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From |
"Scott Merryman" <smerryman@kc.rr.com> |

To |
<statalist@hsphsun2.harvard.edu> |

Subject |
st: RE: -xi- and interactions (was:your stata query) |

Date |
Wed, 22 Jun 2005 19:26:28 -0500 |

I am not sure this is correct. In the 2 variable cross product case, the variable T would be equivalent to T1, variable A would be equal to A1, and the cross product TxA would be equal to T1A1. With these three coefficients and the constant all 4 combinations can be accounted for. T = 0, A = 0 would be the constant T = 0, A = 1 would be A plus the constant T = 1, A = 0 would be T plus the constant T = 1, A = 1 would be TXA plus the constant For the 3 variable case, in the program below, the reference category for the cross product method is T = 0, A = 0, and B= 0; which will be equal to the constant. At the end of the program, a table is produced showing all 8 combinations of the categorical variables for both the "all dummy" and "cross product" method. The results are identical for both methods. +---------------------------------------------------+ | categories all_dummies cross_product | |---------------------------------------------------| | T = 0, A = 0, B = 0 3424.737 3424.737 | | T = 0, A = 0, B = 1 4062.308 4062.308 | | T = 0, A = 1, B = 0 2730.5 2730.5 | | T = 0, A = 1, B = 1 3368.071 3368.071 | |---------------------------------------------------| | T = 1, A = 0, B = 0 2372 2372 | | T = 1, A = 0, B = 1 3420 3420 | | T = 1, A = 1, B = 0 1991.667 1991.667 | | T = 1, A = 1, B = 1 3039.667 3039.667 | +---------------------------------------------------+ Scott ---------------------------------------------------------- sysuse auto, clear qui { gen t = fore mark a if price <4500 mark b if mpg <17 tab t, gen(t) tab a, gen(a) tab b , gen(b) gen t0a0 = t1*a1 gen t1a0 =t2*a1 gen t0a1 = t1*a2 gen t1a1 = t2*a2 gen t0b0= t1*b1 gen t1b0 = t2*b1 gen t0b1 = t1*b2 gen t1b1 = t2*b2 xi i.t*i.a i.t*i.b reg weight t0a0-t1b1, nocon nohead gen all_dummies = . local n = 1 forv i = 0/1 { forv j = 0/1 { forv k = 0/1 { replace all_dummies = _b[t`i'a`j'] + _b[t`i'b`k'] in `n' local n = `n' + 1 } } } gen categories = "" local n = 1 forv i = 0/1 { forv j = 0/1 { forv k = 0/1 { replace cate = "T = `i', A = `j', B = `k'" in `n' local n = `n' + 1 } } } reg weight _I* , nohead gen cross_product = . replace cross_product = _b[_cons] in 1 replace cross_product = _b[_cons] + _b[_Ib_1] in 2 replace cross_product = _b[_cons] + _b[_Ia_1] in 3 replace cross_product = _b[_cons] + _b[_Ib_1]+ _b[_Ia_1] in 4 replace cross_product = _b[_cons] + _b[_It_1] in 5 replace cross_product = _b[_cons] + _b[_It_1]+ _b[_Ib_1] +_b[_ItXb_1_1] in 6 replace cross_product = _b[_cons] + _b[_It_1]+ _b[_Ia_1] +_b[_ItXa_1_1] in 7 replace cross_product = _b[_cons] + _b[_It_1]+ _b[_Ia_1]+ _b[_Ib_1] /// + _b[_ItXa_1_1] + _b[_ItXb_1_1] in 8 } l cate all_dummies cross_product in 1/8, noobs abb(32) sep(4) > -----Original Message----- > From: owner-statalist@hsphsun2.harvard.edu [mailto:owner- > statalist@hsphsun2.harvard.edu] On Behalf Of "Laplante, Benoît" > Sent: Thursday, June 16, 2005 3:22 PM > To: statalist@hsphsun2.harvard.edu > Subject: st: -xi- and interactions (was:your stata query) > > The recent postings on -xi- reminded me of the following riddle about > interactions. > > Say you have two variables, T and A. Each has two values, low and high, > coded 0 and 1 respectively and for both variables. You assume that the > effect of A varies across the values of T, which is a very basic > definition of what an interaction is about. There are (at least) two ways > to deal with the problem. > > The first one is simply to build dummies that represent all the > combinations of values of the two variables and put all of them, minus > one, in the equation. So you would have T0A0, T0A1, T1A0 and T1A1. The > most obvious choice would be to exclude T0A0, but excluding any of the > four variables would provide equivalent results. So you would use three > variables to represent your interaction, say T0A1, T1A0 and T1A1. > > Another method, the one used by -xi-is to compute cross products of the > two variables. The procedure gives you an equation in which you are using > three variables, two of them labelled as the original variables and the > third one as their product. So you are using variables labelled T, A and > TA. Given the original coding scheme and the use of a cross-product, T is > actually equivalent to T1A0, A to T0A1 and TA to T1A1-(T0A1+T1A0). > > Both methods will produce equivalent results. > > Now let's say that you are adding a second variable to your equation, B, > whose effect is also assumed to vary across the values of T. Using the > first method, you would build an equation containing 6 terms, that is > three out of T0A0, T0A1, T1A0 and T1B1, and three out of T0B0, T0B1, T1B0 > and T1B1. > > If you were to use the second method, you would be using 5 terms: T, A, > TA, B, TB. The fit of the two models are not the same and I never found a > reference that dealt with how two such different models could be > equivalent. > > Actually, the second one seems to be equivalent to an equation in which, > using T0A0 and T0B0 as reference categories, T=(T1A0+T1B0)/2, which would > be an unwanted assumption in most cases. > > Anyone has an answer? > > Benoît Laplante, professeur > Université du Québec > Institut National de la Recherche Scientifique > Urbanisation, Culture et Société > http://www.inrs-ucs.uquebec.ca/default.asp?p=lapl > > > > * > * For searches and help try: > * http://www.stata.com/support/faqs/res/findit.html > * http://www.stata.com/support/statalist/faq > * http://www.ats.ucla.edu/stat/stata/ * * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

**References**:**st: -xi- and interactions (was:your stata query)***From:*"Laplante, Benoît" <Benoit_Laplante@UCS.INRS.Ca>

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