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In chapter 11 Keppel shows how to perform comparisons of main effects and simple effects in a two way factorial anova based on the example from chapter 10. On page 234 Keppel shows a comparison comparing group a1 with group a3. We use the char command to set up this comparison and one that is orthogonal to it. You can download the xi3 command by typing findit xi3 (see How can I use the findit command to search for programs and get additional help? for more information about using findit).
use http://www.ats.ucla.edu/stat/stata/examples/da/chap10, clear char a[user] (1 0 -1\1 -2 1)
We now run the anova using xi3. The term _Ia_1 corresponds to the test shown on page 234. We show the test command to illustrate that the F value for this test corresponds to the value shown in the text.
xi3: regress errors u.a*g.b
u.a _Ia_1-3 (naturally coded; _Ia_3 omitted)
g.b _Ib_1-2 (naturally coded; _Ib_1 omitted)
Source | SS df MS Number of obs = 24
-------------+------------------------------ F( 5, 18) = 3.05
Model | 280 5 56 Prob > F = 0.0361
Residual | 330 18 18.3333333 R-squared = 0.4590
-------------+------------------------------ Adj R-squared = 0.3087
Total | 610 23 26.5217391 Root MSE = 4.2817
------------------------------------------------------------------------------
errors | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_Ia_1 | -5 2.140872 -2.34 0.031 -9.497805 -.5021946
_Ia_2 | -3 3.708099 -0.81 0.429 -10.79043 4.790427
_Ib_2 | 2 1.748015 1.14 0.268 -1.672443 5.672443
_Ia1Xb2 | 12 4.281744 2.80 0.012 3.004389 20.99561
_Ia2Xb2 | -1.56e-16 7.416198 -0.00 1.000 -15.58085 15.58085
_cons | 10 .8740074 11.44 0.000 8.163779 11.83622
------------------------------------------------------------------------------
test _Ia_1
( 1) _Ia_1 = 0.0
F( 1, 18) = 5.45
Prob > F = 0.0313
On page 241, Keppel shows how to perform tests of simple main effects testing the effect of a at b1 and b2. We can get the tests of simple main effects using the xi2 command. The second xi2 command with the @ symbol instead of the * gives us the tests of simple effects of A at b1 and b2.
xi3: regress errors g.a@g.b
g.a _Ia_1-3 (naturally coded; _Ia_1 omitted)
g.b _Ib_1-2 (naturally coded; _Ib_1 omitted)
Source | SS df MS Number of obs = 24
-------------+------------------------------ F( 5, 18) = 3.05
Model | 280 5 56 Prob > F = 0.0361
Residual | 330 18 18.3333333 R-squared = 0.4590
-------------+------------------------------ Adj R-squared = 0.3087
Total | 610 23 26.5217391 Root MSE = 4.2817
------------------------------------------------------------------------------
errors | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_Ib_2 | 2 1.748015 1.14 0.268 -1.672443 5.672443
_Ia2Wb1 | 7 3.02765 2.31 0.033 .6391426 13.36086
_Ia2Wb2 | 1 3.02765 0.33 0.745 -5.360857 7.360857
_Ia3Wb1 | 11 3.02765 3.63 0.002 4.639143 17.36086
_Ia3Wb2 | -1 3.02765 -0.33 0.745 -7.360857 5.360857
_cons | 10 .8740074 11.44 0.000 8.163779 11.83622
------------------------------------------------------------------------------
If we perform the test command below, we get all of the effects of b at a1, so this is the overall simple effect of b at a1.
test _Ia2Wb1 _Ia3Wb1
( 1) _Ia2Wb1 = 0
( 2) _Ia3Wb1 = 0
F( 2, 18) = 6.76
Prob > F = 0.0064
Likewise, the test command below tests all of the effects of effects of b at a2, so this is the overall simple effect of b at a2.
test _Ia2Wb2 _Ia3Wb2
( 1) _Ia2Wb2 = 0
( 2) _Ia3Wb2 = 0
F( 2, 18) = 0.22
Prob > F = 0.8061
Page 246 shows how to do a simple comparison comparing groups 1 and 2. Below we use the char command to specify a comparison of groups 1 and 2, and a second comparison (that we don't care about) comparing groups 1 and 2 with group 3. We then use u.a@s.b to indicate we want the effects of a that were user defined at each level of b.
char a[user] (1 -1 0\1 1 -2)
xi3: regress errors u.a@g.b
u.a _Ia_1-3 (naturally coded; _Ia_3 omitted)
g.b _Ib_1-2 (naturally coded; _Ib_1 omitted)
Source | SS df MS Number of obs = 24
-------------+------------------------------ F( 5, 18) = 3.05
Model | 280 5 56 Prob > F = 0.0361
Residual | 330 18 18.3333333 R-squared = 0.4590
-------------+------------------------------ Adj R-squared = 0.3087
Total | 610 23 26.5217391 Root MSE = 4.2817
------------------------------------------------------------------------------
errors | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_Ib_2 | 2 1.748015 1.14 0.268 -1.672443 5.672443
_Ia1Wb1 | -7 3.02765 -2.31 0.033 -13.36086 -.6391426
_Ia1Wb2 | -1 3.02765 -0.33 0.745 -7.360857 5.360857
_Ia2Wb1 | -15 5.244044 -2.86 0.010 -26.01733 -3.982672
_Ia2Wb2 | 3 5.244044 0.57 0.574 -8.017328 14.01733
_cons | 10 .8740074 11.44 0.000 8.163779 11.83622
------------------------------------------------------------------------------
The term _Ia1Wb1 is the 1st comparison on 1 at b1, and the term _Ia1Wb2 is the first comparison on 1 at b2. To see these as F tests, we use the test commands below and see they correspond to the results in the text on page 246.
test _Ia1Wb1
( 1) _Ia1Wb1 = 0
F( 1, 18) = 5.35
Prob > F = 0.0328
test _Ia1Wb2
( 1) _Ia1Wb2 = 0
F( 1, 18) = 0.11
Prob > F = 0.7450
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