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Stata version 10 and earlier
Let's begin by showing the normal anova using a dataset called crf24 to use as a comparison.
use http://www.ats.ucla.edu/stat/stata/faq/crf24, clear
anova y a b 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 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
------------------------------------------------------------------------------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
To get the main-effect for a we will use the the dummy for a plus the
a*b interaction dummies averaged across the four levels of b.
test a1 + (ab1+ab2+ab3)/4 = 0
( 1) a1 + .25 ab1 + .25 ab2 + .25 ab3 = 0
F( 1, 24) = 4.05
Prob > F = 0.0554
test b1 + ab1/2 = 0
( 1) b1 + .5 ab1 = 0
F( 1, 24) = 202.70
Prob > F = 0.0000
test b2 + ab2/2 = 0, accumulate
( 1) b1 + .5 ab1 = 0
( 2) b2 + .5 ab2 = 0
F( 2, 24) = 120.86
Prob > F = 0.0000
test b3 + ab3/2 = 0, accumulate
( 1) b1 + .5 ab1 = 0
( 2) b2 + .5 ab2 = 0
( 3) b3 + .5 ab3 = 0
F( 3, 24) = 84.11
Prob > F = 0.0000So, what's with all of the division, by 4 in the a main-effect and by 2 in the b main-effect. The dummy variable a1 is actually the simple effect of a. To get the "true" main-effect of a we have to combine the simple effect of a with the average of the interaction effects across the four levels of b. Likewise, for the b main-effect we need to combine the simple main-effects of the levels of b with the average interaction effect across the two levels of a.
Stata 11
Here is how the above analyses would look using Stata 11's factor variables.
Stata version 10 and earlier
This method generalizes to more complex designs with multiple factors so let's
consider a 3-factor completely crossed design.
And here are the same analyses using Stata 11.
UCLA Researchers are invited to our Statistical Consulting Services
Once again we will manually create the dummy variables and run the regression
model.
use http://www.ats.ucla.edu/stat/stata/faq/threeway, clear
anova y a b c a*b a*c b*c 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 | 497.833333 11 45.2575758 33.94 0.0000
|
a | 150 1 150 112.50 0.0000
b | .666666667 1 .666666667 0.50 0.4930
c | 127.583333 2 63.7916667 47.84 0.0000
a*b | 160.166667 1 160.166667 120.13 0.0000
a*c | 18.25 2 9.125 6.84 0.0104
b*c | 22.5833333 2 11.2916667 8.47 0.0051
a*b*c | 18.5833333 2 9.29166667 6.97 0.0098
|
Residual | 16 12 1.33333333
-----------+----------------------------------------------------
Total | 513.833333 23 22.3405797
Here is thetest of the three-way a*b*c interaction.
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
------------------------------------------------------------------------------
Next come the two-way interactions with both a*c and b*c using the
accumulate options.
test abc1 abc2
( 1) abc1 = 0
( 2) abc2 = 0
F( 2, 12) = 6.97
Prob > F = 0.0098
Finally, we get to the main-effects.
/* a*b interaction */
test ab + (abc1+abc2)/3 = 0
( 1) ab + .3333333 abc1 + .3333333 abc2 = 0
F( 1, 12) = 120.13
Prob > F = 0.0000
/* a*c interaction) */
test ac1 + abc1/2 = 0
( 1) ac1 + .5 abc1 = 0
F( 1, 12) = 13.55
Prob > F = 0.0031
