Islamic University, Gaza - Palestine. Chapter 3 Experiments with a Single Factor: The Analysis of Variance
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1 Islm Uverst, Gz - Pleste Chpter 3 xpermets wth Sgle Ftor: The Alss of Vre
2 Islm Uverst, Gz - Pleste 3. A xmple Chpter : A sgl-ftor expermet wth two levels of the ftor Cosder sgl-ftor expermets wth levels of the ftor, xmple: The tesle stregth of ew sthet fber. The weght peret of otto Fve levels: 5%, 0%, 5%, 30%, 35% = 5 d = 5
3 Islm Uverst, Gz - Pleste Does hgg the otto weght peret hge the me tesle stregth? Is there optmum level for otto otet? 3
4 Islm Uverst, Gz - Pleste 3. The Alss of Vre levels (tretmets) of ftor d repltes for eh level. j : the jth observto tke uder ftor level or tretmet. 4
5 Islm Uverst, Gz - Pleste Models for the Dt Mes model: j j, j,,...,,,..., j s the j th observto, s the me of the th ftor level, j s rdom error wth me zero, ffets model: j j, j,,...,,,..., 5
6 Islm Uverst, Gz - Pleste Ler sttstl model Oe-w or Sgl-ftor lss of vre model Completel rdomzed desg: the expermets re performed rdom order so tht the evromet whh the tretmet re ppled s s uform s possble. For hpothess testg, the model errors re ssumed to be ormll d depedetl dstrbuted rdom vrbles wth me zero d vre, σ,.e. j ~ N(μ+τ, σ ) Fxed effet model: levels hve bee spefll hose b the expermeter. 6
7 Islm Uverst, Gz - Pleste 3.3 Alss of the Fxed ffets Model Iterested testg the eqult of the tretmet mes, d ( j ) = μ + τ = μ, =,,, H 0: μ = μ = = μ H : μ μ j, for t lest oe pr (, j) Costrt (Restrt): H 0 : τ = τ = = τ = 0 v.s. H : τ 0, for t lest oe 0 7
8 Islm Uverst, Gz - Pleste Nottos:, j j j j /, / N, N : the totl umber of observtos Deomposto of the Totl Sum of Squres Totl vrblt to ts ompoet prts. The totl sum of squres ( mesure of overll vrblt the dt) SS ( ) T j Degree of freedom: = N 8 j..
9 Islm Uverst, Gz - Pleste ( j..) [(...) ( j.)] j j (...) ( j.) j SS SS SS T Tretmets rror SS Tretmet : sum of squres of the dfferees betwee the tretmet verges (sum of squres due to tretmets) d the grd verge, d degree of freedom SS : sum of squres of the dfferees of observtos wth tretmets from the tretmet verge (sum of squres due to error), d N degrees of freedom. 9
10 Islm Uverst, Gz - Pleste SS SS SS T Tretmets A lrge vlue of SS Tretmets reflets lrge dfferees tretmet mes A smll vlue of SS Tretmets lkel dtes o dfferees tretmet mes df Totl = df Tretmet + df rror SS ( ) S ( ) S N ( ) ( ) If there re o dfferees betwee tretmet mes, SS Tretmets ( ) 0
11 Islm Uverst, Gz - Pleste Me squres: MS Tretmets SSTretmets, MS SS N ( MS ( MS ) ( N j j ) ( Tretmets ) /( ) ) 3.3. Sttstl Alss Assumpto: ξ j re ormll d depedetl dstrbuted wth me zero d vre σ
12 Islm Uverst, Gz - Pleste SS T / σ ~ Ch-squre (N ), SS / σ ~ Ch-squre (N ), SS Tretmets /σ ~ Ch-squre ( ), d SS / σ d SS Tretmets / σ re depedet (Theorem 3.) H 0 : τ = τ =. = τ = 0 v.s. H : τ 0, for t lest oe
13 Islm Uverst, Gz - Pleste Rejet H 0 f F 0 > F α, -, N- Rewrte the sum of squres: See pge 7 SS SS SS T Tretmets j j N SS T SS Tretmets N 3
14 Islm Uverst, Gz - Pleste Respose:Stregth ANOVA for Seleted Ftorl Model Alss of vre tble [Prtl sum of squres] Sum of Me F SoureSqures DF Squre Vlue Prob > F Model < A < Pure rror Cor Totl Std. Dev..84 R-Squred Me 5.04 Adj R-Squred C.V Pred R-Squred PRSS 5.88 Adeq Preso
15 Islm Uverst, Gz - Pleste stmto of the Model Prmeters Model: j = µ + τ +ξ j stmtors: Cofdee tervls: ˆ ˆ 5 ˆ MS t MS t MS t MS t N N j j N j N N, /, /, /, / ) /, ( ~
16 Islm Uverst, Gz - Pleste xmple 3.3 (pge 75) Smulteous Cofdee Itervls (Boferro method): Costrut set of r smulteous ofdee tervls o tretmet mes whh s t lest 00(-): 00(-/r) C.I. s Ubled Dt Let observtos be tke uder tretmet, =,,,, N =, ( some of the mesured dt re mssed) SS SS T Tretmets j j N N 6
