TSS = SST + SSE An orthogonal partition of the total SS

Transcription

1 ANOVA: Topc 4. Orhogonal conrass [ST&D p. 183] H 0 : µ 1 = µ =... = µ H 1 : The mean of a leas one reamen group s dfferen To es hs hypohess, a basc ANOVA allocaes he varaon among reamen means (SST) equally across he ( 1) reamen degrees of freedom and asks, "Is hs average poron of SST sgnfcan?" F s among nsy swhn s MST MSE SST dfr SSE df error Bu wha f he varaon among reamen means s no dsrbued equally among he df? Isn' here a beer way o "spend" our reamen degrees of freedom, han jus dvde he n equal pars? TSS = SST + SSE An orhogonal paron of he oal SS SST df SSConras ( df 1) An orhogonal paron of he reamen SS Comparsons o deermne whch specfc reamen means are dfferen can be carred ou by paronng he reamen sum of squares (SST). The orhogonal conras approach o mean separaon s descrbed as planned, sngle degree of freedom F ess. In effec, an expermen can be paroned no ( 1) separae, ndependen expermens, one for each conras. 1

2 4. 1. Defnon of conras and orhogonaly [ST&D p ] A conras s a BALANCED comparson among means A conras (Q) s a lnear sum of erms whose coeffcens sum o 0 Q 1 c Y, wh he consran ha c 0 1 Example: Coeffcens 1 & -1 =0 3 1 hen A conras has a sngle degree of freedom The c s are usually negers. Orhogonaly Now consder a par of wo conrass: Q1 c Y and Q d Y 1 1 These wo conrass are sad o be orhogonal o one anoher f he sum of he producs of her correspondng coeffcens s zero: Orhogonal f c d 0 A se of more han wo conrass s sad o be orhogonal only f each and every par whn he se exhbs parwse orhogonaly, as defned above. In hree ses of conrass A, B, and C, all hree possble pars of comparsons need o be orhogonal: A vs. B A vs. C B vs. C 1

3 Example Treamens, T1, T and T3 (conrol). Two d.f. for reamens. One could es he hypoheses ha T1 and T are no sgnfcanly dfferen from he conrol: 1 = 3 and = 3. 1 = 3 ( = 0) he coeffcens are: c1 = 1, c = 0, c3 = -1 = 3 ( = 0) he coeffcens are: d1 = 0, d = 1, d3= -1 These lnear combnaons of means are conras because 1 1 c 0 ( (-1) = 0) d 0 ( (-1) = 0) These wo conrass are no orhogonal because m 1 c d 0 (c1d1 + cd+ c3d3 = = 1). No every se of hypoheses can be esed usng hs approach. If we es: 1. Is here a sgnfcan average reamen effec (reamens vs. conrol)? 1 H 0 : Is here a dfference beween he wo reamen effecs? H 0 : These are conrass snce: (-) = 0 and 1+ (-1) + 0= 0, and are orhogonal because: c1d1 + cd+ c3d3 = 1 + (-1) + 0 = 0. We wll dscuss wo general knds of lnear combnaons: class comparsons and rend comparsons. 3

4 4.. Class comparsons: ANOVA on groups, or classes. Example: Resuls of an expermen (CRD) o deermne he effec of acd seed reamens on he early growh of rce seedlngs (mg dry wegh). Table 4.1. Toal Treamen Replcaons Mean Y. Y. Conrol HCl Proponc Buyrc Overall Y.. = Y.. = 3.86 Table 4.. ANOVA of daa n Table 4.1. Source of Varaon df Sum of Squares Toal Mean Squares Treamen Exp. error Quesons: 1) Do acd reamens decrease seedlng growh? ) Are organc acds dfferen from norganc acds? 3) Is here a n he effecs of he organc acds? Table 4.3. Orhogonal coeffcens. Conrol HCl Proponc Buyrc Toals Comparsons Means Conrol vs. acd Inorg. vs. org Beween org F 4

5 Rules o consruc coeffcens for class comparsons 1. In comparng he means of wo groups, each conanng he same number of reamens, assgn +1 o he members of one group and -1 o he members of he oher. (Example: Beween org. ).. In comparng groups conanng dfferen numbers of reamens, assgn: 1 s group coeffcens= number of reamens n he second group nd group coeffcens= number of reamens n he frs group, wh oppose sgn. Example: If among 5 reamens, he frs wo are o be compared o he las hree, he coeffcens would be +3, +3, -, -, -. (e.g. conrol vs acds) 3. The coeffcens for any comparson should be reduced o he smalles possble negers for each calculaon. Thus, +4, +4, -, -, -, -. should be reduced o +, +, -1, -1, -1, A mes, a comparson componen may be an neracon of wo oher comparsons. The coeffcens for hs comparson are deermned by mulplyng he correspondng coeffcens of he wo comparsons Example: Expermen wh 4 reamens, levels of N and levels of P. N 0 P 0 N 0 P 1 N 1 P 0 N 1 P 1 Beween N Beween P Ineracon (NxP) Ineracon: N 0 P 0 -N 0 P 1 =N 1 P 0 -N 1 P 1 -> P a N 0 = P a N 1 If comparsons are orhogonal, he concluson drawn for one comparson s ndependen of (no nfluenced by) he ohers. 5

