Lecture Notes: Continuous Data Analysis PDF

연속형 자료분석 이론1 박 범 희 아주대학교 의과대학 의료정보학교실...

연속형 자료분석 이론1 박 범 희 아주대학교 의과대학 의료정보학교실 아주대의료원 의학연구협력센터 의학통계실 [email protected] It is not permitted for you to post recordings of class lectures and course materials online or to distribute them to other people ⦸ unless you have permission from your instructor to do so. (본 강의자료의 무단 복제 및 배포를 금합니다.)  대표적 평균비교 분석 방법 정규성 검정 (정규분포인가?) 예 아니오 Fixed effect only 모수적 방법 비모수적 방법 With one factor 평균: 두 그룹간 비교 두 그룹 이상 비교 중앙값: 두 그룹간 비교 두 그룹 이상 비교 (Independent) Wilcoxon rank Wilcoxon Paired t-test ANOVA Kruskal Wallis test two sample t-test sum test signed rank test = Mann-Whitney U test 사후검정 사후검정 (Fisher 최소유의차이 검정) 한 그룹의 “평균”이 그 집단의 특성을 잘 대표하는가? One sample t-test 두그룹 이상의 비교의 경우, 적어도 한 그룹 이상의 한 그룹의 “중앙값”이 그 집단의 특성을 잘 대표하는가? Wilcoxon signed rank test 평균/중앙값이 유의한지 파악 후 그룹 비교 실시 2  모집단의 분포에 대한 정규성 검정 정규성 검정 (정규분포인가?) 검 정방법 Sh a p ir o– Wil k te st Ko l m og or o v- S m i r no v t es t 표본의 크기 표본의 크기가 작을 때 표본의 크기가 클 때 (연구자가 생각하는) 목적 정규분포의 적합성 임의의 분포에서의 적합성 가설 정규분포를 따른다 임의분포를 따른다 예 아니오 모수적 방법 비모수적 방법 참고: QQ-plot, 왜도, 첨도 등도 대략적인 정규성 검정을 위해 사용됨 3  한 그룹의 평균이 그 집단의 특성을 잘 대표하는가? (귀무가설이 기각한다) 귀무가설이 맞지 않다? distribution under H0 : θ = θ0 H1 : θ ≠ θ0 모수 H1 : θ > θ0 귀무가설이 맞다는 H1 : θ < θ0 가정하의 분포에서 관측되기 어렵다 가설검정 statistically different? θ̂ θ̂ θ0 θ̂ 관측되기 어렵다 관측값으로부터의 귀무가설하에서 −zα 추정량 모수의 값 or (대립가설과 관련) (귀무가설이 맞다는 가정하에서 분포) −zα/2 θ ̂ − θ0 or tθ ̂ = ∼ t-분포 / 표준정규분포 −tα/2(n − 1) 통계량이 의미가 있으려면⋯ SE(θ)̂ (임계값) 통계량 >> 표준오차 ( 통계량 ) 4  2 2 1 /n1 + 2 /n2 두 집단의 평균 비교 (t-검정) @ 0 s 2 1 2 2 =P X̄1 X̄2 z↵/2 + < µ1 µ2 < n1 n2 2 2가정 (assumption) (1) 정규성 1, 2 : unknown , n1 , n2 : small group1 group2 ! + (2) 독립성 "# ~ &((" , * + ) !"# ~ &((" , * ) (3) 등분산성: Levene 검정 + 2 ~ &((+ , 2* + ) + , !+# 2 2 2 !"# ~ &((" , * ) !+# ~ &((+ , * 1) 1 ), N (µ2 , 2 ), N (µ 1 = 2 = + + !-# ~ &((- , * + ) !+# ~ &((+ , * ) !-# ~ &((- , * ) 2 2 1)S(n1 + (n 1)S ˆ2 = (+ vs.." ∶ ("between-group S.p2/=∶ (" = 1 2 ≠ (+ 2 +./ ∶ (" = (+ vs.." ∶ (" ≠ ( !-# ~ &((- , * ) n+1 + n2 2 mean difference + !"# ~ &((!""# , *~ ) &((" , * + ) !"∙ − !+∙ !"∙ − !+∙ !"∙ − !+∙ 3/ = 67!!"∙"∙−−!!+∙+∙ = ∼ 3(:" + :+ − ≪./ ∶ (" = (+ vs.." ∶ (" ≠ (+ 1 1! ~ &(( , * + ) 3/ = X̄1= X̄2 (µ1 µ26) + "# " within-group1 + !