17 Islm Uverst, Gz - Pleste. The test sttst s reltvel sestve to smll deprtures from the ssumpto of equl vre for the tretmets f the smple szes re equl.. The power of the test s mxmzed f the smples re of equl sze. 7
18 Islm Uverst, Gz - Pleste 3.4 Model Adequ Chekg Assumptos: j ~ N(µ+τ, σ ) The exmto of resduls Defto of resdul: The resduls should be struture-less. e j j ˆ j, ˆ j ˆ ˆ ( ) 8
19 Islm Uverst, Gz - Pleste 3.4. The Normlt Assumpto Plot hstogrm of the resduls Plot orml probblt plot of the resduls See Tble 3-6 9
20 Islm Uverst, Gz - Pleste M be Slghtl skewed (rght tl s loger th left tl) Lght tl (the left tl of error s ther th the tl prt of stdrd orml) Outlers The possble uses of outlers: lultos, dt odg, op error,. Sometmes outlers re more formtve th the rest of the dt. 0
21 Islm Uverst, Gz - Pleste Detet outlers: xme the stdrdzed resduls, d j e j MS 3.4. Plot of Resduls Tme Sequee Plottg the resduls tme order of dt olleto s helpful detetg orrelto betwee the resduls. Idepedee ssumpto
22 Islm Uverst, Gz - Pleste R e s d u ls v s. R u Res duls R u N u m b e r
23 Islm Uverst, Gz - Pleste Plot of Resduls Versus Ftted Vlues Plot the resduls versus the ftted vlues Struture-less 5. R e s d u ls v s. P r e d t e d. 9 5 Res duls P r e d t e d 3
24 Islm Uverst, Gz - Pleste Noostt vre: the vre of the observtos reses s the mgtude of the observto rese,.e. j If the ftor levels hvg the lrger vre lso hve smll smple szes, the tul tpe I error rte s lrger th tpted. Vre-stblzg trsformto Posso Squre root trsformto j Logorml Logrthm trsformto log j Boml Ars trsformto rs j
25 Islm Uverst, Gz - Pleste Sttstl Tests for qult Vre: Brtlett s test: 0 oe bove ot true for t lest : H v.s. : H )log ( )log ( S S N q q P Rejet ull hpothess f ) /( ) ( ) ( ) ( ) 3( N S S N p, 0
26 Islm Uverst, Gz - Pleste xmple 3.4: the test sttst s d 0.05, Brtlett s test s sestve to the ormlt ssumpto The modfed Levee test: Use the bsolute devto of the observto eh tretmet from the tretmet med. d j j ~,,,,, j,,, Me devtos re equl => the vre of the observtos ll tretmets wll be the sme. The test sttst for Levee s test s the ANOVA F sttst for testg eqult of mes. 6
27 Islm Uverst, Gz - Pleste xmple 3.5: Four methods of estmtg flood flow freque proedure (see Tble 3.7) ANOVA tble (Tble 3.8) The plot of resduls v.s. ftted vlues (Fgure 3.7) Modfed Levee s test: F 0 = 4.55 wth P-vlue = Rejet the ull hpothess of equl vres. 7
28 Islm Uverst, Gz - Pleste Let () = d Fd * = tht elds ostt vre. * +- Vre-Stblzg Trsformtos * d = - Trsformto * ostt 0 No trsformto * / ½ ½ Squre root * 0 Log * 3/ 3/ -/ Reprol squre root * - Reprol 8
29 Islm Uverst, Gz - Pleste How to fd : S d Use log log log See Fgure 3.8, Tble 3.0 d Fgure 3.9 9
30 Islm Uverst, Gz - Pleste 3.5 Prtl Iterpretto of Results Codut the expermet => perform the sttstl lss => vestgte the uderlg ssumptos => drw prtl oluso 3.5. A Regresso Model Qulttve ftor: ompre the dfferee betwee the levels of the ftors. Qutttve ftor: develop terpolto equto for the respose vrble.
31 Islm Uverst, Gz - Pleste Regresso lss : See Fgure 3. 5 % Fl quto Terms of Atul Ftors: 0.5 Stregth = * Cotto Weght % * Cotto Weght %^ * Cotto Weght %^3 Stre egth 6.5 Ths s emprl model of the expermetl results A: Cotto Weght %
32 Islm Uverst, Gz - Pleste 3.5. Comprsos Amog Tretmet Mes If tht hpothess s rejeted, we do t kow whh spef mes re dfferet Determg whh spef mes dffer followg ANOVA s lled the multple omprsos problem Grphl Comprsos of Mes
33 Islm Uverst, Gz - Pleste Cotrst A otrst: ler ombto of the prmeters of the form H 0 : = 0 v.s. H : 0, 0 Two methods for ths testg. 33
34 Islm Uverst, Gz - Pleste The frst method: N C Vr C (0,) ~, Uder H ) ( The Let N t MS t N ~ Hee the sttst, (0,) ~, Uder H 0 0
35 Islm Uverst, Gz - Pleste The seod method:,n ~F ) ( t F C C C,N SS MS SS MS MS F ~F MS t F 0 0 0, /
36 Islm Uverst, Gz - Pleste The C.I. for otrst, MS t σ Vr(C) C Hee C.I. The. Let Uequl Smple Sze N MS t, / C.I. Hee C MS t 0 3.SS 0..
