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2 ICC Case Case Case ICC ICC11 ICC21 ICC Case1 Case ICC

3 Intraclass correlation coefficients ICC a ICC ICC ICC [1] [2] Fisher [3] ICC Shrout [4] ICC 3 6 Shrout [5][6][7][8] [9] Shrout ICC 3 Case1Case2Case3 6 ICCn1 ICCn k n 123 ICC 6 Bartko [10] ICC One-way ClassificationTwo-way Random ModelTwo-way Mixed Model 3 Case1Case2Case3 Bartko [11] ICC1 ICC2 Two-wayWiner s anchor poin ICC ICC [ ] ICC a item characteristic curve) ICC ICC

4 classical test theory 1950 item response theoryirt test-retest method test-retest reliability 0 b parallel test method parallel form reliability strongly parallel measurement weakly parallel measurement Cronbach α [ ] ICC split-half method odd-even method Spearman-Brown Kuder-Richerdson 20KR-20 Kuder-Richerdson 21KR KR-20 KR α Cronbach s coefficient alpha Cronbach α α c A n n b c

5 ICC3k α X i T i E i 2 1) 0 E(E i ) 0 (2.1) µ(x) E(X i ) E(T i + E i ) E(T i )+E(E i ) µ(t ) (2.2) 2) Cov(T E) ρ(t E) 0 (2.3) σ 2 (X) Var(T )+Var(E)+2Cov(T,E) Var(T )+Var(E) (2.4) ρ ρ(x) = Var(T ) Var(X) (2.5) Spearman-Brown 2 2 X 1 X 2 T 1 T 2 E 1 E 2 X 1 X 2 X X 1 + X 2 /2 T T 1 + T 2 /2 T 1 T 2 Var(X 1 )=Var(X 2 ) (2.6) Var(T 1 )=Var(T 2 ) (2.7) X 1 (= (X 11 X 12 X 1i ))X 2 (= (X 21 X 22 X 2i )) CovX 1 X 2 = E X 1i µ(x 1 ) X 2i µ(x 2 ) = E T µ(t )+E 1i T µ(t )+E2i = E T µ(t ) 2 + E T µ(t ) E2i + E E1i T µ(t ) + E E1i E 2i = Var(T ) (2.8) ρ(x 1,X 2 ) Cov(X 1,X 2 ) Var(X 1 )Var(X 2 ) = Cov(X 1,X 2 ) 2Var(X 1 ) (2.9)

6 ρ(x) = Var(T ) Var(X) Var(T 1 + T 2 ) Var(X 1 + X 2 ) = Var(T 1)+Var(T 2 )+2Cov(T 1,T 2 ) Var(X 1 )+Var(X 2 )+2Cov(X 1,X 2 ) = 2Var(T 1)+2Cov(T 1,T 2 ) 2Var(X 1 )+2Cov(X 1,X 2 ) = 4Var(T 1 ) 2Var(X 1 )[1 + ρ(x 1,X 2 )] = 2 2Cov(X 1,X 2 ) 2Var(X 1 )[1 + ρ(x 1,X 2 )] = 2ρ(X 1,X 2 ) 1+ρ(X 1,X 2 ) = 2ρ(X 1 ) 1+ρ(X 1 ) (2.10) 2.10 Spearman-Brown generalizability theory [2] ICC G generalizability study D decision study A A G ICC11 ICC21 ICC 31 G A A D D ICC1k ICC2k ICC3k D G A D fixed model random effect model mixed model

7 d e 2 f 2 A B C ICC i j x ij i 1 2 nj 1 2 m x 11 x 21 x nm x i x 2 x ij x 2 x i x+x ij x i 2 x ij x i x+x ij x i (2.11) x ij x 2 x i x 2 +x ij x i 2 +2 x i xx ij x i nx mx nx nx mx nx mx x ij x 2 x i x 2 + x ij x i x i xx ij x i (2.12) i=1 j=1 i=1 i=1 j=1 i=1 j= i i 0 g nx i=1 j=1 mx nx x ij x 2 x i x 2 + i=1 nx i=1 j=1 [ ] mx x ij x i 2 (2.13) Case2 Case3 1.1 d e f g x i x x ij i 0

