Incorporating Subjective Characteristics in Product Design and Evaluations. Web Appendix

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1 Incorporatng ubctv Charactrtc n Product Dgn and Evaluaton Lan Luo, P.K. Kannan, and Bran T. Ratchford Wb Appndx A. TEP I MARKOV CHAI MOTE CARLO IMULATIO Our MCMC procdur carrd out by quntally gnratng draw from th followng dtrbuton:. Gnrat th loadng matrx Λ Th loadng matrx a pattrnd matrx wth both fxd and fr lmnt. om of th fxd lmnt ar zro and othr ar on, dpndng on th modl tup and th dntfcaton th rqurmnt. Lt λ dnot th column vctor of th fr lmnt n th loadng matrx, v ~ b th vctor of ndcator varabl that corrpond to th factor loadng, and ~ θ b th corrpondng ub-matrx of th maurmnt rror. Th full condtonal dtrbuton of λ gvn by: f ( λ * Π Π f ( v~ * f ( λ (W = = ~ Whr v~ * ~ MV( λ ( δ + bx, λ Δλ ' + θ and th pror dtrbuton of λ gvn a follow. For ach lmnt λ n λ, w lt λ = ld wth ld ~ Bta( ι, ο. Th tranformd Bta dtrbuton ha an [, ] ntrval for th factor loadng. W t ι = and ο = to nur dffu but propr pror.. Gnrat th factor cor z ( z * MV(, Δ f = ω z z (W Th man ωz and th varanc-covaranc matrx Δ z com from two data ourc. Th frt data ourc th maurmnt quaton v = Λz + ε. Th cond data ourc from th tructural quaton z = δ + B x + μ. Thrfor, th full condtonal dtrbuton for z can b - wrttn a MV ω,δ, whr ω = Δ [Δ (δ + B x + Λ'Θ v ] and Δ Δ + Λ'Θ Λ ( z z 3. Gnrat th maurmnt rror ε f ( ε * f ( v * f ( ε (W3 Whr v * ~ MV(Λ(δ + Bx, ΛΔΛ ' + Θ ε ~ MV(0, Θ 4. Gnrat δ Whr δ ~ MV( κ, Σ 5. Gnrat { b',b',...,b' }' B = J f z z ( δ * f ( v * f ( δ Π = (W4 =. z

2 f (B * Π f (v * f (B (W5 = Whr b ~ MV(β,D for =,..., J 6. Gnrat μ f ( μ * f (v * f (μ (W6 Whr μ ~ MV (0,Δ η = A, γ 7. Gnrat { } f (η * Π f ( y * f (v * f (η (W7 = ' Whr y * ~ MV(γδ + (A + γb x,γδγ + σ η ~ MV( ϕ, Ω 8. Gnrat f ( * f ( y * f (v * f ( (W8 Whr ~ (0, σ 9. Gnrat θ = dag Θ for =,..., K Lt v loadng, and z θ ( =,..., K : ( dnot th corrpondng ndcator varabl, ~ λ b th corrpondng factor rprnt th corrpondng factor cor. Th full condtonal dtrbuton for ~ f ( θ * = IG( ς +,[ ψ + ( v λ z ] (W9 = 0. Gnrat Δ f ( Δ = ' * = W ( ρ +,[ R + μμ ] (W0. Gnrat σ f ( σ * = IG( ϖ +,[ ψ + ( y A x γ z ] (W = = Whr ϖ = and ψ = ar th pror of th Invr Gamma dtrbuton.. Gnrat th hypr-paramtr ς for =,..., K f ( ς * Π f ( θ * f ( ς (W = Whr th pror dtrbuton dfnd a log( ς ~ (0, τ. W u th logtranformaton to nur a potv gn ofς. W tτ = Gnrat th hypr-paramtr ψ for =,..., K f ( ψ * = IG( g + ς,[ h + θ ] (W3 =

3 Whr th pror dtrbuton dfnd a ψ ~ IG( g, h. W t g = 0. 5and h = to nur dffu but propr pror. 4. Gnrat th hypr-paramtr ρ f ( ρ * Π f ( Δ * f ( ρ (W4 = Whr th pror dtrbuton dfnd a log( ρ ~ (0, τ ρ ρ > J. Th pror dtrbuton lctd bad on two crtra: ρ ha to b a potv numbr; and ρ nd to b gratr than th dmnon of th matrx J. W t τ ρ = Gnrat th hypr-paramtr R - f (R * Π f ( Δ * f (R (W5 = Whr th pror dtrbuton dfnd a R ~ W ( ρ0,( ρ0r 0 wth ρ 0 = 5 and R 0 = I ( I a J J dntty matrx. 6. Gnrat th hypr-paramtr κ f κ * = MV(, Γ (W6 ( ϑ κ κ Σ δ = Whr ϑκ = Γκ + Γκ rκ and Γ ( a rκ = 0 and Γ κ 0 = 00I ( I a J J dntty matrx. 7. Gnrat th hypr-paramtr Σ κ = ( Γ Σ κ 0 + f Σ * = W (( + ρ, ( δ κ( δ κ' + R (W7 ( Σ0 Σ0 = Whr th pror ar t a ρσ0 = 5 and R Σ 0 = 5I ( I a J J 8. Gnrat th hypr-paramtr β for =,..., J f β ( ϑ β β * MV(, Γ (W8 b D Whr = ϑ = Γ + Γ β β β 0r β 0 and Γ β = Γ pror a r = β 0 0 and Γ = 00 I β 0 (I am M dntty matrx. 9. Gnrat th hypr-paramtr D for =,..., J f ( D * = W ( ρ, R D D 0 β dntty matrx. + (W9 D. W t th pror. W t th Whr ρ D = + ρ D 0 and R D = ( ( ' R b β b β + D 0. W t th dffu but propr pror a ρ = D 0 0 and R I D 0 = 0 (I am M dntty matrx. 0. Gnrat th hypr-paramtr φ = 3

