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Date:         Wed, 25 Apr 2007 19:32:39 -0700
Reply-To:     David L Cassell <davidlcassell@MSN.COM>
Sender:       "SAS(r) Discussion" <SAS-L@LISTSERV.UGA.EDU>
From:         David L Cassell <davidlcassell@MSN.COM>
Subject:      Re: about weighted least-squares estimation
In-Reply-To:  <1177512597.671577.248740@n35g2000prd.googlegroups.com>
Content-Type: text/plain; format=flowed

shiling99@YAHOO.COM wrote: >On Apr 24, 3:33 am, harry <xuqiyuan1...@gmail.com> wrote: > > The experiment studied the effect of speed(x1), pressure (x2), and > > distance (x3) on a printing machine's ability to apply coloring inks > > on package labels. The following table summarizes the experimental > > results. > > i X1 x2 x3 yi1 yi2 yi3 average of yi > si > > 1 -1 -1 -1 34 10 28 24 12.5 > > 2 0 -1 -1 115 116 130 120.3 8.4 > > 3 1 -1 -1 192 186 263 213.7 42.8 > > 4 -1 0 -1 82 88 88 86 3.7 > > 5 0 0 -1 44 178 188 136.7 80.4 > > 6 1 0 -1 322 350 350 340.7 16.2 > > 7 -1 1 -1 141 110 86 112.3 27.6 > > 8 0 1 -1 259 251 259 256.3 4.6 > > 9 1 1 0 290 280 245 271.7 23.6 > > 10 -1 -1 0 81 81 81 81 0 > > 11 0 -1 0 90 122 93 101.7 17.7 > > 12 1 -1 0 319 376 376 357 32.9 > > 13 -1 0 0 180 180 154 171.3 15 > > 14 0 0 0 372 372 372 372 0 > > 15 1 0 0 541 568 396 501.7 92.5 > > 16 -1 1 0 288 192 312 264 63.5 > > 17 0 1 0 432 336 513 427 88.6 > > 18 1 1 0 713 725 754 730.7 21.1 > > 19 -1 -1 1 364 99 199 220.7 133.8 > > 20 0 -1 1 232 221 266 239.7 23.5 > > 21 1 -1 1 408 415 443 422 18.5 > > 22 -1 0 1 182 233 182 199 29.4 > > 23 0 0 1 507 515 434 485.3 44.6 > > 24 1 0 1 846 535 640 673.7 158.2 > > 25 -1 1 1 236 126 168 176.7 55.5 > > 26 0 1 1 660 440 403 501 138.9 > > 27 1 1 1 878 991 1161 1010 142.5 > > > > Since the data has Heteroscedasticity, weighted ordinary least- > > squares estimation is prefered to construct a Linear Regression model. > > But maybe use the sample variances as the basis for weighted least- > > squares estimation is not the best. > > How to fit a linear model to an appropriate transformation of the > > sample variances and thus to develop a more appropriate weights? > >In a regression model as, > y=a+bx+err for i=1,2,3,...,n > >The heteroscedasticity is defined for err if it exists. If your >regression residual=y-(ahat+bhat*x) having heteroscedasticity, then >GLM should be applied. It is conceptually incorrect because you judge >it from ys in your data. > >It seems to me that you have repeated measures in your data as given > > x1 =-1,x2=-1,x3=-1 > >you have three measures of y ( 34 10 28 ). > >If you believe that each of these three are from the same >distribution, for different xs are from different distribution. Then >proc mix with repeated statement should work for you. > >If you really want weight ols, then proc model will do. You should >look for feasibel generalized least square(FGLS) in literature. > >BTW your data may have problems in i=10,14. It will have a perfect fit >for that groups / no variations. > >Here is a similation data and sas pgm. > >HTH > >data t1; > do x=1 to 3; > i=x; > do j=1 to 50; > y=5+2*x+x*rannor(3450); > output; > end; > end; >run; > >proc reg data=t1; > model y=x; >run; >quit; > > >data t2; > set t1(where=(i=1) rename=(y=y1 x=x1)); > set t1(where=(i=2) rename=(y=y2 x=x2)); > set t1(where=(i=3) rename=(y=y3 x=x3)); >run; > >proc mixed data=t1; > model y = x / s; > repeated / grp=x r=1-3; >run; > >proc model data=t2; > y1=a+b*x1; > y2=a+b*x2; > y3=a+b*x3; > fit y1 y2 y3 /fiml sur ; > run; > quit;

You make some good points here. But I'm interpreting the request differently from you. (Note that I may be wrong here.)

I've seen data like this in SQC (Statistical Quality Control) before, and in that case we're looking at 3 independent observations at each level of a factorial design. So we don't have to worry (well, not a lot) about the repeated measures issue. Unless the experiment was done badly. Which happens way too often.

Of course, there's still a major SQC problem here, in that the need may not be to model Y, but to model the response surface and find out where Y is a max, or a min, or most stable, or least variable, or a combination of some of these. Since we didn't get a decent answer on *that* part, we may never know what the teacher was actually asking for.

David -- David L. Cassell mathematical statistician Design Pathways 3115 NW Norwood Pl. Corvallis OR 97330

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