Markov s s & Chebyshev s Inequalities. Chebyshev s Theorem. Coefficient of Variation an example. Coefficient of Variation

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1 Markov s s & Chbyshv s Iqualitis Markov's iquality: (Markov was a studt of (Markov was a studt of Chbyshv) If Y & d > E( Y ) P( Y d) d Sic, if d, if Y d X,, othrwis Not Y, X Th : E( Y ) E( X ) d P Y { d} Lt Y X E(X) ad d k with k > E ( ) ( ) ( X E(X) ) Y d P X E(X) k P Var(X) ( X - E(X) k) P ( X - E(X) k ) P k k k k Lt k k/ kk Chbyshv s Thorm Applis to all distributios, whr ma Numbr xists (,( ofμ< ) Stadard Distac from Dviatios th Ma μ± μ± 3 Miimum Proportio of Valus Fallig Withi Distac K -/.75 K 3 -/3.89 K 4 μ± 4 -/4.94 Cofficit of Variatio Ratio of th stadard dviatio to th ma, xprssd as a prctag Masurmt of rlativ disprsio Cofficit of Variatio a xampl μ C. V. ( ) CV.. ( ) CV.. ( ) μ 46. ( ) 9 ( ) μ μ 84 μ Outli Probability Thory Axioms Basic Pricipls for probability modlig ad computatio Law of Total Probability & Baysia Thorm Data Summaris ad EDA Distributios ( Exprimts & Dmos ( Statistical Ifrc Paramtr Estimatio Hypothsis Tstig & Cofidc itrvals Paramtric vs. No-paramtric ifrc ( CLT Liar modlig Simpl liar rgrssio, Multipl liar rgrssio ANOVA & GLM Paramtrs, Estimators, Estimats A paramtr is a charactristic of procss, populatio or distributio E.g., ma, st quartil, SD, mi, max, rag, skwss,, 97 th prctil, tc. A stimator is a abstract rul for calculatig a quatity (or paramtr) from sampl data. A stimat is th valu obtaid wh ral data ar pluggd-i th stimator rul.

2 Paramtrs, Estimators, Estimats E.g., W ar itrstd i th populatio ma rspos tim (paramtr)) of a cogitiv xprimt. Th sampl- avrag formula rprsts a stimator w ca us, whr as th (valu of th) sampl avrag for a particular datast is th stimat (for th ma paramtr). N paramtr μ Y ; stimator Y Yk N Data : Y stimat y.93. k {.896,.93,.9} y ( ) 3 How about y ( ) 3 Paramtr (Poit) Estimatio Two Ways of Proposig Poit Estimators Mthod of Momts (MOMs): St your k paramtrs qual to your first k momts. Solv. (.g., Biomial, Expotial ad Normal) Mthod of Maximum Liklihood (MLEs):. Writ out liklihood for sampl of siz.. Tak atural log of th liklihood. 3. Tak partial drivativs with rspct to your k paramtrs. 4. Tak scod drivativs to chck that a maximum xists(f >). 5. St st drivativs qual to zro ad solv for MLEs..g., Biomial, Expotial ad Normal Suppos w flip a coi 8 tims ad obsrv {T,H,T,H,H,T,H,H}. Estimat th valu p P(H). Mthod of Momts Estimat p^: St your k paramtrs qual to your first k momts. Lt X {# H s} p8pe(x) Sampl#H s 5 p^5/8. Mthod of Maximum Liklihood Estimat p^:. f(x p) Paramtr (Poit) Estimatio 8 5 p ( 5 3 p) liklihood fuctio. 8 8 l p p ( p) + ( p). 5 ( ) 3 l + 5 l 3 l d l 5 l( ) 3 l( ) p + p 5 3 d p p p 5( p) 3p p 5 8 Exampl Maximum Liklihood Estimat Lt {X,, X }{.5,.3,.6,.,.}, wights, b IID N(μ, ) f(x;μ). Joit dsity is f(x,,x ; μ)f(x ;μ)x xf(x ;μ). Th liklihood fuctio L(p) f(x,,x ; p) L( μ) λ( x,..., x ) l( L) 5 (.5 μ) + (.3 μ) + (.6 μ) + (. μ ) + (. μ) ( )[ / (.5 μ) + (.3 μ) + (.6 μ) + (. μ) + (. μ) ] d l( L) (.5 μ) + (.3 μ) + (.6 μ) + (. μ) + (. μ) d μ μ +.7 μ.34 d l( L) 5 L( d μ μ.34 ) max (Log)Liklihood Fuctio Suppos w hav a sampl {X,, X } IID D(θ) with probability dsity fuctio p p(x θ). Th th joit dsity p({x,, X } θ) is a fuctio of th (ukow) paramtr θ. Liklihood fuctio l(θ {X,, X }) p({x,,x } θ) Log-liklihood L(θ {X,, X })Log l(θ {X,, X }) Maximum-liklihood stimatio (MLE): Suppos {X,, X } IID N(μ, ), μ is ukow. W stimat it by:mle(μ)μ^argmax μ L(μ ({X,,X }) (Log)Liklihood Fuctio Suppos {X,, X } IID N(μ, ), μ is ukow. W stimat it by:mle(μ)μ^argmax μ L(μ ({X,,X }) MLE( μ) Log L' ( ˆ) μ i ( π ) Similarly show that : i ( x μ ) π ( i ( x ˆ) μ ˆ μ i i i x ˆ μ ) xi. L( μ) MLE( ) ˆ i i ( xi ˆ) μ ( xi μ). i

