Chapter 5 Supplemental Text Material R S T. ij i j ij ijk

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1 Chpter 5 Supplementl Text Mterl 5-. Expected Men Squres n the Two-fctor Fctorl Consder the two-fctor fxed effects model y = µ + τ + β + ( τβ) + ε k R S T =,,, =,,, k =,,, n gven s Equton (5-) n the textook. We lst the expected men squres for ths model, ut do not develop them. It s reltvely esy to develop the expected men squres from drect pplcton of the expectton opertor. Consder fndng E MS E SS A I ( A) = ESS ( A) HG K J = where SS A s the sum of squres for the row fctor. Snce SS A ESS ( ) F y... = y.. n n A n E y E y..... n = F H G I K J Recll tht τ. = 0, β. = 0,( τβ). = 0,( τβ). = 0, nd ( τβ).. = 0, where the dot suscrpt mples summton over tht suscrpt. Now nd n y = yk = nµ + nτ + nβ + n( τβ) + ε = k = = nµ + nτ + ε.. n E y = E ( nµ ) + ( n) τ + ε + ( n) µτ + nµε + nτε n L O = ( nµ ) + ( n) τ + nσ n NM n n = µ + τ + σ Furthermore, we cn esly show tht y so = nµ + ε QP

2 Therefore n E ( y n E n... ) = ( µ + ε... ) = n E ( n µ ) + ε n... + µε... ) = ( nµ ) + nσ n = nµ + σ F HG I K J E MS E SS A ( A) = = ESS ( A) L nµ + nτ + σ ( nµ + σ ) NM L O = σ ( ) + n τ NM n τ = σ + whch s the result gven n the textook. The other expected men squres re derved smlrly. QP O QP 5-. The Defnton of Intercton In Secton 5- we ntroduced oth the effects model nd the mens model for the twofctor fctorl experment. If there s no ntercton n the two-fctor model, then Defne the row nd column mens s µ = µ + τ + β Then f there s no ntercton, µ µ.. = = = µ µ µ = µ + µ µ..

3 = = where µ µ µ. /. /. It cn lso e shown tht f there s no ntercton, ech cell men cn e expressed n terms of three other cell mens: µ = µ + µ µ Ths llustrtes why model wth no ntercton s sometmes clled n ddtve model, or why we sy the tretment effects re ddtve. When there s ntercton, the ove reltonshps do not hold. Thus the ntercton term ( τβ) cn e defned s or equvlently, ( τβ ) = µ ( µ + τ + β ) ( τβ ) = µ ( µ + µ µ ) = µ µ µ + µ Therefore, we cn determne whether there s ntercton y determnng whether ll the cell mens cn e expressed s µ = µ + τ + β. Sometmes nterctons re result of the scle on whch the response hs een mesured. Suppose, for exmple, tht fctor effects ct n multplctve fshon, µ = µτβ If we were to ssume tht the fctors ct n n ddtve mnner, we would dscover very quckly tht there s ntercton present. Ths ntercton cn e removed y pplyng log trnsformton, snce log µ = log µ + logτ + log β Ths suggests tht the orgnl mesurement scle for the response ws not the est one to use f we wnt results tht re esy to nterpret (tht s, no ntercton). The log scle for the response vrle would e more pproprte. Fnlly, we oserve tht t s very possle for two fctors to nterct ut for the mn effects for one (or even oth) fctor s smll, ner zero. To llustrte, consder the twofctor fctorl wth ntercton n Fgure 5- of the textook. We hve lredy noted tht the ntercton s lrge, AB = -9. However, the mn effect of fctor A s A =. Thus, the mn effect of A s so smll s to e neglgle. Now ths stuton does not occur ll tht frequently, nd typclly we fnd tht ntercton effects re not lrger thn the mn effects. However, lrge two-fctor nterctons cn msk one or oth of the mn effects. A prudent expermenter needs to e lert to ths posslty Estmle Functons n the Two-fctor Fctorl Model The lest squres norml equtons for the two-fctor fctorl model re gven n Equton (5-4) n the textook s:

4 n µ + n τ + n β + ( τβ ) = y = = n µ + n τ + n β + n ( τβ ) = y, =,,, n µ + n τ + n β + n ( τβ ) = y, =,,, n µ + n τ + n β + n( τβ ) = y,.. = =. R S T.. =,,, =,,, Recll tht n generl n estmle functon must e lner comnton of the left-hnd sde of the norml equtons. Consder contrst comprng the effects of row tretments nd. The contrst s τ τ + ( τβ) ( τβ).. Snce ths s ust the dfference etween two norml equtons, t s n estmle functon. Notce tht the dfference n ny two levels of the row fctor lso ncludes the dfference n verge ntercton effects n those two rows. Smlrly, we cn show tht the dfference n ny pr of column tretments lso ncludes the dfference n verge ntercton effects n those two columns. An estmle functon nvolvng nterctons s ( τβ) ( τβ) ( τβ) + ( τβ).... It turns out tht the only hypotheses tht cn e tested n n effects model must nvolve estmle functons. Therefore, when we test the hypothess of no ntercton, we re relly testng the null hypothess H0:( τβ) ( τβ). ( τβ). + ( τβ).. = 0 for ll, When we test hypotheses on mn effects A nd B we re relly testng the null hypotheses nd τ + τβ = τ + τβ = = τ + τβ H0: ( ). ( ). ( ). β + τβ = β + τβ = = β + τβ H0: ( ). ( ). ( ). Tht s, we re not relly testng hypothess tht nvolves only the equlty of the tretment effects, ut nsted hypothess tht compres tretment effects plus the verge ntercton effects n those rows or columns. Clerly, these hypotheses my not e of much nterest, or much prctcl vlue, when ntercton s lrge. Ths s why n the textook (Secton 5-) tht when ntercton s lrge, mn effects my not e of much prctcl vlue. Also, when ntercton s lrge, the sttstcl tests on mn effects my not relly tell us much out the ndvdul tretment effects. Some sttstcns do not even conduct the mn effect tests when the no-ntercton null hypothess s reected. It cn e shown [see Myers nd Mlton (99)] tht the orgnl effects model...

5 µ τ β τβ ε y k = ( ) + k cn e re-expressed s or = [ µ + τ + β+ ( τβ)] + [ τ τ + ( τβ) ( τβ)] + y k. [ β β + ( τβ) ( τβ)] + [( τβ) ( τβ) ( τβ) + ( τβ)] + ε... k = µ + τ + β + τβ + ε * * * * y k ( ) k * * * * It cn e shown tht ech of the new prmeters µ, τ, β, nd ( τβ) s estmle. Therefore, t s resonle to expect tht the hypotheses of nterest cn e expressed smply n terms of these redefned prmeters. It prtculr, t cn e shown tht there s no ntercton f nd only f ( τβ) * = 0. Now n the text, we presented the null hypothess of no ntercton s H0:( τβ ) = 0 for ll nd. Ths s not ncorrect so long s t s understood tht t s the model n terms of the redefned (or strred ) prmeters tht we re usng. However, t s mportnt to understnd tht n generl ntercton s not prmeter tht refers only to the ()th cell, ut t contns nformton from tht cell, the th row, the th column, nd the overll verge response. One fnl pont s tht s consequence of defnng the new strred prmeters, we hve ncluded certn restrctons on them. In prtculr, we hve * * * * * τ = 0, β = 0, ( τβ) = 0, ( τβ) = 0 nd ( τβ) = 0... These re the usul constrnts mposed on the norml equtons. Furthermore, the tests on mn effects ecome nd. * * * H : τ = τ = = τ = 0 0 * * * H : β = β = = β = 0 0 Ths s the wy tht these hypotheses re stted n the textook, ut of course, wthout the strs Regresson Model Formulton of the Two-fctor Fctorl We noted n Chpter 3 tht there ws close reltonshp etween ANOVA nd regresson, nd n the Supplementl Text Mterl for Chpter 3 we showed how the sngle-fctor ANOVA model could e formulted s regresson model. We now show how the two-fctor model cn e formulted s regresson model nd stndrd multple regresson computer progrm employed to perform the usul ANOVA.

6 We wll use the ttery lfe experment of Exmple 5- to llustrte the procedure. Recll tht there re three mterl types of nterest (fctor A) nd three tempertures (fctor B), nd the response vrle of nterest s ttery lfe. The regresson model formulton of n ANOVA model uses ndctor vrles. We wll defne the ndctor vrles for the desgn fctors mterl types nd temperture s follows: Mterl type X X Temperture X 3 X The regresson model s y = β + β x + β x + β x + β x k 0 k k 3 k 3 4 k 4 + β x x + β x x + β x x + β x x + ε 5 k k 3 6 k k 4 7 k k 3 8 k k 4 k () where, =,,3 nd the numer of replctes k =,,3,4. In ths model, the terms β x + β x represent the mn effect of fctor A (mterl type), nd the terms k k β3xk 3+ β4xk 4 represent the mn effect of temperture. Ech of these two groups of terms contns two regresson coeffcents, gvng two degrees of freedom. The terms β5xkxk 3 + β6xkxk 4 + β7xk xk 3 + β8xk xk 4 n Equton () represent the AB ntercton wth four degrees of freedom. Notce tht there re four regresson coeffcents n ths term. Tle shows the dt from ths experment, orgnlly presented n Tle 5- of the text. In Tle, we hve shown the ndctor vrles for ech of the 36 trls of ths experment. The notton n ths tle s X = x,,,3,4 for the mn effects n the ove regresson model nd X 5 = x x 3, X 6 = x x 4,, X 7 = x x 3, nd X 8 = x x 4, for the ntercton terms n the model.