+# ~ &((!++#, * ~)within-group2 + &((+ , * ) 67 !"∙T−=!+∙ p 1 1 8⇠ t: n"1 +n2:+2 variance S 6 1/n + + 1/n combined !"∙ − !+∙ !"∙ − !+∙ variance p 8 :" :+ 1 2 within-group SE + 3/ = = + ! +# + ~ &(( + , * ) 67 !"∙ − !-# + + : − ! 1 ~6 + &(( : , * +− 1 6 ) +∙ ~ &(( !--#1 , * ~) 1&((- , * ) * + = 6 + = " "# " " + + 68 + : − 1 6 + 8 + : − :1" −6 +1 + : − 1 :" :+ + + " " + + + + X̄1! X̄~ 2 &(( (µ-1, * + )µ * = 68 = 1 ↵ = P t↵/2 (n1 + ! ~ +# n2 &(( , 2)+ : 3 −(:1 + : − Reject 2). / , if 3 :/ −>1 3+ =/+:(: −" 1+ : + s− 2) s 2 2 (Decision rule)."///, ifif 33/// > + (: " + : + − 2) =/+ 33=/+ " + n + n 1 2 Reject Reject. Reject, if 3. > >, if3 3 =/+ (: (:"+ " > + 3 ::++ (: − − 2) 2) + : − 2) X̄1 X̄2 ⇠ tv , v = ✓ ◆ X̄✓1 ◆ X̄2 1 2 / / / ?/+/ " ?/++ " + t= q s2 (µ 2 s2 2 1 ↵Reject = P./ , ift↵/2 3/n (n s >+ 3n1=/+ s 2 1 2 +(:n"2+ :+2) 2 < − 2) p 1 2 Reject./ , if 3/ > 3=/+ (:" + :+ − 2) 1 2 n 1 + n n1 Sp1 1/n1 + 1 n2 2 6 ✓ H0 : µ 1 = µ 2 vs. H1 : µ1 6= µ2 r  평균 비교와 변동 grand mean group1 group2 !"# ~ &((" , * + ) !"# ~ &((" , * + ) !+# ~ &((+ , * + ) Basic Idea !+# ~ &((+ , * + ) !-# ~ &((- , * + ) between-groups distribution variance !-# ~ &((- , * + )./ ∶ (" = (under + vs.." ∶ (" ≠ (+ + !"# ~ &((!""# , *~ ) &((" , * + ) >>./ ∶ (" = (+ vs.." ∶ (" ≠ (+ !"∙ − !+∙ !"∙ − !+∙ + +3/ = 67 ! − ! = within-group1 within-group2 !+# ~ &((!++#, * ~) &((+ , * ) "∙ +∙ 1 1 + 68 + !"∙ − !+∙ !"∙ − !+∙ :" variance :+ variance 3/ = = + 67 !"∙ − !-# +∙ ~ &(( !--#1 , *~ ) 1&((- , * + ) 68 + : − 1 6 + + : − 1 6 + :" :+ " " + + * + = 68+ =./ ∶ ("+ = (.+/ ∶vs. (". ≠ (.+" ∶ (" ≠ (:+" − 1 + :+ − 1 ="+(∶+("vs. + + :" − 1 6" + :+ − 1 6+ * = 68 =. 3 > 3 (: + : Reject − 2) H0 : µ1 = µ2 vs. H :" − 1 !+"∙ − :+!− Within-group2 +∙ 1 !"∙ − !! Reject , if +∙"∙ − !+∙ /!"∙ − ! / +∙ =/+ " + 3/ = 3 = = = variance ∼ 3(:" + :∼+ − 3(:2) " + :+ − 2) 67 !"∙/− !67 ! − !+∙1 1 1 1 + +∙ :+ −"∙2) Reject./ , if 3/ > 3=/+ (:"between-groups 68 +6 + :" :8+ :" :+ variance Y¯1 Y¯2 + z0 = q 2 = 1 + + : + " − 1 6 + " + variance + total : + " − : 1 + −6"1 + 6+ :+ −1 6++ s1 2 s2 * = 68 =* = 68 = n1 + n2 :" − 1 + ::" +−−117+ :+ −1  평균 비교와 변동 !"# ~ &((" , * + ) group1 group2 ! !++# ~ &((+ , * + ) "# ~ &((" , * ) + !"