs = 24
Model VCE : OLS

Expression : Linear prediction, predict()
dy/dx w.r.t. : 2.b
over : c

——————————————————————————
| Delta-method
| dy/dx Std. Err. t P>|t| [95% Conf. Interval]
————-+—————————————————————-
1.b | (base outcome)
————-+—————————————————————-
2.b |
c |
1 | 2.75 .8164966 3.37 0.006 .9710068 4.528993
2 | .25 .8164966 0.31 0.765 -1.528993 2.028993
3 | -2 .8164966 -2.45 0.031 -3.778993 -.2210068
——————————————————————————
Note: dy/dx for factor levels is the discrete change from the base level.

test ([2.b]1.c=[2.b]2.c)([2.b]1.c=[2.b]3.c) /* test of b#c interaction */

( 1) [2.b]1bn.c – [2.b]2.c = 0
( 2) [2.b]1bn.c – [2.b]3.c = 0

F( 2, 12) = 8.47
Prob > F = 0.0051

Finally, we will compute the main effects using testparm with method 2 as shown above.

estimates restore m1

margins, dydx(a b c) asbalanced post /* margins command for main effects */

Conditional marginal effects Number of obs = 24
Model VCE : OLS

Expression : Linear prediction, predict()
dy/dx w.r.t. : 2.a 2.b 2.c 3.c
at : a (asbalanced)
b (asbalanced)
c (asbalanced)

——————————————————————————
| Delta-method
| dy/dx Std. Err. t P>|t| [95% Conf. Interval]
————-+—————————————————————-
2.a | 5 .4714045 10.61 0.000 3.972898 6.027102
2.b | .3333333 .4714045 0.71 0.493 -.6937689 1.360436
|
c |
2 | 3.25 .5773503 5.63 0.000 1.992062 4.507938
3 | 5.625 .5773503 9.74 0.000 4.367062 6.882938
——————————————————————————
Note: dy/dx for factor levels is the discrete change from the base level.

testparm i.a /* a main-effect */

( 1) 2.a = 0

F( 1, 12) = 112.50
Prob > F = 0.0000

testparm i.b /* b main-effect */

( 1) 2.b = 0

F( 1, 12) = 0.50
Prob > F = 0.4930

testparm i.c /* c main-effect */

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

F( 2, 12) = 47.84
Prob > F = 0.0000