Stata FAQ
How can I get anova simple main effects with the margins command? (Stata 11)

Now that Stata 11 has been released, you be wondering if there is an easy way to compute tests of simple main effects? Yes there is and this FAQ will show you how to get tests of simple main effects using the new margins command.

Consider a two-factor design using a dataset called crf24.

use http://www.ats.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  64.8333333      84.11     0.0000

                     a#b |      19.375     3  6.45833333       8.38     0.0006

                         |

                Residual |        18.5    24  .770833333   

              -----------+----------------------------------------------------

                   Total |       235.5    31  7.59677419   
As you can see the a#b interaction is statistically significant.

Next we will use the margins command on the a#b interaction. We will include the post option and if the design were unbalanced or included a covariate we would also include the asbalanced option.

margins a#b, post

	

Adjusted predictions                              Number of obs   =         32



Expression   : Linear prediction, predict()



------------------------------------------------------------------------------

             |            Delta-method

             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]

-------------+----------------------------------------------------------------

         a#b |

        1 1  |       3.75   .4389856     8.54   0.000     2.889604    4.610396

        1 2  |          4   .4389856     9.11   0.000     3.139604    4.860396

        1 3  |          7   .4389856    15.95   0.000     6.139604    7.860396

        1 4  |          8   .4389856    18.22   0.000     7.139604    8.860396

        2 1  |       1.75   .4389856     3.99   0.000     .8896041    2.610396

        2 2  |          3   .4389856     6.83   0.000     2.139604    3.860396

        2 3  |        5.5   .4389856    12.53   0.000     4.639604    6.360396

        2 4  |         10   .4389856    22.78   0.000     9.139604     10.8604

------------------------------------------------------------------------------
The margins command has given us a list of the cell means and their standard errors. We can use combinations of these means to compute the test of simple maun effects. For example, a1 versus a2 at b1 would test cell a1,b1 versus cell a2,b1. Here is what the tests of simple main effects for a each level of b would look like.
test 1.a#1.b == 2.a#1.b



 ( 1)  1bn.a#1bn.b - 2.a#1bn.b = 0



           chi2(  1) =   10.38

         Prob > chi2 =    0.0013



test 1.a#2.b == 2.a#2.b



 ( 1)  1bn.a#2.b - 2.a#2.b = 0



           chi2(  1) =    2.59

         Prob > chi2 =    0.1072



test 1.a#3.b == 2.a#3.b



 ( 1)  1bn.a#3.b - 2.a#3.b = 0



           chi2(  1) =    5.84

         Prob > chi2 =    0.0157



test 1.a#4.b == 2.a#4.b



 ( 1)  1bn.a#4.b - 2.a#4.b = 0



           chi2(  1) =   10.38

         Prob > chi2 =    0.0013
Each one of the above tests used only one degree of freedom so the terms in the test were simple. We will next move on to test of b at each level of a. Each of these tests have three degrees of freedom and so, will will involve a more complex test command. We will start off showing the accum approach and then show a cleaner way to get the result with a single test command.

/* test of b at a==1 */



test 1.a#1.b == 1.a#2.b

test 1.a#1.b == 1.a#3.b, accum

test 1.a#1.b == 1.a#4.b, accum



 ( 1)  1bn.a#1bn.b - 1bn.a#2.b = 0

 ( 2)  1bn.a#1bn.b - 1bn.a#3.b = 0

 ( 3)  1bn.a#1bn.b - 1bn.a#4.b = 0



           chi2(  3) =   70.95

         Prob > chi2 =    0.0000

  

/* the same test using a single test command */



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



 ( 1)  1bn.a#1bn.b - 1bn.a#2.b = 0

 ( 2)  1bn.a#1bn.b - 1bn.a#3.b = 0

 ( 3)  1bn.a#1bn.b - 1bn.a#4.b = 0



           chi2(  3) =   70.95

         Prob > chi2 =    0.0000



/* convert chi2 to approximate F-value  */



display r(chi2)/r(df)



23.648649
We will finish up with tests of b at a==2 just by changing all of the 1.a terms to 2.a

/* test of b at a==1 */



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



 ( 1)  2.a#1bn.b - 2.a#2.b = 0

 ( 2)  2.a#1bn.b - 2.a#3.b = 0

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



           chi2(  3) =  206.51

         Prob > chi2 =    0.0000



/* convert chi2 to approximate F-value  */



display r(chi2)/r(df)



68.837838
This FAQ only covers the computation of the tests of simple main effects using the margins command. 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 (findit smecriticalvalue).

