Assessing Normality. Assessing Normality. Assessing Normality. Assessing Normality. Normal Probability Plot for Normal Distribution.

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1 Assessg Normaly No All Couous Radom Varables are Normally Dsrbued I s Impora o Evaluae how Well he Daa Se Seems o be Adequaely Approxmaed by a Normal Dsrbuo Cosruc Chars Assessg Normaly For small- or moderae-szed daa ses, does boxad-whsker plo look symmerc? Does he hsogram look symmerc? Compue Descrpve Summary Measures Do he mea, meda ad mode have smlar values? Is he erquarle rage approxmaely.33 σ? Is he rage approxmaely 6 σ? p.0, Exhb 4.5 (coued) 00 Prece-Hall, Ic. Chap 6-00 Prece-Hall, Ic. Chap 6- Assessg Normaly Observe he Dsrbuo of he Daa Se Do approxmaely /3 of he observaos le bewee mea ± sadard devao? Do approxmaely 4/5 of he observaos le bewee mea ±.8 sadard devaos? Do approxmaely 9/0 of he observaos le bewee mea ± sadard devaos? Evaluae Normal Probably Plo p., Exhb 4.5 (coued) Do he pos le o or close o a sragh le wh posve slope? Assessg Normaly Normal Probably Plo for Normal Dsrbuo X p.5 (coued)

2 Normal Probably Plo Obag Normal Probably Plo PHSa Lef-Skewed 90 Rgh-Skewed 90 PHSa Probably Dsrbuos Normal Probably Plo X Recagular X U-Shaped Eer he rage of he cells ha coa he daa he Varable Cell Rage wdow Daa o NFL Scores Daa o Body Fa X X Prece-Hall, Ic. Chap Prece-Hall, Ic. Chap 6-6 AOV Assumpo Checkg ad Trasformaos ( 8.4) How do we check he Normaly of resduals assumpo AOV? How do we check he Homogeey of varaces assumpo AOV? ( 7.4) Wha o do f hese assumpos are o me? Model Assumpos Homoscedascy (commo group varaces). Normaly of resduals. Idepedece of resduals. (Hopefully acheved hrough radomzao.) Effec addvy. (Oly a ssue mul-way AOV; laer).

3 H : 0 Checkg he Equal Varace Assumpo σ = σ = L = σ H A : some of he varaces are dffere from each oher More work bu beer power Barle s Tes Barle s Tes: Allows uequal replcao, bu requres ormaly. Harley s Tes: A logcal exeso of he F es for =. Requres equal replcao,, amog groups. Requres ormaly. s F max = s max m Lle work bu lle power T.S. C = ( ) loge s ( )log = = If C > χ (-),α he apply he correco erm CF = + 3( ) ( ) = ( = ) e s s = = s Rejec f F max > F α,,-, abulaed Table. R.R. Rejec f C/CF > χ (-),α 00 Prece-Hall, Ic. Chap Prece-Hall, Ic. Chap 6-0 More work bu powerful resul. T.S. R.R. L = = = j= ( z ( z Levee s Tes Le zj = yj y% Rejec H 0 f j z z ) ) /( ) /( T L F α,df,df ) Esseally a AOV o he z j y% = sample meda of -h group T = = df = - df = T - Tes for Equal Varaces Respose Ress Facors Sad CofLvl Boferro cofdece ervals for sadard devaos Lower Sgma Upper N Facor Levels Barle's Tes (ormal dsrbuo) Tes Sasc:.890 P-Value : Levee's Tes (ay couous dsrbuo) Tes Sasc: P-Value : 0.76 Mab Sa > ANOVA > Tes for Equal Varaces Mab Help Use Barle s es whe he daa come from ormal dsrbuos; Barle s es s o robus o deparures from ormaly. Use Levee s es whe he daa come from couous, bu o ecessarly ormal, dsrbuos. The compuaoal mehod for Levee s Tes s a modfcao of Levee s procedure [0] developed by []. Ths mehod cosders he dsaces of he observaos from her sample meda raher ha her sample mea. Usg he sample meda raher ha he sample mea makes he es more robus for smaller samples. Do o rejec H 0 sce p-value > 0.05 (radoal α)

