under contrast effects for interactions

[Doug Hess]

I have a pretty basic question about interpreting the results from using the -contrast- command for interaction effects. In the example below, why does the interaction in the results from contrast give the same as in the regression, but the two effects on their own do not match the regression? (Using Stat 15)

sysuse auto, clear
tab for rep
drop if rep==1
drop if rep==2
reg mpg head trunk weight length i.for##i.rep
contrast i.for##i.rep, effect nowald

[Joseph Coveney]

-contrast- will use ANOVA contrasts by default. With factorial ANOVA, the parameterization of the contrasts is such that the main effects will differ from what you get with linear regression using indicator variables, but the interaction term will be the same.

[Doug Hess]

A better example might be this:

use http://www.stata-press.com/data/r15/cholesterol, clear
regress chol i.agegrp##i.race
contrast i.age##i.race, effects nowald

Why do the age groups have different coeffs in the regression and the contrast effects?

[Joseph Coveney]

Here's a better example. You'll see why the coefficients differ for the main effects. (Answer: they test different hypotheses. See the -lincom- output.)

.ÿ
.ÿversionÿ15.1

.ÿ
.ÿclearÿ*

.ÿ
.ÿsetÿseedÿ`=strreverse("1528380")'

.ÿquietlyÿsetÿobsÿ20

.ÿ
.ÿgenerateÿbyteÿAÿ=ÿmod(_n,ÿ2)

.ÿgenerateÿbyteÿBÿ=ÿ_nÿ>=ÿ_Nÿ/ÿ2

.ÿgenerateÿbyteÿoutÿ=ÿAÿ+ÿBÿ+ÿAÿ*ÿBÿ+ÿrnormal()

.ÿ
.ÿ*
.ÿ*ÿBeginÿhere
.ÿ*
.ÿregressÿoutÿi.(A##B)

ÿÿÿÿÿÿSourceÿ|ÿÿÿÿÿÿÿSSÿÿÿÿÿÿÿÿÿÿÿdfÿÿÿÿÿÿÿMSÿÿÿÿÿÿNumberÿofÿobsÿÿÿ=ÿÿÿÿÿÿÿÿ20
-------------+----------------------------------ÿÿÿF(3,ÿ16)ÿÿÿÿÿÿÿÿ=ÿÿÿÿÿÿ4.49
ÿÿÿÿÿÿÿModelÿ|ÿÿÿÿÿÿÿÿ6.65ÿÿÿÿÿÿÿÿÿ3ÿÿ2.21666667ÿÿÿProbÿ>ÿFÿÿÿÿÿÿÿÿ=ÿÿÿÿ0.0181
ÿÿÿÿResidualÿ|ÿÿÿÿÿÿÿÿÿ7.9ÿÿÿÿÿÿÿÿ16ÿÿÿÿÿÿ.49375ÿÿÿR-squaredÿÿÿÿÿÿÿ=ÿÿÿÿ0.4570
-------------+----------------------------------ÿÿÿAdjÿR-squaredÿÿÿ=ÿÿÿÿ0.3552
ÿÿÿÿÿÿÿTotalÿ|ÿÿÿÿÿÿÿ14.55ÿÿÿÿÿÿÿÿ19ÿÿ.765789474ÿÿÿRootÿMSEÿÿÿÿÿÿÿÿ=ÿÿÿÿ.70267

------------------------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿoutÿ|ÿÿÿÿÿÿCoef.ÿÿÿStd.ÿErr.ÿÿÿÿÿÿtÿÿÿÿP>|t|ÿÿÿÿÿ[95%ÿConf.ÿInterval]
-------------+----------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿ1.Aÿ|ÿÿÿÿÿÿÿÿÿ.4ÿÿÿ.4713677ÿÿÿÿÿ0.85ÿÿÿ0.409ÿÿÿÿ-.5992549ÿÿÿÿ1.399255
ÿÿÿÿÿÿÿÿÿ1.Bÿ|ÿÿÿÿÿÿÿÿÿ.5ÿÿÿ.4535738ÿÿÿÿÿ1.10ÿÿÿ0.287ÿÿÿÿ-.4615334ÿÿÿÿ1.461533
ÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿA#Bÿ|
ÿÿÿÿÿÿÿÿ1ÿ1ÿÿ|ÿÿÿÿÿÿÿÿÿ.7ÿÿÿ.6350033ÿÿÿÿÿ1.10ÿÿÿ0.287ÿÿÿÿ-.6461468ÿÿÿÿ2.046147
ÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿ_consÿ|ÿÿ-6.66e-16ÿÿÿ.3513367ÿÿÿÿ-0.00ÿÿÿ1.000ÿÿÿÿ-.7448006ÿÿÿÿ.7448006
------------------------------------------------------------------------------