3 (Log)Liklihood Fuctio Suppos {X,, X } IID Poisso(λ), λ is ukow. Estimat λ by:mle(λ)λ^argmax λ L(λ ({X,,X }) MLE( λ) Log L'( ˆ) λ λ Log λ xi λ L( λ) i ( xi )! λ λ λ ( xi )! i ( ( ) ) ˆ x Log x λ + λ + x i i λ. i i i xi i i λ Hypothss Guidig pricipls W caot rul i a hypothsizd valu for a paramtr, w ca oly dtrmi whthr thr is vidc to rul out a hypothsizd valu. Th ull hypothsis tstd is typically a skptical ractio to a rsarch hypothsis Hypothsis Tstig th Liklihood Ratio Pricipl Lt {X,, X } b a radom sampl from a dsity f(x; p), whr p is som populatio paramtr. Th th joit dsity is f(x,, x ; p) f(x ; p)x xf(x ; p). Th liklihood fuctio L(p) f(x,, X ; p) Tstig: H o : p is i Ω vs. H a : p is i Ω a, whr ΩI Ω a Fid max of L(p) i Ω. max L( p) Fid max of L(p) i Ω a. p Ω λ( x,..., x) max L( p) Fid liklihood ratio p Ω a Rjct H o if liklihood-ratio statistics λ is small (λ<k) Hypothsis Tstig th Liklihood Ratio Pricipl Exampl Lt {X,, X }{.5,.3,.6,.,.} b IID N(μ, ) f(x;μ). Th joit dsity is f(x,,x ; μ)f(x ;μ)x xf(x ;μ). Th liklihood fuctio L(p) f(x,,x ; p) ( x μ) f ( x) Tstig: H o : μ> is i Ω vs H a : μ<. Rjct H o if liklihood-ratio statistics λ is small (λ<k) l(umr) quadratic i μ! max L( p) p Ω l(do) quadratic i μ! λ λ( x,..., x) max L( p) Maximiz both fid ratio max μ> max μ p Ω a (.5 μ ) + (.3 μ ) + (.6 μ ) + (. μ ) + (. μ ) (.5 μ ) + (.3 μ ) + (.6 μ ) + (. μ ) + (. μ ) Lt P(Typ I) α t o ~/λ ο ~ t α,df4 o-sampl T-tst Hypothsis Tstig th Liklihood Ratio Pricipl Exampl Tstig: H o : μ> is i Ω vs H a : μ<. Rjct H o if liklihood-ratio statistics λ is small (λ<k) max L( p) p Ω λ (,..., ) λ Lt P(Typ I) α x x max L( p) p Ω t a o ~/λ ο ~ t α,df4 (.5 μ) + (.3 μ) + (.6 μ) + (. μ ) + (. μ) o-sampl T-tst max μ > (.5 μ) + (.3 μ) + (.6 μ) + (. μ ) + (. μ) max μ (.5 μ) + (.3 μ) + (.6 μ) + (. μ ) + (. μ).86 μ.34 (.5 μ) + (.3 μ) + (.6 μ) + (. μ ) + (. μ) μ Ifrc ad Hypothsis Tstig. Idtify your dsig & appropriat statistical tchiqu Validat your Data/Modl Assumptios 3. Calculat a Tst Statistic (Exampl: z o ) z 4. Spcify a Rjctio Rgio (Exampl: o > z α ) 5. Ifrc: Th ull hypothsis is rjctd iff th computd valu for th statistic falls i th rjctio rgio