4 SAS Program proc glm daa=sress; class sad; model ressace = sad / soluo; meas sad / hoves=barle; meas sad / hoves=levee(ype=abs); meas sad / hoves=levee(ype=square); meas sad / hoves=bf; /* Brow ad Forsyhe mod of Levee */ le 'Compresso ressace cocree beams as'; le ' a fuco of perce sad he mx'; ru; Hoves oly works whe oe facor (rgh had sde) model. 00 Prece-Hall, Ic. Chap Prece-Hall, Ic. Chap 6-4 hoves=barle; Barle's Tes for Homogeey of ressace Varace Source DF Ch-Square Pr > ChSq sad SAS SPSS Levee's Tes for Homogeey of ressace Varace ANOVA of Absolue Devaos from Group Meas Sum of Mea hoves=levee(ype=abs); Source DF Squares Square F Value Pr > F sad Error Levee's Tes for Homogeey of ressace Varace ANOVA of Squared Devaos from Group Meas hoves=levee(ype=square); Sum of Mea Source DF Squares Square F Value Pr > F sad Error Brow ad Forsyhe's Tes for Homogeey of ressace Varace ANOVA of Absolue Devaos from Group Medas Sum of Mea hoves=bf; Source DF Squares Square F Value Pr > F sad RESIST Tes of Homogeey of Varaces Levee Sasc df df Sg Sce he p-value (0.457) s greaer ha our (ypcal) α =0.05 Type I error rsk level, we do o rejec he ull hypohess. Ths s Levee s orgal es whch he z j are ceered o group meas ad o medas.

5 R Tess of Homogeey of Varaces barle.es(): Barle s Tes. flger.es(): Flger-Kllee Tes (oparamerc). Checkg for Normaly Remder: Normaly of he RESIDUALS s assumed. The orgal daa are assumed ormal also, bu each group may have a dffere mea f H A s rue. Pracce s o frs f he model, THEN oupu he resduals, he es for ormaly of he resduals. Ths APPROACH s always correc. TOOLS. Hsogram ad/or boxplo of all resduals (ε j ).. Normal probably (Q-Q) plo. 3. Formal es for ormaly. 00 Prece-Hall, Ic. Chap Prece-Hall, Ic. Chap 6-8 Hsogram of Resduals proc glm daa=sress; class sad; model ressace = sad / soluo; oupu ou=resd r=r_ress p=p_ress ; le 'Compresso ressace cocree beams as'; le ' a fuco of perce sad he mx'; ru; proc capably daa=resd; hsogram r_ress / ormal; ppplo r_ress / ormal square ; ru; Probably Plos (QQ-Plos) A scaer plo of he perceles of he resduals agas he perceles of a sadard ormal dsrbuo. The basc dea s ha f he resduals came from a ormal dsrbuo, values for hese perceles should le o a sragh le. Compue ad sor he resduals (), (),, (). Assocae wh each resdual a sadard ormal percele: z () = F - ((-.5)/). Plo z () versus (). Compare o sragh le (do

6 Sofware EXCEL: Use AddLe opo. Percele p = (-0.5)/ Normal percele =NORMSINV(p ) MTB: Graph -> Probably Plo R: wh resduals y qqorm(y) qqle(y) 00 Prece-Hall, Ic. Chap 6- Normal Perceles Excel Probably Plo Probably Plo - Perce Sad Daa Daa Perceles 00 Prece-Hall, Ic. Chap 6- Mab SAS (oe axes chaged) These look ormal! Probably Plo Formal Tess of Normaly May, may ess (a favore pass-me of sascas s developg ew ess for ormaly.) Kolmogorov-Smrov es. Shapro-Wlks es ( < 50). D Agoso s es (>=50) All que coservave hey fal o rejec he ull hypohess of ormaly more ofe ha hey should.