7 Tle. Dt from Exmple 5- n Regresson Model Form Y X X X 3 X 4 X 5 X 6 X 7 X Ths tle ws used s nput to the Mnt regresson procedure, whch produced the followng results for fttng Equton (): Regresson Anlyss The regresson equton s y = x + 9. x x3-77. x x5-9.0 x x x8

8 mnt Output (Contnued) Predctor Coef StDev T P Constnt x x x x x x x x S = 5.98 R-Sq = 76.5% R-Sq(d) = 69.6% Anlyss of Vrnce Source DF SS MS F P Regresson Resdul Error Totl Source DF Seq SS x 4.7 x x3 76. x x x x x Frst exmne the Anlyss of Vrnce nformton n the ove dsply. Notce tht the regresson sum of squres wth 8 degrees of freedom s equl to the sum of the sums of squres for the mn effects mterl types nd temperture nd the ntercton sum of squres from Tle 5-5 n the textook. Furthermore, the numer of degrees of freedom for regresson (8) s the sum of the degrees of freedom for mn effects nd ntercton ( + + 4) from Tle 5-5. The F-test n the ove ANOVA dsply cn e thought of s testng the null hypothess tht ll of the model coeffcents re zero; tht s, there re no sgnfcnt mn effects or ntercton effects, versus the lterntve tht there s t lest one nonzero model prmeter. Clerly ths hypothess s reected. Some of the tretments produce sgnfcnt effects. Now consder the sequentl sums of squres t the ottom of the ove dsply. Recll tht X nd X represent the mn effect of mterl types. The sequentl sums of squres re computed sed on n effects dded n order pproch, where the n order refers to the order n whch the vrles re lsted n the model. Now

9 SS = SS ( X ) + MterlTypes SS ( X X ) =. +. = whch s the sum of squres for mterl types n tle 5-5. The notton SS( X X) ndctes tht ths s sequentl sum of squres; tht s, t s the sum of squres for vrle X gven tht vrle X s lredy n the regresson model. Smlrly, SS = SS ( X X, X ) + Temperture SS ( X X, X, X ) =. +. = whch closely grees wth the sum of squres for temperture from Tle 5-5. Fnlly, note tht the ntercton sum of squres from Tle 5-5 s SS = Intercton SS( X X X X X + 5,, 3, 4) SS( X6 X, X, X3, X4, X5) + SS( X7 X, X, X3, X4, X5, X6) + SS( X8 X, X, X3, X4, X5, X6, X7) = = When the desgn s lnced, tht s, we hve n equl numer of oservtons n ech cell, we cn show tht ths model regresson pproch usng the sequentl sums of squres produces results tht re exctly dentcl to the usul ANOVA. Furthermore, ecuse of the lnced nture of the desgn, the order of the vrles A nd B does not mtter. The effects dded n order prttonng of the overll model sum of squres s sometmes clled Type nlyss. Ths termnology s prevlent n the SAS sttstcs pckge, ut other uthors nd softwre systems lso use t. An lterntve prttonng s to consder ech effect s f t were dded lst to model tht contns ll the others. Ths effects dded lst pproch s usully clled Type 3 nlyss. There s nother wy to use the regresson model formulton of the two-fctor fctorl to generte the stndrd F-tests for mn effects nd ntercton. Consder fttng the model n Equton (), nd let the regresson sum of squres n the Mnt output ove for ths model e the model sum of squres for the full model. Thus, SS Model ( FM ) = wth 8 degrees of freedom. Suppose we wnt to test the hypothess tht there s no ntercton. In terms of model (), the no-ntercton hypothess s H0: β5 = β6 = β7 = β8 = 0 H : t lest one β 0, = 5678,,, 0 When the null hypothess s true, reduced model s y = β + β x + β x + β x + β x + ε k 0 k k 3 k 3 4 k 4 k (3) Fttng Equton () usng Mnt produces the followng: () The regresson equton s y = + 5. x x x x4

10 Predctor Coef StDev T P Constnt x x x x S = 9.97 R-Sq = 64.% R-Sq(d) = 59.5% Anlyss of Vrnce Source DF SS MS F P Regresson Resdul Error Totl The model sum of squres for ths reduced model s SS Model (RM) = wth 4 degrees of freedom. The test of the no-ntercton hypotheses () s conducted usng the extr sum of squres SS Model ( Intercton) = SS Model ( FM) SS Model ( RM) = 59, , 8. 0 = 9, 604. wth 8 4 = 4 degrees of freedom. Ths quntty s, prt from round-off errors n the wy the results re reported n Mnt, the ntercton sum of squres for the orgnl nlyss of vrnce n Tle 5-5 of the text. Ths s mesure of fttng ntercton fter fttng the mn effects. Now consder testng for no mn effect of mterl type. In terms of equton (), the hypotheses re H0: β = β = 0 (4) H : t lest one β 0, =, 0 Becuse we re usng lnced desgn, t turns out tht to test ths hypothess ll we hve to do s to ft the model y = β + β x + β x + ε Fttng ths model n Mnt produces Regresson Anlyss The regresson equton s y = x x k 0 k k k (5) Predctor Coef StDev T P Constnt x x