37 Islm Uverst, Gz - Pleste Orthogol Cotrst Two otrsts wth oeffets, { } d {d }, re orthogol f d = 0 For tretmets, the set of orthogol otrsts prtto the sum of squres due to tretmets to depedet sgle-degree-of-freedom ompoets. Thus, tests performed o orthogol otrsts re depedet. See xmple 3.6 (Pge 94) 37
38 Islm Uverst, Gz - Pleste Sheffe s Method for Comprg All Cotrsts Sheffe (953) proposed method for omprg d ll possble otrsts betwee tretmet mes.,,,, Suppose u u u m u See Pge 95 d 96 0 : the rejet H, If ) ( vlue : The rtl ) / ( d 0,,,, u u u N C u u C u u S C F S S MS S C u u
39 Islm Uverst, Gz - Pleste Comprg Prs of Tretmet Mes Compre ll prs of tretmet mes Tuke s Test: The studetzed rge sttst: q or T mx MS See xmple 3.7 m / (,, f ) mx The rtl pot s T d (/ m (, re the lrgest d smllest smple mes out of group of p smple mes q MS q f ) / j MS )
40 Islm Uverst, Gz - Pleste Sometmes overll F test from ANOVA s sgft, but the prwse omprso of me fls to revel sgft dfferees. The F test s smulteousl osderg ll possble otrsts volvg the tretmet mes, ot just prwse omprsos. The Fsher Lest Sgft Dfferee (LSD) Method For H 0 : = j t 0 MS (/ j / j )
41 Islm Uverst, Gz - Pleste The lest sgft dfferee (LSD): See xmple 3.8 LSD t /, N MS j Du s Multple Rge Test The tretmet verges re rrged sedg order, d the stdrd error of eh verge s determed s S MS h, h /
42 Islm Uverst, Gz - Pleste Assume equl smple sze, the sgft rges re Totl (-)/ prs xmple 3.9 p, f S, p,3, RP r, The Newm-Keuls Test Smlr s Du s multple rge test The rtl vlues: K q ( p, f ) S P 4
43 Islm Uverst, Gz - Pleste Comprg Tretmet Mes wth Cotrol Assume oe of the tretmets s otrol, d the lst s terested omprg eh of the other tretmet mes wth the otrol. Test H 0 : = v.s. H : :, =,,, Duett (964) Compute,,,, Rejet H 0 f d (, f ) MS xmple
44 Islm Uverst, Gz - Pleste 3.7 Determg Smple Sze Determe the umber of repltes to ru 3.7. Opertg Chrterst Curves (OC Curves) OC urves: plot of tpe II error probblt of sttstl test, P P( F Rejet H 0 F 0 H 0,, N s flse H 0 s flse) 44
45 Islm Uverst, Gz - Pleste If H 0 s flse, the F 0 = MS Tretmet / MS ~ oetrl F wth degree of freedom d N d oetrlt prmeter Chrt V of the Appedx Determe Let be the spefed tretmets. The estmtes of : For, from pror experee, prevous expermet or prelmr test or judgmet estmte., / 45
46 Islm Uverst, Gz - Pleste xmple 3. Dffult: How to selet set of tretmet mes o whh the smple sze deso should be bsed. Aother pproh: Selet smple sze suh tht f the dfferee betwee two tretmet mes exeeds spefed vlue the ull hpothess should be rejeted. D
47 Islm Uverst, Gz - Pleste 3.7. Spefg Stdrd Devto Irese Let P be peretge for rese stdrd devto of observto. The / / For exmple (Pge 0): If P = 0, the 0.0P
48 Islm Uverst, Gz - Pleste Cofdee Itervl stmto Method Use Cofdee tervl. j t MS /, N j j /, N t MS For exmple: we wt 95% C.I. o the dfferee me tesle stregth for two otto weght peretges to be 5 ps d = 3. See Pge 0. 48
49 Islm Uverst, Gz - Pleste 3.9 The Regresso Approh to the Alss of Vre Model: j = + + j L L j j L 0,,,, j ˆ ˆ ˆ ˆ 0 & j 0,,,, j j j j
50 Islm Uverst, Gz - Pleste The orml equtos N ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Appl the ostrt The estmtos re Regresso sum of squres (the reduto due to fttg the full model) ˆ ˆ ˆ, ˆ R ˆ ˆ ), (
51 Islm Uverst, Gz - Pleste The error sum of squres: SS j j R, Fd the sum of squres resultg from the tretmet effets: R( ) R(, ) R( ) R(Full Model) - / N R(Redued Model) 5
52 Islm Uverst, Gz - Pleste The testg sttst for H 0 : = = F R( ) /( ) 0 ~ j R (, ) /( N ) j F, N 5
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