4 Typ I ad Typ II Errors α β Pr{ Rjct H H is tru} Pr{ Fail to Rjct H H is Fals} Th valu of α is spcifid by th xprimtr Th valu of β is a fuctio of α,, ad δ (th diffrc btw th ull hypothsizd ma ad th tru ma). For a two sidd hypothsis tst of a ormally distributd populatio β Φ Z α δ + Φ Z δ + It is ot tru that α - β (RHSthis is th tst powr!) α Typ I, Typ II Errors & Powr of Tsts Suppos th tru MMSE scor for AD subjcts is ~ N(3, ). A w cogitiv tst is proposd, ad it s s assumd that its valus ar ~ N(5, ). A sampl of AD subjcts tak th w tst. Hypothss ar: H o : μ tst 5 vs. H a : μ tst <5 (o-sidd, mor powr) α P(fals-positiv, Typ I, rror).5. Critical Valu for α is Z scor Thus, X avg critical Z critical *+μ X avg critical , Ad our coclusio, from {X,,, X } which yilds X avg will b rjct H o, if X avg < 3.4. βp(fail to rjct H o H o is fals)p(x avg >3.4 X avg ~N(3, /)) Not: X avg ~N(3, /)), wh it s s giv that X ~N(3, )) Stadardiz: Z (3.4 3)/(/) 4. Typ I, Typ II Errors & Powr of Tsts Suppos th tru MMSE scor for AD subjcts is ~ N(3, ). A w cogitiv tst is proposd, ad it s s assumd that its valus ar ~ N(5, ). A sampl of AD subjcts tak th w tst. βp(fail to rjct H o H o is fals)p(x avg >3.4 X avg ~N(3, /)) Not: X avg ~N(3, /)) wh it s s giv that X ~N(3, )) Stadardiz: Z (3.4 3)/(/) 4.. βp(fail to rjct H o H o is fals)p(z>4.).3 Powr (Nw Tst) How dos Powr(Tst) dpd o: Sampl siz, : -icras powr icras diffrt β for ach diffrt α, tru ma μ, altrativ H a Siz-of of-studid-ffct: ffct: ffct-siz icras powr icras Typ of Altrativ hypothsis: -sizd tsts ar mor powrful Aothr Exampl Typ I ad II Errors & Powr About 75% of all 8 yar old humas ar fr of amyloid plaqus ad tagls, markrs of AD. A w AD vacci is proposd that is supposd to icras this proportio. Lt p b th w proportio of subjcts with o AD charactristics followig vacciatio. H o : p.75, H : p>.75. X umbr of AD tsts with o pathology fidigs i 8-y/o vacciatd subjcts. Udr H o w xpct to gt about *p5 o AD rsults. Suppos w d d ivst i th w vacci if w gt > 8 o AD tsts rjctio rgio R{8, 9, }. Fid α ad β.. How powrful is this tst? Aothr Exampl Typ I ad Typ II Errors H o : p.75, H : p>.75. X umbr of tst with o AD fidigs i xprimts. X~Biomial(,.75). Rjctio rgio R{8, 9, }. Fid α P(Typ I) P(X>8 wh X~Biomial(,.75)). Us SOCR rsourc α How dos Powr(Tst) Fid β(p.85) P (Typ II) dpd o, ffct-siz? P(fail to rjct H o X~Biomial(,.85))P(X<8 X~Biomial(,.85)) Us SOCR rsourc β.595 Powr Powr of tst - β.45 Fid β(p.95) P (Typ II) P(fail to rjct H o X~Biomial(,.95))P(X<8 X~Biomial(,.95)) Us SOCR rsourc β.76 Powr Powr of tst - β.94 A 95% cofidc itrval A typ of itrval that cotais th tru valu of a paramtr for 95% of sampls tak is calld a 95% cofidc itrval for that paramtr, th ds of th CI ar calld cofidc limits. (For th situatios w dal with) a cofidc itrval (CI) for th tru valu of a paramtr is giv by stimat ± t stadard rrors (SE) TABLE 8.. Valu of th Multiplir, t, for a 95% CI df : t : df : t :