# ~ &((" , * ) + !+# !+ -# ~ &((+ , * ) ~ &(( - , * ) !+# ~ &((+ , * + ).+/) ∶ (" = (+ vs.." ∶ (" ≠ (+ !-# ~ &((- , * between-group mean difference !-# ~ &((- , * + )./ ∶ (" = (+ vs.." ∶!(" ≠ ( "∙ − !++∙ !"∙ − !! +∙"# ~ &((" , + + !"# ~ &((!""# , *~ ) &((" , * ) 3/ = = ∼ 3(:" >>./ ∶ (" = (+ vs.." ∶ (" ≠ (+ !"∙ − !+∙ 67 ! !"∙ − !!+∙+∙ − + 1 1 = !"# ~ &((" , 6 *8combined ) + "∙ 3/ = : within-group1 + !+# ~ &((!++#, * ~)within-group2 + &((+ , * ) 67 !"∙ − !+∙ 1 1 " !: +#+ ~ &((+ , 68 + within-group SE !"∙ − !+∙variation !"∙ − !+∙ variation :" :+ + 3/ = = !+# ~ &(( ++ ) , * + 67 !"∙ − !-# +∙ ~ &(( !--#1 + + , * ~) 1&((- , * ) + + : " − 1 6 " + :+ − !1 6 -# +~ &((- , 68 + :" :+ :* =6 = + " − 1 6"8 + :+ − 1 6+ + + * = 68 = + ! ~ :&(( " − -1, *+ + :+ − 1 ) : − 1 + -#: − 1./ ∶ ("+ = (.+/ ∶vs. (".="+(∶+("vs. ≠ (.+" ∶ (" ≠ (+" + Reject./ ∶ (" = (+ :" − 1 6" + :+ − 1 6+ + + * = 68 =. Reject 3 > 3.. , if∶ +( 3:/=−> /(: (2)3?/+ vs. (:." +∶ :+≠−(2) ( :" − 1between-group + : − 1 Reject / , !"∙ −+!+∙ !"∙ − !!+∙"∙ − !+∙ !"∙ − !+∙if / =/+ / " " + + " " + 3/ = mean3/difference = = = ∼ 3(:" + :∼+ − 3(:2) " + :+ − 2) !"∙ − 67 !"∙ − !67+∙ !"∙ − !+∙1 1 1 1 3/ = Reject./ , if 3/ > 3=/+ (:" + :+ − 2) 68 + 68 + !"∙ − !+∙ !"∙ −67 !+∙!"∙ − ] [ :" :+between-group :" :+ 3/ = 2 = directionality exists mean difference 67 !"∙ − !+∙ 1 1 68 + + + : − 1 6 :+ −: 1−61 +6 : + −1 6++ :" :+ + + " " " + " + + * = 68+ =* = 68+ = : :" − 1 + ::" +−−118+ :+ − 1 + * := : − 1 6+ + 6−8+1=6 +  평균 비교와 변동 group1 group2 ! + !"# ~ &((" , * + ) "# ~ &((" , * ) !+# ~ &((+ , * + ) !"# ~ &((" , * + ) !+# ~ &((+ , * + ) !-# ~ &((- , * + ) between-group !+# ~ &((+ , * ) + !-# ~ &((- , * + )./ ∶ (" = (+variation vs.." ∶ (" ≠ (+ >> !-# ~ &((- , * + )./ ∶ (" = (+ vs.." ∶ ("3/≠=(+ !"∙ − !+∙ = !"∙ − !+∙ !"# ~ &((!""# + , *~ ) &((" , * + ) 67 !"∙ − !+∙ 1 1 combined within-group 68 + :" :+./ ∶ (" = (+ vs.." ∶ (" ≠ (+ !"∙ − ! +∙ !"∙ − !+∙ variation ! ~ &(( , * + ) 3 = = "# " within-group1 !+# ~ &((!++#, * ~ + &((+ , * + )/ 67 !"∙ − !+∙ )within-group2 +1 + 1 " : − 1 6"+ + :+ − 1 6++ variation !"∙ − !+∙ !"∙ − !+∙ variation 6 8 * = 6+8 = :" :+ :" − 1 !++# :~ + −&(( 1 + , * +) 3/ = = + 67 !"∙ − !-#+∙ ~ &(( !