6 COMPUTATION Sum of squares for a sngle degree of freedom F es for lnear combnaons of reamen means ( SS Q) MS( Q) ( cy. ) ( or c ) / r c Y. ( ( c ) / r ) for unbalanced desgns SS1 (conrol vs. acd) = [3(4.19) ] / [(1)/5] = 0.74 SS1 (Inorg. vs. org.) = [ (3.87)] / [(6)/5] = 0.11 SS1 (beween org.) = [ ] / [()/5] = 0.0 ST&D: Q formulas p. 184: for reamen oals (r*c ), no reamen means. Table 4.5. Orhogonal paronng of reamens of Table 4.. Source of Var. df SS MS F Toal Treamen ** Conrol vs. acd ** Inorg. vs. Org ** Beween Org NS Error Noe ha =0.87! We conclude ha On average acds sgnfcanly reduce seedlng growh (P<0.01) On average organc acds cause more reducon han he norganc acd (P<0.01) The dfference beween he organc acds s no sgnfcan (P>0.05). When he ndvdual comparsons are orhogonal: SS of conrass add up o he SST The maxmum number of orhogonal comparsons s -1 The SS for one comparson does no conan any par of he SS of anoher comparson. The conclusons are ndependen from each oher Powerful as hey are, conrass are no always approprae. If you have o choose, meanngful hypoheses are more desrable han orhogonal ones! 6

7 4. 3. Trend comparsons Obj.: sudy he effec of changng levels of a facor on a response varable The expermener s neresed n he dose response relaonshp. The sascal analyss should no be concerned wh parwse comparsons. Examples for orhogonal conrass. Genec examples. (Table 4.6. noes) An expermen s conduced o deermne he effec of a ceran allele on he N conen of seeds. The expermen nvolves a sngle facor (allele A) a 3 levels: 0 dose of allele A (homozygous BB ndvduals) 1 dose of allele A (heerozygous AB ndvduals) doses of allele A (homozygous AA ndvduals) Daa conras; Inpu Geno Nrogen Flowerng; Cards; AB-[(AA+BB)/] =BB 1=AB =AA Lneal conras: BB vs. AA Quadrac conras: AB vs. average of AA + BB ; proc glm; class geno; model Nrogen Flowerng= geno; conras 'Lneal' geno ; conras 'Quadrac' geno 1-1; run; qu; 7

8 ANOVA dependen varable: Nrogen Source DF SS MS F Value Pr > F Model Error Correced Toal Conras DF Conras SS MS F Value Pr > F Lneal Quadrac SS lneal SS quadrac. + = MS Model The ANOVA MS Model s he average of he wo effecs. ANOVA dependen varable: Flowerng Source DF SS MS F Value Pr > F Model Error Correced Toal Conras DF Conras SS MS F Value Pr > F Lneal Quadrac AB-[(AA+BB)/] 0=BB 1=AB =AA Regresson analyss Source DF SS MS F Value Pr > F Model Error Toal =

9 Example ST&D Table Page 387 Yeld of Oawa Mandarn soybeans grown n MN, n bushels per acre. Row spacng (n nches) Rep.* Means * Orgnal example wh blocks, reaed as reps n hs example Coeffcens for rend comparsons for equally spaced reamens ST&D p390 and PLS05 WEB page Lecure No. of reamens Degree polynom. T1 T T3 T4 T5 T c ( c Y.) ( ( c Y ). c ) / r ** ** NS NS T1 T5-9

10 Unequally spaced reamens usng mulple regresson daa sp387reg; le 'Mulple regresson CRD'; npu S yeld; cards; ; proc glm; model yeld= S S*S S*S*S S*S*S*S; run; qu; (noe he absence of a class saemen n he regresson analyss) Dependen Varable: yeld Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Correced Toal Source DF Type I SS Mean Square F Value Pr > F S <.0001 S*S S*S*S S*S*S*S Same as daa sp387reg; le 'Conras CRD'; npu S yeld; cards; ; proc glm; class S; model yeld=s; conras 'lnear' S ; conras 'Quadrac' S ; conras 'Cubc' S ; conras 'Quarc' S ; run; qu; Dependen Varable: yeld Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Correced Toal Source DF Type I SS Mean Square F Value Pr > F Lnear <.0001 Quadrac Cubc Quarc Same resul n boh analyses. The mulple regresson analyss can be used wh unequally spaced reamens, bu he Conras analyss no. 10