8 a. A B A B b. c. A B 1 A 2 B BA AB ICC ICC 1.1 ICC 1.1 [4] df M S n 1 BMS kσt 2 + σ2 W kσt 2 + σ2 I + σ2 E kσt 2 + σ2 E n(k 1) WMS σw 2 σj 2 + σ2 I + σ2 E θj 2 + fσ2 I + σ2 E (k 1) JMS nσj 2 + σ2 I + σ2 E nθj 2 + fσ2 I + σ2 E (n 1)(k 1) EMS σi 2 + σ2 E fσi 2 + σ2 E P f k/(k 1) θ2 J a 2 i /k 1 n k MS BMSWMSJMSEMS ICC 3

9 f θ 1.1 [ ] h Bartko [10] 1.1 h EMS θ J 2 σ2 J σ2 I f

10 ICC ICC Case Case a. Case1 1.2 b. Case2 2 a. ICC Case1 ICC Case2b. c. ICC Case3

11 1.3. ICC c. Case3 Case2 i Case2 Case3 A B C Case3 3 3 Case3 A a b a 85 b 75 a b 10 Case3 Case Case1 1 Intra-rater reliability Case1 1.1 ICC11 ICC1k 1 r k x ij x ij µ + T i + W ij (3.14) µ T i i W ij ij i12 rj12 k σx 2 σ2 T + σw 2 (3.15) σt 2 + σ2 W σ2 T ρ σ2 T σ 2 T + σ2 W (3.16) 3.16 ICC BMS WMS Case1 BMSWMS BMS kσ 2 T + σ 2 W (3.17) WMS σ 2 W (3.18) σt 2 BMS WMS] /k 3.16 i BMS WMS ICC1 /k BMS WMS /k + WMS

12 10 1 BMS WMS BMS WMS + kwms BMS WMS ICC1 BMS +k 1WMS (3.19) A ICC11 A 1 r m x 11 ( x x x 11 m )/m 3.16 ρ σ2 T σt 2 + σ2 W /k (3.20) σw 2 /k k 3.20 ICC1k BMS WMS ICC1 k BMS (3.21) [4] ICC11 100(1 α) j F 0 F U BMS/WMS F 0 F 1 α/2n(k 1)n 1 F L = F 0 /F 1 α/2n 1n(k 1) F L 1 F L +(k 1) ρ ICC(1,1) F U 1 (3.22) F U +(k 1) F 1 α/2nk 1 n 1 α0.05 df 1=nk 1 df 2 n / F ICC1k 100(1 α) 1 1 F L ρ ICC(1,k) 1 1 F U (3.23) Case2 Inter-rater reliability Case2 Case3 [1.2.4 ] Case2 ICC21 ICC2 k k n Case1 x ij µ + T i + J j + I ij + E ij (3.24) j 1 1 [13]

13 1.3. ICC 11 µ T i i J j j I ij E ij i j i 1 2 nj 1 2 k 3.14 J j I ij J j I ij A a b c B a b c C a b c C Case1 σt 2 σ 2 T ρ σt 2 + σ2 J + σ2 I + σ2 E (3.25) ICC BMS kσ 2 T + σ 2 I + σ 2 E (3.26) JMS nσ 2 J + σ 2 I + σ 2 E (3.27) EMS σ 2 I + σ 2 E (3.28) σ 2 T σ 2 J BMS EMS /k (3.29) JMS EMS /n (3.30) 3.25 BMS EMS /k ICC2 1 BMS EMS /k +JMS EMS /n + EMS BMS EMS BMS EMS + k JMS EMS /n + kems BMS EMS ICC2 1 BMS +k 1EMS + k JMS EMS /n (3.31) Case1 A k n m 3.25 σ 2 T ICC2 k σt 2 +σ2 J + σ2 I + σ2 E /k BMS EMS /k BMS EMS /k +JMS EMS /n + EMS /k BMS EMS ICC2 k BMS +JMS EMS /n (3.32) ICC21 100(1 α) ˆρ ICC2 1