--#1 , *~ ) 1&((- , * + ) Reject./ , if+ 3/ > 3=/+ (:" + :+ − 2) 68 + + :" − 1 6" + :+ − 1 6+ :" :+ + + !-# ~ &((- , * + ) * = 68 =./ ∶ ("+ = (.+/ ∶vs. (". ≠ (.+" ∶ (" ≠ (:+" − 1 + :+ − 1 ="+(∶+("vs. + + :" − 1 6" + :+ − 1 6+ Reject./ ∶ (" = (+ vs.." ∶ ( * = 68 =. , if 3/ > 3=/+ (:" + :+ − 2) :between-group " − 1 !+ :+!− "∙ − !"∙ − !!+∙Reject +∙ 1variation "∙ − !+∙ /!"∙ − ! +∙ 3/ = 3/ = = = ∼ 3(:" + :∼+ − 3(:2) " + :+ − 2) !"∙ − !+∙ 67 !"∙ − !67 ! − ! +∙1 1 1 1 3/ = = Reject./ , if 3/ > 3=/+ (:" + +∙ :+ −"∙2) 68 + 68 + 67 ! − ! :" :+ :" :+ "∙ +∙ 68 9 + + + +  예제: 독립 이표본 t-검정 부모의 흡연 여부가 신생아의 건강에 미치는 영향을 살펴보기 H0 : µ 1 = µ 2 vs. H1 : µ1 6= µ2 위해 신생아의 요중 코티닌(니코틴의 대사로 인한 부산물) 농 도를 조사해 보았다. 관측은 흡연에 노출된 8가구와 그렇지 않 Y¯1 = 116, Y¯2 = 31.29, S12 = 59.992 , S22 = 37.072 은 7가구를 대상으로 이루어졌으며, 그에 대한 결과는 아래의 표와 같다. 유의수준 1%에서 부모의 흡연이 신생아의 건강에 (n1 1) S12 + (n2 1) S22 Sp2 = = 50.722 n1 + n2 2 영향을 미치는지 검정하여라. (단, 두 표본의 모집단은 각각 정규분포를 따르며, 공통분산을 p SE Y¯1 Y¯2 = 50.72 1/8 + 1/7 = 26.25 가진다고 가정하자.) 116 31.29 t0 = = 3.23 26.25 흡연 가구 35 56 83 92 128 150 176 208 |t0 | = 3.23 > t0.005 (13) = 3.012 ) Reject H0 비흡연 가구 8 11 12 14 20 43 111 CIµ1 µ2 = Y¯1 Y¯2 ± t0.005 (13) ⇥ SE Y¯1 Y¯2 = (5.64, 163.78) Y¯1 Y¯2 z0 = q 2 = 1.42 s1 s22 n1 + n2 |z0 | = 1.42 < z0.025 = 1.96 ) Do not reject H0 10  예제: 대응 표본 26, September t-검정 2016 September 26, 2016 임의로 추출된 10명의 비만 여성에 대해서 음식 조절법을 실시한 전후의 체중이 다음과 같다. 체중이 정규분포를 따른다고 (i) H0 : µX = µY vs H1 : µX > µY 할 때, 음식 조절법의 효과를 유의수준 5%에서September 검정하시오. 26, 2016 (ii) H0 : µX = µY vs H1 : µ X < µ Y (i) H0 : µX = µY vs H1 : µ X > µ Y (iii) H0 : µX1 = µY2 vs3 H14: µX 56= µY6 7 8 9 10 (ii) 구분 H0 : µX = µY vs H1 : µX < µY (i) H(iii) : µHX0 :=µµXY82.1 vs 78.1 H :H µX :> vs1 86.2 µY6= 95.2 1 84.8 µY 91.6 75.3 78.5 83.0 83.5 0음식조절전 =µ µX Y (ii) H(i*) 0 : H : µ80.7 µX0 = 음식조절후 Y =vs D 078.1H vs183.9 HX1 83.5 :µ : µµDY > < 0 91.2 72.6 76.2 81.6 81.2 91.2 (iii) H 0 : Hµ (i*) (ii*) H : = µ µY= 0 vsvs HH 1 :µ :µX 6=>µ0Y 0X0 : DµD = 0 vs 1 H1D: µD < 0 (iii*)차이 (ii*) HH 0 0 = 음식 조절전 - 음식조절후 : :µDµD = 0= 0vs vs H1 :Hµ1D: < µD0 6= 0 (i*) H 0 : µ (iii*) HD0 :=µ0D = vs0 H vs1 : H µD1 :>µD 0 6= 0 (ii*) H0 : D̄µD =µD 0 vs H1 : µD < 0 T = D̄ µp ⇠ t(n 1) (iii*)TH=0 : S µDp=n0 ⇠vs t(nH1 1) D D : µD 6= 0 SD n ¯ dD 