5 (Gral) Cofidc Itrval (CI) A lvl L cofidc itrval for a paramtr (θ),( is a itrval (θ( ^ ^, θ ^), whr θ ^ ^ & θ ^, ar stimators of θ,, such that P(θ ^ ^ < θ < θ ^) L. E.g., C+E modl: : Y μ+ε.. Whr ε Ν(, ), th by CLT w hav Y_bar ~ Ν(μ, /) Ara? ½ (Y_bar - μ)/ )/ ~ Ν(, ). L P ( z (-L)/ < ½ (Y_bar - μ)/ )/ < z (+L)/ ), whr z q is th q th quartil. E.g.,.95 P ( z.5 < ½ (Y_bar - μ)/ )/ < z.975 ),.975 ), CI ar costructd usig th sampl ad sse. But diffrt sampls yild diffrt stimats ad diff. CI s?!? Blow is a computr simulatio showig how th procss of takig sampls ffcts th stimats ad th CI s. Sampl st d 3rd 999th th Tru ma x ± t SE( x) x ±.6SE( x) Tru ma almost always capturd i th CI. Tru ma Figur 8.. x o 4.84 Covrag to dat Sampls of siz from a Normal(µ4.83, s.5) distributio ad thir 95% cofidc itrvals for µ.. % % % 95.% 95.% Cofidc Itrval Cofidc Itrval (CI) Exprimt CI Activity: Comparig two mas for idpdt sampls Suppos w hav sampls/mas/distributios as x, N( μ, ) x, N( μ, ) follows: { } ad { }. W v s bfor that to mak ifrc about μ w ca μ us a T-tst for H : μ μ with ( x x ) t Ad CI( μ μ ) ( ) x SE x x x ± t SE( x x ) If th sampls ar idpdt w us th SE formula SE s / + s / with df Mi( ;. ) This givs a cosrvativ approach for had calculatio of a approximatio to th what is kow as th Wlch procdur,, which has a complicatd xact formula. Mas for idpdt sampls qual or uqual variacs? Poold T-tstT is usd for sampls with assumd qual variacs. Udr data Normal assumptios ad qual variacs of ( ) ( ) x x / SE x x,whr ( ) ( ) ; s + s SE sp / + / s p + df ( + ) is xactly Studt s s t distributd with Hr s p is calld th poold stimat of th variac,, sic it pools ifo from th sampls to form a combid stimat of th sigl variac. Th book rcommds routi us of th Wlch uqual variac mthod. Sigl Sampl: Tstig/CI Exampl: Suppos a rsarchr is itrstd i studyig th ffct of aspiri i rducig hart attacks.. H radomly rcruits 5 subjcts with vidc of arly hart disas ad has thm tak o aspiri daily for two yars. At th d of th two yars h fids that durig th study oly 7 subjcts had a hart attack. Calculat Calculat a 95% cofidc itrval for th tru proportio of subjcts with arly hart disas that hav a hart attack whil takig aspiri daily. 4

6 Sigl Sampl: Tstig/CI Exampl: Hart Attacks (cot ) First, w d to fid z α/ bcaus this is a 95% CI, this mas that α will b.5 ad z α/ will b z.5 i this cas z α/ Z.5.95 Z.5.5 Z Nxt, solv for y +.5 z ~ p + z Sigl Sampl: Tstig/CI α α that s s just th formula for actually hav to fid p~ p~ y z ( z.5 ) y +.5(.96 ).5 ~ p Oft roudd to y p~, ow w y Nxt, solv for SE p SE~ p (.38)(.96) ~ Fially th 95% CI for p ~ p Sigl Sampl: Tstig/CI z.85 ( SE ~ ).38 ±.96(.85) ± α p.38 ±.67 (.3,.547) Sigl Sampl: Tstig/CI What is our itrprtatio of this itrval? CONCLUSION: W ar highly cofidt, at th.5 lvl (95% cofidc), that th tru proportio of subjcts with arly hart disas who hav a hart attack aftr takig aspiri daily is btw.3 ad.547. Is this maigful? Compariso of Two Idpdt Sampls Two Approachs for Compariso What sms lik a rasoabl way to compar two groups? What paramtr ar w tryig to stimat? Compariso of Two Idpdt Sampls RECALL: Th samplig distributio of y was ctrd at μ,, ad had a stadard dviatio of W ll start by dscribig th samplig distributio of Ma: μ μ Stadard dviatio of + y y What sms lik appropriat stimats for ths quatitis? 5

7 Stadard Error of y y W kow y y stimats μ μ What w d to dscrib xt is th prcisio of our stimat, SE( ) y y s s SE( y y ) + SE + SE Stadard Error of y y Exampl: : A study is coductd to quatify th bfits of a w cholstrol lowrig mdicatio. Two groups of subjcts ar compard, thos who took th mdicatio twic a day for 3 yars, ad thos who took a placbo. Assum subjcts wr radomly assigd to ithr group ad that both groups data ar ormally distributd. Rsults from th study ar show blow: Mdicatio Placbo y s SE Stadard Error of y y Poold vs. Upoold Exampl: : Cholstrol mdici (cot ) (.g., ftp://ftp.ist.gov/pub/dataplot/othr/rfrc/cholest.dat) Calculat a stimat of th tru ma diffrc btw tratmt groups ad this stimat s s prcisio. First, dot mdicatio as group ad placbo as group ( y y ) s s SE( y ). 4 y + + Mdicatio Placbo y s SE s is kow as a upoold vrsio of th s + stadard rror thr is also a poold SE First w dscrib a poold variac, which ca b thought of as a wightd avrag of s ad s s poold ( ) s + ( ) + s Poold vs. Upoold Th w us th poold variac to calculat th poold vrsio of th stadard rror SE + poold s poold NOTE: If ( ) ad (s s ) th poold ad upoold will giv th sam aswr for SE( y y ) It is wh that w d to dcid whthr to us poold or upoold: if s s th us poold (upoold( will giv similar aswr) if s s th us upoold (poold will NOT giv similar aswr) Poold vs. Upoold RESULT: Bcaus both mthods ar similar wh s s ad, ad th poold vrsio is ot valid wh Why all th tortur? This will com up agai i chaptr. Bcaus Bcaus th df icrass a grat dal wh w do pool th variac. 6