7 Shapro-Wlk s W es,,, represe daa raked from smalles o larges. Coeffces H 0 : The populao has a ormal dsrbuo. H A : The populao does o have a ormal dsrbuo. k T.S. W= d a ( ε + j εj) d = ( ε ε) j = = Coeffces a come from a able. k = If s eve R.R. Rejec H 0 f W < W 0.05 ( ) If s odd. k = Crcal values of W come from a able. 00 Prece-Hall, Ic. Chap Prece-Hall, Ic. Chap 6-6 Shapro-Wlk Coeffces Shapro-Wlk W Table

8 D Agoso s Tes,,, represe daa raked from smalles o larges. H 0 : The populao has a ormal dsrbuo. H A : The populao does o have a ormal dsrbuo. T.S. (D ) Y = R.R. (wo sded es) Rejec H 0 f Y< Y0.05 or Y> Y0.975 s = ( εj ε) D = j = j = s [j (+ )] ε Y 0.05 ad Y come from a able of perceles of he Y sasc. j 00 Prece-Hall, Ic. Chap Prece-Hall, Ic. Chap 6-30 Trasformaos o Acheve Homoscedascy Wha ca we do f he homoscedascy (equal varaces) assumpo s rejeced?. Declare ha he AOV model s o a adequae model for he daa. Look for alerave models. (Laer.). Try o chea by forcg he daa be homoscedasc hrough a rasformao of he respose varable Y. (Varace Sablzg Trasformaos.) Square Roo Trasformao Respose s posve ad couous. z = y Ths rasformao works whe we oce he varace chages as a lear fuco of he mea. Useful for cou daa (Posso Dsrbued). For small values of Y, use Y+.5. Typcal use: Cous of ems whe cous are bewee 0 ad 0. σ = kµ Sample Varace k>0 0.00

9 Logarhmc Trasformao Respose s posve ad couous. = l( Y ) Ths rasformao eds o work whe he varace s a lear fuco of he square of he mea Replace Y by Y+ f zero occurs. Useful f effecs are mulplcave (laer). Useful If here s cosderable heerogeey he daa. Typcal use: Growh over me. Coceraos. Cous of mes whe cous are greaer ha 0. σ= k µ 00 Prece-Hall, Ic. Chap 6-33 Sample Varace k> Sample Mea Respose s a proporo. = s Y = arcs Wh proporos, he varace s a lear fuco of he mea mes (-mea) where he sample mea s he expeced proporo. Y s a proporo (decmal bewee 0 ad ). ero cous should be replaced by /4, ad N by N-/4 before coverg o perceages Typcal use: Proporo of seeds germag. Proporo respodg. ARCSINE SQUARE ROOT Sample Mea 00 Prece-Hall, Ic. Chap 6-34 Y ( ) σ = kµ µ Respose s posve ad couous. Ths rasformao works whe he varace s a lear fuco of he fourh roo of he mea. Use Y+ f zero occurs. Useful f he recprocal of he orgal scale has meag. Typcal use: Survval me. Recprocal Trasformao = Y σ = kµ Power Famly of Trasformaos () Suppose we apply he power rasformao: Suppose he rue suao s ha he varace s proporoal o he κ h power of he mea. I he rasformed varable we wll have: σ σ z = If p s ake as -κ, he he varace of wll o deped o he mea,.e. he varace wll be cosa. Ths s a Varace = y p kµ κ p + κ µ

10 Power Famly of Trasformaos () Box/Cox Trasformaos (advaced) Wh replcaed daa, κ ca somemes be foud emprcally by fg: σ ˆ ( ) Esmae: = yj y j= µ ˆ = y κ ca be esmaed by leas squares (regresso Nex U). p= ˆ κˆ pˆ If s zero use he logarhmc rasformao. l( ˆ σ ) l( ˆ = C +κ µ ) 00 Prece-Hall, Ic. Chap Sample Mea suggesed rasformao geomerc mea of he orgal daa. ω = λ y λω λ ω l y ( y ) 00 Prece-Hall, Ic. Chap 6-38 z = exp = λ λ = Expoe, λ, s ukow. Hece he model ca be vewed as havg a addoal parameer whch mus be esmaed (choose he value of λ ha mmzes he resdual sum of squares). l 0 0 Hadlg Heerogeey Trasformaos o Acheve Normaly Regresso? o ANOVA Regresso? yes o ANOVA F lear model Plo resduals yes No OK Trasform F Effec Model Tes for Homoscedascy rejec accep OK F lear model Resduals Normal? Esmae group meas yes Probably plo Formal Tess OK OK Tradoal Box/Cox Famly Power Famly Trasform o Dffere Model