14 12 1 F J ν F JMS/EMS (k 1)(n 1)kˆρF J + n[1 + (k 1)ˆρ] kˆρ 2 (n 1)k 2 ˆρ 2 FJ 2 +n[1 + (k 1)ˆρ] kˆρ 2 F 1 α/2 (n 1) ν F = F 1 α/2 ν (n 1) n(bms F EMS) F [kjms +(kn k n)ems]+nbms ρ n(f BMS EMS) ICC(2,1) kjms +(kn k n)ems + nf BMS (3.33) ICC2k 100(1 α) ρ L = ρ U = kρ L 1+(k 1)ρ L kρ U 1+(k 1)ρ U (3.34) (3.35) ρ ρ L ρ ICC(2,k) ρ U (3.36) L ρ U 3.33 ICC Case3 ICC21 ICC2k [1.2.4 ] ICC31 ICC3k 3.24 ρ σ2 T σ2 I /k 1 σt 2 + σ2 I + σ2 E 3.37 σt 2 σ2 I /k 1 σj 2 k 1.3 Case1Case2 1.1 BMSEMS (3.37) σt 2 (BMS σ E 2 )/k (3.38) σi 2 (k 1)(EMS σe)/k 2 (3.39) 3.37 ICC31 ICC3 1 (BMS σ2 E )/k (k 1)(EMS σ2 E )/k1/k 1 (BMS σ E 2 )/k +(k 1)(EMS σ2 E )/k + σ2 E BMS σ2 E EMS + σ2 E BMS σe 2 +(k 1)(EMS σ2 E )+k σ2 E BMS EMS ICC3 1 BMS +k 1EMS (3.40) k σi 2 0

15 1.3. ICC 13 Case1Case2 m ICC3k 3.37 σ2 T ρ σt 2 + σ2 E /k (3.41) 1.1 /k 1 σ 2 I BMS kσ 2 T + σ 2 E (3.42) EMS σ 2 E (3.43) a a b b a a 2 2 b 2 b BMS EMS ICC3 k BMS ICC31 (3.44) F 0 BMS/EMS (3.45) F L F 0 /F 1 α/2 (n 1),(n 1)(k 1) (3.46) F U = F 0 F 1 α/2 (n 1)(k 1),(n 1) (3.47)

16 14 1 F L 1 F L +(k 1) ρ ICC(3,1) F U 1 F U +(k 1) ICC3k (3.48) 1 1 F L ρ ICC(3,k) 1 1 F U (3.49) k n [ ] I ij V ( kx j=1 I ij ) kv (I ij )+kk 1COV (I ij I i0 j) (3.50) V (I ij ) V (I i0 j) i i 0 COV (I ij I i0 j)=v (I ij ) COV (I ij I i0 j)= σi 2 /k Bartko [10] 3.37 σ2 T ρ σt 2 + σ2 I + σ2 E (3.51) 1.1 EMS EMS σ I 2 + σ2 E EMS BMS 3.51 ICC3 1 BMS EMS + σ2 I BMS +k 1EMS + σi 2 (3.52) ICC3 1 BMS EMS = BMS +k 1EMS (3.53) 3.40 EMS σi 2 EMS σ2 I ICC3 1 BMS BMS + kems (3.54) ICC ICC

17 1.4. ICC ICC ICC11 ICC21 ICC31 ICC11 ICC21 ICC31 ICC11 ICC21 ICC31 [8] 0 ICC11 ICC21 ICC Case Case Case1 Case2 Case1 [1.3.1 ] [4] [8] ICC(11) ICC21 ICC11 BMS BMS kσt 2 + σw ICC21 BMS BMS kσt 2 + σi 2 + σe σw σw 2 σ2 J + σi 2 + σe 2 (4.55) ICC21 BMS ICC11 σt 2 σ 2 T BMS ([JMS EMS]/n + EMS) = BMS EMS (JMS EMS/n) (4.56) (4.55) (4.56) ICC11 σt 2 ICC(1 1) σ2 J σt 2 + σ2 J + σ2 I + σ2 E σ2 J (4.57) ICC21 σj 2 (= (JMS EMS)/n) ICC11 l ICC ICC ICC 1.2 Landis [12] Kappa ICC l