8 CI for μ - μ RECALL: W dscribd a CI arlir as: th stimat + (a appropriat multiplir)x(se) A (- α)% cofidc itrval for μ - μ (p.7) y y ± t( df α SE y y whr df SE 4 ( ) ( ) ) ( SE + SE ) 4 ( ) ( ) SE + Exampl: Cholstrol mdicatio (cot ) Calculat a 95% cofidc itrval for μ - μ W kow y y ad SE( y y ) from th prvious slids. Now w d to fid th t multiplir ( ) CI for μ - μ df ( ) ( ) Roud dow to b cosrvativ NOTE: Calculatig that df is ot rally that fu, a quick rul of thumb for chckig your work is: + - ( y y ) ± t( df ) ( SE ) α y y 4.5 ± t(7) 4.5 ±.(.4) ( 57.,.5 8.) CI for μ - μ (.4) CONCLUSION: W ar highly cofidt at th.5 lvl, that th tru ma diffrc i cholstrol btw th mdicatio ad placbo groups is btw -57. ad 8. mg/dl. Not th chag i th coclusio of th paramtr that w ar stimatig. Still lookig for th 5 basic parts of a CI coclusio (s slid 38 of lctur st 5). CI for μ - μ What s s so grat about this typ of cofidc itrval? I th prvious xampl our CI cotaid zro This itrval is't tllig us much bcaus: th tru ma diffrc could b mor tha zro (i which cas th ma of group is largr tha th ma of group ) or th tru ma diffrc could b lss tha zro (i which cas th ma of group is smallr tha th ma of group ) or th tru ma diffrc could v b zro! Th ZERO RULE! Suppos th CI cam out to b (5., 8.), would this idicat a tru ma diffrc? Hypothsis Tstig: Th idpdt t tst Th ida of a hypothsis tst is to formulat a hypothsis that othig is goig o ad th to s if collctd data is cosistt with this hypothsis (or if th data shows somthig diffrt) Lik ioct util prov guilty Thr ar four mai parts to a hypothsis tst: hypothss tst statistic p-valu coclusio Hypothsis Tstig: # Th Hypothss Thr ar two hypothss: Null hypothsis (aka th status quo hypothsis) dotd by H o Altrativ hypothsis (aka th rsarch hypothsis) dotd by H a 7

9 Hypothsis Tstig: # Th Hypothss Hypothsis Tstig: # Th Hypothss If w ar comparig two group mas othig goig o would imply o diffrc th mas ar "th sam" ( μ μ ) For th idpdt t-tst t tst th hypothss ar: H o : ( μ μ ) (o statistical ( μ μ ) diffrc i th populatio mas) H a : ( μ μ ) (a statistical diffrc i th populatio mas) Exampl: Cholstrol mdicatio (cot ) Suppos w wat to carry out a hypothsis tst to s if th data show that thr is ough vidc to support a diffrc i tratmt mas. Fid th appropriat ull ad altrativ hypothss. ( ) H o : μ μ (o statistical diffrc th tru mas of th mdicatio ad placbo groups) H a : ( μ μ ) (a statistical diffrc i th tru mas of th mdicatio ad placbo groups, mdicatio has a ffct o cholstrol) Hypothsis Tstig: # Tst Statistic Hypothsis Tstig: # Tst Statistic A tst statistic is calculatd from th sampl data it masurs th disagrmt btw th data ad th ull hypothsis if thr is a lot of disagrmt th w would thik that th data provid vidc that th ull hypothsis is fals if thr is littl to o disagrmt th w would thik that th data do ot provid vidc that th ull hypothsis is fals t s ( y y ) SE y y subtract bcaus th ull says th diffrc is zro O a t distributio t s could fall aywhr If th tst statistic is clos to, this shows that th data ar a compatibl with H o (o diffrc) th dviatio ca b attributd to chac If th tst statistic is far from (i th tails, uppr or lowr), this shows that th data ar icompatibl to H o (thr is a diffrc) dviatio dos ot appar to b attributd to chac (i( i.. If H o is tru th it is ulikly that t s would fall so far from ) t s t s Hypothsis Tstig: # Tst Statistic Exampl: Cholstrol mdicatio (cot ) Calculat th tst statistic ( y y) ts SE y y ( ).4.76 Grat, what dos this ma? y ad diffr by about.7 SE's y this is bcaus t s is th masur of diffrc btw th sampl mas xprssd i trms of th SE of th diffrc Hypothsis Tstig: # Tst Statistic How do w us this iformatio to dcid if th data support H o? Prfct agrmt btw th mas would idicat that t s,, but logically w xpct th mas do diffr by at last a littl bit. Th qustio is how much diffrc is statistically sigificat? If H o is tru, it is ulikly that t s would fall i ithr of th far tails If H o is fals it is ulikly that t s would fall ar 8

