ST 524 NCSU - Fall 2008 One way Analysis of variance Variances not homogeneous

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1 ST 54 NCSU - Fall 008 On way Analyss of varanc Varancs not homognous On way Analyss of varanc Exampl (Yandll, 997) A plant scntst masurd th concntraton of a partcular vrus n plant sap usng ELISA (nzym-lnkd mmunosorbnt assay) (Novy 99 ). A subst of th study s prsntd hr. Th scntst wants to undrstand th rsstanc to th vrus among th thr slctd potato clons. Plant sap was takn from 5 noculatd plants of ach clon, for a total of 5 (3 Clons * 5 Rp) masurmnts of ttr. Data Lnar Modl yj = μ + α + j, =,, 3, j =,..., 5 whr j dn( 0, σ ) whr μ = μ+ α s th populaton man for th clon., and yj dn( μ, σ ), 3 = μ + = μ+ α + 59 = μ + = μ+ α = μ + = μ+ α Matrx rprsntaton of data accordng to lnar modl Novy RG (99) ``Charactrzaton of somatc hybrds btwn Solanum tubrosum and dplod, tubr-barng Solanums. PhD Dssrtaton, Dpartmnt of Plant Pathology, UW-Madson. 3 pp. Tusday August 6, 008

2 ST 54 NCSU - Fall 008 On way Analyss of varanc Varancs not homognous μ 69 0 α = α = μ + α + 59 = μ+ α = μ+ 0 α + 0 α + = μ+ 3 3 Y=Xβ + H : μ = μ = μ = μ o 3 Analyss of varanc tabl - Last Squars stmaton H : at last on μ s dffrnt Brown-Forsyth mognty tst : σ = σ = σ3 = σ p valu = Do not rjct H 0 at a sgnfcanc lvl of 5% H s dffrnt Concluson: W can assum that rsdual varancs for ach group ar not sgnfcantly dffrnt from ach othr. Error Man Squars (Error MS = ) s th last squars stmat of th common rsdual varanc σ, Tusday August 6, 008

3 ST 54 NCSU - Fall 008 On way Analyss of varanc Varancs not homognous Bartltt s Tst for mognty of varancs : σ = σ = σ3 = σ p valu =<.000 Rjct H 0 at a sgnfcanc lvl of 5% H s dffrnt Concluson: W can assum that at last on rsdual varancs for ach group s sgnfcantly dffrnt from othrs. Assumpton of Varanc qualty Error Man Squars (Error MS = ) s th last squars stmat of th common rsdual varanc σ, mognty of varanc tst (SAS Manual) On of th usual assumptons for th GLM procdur s that th undrlyng rrors ar all uncorrlatd wth homognous varancs. You can tst ths assumpton n PROC GLM by usng th HOVTEST opton n th MEANS statmnt, rqustng a homognty of varanc tst. Ths scton dscusss th computatonal dtals bhnd ths tsts. Not that th GLM procdur allows homognty of varanc tstng for smpl on-way modls only. Bartltt (937) proposs a tst for qual varancs that s a modfcaton of th normal-thory lklhood rato tst (th HOVTEST=BARTLETT opton). Whl Bartltt's tst has accurat Typ I rror rats and optmal powr whn th undrlyng dstrbuton of th data s normal, t can b vry naccurat f that dstrbuton s vn slghtly nonnormal (Box 953). An approach that lads to tsts that ar much mor robust to th undrlyng dstrbuton s to transform th orgnal valus of th dpndnt varabl to drv a dsprson varabl and thn to prform analyss of varanc on ths varabl. Th sgnfcanc lvl for th tst of homognty of varanc s th p-valu for th ANOVA F-tst on th dsprson varabl. Brown and Forsyth (974) suggst usng th absolut dvatons from th group mdans: BF zj = yj m, whr m s th mdan of th th group. You can us th HOVTEST=BF opton to spcfy ths tst. Smulaton rsults show that th Brown-Forsyth tst sms bst at provdng powr to dtct varanc dffrncs whl protctng th probablty of a Typ I rror If on of ths tsts rjcts th assumpton of homognty of varanc, you should us Wlch's ANOVA nstad of th usual ANOVA to tst for dffrncs btwn group mans. Unlss th group varancs ar xtrmly dffrnt or th numbr of groups s larg, th usual ANOVA tst s rlatvly robust whn th groups ar all about th sam sz. Rsdual plot Tusday August 6, 008 3