11 Anlyss of Vrnce Source DF SS MS F P Regresson Resdul Error Totl Notce tht the regresson sum of squres for ths model [Equton (5)] s essentlly dentcl to the sum of squres for mterl types n tle 5-5 of the text. Smlrly, testng tht there s no temperture effect s equvlent to testng H0: β3 = β4 = 0 H : t lest one β 0, = 34, To test the hypotheses n (6), ll we hve to do s ft the model y = β + β x + β x + ε The Mnt regresson output s Regresson Anlyss The regresson equton s y = x x4 0 k 0 3 k 3 4 k 4 k (7) Predctor Coef StDev T P Constnt x x S = 34.7 R-Sq = 50.4% R-Sq(d) = 47.4% Anlyss of Vrnce Source DF SS MS F P Regresson Resdul Error Totl Notce tht the regresson sum of squres for ths model, Equton (7), s essentlly equl to the temperture mn effect sum of squres from Tle 5-5. (6) 5-5. Model Herrchy In Exmple 5-4 we used the dt from the ttery lfe experment (Exmple 5-) to demonstrte fttng response curves when one of the fctors n two-fctor fctorl experment ws quntttve nd the other ws qulttve. In ths cse the fctors re temperture (A) nd mterl type (B). Usng the Desgn-Expert softwre pckge, we ft model tht the mn effect of mterl type, the lner nd qudrtc effects of temperture, the mterl type y lner effect of temperture ntercton, nd the mterl type y qudrtc effect of temperture ntercton. Refer to Tle 5-5 n the textook. From exmnng ths tle, we oserved tht the qudrtc effect of temperture nd the

12 mterl type y lner effect of temperture ntercton were not sgnfcnt; tht s, they hd frly lrge P-vlues. We left these non-sgnfcnt terms n the model to preserve herrchy. The herrchy prncpl sttes tht f model contns hgher-order term, then t should lso contn ll the terms of lower-order tht comprse t. So, f second-order term, such s n ntercton, s n the model then ll mn effects nvolved n tht ntercton s well s ll lower-order nterctons nvolvng those fctors should lso e ncluded n the model. There re tmes tht herrchy mkes sense. Generlly, f the model s gong to e used for explntory purposes then herrchcl model s qute logcl. On the other hnd, there my e stutons where the non-herrchcl model s much more logcl. To llustrte, consder nother nlyss of Exmple 5-4 n Tle, whch ws otned from Desgn-Expert. We hve selected non-herrchcl model n whch the qudrtc effect of temperture ws not ncluded (t ws n ll lkelhood the wekest effect), ut oth two-degree-of-freedom components of the temperture-mterl type ntercton re n the model. Notce from Tle tht the resdul men squre s smller for the nonherrchcl model (653.8 versus 675. from Tle 5-5). Ths s mportnt, ecuse the resdul men squre cn e thought of s the vrnce of the unexplned resdul vrlty, not ccounted for y the model. Tht s, the non-herrchcl model s ctully etter ft to the expermentl dt. Notce lso tht the stndrd errors of the model prmeters re smller for the nonherrchcl model. Ths s n ndcton tht he prmeters re estmted wth etter precson y levng out the nonsgnfcnt terms, even though t results n model tht does not oey the herrchy prncpl. Furthermore, note tht the 95 percent confdence ntervls for the model prmeters n the herrchcl model re lwys longer thn ther correspondng confdence ntervls n the non-herrchcl model. The non-herrchcl model, n ths exmple, does ndeed provde etter estmtes of the fctor effects tht otned from the herrchcl model Tle. Desgn-Expert Output for Non-herrchcl Model, Exmple 5-4. ANOVA for Response Surfce Reduced Cuc Model Anlyss of vrnce tle [Prtl sum of squres] Sum of Men F Source Squres DF Squre Vlue Pro > F Model < A B < AB AB Resdul Lck of Ft Pure Error Cor Totl

13 Std. Dev R-Squred Men Ad R-Squred C.V. 4.3 Pred R-Squred PRESS Adeq Precson 8.85 Coeffcent Stndrd 95% CI 95% CI Term Estmte DF Error Low Hgh Intercept A[] A[] B-Temp A[]B A[]B A[]B A[]B Supplementl Reference Myers, R. H. nd Mlton, J. S. (99), A Frst Course n the Theory of the Lner Model, PWS-Kent, Boston, MA.