10 Hypothsis Tstig: #3 P-valuP How far is far? For a two taild tst (i.. H a : ( μ μ ) ) Th p-valu p of th tst is th ara udr th Studt's T distributio i th doubl tails byod -t ad t s s. Hypothsis Tstig: #3 P-valu P What this mas is that w ca thik of th p-valu p as a masur of compatibility btw th data ad H o a larg p-valup (clos to ) idicats that t s is ar th ctr (data support H o ) -t s t s Dfiitio (p. 38): Th p-valu p for a hypothsis tst is th probability, computd udr th coditio that th ull hypothsis is tru, of th tst statistic big at last as xtrm m or mor xtrm as th valu of th tst statistic that was actually obtaid [from th data]. a small p-valup (clos to ) idicats that t s is i th tail (data do ot support H o ) Hypothsis Tstig: #3 P-valuP Hypothsis Tstig: #3 P-valuP Whr do w draw th li? how small is small for a p-valu? p Th thrshold valu o th p-valu p scal is calld th sigificac lvl, ad is dotd by a Th sigificac lvl is chos by whomvr is makig th dcisio (BEFORE THE DATA ARE COLLECTED!) Commo valus for iclud.,.5 ad. Ruls for makig a dcisio: If p < a th rjct H o (statistical sigificac) If p > a th fail to rjct H o (o statistical sigificac) Exampl: : Cholstrol mdicatio (cot ) Fid th p-valu p that corrspods to th rsults of th cholstrol lowrig mdicatio xprimt W kow from th prvious slids that t -.76 (which is clos to ) This mas that th p-valu p is th ara udr th curv byod +.76 with 8 df. Hypothsis Tstig: #3 P-valuP Exampl: Cholstrol mdicatio (cot ) Usig SOCR w ca fid th ara udr th curv byod +.76 with 8 df to b: p > (.).4 NOTE, wh H a is, th p-valu is th ara byod th tst statistic i BOTH tails. Hypothsis Tstig: #4 Coclusio Exampl: Cholstrol mdicatio (cot ) Suppos th rsarchrs had st α.5 Our dcisio would b to fail to rjct Ho bcaus p >.4 which is >.5 (#4) CONCLUSION: Basd o this data thr is o statistically sigificat diffrc btw tru ma cholstrol of th mdicatio ad placbo groups (p >.4). I othr words th cholstrol lowrig mdicatio dos ot sm to hav a sigificat ffct o cholstrol. Kp i mid, w ar sayig that w could't provid sufficit vidc to show that thr is a sigificat diffrc btw th two populatio mas. 9

11 Comparig two mas for idpdt sampls. How ssitiv is th two-sampl t-tst tst to o- Normality i th data? (Th -sampl T-tsts T ad CI s s ar v mor robust tha th -sampl tsts, agaist o-normality, particularly wh th shaps of th distributios ar similar ad, v for small, rmmbr df Ar thr oparamtric altrativs to th two- sampl t-tstt tst? (Wilcoxo rak-sum sum-tst, Ma-Wity tst, quivalt tsts, sam P-valus.) P 4. What diffrc is thr btw th quatitis tstd ad stimatd by th two-sampl t- procdurs ad th oparamtric quivalt? (No-paramtric tsts ar basd o ordrig, ot siz, of th data ad hc us mdia, ot ma, for th avrag. Th quality of mas is tstd ad CI(μ ~ - μ ~ ). Paird Comparisos A fmri study of N subjcts: : Th poit i th tim cours of maximal activatio i th rostral ad caudal mdial prmotor cortx was idtifid, ad th prctag chags i rspos to th go ad o-go tasks from th rst stat masurd. Similarly th poits of maximal activity durig th go ad o-go task wr idtifid i th primary motor cortx. Paird t-tst t tst comparisos btw th go ad o-go prctag chags wr prformd across subjcts for ths rgios of maximum activity. Paird data W hav to distiguish btw idpdt ad rlatd sampls bcaus thy rquir diffrt mthods of aalysis. Paird data is a xampl of rlatd data. With paird data, w aalyz th diffrcs this covrts th iitial problm ito a o- sampl problm. Th sig tst ad Wilcoxo rak-sum tst ar oparamtric altrativs to th ad paird t- tst,, ad idpdt t-tst, rspctivly. Th Wilcoxo-Ma Ma-Whity Also kow as th rak sum tst This hypothsis tst is also usd to compar two idpdt sampls This procdur is diffrt from th idpdt t tst bcaus it is valid v if th populatio distributios ar ot ormal I othr words, w ca us this tst as a fair substitut wh w caot ot mt th rquird ormality assumptio of th t tst WMW is calld a distributio-fr typ of tst or a o- paramtric tst This tst dos't focus o a paramtr lik th ma, istad it xamis th distributios of th two groups Th Wilcoxo-Ma Ma-Whity Kp i mid that this is aothr hypothsis tst, thr ar four r major parts to cosidr # Th hypothss: H o : Th populatio distributios of Y ad Y ar th sam H a : Th populatio distributios of Y ad Y ar th diffrt This could also b dirctioal: distributio of Y is lss tha Y ; OR distributio of Y is gratr tha Y # Th tst statistic: dotd by U s masurs th dgr of sparatio btw th two sampls a larg valu of U s idicats that th two sampls ar wll sparatd with littl ovrlap a small valu of U s idicats that th two sampls ar ot wll sparatd with much ovrlap Th Wilcoxo-Ma Ma-Whity #3 Th p-valu: p mtabl.html Mthod vry similar to usig th t tabl fid th appropriat row ad th sarch for a umbr closst to th tst statistic do t t d to worry about doublig th p-valu p for a two- taild tst (assumig w go to th right row hadr) #4 Coclusio: Similar to th coclusio of a idpdt t tst, but ot likd to ay paramtr (for xampl th diffrc i mas)