18 16 1 [6] [9] ICC 0.7 ICC ICC 1.2 [12] ICC slight fair moderate substantial almost perfect D ICCn kn =1, 2, 3 ICC m k ρ 11 ρ 2 ρ 2 1 ρ 1 (4.58) ρ 1 ρ k (4.59) Eliasziw [13] ICC Eliasziw [13] n H 0 ρ H 0 ρ 0.6 sabstantial m Spearman-Brown [1.2.2 ] n

19 1.4. ICC SEM ICC Case SEM SEM q SEM σtotal 2 (1 ρ) (4.60) σtotal 2 ICC11 SEM 1 SEM1 2 σ2 W WMS (4.61) ICC12 ICC13 SEM 2 SEM 3 SEM2 2 = σj 2 + σi 2 + σe 2 JMS EMS /n + EMS (4.62) SEM3 2 = σi 2 + σe 2 EMS (4.63) SEM Stratford [14] SEM " # WMS WMS, (4.64) χ 2 χ 2 α/2,df1 α/2,df χ 2 α,df df α/2 χ2 2 SEM SEM A SEM B H 0 2 df A df B F F df Adf B SEMA 2 /SEM B 2

20 ICC ICC ICC Case2 T i N(0, σt 2 ),J jn(0, σj 2) I ijn(0, σi 2),E ijn(0, σe 2 ) A k n k ICC ICC21 J j T i X (x 11 x 12 x ij )(i =1, 2,,nj =1, 2,,k) x ij µ + T i + J j +(TJ) ij + e ij (5.65) (TJ) ij µ 0XN(µ X σx 2 ) X x n1 x n2 x nk k k i i r r k k R R AA t A n k DN0 1 A t D X ICC21 1, n k ricc 0 Fisher [3]Shrout [4] o ICC ICC 0 ICC 1 o Fisher + 1 n0 log 2 n n 0 1 n0 n +2

21 k=5 n20 r0 n20 r0.5 n20 r0.9 n20 r0 n20 r0.5 n20 r0.9 n10 r0 n10 r0.5 n10 r0.9 n20 r0 n20 r0.5 n20 r0.9 n30 r0 n30 r0.5 n30 r0.9 k=10 n10 r0 n10 r0.5 n10 r0.9 n30 r0 n30 r0.5 n30 r0.9 k=20 n10 r0 n10 r0.5 n10 r0.9 n30 r0 n30 r0.5 n30 r ICC21

22 20 1 a. r0 95 b. r c. r ICC X A t X ICC21 95 k 5n 20 r r Eliasziw [13] ICC ICC 1.6 ICC11 ICC21 ICC a. AD ad ICC 1 b. A d +2 ICC A d c.

23 k 4 n 4 A B C D ICC(1,1)= a ICC(2,1)= b ICC(3,1)= c SEM1= d SEM2= SEM3= A B C D ICC(1,1)= a ICC(2,1)= b ICC(3,1)= c SEM1= d SEM2= SEM3= A B C D ICC(1,1)= A B C D ICC(1,1)= a ICC(2,1)= a ICC(2,1)= b ICC(3,1)= b ICC(3,1)= c SEM1= c SEM1= d SEM2= d SEM2= SEM3= SEM3= A B C D ICC(1,1)= A B C D ICC(1,1)= a ICC(2,1)= a ICC(2,1)= b ICC(3,1)= b ICC(3,1)= c SEM1= c SEM1= d SEM2= d SEM2= SEM3= SEM3= A B C D ICC(1,1)= - A B C D ICC(1,1)= a ICC(2,1)= a ICC(2,1)= b ICC(3,1)= b ICC(3,1)= c SEM1= c SEM1= d SEM2= d SEM2= SEM3= SEM3= A B C D ICC(1,1)= A B C D ICC(1,1)= a ICC(2,1)= a ICC(2,1)= b ICC(3,1)= b ICC(3,1)= c SEM1= c SEM1= d SEM2= d SEM2= SEM3= SEM3= 1.6 ICC b. ICC c d. ICC a. b. A d +2 A d +20 ICC ICC SEM 1.6 b. c. d. SEM 1.6-e. ICC ICC31 1 e. 10 f. ICC