0 D̄ = µ¯ T = t 0 =p pp0 ⇠ under t d under t(n H1)H0 0SD s n snd n 0 d ¯ d 0tt (n t0 = 00 p ↵↵ (n t t 1)1) H0 under 11 sd n  실제 연구에서 two-sample t-test 사례 J ALLERGY CLIN IMMUNOL PRACT LEE ET AL 1289 VOLUME 9, NUMBER 3 TABLE III. Baseline comorbidities of severe and nonsevere asthmatics and the highly exacerbation-prone and exacerbation-prone groups Severe asthmatics HiEP group EP group Nonsevere asthmatics Comorbidity Total (n [ 567) (n [ 189) (n [ 378) (n [ 1337) P value* P value† Atopy 420 (74.1) 139 (73.5) 281 (74.3) 1108 (82.9) /+. (9/ "∶ (+" 9=+(− vs.." ∶ (" ≠ (+ + 2) combined within-group ! ~ &(( , * + ) !"#!"#~ ~&(( &(( ", " *, *) ) +# + variation !"∙ − !+∙ !"∙ − !+∙ 3/ = = !+#!+#~ ~&(( &(( *,+*)+!)"# ~ &((" , * + ) +, + within-group 67 !"∙ − !+∙ 68 1 + 1 !-# ~ &((- , * + ) :" :+ variation !-#!-#~ ~&(( &(( *,+*)+!)+# ~ &((+ , * + ) -, - : − 1 6 + + : − 1Reject 6 +./ ∶ (" = (+ vs " " + + * + = 68+ =./.∶/ (∶"(= (++ ) :" − 1 + :+ − 1 " = (+(+vs.vs..".~ !-# ∶" (∶&(( "(≠ " ≠ -(,+* !"∙ − !+∙ ! ! − − ! ! ! ! − − ! ! Reject./ , if 3/ > 3=/+ (:" + :+ − 2) 3/ = 3/3=between-group / = "∙ "∙ +∙./=variation +∙ ∶=(" = (+ vs.."∼∶ ∼(3(: "∙ "∙ +∙ +∙ ≠+ " 3(:(+: :− − " "+ + + 2)2) 67 !"∙ − !+ 6767!"∙!"∙−−!+∙!+∙ 11 11 686!8 − + !+∙+ !"∙ − !+∙ "∙:":" :+:+ 3/ = 14 = ∼ 3(:" + :+ − 2) 67 ! − !  5 !?# = (? !?# @ = 1, 2, ⋯ , B C = 1, 2, ⋯ , 8 One-way Analysis ! ~ @@I & ( , * of Variance (ANOVA) ?# ? + E 1 !!?∙?#=− !∙∙ =!?#!?∙ − !∙∙ + !?# − !?∙ 8 #F" Case of same # replicates !?# @ = 1, 2, ⋯ , B C = 1, 2, ⋯ , 8 G E G E G E G E + + + (Data structure) !?# 1@!?#= −1,!2,∙∙ ⋯=, B C = 1,!?∙2,−⋯!∙∙, 8 + E !?# − !?∙ !∙∙ = ?F" #F" !?# ?F" #F" 1 ?F" #F" B8E #groups !?∙ = #replicates !?# ?F" #F" 8 G1 G E #F" !?∙ = !?# + = 8 8 ! − ! + + !?# ! @−=!1,+ 2, ⋯ , B C = 1, 2, ⋯ , 8 (Model) !?# = (#F" ? +?∙ H ?# , ∙∙ H ?# ~ @@I & ?# 0, * ?∙ G E ?F" ?F" #F" 1 E !∙∙ = !?# G E 1 B8 G 1 !?∙ = !?# ?F" #F" (Hypothesis). ∶ (" = (+ = ⋯ !!∙∙/ = ! = ( vs..8 ∶ not.+ ?# =B8 (@ + =K +⋯H?#?#, B, HC?#=~1,@@I ⋯&, 80, * , K? = 0 G " / ?# 1,? 2, 2, #F" ?F" #F" + 442MNO2/(B − 1) P42MNO2G E !?# = ( ? + ?F"H?# , H ?# ~ @@I & 0, * E L/ = 1 = 1 + ~ L(B − 1, 8 − B) (Means) !!?# 445/(& = (? + !H?#?# , ?∙ = − B) H?# ~ @@I P45 ! =& 0, * ! 