.ÿcontrastÿAÿBÿA#B

Contrastsÿofÿmarginalÿlinearÿpredictions

Marginsÿÿÿÿÿÿ:ÿasbalanced

------------------------------------------------
ÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿÿÿÿÿÿÿÿÿdfÿÿÿÿÿÿÿÿÿÿÿFÿÿÿÿÿÿÿÿP>F
-------------+----------------------------------
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ÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿA#Bÿ|ÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿ1.22ÿÿÿÿÿ0.2866
ÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿDenominatorÿ|ÿÿÿÿÿÿÿÿÿ16
------------------------------------------------

.ÿ
.ÿanovaÿoutÿA##B

ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿNumberÿofÿobsÿ=ÿÿÿÿÿÿÿÿÿ20ÿÿÿÿR-squaredÿÿÿÿÿ=ÿÿ0.4570
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿRootÿMSEÿÿÿÿÿÿ=ÿÿÿÿ.702673ÿÿÿÿAdjÿR-squaredÿ=ÿÿ0.3552

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ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ-----------+----------------------------------------------------
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿModelÿ|ÿÿÿÿÿÿÿ6.65ÿÿÿÿÿÿÿÿÿÿ3ÿÿÿ2.2166667ÿÿÿÿÿÿ4.49ÿÿ0.0181
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ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿA#Bÿ|ÿÿÿÿÿÿÿÿÿ.6ÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿÿÿ.6ÿÿÿÿÿÿ1.22ÿÿ0.2866
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
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ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ-----------+----------------------------------------------------
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.ÿ
.ÿ*
.ÿ*ÿExplanation
.ÿ*
.ÿquietlyÿregressÿoutÿi.(A##B)

.ÿ//ÿlinear-regressionÿindicator-variableÿcontrastÿ(matchesÿ-regress-)
.ÿlincomÿ_b[1.A]

ÿ(ÿ1)ÿÿ1.Aÿ=ÿ0

------------------------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿoutÿ|ÿÿÿÿÿÿCoef.ÿÿÿStd.ÿErr.ÿÿÿÿÿÿtÿÿÿÿP>|t|ÿÿÿÿÿ[95%ÿConf.ÿInterval]
-------------+----------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿ(1)ÿ|ÿÿÿÿÿÿÿÿÿ.4ÿÿÿ.4713677ÿÿÿÿÿ0.85ÿÿÿ0.409ÿÿÿÿ-.5992549ÿÿÿÿ1.399255
------------------------------------------------------------------------------

.ÿ
.ÿ//ÿANOVAÿcontrastÿforÿmainÿeffectsÿ(matchesÿ-constrast-ÿandÿ-anova-)
.ÿlincomÿ_b[1.A]ÿ+ÿ_b[1.A#1.B]ÿ/ÿ2

ÿ(ÿ1)ÿÿ1.Aÿ+ÿ.5*1.A#1.Bÿ=ÿ0

------------------------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿoutÿ|ÿÿÿÿÿÿCoef.ÿÿÿStd.ÿErr.ÿÿÿÿÿÿtÿÿÿÿP>|t|ÿÿÿÿÿ[95%ÿConf.ÿInterval]
-------------+----------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿ(1)ÿ|ÿÿÿÿÿÿÿÿ.75ÿÿÿ.3175016ÿÿÿÿÿ2.36ÿÿÿ0.031ÿÿÿÿÿ.0769266ÿÿÿÿ1.423073
------------------------------------------------------------------------------

.ÿ
.ÿexit

endÿofÿdo-file


.ÿ

[Joseph Coveney]

Should have been a floating-point variable for the outcome, but the basic message is the same.

Here's the code below if you want to run it.

version 15.1

clear *

set seed `=strreverse("1528380")'
quietly set obs 20

generate byte A = mod(_n, 2)
generate byte B = _n >= _N / 2
generate double out = A + B + A * B + rnormal()

*
* Begin here
*
regress out i.(A##B)
contrast A B A#B

anova out A##B

*
* Explanation
*
quietly regress out i.(A##B)
// linear-regression indicator-variable contrast (matches -regress-)
lincom _b[1.A]

// ANOVA contrast for main effects (matches -constrast- and -anova-)
lincom _b[1.A] + _b[1.A#1.B] / 2

exit

[Doug Hess]

Thanks, Joseph. I hope you don't mind, but I made a few more tweaks to make it clearer, to me at least & perhaps students, what was going on.
-DH

clear *

set seed `=strreverse("1528380")'
quietly set obs 20

generate byte A = mod(_n, 2)
generate byte B = _n >= _N / 2
generate double out = (A + B) + (A * B) + abs(rnormal())

*
* Begin here
*
anova out A##B

regress out i.(A##B)

contrast A##B, effects 

*
* Explanation
*
// linear-regression indicator-variable contrast (matches -regress-)
lincom _b[1.A]

// ANOVA contrast for main effects (matches -constrast- and -anova-)
lincom _b[1.A] + _b[1.A#1.B] / 2

* Show that Coeff for regression transforms into ANOVA contrast main effects    
di .8677731 + (1.522519/2)

* I.e., the coeff for A in the regression is equal to the coeff in 
**     ANOVA minus one-half the effect of the interaction
di 1.629033-(1.522519/2)