12 Th Wilcoxo-Ma Ma-Whity Th Mthod: Stp : Arrag th data i icrasig ordr Stp : Dtrmi K ad K K : for ach obsrvatio i group, cout th umbr of obsrvatios i th scod group that ar smallr. Us / for tid obsrvatios. K is th sum of ths raks. K : for ach obsrvatio i group, cout th umbr of obsrvatios i th first group that ar smallr. Us / for tid t obsrvatios. K is th sum of ths raks. CHECK: if you hav do th procdur corrctly K + K Stp 3: If th tst is o-dirctioal th U s is th largr of K ad K. If th tst is dirctioal th U s is th K that jivs with th dirctio of H a (if H a is Y >Y th U s K, if H a is Y <Y th U s K ) Stp 4: Dtrmi th critical valu largr of ad ' smallr of ad Stp 5: Brackt th p-valup Th Wilcoxo-Ma Ma-Whity Exampl: Th uriary fluorid coctratio (ppm( ppm) ) was masurd both for a sampl of livstock grazig i a ara prviously xposd to fluorid pollutio ad also for a similar sampl of livstock grazig i a upollutd ara. Pollutd Upollutd Dos th data suggst that th fluorid coctratio for livstock grazig i th pollutd rgio is largr that for th upollutd rgio? Tst usig α.. Th Wilcoxo-Ma Ma-Whity Th Wilcoxo-Ma Ma-Whity Prct Probability Plot of Pollutd Normal Chck Normality: Ma 8.6 StDv N 7 AD.6 P-Valu.6 Prct Probability Plot of Upollutd Normal Ma 6.8 StDv N 5 AD.6 P-Valu.49 Coditios for th WMW: Data ar from radom sampls Obsrvatios ar idpdt Sampls ar idpdt Rmmbr: ormality will ot mattr for this tst Pollutd Upollutd 5 Wilcoxo-Ma Ma-Whity vs. Idpdt T-TstT Tst Both try to aswr th sam qustio, but trat data diffrtly. W-M-W W uss rak ordrig Pro: dos t t dpd o ormality or populatio paramtrs Co: distributio fr lacks powr bcaus it dos't us all th t ifo i th data T-tst uss actual Y valus Pro : Icorporats all of th data ito calculatios Co : Must mt ormality assumptio ithr is suprior So If your data ar ormally distributd us th t-tstt tst If your data ar ot ormal us th WMW tst Th Sig Tst Th sig tst is a o-paramtric altrativ of th paird t tst W us th sig tst wh pairig is appropriat, but w ca t t mt th ormality assumptio rquird for th t tst Th sig tst is ot vry sophisticatd ad thrfor quit asy to udrstad Sig tst is also basd o diffrcs d Y Y Th iformatio usd by th sig tst from this diffrc is th sig of d (+ or -)