5 !?#8 = (? ∙∙ B8 ?# G #F" ?F" #F" + Reject./ , if L/ > L (B − 1, 8 − !B) ?# = ( G+ K ? + H ?# , H ?# ~ @@I & 0, * , K? + =, ! ?# !for ~ = @@I ( +G& E(? , * K + H , H ~ ! @@I = & ( + 0, H * total +, , H K ~ = @@I0& 0, * + ?F" ?# each 1 group? ?# ?# ?# ? ?# ?# ? !∙∙ = !?# B8 !?# − !?F" 5 !?# ?F" = (? ∙∙ =#F" !?∙ − !∙∙ + !?# − !?∙ G (Assumption) (1) ! 5 Linearity ?# = (? !?# = ( + K? + H?# , H?# ~ @@I & 0, * + , K? = 0 + G &E 0, * !?# ~ @@I & G(? , E * + !G?# Normality (2) =E(? + H?# , H?# ~ @@I ?F" (3) Independence ++ + + !?# ~ @@I!&?# (−? ,!*∙∙ = 그룹간 !등분산/독립, ?∙ − !∙∙ G + 그룹내 정규분포 !?# 가정 − !?∙ (4) Equal Variance 5 !?# = (! ? ?# − !∙∙ = !?∙ − !∙∙ + !?# − !?∙ ?F" #F" ?F" #F" + ?F" #F" !?# = ( + K? + H?# , H?# ~ @@I & 0, * , K? = 0 !?# − !∙∙ = !?∙ − !∙∙ + !?# − !?∙ ?F"+ G G!?# E ~ @@I &G(? , E* G E G E + + 15 G +! − ! + =G 8E ! −! + E ! −! G?# E ∙∙ = !?∙ − !∙∙ +  !?# = (1 + K? + H?# ,!∙∙ H= ?# ~ @@I & 0,!*?#+ , K? = 0 !∙∙ = !?# B8 B8 ?F" #F" ?F" 변동의 분해 ?F" #F" + 8+ − 2) + 8+ − 2) ∼ 2(8" + 8+ − 2) 5 !?# = (? ! =( +H , ?# * + H?# ~ @@I & 0, * + !?# = (? + H?# , H?#?# ~ @@I? & 0, " !?# ~ @@I & (? , * + G ※ SS: sum of squares " G ∼ 2(8 ∼ 2(8 !?# = ( + K? + H?# +, H?# ~ @@I & 0, * + , K? = 0 !?# = ( + K? + H?# , H?# ~ @@I & 0, * , K? = 0 − 2) " + 8+ − 2) > 2=/+ (8" + 8+ − 2) !?# − !∙∙ = !?∙ − !∙∙ + !?# − !?∙ ?F" ?F" + + + + 8+ − 1 4+ 8+ − 1 4+ / /." ∶ not./ 8 − 1 4 + 8 − 1 4 + ∶ ( = ( = ( vs.. ∶ not. + + 8. + + 8+ 5 !G?# E= (?mean of groupGi E 1 1 1 8 − 1 + 8 − 1 8+ − 1 8+ − 1 G E 8 8 obs 5 !?# total = (mean 2.(8∶ not + + !" − !+ ? + + + + + ! − ! ! − ! !?# − !∙∙ = !?∙ − !∙∙ + + !?# − !?∙ + + squaring " " " 8" 1 1 1 + > 2=/+ (8 ! ~ @@I & (? , * 8 8 both sides !?#Total ~ @@I & (?between- , * + ?#?F" #F" " " " =/+ " ?F" #F" ?F" #F" within-group + + + + & deviation group deviation 6 6 6 1 Reject. , if 2 >vs. summing / ∶ (" = (+ = (- vs. + + + deviationG " 8" − 1 4" " 4 4 4 8" − 1 8" − 1 G !?# − E!∙∙ = !?∙ − !∙∙ + !?# − !?∙ 8" 8+ * = 4 = 8 − 1 4 across all obs =~ &(( , * ) = = !?# − !∙∙ = !?∙+− !∙∙ + !?# − !?∙ + = ) 2) 2) =8 !?∙ − !∙∙ + !?# − !?∙ !− 2) - /- " ( 2/ 2/ +− +− ~ &((+ , * + ) ~ &((- , * + ) ~ &(( , * ) = ,* ) ~ &((+ , * + ) = -, * +) ) + + + + + + + ++ 45 ! − ! 