24 22 1 g. ICC11 ICC21 ICC31 ICC11ICC21 ICC h. d +10 ICC 11 ICC21 AD ICC i. d +5 d 1.6-h. d ICC11 ICC21 h. i. h. i. j. ICC11 ICC21 ICC SEM ICC ICC ICC 0 ICC

25 AD 4 A B C D A B C D df M S BMS JMS EMS BMS JMS EMS p 0.01 p 0.05 BMSJMSEMS ICC21

26 24 1 BMS EMS ICC2 1 BMS +k 1EMS + k JMS EMS /n = (4 1) / ICC [1.3.2 ] ˆρ F J ν ICC JMS/EMS 0.89 (k 1)(n 1)kˆρF J + n[1 + (k 1)ˆρ] kˆρ 2 (n 1)k 2 ˆρ 2 FJ 2 +n[1 + (k 1)ˆρ] kˆρ 2 F (4 1) (10 1) [1 + (4 1) 0.909] = (10 1) [1 + (4 1) 0.909] F 1 α/2 (n 1) ν = F ( ) F = F 1 α/2 ν (n 1) F ( ) n(bms F EMS) F [kjms +(kn k n)ems]+nbms ρ n(f BMS EMS) ICC(2,1) kjms +(kn k n)ems + nf BMS 10 ( ) [ ( ) 28.4] ρ ICC(2,1) 10 ( ) ( ) ρ ICC(2,1) q q SEM = JMS EMS /n + EMS / ρ0 ρ ICC ρ ICC(2,1) SEM 1.43 H 0 ρ SEM

27 A B C D A B C D a. c. d box plot a. b.c. d. a. b. SEM SEM

28 D ICC ICC ICC ρ ρ

29 FAX E mailpteiki@cc.hirosaki-u.ac.jp URLhttp://

30

31 29 [1] [2] [3] Fisher RA 1975p [4] Shrout PE,Fleiss JLIntraclass Correlations:Uses in Assessing Rater ReliabilityPsychol Bull [5] James WY,Connie LB,et.al.:Reliability of goniometric measurements and visual estimates of ankle joint active range of motion obtained in a clinical setting.arch Phys Med Rehabil 74: ,1993. [6] Reliability. 41: ,1993. [7]. 12: ,1997. [8] Gabrielle R,Maria S:Reliability of assessment tools in rehabilitation:an illustration of appropriate statistical analyses.clinical Rehabilitation 12: ,1998. [9] Portney LG,Watkins MP:Foundations of clinical research-applications to practice-,appleton Lange,USA,1993,p [10] Bartko JJThe intraclass correlation coefficient as a measure of reliability.psychological Reports [11] Bartko JJOn various intraclass correlation reliability coefficients.psychological Bulletin [12] Landis JR,Koch GG:The measurement of observer agreement for categorical data.biometrics. 33, ,1977. [13] Eliasziw M,Young SL,et al.:statistical methodology for the concurrent assessment of interrater and intrarater reliability:using goniometric measurements as an example.physical Therapy 74: ,1994. [14] Stratford PW,Goldsmith CH:Use of the error as a reliability index of interest:an applied eample using elbow flexor strength data.physical Therapy 77: ,1997.

2 豊橋創造大学紀要第 17 号 テストの信頼性は, 測定の対象がたとえば血清中のNa 濃度のように検体として取り出せるものであれば, 標準となる試料を用意し, 何回か測定を繰り返して得られた一連の測定値のばらつきの大小によって評価することができる. しかし, 筋力テストのように人間を対象として測定

2 豊橋創造大学紀要第 17 号 テストの信頼性は, 測定の対象がたとえば血清中のNa 濃度のように検体として取り出せるものであれば, 標準となる試料を用意し, 何回か測定を繰り返して得られた一連の測定値のばらつきの大小によって評価することができる. しかし, 筋力テストのように人間を対象として測定 Bulletin of Toyohashi Sozo University 信頼性係数による筋力テストの信頼性評価 2013, No. 17, 1 8 1 筋力テストの信頼性評価を使用した Shrout と Eliasziw の 信頼性係数の比較 1)2) 柴田賢一 2) 宮原英夫 1) 森嶋直人 抄録筋力テストの信頼性の評価法としてShroutらの日内信頼性係数 (ICC(1, 1)) と日間信頼性係数

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