test ac2 + abc2/2 = 0, accumulate
( 1) ac1 + .5 abc1 = 0
( 2) ac2 + .5 abc2 = 0
F( 2, 12) = 6.84
Prob > F = 0.0104
/* b*c interaction */
test bc1 + abc1/2 = 0
( 1) bc1 + .5 abc1 = 0
F( 1, 12) = 16.92
Prob > F = 0.0014
test bc2 + abc2/2 = 0, accumulate
( 1) bc1 + .5 abc1 = 0
( 2) bc2 + .5 abc2 = 0
F( 2, 12) = 8.47
Prob > F = 0.0051
Stata 11/* a main-effect */
test a + ab/2 + (ac1+ac2)/3 + (abc1+abc2)/6 = 0
( 1) a + .5 ab + .3333333 ac1 + .3333333 ac2 + .1666667 abc1 + .1666667 abc2 = 0
F( 1, 12) = 112.50
Prob > F = 0.0000
/* b main-effect */
test b + ab/2 + (bc1+bc2)/3 + (abc1+abc2)/6 = 0
( 1) b + .5 ab + .3333333 bc1 + .3333333 bc2 + .1666667 abc1 + .1666667 abc2 = 0
F( 1, 12) = 0.50
Prob > F = 0.4930
/* c main-effect */
test c1 + ac1/2 + bc1/2 + abc1/4 = 0
( 1) c1 + .5 ac1 + .5 bc1 + .25 abc1 = 0
F( 1, 12) = 94.92
Prob > F = 0.0000
test c2 + ac2/2 + bc2/2 + abc2/4 = 0, accumulate
( 1) c1 + .5 ac1 + .5 bc1 + .25 abc1 = 0
( 2) c2 + .5 ac2 + .5 bc2 + .25 abc2 = 0
F( 2, 12) = 47.84
Prob > F = 0.0000
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 | 497.833333 11 45.2575758 33.94 0.0000
|
a | 150 1 150 112.50 0.0000
b | .666666667 1 .666666667 0.50 0.4930
a#b | 160.166667 1 160.166667 120.12 0.0000
c | 127.583333 2 63.7916667 47.84 0.0000
a#c | 18.25 2 9.125 6.84 0.0104
b#c | 22.5833333 2 11.2916667 8.47 0.0051
a#b#c | 18.5833333 2 9.29166667 6.97 0.0098
|
Residual | 16 12 1.33333333
-----------+----------------------------------------------------
Total | 513.833333 23 22.3405797
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 | -1.10e-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
------------------------------------------------------------------------------
/* abc interaction */
test 2.a#2.b#2.c 2.a#2.b#3.c
( 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
/* ab interaction */
test 2.a#2.b + (2.a#2.b#2.c+2.a#2.b#3.c)/3 == 0
( 1) 2.a#2.b + .3333333*2.a#2.b#2.c + .3333333*2.a#2.b#3.c = 0
F( 1, 12) = 120.13
Prob > F = 0.0000
/* ac interaction) */
test (2.a#2.c + 2.a#2.b#2.c/2 == 0)(2.a#3.c + 2.a#2.b#3.c/2 == 0)
( 1) 2.a#2.c + .5*2.a#2.b#2.c = 0
( 2) 2.a#3.c + .5*2.a#2.b#3.c = 0
F( 2, 12) = 6.84
Prob > F = 0.0104
/* bc interaction */
test (2.b#2.c + 2.a#2.b#2.c/2 == 0)(2.b#3.c + 2.a#2.b#3.c/2 == 0)
( 1) 2.b#2.c + .5*2.a#2.b#2.c = 0
( 2) 2.b#3.c + .5*2.a#2.b#3.c = 0
F( 2, 12) = 8.47
Prob > F = 0.0051
/* a main-effect */
test 2.a + 2.a#2.b/2 + (2.a#2.c+2.a#3.c)/3 + (2.a#2.b#2.c+2.a#2.b#3.c)/6 == 0
( 1) 2.a + .5*2.a#2.b + .3333333*2.a#2.c + .3333333*2.a#3.c + .1666667*2.a#2.b#2.c + .1666667*2.a#2.b#3.c = 0
F( 1, 12) = 112.50
Prob > F = 0.0000
/* b main-effect */
test 2.b + 2.a#2.b/2 + (2.b#2.c+2.b#3.c)/3 + (2.a#2.b#2.c+2.a#2.b#3.c)/6 == 0
( 1) 2.b + .5*2.a#2.b + .3333333*2.b#2.c + .3333333*2.b#3.c + .1666667*2.a#2.b#2.c + .1666667*2.a#2.b#3.c = 0
F( 1, 12) = 0.50
Prob > F = 0.4930
/* c main-effect */
test (2.c + 2.a#2.c/2 + 2.b#2.c/2 + 2.a#2.b#2.c/4 == 0)(3.c + 2.a#3.c/2 + 2.b#3.c/2 + 2.a#2.b#3.c/4 == 0)
( 1) 2.c + .5*2.a#2.c + .5*2.b#2.c + .25*2.a#2.b#2.c = 0
( 2) 3.c + .5*2.a#3.c + .5*2.b#3.c + .25*2.a#2.b#3.c = 0
F( 2, 12) = 47.84
Prob > F = 0.0000
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