45 ! − ! G?F"E#F" G E G E " ( = + + " " + G ?F" ,* * ! − ! ! − ! !" − !+ 2 = ! − ! E G E G E 45 !+ − "+8 8 8 + Reject./ , if Reject./ , if , + + !?# − !∙∙ = !?∙ − !∙∙ + ++ !?# − !?∙ + - " "" " + + " " 2)=~4&(( &(( + &(( &(( !?# − !∙∙ = !?∙ − !∙∙ + !?# − !?∙ / " " " =./ ∶ (" = (+ = ⋯ = (G#F"vs.." ∶ not.?F" " ?F" / #F" ?F" #F" ∼ ~2(8 2(8 2) 2(8 + + + 6 ~46 6 ?F" #F" ?F" #F" ?F" #F" " ( " ~ *-# = = f 2/ > 2=/+ (8" + 8+ − 2) ∶ 442MNO2/(B − 1)G G P42MNO2 G E ∼ > 2=/+ (8" + 8+ − ∼ !+# !-# +# -# !+# !"# "# "# / / + + 8+ − 1 4+++ + G E L/ = = ~ +L(B − 1, 8 − B) / / / 8" − 1 4"+ + 8+ − 1 4.. + " 8+ − 1. + 8" − 1 4"+ + 8+ − 1 4++ 8+ − + * ! ! ! 8" 8+ ! ! = ("- vs.." ∶+ not./. ∶ not./ 2 2 + =8 !?∙!− − !∙∙ + !?# − !?∙ = 8 445/(& ! − ! −+B) + P45 ! 8" 8+ 1 1 1 8" − 1 + 8+ − 1 ?∙ ∙∙ ?# ?∙ ?F" ?F" #F" !" − !+ + − "+ + + + ! ?F" ?F" #F" Reject./ , if L/ > B) L=, (B − 1, 8 − f! 2/ > 2=/+!(8− 1 1 1 8. ∶ ( SStreat = (+ = ⋯ = (G vs.SSE." ∶ not./ "./ ∶ ("SST = (+ = ⋯ =/(G "vs.." ∶ not./ + between-group within-group 46 46 46 Total variation = (- vs. 4"+ variation variation 1 8" − 1 442MNO2/(B − 1) P42MNO2 = = f 2/ = 442MNO2/(B −L/1)= P42MNO2 − = ~ L(B − 1, 8 − B) 8" − 1 L/ = = 445/(& −~B)L(B − 1, P458 − B) ,* ) , * +) , *! ) , * +) ,* ) !++ + !+ 445/(& − B) 16 P45 8 + + !+ + !+  ?F" #F" !?# = (? + H?# , H?# ~ @@I & 0, * + ANOVA table & F-test G !?# = ( + K? + H?# , H?# ~ @@I & 0, * + , K? = 0 ?F" 5 !?# = (? Source SS DF + MS F-statistic !?# ~ @@I & (? , * SStreat / (k − 1) MStreat Treat SStreat k − 1 !?# − !∙∙ = !?∙ − !∙∙ + !?# − !?∙ = MStreat MSE G E G E SSE / (N −G k)E Error SSE N − k+ + + !?# − !∙∙ = !?∙ − != ∙∙ MSE + !?# − !?∙ ?F" #F" ?F" #F" ?F" #F" Total SST N−1 G G E + + =8 !?∙ − !∙∙ + !?# − !?∙ ※N=nxk ?F" ?F" #F" (Hypothesis)./ ∶ (" = (+ = ⋯ = (G vs.." ∶ not./ 442MNO2/(B − 1) P42MNO2 (Test statistic) L/ = = ~ L(B − 1, 8 − B) 445/(& − B) P45 (Decision rule) Reject./ , if L/ > L=, (B − 1, 8 − B) R 2Q = 17 SQ+ /T  예제1 아래 6개의 그룹에 대해 노력성 호기 중간 유량 (forced mid-expiratory flow, FEF)를 측정하여, 흡연의 형태에 따라 폐 건강의 차이가 있는지 알아본다고 하자. 1. Nonsmokers (NS) 2. Passive smokers (PS) 3. Noninhaling smokers (NI) 4. Light smokers (LS) 5. Moderate smokers (MS) 6. Heavy smokers (HS) 18  예제1 k 2 ni yi k