4 ST 54 NCSU - Fall 008 On way Analyss of varanc Varancs not homognous Rsdual plot shows dsprson s not homognous among clons, clon 7 shows a largr varablty than clon, wth clon 3 havng modrat varablty. Hartly s Max F tst for homognty of k varancs. H : σ = = σ H o k s dffrnt Tst-statstc: F max s =, whr k s th numbr of groups and ν s th numbr max F Hartly, k, ν mn s of dgrs of frdom for ach sampl, ν = r-, Assumptons Random samplng wthn ach group Equal sampl sz for all k groups Normalty of obsrvatons. Hghly snstv to dvatons from normalty. Exampl : σ = σ = σ3 = σ calculatd Fmax = = H s dffrnt F = 4.8 Hartly,3,4,0.05 Concluson: Rjct H o Tusday August 6, 008 4

5 ST 54 NCSU - Fall 008 On way Analyss of varanc Varancs not homognous Mxd Modl approach to htrognty of varancs. Ft lnar modl for tttr wth common rsdual varanc. Ft Lnar modl for tttr allowng sparat rsdual varanc wthn ach group 3. Compar both fttng: lklhood rato tst, AIC, AICC, BIC. Lnar modl yj = μ + α + j, =,, 3, j =,..., 5 whr j dn ( 0, σ ), and y j (, ) a. Y=Xβ + dn ( 0, σ I) dn μ σ b. Rsults from PROC MIXED, as prsntd abov ˆ σ = Error MS GLM H : μ = μ = μ = μ o 3 H : at last on μ s dffrnt. Lnar modl j j y = μ + α +, =,, 3, j =,..., 5 whr j dn ( 0, σ ) a. Y=Xβ + dn ( 0, R) b. Rsults from PROC MIXED, and yj dn( μ, σ ) Tusday August 6, 008 5

6 ST 54 NCSU - Fall 008 On way Analyss of varanc Varancs not homognous Lklhood rato tst: LRT s usd to tst whthr a modl wth common varanc should b prfrrd to a modl wth thr sparat varancs. H o : σ = σ = σ3 = σ, H : σ σ σ 3 Null hypothss ft a common rsdual varanc for all thr groups whl th altrnatv hypothss ft a sparat rsdual varanc for ach group. -RsLogL (common rsdual varanc) = RsLogL (thr rsdual varancs) = 57.9 (ft null hypothss, varanc paramtr) (ft altrnatv hypothss, 3 varanc paramtr) Dffrnc = [-RsLogL rducd modl ] [-RsLogL full modl ] = =.9 Undr null hypothss Dffrnc of s dstrbutd as a Ch-squar random varabl wth (3 ) dgrs of frdom. Crtcal valu at a 0.05 sgnfcanc lvl, χ = df, ( df.85) P χ > Concluson: Rjct H o, thr s nough statstcal vdnc, at 5% sgnfcanc lvl, to conclud that varancs ar not qual. Tst of hypothss for fxd ffcts : μ = μ = μ3 = μ H : at last on μ s dffrnt Dn DF = 3 ( 5 ) = Thus, Sattrthwat approxmaton for calculaton of dgrs of frdom whn varancs ar not th sam Dnomnator dgrs of frdom n tst of hypothss for fxd ffcts df [ MS+ MS + MS3] ( ) + ( ) + ( ) sattrthwat = MS df MS df MS3 df3 Tusday August 6, 008 6

7 ST 54 NCSU - Fall 008 On way Analyss of varanc Varancs not homognous df sattrthwat [ ] ( ) + ( ) + ( ) = = Rsults from PROC MIXED Concluson: Thr s statstcal vdnc (p-valu=0.0064) that at last on clon tttr man s dffrnt from othrs, at 0.05 sgnfcanc lvl. Last squars mans > sqrt(054.3/5) [] 4.50 > sqrt(53/5) [] > sqrt(35756/5) [] Dffrncs btwn pars of last squars mans > sqrt(054.3/5+53/5) [] > sqrt(054.3/ /5) [] > sqrt(53/ /5) [] Rsdual plot Important: Analyss of varanc F tst s robust to small varanc htrognty whn sampl szs ar qual. It s snstv to dvatons from normalty. Quston: Is thr a rlatonshp btwn man and varanc, should a transformaton b usd? Tusday August 6, 008 7

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