13 Th Sig Tst # Hypothss: H o : th distributios of th two groups is th sam H a : th distributios of th two groups is diffrt or H a : th distributio of group is lss tha group or H a : th distributio of group is gratr tha group # Tst Statistic B s Th Sig Tst - Mthod # Tst Statistic B s :. Fid th sig of th diffrcs. Calculat N + ad N - 3. If H a is o-dirctioal, B s is th largr of N + ad N - If H a is dirctioal, B s is th N that jivs with th dirctio of Ha: if H a : Y <Y th w xpct a largr N -, if H a : Y >Y th w xpct a largr N +. NOTE: If w hav a diffrc of zro it is ot icludd i N or N + -, thrfor d ds to b adjustd Th Sig Tst #3 p-valu: p Similar to th WMW Us th umbr of pairs with quality iformatio #4 Coclusio: Similar to th Wilcoxo-Ma Ma-Whity Tst Do NOT mtio ay paramtrs! Exampl:: sts of idtical twis ar giv psychological tsts to dtrmi whthr th first bor of th st tds to b mor aggrssiv tha th scod bor. Each twi is scord accordig to aggrssivss, a highr scor idicats gratr aggrssivss. Bcaus of th atural pairig i a st of twis ths data ca b cosidrd paird. Th Sig Tst St st bor d bor Sig of d Drop Th Sig Tst (cot ) Do Do th data provid sufficit vidc to idicat that th first bor of a st of twis is mor aggrssiv tha th scod? Tst usig α.5. H o : Th aggrssivss is th sam for st bor ad d bor twis H a : Th aggrssivss of th st bor twi tds to b mor tha d bor. NOTE: Dirctioal Ha (w r xpctig highr scors for th st bor twi), this mas w prdict that most of th diffrcs will b positiv N + umbr of positiv 7 N - umbr of gativ 4 d umbr of pairs with usful ifo Th Sig Tst B s N + 7 (bcaus of dirctioal altrativ) P >., Fail to rjct H o CONCLUSION: Ths data show that th aggrssivss of st bor twis is ot sigificatly gratr tha th d bor twis (P >.). X~B(,.5) P(X>7) (Biomial Distributio)

14 Phoological vs. Rapid-Namig diffrcs usig fmri β uiqu cotributio of PHONO Y β o + β X + β X + β 3 X X + ε Thickss Δ PHONO Δ RAN Δ itractio β uiqu cotributio of RAN Approximatio of th Fishr Sig Tst usig th ormal distributio Lft ROIs Latral dorsofrotal Latral vtrofrotal Latral parital Latral occipital Tmporal Mdial dorsofrotal Mdial vtrofrotal Mdial parital Mdial occipital Pos 7,6,934 7,659,95 3,83 3,484 3,864 3, Ng , Total 8,8 4,3 9,36 3,38 3,336 3,5 4,67 3,568 6 Z p <. CLT Samplig Distributio of th Sampl Ma Lu, L.H., Loard, C.M., Diov, I.D., Thompso, P.M., Ka, E., Jolly, J., Toga, A.W., & Sowll, E.R. (6, Fbruary). Diffrtiatig btw phoological procssig ad rapid amig usig structural MRI. Papr prstd at th 34th Aual Mtig of th Itratioal Nuropsychological Socity, Bosto, MA. Usig th Sampl Ma Lt X,, X b a radom sampl from a distributio with ma valu μ ad stadard dviatio.th ( ) ( ). E X. V X μ μ X X I additio, with T o X + + X, E T μ, V T,ad. ( ) ( ) o o T o Normal Populatio Distributio Lt X,, X b a radom sampl from a ormal distributio with ma valu μ ad stadard dviatio Th for ay, X is ormally distributd, as is T o. 3

15 Th Ctral Limit Thorm Lt X,, X b a radom sampl from a distributio with ma valu μ ad variac. Th if sufficitly larg, X has approximatly a ormal distributio with μ μ ad, X X ad T o also has approximatly a ormal distributio with μt μ,. o T Th largr th valu of o, th bttr th approximatio. Populatio distributio Th Ctral Limit Thorm X small to modrat μ X larg X Ctral Limit Thorm huristic formulatio Ctral Limit Thorm: Wh samplig from almost ay distributio, is approximatly Normally distributd i larg sampls. Show Samplig Distributio Simulatio Applt: { } Lt X,X,...,X,... b a squc of idpdt k obsrvatios from o spcific radom procss. Lt ad E(X ) μ ad SD(X ) ad both b fiit ( < < ; μ < ). If X X, sampl-avg, X Ctral Limit Thorm thortical formulatio k k Th has a distributio which approachs N(μ, /), as. Rcall w lookd at th samplig distributio of X For th sampl ma calculatd from a radom sampl, E( X ) μ ad SD( X ), providd X (X +X + + X )/,, ad X k ~N(μ,, ). Th X ~ N(μ, ). Ad variability from sampl to sampl i th sampl-mas mas is giv by th variability of th idividual obsrvatios dividd by th squar root of th sampl-siz. siz. I a way, avragig dcrass variability. Law of Larg Numbrs (LLN) Th wak law of larg umbrs stats that if X, X, X 3,... is a ifiit squc of radom variabls, whr all th radom variabls hav th sam xpctd valu μ ad variac ; ad ar ucorrlatd (i.., th corrlatio btw ay two of thm is zro), th th sampl avrag X + X X X covrgs i probability to μ. Somwhat lss trsly: For ay positiv umbr ε, o mattr how small, w hav Proof by Chbyshv s iquality! ( X μ < ε ) Lim P LLN Activity: 4