i 1 k y..2 Between SS ni yi2 ni yi2 i 1 n i 1 n k Within SS (ni 1)si2 i 1 where y.. sum of the observations across all groups—i.e., the grand total of all observations over all groups—and n total number of observations over all groups. Compute the Within SS and Between SS for the FEF data in Table 12.1. We use Equation 12.5 as follows: Between SS 200(3.78)2 200(3.30 )2... 200(2.59)2 2 200(3.78) 200(3.30 )... 200(2.59) 1050 10, 505.58 3292 2/1050 10, 505.58 10, 321.20 184.38 Within SS 199( 0.79)2 199( 0.77)2 49( 0.86)2 199( 0.78)2 199( 0.81)2 199( 0.82 )2 124.20 117.99 36.24 121.07 130.56 133.81 663.87 19 Finally, the following definitions are important.  예제1 ANOVA table 20  dialysis, and kidney transplant groups. The most commonly observed cancer sites in ESKD patients were the colorectum, lung, and liver. The incidence of cancer increased progressively among patients undergoing kidney transplant, peritoneal dialysis, and hemodialysis in that order. Hemodialysis 실제 연구에서 ANOVA 사례 patients were found to have an increased risk of digestive tract cancer compared with kidney www.nature.com/scientificreports transplant patients (adjusted hazard 1 ratio = 1.9; 95% confidence interval: 1.31–2.81; P < 0.001). The Department of Nephrology, Ajou University School of Medicine, Suwon, Republic of Korea. 2Department of study findings may be a useful reference for cancer‑screening guidelines. 3 Biomedical Informatics, Ajou University School of Medicine, Suwon, Republic of Korea. Office of Biostatistics, Medical Research Collaborating Center, Ajou Research Institute for Innovative Medicine, Ajou University Medical The incidence of cancer has been reported to be Center, higher in Suwon, patients with Republic end-stage of Korea. 4 kidney disease Department of (ESKD) Medicalthan in Sciences, Biomedical Informatics, Graduate School 1,2 the general population. Also, as the of worldwide incidence of Ajou University, chronicRepublic Suwon, kidney disease (CKD) of Korea. 5 and cancer increases These authors contributed equally: Min-Jeong Lee and Eunyoung www.nature.com/scientificreports/ with the increase in life span3,4, cancer occurrence Lee. * email: in patients with [email protected]