Let's try a little more challenging example, one that is unbalanced with a covariate.

use http://www.ats.ucla.edu/stat/data/hsbdemo, clear



anova write female##prog c.read





                           Number of obs =     200     R-squared     =  0.4889

                           Root MSE      = 6.88105     Adj R-squared =  0.4730



                  Source |  Partial SS    df       MS           F     Prob > F

             ------------+----------------------------------------------------

                   Model |  8740.54801     6    1456.758      30.77     0.0000

                         |

                  female |  1680.08743     1  1680.08743      35.48     0.0000

                    prog |  632.133455     2  316.066727       6.68     0.0016

             female#prog |  301.770797     2  150.885399       3.19     0.0435

                    read |  4110.18709     1  4110.18709      86.81     0.0000

                         |

                Residual |  9138.32699   193  47.3488445   

             ------------+----------------------------------------------------

                   Total |   17878.875   199   89.843593 
The female#prog interaction is significant but we'll go ahead and compute the tests of simple main effects. This time because the model is both unbalanced and includes a covariate we will include the asbalanced option which is equivalent to least squares means (lsmeans) or expected marginal means (emmeans) in SAS and SPSS.
margins female#prog, asbalanced post



Predictive margins                                Number of obs   =        200



Expression   : Linear prediction, predict()



------------------------------------------------------------------------------

             |            Delta-method

             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]

-------------+----------------------------------------------------------------

 female#prog |

        0 1  |    48.7854   1.502058    32.48   0.000     45.84142    51.72938

        0 2  |   52.61465   1.026455    51.26   0.000     50.60284    54.62647

        0 3  |   45.08098   1.476716    30.53   0.000     42.18667    47.97529

        1 1  |   55.85857   1.432221    39.00   0.000     53.05146    58.66567

        1 2  |   55.68658    .926245    60.12   0.000     53.87117    57.50199

        1 3  |   53.71585   1.356821    39.59   0.000     51.05653    56.37517

------------------------------------------------------------------------------



/* tests of female at each level of prog */



test 0.female#1.prog == 1.female#1.prog



 ( 1)  0bn.female#1bn.prog - 1.female#1bn.prog = 0



           chi2(  1) =   11.56

         Prob > chi2 =    0.0007



test 0.female#2.prog == 1.female#2.prog



 ( 1)  0bn.female#2.prog - 1.female#2.prog = 0



           chi2(  1) =    5.17

         Prob > chi2 =    0.0229



test 0.female#3.prog == 1.female#3.prog



 ( 1)  0bn.female#3.prog - 1.female#3.prog = 0



           chi2(  1) =   19.54

         Prob > chi2 =    0.0000



/* tests of prog at each level of female */



test (0.female#1.prog == 0.female#2.prog)(0.female#1.prog == 0.female#3.prog)



 ( 1)  0bn.female#1bn.prog - 0bn.female#2.prog = 0

 ( 2)  0bn.female#1bn.prog - 0bn.female#3.prog = 0



           chi2(  2) =   17.41

         Prob > chi2 =    0.0002

         

/* convert chi2 to approximate F-value  */



display r(chi2)/r(df)



8.7026677



test (1.female#1.prog == 1.female#2.prog)(1.female#1.prog == 1.female#3.prog)



 ( 1)  1.female#1bn.prog - 1.female#2.prog = 0

 ( 2)  1.female#1bn.prog - 1.female#3.prog = 0



           chi2(  2) =    1.69

         Prob > chi2 =    0.4287



/* convert chi2 to approximate F-value  */



display r(chi2)/r(df)



.84693839
So that's how to compute tests of simple main effects using margins in Stata 11.

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