Many researchers like to do their anova using regression with dummy coding but wonder
how to get tests of simple main effects. This FAQ
will show you how to get tests of simple main effects when using dummy coding.

We will begin with a two-factor design using a dataset called crf24.
We will manually compute the various dummy variables and run the regression model.

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

tab a, gen(a)
tab b, gen(b)
generate ab1 = a1*b1
generate ab2 = a1*b2
generate ab3 = a1*b3

regress y a1 b1 b2 b3 ab1 ab2 ab3

      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]
-------------+----------------------------------------------------------------
          a1 |         -2   .6208194    -3.22   0.004    -3.281308   -.7186918
          b1 |      -8.25   .6208194   -13.29   0.000    -9.531308   -6.968692
          b2 |         -7   .6208194   -11.28   0.000    -8.281308   -5.718692
          b3 |       -4.5   .6208194    -7.25   0.000    -5.781308   -3.218692
         ab1 |          4   .8779711     4.56   0.000     2.187957    5.812043
         ab2 |          3   .8779711     3.42   0.002     1.187957    4.812043
         ab3 |        3.5   .8779711     3.99   0.001     1.687957    5.312043
       _cons |         10   .4389856    22.78   0.000     9.093978    10.90602
------------------------------------------------------------------------------

For this model a2 is the reference level for a and b4 is the
reference level for b, i.e., they are the omitted levels.

Here is the test of the a*b interaction.

test ab1 ab2 ab3

 ( 1)  ab1 = 0
 ( 2)  ab2 = 0
 ( 3)  ab3 = 0

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

Since the a*b interaction is statistically significant we will procede with
tests of simple main effects. First, we will do the tests of b at
a2 which is the reference level for a.

test b1 b2 b3 

 ( 1)  b1 = 0
 ( 2)  b2 = 0
 ( 3)  b3 = 0

       F(  3,    24) =   68.84
            Prob > F =    0.0000

We will follow this with tests of b at
a1 which will require that we combine the b dummies
with their interaction terms and use the accumulate option to get
a test with the correct degrees of freedom.

test b1 + ab1 = 0

 ( 1)  b1 + ab1 = 0

       F(  1,    24) =   46.86
            Prob > F =    0.0000

test b2 + ab2 = 0, accumulate

 ( 1)  b1 + ab1 = 0
 ( 2)  b2 + ab2 = 0

       F(  2,    24) =   29.51
            Prob > F =    0.0000

test b3 + ab3 = 0, accumulate

 ( 1)  b1 + ab1 = 0
 ( 2)  b2 + ab2 = 0
 ( 3)  b3 + ab3 = 0

       F(  3,    24) =   23.65
            Prob > F =    0.0000

The last test command has the F-ratio = 23.65 for b at a1.

We could have done the tests of simple main effects for differences in a
for each level of b. Thoses tests would look like this:

/* test of a at b=4 (the reference group) */

test a1

 ( 1)  a1 = 0

       F(  1,    24) =   10.38
            Prob > F =    0.0036

/* test of a at b=1 */

test a1 + ab1 = 0

 ( 1)  a1 + ab1 = 0

       F(  1,    24) =   10.38
            Prob > F =    0.0036

/* test of a at b=2 */

test a1 + ab2 = 0

 ( 1)  a1 + ab2 = 0

       F(  1,    24) =    2.59
            Prob > F =    0.1203

/* test of a at b=3 */

test a1 + ab3 = 0

 ( 1)  a1 + ab3 = 0

       F(  1,    24) =    5.84
            Prob > F =    0.0237

As you can see, testing an effect, say a, by itself tests the simple
main effect for the reference group. To get the simple main effects for the
other levels, you combine the test of a with the appropriate interaction
term.

This FAQ only covers the computation of the tests of simple main effects
using dummy coding. The FAQ does not cover computing the critical values of these
tests. There is a user written ado-program smecriticalvalue which can
assist in this process (search smecriticalvalue).

This method of using dummy coding to compute tests of simple main effects generalizes
to more complex designs with multiple factors. Let’s try it with
a three-factor completely crossed design. Once again we will manually create the dummy variables
and run the regression model.

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

recode a (1=0)(2=1)
recode b (1=0)(2=1)
tab c, gen(c)
gen ab=a*b
gen ac1=a*c1
gen ac2=a*c2
gen bc1=b*c1
gen bc2=b*c2
gen abc1=a*b*c1
gen abc2=a*b*c2

regress y a b c1 c2 ab ac1 ac2 bc1 bc2 abc1 abc2

      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]
-------------+----------------------------------------------------------------
           a |        -.5   1.154701    -0.43   0.673    -3.015876    2.015876
           b |       -9.5   1.154701    -8.23   0.000    -12.01588   -6.984124
          c1 |         -8   1.154701    -6.93   0.000    -10.51588   -5.484124
          c2 |         -4   1.154701    -3.46   0.005    -6.515876   -1.484124
          ab |         15   1.632993     9.19   0.000     11.44201    18.55799
         ac1 |   6.39e-14   1.632993     0.00   1.000    -3.557986    3.557986
         ac2 |          1   1.632993     0.61   0.552    -2.557986    4.557986
         bc1 |          9   1.632993     5.51   0.000     5.442014    12.55799
         bc2 |          5   1.632993     3.06   0.010     1.442014    8.557986
        abc1 |       -8.5   2.309401    -3.68   0.003    -13.53175   -3.468247
        abc2 |       -5.5   2.309401    -2.38   0.035    -10.53175   -.4682473
       _cons |         19   .8164966    23.27   0.000     17.22101    20.77899
------------------------------------------------------------------------------

In this model a1, b1 and c3 are the reference groups for
variables a, b and c, respectively.

Here is the test of the three-way a*b*c interaction.

test abc1 abc2 abc2

 ( 1)  abc1 = 0
 ( 2)  abc2 = 0
 ( 3)  abc2 = 0
       Constraint 3 dropped

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

Since the test of the three-way interaction is statistically significant it means that
there is one or more two-way interactions that are significant at different levels of
a third variable. We have reason to believe that looking at the b*c interaction
could be productive, therefore we will test the b*c interactions for each
level of a.

/* test b*c at a1 (the reference level) */ 

test bc1 bc2 

( 1) bc1 = 0 
( 2) bc2 = 0 

F( 2, 12) = 15.25 
Prob > F = 0.0005 

/* test b*c at a2 */ 

test bc1+abc1=0 
test bc2+abc2=0, accumulate

( 1) bc1 + abc1 = 0 
( 2) bc2 + abc2 = 0 

F( 2, 12) = 0.1875 
Prob > F = 0.8314 

We see that b*c at a1 has a pretty large F-ratio and that the
F-ratio for b*c at a2 is less than one.

So we will follow up this analysis by looking for differences in c for
eacl level of b all within a1.

/* test for c at b==1 & a==1 */ 

test c1 c2 

( 1) c1 = 0 
( 2) c2 = 0 

F( 2, 12) = 24.00 
Prob > F = 0.0001 

/* test for c at b==2 & a==1 */ 

test c1+bc1=0 
test c2+bc2=0, accum 

( 1) c1 + bc1 = 0 
( 2) c2 + bc2 = 0 

F( 2, 12) = 0.50 
Prob > F = 0.6186 

Dummy coding provides a fast and easy way to compute tests of simple
main effects.