4 Ω η = Whr ϑ ϕ = Γ ϕ + Γ ϕ 0 rϕ 0 = 0and Γ = 00 ϕ 0 I.. Gnrat th hypr-paramtr Ω f φ * = MV(, Γ (W0 ( ϑ ϕ ϕ r ϕ 0 and Γ ϕ = Γ ϕ 0 Ω + f ( Ω * = W ( ρ Ω0 +, (η φ(η φ' + R Ω0 (W = Whr th pror ar t at ρω0 = 5 and R 0 = 5I Ω.. W t th pror a B. UPPLEMETAL IFORMATIO O TUDY OE: THE DEIG OF A HADHELD POWER TOOL In th tudy, th obctv attrbut wr lctd bad on: thr mportanc to th nd ur, and thr rlvanc to th ubctv charactrtc. For xampl, motor typ wa not ncludd bcau t lcton do not nflunc th motonal or phycal appal of th tool. Howvr, th lmtaton of xcludng motor powr a an obctv attrbut that th propod modl ncrmntal goodn-of-ft and prdctv powr ovr th bnchmar conont modl may b nflatd to om xtnt. Our xprmntal dgn among th calbraton profl provd a D-ffcncy of Th D-ffcncy ndx calculatd ung th formula wth rprntng th / M ( X ' X numbr of calbraton profl, X bng th dgn matrx ung ffct-typ dummy varabl codng, and M bng th dmnon of th dgn matrx X (Kuhfld, Toba, and Garratt 994. Th avrag ratng of ach prototyp on prcvd powr rangd from 3.49 (prototyp #5 to 5.9 (prototyp # and, on prcvd comfort, t vard from.768 (prototyp #3 to 5.89 (prototyp #9 on a 7-pont cal. Th ndcat that thr wa a condrabl amount of varaton n prcvd powr and prcvd comfort acro th prototyp. Th factor loadng from th propod modl ar prntd n th thrd column of Tabl W. In ordr to valuat how wll th valu of th obctv attrbut and th potror dtrbuton of th rlvant modl paramtr can prdct th actual ubctv ratng, w calculatd th Pudo R maur (lat column of Tabl W. In gnral, our fndng uggt that th ndcator varabl wr rlabl maur of th undrlyng latnt contruct and our modl wa abl to xplan a raonabl amount of varanc n th ubctv ratng. 4

5 TABLE W: IDICATOR VARIABLE ETIMATE: POWER TOOL TUDY Latnt Contruct Indcator Factor Loadng Pudo R Prcvd Powr pwr pwr 0.78 ( pwr ( Prcvd Comfort cft cft ( cft ( cft ( Potror tandard dvaton ar n parnth. Pudo R for rgron of ndcator on obctv attrbut (.. vˆ = Λ(δ B x. + In Tabl W, w provd a ummary of th comparon btwn our tructural quaton modl (EM and th path modl. W only rport th mpact of th ubctv charactrtc on purcha llhood n th tabl bcau th othr paramtr tmat from th EM modl and th path modl ar hghly mlar. TABLE W: COMPARIO BETWEE EM MODEL AD PATH MODEL: POWER TOOL TUDY EM Path Prcvd Powr* Prcvd Comfort In-ampl Ft - Pudo R RMD Prdctv Powr -MAE 0.79%.0% -RME.3%.8% C. UPPLEMETAL IFORMATIO O TUDY TWO: THE DEIG OF A TOOTHBRUH W cho th toothbruh catgory n our tudy for th followng raon. Frt, our plot tudy uggtd that th vat maorty of collg tudnt ar at lat omwhat concrnd about both dntal hygn and th prcvd comfort of thr toothbruh. cond, w blvd that th larg varty of toothbruh dgn on th mart an ndcaton of conumr htrognou prfrnc. Th D-Effcncy for our xprmntal dgn among th calbraton profl. Bad on th data collctd n Condton, th avrag ratng of ach toothbruh on prcvd ffctvn vard from a mnmum of.507 (toothbruh #5 to a maxmum of (prototyp # and on prcvd comfort rangng from (toothbruh #3 to 5.86 (toothbruh #4. 5

6 Tabl W3 gv th factor loadng tmatd from th propod modl. Th ndcator varabl appar to b good maur of th latnt contruct. Th Pudo R maur alo ndcatd that a raonabl amount of varanc n th ubctv ratng xpland by our modl. TABLE W3: IDICATOR VARIABLE ETIMATE: TOOTHBRUH TUDY Latnt Contruct Indcator Factor Loadng Pudo R Prcvd Effctvn ff ff 0.88( ff ( ff4 0.85( Prcvd Comfort cft 0.60 cft 0.705( cft ( Potror tandard dvaton ar n parnth. Pudo R for rgron of ndcator on obctv attrbut (.. vˆ = Λ(δ B x + In Tabl W4, w provd a ummary of th comparon btwn our tructural quaton modl (EM and th path modl. W only rport th mpact of th ubctv charactrtc on purcha llhood n th tabl bcau th othr paramtr tmat from th EM modl and th path modl ar hghly mlar. TABLE W4: COMPARIO BETWEE EM MODEL AD PATH MODEL: TOOTHBRUH TUDY EM Path Prcvd Effctvn Prcvd Comfort In-ampl Ft - Pudo R RMD Prdctv Powr -MAE 9.6% 0.5% -RME.74%.8% Rfrnc: Kuhfld, Warrn F., Randall D. Toba, and Mar Garratt (994, Effcnt Exprmntal Dgn wth Martng Rarch Applcaton, Journal of Martng Rarch, Vol. 3, o. 4,